An analytical approach to bistable biological circuit discrimination using real algebraic geometry.

Science.gov (United States)

Siegal-Gaskins, Dan; Franco, Elisa; Zhou, Tiffany; Murray, Richard M

2015-07-06

Biomolecular circuits with two distinct and stable steady states have been identified as essential components in a wide range of biological networks, with a variety of mechanisms and topologies giving rise to their important bistable property. Understanding the differences between circuit implementations is an important question, particularly for the synthetic biologist faced with determining which bistable circuit design out of many is best for their specific application. In this work we explore the applicability of Sturm's theorem--a tool from nineteenth-century real algebraic geometry--to comparing 'functionally equivalent' bistable circuits without the need for numerical simulation. We first consider two genetic toggle variants and two different positive feedback circuits, and show how specific topological properties present in each type of circuit can serve to increase the size of the regions of parameter space in which they function as switches. We then demonstrate that a single competitive monomeric activator added to a purely monomeric (and otherwise monostable) mutual repressor circuit is sufficient for bistability. Finally, we compare our approach with the Routh-Hurwitz method and derive consistent, yet more powerful, parametric conditions. The predictive power and ease of use of Sturm's theorem demonstrated in this work suggest that algebraic geometric techniques may be underused in biomolecular circuit analysis.

Methods of algebraic geometry in control theory

CERN Document Server

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...

Formalization and Implementation of Algebraic Methods in Geometry

Directory of Open Access Journals (Sweden)

Filip Marić

2012-02-01

Full Text Available We describe our ongoing project of formalization of algebraic methods for geometry theorem proving (Wu's method and the Groebner bases method, their implementation and integration in educational tools. The project includes formal verification of the algebraic methods within Isabelle/HOL proof assistant and development of a new, open-source Java implementation of the algebraic methods. The project should fill-in some gaps still existing in this area (e.g., the lack of formal links between algebraic methods and synthetic geometry and the lack of self-contained implementations of algebraic methods suitable for integration with dynamic geometry tools and should enable new applications of theorem proving in education.

CIME-CIRM course Rationality Problems in Algebraic Geometry

CERN Document Server

Pirola, Gian

2016-01-01

Providing an overview of the state of the art on rationality questions in algebraic geometry, this volume gives an update on the most recent developments. It offers a comprehensive introduction to this fascinating topic, and will certainly become an essential reference for anybody working in the field. Rationality problems are of fundamental importance both in algebra and algebraic geometry. Historically, rationality problems motivated significant developments in the theory of abelian integrals, Riemann surfaces and the Abel–Jacobi map, among other areas, and they have strong links with modern notions such as moduli spaces, Hodge theory, algebraic cycles and derived categories. This text is aimed at researchers and graduate students in algebraic geometry.

Elementary algebraic geometry

CERN Document Server

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

Hopf algebras in noncommutative geometry

International Nuclear Information System (INIS)

Varilly, Joseph C.

2001-10-01

We give an introductory survey to the use of Hopf algebras in several problems of non- commutative geometry. The main example, the Hopf algebra of rooted trees, is a graded, connected Hopf algebra arising from a universal construction. We show its relation to the algebra of transverse differential operators introduced by Connes and Moscovici in order to compute a local index formula in cyclic cohomology, and to the several Hopf algebras defined by Connes and Kreimer to simplify the combinatorics of perturbative renormalization. We explain how characteristic classes for a Hopf module algebra can be obtained from the cyclic cohomology of the Hopf algebra which acts on it. Finally, we discuss the theory of non- commutative spherical manifolds and show how they arise as homogeneous spaces of certain compact quantum groups. (author)

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...

Algebraic geometry and theta functions

CERN Document Server

Coble, Arthur B

1929-01-01

This book is the result of extending and deepening all questions from algebraic geometry that are connected to the central problem of this book: the determination of the tritangent planes of a space curve of order six and genus four, which the author treated in his Colloquium Lecture in 1928 at Amherst. The first two chapters recall fundamental ideas of algebraic geometry and theta functions in such fashion as will be most helpful in later applications. In order to clearly present the state of the central problem, the author first presents the better-known cases of genus two (Chapter III) and

Analytic real algebras.

Science.gov (United States)

Seo, Young Joo; Kim, Young Hee

2016-01-01

In this paper we construct some real algebras by using elementary functions, and discuss some relations between several axioms and its related conditions for such functions. We obtain some conditions for real-valued functions to be a (edge) d -algebra.

Differential geometry on Hopf algebras and quantum groups

International Nuclear Information System (INIS)

Watts, P.

1994-01-01

The differential geometry on a Hopf algebra is constructed, by using the basic axioms of Hopf algebras and noncommutative differential geometry. The space of generalized derivations on a Hopf algebra of functions is presented via the smash product, and used to define and discuss quantum Lie algebras and their properties. The Cartan calculus of the exterior derivative, Lie derivative, and inner derivation is found for both the universal and general differential calculi of an arbitrary Hopf algebra, and, by restricting to the quasitriangular case and using the numerical R-matrix formalism, the aforementioned structures for quantum groups are determined

International conference on Algebraic and Complex Geometry

CERN Document Server

Kloosterman, Remke; Schütt, Matthias

2014-01-01

Several important aspects of moduli spaces and irreducible holomorphic symplectic manifolds were highlighted at the conference “Algebraic and Complex Geometry” held September 2012 in Hannover, Germany. These two subjects of recent ongoing progress belong to the most spectacular developments in Algebraic and Complex Geometry. Irreducible symplectic manifolds are of interest to algebraic and differential geometers alike, behaving similar to K3 surfaces and abelian varieties in certain ways, but being by far less well-understood. Moduli spaces, on the other hand, have been a rich source of open questions and discoveries for decades and still continue to be a hot topic in itself as well as with its interplay with neighbouring fields such as arithmetic geometry and string theory. Beyond the above focal topics this volume reflects the broad diversity of lectures at the conference and comprises 11 papers on current research from different areas of algebraic and complex geometry sorted in alphabetic order by the ...

Vanishing theorems and effective results in algebraic geometry

International Nuclear Information System (INIS)

Demailly, J.P.; Goettsche, L.; Lazarsfeld, R.

2001-01-01

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

Vanishing theorems and effective results in algebraic geometry

Energy Technology Data Exchange (ETDEWEB)

Demailly, J P [Universite de Grenoble (France); Goettsche, L [Abdus Salam International Centre for Theoretical Physics, Trieste (Italy); Lazarsfeld, R [University of Michigan (United States)

2001-12-15

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.

Symplectic and Poisson Geometry in Interaction with Analysis, Algebra and Topology & Symplectic Geometry, Noncommutative Geometry and Physics

CERN Document Server

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 ...

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...... and isolation with rational univariate representations. This has the advantage of avoiding computations with polynomials with algebraic coefficients, even in non-generic positions. Our algorithm isolates critical points in boxes and computes a decomposition of the plane by rectangular boxes. This decomposition...

Real division algebras and other algebras motivated by physics

International Nuclear Information System (INIS)

Benkart, G.; Osborn, J.M.

1981-01-01

In this survey we discuss several general techniques which have been productive in the study of real division algebras, flexible Lie-admissible algebras, and other nonassociative algebras, and we summarize results obtained using these methods. The principal method involved in this work is to view an algebra A as a module for a semisimple Lie algebra of derivations of A and to use representation theory to study products in A. In the case of real division algebras, we also discuss the use of isotopy and the use of a generalized Peirce decomposition. Most of the work summarized here has appeared in more detail in various other papers. The exceptions are results on a class of algebras of dimension 15, motivated by physics, which admit the Lie algebra sl(3) as an algebra of derivations

Geometry of time-spaces non-commutative algebraic geometry, applied to quantum theory

CERN Document Server

Landau, Olav Arnfinn

2011-01-01

This is a monograph about non-commutative algebraic geometry, and its application to physics. The main mathematical inputs are the non-commutative deformation theory, moduli theory of representations of associative algebras, a new non-commutative theory o

Classical versus Computer Algebra Methods in Elementary Geometry

Science.gov (United States)

Pech, Pavel

2005-01-01

Computer algebra methods based on results of commutative algebra like Groebner bases of ideals and elimination of variables make it possible to solve complex, elementary and non elementary problems of geometry, which are difficult to solve using a classical approach. Computer algebra methods permit the proof of geometric theorems, automatic…

Experimental and Theoretical Methods in Algebra, Geometry and Topology

CERN Document Server

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...

A study in derived algebraic geometry volume I : correspondences and duality

CERN Document Server

Gaitsgory, Dennis

2017-01-01

Derived algebraic geometry is a far-reaching generalization of algebraic geometry. It has found numerous applications in various parts of mathematics, most prominently in representation theory. This volume develops the theory of ind-coherent sheaves in the context of derived algebraic geometry. Ind-coherent sheaves are a "renormalization" of quasi-coherent sheaves and provide a natural setting for Grothendieck-Serre duality as well as geometric incarnations of numerous categories of interest in representation theory. This volume consists of three parts and an appendix. The first part is a survey of homotopical algebra in the setting of \\infty-categories and the basics of derived algebraic geometry. The second part builds the theory of ind-coherent sheaves as a functor out of the category of correspondences and studies the relationship between ind-coherent and quasi-coherent sheaves. The third part sets up the general machinery of the \\mathrm{(}\\infty, 2\\mathrm{)}-category of correspondences needed for the sec...

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...

Bicomplex holomorphic functions the algebra, geometry and analysis of bicomplex numbers

CERN Document Server

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. ...

Semi-algebraic function rings and reflectors of partially ordered rings

CERN Document Server

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.

Geometry, algebra and applications from mechanics to cryptography

CERN Document Server

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.

Conference on Algebraic Geometry for Coding Theory and Cryptography

CERN Document Server

Lauter, Kristin; Walker, Judy

2017-01-01

Covering topics in algebraic geometry, coding theory, and cryptography, this volume presents interdisciplinary group research completed for the February 2016 conference at the Institute for Pure and Applied Mathematics (IPAM) in cooperation with the Association for Women in Mathematics (AWM). The conference gathered research communities across disciplines to share ideas and problems in their fields and formed small research groups made up of graduate students, postdoctoral researchers, junior faculty, and group leaders who designed and led the projects. Peer reviewed and revised, each of this volume's five papers achieves the conference’s goal of using algebraic geometry to address a problem in either coding theory or cryptography. Proposed variants of the McEliece cryptosystem based on different constructions of codes, constructions of locally recoverable codes from algebraic curves and surfaces, and algebraic approaches to the multicast network coding problem are only some of the topics covered in this vo...

Connecting Functions in Geometry and Algebra

Science.gov (United States)

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…

The $K$-theory of real graph $C*$-algebras

OpenAIRE

Boersema, Jeffrey L.

2014-01-01

In this paper, we will introduce real graph algebras and develop the theory to the point of being able to calculate the $K$-theory of such algebras. The $K$-theory situation is significantly more complicated than in the case for complex graph algebras. To develop the long exact sequence to compute the $K$-theory of a real graph algebra, we need to develop a generalized theory of crossed products for real C*-algebras for groups with involution. We also need to deal with the additional algebrai...

Algebra and Geometry of Hamilton's Quaternions

Indian Academy of Sciences (India)

2016-08-26

Aug 26, 2016 ... ... Public Lectures · Lecture Workshops · Refresher Courses · Symposia. Home; Journals; Resonance – Journal of Science Education; Volume 21; Issue 6. Algebra and Geometry of Hamilton's Quaternions: 'Well, Papa, Can You Multiply Triplets?' General Article Volume 21 Issue 6 June 2016 pp 529-544 ...

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...

Quantum algebras and Poisson geometry in mathematical physics

CERN Document Server

Karasev, M V

2005-01-01

This collection presents new and interesting applications of Poisson geometry to some fundamental well-known problems in mathematical physics. The methods used by the authors include, in addition to advanced Poisson geometry, unexpected algebras with non-Lie commutation relations, nontrivial (quantum) Kählerian structures of hypergeometric type, dynamical systems theory, semiclassical asymptotics, etc.

Ideals, varieties, and algorithms an introduction to computational algebraic geometry and commutative algebra

CERN Document Server

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...

From classical to modern algebraic geometry Corrado Segre's mastership and legacy

CERN Document Server

Conte, Alberto; Gatto, Letterio; Giacardi, Livia; Marchisio, Marina; Verra, Alessandro

2016-01-01

This book commemorates the 150th birthday of Corrado Segre, one of the founders of the Italian School of Algebraic Geometry and a crucial figure in the history of Algebraic Geometry. It is the outcome of a conference held in Turin, Italy. One of the book's most unique features is the inclusion of a previously unpublished manuscript by Corrado Segre, together with a scientific commentary. Representing a prelude to Segre's seminal 1894 contribution on the theory of algebraic curves, this manuscript and other important archival sources included in the essays shed new light on the eminent role he played at the international level. Including both survey articles and original research papers, the book is divided into three parts: section one focuses on the implications of Segre's work in a historic light, while section two presents new results in his field, namely Algebraic Geometry. The third part features Segre's unpublished notebook: Sulla Geometria Sugli Enti Algebrici Semplicemente Infiniti (1890-1891). This v...

On Ancient Babylonian Algebra and Geometry

Indian Academy of Sciences (India)

Home; Journals; Resonance – Journal of Science Education; Volume 8; Issue 8. On Ancient Babylonian Algebra and Geometry. Rahul Roy. General Article Volume 8 Issue 8 August 2003 pp 27-42. Fulltext. Click here to view fulltext PDF. Permanent link: https://www.ias.ac.in/article/fulltext/reso/008/08/0027-0042. Keywords.

Selected papers on number theory and algebraic geometry

CERN Document Server

Nomizu, Katsumi

1996-01-01

This book presents papers that originally appeared in the Japanese journal Sugaku from the Mathematical Society of Japan. The papers explore the relationship between number theory and algebraic geometry.

Combinatorial algebraic geometry selected papers from the 2016 apprenticeship program

CERN Document Server

Sturmfels, Bernd

2017-01-01

This volume consolidates selected articles from the 2016 Apprenticeship Program at the Fields Institute, part of the larger program on Combinatorial Algebraic Geometry that ran from July through December of 2016. Written primarily by junior mathematicians, the articles cover a range of topics in combinatorial algebraic geometry including curves, surfaces, Grassmannians, convexity, abelian varieties, and moduli spaces. This book bridges the gap between graduate courses and cutting-edge research by connecting historical sources, computation, explicit examples, and new results.

Computational algebraic geometry of epidemic models

Science.gov (United States)

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.

Real space process algebra

NARCIS (Netherlands)

Bergstra, J.A.; Baeten, J.C.M.

1993-01-01

The real time process algebra of Baeten and Bergstra [Formal Aspects of Computing, 3, 142-188 (1991)] is extended to real space by requiring the presence of spatial coordinates for each atomic action, in addition to the required temporal attribute. It is found that asynchronous communication

Algorithmic algebraic geometry and flux vacua

International Nuclear Information System (INIS)

Gray, James; He Yanghui; Lukas, Andre

2006-01-01

We develop a new and efficient method to systematically analyse four dimensional effective supergravities which descend from flux compactifications. The issue of finding vacua of such systems, both supersymmetric and non-supersymmetric, is mapped into a problem in computational algebraic geometry. Using recent developments in computer algebra, the problem can then be rapidly dealt with in a completely algorithmic fashion. Two main results are (1) a procedure for calculating constraints which the flux parameters must satisfy in these models if any given type of vacuum is to exist; (2) a stepwise process for finding all of the isolated vacua of such systems and their physical properties. We illustrate our discussion with several concrete examples, some of which have eluded conventional methods so far

PREFACE: Algebra, Geometry, and Mathematical Physics 2010

Science.gov (United States)

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

Torsional Newton-Cartan geometry and the Schrodinger algebra

NARCIS (Netherlands)

Bergshoeff, Eric A.; Hartong, Jelle; Rosseel, Jan

2015-01-01

We show that by gauging the Schrodinger algebra with critical exponent z and imposing suitable curvature constraints, that make diffeomorphisms equivalent to time and space translations, one obtains a geometric structure known as (twistless) torsional Newton-Cartan geometry (TTNC). This is a version

The algebra and geometry of SU(3) matrices

OpenAIRE

Mallesh, KS; Mukunda, N

1997-01-01

We give an elementary treatment of the defining representation and Lie algebra of the three-dimensional unitary unimodular group SU(3). The geometrical properties of the Lie algebra, which is an eight dimensional real Linear vector space, are developed in an SU(3) covariant manner. The f and d symbols of SU(3) lead to two ways of 'multiplying' two vectors to produce a third, and several useful geometric and algebraic identities are derived. The axis-angle parametrization of SU(3) is developed...

SAGA advances in ShApes, Geometry, and Algebra : results from the Marie Curie initial training network

CERN Document Server

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.

Quantum groups and algebraic geometry in conformal field theory

International Nuclear Information System (INIS)

Smit, T.J.H.

1989-01-01

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

Moduli spaces in algebraic geometry

International Nuclear Information System (INIS)

Goettsche, L.

2000-01-01

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

Classification of non-simple C*-algebras of real rank zero

DEFF Research Database (Denmark)

Arklint, Sara Esther

for real rank zero graph algebras over primitive ideal spaces that admit classification. As a consequence of completeness of filtered K-theory combined with this range result, one can conclude that real rank zero extensions of stabilized Cuntz-Krieger algebras are stabilized Cuntz-Krieger algebras......This thesis deals with classification of nonsimple C-algebras of real rank zero, and whether filtered K-theory is a suitable invariant for this purpose. As a consequence of the result of E. Kirchberg for purely infinite, nuclear C-algebras with a finite primitive ideal space, it suffices to lift...... the thesis is the following: is it possible to achieve the desired classification result for arbitrary finite primitive ideal spaces by restricting to C-algebras of real rank zero that possibly satisfy further restrictions on K-theory? The thesis consists of an account of the relevant theory and the relevant...

Some homological properties of skew PBW extensions arising in non-commutative algebraic geometry

Directory of Open Access Journals (Sweden)

Lezama Oswaldo

2017-06-01

Full Text Available In this short paper we study for the skew PBW (Poincar-Birkhoff-Witt extensions some homological properties arising in non-commutative algebraic geometry, namely, Auslander-Gorenstein regularity, Cohen-Macaulayness and strongly noetherianity. Skew PBW extensions include a considerable number of non-commutative rings of polynomial type such that classical PBW extensions, quantum polynomial rings, multiplicative analogue of the Weyl algebra, some Sklyanin algebras, operator algebras, diffusion algebras, quadratic algebras in 3 variables, among many others. Parametrization of the point modules of some examples is also presented.

Foliation theory in algebraic geometry

CERN Document Server

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...

Linking geometry and algebra in the school mathematics curriculum

OpenAIRE

Jones, Keith

2010-01-01

This chapter focuses on the linking of geometry and algebra in the teaching and learning of mathematics - and how, through such linking, the mathematics curriculum might be strengthened. Through reviewing the case of the school mathematics curriculum in England, together with examples of how the power of geometry can bring contemporary mathematics to life in the classroom, the chapter argues for greater concinnity in the mathematics curriculum, especially in terms of the harmonious/purposeful...

The algebra and geometry of SU(3) matrices

International Nuclear Information System (INIS)

Mallesh, K.S.; Mukunda, N.

1997-01-01

We give an elementary treatment of the defining representation and Lie algebra of the three-dimensional unitary unimodular group SU(3). The geometrical properties of the Lie algebra, which is an eight dimensional real linear vector space, are developed in an SU(3) covariant manner. The f and d symbols of SU(3) lead to two ways of multiplying two vectors to produce a third, and several useful geometric and algebraic identities are derived. The axis-angle parametrization of SU(3) is developed as a generalization of that for SU(2), and the specifically new features are brought out. Application to the dynamics of three-level system is outlined. (author)

Non commutative geometry methods for group C*-algebras

International Nuclear Information System (INIS)

Do Ngoc Diep.

1996-09-01

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

Topology of real algebraic varieties and related topics

CERN Document Server

Kharlamov, V M; Polotovskiĭ, G M; Viro, Oleg Ya

1996-01-01

This volume is dedicated to the memory of the Russian mathematician D. A. Gudkov. It contains papers written by his friends, students, and collaborators and is devoted mainly to the areas where D. A. Gudkov made important contributions. The main topic is the topology of real algebraic varieties. Several papers include new results on the topology of real plane algebraic curves (the Hilbert 16th problem).

Brain activity associated with translation from a visual to a symbolic representation in algebra and geometry.

Science.gov (United States)

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.

Nilpotent algebras of the generalized differential forms and the geometry of superfield theories

International Nuclear Information System (INIS)

Zupnik, B.M.

1991-01-01

We consider a new algebraic approach in the geometry of supergauge theories and supergravity. An introduction of nilpotent algebras simplifies significantly the analysis of D = 3, 4, N = 1 supergravity constraints. Different terms in the invariant action functionals of SG- and SYM-theories are constructed as the integrals of corresponding generalized differential forms. (orig.)

Difference sets connecting algebra, combinatorics, and geometry

CERN Document Server

Moore, Emily H

2013-01-01

Difference sets belong both to group theory and to combinatorics. Studying them requires tools from geometry, number theory, and representation theory. This book lays a foundation for these topics, including a primer on representations and characters of finite groups. It makes the research literature on difference sets accessible to students who have studied linear algebra and abstract algebra, and it prepares them to do their own research. This text is suitable for an undergraduate capstone course, since it illuminates the many links among topics that the students have already studied. To this end, almost every chapter ends with a coda highlighting the main ideas and emphasizing mathematical connections. This book can also be used for self-study by anyone interested in these connections and concrete examples. An abundance of exercises, varying from straightforward to challenging, invites the reader to solve puzzles, construct proofs, and investigate problems--by hand or on a computer. Hints and solutions are...

Chiral topological insulator on Nambu 3-algebraic geometry

Directory of Open Access Journals (Sweden)

Kazuki Hasebe

2014-09-01

Full Text Available 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−12+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.

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...

Algebraic structure of Robinson–Trautman and Kundt geometries in arbitrary dimension

International Nuclear Information System (INIS)

Podolský, J; Švarc, R

2015-01-01

We investigate the Weyl tensor algebraic structure of a fully general family of D-dimensional geometries that admit a non-twisting and shear-free null vector field k. From the coordinate components of the curvature tensor we explicitly derive all Weyl scalars of various boost weights. This enables us to give a complete algebraic classification of the metrics in the case when the optically privileged null direction k is a (multiple) Weyl aligned null direction (WAND). No field equations are applied, so the results are valid not only in Einstein's gravity, including its extension to higher dimensions, but also in any metric gravitation theory that admits non-twisting and shear-free spacetimes. We prove that all such geometries are of type I(b), or more special, and we derive surprisingly simple necessary and sufficient conditions under which k is a double, triple or quadruple WAND. All possible algebraically special types, including the refinement to subtypes, are thus identified, namely II(a), II(b), II(c), II(d), III(a), III(b), N, O, II i , III i , D, D(a), D(b), D(c), D(d), and their combinations. Some conditions are identically satisfied in four dimensions. We discuss both important subclasses, namely the Kundt family of geometries with the vanishing expansion (Θ=0) and the Robinson–Trautman family (Θ ≠ 0, and in particular Θ=1/r). Finally, we apply Einstein's field equations and obtain a classification of all Robinson–Trautman vacuum spacetimes. This reveals fundamental algebraic differences in the D>4 and D=4 cases, namely that in higher dimensions there only exist such spacetimes of types D(a) ≡ D(abd), D(c) ≡ D(bcd) and O. (paper)

Non commutative geometry methods for group C{sup *}-algebras

Energy Technology Data Exchange (ETDEWEB)

Diep, Do Ngoc

1996-09-01

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{sup *}-algebras: started in the elementary part, with one example of description of the structure of C{sup *}-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{sup *}-algebras were created and developed. (author). Refs.

Vertex algebras and algebraic curves

CERN Document Server

Frenkel, Edward

2004-01-01

Vertex algebras are algebraic objects that encapsulate the concept of operator product expansion from two-dimensional conformal field theory. Vertex algebras are fast becoming ubiquitous in many areas of modern mathematics, with applications to representation theory, algebraic geometry, the theory of finite groups, modular functions, topology, integrable systems, and combinatorics. This book is an introduction to the theory of vertex algebras with a particular emphasis on the relationship with the geometry of algebraic curves. The notion of a vertex algebra is introduced in a coordinate-independent way, so that vertex operators become well defined on arbitrary smooth algebraic curves, possibly equipped with additional data, such as a vector bundle. Vertex algebras then appear as the algebraic objects encoding the geometric structure of various moduli spaces associated with algebraic curves. Therefore they may be used to give a geometric interpretation of various questions of representation theory. The book co...

Distribution theory of algebraic numbers

CERN Document Server

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 analytic geometry for physical sciences

CERN Document Server

Landi, Giovanni

2018-01-01

A self-contained introduction to finite dimensional vector spaces, matrices, systems of linear equations, spectral analysis on euclidean and hermitian spaces, affine euclidean geometry, quadratic forms and conic sections. The mathematical formalism is motivated and introduced by problems from physics, notably mechanics (including celestial) and electro-magnetism, with more than two hundreds examples and solved exercises. Topics include: The group of orthogonal transformations on euclidean spaces, in particular rotations, with Euler angles and angular velocity. The rigid body with its inertia matrix. The unitary group. Lie algebras and exponential map. The Dirac’s bra-ket formalism. Spectral theory for self-adjoint endomorphisms on euclidean and hermitian spaces. The Minkowski spacetime from special relativity and the Maxwell equations. Conic sections with the use of eccentricity and Keplerian motions. An appendix collects basic algebraic notions like group, ring and field; and complex numbers and integers m...

Fast Erasure and Error decoding of Algebraic Geometry Codes up to the Feng-Rao Bound

DEFF Research Database (Denmark)

Jensen, Helge Elbrønd; Sakata, S.; Leonard, D.

1996-01-01

This paper gives an errata(that is erasure-and error-) decoding algorithm of one-point algebraic geometry codes up to the Feng-Rao designed minimum distance using Sakata's multidimensional generalization of the Berlekamp-massey algorithm and the votin procedure of Feng and Rao.......This paper gives an errata(that is erasure-and error-) decoding algorithm of one-point algebraic geometry codes up to the Feng-Rao designed minimum distance using Sakata's multidimensional generalization of the Berlekamp-massey algorithm and the votin procedure of Feng and Rao....

Algorithmic and experimental methods in algebra, geometry, and number theory

CERN Document Server

Decker, Wolfram; Malle, Gunter

2017-01-01

This book presents state-of-the-art research and survey articles that highlight work done within the Priority Program SPP 1489 “Algorithmic and Experimental Methods in Algebra, Geometry and Number Theory”, which was established and generously supported by the German Research Foundation (DFG) from 2010 to 2016. The goal of the program was to substantially advance algorithmic and experimental methods in the aforementioned disciplines, to combine the different methods where necessary, and to apply them to central questions in theory and practice. Of particular concern was the further development of freely available open source computer algebra systems and their interaction in order to create powerful new computational tools that transcend the boundaries of the individual disciplines involved.  The book covers a broad range of topics addressing the design and theoretical foundations, implementation and the successful application of algebraic algorithms in order to solve mathematical research problems. It off...

Algebraic geometry and Bethe ansatz. Part I. The quotient ring for BAE

Science.gov (United States)

Jiang, Yunfeng; Zhang, Yang

2018-03-01

In this paper and upcoming ones, we initiate a systematic study of Bethe ansatz equations for integrable models by modern computational algebraic geometry. We show that algebraic geometry provides a natural mathematical language and powerful tools for understanding the structure of solution space of Bethe ansatz equations. In particular, we find novel efficient methods to count the number of solutions of Bethe ansatz equations based on Gröbner basis and quotient ring. We also develop analytical approach based on companion matrix to perform the sum of on-shell quantities over all physical solutions without solving Bethe ansatz equations explicitly. To demonstrate the power of our method, we revisit the completeness problem of Bethe ansatz of Heisenberg spin chain, and calculate the sum rules of OPE coefficients in planar N=4 super-Yang-Mills theory.

Non-geometric flux vacua, S-duality and algebraic geometry

International Nuclear Information System (INIS)

Guarino, Adolfo; Weatherill, George James

2009-01-01

The four dimensional gauged supergravities descending from non-geometric string compactifications involve a wide class of flux objects which are needed to make the theory invariant under duality transformations at the effective level. Additionally, complex algebraic conditions involving these fluxes arise from Bianchi identities and tadpole cancellations in the effective theory. In this work we study a simple T and S-duality invariant gauged supergravity, that of a type IIB string compactified on a T 6 /Z 2 x Z 2 orientifold with O3/O7-planes. We build upon the results of recent works and develop a systematic method for solving all the flux constraints based on the algebra structure underlying the fluxes. Starting with the T-duality invariant supergravity, we find that the fluxes needed to restore S-duality can be simply implemented as linear deformations of the gauge subalgebra by an element of its second cohomology class. Algebraic geometry techniques are extensively used to solve these constraints and supersymmetric vacua, centering our attention on Minkowski solutions, become systematically computable and are also provided to clarify the methods.

Prime factorization using quantum annealing and computational algebraic geometry

Science.gov (United States)

Dridi, Raouf; Alghassi, Hedayat

2017-02-01

We investigate prime factorization from two perspectives: quantum annealing and computational algebraic geometry, specifically Gröbner bases. We present a novel autonomous algorithm which combines the two approaches and leads to the factorization of all bi-primes up to just over 200000, the largest number factored to date using a quantum processor. We also explain how Gröbner bases can be used to reduce the degree of Hamiltonians.

Prime factorization using quantum annealing and computational algebraic geometry

OpenAIRE

Dridi, Raouf; Alghassi, Hedayat

2017-01-01

We investigate prime factorization from two perspectives: quantum annealing and computational algebraic geometry, specifically Gr?bner bases. We present a novel autonomous algorithm which combines the two approaches and leads to the factorization of all bi-primes up to just over 200000, the largest number factored to date using a quantum processor. We also explain how Gr?bner bases can be used to reduce the degree of Hamiltonians.

Filtrated K-theory for real rank zero C*-algebras

DEFF Research Database (Denmark)

Arklint, Sara Esther; Restorff, Gunnar; Ruiz, Efren

2012-01-01

The smallest primitive ideal spaces for which there exist counterexamples to the classification of non-simple, purely infinite, nuclear, separable C*-algebras using filtrated K-theory, are four-point spaces. In this article, we therefore restrict to real rank zero C*-algebras with four-point prim...

From geometry to algebra and vice versa: Realistic mathematics education principles for analyzing geometry tasks

Science.gov (United States)

Jupri, Al

2017-04-01

In this article we address how Realistic Mathematics Education (RME) principles, including the intertwinement and the reality principles, are used to analyze geometry tasks. To do so, we carried out three phases of a small-scale study. First we analyzed four geometry problems - considered as tasks inviting the use of problem solving and reasoning skills - theoretically in the light of the RME principles. Second, we tested two problems to 31 undergraduate students of mathematics education program and other two problems to 16 master students of primary mathematics education program. Finally, we analyzed student written work and compared these empirical to the theoretical results. We found that there are discrepancies between what we expected theoretically and what occurred empirically in terms of mathematization and of intertwinement of mathematical concepts from geometry to algebra and vice versa. We conclude that the RME principles provide a fruitful framework for analyzing geometry tasks that, for instance, are intended for assessing student problem solving and reasoning skills.

Model Checking Process Algebra of Communicating Resources for Real-time Systems

DEFF Research Database (Denmark)

Boudjadar, Jalil; Kim, Jin Hyun; Larsen, Kim Guldstrand

2014-01-01

This paper presents a new process algebra, called PACOR, for real-time systems which deals with resource constrained timed behavior as an improved version of the ACSR algebra. We define PACOR as a Process Algebra of Communicating Resources which allows to express preemptiveness, urgent ness...

The algebraic geometry of Harper operators

International Nuclear Information System (INIS)

Li, Dan

2011-01-01

Following an approach developed by Gieseker, Knoerrer and Trubowitz for discretized Schroedinger operators, we study the spectral theory of Harper operators in dimensions 2 and 1, as a discretized model of magnetic Laplacians, from the point of view of algebraic geometry. We describe the geometry of an associated family of Bloch varieties and compute their density of states. Finally, we also compute some spectral functions based on the density of states. We discuss the difference between the cases with rational or irrational parameters: for the two-dimensional Harper operator, the compactification of the Bloch variety is an ordinary variety in the rational case and an ind-pro-variety in the irrational case. This gives rise, at the algebro-geometric level of Bloch varieties, to a phenomenon similar to the Hofstadter butterfly in the spectral theory. In dimension 2, the density of states can be expressed in terms of period integrals over Fermi curves, where the resulting elliptic integrals are independent of the parameters. In dimension 1, for the almost Mathieu operator, with a similar argument, we find the usual dependence of the spectral density on the parameter, which gives rise to the well-known Hofstadter butterfly picture. (paper)

Model checking process algebra of communicating resources for real-time systems

DEFF Research Database (Denmark)

Boudjadar, Jalil; Kim, Jin Hyun; Larsen, Kim Guldstrand

2014-01-01

This paper presents a new process algebra, called PACoR, for real-time systems which deals with resource- constrained timed behavior as an improved version of the ACSR algebra. We define PACoR as a Process Algebra of Communicating Resources which allows to explicitly express preemptiveness...

Algebraic methods in random matrices and enumerative geometry

CERN Document Server

Eynard, Bertrand

2008-01-01

We review the method of symplectic invariants recently introduced to solve matrix models loop equations, and further extended beyond the context of matrix models. For any given spectral curve, one defined a sequence of differential forms, and a sequence of complex numbers Fg . We recall the definition of the invariants Fg, and we explain their main properties, in particular symplectic invariance, integrability, modularity,... Then, we give several example of applications, in particular matrix models, enumeration of discrete surfaces (maps), algebraic geometry and topological strings, non-intersecting brownian motions,...

The algebraic geometry of Harper operators

Science.gov (United States)

Li, Dan

2011-10-01

Following an approach developed by Gieseker, Knörrer and Trubowitz for discretized Schrödinger operators, we study the spectral theory of Harper operators in dimensions 2 and 1, as a discretized model of magnetic Laplacians, from the point of view of algebraic geometry. We describe the geometry of an associated family of Bloch varieties and compute their density of states. Finally, we also compute some spectral functions based on the density of states. We discuss the difference between the cases with rational or irrational parameters: for the two-dimensional Harper operator, the compactification of the Bloch variety is an ordinary variety in the rational case and an ind-pro-variety in the irrational case. This gives rise, at the algebro-geometric level of Bloch varieties, to a phenomenon similar to the Hofstadter butterfly in the spectral theory. In dimension 2, the density of states can be expressed in terms of period integrals over Fermi curves, where the resulting elliptic integrals are independent of the parameters. In dimension 1, for the almost Mathieu operator, with a similar argument, we find the usual dependence of the spectral density on the parameter, which gives rise to the well-known Hofstadter butterfly picture.

Geometric invariant theory over the real and complex numbers

CERN Document Server

Wallach, Nolan R

2017-01-01

Geometric Invariant Theory (GIT) is developed in this text within the context of algebraic geometry over the real and complex numbers. This sophisticated topic is elegantly presented with enough background theory included to make the text accessible to advanced graduate students in mathematics and physics with diverse backgrounds in algebraic and differential geometry.  Throughout the book, examples are emphasized. Exercises add to the reader’s understanding of the material; most are enhanced with hints. The exposition is divided into two parts. The first part, ‘Background Theory’, is organized as a reference for the rest of the book. It contains two chapters developing material in complex and real algebraic geometry and algebraic groups that are difficult to find in the literature. Chapter 1 emphasizes the relationship between the Zariski topology and the canonical Hausdorff topology of an algebraic variety over the complex numbers. Chapter 2 develops the interaction between Lie groups and algebraic ...

Geometry of Spin: Clifford Algebraic Approach

Indian Academy of Sciences (India)

Then the various algebraic properties of Pauli matricesare studied as properties of matrix algebra. What has beenshown in this article is that Pauli matrices are a representationof Clifford algebra of spin and hence all the propertiesof Pauli matrices follow from the underlying algebra. Cliffordalgebraic approach provides a ...

Asynchronous communication in real space process algebra

NARCIS (Netherlands)

Baeten, J.C.M.; Bergstra, J.A.

1991-01-01

A version of classical real space process algebra is given in which messages travel with constant speed through a three-dimensional medium. It follows that communication is asynchronous and has a broadcasting character. A state operator is used to describe asynchronous message transfer and a

Asynchronous communication in real space process algebra

NARCIS (Netherlands)

Bergstra, J.A.; Baeten, J.C.M.

1992-01-01

A version of classical real space process algebra is given in which messages travel with constant speed through a three-dimensional medium. It follows that communication is asynchronous and has a broadcasting character. A state operator is used to describe asynchronous message transfer and a

Quadratic algebras

CERN Document Server

Polishchuk, Alexander

2005-01-01

Quadratic algebras, i.e., algebras defined by quadratic relations, often occur in various areas of mathematics. One of the main problems in the study of these (and similarly defined) algebras is how to control their size. A central notion in solving this problem is the notion of a Koszul algebra, which was introduced in 1970 by S. Priddy and then appeared in many areas of mathematics, such as algebraic geometry, representation theory, noncommutative geometry, K-theory, number theory, and noncommutative linear algebra. The book offers a coherent exposition of the theory of quadratic and Koszul algebras, including various definitions of Koszulness, duality theory, Poincar�-Birkhoff-Witt-type theorems for Koszul algebras, and the Koszul deformation principle. In the concluding chapter of the book, they explain a surprising connection between Koszul algebras and one-dependent discrete-time stochastic processes.

Vladimir I. Arnold collected works : hydrodynamics, bifurcation theory, algebraic geometry : 1965-1972

CERN Document Server

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.

Principles of algebraic geometry

CERN Document Server

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

Multiplier ideal sheaves and analytic methods in algebraic geometry

International Nuclear Information System (INIS)

Demailly, J.-P.

2001-01-01

Our main purpose here is to describe a few analytic tools which are useful to study questions such as linear series and vanishing theorems for algebraic vector bundles. One of the early successes of analytic methods in this context is Kodaira's use of the Bochner technique in relation with the theory of harmonic forms, during the decade 1950-60.The idea is to represent cohomology classes by harmonic forms and to prove vanishing theorems by means of suitable a priori curvature estimates. We pursue the study of L2 estimates, in relation with the Nullstellenstatz and with the extension problem. We show how subadditivity can be used to derive an approximation theorem for (almost) plurisubharmonic functions: any such function can be approximated by a sequence of (almost) plurisubharmonic functions which are smooth outside an analytic set, and which define the same multiplier ideal sheaves. From this, we derive a generalized version of the hard Lefschetz theorem for cohomology with values in a pseudo-effective line bundle; namely, the Lefschetz map is surjective when the cohomology groups are twisted by the relevant multiplier ideal sheaves. These notes are essentially written with the idea of serving as an analytic tool- box for algebraic geometers. Although efficient algebraic techniques exist, our feeling is that the analytic techniques are very flexible and offer a large variety of guidelines for more algebraic questions (including applications to number theory which are not discussed here). We made a special effort to use as little prerequisites and to be as self-contained as possible; hence the rather long preliminary sections dealing with basic facts of complex differential geometry

Multiplier ideal sheaves and analytic methods in algebraic geometry

Energy Technology Data Exchange (ETDEWEB)

Demailly, J -P [Universite de Grenoble I, Institut Fourier, Saint-Martin d' Heres (France)

2001-12-15

Our main purpose here is to describe a few analytic tools which are useful to study questions such as linear series and vanishing theorems for algebraic vector bundles. One of the early successes of analytic methods in this context is Kodaira's use of the Bochner technique in relation with the theory of harmonic forms, during the decade 1950-60.The idea is to represent cohomology classes by harmonic forms and to prove vanishing theorems by means of suitable a priori curvature estimates. We pursue the study of L2 estimates, in relation with the Nullstellenstatz and with the extension problem. We show how subadditivity can be used to derive an approximation theorem for (almost) plurisubharmonic functions: any such function can be approximated by a sequence of (almost) plurisubharmonic functions which are smooth outside an analytic set, and which define the same multiplier ideal sheaves. From this, we derive a generalized version of the hard Lefschetz theorem for cohomology with values in a pseudo-effective line bundle; namely, the Lefschetz map is surjective when the cohomology groups are twisted by the relevant multiplier ideal sheaves. These notes are essentially written with the idea of serving as an analytic tool- box for algebraic geometers. Although efficient algebraic techniques exist, our feeling is that the analytic techniques are very flexible and offer a large variety of guidelines for more algebraic questions (including applications to number theory which are not discussed here). We made a special effort to use as little prerequisites and to be as self-contained as possible; hence the rather long preliminary sections dealing with basic facts of complex differential geometry.

Euclidean distance degrees of real algebraic groups

NARCIS (Netherlands)

Baaijens, J.A.; Draisma, J.

2015-01-01

We study the problem of finding, in a real algebraic matrix group, the matrix closest to a given data matrix. We do so from the algebro-geometric perspective of Euclidean distance degrees. We recover several classical results; and among the new results that we prove is a formula for the Euclidean

Euclidean distance degrees of real algebraic groups

NARCIS (Netherlands)

Baaijens, J.A.; Draisma, J.

2014-01-01

We study the problem of finding, in a real algebraic matrix group, the matrix closest to a given data matrix. We do so from the algebro-geometric perspective of Euclidean distance degrees. We recover several classical results; and among the new results that we prove is a formula for the Euclidean

Invertibility-preserving maps of C∗-algebras with real rank zero

Directory of Open Access Journals (Sweden)

Istvan Kovacs

2005-01-01

Full Text Available In 1996, Harris and Kadison posed the following problem: show that a linear bijection between C∗-algebras that preserves the identity and the set of invertible elements is a Jordan isomorphism. In this paper, we show that if A and B are semisimple Banach algebras and Φ:A→B is a linear map onto B that preserves the spectrum of elements, then Φ is a Jordan isomorphism if either A or B is a C∗-algebra of real rank zero. We also generalize a theorem of Russo.

Scattering Amplitudes via Algebraic Geometry Methods

CERN Document Server

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 ...

Geometry of Spin: Clifford Algebraic Approach

Indian Academy of Sciences (India)

of Pauli matrices follow from the underlying algebra. Clif- ford algebraic approach provides a geometrical and hence intuitive way to understand quantum theory of spin, and is a natural formalism to study spin. Clifford algebraic formal- ism has lot of applications in every field where spin plays an important role. Introduction.

Descartes on the Unification of Arithmetic, Algebra and Geometry Via the Theory of Proportions

Czech Academy of Sciences Publication Activity Database

Crippa, Davide

2017-01-01

Roč. 73, č. 3/4 (2017), s. 1239-1258 ISSN 0870-5283 Institutional support: RVO:67985955 Keywords : algebra * Descartes * Euclid * geometry * multiplication * proportion theory * structure Subject RIV: AA - Philosophy ; Religion OBOR OECD: Philosophy, History and Philosophy of science and technology

The homogeneous geometries of real hyperbolic space

DEFF Research Database (Denmark)

Castrillón López, Marco; Gadea, Pedro Martínez; Swann, Andrew Francis

We describe the holonomy algebras of all canonical connections of homogeneous structures on real hyperbolic spaces in all dimensions. The structural results obtained then lead to a determination of the types, in the sense of Tricerri and Vanhecke, of the corresponding homogeneous tensors. We use...... our analysis to show that the moduli space of homogeneous structures on real hyperbolic space has two connected components....

Linear algebra meets Lie algebra: the Kostant-Wallach theory

OpenAIRE

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.

Lectures on algebraic statistics

CERN Document Server

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.

Algebraic geometry and effective lagrangians

International Nuclear Information System (INIS)

Martinec, E.J.; Chicago Univ., IL

1989-01-01

N=2 supersymmetric Landau-Ginsburg fixed points describe nonlinear models whose target spaces are algebraic varieties in certain generalized projective spaces; the defining equation is precisely the zero set of the superpotential, considered as a condition in the projective space. The ADE classification of modular invariants arises as the classification of projective descriptions of P 1 ; in general, the hierarchy of fixed points is conjectured to be isomorphic to the classification of quasihomogeneous singularities. The condition of vanishing first Chern class is an integrality condition on the Virasoro central charge; the central charge is determined by the superpotential. The operator algebra is given by the algebra of Wick contractions of perturbations of the superpotential. (orig.)

Open algebraic surfaces

CERN Document Server

Miyanishi, Masayoshi

2000-01-01

Open algebraic surfaces are a synonym for algebraic surfaces that are not necessarily complete. An open algebraic surface is understood as a Zariski open set of a projective algebraic surface. There is a long history of research on projective algebraic surfaces, and there exists a beautiful Enriques-Kodaira classification of such surfaces. The research accumulated by Ramanujan, Abhyankar, Moh, and Nagata and others has established a classification theory of open algebraic surfaces comparable to the Enriques-Kodaira theory. This research provides powerful methods to study the geometry and topology of open algebraic surfaces. The theory of open algebraic surfaces is applicable not only to algebraic geometry, but also to other fields, such as commutative algebra, invariant theory, and singularities. This book contains a comprehensive account of the theory of open algebraic surfaces, as well as several applications, in particular to the study of affine surfaces. Prerequisite to understanding the text is a basic b...

Fast Erasure-and error decoding of algebraic geometry codes up to the Feng-Rao bound

DEFF Research Database (Denmark)

Høholdt, Tom; Jensen, Helge Elbrønd; Sakata, Shojiro

1998-01-01

This correspondence gives an errata (that is erasure-and error-) decoding algorithm of one-point algebraic-geometry codes up to the Feng-Rao designed minimum distance using Sakata's multidimensional generalization of the Berlekamp-Massey algorithm and the voting procedure of Feng and Rao....

Representations of fundamental groups of algebraic varieties

CERN Document Server

Zuo, Kang

1999-01-01

Using harmonic maps, non-linear PDE and techniques from algebraic geometry this book enables the reader to study the relation between fundamental groups and algebraic geometry invariants of algebraic varieties. The reader should have a basic knowledge of algebraic geometry and non-linear analysis. This book can form the basis for graduate level seminars in the area of topology of algebraic varieties. It also contains present new techniques for researchers working in this area.

Chiral-Yang-Mills theory, non commutative differential geometry, and the need for a Lie super-algebra

International Nuclear Information System (INIS)

Thierry-Mieg, Jean

2006-01-01

In Yang-Mills theory, the charges of the left and right massless Fermions are independent of each other. We propose a new paradigm where we remove this freedom and densify the algebraic structure of Yang-Mills theory by integrating the scalar Higgs field into a new gauge-chiral 1-form which connects Fermions of opposite chiralities. Using the Bianchi identity, we prove that the corresponding covariant differential is associative if and only if we gauge a Lie-Kac super-algebra. In this model, spontaneous symmetry breakdown naturally occurs along an odd generator of the super-algebra and induces a representation of the Connes-Lott non commutative differential geometry of the 2-point finite space

Asynchronous communication in real space process algebra

OpenAIRE

Baeten, JCM Jos; Bergstra, JA Jan

1990-01-01

A version of classical real space process algebra is given in which messages travel with constant speed through a three-dimensional medium. It follows that communication is asynchronous and has a broadcasting character. A state operator is used to describe asynchronous message transfer and a priority mechanism allows to express the broadcasting mechanism. As an application, a protocol is specified in which the receiver moves with respect to the sender.

Real solutions to equations from geometry

CERN Document Server

Sottile, Frank

2011-01-01

Understanding, finding, or even deciding on the existence of real solutions to a system of equations is a difficult problem with many applications outside of mathematics. While it is hopeless to expect much in general, we know a surprising amount about these questions for systems which possess additional structure often coming from geometry. This book focuses on equations from toric varieties and Grassmannians. Not only is much known about these, but such equations are common in applications. There are three main themes: upper bounds on the number of real solutions, lower bounds on the number of real solutions, and geometric problems that can have all solutions be real. The book begins with an overview, giving background on real solutions to univariate polynomials and the geometry of sparse polynomial systems. The first half of the book concludes with fewnomial upper bounds and with lower bounds to sparse polynomial systems. The second half of the book begins by sampling some geometric problems for which all ...

Algebra

CERN Document Server

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.

A zero-dimensional approach to compute real radicals

Directory of Open Access Journals (Sweden)

Silke J. Spang

2008-04-01

Full Text Available The notion of real radicals is a fundamental tool in Real Algebraic Geometry. It takes the role of the radical ideal in Complex Algebraic Geometry. In this article I shall describe the zero-dimensional approach and efficiency improvement I have found during the work on my diploma thesis at the University of Kaiserslautern (cf. [6]. The main focus of this article is on maximal ideals and the properties they have to fulfil to be real. New theorems and properties about maximal ideals are introduced which yield an heuristic prepare_max which splits the maximal ideals into three classes, namely real, not real and the class where we can't be sure whether they are real or not. For the latter we have to apply a coordinate change into general position until we are sure about realness. Finally this constructs a randomized algorithm for real radicals. The underlying theorems and algorithms are described in detail.

Multilinear Computing and Multilinear Algebraic Geometry

Science.gov (United States)

2016-08-10

algebra : linear systems, least squares, eigevalue problems, singular value problems, determinant evaluation, low-rank approximations, etc — problems...intractability to move beyond linear algebra , substantiating what the PI had proposed. High-resolution MRI with tensors: In another piece of work... applications . One reason is that we found out that many statistical estimation problems ( linear regression, errors-in-variables regression, principal components

Cylindric-like algebras and algebraic logic

CERN Document Server

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.

A concrete approach to abstract algebra from the integers to the insolvability of the quintic

CERN Document Server

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

Pseudo-Riemannian Novikov algebras

Energy Technology Data Exchange (ETDEWEB)

Chen Zhiqi; Zhu Fuhai [School of Mathematical Sciences and LPMC, Nankai University, Tianjin 300071 (China)], E-mail: chenzhiqi@nankai.edu.cn, E-mail: zhufuhai@nankai.edu.cn

2008-08-08

Novikov algebras were introduced in connection with the Poisson brackets of hydrodynamic-type and Hamiltonian operators in formal variational calculus. Pseudo-Riemannian Novikov algebras denote Novikov algebras with non-degenerate invariant symmetric bilinear forms. In this paper, we find that there is a remarkable geometry on pseudo-Riemannian Novikov algebras, and give a special class of pseudo-Riemannian Novikov algebras.

Standard integral table algebras generated by non-real element of small degree

CERN Document Server

Muzychuk, Mikhail

2002-01-01

This book is addressed to the researchers working in the theory of table algebras and association schemes. This area of algebraic combinatorics has been rapidly developed during the last decade. The volume contains further developments in the theory of table algebras. It collects several papers which deal with a classification problem for standard integral table algebras (SITA). More precisely, we consider SITA with a faithful non-real element of small degree. It turns out that such SITA with some extra conditions may be classified. This leads to new infinite series of SITA which has interesting properties. The last section of the book uses a part of obtained results in the classification of association schemes. This volume summarizes the research which was done at Bar-Ilan University in the academic year 1998/99.

Current algebra and differential geometry

International Nuclear Information System (INIS)

Alekseev, Anton; Strobl, Thomas

2005-01-01

We show that symmetries and gauge symmetries of a large class of 2-dimensional sigma models are described by a new type of a current algebra. The currents are labeled by pairs of a vector field and a 1-form on the target space of the sigma model. We compute the current-current commutator and analyse the anomaly cancellation condition, which can be interpreted geometrically in terms of Dirac structures, previously studied in the mathematical literature. Generalized complex structures correspond to decompositions of the current algebra into pairs of anomaly free subalgebras. Sigma models that we can treat with our method include both physical and topological examples, with and without Wess-Zumino type terms. (author)

An Algorithm for Isolating the Real Solutions of Piecewise Algebraic Curves

Directory of Open Access Journals (Sweden)

Jinming Wu

2011-01-01

to compute the real solutions of two piecewise algebraic curves. It is primarily based on the Krawczyk-Moore iterative algorithm and good initial iterative interval searching algorithm. The proposed algorithm is relatively easy to implement.

Strong result for real zeros of random algebraic polynomials

Directory of Open Access Journals (Sweden)

T. Uno

2001-01-01

Full Text Available An estimate is given for the lower bound of real zeros of random algebraic polynomials whose coefficients are non-identically distributed dependent Gaussian random variables. Moreover, our estimated measure of the exceptional set, which is independent of the degree of the polynomials, tends to zero as the degree of the polynomial tends to infinity.

Euclidean distance geometry an introduction

CERN Document Server

Liberti, Leo

2017-01-01

This textbook, the first of its kind, presents the fundamentals of distance geometry:  theory, useful methodologies for obtaining solutions, and real world applications. Concise proofs are given and step-by-step algorithms for solving fundamental problems efficiently and precisely are presented in Mathematica®, enabling the reader to experiment with concepts and methods as they are introduced. Descriptive graphics, examples, and problems, accompany the real gems of the text, namely the applications in visualization of graphs, localization of sensor networks, protein conformation from distance data, clock synchronization protocols, robotics, and control of unmanned underwater vehicles, to name several.  Aimed at intermediate undergraduates, beginning graduate students, researchers, and practitioners, the reader with a basic knowledge of linear algebra will gain an understanding of the basic theories of distance geometry and why they work in real life.

Pole-placement Predictive Functional Control for under-damped systems with real numbers algebra.

Science.gov (United States)

Zabet, K; Rossiter, J A; Haber, R; Abdullah, M

2017-11-01

This paper presents the new algorithm of PP-PFC (Pole-placement Predictive Functional Control) for stable, linear under-damped higher-order processes. It is shown that while conventional PFC aims to get first-order exponential behavior, this is not always straightforward with significant under-damped modes and hence a pole-placement PFC algorithm is proposed which can be tuned more precisely to achieve the desired dynamics, but exploits complex number algebra and linear combinations in order to deliver guarantees of stability and performance. Nevertheless, practical implementation is easier by avoiding complex number algebra and hence a modified formulation of the PP-PFC algorithm is also presented which utilises just real numbers while retaining the key attributes of simple algebra, coding and tuning. The potential advantages are demonstrated with numerical examples and real-time control of a laboratory plant. Copyright © 2017 ISA. All rights reserved.

Real-Time Algebraic Derivative Estimations Using a Novel Low-Cost Architecture Based on Reconfigurable Logic

Science.gov (United States)

Morales, Rafael; Rincón, Fernando; Gazzano, Julio Dondo; López, Juan Carlos

2014-01-01

Time derivative estimation of signals plays a very important role in several fields, such as signal processing and control engineering, just to name a few of them. For that purpose, a non-asymptotic algebraic procedure for the approximate estimation of the system states is used in this work. The method is based on results from differential algebra and furnishes some general formulae for the time derivatives of a measurable signal in which two algebraic derivative estimators run simultaneously, but in an overlapping fashion. The algebraic derivative algorithm presented in this paper is computed online and in real-time, offering high robustness properties with regard to corrupting noises, versatility and ease of implementation. Besides, in this work, we introduce a novel architecture to accelerate this algebraic derivative estimator using reconfigurable logic. The core of the algorithm is implemented in an FPGA, improving the speed of the system and achieving real-time performance. Finally, this work proposes a low-cost platform for the integration of hardware in the loop in MATLAB. PMID:24859033

The algebraic geometry of multimonopoles

International Nuclear Information System (INIS)

Nahm, W.

1982-11-01

Multimonopole solutions of the Bogomolny equation are treated by a transform to an ordinary differential equation. The solution of this equation yields algebraic curves and holomorphic line bundles over them. (orig.)

Higher geometry an introduction to advanced methods in analytic geometry

CERN Document Server

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

On the Invariant Uniform Roe Algebra as Crossed Product

OpenAIRE

Kankeyanathan Kannan

2013-01-01

The uniform Roe C*-algebra (also called uniform translation)C^*- algebra provides a link between coarse geometry and C^*- algebra theory. The uniform Roe algebra has a great importance in geometry, topology and analysis. We consider some of the elementary concepts associated with coarse spaces.

Algebraic K-theory

CERN Document Server

Srinivas, V

1996-01-01

Algebraic K-Theory has become an increasingly active area of research. With its connections to algebra, algebraic geometry, topology, and number theory, it has implications for a wide variety of researchers and graduate students in mathematics. The book is based on lectures given at the author's home institution, the Tata Institute in Bombay, and elsewhere. A detailed appendix on topology was provided in the first edition to make the treatment accessible to readers with a limited background in topology. The second edition also includes an appendix on algebraic geometry that contains the required definitions and results needed to understand the core of the book; this makes the book accessible to a wider audience. A central part of the book is a detailed exposition of the ideas of Quillen as contained in his classic papers "Higher Algebraic K-Theory, I, II." A more elementary proof of the theorem of Merkujev--Suslin is given in this edition; this makes the treatment of this topic self-contained. An application ...

Algebraic theory of numbers

CERN Document Server

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

Categorical Algebra and its Applications

CERN Document Server

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.

Big Bang, Blowup, and Modular Curves: Algebraic Geometry in Cosmology

Science.gov (United States)

Manin, Yuri I.; Marcolli, Matilde

2014-07-01

We introduce some algebraic geometric models in cosmology related to the ''boundaries'' of space-time: Big Bang, Mixmaster Universe, Penrose's crossovers between aeons. We suggest to model the kinematics of Big Bang using the algebraic geometric (or analytic) blow up of a point x. This creates a boundary which consists of the projective space of tangent directions to x and possibly of the light cone of x. We argue that time on the boundary undergoes the Wick rotation and becomes purely imaginary. The Mixmaster (Bianchi IX) model of the early history of the universe is neatly explained in this picture by postulating that the reverse Wick rotation follows a hyperbolic geodesic connecting imaginary time axis to the real one. Penrose's idea to see the Big Bang as a sign of crossover from ''the end of previous aeon'' of the expanding and cooling Universe to the ''beginning of the next aeon'' is interpreted as an identification of a natural boundary of Minkowski space at infinity with the Big Bang boundary.

Algebraic K-theory of generalized schemes

DEFF Research Database (Denmark)

Anevski, Stella Victoria Desiree

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......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......, appearing in the construction of \\Spec Z....

Algebraic and computational aspects of real tensor ranks

CERN Document Server

Sakata, Toshio; Miyazaki, Mitsuhiro

2016-01-01

This book provides comprehensive summaries of theoretical (algebraic) and computational aspects of tensor ranks, maximal ranks, and typical ranks, over the real number field. Although tensor ranks have been often argued in the complex number field, it should be emphasized that this book treats real tensor ranks, which have direct applications in statistics. The book provides several interesting ideas, including determinant polynomials, determinantal ideals, absolutely nonsingular tensors, absolutely full column rank tensors, and their connection to bilinear maps and Hurwitz-Radon numbers. In addition to reviews of methods to determine real tensor ranks in details, global theories such as the Jacobian method are also reviewed in details. The book includes as well an accessible and comprehensive introduction of mathematical backgrounds, with basics of positive polynomials and calculations by using the Groebner basis. Furthermore, this book provides insights into numerical methods of finding tensor ranks through...

Real forms of complex quantum anti-de-Sitter algebra Uq(Sp(4;C)) and their contraction schemes

International Nuclear Information System (INIS)

Lukierski, J.; Nowicki, A.; Ruegg, H.

1991-01-01

We describe four types of inner involutions of the Cartan-Weyl basis providing (for vertical strokeqvertical stroke=1 and q real) three types of real quantum Lie algebras: U q (O(3, 2)) (quantum D=4 anti-de-Sitter), U q (O(4, 1)) (quantum D=4 de-Sitter) and U q (O(5)). We give also two types of inner involutions of the Cartan-Chevalley basis of U q (Sp(4; C)) which cannot be extended to inner involutions of the Cartan-Weyl basis. We outline twelve contraction schemes for quantum D=4 anti-de-Sitter algebra. All these contractions provide four commuting translation generators, but only two (one for vertical strokeqvertical stroke=1, the second for q real) lead to the quantum Poincare algebra with an undeformed space rotation O(3) subalgebra. (orig.)

Algebraic Methods in Plane Geometry

Indian Academy of Sciences (India)

Home; Journals; Resonance – Journal of Science Education; Volume 13; Issue 10. Algebraic Methods in ... General Article Volume 13 Issue 10 October 2008 pp 916-928 ... Keywords. Conics; family of curves; Pascal's theorem; homogeneous coordinates; Butterfly theorem; abelian group; associativity of addition; group law.

Answers to selected problems in multivariable calculus with linear algebra and series

CERN Document Server

Trench, William F

1972-01-01

Answers to Selected Problems in Multivariable Calculus with Linear Algebra and Series contains the answers to selected problems in linear algebra, the calculus of several variables, and series. Topics covered range from vectors and vector spaces to linear matrices and analytic geometry, as well as differential calculus of real-valued functions. Theorems and definitions are included, most of which are followed by worked-out illustrative examples.The problems and corresponding solutions deal with linear equations and matrices, including determinants; vector spaces and linear transformations; eig

Algebraic groups and their birational invariants

CERN Document Server

Voskresenskiĭ, V E

2011-01-01

Since the late 1960s, methods of birational geometry have been used successfully in the theory of linear algebraic groups, especially in arithmetic problems. This book--which can be viewed as a significant revision of the author's book, Algebraic Tori (Nauka, Moscow, 1977)--studies birational properties of linear algebraic groups focusing on arithmetic applications. The main topics are forms and Galois cohomology, the Picard group and the Brauer group, birational geometry of algebraic tori, arithmetic of algebraic groups, Tamagawa numbers, R-equivalence, projective toric varieties, invariants of finite transformation groups, and index-formulas. Results and applications are recent. There is an extensive bibliography with additional comments that can serve as a guide for further reading.

The algebraic approach to space-time geometry

International Nuclear Information System (INIS)

Heller, M.; Multarzynski, P.; Sasin, W.

1989-01-01

A differential manifold can be defined in terms of smooth real functions carried by it. By rejecting the postulate, in such a definition, demanding the local diffeomorphism of a manifold to the Euclidean space, one obtains the so-called differential space concept. Every subset of R n turns out to be a differential space. Extensive parts of differential geometry on differential spaces, developed by Sikorski, are reviewed and adapted to relativistic purposes. Differential space as a new model of space-time is proposed. The Lorentz structure and Einstein's field equations on differential spaces are discussed. 20 refs. (author)

Cartan calculus on quantum Lie algebras

International Nuclear Information System (INIS)

Schupp, P.; Watts, P.; Zumino, B.

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 au into one big algebra, the ''Cartan Calculus.''

Algebraic partial Boolean algebras

International Nuclear Information System (INIS)

Smith, Derek

2003-01-01

Partial Boolean algebras, first studied by Kochen and Specker in the 1960s, provide the structure for Bell-Kochen-Specker theorems which deny the existence of non-contextual hidden variable theories. In this paper, we study partial Boolean algebras which are 'algebraic' in the sense that their elements have coordinates in an algebraic number field. Several of these algebras have been discussed recently in a debate on the validity of Bell-Kochen-Specker theorems in the context of finite precision measurements. The main result of this paper is that every algebraic finitely-generated partial Boolean algebra B(T) is finite when the underlying space H is three-dimensional, answering a question of Kochen and showing that Conway and Kochen's infinite algebraic partial Boolean algebra has minimum dimension. This result contrasts the existence of an infinite (non-algebraic) B(T) generated by eight elements in an abstract orthomodular lattice of height 3. We then initiate a study of higher-dimensional algebraic partial Boolean algebras. First, we describe a restriction on the determinants of the elements of B(T) that are generated by a given set T. We then show that when the generating set T consists of the rays spanning the minimal vectors in a real irreducible root lattice, B(T) is infinite just if that root lattice has an A 5 sublattice. Finally, we characterize the rays of B(T) when T consists of the rays spanning the minimal vectors of the root lattice E 8

Real tunneling geometries and the large-scale topology of the universe

International Nuclear Information System (INIS)

Gibbons, G.W.; Hartle, J.B.

1990-01-01

If the topology and geometry of spacetime are quantum-mechanically variable, then the particular classical large-scale topology and geometry observed in our universe must be statistical predictions of its initial condition. This paper examines the predictions of the ''no boundary'' initial condition for the present large-scale topology and geometry. Finite-action real tunneling solutions of Einstein's equation are important for such predictions. These consist of compact Riemannian (Euclidean) geometries joined to a Lorentzian cosmological geometry across a spacelike surface of vanishing extrinsic curvature. The classification of such solutions is discussed and general constraints on their topology derived. For example, it is shown that, if the Euclidean Ricci tensor is positive, then a real tunneling solution can nucleate only a single connected Lorentzian spacetime (the unique conception theorem). Explicit examples of real tunneling solutions driven by a cosmological constant are exhibited and their implications for cosmic baldness described. It is argued that the most probable large-scale spacetime predicted by the real tunneling solutions of the ''no-boundary'' initial condition has the topology RxS 3 with the de Sitter metric

Leavitt path algebras

CERN Document Server

Abrams, Gene; Siles Molina, Mercedes

2017-01-01

This book offers a comprehensive introduction by three of the leading experts in the field, collecting fundamental results and open problems in a single volume. Since Leavitt path algebras were first defined in 2005, interest in these algebras has grown substantially, with ring theorists as well as researchers working in graph C*-algebras, group theory and symbolic dynamics attracted to the topic. Providing a historical perspective on the subject, the authors review existing arguments, establish new results, and outline the major themes and ring-theoretic concepts, such as the ideal structure, Z-grading and the close link between Leavitt path algebras and graph C*-algebras. The book also presents key lines of current research, including the Algebraic Kirchberg Phillips Question, various additional classification questions, and connections to noncommutative algebraic geometry. Leavitt Path Algebras will appeal to graduate students and researchers working in the field and related areas, such as C*-algebras and...

Reductive Lie-admissible algebras applied to H-spaces and connections

International Nuclear Information System (INIS)

Sagle, A.A.

1982-01-01

An algebra A with multiplication xy is Lie-admissible if the vector space A with new multiplication [x,y] = xy-yx is a Lie algebra; we denote this Lie algebra by A - . Thus, an associative algebra is Lie-admissible but a Cayley algebra is not Lie-admissible. In this paper we show how Lie-admissible algebras arise from Lie groups and their application to differential geometry on Lie groups via the following theorem. Let A be an n-dimensional Lie-admissible algebra over the reals. Let G be a Lie group with multiplication function μ and with Lie algebra g which is isomorphic to A - . Then there exiss a corrdinate system at the identify e in G which represents μ by a function F:gxg→g defined locally at the origin, such that the second derivative, F 2 , at the origin defines on the vector space g the structure of a nonassociative algebra (g, F 2 ). Furthermore this algebra is isomorphic to A and (g, F 2 ) - is isomorphic to A - . Thus roughly, any Lie-admissible algebra is isomorphic to an algebra obtained from a Lie algebra via a change of coordinates in the Lie group. Lie algebras arise by using canonical coordinates and the Campbell-Hausdorff formula. Applications of this show that any G-invariant psuedo-Riemannian connection on G is completely determined by a suitable Lie-admissible algebra. These results extend to H-spaces, reductive Lie-admissible algebras and connections on homogeneous H-spaces. Thus, alternative and other non-Lie-admissible algebras can be utilized

Topological properties of real algebraic varieties: du cote de chez Rokhlin

Energy Technology Data Exchange (ETDEWEB)

Degtyarev, A I [Bilkent University, Ankara (Turkey); Kharlamov, V M [Institut de Recherche Matematique Avanee, Universite Louis Pasteur et CNRS 7, rue Rene Descartes (France)

2000-08-31

The survey covers results in the topology of real algebraic varieties in the direction initiated in the early seventies by V.I. Arnol'd and V.A. Rokhlin. We make an attempt to systematize the principal achievements in the subject. After presenting general tools and results, we pay special attention to surfaces and curves on surfaces.

Topological properties of real algebraic varieties: du cote de chez Rokhlin

International Nuclear Information System (INIS)

Degtyarev, A I; Kharlamov, V M

2000-01-01

The survey covers results in the topology of real algebraic varieties in the direction initiated in the early seventies by V.I. Arnol'd and V.A. Rokhlin. We make an attempt to systematize the principal achievements in the subject. After presenting general tools and results, we pay special attention to surfaces and curves on surfaces

Noncommutative geometry and twisted conformal symmetry

International Nuclear Information System (INIS)

Matlock, Peter

2005-01-01

The twist-deformed conformal algebra is constructed as a Hopf algebra with twisted coproduct. This allows for the definition of conformal symmetry in a noncommutative background geometry. The twisted coproduct is reviewed for the Poincare algebra and the construction is then extended to the full conformal algebra. The case of Moyal-type noncommutativity of the coordinates is considered. It is demonstrated that conformal invariance need not be viewed as incompatible with noncommutative geometry; the noncommutativity of the coordinates appears as a consequence of the twisting, as has been shown in the literature in the case of the twisted Poincare algebra

Head First Algebra A Learner's Guide to Algebra I

CERN Document Server

Pilone, Tracey

2008-01-01

Having trouble understanding algebra? Do algebraic concepts, equations, and logic just make your head spin? We have great news: Head First Algebra is designed for you. Full of engaging stories and practical, real-world explanations, this book will help you learn everything from natural numbers and exponents to solving systems of equations and graphing polynomials. Along the way, you'll go beyond solving hundreds of repetitive problems, and actually use what you learn to make real-life decisions. Does it make sense to buy two years of insurance on a car that depreciates as soon as you drive i

Algebra II textbook for students of mathematics

CERN Document Server

Gorodentsev, Alexey L

2017-01-01

This book is the second volume of an intensive “Russian-style” two-year undergraduate course in abstract algebra, and introduces readers to the basic algebraic structures – fields, rings, modules, algebras, groups, and categories – and explains the main principles of and methods for working with them. The course covers substantial areas of advanced combinatorics, geometry, linear and multilinear algebra, representation theory, category theory, commutative algebra, Galois theory, and algebraic geometry – topics that are often overlooked in standard undergraduate courses. This textbook is based on courses the author has conducted at the Independent University of Moscow and at the Faculty of Mathematics in the Higher School of Economics. The main content is complemented by a wealth of exercises for class discussion, some of which include comments and hints, as well as problems for independent study.

Algebra I textbook for students of mathematics

CERN Document Server

Gorodentsev, Alexey L

2016-01-01

This book is the first volume of an intensive “Russian-style” two-year undergraduate course in abstract algebra, and introduces readers to the basic algebraic structures – fields, rings, modules, algebras, groups, and categories – and explains the main principles of and methods for working with them. The course covers substantial areas of advanced combinatorics, geometry, linear and multilinear algebra, representation theory, category theory, commutative algebra, Galois theory, and algebraic geometry – topics that are often overlooked in standard undergraduate courses. This textbook is based on courses the author has conducted at the Independent University of Moscow and at the Faculty of Mathematics in the Higher School of Economics. The main content is complemented by a wealth of exercises for class discussion, some of which include comments and hints, as well as problems for independent study.

Algebraic Methods in Plane Geometry

Indian Academy of Sciences (India)

Srimath

rally, a1 ;a2 ;a3 ;a4 m ust not all be 0.) If w e m ultiply all th e ai's by a non-zero constant w e get the sam e cubic. .... B ut a sm all m odi¯cation of the operation changes the ..... Robert Bix, Conics and Cubics: A Concrete Introduction to Algebraic ... Joseph H Silverman and John Torrence Tate, Rational Points on Elliptic.

From the topological development of matrix models to the topological string theory: arrangement of surfaces through algebraic geometry

International Nuclear Information System (INIS)

Orantin, N.

2007-09-01

The 2-matrix model has been introduced to study Ising model on random surfaces. Since then, the link between matrix models and arrangement of discrete surfaces has strongly tightened. This manuscript aims to investigate these deep links and extend them beyond the matrix models, following my work's evolution. First, I take care to define properly the hermitian 2 matrix model which gives rise to generating functions of discrete surfaces equipped with a spin structure. Then, I show how to compute all the terms in the topological expansion of any observable by using algebraic geometry tools. They are obtained as differential forms on an algebraic curve associated to the model: the spectral curve. In a second part, I show how to define such differentials on any algebraic curve even if it does not come from a matrix model. I then study their numerous symmetry properties under deformations of the algebraic curve. In particular, I show that these objects coincide with the topological expansion of the observable of a matrix model if the algebraic curve is the spectral curve of this model. Finally, I show that the fine tuning of the parameters ensures that these objects can be promoted to modular invariants and satisfy the holomorphic anomaly equation of the Kodaira-Spencer theory. This gives a new hint that the Dijkgraaf-Vafa conjecture is correct. (author)

Arithmetic noncommutative geometry

CERN Document Server

Marcolli, Matilde

2005-01-01

Arithmetic noncommutative geometry denotes the use of ideas and tools from the field of noncommutative geometry, to address questions and reinterpret in a new perspective results and constructions from number theory and arithmetic algebraic geometry. This general philosophy is applied to the geometry and arithmetic of modular curves and to the fibers at archimedean places of arithmetic surfaces and varieties. The main reason why noncommutative geometry can be expected to say something about topics of arithmetic interest lies in the fact that it provides the right framework in which the tools of geometry continue to make sense on spaces that are very singular and apparently very far from the world of algebraic varieties. This provides a way of refining the boundary structure of certain classes of spaces that arise in the context of arithmetic geometry, such as moduli spaces (of which modular curves are the simplest case) or arithmetic varieties (completed by suitable "fibers at infinity"), by adding boundaries...

Introduction to tropical geometry

CERN Document Server

Maclagan, Diane

2015-01-01

Tropical geometry is a combinatorial shadow of algebraic geometry, offering new polyhedral tools to compute invariants of algebraic varieties. It is based on tropical algebra, where the sum of two numbers is their minimum and the product is their sum. This turns polynomials into piecewise-linear functions, and their zero sets into polyhedral complexes. These tropical varieties retain a surprising amount of information about their classical counterparts. Tropical geometry is a young subject that has undergone a rapid development since the beginning of the 21st century. While establishing itself as an area in its own right, deep connections have been made to many branches of pure and applied mathematics. This book offers a self-contained introduction to tropical geometry, suitable as a course text for beginning graduate students. Proofs are provided for the main results, such as the Fundamental Theorem and the Structure Theorem. Numerous examples and explicit computations illustrate the main concepts. Each of t...

Complex analysis and geometry

CERN Document Server

Silva, Alessandro

1993-01-01

The papers in this wide-ranging collection report on the results of investigations from a number of linked disciplines, including complex algebraic geometry, complex analytic geometry of manifolds and spaces, and complex differential geometry.

Real forms of non-linear superconformal and quasi-superconformal algebras and their unified realization

International Nuclear Information System (INIS)

Bina, B.; Guenaydin, M.

1997-01-01

We give a complete classification of the real forms of simple non-linear superconformal algebras (SCA) and quasi-superconformal algebras (QSCA) and present a unified realization of these algebras with simple symmetry groups. This classification is achieved by establishing a correspondence between simple non-linear QSCA's and SCA's and quaternionic and super-quaternionic symmetric spaces of simple Lie groups and Lie supergroups, respectively. The unified realization we present involves a dimension zero scalar field (dilaton), dimension-1 symmetry currents, and dimension-1/2 free bosons for QSCA's and dimension-1/2 free fermions for SCA's. The free bosons and fermions are associated with the quaternionic and super-quaternionic symmetric spaces of corresponding Lie groups and Lie supergroups, respectively. We conclude with a discussion of possible applications of our results. (orig.)

UCSMP Algebra. What Works Clearinghouse Intervention Report

Science.gov (United States)

What Works Clearinghouse, 2007

2007-01-01

"University of Chicago School Mathematics Project (UCSMP) Algebra," designed to increase students' skills in algebra, is appropriate for students in grades 7-10, depending on the students' incoming knowledge. This one-year course highlights applications, uses statistics and geometry to develop the algebra of linear equations and inequalities, and…

Limit algebras of differential forms in non-commutative geometry

Indian Academy of Sciences (India)

The holomorphic functional calculus closure of Connes' non- commutative de Rham algebra. ∗. D. (p. 549 of [C]) leads to a couple of operator algebras which are briefly discussed in this section. In §5, which contains the main contributions of the paper, quantized integrals are constructed on ∞A by using Dixmier trace ...

Computational aspects of algebraic curves

CERN Document Server

Shaska, Tanush

2005-01-01

The development of new computational techniques and better computing power has made it possible to attack some classical problems of algebraic geometry. The main goal of this book is to highlight such computational techniques related to algebraic curves. The area of research in algebraic curves is receiving more interest not only from the mathematics community, but also from engineers and computer scientists, because of the importance of algebraic curves in applications including cryptography, coding theory, error-correcting codes, digital imaging, computer vision, and many more.This book cove

Local analytic geometry

CERN Document Server

Abhyankar, Shreeram Shankar

1964-01-01

This book provides, for use in a graduate course or for self-study by graduate students, a well-motivated treatment of several topics, especially the following: (1) algebraic treatment of several complex variables; (2) geometric approach to algebraic geometry via analytic sets; (3) survey of local algebra; (4) survey of sheaf theory. The book has been written in the spirit of Weierstrass. Power series play the dominant role. The treatment, being algebraic, is not restricted to complex numbers, but remains valid over any complete-valued field. This makes it applicable to situations arising from

Residues and duality for projective algebraic varieties

CERN Document Server

Kunz, Ernst; Dickenstein, Alicia

2008-01-01

This book, which grew out of lectures by E. Kunz for students with a background in algebra and algebraic geometry, develops local and global duality theory in the special case of (possibly singular) algebraic varieties over algebraically closed base fields. It describes duality and residue theorems in terms of K�hler differential forms and their residues. The properties of residues are introduced via local cohomology. Special emphasis is given to the relation between residues to classical results of algebraic geometry and their generalizations. The contribution by A. Dickenstein gives applications of residues and duality to polynomial solutions of constant coefficient partial differential equations and to problems in interpolation and ideal membership. D. A. Cox explains toric residues and relates them to the earlier text. The book is intended as an introduction to more advanced treatments and further applications of the subject, to which numerous bibliographical hints are given.

Algebraic Hyperstructures and Fuzzy Logic in the Treatment of Uncertainty

OpenAIRE

Antonio Maturo; Annamaria Porreca

2016-01-01

This study presents some fundamental aspects of recent theories  on algebraic Hyperstructures, an important tool for an interdisciplinary vision of Geometry and Algebra. We examine some hypergroupoids of events, useful for a new algebraic-geometry perspective in the study and issues of probability applications. This paper considers some fundamental aspects of fuzzy classifications and their applications to problems of evaluation and decision in Architecture and Economics. Finally, we present ...

Algebra of pseudo-differential operators over C*-algebra

International Nuclear Information System (INIS)

Mohammad, N.

1982-08-01

Algebras of pseudo-differential operators over C*-algebras are studied for the special case when in Hormander class Ssub(rho,delta)sup(m)(Ω) Ω = Rsup(n); rho = 1, delta = 0, m any real number, and the C*-algebra is infinite dimensional non-commutative. The space B, i.e. the set of A-valued C*-functions in Rsup(n) (or Rsup(n) x Rsup(n)) whose derivatives are all bounded, plays an important role. A denotes C*-algebra. First the operator class Ssub(phi,0)sup(m) is defined, and through it, the class Lsub(1,0)sup(m) of pseudo-differential operators. Then the basic asymptotic expansion theorems concerning adjoint and product of operators of class Ssub(1,0)sup(m) are stated. Finally, proofs are given of L 2 -continuity theorem and the main theorem, which states that algebra of all pseudo-differential operators over C*-algebras is itself C*-algebra

q-deformed Poincare algebra

International Nuclear Information System (INIS)

Ogievetsky, O.; Schmidke, W.B.; Wess, J.; Muenchen Univ.; Zumino, B.; Lawrence Berkeley Lab., CA

1992-01-01

The q-differential calculus for the q-Minkowski space is developed. The algebra of the q-derivatives with the q-Lorentz generators is found giving the q-deformation of the Poincare algebra. The reality structure of the q-Poincare algebra is given. The reality structure of the q-differentials is also found. The real Laplaacian is constructed. Finally the comultiplication, counit and antipode for the q-Poincare algebra are obtained making it a Hopf algebra. (orig.)

Hecke algebras with unequal parameters

CERN Document Server

Lusztig, G

2003-01-01

Hecke algebras arise in representation theory as endomorphism algebras of induced representations. One of the most important classes of Hecke algebras is related to representations of reductive algebraic groups over p-adic or finite fields. In 1979, in the simplest (equal parameter) case of such Hecke algebras, Kazhdan and Lusztig discovered a particular basis (the KL-basis) in a Hecke algebra, which is very important in studying relations between representation theory and geometry of the corresponding flag varieties. It turned out that the elements of the KL-basis also possess very interesting combinatorial properties. In the present book, the author extends the theory of the KL-basis to a more general class of Hecke algebras, the so-called algebras with unequal parameters. In particular, he formulates conjectures describing the properties of Hecke algebras with unequal parameters and presents examples verifying these conjectures in particular cases. Written in the author's precise style, the book gives rese...

Tensor spaces and exterior algebra

CERN Document Server

Yokonuma, Takeo

1992-01-01

This book explains, as clearly as possible, tensors and such related topics as tensor products of vector spaces, tensor algebras, and exterior algebras. You will appreciate Yokonuma's lucid and methodical treatment of the subject. This book is useful in undergraduate and graduate courses in multilinear algebra. Tensor Spaces and Exterior Algebra begins with basic notions associated with tensors. To facilitate understanding of the definitions, Yokonuma often presents two or more different ways of describing one object. Next, the properties and applications of tensors are developed, including the classical definition of tensors and the description of relative tensors. Also discussed are the algebraic foundations of tensor calculus and applications of exterior algebra to determinants and to geometry. This book closes with an examination of algebraic systems with bilinear multiplication. In particular, Yokonuma discusses the theory of replicas of Chevalley and several properties of Lie algebras deduced from them.

Cubic forms algebra, geometry, arithmetic

CERN Document Server

Manin, Yu I

1986-01-01

Since this book was first published in English, there has been important progress in a number of related topics. The class of algebraic varieties close to the rational ones has crystallized as a natural domain for the methods developed and expounded in this volume. For this revised edition, the original text has been left intact (except for a few corrections) and has been brought up to date by the addition of an Appendix and recent references.The Appendix sketches some of the most essential new results, constructions and ideas, including the solutions of the Luroth and Zariski problems, the th

Exponentiation and deformations of Lie-admissible algebras

International Nuclear Information System (INIS)

Myung, H.C.

1982-01-01

The exponential function is defined for a finite-dimensional real power-associative algebra with unit element. The application of the exponential function is focused on the power-associative (p,q)-mutation of a real or complex associative algebra. Explicit formulas are computed for the (p,q)-mutation of the real envelope of the spin 1 algebra and the Lie algebra so(3) of the rotation group, in light of earlier investigations of the spin 1/2. A slight variant of the mutated exponential is interpreted as a continuous function of the Lie algebra into some isotope of the corresponding linear Lie group. The second part of this paper is concerned with the representation and deformation of a Lie-admissible algebra. The second cohomology group of a Lie-admissible algebra is introduced as a generalization of those of associative and Lie algebras in the Hochschild and Chevalley-Eilenberg theory. Some elementary theory of algebraic deformation of Lie-admissible algebras is discussed in view of generalization of that of associative and Lie algebras. Lie-admissible deformations are also suggested by the representation of Lie-admissible algebras. Some explicit examples of Lie-admissible deformation are given in terms of the (p,q)-mutation of associative deformation of an associative algebra. Finally, we discuss Lie-admissible deformations of order one

College algebra

CERN Document Server

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

Numerical algebraic geometry for model selection and its application to the life sciences

KAUST Repository

Gross, Elizabeth

2016-10-12

Researchers working with mathematical models are often confronted by the related problems of parameter estimation, model validation and model selection. These are all optimization problems, well known to be challenging due to nonlinearity, non-convexity and multiple local optima. Furthermore, the challenges are compounded when only partial data are available. Here, we consider polynomial models (e.g. mass-action chemical reaction networks at steady state) and describe a framework for their analysis based on optimization using numerical algebraic geometry. Specifically, we use probability-one polynomial homotopy continuation methods to compute all critical points of the objective function, then filter to recover the global optima. Our approach exploits the geometrical structures relating models and data, and we demonstrate its utility on examples from cell signalling, synthetic biology and epidemiology.

The Work of Lagrange in Number Theory and Algebra

Indian Academy of Sciences (India)

GENERAL I ARTICLE. The Work of Lagrange in Number Theory and Algebra. D P Patil, C R Pranesachar and Renuka RafJindran. (left) D P Patil got his Ph.D from the School of Math- ematics, TIFR and joined. IISc in 1992. His interests are commutative algebra, algebraic geometry and algebraic number theory. (right) C R ...

On some methods of achieving a continuous and differentiated assessment in Linear Algebra and Analytic and Differential Geometry courses and seminars

Directory of Open Access Journals (Sweden)

M. A.P. PURCARU

2017-12-01

Full Text Available This paper aims at highlighting some aspects related to assessment as regards its use as a differentiated training strategy for Linear Algebra and Analytic and Differential Geometry courses and seminars. Thus, the following methods of continuous differentiated assessment are analyzed and exemplified: the portfolio, the role play, some interactive methods and practical examinations.

Complex differential geometry

CERN Document Server

Zheng, Fangyang

2002-01-01

The theory of complex manifolds overlaps with several branches of mathematics, including differential geometry, algebraic geometry, several complex variables, global analysis, topology, algebraic number theory, and mathematical physics. Complex manifolds provide a rich class of geometric objects, for example the (common) zero locus of any generic set of complex polynomials is always a complex manifold. Yet complex manifolds behave differently than generic smooth manifolds; they are more coherent and fragile. The rich yet restrictive character of complex manifolds makes them a special and interesting object of study. This book is a self-contained graduate textbook that discusses the differential geometric aspects of complex manifolds. The first part contains standard materials from general topology, differentiable manifolds, and basic Riemannian geometry. The second part discusses complex manifolds and analytic varieties, sheaves and holomorphic vector bundles, and gives a brief account of the surface classifi...

Computational synthetic geometry

CERN Document Server

Bokowski, Jürgen

1989-01-01

Computational synthetic geometry deals with methods for realizing abstract geometric objects in concrete vector spaces. This research monograph considers a large class of problems from convexity and discrete geometry including constructing convex polytopes from simplicial complexes, vector geometries from incidence structures and hyperplane arrangements from oriented matroids. It turns out that algorithms for these constructions exist if and only if arbitrary polynomial equations are decidable with respect to the underlying field. Besides such complexity theorems a variety of symbolic algorithms are discussed, and the methods are applied to obtain new mathematical results on convex polytopes, projective configurations and the combinatorics of Grassmann varieties. Finally algebraic varieties characterizing matroids and oriented matroids are introduced providing a new basis for applying computer algebra methods in this field. The necessary background knowledge is reviewed briefly. The text is accessible to stud...

Introduction to applied algebraic systems

CERN Document Server

Reilly, Norman R

2009-01-01

This upper-level undergraduate textbook provides a modern view of algebra with an eye to new applications that have arisen in recent years. A rigorous introduction to basic number theory, rings, fields, polynomial theory, groups, algebraic geometry and elliptic curves prepares students for exploring their practical applications related to storing, securing, retrieving and communicating information in the electronic world. It will serve as a textbook for an undergraduate course in algebra with a strong emphasis on applications. The book offers a brief introduction to elementary number theory as

Coxeter groups and Hopf algebras

CERN Document Server

Aguiar, Marcelo

2011-01-01

An important idea in the work of G.-C. Rota is that certain combinatorial objects give rise to Hopf algebras that reflect the manner in which these objects compose and decompose. Recent work has seen the emergence of several interesting Hopf algebras of this kind, which connect diverse subjects such as combinatorics, algebra, geometry, and theoretical physics. This monograph presents a novel geometric approach using Coxeter complexes and the projection maps of Tits for constructing and studying many of these objects as well as new ones. The first three chapters introduce the necessary backgrou

Linear operators in Clifford algebras

International Nuclear Information System (INIS)

Laoues, M.

1991-01-01

We consider the real vector space structure of the algebra of linear endomorphisms of a finite-dimensional real Clifford algebra (2, 4, 5, 6, 7, 8). A basis of that space is constructed in terms of the operators M eI,eJ defined by x→e I .x.e J , where the e I are the generators of the Clifford algebra and I is a multi-index (3, 7). In particular, it is shown that the family (M eI,eJ ) is exactly a basis in the even case. (orig.)

INdAM Meeting on Homological and Computational Methods in Commutative Algebra

CERN Document Server

Gubeladze, Joseph; Römer, Tim

2017-01-01

This volume collects contributions by leading experts in the area of commutative algebra related to the  INdAM meeting “Homological and Computational Methods in Commutative Algebra” held in Cortona (Italy) from May 30 to  June 3, 2016 . The conference and this volume are dedicated to Winfried Bruns on the occasion of his 70th birthday. In particular, the topics of this book strongly reflect the variety of Winfried Bruns’ research interests and his great impact on commutative algebra as well as its applications to related fields. The authors discuss recent and relevant developments in algebraic geometry, commutative algebra, computational algebra, discrete geometry and homological algebra. The book offers a unique resource, both for young and more experienced researchers seeking comprehensive overviews and extensive bibliographic references.

Representation Theory of Algebraic Groups and Quantum Groups

CERN Document Server

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

Clifford algebras and the minimal representations of the 1D N-extended supersymmetry algebra

International Nuclear Information System (INIS)

Toppan, Francesco

2008-01-01

The Atiyah-Bott-Shapiro classification of the irreducible Clifford algebra is used to derive general properties of the minimal representations of the 1D N-Extended Supersymmetry algebra (the Z 2 -graded symmetry algebra of the Supersymmetric Quantum Mechanics) linearly realized on a finite number of fields depending on a real parameter t, the time. (author)

Geometries

CERN Document Server

Sossinsky, A B

2012-01-01

The book is an innovative modern exposition of geometry, or rather, of geometries; it is the first textbook in which Felix Klein's Erlangen Program (the action of transformation groups) is systematically used as the basis for defining various geometries. The course of study presented is dedicated to the proposition that all geometries are created equal--although some, of course, remain more equal than others. The author concentrates on several of the more distinguished and beautiful ones, which include what he terms "toy geometries", the geometries of Platonic bodies, discrete geometries, and classical continuous geometries. The text is based on first-year semester course lectures delivered at the Independent University of Moscow in 2003 and 2006. It is by no means a formal algebraic or analytic treatment of geometric topics, but rather, a highly visual exposition containing upwards of 200 illustrations. The reader is expected to possess a familiarity with elementary Euclidean geometry, albeit those lacking t...

Aspects of differential geometry II

CERN Document Server

Gilkey, Peter

2015-01-01

Differential Geometry is a wide field. We have chosen to concentrate upon certain aspects that are appropriate for an introduction to the subject; we have not attempted an encyclopedic treatment. Book II deals with more advanced material than Book I and is aimed at the graduate level. Chapter 4 deals with additional topics in Riemannian geometry. Properties of real analytic curves given by a single ODE and of surfaces given by a pair of ODEs are studied, and the volume of geodesic balls is treated. An introduction to both holomorphic and Kähler geometry is given. In Chapter 5, the basic properties of de Rham cohomology are discussed, the Hodge Decomposition Theorem, Poincaré duality, and the Künneth formula are proved, and a brief introduction to the theory of characteristic classes is given. In Chapter 6, Lie groups and Lie algebras are dealt with. The exponential map, the classical groups, and geodesics in the context of a bi-invariant metric are discussed. The de Rham cohomology of compact Lie groups an...

Extended conformal algebras

International Nuclear Information System (INIS)

Goddard, Peter

1990-01-01

The algebra of the group of conformal transformations in two dimensions consists of two commuting copies of the Virasoro algebra. In many mathematical and physical contexts, the representations of ν which are relevant satisfy two conditions: they are unitary and they have the ''positive energy'' property that L o is bounded below. In an irreducible unitary representation the central element c takes a fixed real value. In physical contexts, the value of c is a characteristic of a theory. If c < 1, it turns out that the conformal algebra is sufficient to ''solve'' the theory, in the sense of relating the calculation of the infinite set of physically interesting quantities to a finite subset which can be handled in principle. For c ≥ 1, this is no longer the case for the algebra alone and one needs some sort of extended conformal algebra, such as the superconformal algebra. It is these algebras that this paper aims at addressing. (author)

Certain algebraic structures and their applications to physics

International Nuclear Information System (INIS)

Salingaros, N.A.

1978-01-01

The aim of this thesis is to understand internal and external symmetries in Physics as arising from the same algebra by different processes, while the algebra itself arises out of the geometry of space-time. The result obtained is the Associative Generalized Algebra of Tensor Types. This algebra is constructed from the differential forms of spacetime, and is an algebra in the mathematical sense, describing all tensor types together. It is associative, and therefore very easy to use. A calculational formalism is developed that simplifies algebraic manipulations. The construction allows a classification of algebras that appear useful in Physics. The geometry excludes self-dual Minkowski bivector fields, but allows self-dual Euclidean bivector fields, a result, with important consequences in the theory of solutions of Yang-Mills gauge fields are demonstrated. There is only one bivector field, and every other bivector field, such as the electromagnetic field, is isomorphic to it. An exhaustive classification of the transformations of all fields in space-time yields the result that the only transformations of the electromagnetic field are the Lorentz transformations and the duality rotation. A fundamental asymmetry between the electric and magnetic fields are demonstrated. The derivative in the algebra is associative, and combines the Cartan exterior derivative with the coderivative of Hodge. The simplest derivative equations satisfied by a field in flat space-time are precisely the Maxwell equations

Contemporary developments in algebraic K-theory

International Nuclear Information System (INIS)

Karoubi, M.; Kuku, A.O.; Pedrini, C.

2003-01-01

The School and Conference on Algebraic K-theory which took place at ICTP July 8-26, 2002 was a follow-up to the earlier one in 1997, and like its predecessor, the 2002 meeting endeavoured to emphasise the multidisciplinary aspects of the subject. However, one special feature of the 2002 School and Conference is that the whole activity was dedicated to H. Bass, one of the founders of Algebraic K-theory, on the occasion of his seventieth birthday. The School during the first two weeks, July 8 to 19 was devoted to expository lectures meant to explore and highlight connections between K-theory and several other areas of mathematics - Algebraic Topology, Number theory, Algebraic Geometry, Representation theory, and Non-commutative Geometry. This volume, constituting the Proceedings of the School, is dedicated to H. Bass. The Proceedings of the Conference during the last week July 22 - 26, which will appear in Special issues of K-theory, is also dedicated to H. Bass. The opening contribution by M. Karoubi to this volume consists of a comprehensive survey of developments in K-theory in the last forty-five years, and covers a very broad spectrum of the subject, including Topological K-theory, Atiyah-Singer index theorem, K-theory of Banach algebras, Higher Algebraic K-theory, Cyclic Homology etc. J. Berrick's contribution on 'Algebraic K-theory and Algebraic Topology' treats the various topological constructions of Algebraic K-theory together with the underlying homotopy theory. Topics covered include the plus construction together with its various ramifications and applications, Topological Hochschild and Cyclic Homology as well as K-theory of the ring of integers. The contributions by M. Kolster titled 'K-theory and Arithmetics' includes such topics as values of zeta functions and relations to K-theory, K-theory of integers in number fields and associated conjectures, Etale cohomology, Iwasawa theory etc. A.O. Kuku's contributions on 'K-theory and Representation theory

Contemporary developments in algebraic K-theory

Energy Technology Data Exchange (ETDEWEB)

Karoubi, M [Univ. Paris (France); Kuku, A O [Abdus Salam International Centre for Theoretical Physics, Trieste (Italy); Pedrini, C [Univ. Genova (Italy)

2003-09-15

The School and Conference on Algebraic K-theory which took place at ICTP July 8-26, 2002 was a follow-up to the earlier one in 1997, and like its predecessor, the 2002 meeting endeavoured to emphasise the multidisciplinary aspects of the subject. However, one special feature of the 2002 School and Conference is that the whole activity was dedicated to H. Bass, one of the founders of Algebraic K-theory, on the occasion of his seventieth birthday. The School during the first two weeks, July 8 to 19 was devoted to expository lectures meant to explore and highlight connections between K-theory and several other areas of mathematics - Algebraic Topology, Number theory, Algebraic Geometry, Representation theory, and Non-commutative Geometry. This volume, constituting the Proceedings of the School, is dedicated to H. Bass. The Proceedings of the Conference during the last week July 22 - 26, which will appear in Special issues of K-theory, is also dedicated to H. Bass. The opening contribution by M. Karoubi to this volume consists of a comprehensive survey of developments in K-theory in the last forty-five years, and covers a very broad spectrum of the subject, including Topological K-theory, Atiyah-Singer index theorem, K-theory of Banach algebras, Higher Algebraic K-theory, Cyclic Homology etc. J. Berrick's contribution on 'Algebraic K-theory and Algebraic Topology' treats the various topological constructions of Algebraic K-theory together with the underlying homotopy theory. Topics covered include the plus construction together with its various ramifications and applications, Topological Hochschild and Cyclic Homology as well as K-theory of the ring of integers. The contributions by M. Kolster titled 'K-theory and Arithmetics' includes such topics as values of zeta functions and relations to K-theory, K-theory of integers in number fields and associated conjectures, Etale cohomology, Iwasawa theory etc. A.O. Kuku's contributions on 'K-theory and Representation theory

Perceptions of 9th and 10th Grade Students on How Their Environment, Cognition, and Behavior Motivate Them in Algebra and Geometry Courses

Science.gov (United States)

Harootunian, Alen

2012-01-01

In this study, relationships were examined between students' perception of their cognition, behavior, environment, and motivation. The purpose of the research study was to explore the extent to which 9th and 10th grade students' perception of environment, cognition, and behavior can predict their motivation in Algebra and Geometry courses. A…

Real and étale cohomology

CERN Document Server

Scheiderer, Claus

1994-01-01

This book makes a systematic study of the relations between the étale cohomology of a scheme and the orderings of its residue fields. A major result is that in high degrees, étale cohomology is cohomology of the real spectrum. It also contains new contributions in group cohomology and in topos theory. It is of interest to graduate students and researchers who work in algebraic geometry (not only real) and have some familiarity with the basics of étale cohomology and Grothendieck sites. Independently, it is of interest to people working in the cohomology theory of groups or in topos theory.

Higher dimensional uniformisation and W-geometry

International Nuclear Information System (INIS)

Govindarajan, S.

1995-01-01

We formulate the uniformisation problem underlying the geometry of W n -gravity using the differential equation approach to W-algebras. We construct W n -space (analogous to superspace in supersymmetry) as an (n-1)-dimensional complex manifold using isomonodromic deformations of linear differential equations. The W n -manifold is obtained by the quotient of a Fuchsian subgroup of PSL(n,R) which acts properly discontinuously on a simply connected domain in bfCP n-1 . The requirement that a deformation be isomonodromic furnishes relations which enable one to convert non-linear W-diffeomorphisms to (linear) diffeomorphisms on the W n -manifold. We discuss how the Teichmueller spaces introduced by Hitchin can then be interpreted as the space of complex structures or the space of projective structures with real holonomy on the W n -manifold. The projective structures are characterised by Halphen invariants which are appropriate generalisations of the Schwarzian. This construction will work for all ''generic'' W-algebras. (orig.)

Representations of the algebra Uq'(son) related to quantum gravity

International Nuclear Information System (INIS)

Klimyk, A.U.

2002-01-01

The aim of this paper is to review our results on finite dimensional irreducible representations of the nonstandard q-deformation U q ' (so n ) of the universal enveloping algebra U(so(n)) of the Lie algebra so(n) which does not coincide with the Drinfeld-Jimbo quantum algebra U q (so n ).This algebra is related to algebras of observables in quantum gravity and to algebraic geometry.Irreducible finite dimensional representations of the algebra U q ' (so n ) for q not a root of unity and for q a root of unity are given

Kinematic algebras, groups for elementary particles, and the geometry of momentum space

International Nuclear Information System (INIS)

Izmest'ev, A.A.

1986-01-01

It is shown that to each n-dimensional (n≥2) homogeneous isotropic Riemannian momentum (coordinate) space there corresponds a definite kinematic local algebra of operators N/sub a/, M/sub a//sub b/, P/sub a//sub ,/ ω(a,b = 1,2,...,n). In the three-dimensional case this gives the possibility of classifying particles in accordance with the algebras of the types of momentum space. The approach developed also makes it possible to obtain generalized equations describing particles of the different types. The operators under consideration satisfy not only the relevant algebra but also relations independent of the algebra that coincide in form with the Maxwell equations

Geometry Euclid and beyond

CERN Document Server

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 ...

Higher regulators, algebraic

CERN Document Server

Bloch, Spencer J

2000-01-01

This book is the long-awaited publication of the famous Irvine lectures. Delivered in 1978 at the University of California at Irvine, these lectures turned out to be an entry point to several intimately-connected new branches of arithmetic algebraic geometry, such as regulators and special values of L-functions of algebraic varieties, explicit formulas for them in terms of polylogarithms, the theory of algebraic cycles, and eventually the general theory of mixed motives which unifies and underlies all of the above (and much more). In the 20 years since, the importance of Bloch's lectures has not diminished. A lucky group of people working in the above areas had the good fortune to possess a copy of old typewritten notes of these lectures. Now everyone can have their own copy of this classic work.

Real symplectic formulation of local special geometry

CERN Document Server

Ferrara, Sergio; Ferrara, Sergio; Macia, Oscar

2006-01-01

We consider a formulation of local special geometry in terms of Darboux special coordinates $P^I=(p^i,q_i)$, $I=1,...,2n$. A general formula for the metric is obtained which is manifestly $\\mathbf{Sp}(2n,\\mathbb{R})$ covariant. Unlike the rigid case the metric is not given by the Hessian of the real function $S(P)$ which is the Legendre transform of the imaginary part of the holomorphic prepotential. Rather it is given by an expression that contains $S$, its Hessian and the conjugate momenta $S_I=\\frac{\\partial S}{\\partial P^I}$. Only in the one-dimensional case ($n=1$) is the real (two-dimensional) metric proportional to the Hessian with an appropriate conformal factor.

Vertex algebras and mirror symmetry

International Nuclear Information System (INIS)

Borisov, L.A.

2001-01-01

Mirror Symmetry for Calabi-Yau hypersurfaces in toric varieties is by now well established. However, previous approaches to it did not uncover the underlying reason for mirror varieties to be mirror. We are able to calculate explicitly vertex algebras that correspond to holomorphic parts of A and B models of Calabi-Yau hypersurfaces and complete intersections in toric varieties. We establish the relation between these vertex algebras for mirror Calabi-Yau manifolds. This should eventually allow us to rewrite the whole story of toric mirror symmetry in the language of sheaves of vertex algebras. Our approach is purely algebraic and involves simple techniques from toric geometry and homological algebra, as well as some basic results of the theory of vertex algebras. Ideas of this paper may also be useful in other problems related to maps from curves to algebraic varieties.This paper could also be of interest to physicists, because it contains explicit description of holomorphic parts of A and B models of Calabi-Yau hypersurfaces and complete intersections in terms of free bosons and fermions. (orig.)

From groups to geometry and back

CERN Document Server

Climenhaga, Vaughn

2017-01-01

Groups arise naturally as symmetries of geometric objects, and so groups can be used to understand geometry and topology. Conversely, one can study abstract groups by using geometric techniques and ultimately by treating groups themselves as geometric objects. This book explores these connections between group theory and geometry, introducing some of the main ideas of transformation groups, algebraic topology, and geometric group theory. The first half of the book introduces basic notions of group theory and studies symmetry groups in various geometries, including Euclidean, projective, and hyperbolic. The classification of Euclidean isometries leads to results on regular polyhedra and polytopes; the study of symmetry groups using matrices leads to Lie groups and Lie algebras. The second half of the book explores ideas from algebraic topology and geometric group theory. The fundamental group appears as yet another group associated to a geometric object and turns out to be a symmetry group using covering space...

Teaching Quantitative Reasoning: A Better Context for Algebra

OpenAIRE

Eric Gaze

2014-01-01

This editorial questions the preeminence of algebra in our mathematics curriculum. The GATC (Geometry, Algebra, Trigonometry, Calculus) sequence abandons the fundamental middle school math topics necessary for quantitative literacy, while the standard super-abundance of algebra taught in the abstract fosters math phobia and supports a culturally acceptable stance that math is not relevant to everyday life. Although GATC is seen as a pipeline to STEM (Science, Technology, Engineering, Mathemat...

Graded geometry and Poisson reduction

OpenAIRE

Cattaneo, A S; Zambon, M

2009-01-01

The main result of [2] extends the Marsden-Ratiu reduction theorem [4] in Poisson geometry, and is proven by means of graded geometry. In this note we provide the background material about graded geometry necessary for the proof in [2]. Further, we provide an alternative algebraic proof for the main result. ©2009 American Institute of Physics

Factorization algebras in quantum field theory

CERN Document Server

Costello, Kevin

2017-01-01

Factorization algebras are local-to-global objects that play a role in classical and quantum field theory which is similar to the role of sheaves in geometry: they conveniently organize complicated information. Their local structure encompasses examples like associative and vertex algebras; in these examples, their global structure encompasses Hochschild homology and conformal blocks. In this first volume, the authors develop the theory of factorization algebras in depth, but with a focus upon examples exhibiting their use in field theory, such as the recovery of a vertex algebra from a chiral conformal field theory and a quantum group from Abelian Chern-Simons theory. Expositions of the relevant background in homological algebra, sheaves and functional analysis are also included, thus making this book ideal for researchers and graduates working at the interface between mathematics and physics.

Zorn algebra in general relativity

International Nuclear Information System (INIS)

Oliveira, C.G.; Maia, M.D.

The covariant differential properties of the split Cayley subalgebra of local real quaternion tetrads is considered. Referred to this local quaternion tetrad several geometrical objects are given in terms of Zorn-Weyl matrices. Associated to a pair of real null vectors we define two-component spinor fields over the curved space and the associated Zorn-Weyl matrices which satisfy the Dirac equation written in terms of the Zorn algebra. The formalism is generalized by considering a field of complex tetrads defining a Hermitian second rank tensor. The real part of this tensor describes the gravitational potentials and the imaginary part the electromagnetic potentials in the Lorentz gauge. The motion of a charged spin zero test body is considered. The Zorn-Weyl algebra associated to this generalized formalism has elements belonging to the full octonion algebra [pt

Basic algebraic topology and its applications

CERN Document Server

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...

Non-Hermitian systems of Euclidean Lie algebraic type with real energy spectra

International Nuclear Information System (INIS)

Dey, Sanjib; Fring, Andreas; Mathanaranjan, Thilagarajah

2014-01-01

We study several classes of non-Hermitian Hamiltonian systems, which can be expressed in terms of bilinear combinations of Euclidean–Lie algebraic generators. The classes are distinguished by different versions of antilinear (PT)-symmetries exhibiting various types of qualitative behaviour. On the basis of explicitly computed non-perturbative Dyson maps we construct metric operators, isospectral Hermitian counterparts for which we solve the corresponding time-independent Schrödinger equation for specific choices of the coupling constants. In these cases general analytical expressions for the solutions are obtained in the form of Mathieu functions, which we analyze numerically to obtain the corresponding energy spectra. We identify regions in the parameter space for which the corresponding spectra are entirely real and also domains where the PT symmetry is spontaneously broken and sometimes also regained at exceptional points. In some cases it is shown explicitly how the threshold region from real to complex spectra is characterized by the breakdown of the Dyson maps or the metric operator. We establish the explicit relationship to models currently under investigation in the context of beam dynamics in optical lattices. -- Highlights: •Different PT-symmetries lead to qualitatively different systems. •Construction of non-perturbative Dyson maps and isospectral Hermitian counterparts. •Numerical discussion of the eigenvalue spectra for one of the E(2)-systems. •Established link to systems studied in the context of optical lattices. •Setup for the E(3)-algebra is provided

Real zeros of classes of random algebraic polynomials

Directory of Open Access Journals (Sweden)

K. Farahmand

2003-01-01

Full Text Available There are many known asymptotic estimates for the expected number of real zeros of an algebraic polynomial a0+a1x+a2x2+⋯+an−1xn−1 with identically distributed random coefficients. Under different assumptions for the distribution of the coefficients {aj}j=0n−1 it is shown that the above expected number is asymptotic to O(logn. This order for the expected number of zeros remains valid for the case when the coefficients are grouped into two, each group with a different variance. However, it was recently shown that if the coefficients are non-identically distributed such that the variance of the jth term is (nj the expected number of zeros of the polynomial increases to O(n. The present paper provides the value for this asymptotic formula for the polynomials with the latter variances when they are grouped into three with different patterns for their variances.

Index theory for locally compact noncommutative geometries

CERN Document Server

Carey, A L; Rennie, A; Sukochev, F A

2014-01-01

Spectral triples for nonunital algebras model locally compact spaces in noncommutative geometry. In the present text, the authors prove the local index formula for spectral triples over nonunital algebras, without the assumption of local units in our algebra. This formula has been successfully used to calculate index pairings in numerous noncommutative examples. The absence of any other effective method of investigating index problems in geometries that are genuinely noncommutative, particularly in the nonunital situation, was a primary motivation for this study and the authors illustrate this point with two examples in the text. In order to understand what is new in their approach in the commutative setting the authors prove an analogue of the Gromov-Lawson relative index formula (for Dirac type operators) for even dimensional manifolds with bounded geometry, without invoking compact supports. For odd dimensional manifolds their index formula appears to be completely new.

q-deformations of noncompact Lie (super-) algebras: The examples of q-deformed Lorentz, Weyl, Poincare' and (super-) conformal algebras

International Nuclear Information System (INIS)

Dobrev, V.K.

1992-01-01

We review and explain a canonical procedure for the q-deformation of the real forms G of complex Lie (super-) algebras associated with (generalized) Cartan matrices. Our procedure gives different q-deformations for the non-conjugate Cartan subalgebras of G. We give several in detail the q-deformed Lorentz and conformal (super-) algebras. The q-deformed conformal algebra contains as a subalgebra a q-deformed Poincare algebra and as Hopf subalgebras two conjugate 11-generator q-deformed Weyl algebras. The q-deformed Lorentz algebra in Hopf subalgebra of both Weyl algebras. (author). 24 refs

An algebraic approach to the scattering equations

Energy Technology Data Exchange (ETDEWEB)

Huang, Rijun; Rao, Junjie [Zhejiang Institute of Modern Physics, Zhejiang University,Hangzhou, 310027 (China); Feng, Bo [Zhejiang Institute of Modern Physics, Zhejiang University,Hangzhou, 310027 (China); Center of Mathematical Science, Zhejiang University,Hangzhou, 310027 (China); He, Yang-Hui [School of Physics, NanKai University,Tianjin, 300071 (China); Department of Mathematics, City University,London, EC1V 0HB (United Kingdom); Merton College, University of Oxford,Oxford, OX14JD (United Kingdom)

2015-12-10

We employ the so-called companion matrix method from computational algebraic geometry, tailored for zero-dimensional ideals, to study the scattering equations. The method renders the CHY-integrand of scattering amplitudes computable using simple linear algebra and is amenable to an algorithmic approach. Certain identities in the amplitudes as well as rationality of the final integrand become immediate in this formalism.

An algebraic approach to the scattering equations

International Nuclear Information System (INIS)

Huang, Rijun; Rao, Junjie; Feng, Bo; He, Yang-Hui

2015-01-01

We employ the so-called companion matrix method from computational algebraic geometry, tailored for zero-dimensional ideals, to study the scattering equations. The method renders the CHY-integrand of scattering amplitudes computable using simple linear algebra and is amenable to an algorithmic approach. Certain identities in the amplitudes as well as rationality of the final integrand become immediate in this formalism.

Supersymmetrization schemes of D=4 Maxwell algebra

International Nuclear Information System (INIS)

Kamimura, Kiyoshi; Lukierski, Jerzy

2012-01-01

The Maxwell algebra, an enlargement of Poincaré algebra by Abelian tensorial generators, can be obtained in arbitrary dimension D by the suitable contraction of O(D-1,1)⊕O(D-1,2) (Lorentz algebra ⊕ AdS algebra). We recall that in D=4 the Lorentz algebra O(3,1) is described by the realification Sp R (2|C) of complex algebra Sp(2|C)≃Sl(2|C) and O(3,2)≃Sp(4). We study various D=4N-extended Maxwell superalgebras obtained by the contractions of real superalgebras OSp R (2N-k;2|C)⊕OSp(k;4) (k=0,1,2,…,2N); (extended Lorentz superalgebra ⊕ extended AdS superalgebra). If N=1 (k=0,1,2) one arrives at three different versions of simple Maxwell superalgebra. For any fixed N we get 2N different superextensions of Maxwell algebra with n-extended Poincaré superalgebras (1⩽n⩽N) and the internal symmetry sectors obtained by suitable contractions of the real algebra O R (2N-k|C)⊕O(k). Finally the comments on possible applications of Maxwell superalgebras are presented.

Essential linear algebra with applications a problem-solving approach

CERN Document Server

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.    ...

Finding all real roots of a polynomial by matrix algebra and the Adomian decomposition method

Directory of Open Access Journals (Sweden)

Hooman Fatoorehchi

2014-10-01

Full Text Available In this paper, we put forth a combined method for calculation of all real zeroes of a polynomial equation through the Adomian decomposition method equipped with a number of developed theorems from matrix algebra. These auxiliary theorems are associated with eigenvalues of matrices and enable convergence of the Adomian decomposition method toward different real roots of the target polynomial equation. To further improve the computational speed of our technique, a nonlinear convergence accelerator known as the Shanks transform has optionally been employed. For the sake of illustration, a number of numerical examples are given.

On Dunkl angular momenta algebra

Energy Technology Data Exchange (ETDEWEB)

Feigin, Misha [School of Mathematics and Statistics, University of Glasgow,15 University Gardens, Glasgow G12 8QW (United Kingdom); Hakobyan, Tigran [Yerevan State University,1 Alex Manoogian, 0025 Yerevan (Armenia); Tomsk Polytechnic University,Lenin Ave. 30, 634050 Tomsk (Russian Federation)

2015-11-17

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 subalgebra 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.

Algebraic curves and cryptography

CERN Document Server

Murty, V Kumar

2010-01-01

It is by now a well-known paradigm that public-key cryptosystems can be built using finite Abelian groups and that algebraic geometry provides a supply of such groups through Abelian varieties over finite fields. Of special interest are the Abelian varieties that are Jacobians of algebraic curves. All of the articles in this volume are centered on the theme of point counting and explicit arithmetic on the Jacobians of curves over finite fields. The topics covered include Schoof's \\ell-adic point counting algorithm, the p-adic algorithms of Kedlaya and Denef-Vercauteren, explicit arithmetic on

Exceptional quantum geometry and particle physics

Directory of Open Access Journals (Sweden)

Michel Dubois-Violette

2016-11-01

Full Text Available Based on an interpretation of the quark–lepton symmetry in terms of the unimodularity of the color group SU(3 and on the existence of 3 generations, we develop an argumentation suggesting that the “finite quantum space” corresponding to the exceptional real Jordan algebra of dimension 27 (the Euclidean Albert algebra is relevant for the description of internal spaces in the theory of particles. In particular, the triality which corresponds to the 3 off-diagonal octonionic elements of the exceptional algebra is associated to the 3 generations of the Standard Model while the representation of the octonions as a complex 4-dimensional space C⊕C3 is associated to the quark–lepton symmetry (one complex for the lepton and 3 for the corresponding quark. More generally it is suggested that the replacement of the algebra of real functions on spacetime by the algebra of functions on spacetime with values in a finite-dimensional Euclidean Jordan algebra which plays the role of “the algebra of real functions” on the corresponding almost classical quantum spacetime is relevant in particle physics. This leads us to study the theory of Jordan modules and to develop the differential calculus over Jordan algebras (i.e. to introduce the appropriate notion of differential forms. We formulate the corresponding definition of connections on Jordan modules.

Using Dynamic Geometry and Computer Algebra Systems in Problem Based Courses for Future Engineers

Science.gov (United States)

Tomiczková, Svetlana; Lávicka, Miroslav

2015-01-01

It is a modern trend today when formulating the curriculum of a geometric course at the technical universities to start from a real-life problem originated in technical praxis and subsequently to define which geometric theories and which skills are necessary for its solving. Nowadays, interactive and dynamic geometry software plays a more and more…

Finite quantum physics and noncommutative geometry

International Nuclear Information System (INIS)

Balachandran, A.P.; Ercolessi, E.; Landi, G.; Teotonio-Sobrinho, P.; Lizzi, F.; Sparano, G.

1994-04-01

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

Basic linear algebra

CERN Document Server

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:...

Complex Algebraic Varieties

CERN Document Server

Peternell, Thomas; Schneider, Michael; Schreyer, Frank-Olaf

1992-01-01

The Bayreuth meeting on "Complex Algebraic Varieties" focussed on the classification of algebraic varieties and topics such as vector bundles, Hodge theory and hermitian differential geometry. Most of the articles in this volume are closely related to talks given at the conference: all are original, fully refereed research articles. CONTENTS: A. Beauville: Annulation du H(1) pour les fibres en droites plats.- M. Beltrametti, A.J. Sommese, J.A. Wisniewski: Results on varieties with many lines and their applications to adjunction theory.- G. Bohnhorst, H. Spindler: The stability of certain vector bundles on P(n) .- F. Catanese, F. Tovena: Vector bundles, linear systems and extensions of (1).- O. Debarre: Vers uns stratification de l'espace des modules des varietes abeliennes principalement polarisees.- J.P. Demailly: Singular hermitian metrics on positive line bundles.- T. Fujita: On adjoint bundles of ample vector bundles.- Y. Kawamata: Moderate degenerations of algebraic surfaces.- U. Persson: Genus two fibra...

Geometry through history Euclidean, hyperbolic, and projective geometries

CERN Document Server

Dillon, Meighan I

2018-01-01

Presented as an engaging discourse, this textbook invites readers to delve into the historical origins and uses of geometry. The narrative traces the influence of Euclid’s system of geometry, as developed in his classic text The Elements, through the Arabic period, the modern era in the West, and up to twentieth century mathematics. Axioms and proof methods used by mathematicians from those periods are explored alongside the problems in Euclidean geometry that lead to their work. Students cultivate skills applicable to much of modern mathematics through sections that integrate concepts like projective and hyperbolic geometry with representative proof-based exercises. For its sophisticated account of ancient to modern geometries, this text assumes only a year of college mathematics as it builds towards its conclusion with algebraic curves and quaternions. Euclid’s work has affected geometry for thousands of years, so this text has something to offer to anyone who wants to broaden their appreciation for the...

Braided affine geometry and q-analogs of wave operators

International Nuclear Information System (INIS)

Gurevich, Dimitri; Saponov, Pavel

2009-01-01

The main goal of this review is to compare different approaches to constructing the geometry associated with a Hecke type braiding (in particular, with that related to the quantum group U q (sl(n))). We place emphasis on the affine braided geometry related to the so-called reflection equation algebra (REA). All objects of such a type of geometry are defined in the spirit of affine algebraic geometry via polynomial relations on generators. We begin by comparing the Poisson counterparts of 'quantum varieties' and describe different approaches to their quantization. Also, we exhibit two approaches to introducing q-analogs of vector bundles and defining the Chern-Connes index for them on quantum spheres. In accordance with the Serre-Swan approach, the q-vector bundles are treated as finitely generated projective modules over the corresponding quantum algebras. Besides, we describe the basic properties of the REA used in this construction and compare different ways of defining q-analogs of partial derivatives and differentials on the REA and algebras close to them. In particular, we present a way of introducing a q-differential calculus via Koszul type complexes. The elements of the q-calculus are applied to defining q-analogs of some relativistic wave operators. (topical review)

Classification of real Lie superalgebras based on a simple Lie algebra, giving rise to interesting examples involving {mathfrak {su}}(2,2)

Science.gov (United States)

Guzzo, H.; Hernández, I.; Sánchez-Valenzuela, O. A.

2014-09-01

Finite dimensional semisimple real Lie superalgebras are described via finite dimensional semisimple complex Lie superalgebras. As an application of these results, finite dimensional real Lie superalgebras mathfrak {m}=mathfrak {m}_0 oplus mathfrak {m}_1 for which mathfrak {m}_0 is a simple Lie algebra are classified up to isomorphism.

Vector geometry

CERN Document Server

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

International Conference on Analytic and Algebraic Geometry held at the Tata Institute of Fundamental Research and the University of Hyderabad

CERN Document Server

Biswas, Indranil; Morye, Archana; Parameswaran, A

2017-01-01

This volume is an outcome of the International conference held in Tata Institute of Fundamental Research and the University of Hyderabad. There are fifteen articles in this volume. The main purpose of the articles is to introduce recent and advanced techniques in the area of analytic and algebraic geometry. This volume attempts to give recent developments in the area to target mainly young researchers who are new to this area. Also, some research articles have been added to give examples of how to use these techniques to prove new results.

Nonabelian Jacobian of projective surfaces geometry and representation theory

CERN Document Server

Reider, Igor

2013-01-01

The Jacobian of a smooth projective curve is undoubtedly one of the most remarkable and beautiful objects in algebraic geometry. This work is an attempt to develop an analogous theory for smooth projective surfaces - a theory of the nonabelian Jacobian of smooth projective surfaces. Just like its classical counterpart, our nonabelian Jacobian relates to vector bundles (of rank 2) on a surface as well as its Hilbert scheme of points. But it also comes equipped with the variation of Hodge-like structures, which produces a sheaf of reductive Lie algebras naturally attached to our Jacobian. This constitutes a nonabelian analogue of the (abelian) Lie algebra structure of the classical Jacobian. This feature naturally relates geometry of surfaces with the representation theory of reductive Lie algebras/groups. This work’s main focus is on providing an in-depth study of various aspects of this relation. It presents a substantial body of evidence that the sheaf of Lie algebras on the nonabelian Jacobian is an effic...

Comparison between two differential graded algebras in ...

Indian Academy of Sciences (India)

76

A differential calculus on a “space” means the specification of a differential graded algebra (dga), often interpreted as space of forms. In classical geometry the “space” is a manifold and we have the de-Rham dga, whereas in noncommutative geometry a “space” is described by a triple called spectral triple. A spectral triple is ...

Algebra success in 20 minutes a day

CERN Document Server

LearningExpress, LLC

2014-01-01

Stripped of unnecessary math jargon but bursting with algebra essentials, this handy guide covers vital algebra skills that apply to real-world scenarios. Whether you're new to algebra or just looking for a refresher, Algebra Success in 20 Minutes a Day offers a lesson plan that provides quick and thorough instruction in practical, critical skills. All lessons can be completed in just 20 minutes a day, for a manageable and non-intimidating learning experience.

An introduction to Clifford algebras and spinors

CERN Document Server

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...

Invariants of triangular Lie algebras

International Nuclear Information System (INIS)

Boyko, Vyacheslav; Patera, Jiri; Popovych, Roman

2007-01-01

Triangular Lie algebras are the Lie algebras which can be faithfully represented by triangular matrices of any finite size over the real/complex number field. In the paper invariants ('generalized Casimir operators') are found for three classes of Lie algebras, namely those which are either strictly or non-strictly triangular, and for so-called special upper triangular Lie algebras. Algebraic algorithm of Boyko et al (2006 J. Phys. A: Math. Gen.39 5749 (Preprint math-ph/0602046)), developed further in Boyko et al (2007 J. Phys. A: Math. Theor.40 113 (Preprint math-ph/0606045)), is used to determine the invariants. A conjecture of Tremblay and Winternitz (2001 J. Phys. A: Math. Gen.34 9085), concerning the number of independent invariants and their form, is corroborated

Classification of digital affine noncommutative geometries

Science.gov (United States)

Majid, Shahn; Pachoł, Anna

2018-03-01

It is known that connected translation invariant n-dimensional noncommutative differentials dxi on the algebra k[x1, …, xn] of polynomials in n-variables over a field k are classified by commutative algebras V on the vector space spanned by the coordinates. These data also apply to construct differentials on the Heisenberg algebra "spacetime" with relations [xμ, xν] = λΘμν, where Θ is an antisymmetric matrix, as well as to Lie algebras with pre-Lie algebra structures. We specialise the general theory to the field k =F2 of two elements, in which case translation invariant metrics (i.e., with constant coefficients) are equivalent to making V a Frobenius algebra. We classify all of these and their quantum Levi-Civita bimodule connections for n = 2, 3, with partial results for n = 4. For n = 2, we find 3 inequivalent differential structures admitting 1, 2, and 3 invariant metrics, respectively. For n = 3, we find 6 differential structures admitting 0, 1, 2, 3, 4, 7 invariant metrics, respectively. We give some examples for n = 4 and general n. Surprisingly, not all our geometries for n ≥ 2 have zero quantum Riemann curvature. Quantum gravity is normally seen as a weighted "sum" over all possible metrics but our results are a step towards a deeper approach in which we must also "sum" over differential structures. Over F2 we construct some of our algebras and associated structures by digital gates, opening up the possibility of "digital geometry."

The Algebra of Complex Numbers.

Science.gov (United States)

LePage, Wilbur R.

This programed text is an introduction to the algebra of complex numbers for engineering students, particularly because of its relevance to important problems of applications in electrical engineering. It is designed for a person who is well experienced with the algebra of real numbers and calculus, but who has no experience with complex number…

Classical theory of algebraic numbers

CERN Document Server

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...

Kaehler geometry and SUSY mechanics

International Nuclear Information System (INIS)

Bellucci, Stefano; Nersessian, Armen

2001-01-01

We present two examples of SUSY mechanics related with Kaehler geometry. The first system is the N = 4 supersymmetric one-dimensional sigma-model proposed in hep-th/0101065. Another system is the N = 2 SUSY mechanics whose phase space is the external algebra of an arbitrary Kaehler manifold. The relation of these models with antisymplectic geometry is discussed

From Cayley-Dickson Algebras to Combinatorial Grassmannians

Czech Academy of Sciences Publication Activity Database

Saniga, M.; Holweck, F.; Pracna, Petr

2015-01-01

Roč. 3, č. 4 (2015), s. 1192-1221 ISSN 2227-7390 Institutional support: RVO:61388955 Keywords : Cayley-Dickson algebras * Veldkamp spaces * finite geometries Subject RIV: CF - Physical ; Theoretical Chemistry

Differential forms and the geometry of general relativity

CERN Document Server

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

On left Hopf algebras within the framework of inhomogeneous quantum groups for particle algebras

Energy Technology Data Exchange (ETDEWEB)

Rodriguez-Romo, Suemi [Facultad de Estudios Superiores Cuautitlan, Universidad Nacional Autonoma de Mexico (Mexico)

2012-10-15

We deal with some matters needed to construct concrete left Hopf algebras for inhomogeneous quantum groups produced as noncommutative symmetries of fermionic and bosonic creation/annihilation operators. We find a map for the bidimensional fermionic case, produced as in Manin's [Quantum Groups and Non-commutative Hopf Geometry (CRM Univ. de Montreal, 1988)] seminal work, named preantipode that fulfills all the necessary requirements to be left but not right on the generators of the algebra. Due to the complexity and importance of the full task, we consider our result as an important step that will be extended in the near future.

INdAM conference "CoMeTA 2013 - Combinatorial Methods in Topology and Algebra"

CERN Document Server

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.

Bundles over Quantum RealWeighted Projective Spaces

Directory of Open Access Journals (Sweden)

Tomasz Brzeziński

2012-09-01

Full Text Available The algebraic approach to bundles in non-commutative geometry and the definition of quantum real weighted projective spaces are reviewed. Principal U(1-bundles over quantum real weighted projective spaces are constructed. As the spaces in question fall into two separate classes, the negative or odd class that generalises quantum real projective planes and the positive or even class that generalises the quantum disc, so do the constructed principal bundles. In the negative case the principal bundle is proven to be non-trivial and associated projective modules are described. In the positive case the principal bundles turn out to be trivial, and so all the associated modules are free. It is also shown that the circle (coactions on the quantum Seifert manifold that define quantum real weighted projective spaces are almost free.

Introduction to modern algebra and matrix theory

CERN Document Server

Schreier, O; David, Martin

2011-01-01

This unique text provides students with a basic course in both calculus and analytic geometry. It promotes an intuitive approach to calculus and emphasizes algebraic concepts. Minimal prerequisites. Numerous exercises. 1951 edition.

Infinite-dimensional Lie algebras in 4D conformal quantum field theory

International Nuclear Information System (INIS)

Bakalov, Bojko; Nikolov, Nikolay M; Rehren, Karl-Henning; Todorov, Ivan

2008-01-01

The concept of global conformal invariance (GCI) opens the way of applying algebraic techniques, developed in the context of two-dimensional chiral conformal field theory, to a higher (even) dimensional spacetime. In particular, a system of GCI scalar fields of conformal dimension two gives rise to a Lie algebra of harmonic bilocal fields, V M (x, y), where the M span a finite dimensional real matrix algebra M closed under transposition. The associative algebra M is irreducible iff its commutant M' coincides with one of the three real division rings. The Lie algebra of (the modes of) the bilocal fields is in each case an infinite-dimensional Lie algebra: a central extension of sp(∞,R) corresponding to the field R of reals, of u(∞, ∞) associated with the field C of complex numbers, and of so*(4∞) related to the algebra H of quaternions. They give rise to quantum field theory models with superselection sectors governed by the (global) gauge groups O(N), U(N) and U(N,H)=Sp(2N), respectively

On extensions of superconformal algebras

International Nuclear Information System (INIS)

Nagi, Jasbir

2005-01-01

Starting from vector fields that preserve a differential form on a Riemann sphere with Grassmann variables, one can construct a superconformal algebra by considering central extensions of the algebra of vector fields. In this paper, the N=4 case is analyzed closely, where the presence of weight zero operators in the field theory forces the introduction of noncentral extensions. How this modifies the existing field theory, representation theory, and Gelfand-Fuchs constructions is discussed. It is also discussed how graded Riemann sphere geometry can be used to give a geometrical description of the central charge in the N=1 theory

Lie n-algebras of BPS charges

Energy Technology Data Exchange (ETDEWEB)

Sati, Hisham [University of Pittsburgh,Pittsburgh, PA, 15260 (United States); Mathematics Program, Division of Science and Mathematics, New York University Abu Dhabi,Saadiyat Island, Abu Dhabi (United Arab Emirates); Schreiber, Urs [Mathematics Institute of the Academy,Žitna 25, Praha 1, 115 67 (Czech Republic)

2017-03-16

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 charges as they are lifted from ordinary cohomology to twisted K-theory. This supports the proposal that M-brane charges live in a twisted cohomology theory.

Matrix De Rham Complex and Quantum A-infinity algebras

Science.gov (United States)

Barannikov, S.

2014-04-01

I establish the relation of the non-commutative BV-formalism with super-invariant matrix integration. In particular, the non-commutative BV-equation, defining the quantum A ∞-algebras, introduced in Barannikov (Modular operads and non-commutative Batalin-Vilkovisky geometry. IMRN, vol. 2007, rnm075. Max Planck Institute for Mathematics 2006-48, 2007), is represented via de Rham differential acting on the supermatrix spaces related with Bernstein-Leites simple associative algebras with odd trace q( N), and gl( N| N). I also show that the matrix Lagrangians from Barannikov (Noncommutative Batalin-Vilkovisky geometry and matrix integrals. Isaac Newton Institute for Mathematical Sciences, Cambridge University, 2006) are represented by equivariantly closed differential forms.

On MV-algebras of non-linear functions

Directory of Open Access Journals (Sweden)

Antonio Di Nola

2017-01-01

Full Text Available In this paper, the main results are:a study of the finitely generated MV-algebras of continuous functions from the n-th power of the unit real interval I to I;a study of Hopfian MV-algebras; anda category-theoretic study of the map sending an MV-algebra as above to the range of its generators (up to a suitable form of homeomorphism.

On MV-algebras of non-linear functions

Directory of Open Access Journals (Sweden)

Antonio Di Nola

2017-01-01

Full Text Available In this paper, the main results are: a study of the finitely generated MV-algebras of continuous functions from the n-th power of the unit real interval I to I; a study of Hopfian MV-algebras; and a category-theoretic study of the map sending an MV-algebra as above to the range of its generators (up to a suitable form of homeomorphism.

University of Chicago School Mathematics Project (UCSMP) Algebra. WWC Intervention Report

Science.gov (United States)

What Works Clearinghouse, 2009

2009-01-01

University of Chicago School Mathematics Project (UCSMP) Algebra is a one-year course covering three primary topics: (1) linear and quadratic expressions, sentences, and functions; (2) exponential expressions and functions; and (3) linear systems. Topics from geometry, probability, and statistics are integrated with the appropriate algebra.…

Linear algebra

CERN Document Server

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...

Geometry and Hamiltonian mechanics on discrete spaces

International Nuclear Information System (INIS)

Talasila, V; Clemente-Gallardo, J; Schaft, A J van der

2004-01-01

Numerical simulation is often crucial for analysing the behaviour of many complex systems which do not admit analytic solutions. To this end, one either converts a 'smooth' model into a discrete (in space and time) model, or models systems directly at a discrete level. The goal of this paper is to provide a discrete analogue of differential geometry, and to define on these discrete models a formal discrete Hamiltonian structure-in doing so we try to bring together various fundamental concepts from numerical analysis, differential geometry, algebraic geometry, simplicial homology and classical Hamiltonian mechanics. For example, the concept of a twisted derivation is borrowed from algebraic geometry for developing a discrete calculus. The theory is applied to a nonlinear pendulum and we compare the dynamics obtained through a discrete modelling approach with the dynamics obtained via the usual discretization procedures. Also an example of an energy-conserving algorithm on a simple harmonic oscillator is presented, and its effect on the Poisson structure is discussed

Equivalency of two-dimensional algebras

International Nuclear Information System (INIS)

Santos, Gildemar Carneiro dos; Pomponet Filho, Balbino Jose S.

2011-01-01

Full text: Let us consider a vector z = xi + yj over the field of real numbers, whose basis (i,j) satisfy a given algebra. Any property of this algebra will be reflected in any function of z, so we can state that the knowledge of the properties of an algebra leads to more general conclusions than the knowledge of the properties of a function. However structural properties of an algebra do not change when this algebra suffers a linear transformation, though the structural constants defining this algebra do change. We say that two algebras are equivalent to each other whenever they are related by a linear transformation. In this case, we have found that some relations between the structural constants are sufficient to recognize whether or not an algebra is equivalent to another. In spite that the basis transform linearly, the structural constants change like a third order tensor, but some combinations of these tensors result in a linear transformation, allowing to write the entries of the transformation matrix as function of the structural constants. Eventually, a systematic way to find the transformation matrix between these equivalent algebras is obtained. In this sense, we have performed the thorough classification of associative commutative two-dimensional algebras, and find that even non-division algebra may be helpful in solving non-linear dynamic systems. The Mandelbrot set was used to have a pictorial view of each algebra, since equivalent algebras result in the same pattern. Presently we have succeeded in classifying some non-associative two-dimensional algebras, a task more difficult than for associative one. (author)

Quantum potential physics, geometry and algebra

CERN Document Server

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.

The three-dimensional origin of the classifying algebra

International Nuclear Information System (INIS)

Fuchs, Juergen; Schweigert, Christoph; Stigner, Carl

2010-01-01

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 structure constants of the classifying algebra as invariants of ribbon graphs in the three-manifold S 2 xS 1 . Our result unravels a precise relation between intertwiners of the action of the mapping class group on spaces of conformal blocks and boundary conditions in rational conformal field theories.

An introduction to differential geometry

CERN Document Server

Willmore, T J

2012-01-01

This text employs vector methods to explore the classical theory of curves and surfaces. Topics include basic theory of tensor algebra, tensor calculus, calculus of differential forms, and elements of Riemannian geometry. 1959 edition.

Linear algebra a first course with applications to differential equations

CERN Document Server

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.

The interplay between differential geometry and differential equations

CERN Document Server

Lychagin, V V

1995-01-01

This work applies symplectic methods and discusses quantization problems to emphasize the advantage of an algebraic geometry approach to nonlinear differential equations. One common feature in most of the presentations in this book is the systematic use of the geometry of jet spaces.

A first course in geometry

CERN Document Server

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

Non-Hermitian systems of Euclidean Lie algebraic type with real energy spectra

Science.gov (United States)

Dey, Sanjib; Fring, Andreas; Mathanaranjan, Thilagarajah

2014-07-01

We study several classes of non-Hermitian Hamiltonian systems, which can be expressed in terms of bilinear combinations of Euclidean-Lie algebraic generators. The classes are distinguished by different versions of antilinear (PT)-symmetries exhibiting various types of qualitative behaviour. On the basis of explicitly computed non-perturbative Dyson maps we construct metric operators, isospectral Hermitian counterparts for which we solve the corresponding time-independent Schrödinger equation for specific choices of the coupling constants. In these cases general analytical expressions for the solutions are obtained in the form of Mathieu functions, which we analyze numerically to obtain the corresponding energy spectra. We identify regions in the parameter space for which the corresponding spectra are entirely real and also domains where the PT symmetry is spontaneously broken and sometimes also regained at exceptional points. In some cases it is shown explicitly how the threshold region from real to complex spectra is characterized by the breakdown of the Dyson maps or the metric operator. We establish the explicit relationship to models currently under investigation in the context of beam dynamics in optical lattices.

College Algebra I.

Science.gov (United States)

Benjamin, Carl; And Others

Presented are student performance objectives, a student progress chart, and assignment sheets with objective and diagnostic measures for the stated performance objectives in College Algebra I. Topics covered include: sets; vocabulary; linear equations; inequalities; real numbers; operations; factoring; fractions; formulas; ratio, proportion, and…

Path operator algebras in conformal quantum field theories

International Nuclear Information System (INIS)

Roesgen, M.

2000-10-01

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.)

Universal algebra

CERN Document Server

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...

C*-algebras and operator theory

CERN Document Server

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.

Lattice and algebra homomorphisms on C (X) in Zermelo-Fraenkel ...

African Journals Online (AJOL)

Let C (X) denote the lattice-ordered algebra of all real-valued continuous functions on a topological space X. This paper discusses in Zermelo-Fraenkel Set Theory the equivalence on C (X) between algebra homomorphisms, lattice homo- morphisms, and point evaluations. Keywords: Algebra homomorphism, lattice ...

Conditions of the existence of the exact symmetric representation for some algebra classes with involution of the fields of real and complex numbers

International Nuclear Information System (INIS)

Ivanov, V.P.

1980-01-01

Necessary and sufficient conditions for existence of the exact symmetric representation of algebras with involution called sometimes regular of the fields of real and complex numbers are formulated in the paper

Commutative and Non-commutative Parallelogram Geometry: an Experimental Approach

OpenAIRE

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...

Hilbert schemes of points and infinite dimensional Lie algebras

CERN Document Server

Qin, Zhenbo

2018-01-01

Hilbert schemes, which parametrize subschemes in algebraic varieties, have been extensively studied in algebraic geometry for the last 50 years. The most interesting class of Hilbert schemes are schemes X^{[n]} of collections of n points (zero-dimensional subschemes) in a smooth algebraic surface X. Schemes X^{[n]} turn out to be closely related to many areas of mathematics, such as algebraic combinatorics, integrable systems, representation theory, and mathematical physics, among others. This book surveys recent developments of the theory of Hilbert schemes of points on complex surfaces and its interplay with infinite dimensional Lie algebras. It starts with the basics of Hilbert schemes of points and presents in detail an example of Hilbert schemes of points on the projective plane. Then the author turns to the study of cohomology of X^{[n]}, including the construction of the action of infinite dimensional Lie algebras on this cohomology, the ring structure of cohomology, equivariant cohomology of X^{[n]} a...

Workshop on Non-Associative & Non-Commutative Algebra and Operator Theory

CERN Document Server

Molina, Mercedes

2016-01-01

Presenting the collaborations of over thirty international experts in the latest developments in pure and applied mathematics, this volume serves as an anthology of research with a common basis in algebra, functional analysis and their applications. Special attention is devoted to non-commutative algebras, non-associative algebras, operator theory and ring and module theory. These themes are relevant in research and development in coding theory, cryptography and quantum mechanics. The topics in this volume were presented at the Workshop on Non-Associative & Non-Commutative Algebra and Operator Theory, held May 23—25, 2014 at Cheikh Anta Diop University in Dakar, Senegal in honor of Professor Amin Kaidi. The workshop was hosted by the university's Laboratory of Algebra, Cryptology, Algebraic Geometry and Applications, in cooperation with the University of Almería and the University of Málaga. Dr. Kaidi's work focuses on non-associative rings and algebras, operator theory and functional analysis, and he...

A vector space approach to geometry

CERN Document Server

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.

Automatic Deduction in Dynamic Geometry using Sage

Directory of Open Access Journals (Sweden)

Francisco Botana

2012-02-01

Full Text Available We present a symbolic tool that provides robust algebraic methods to handle automatic deduction tasks for a dynamic geometry construction. The main prototype has been developed as two different worksheets for the open source computer algebra system Sage, corresponding to two different ways of coding a geometric construction. In one worksheet, diagrams constructed with the open source dynamic geometry system GeoGebra are accepted. In this worksheet, Groebner bases are used to either compute the equation of a geometric locus in the case of a locus construction or to determine the truth of a general geometric statement included in the GeoGebra construction as a boolean variable. In the second worksheet, locus constructions coded using the common file format for dynamic geometry developed by the Intergeo project are accepted for computation. The prototype and several examples are provided for testing. Moreover, a third Sage worksheet is presented in which a novel algorithm to eliminate extraneous parts in symbolically computed loci has been implemented. The algorithm, based on a recent work on the Groebner cover of parametric systems, identifies degenerate components and extraneous adherence points in loci, both natural byproducts of general polynomial algebraic methods. Detailed examples are discussed.

Lie n-algebras of BPS charges

Czech Academy of Sciences Publication Activity Database

Sati, H.; Schreiber, Urs

2017-01-01

Roč. 2017, č. 3 (2017), č. článku 87. ISSN 1126-6708 Institutional support: RVO:67985840 Keywords : Differential and Algebra ic Geometry * p-branes Subject RIV: BA - General Mathematics OBOR OECD: Applied mathematics http://link.springer.com/article/10.1007%2FJHEP03%282017%29087

Algebraic Hyperstructures and Fuzzy Logic in the Treatment of Uncertainty

Directory of Open Access Journals (Sweden)

Antonio Maturo

2016-09-01

Full Text Available This study presents some fundamental aspects of recent theories  on algebraic Hyperstructures, an important tool for an interdisciplinary vision of Geometry and Algebra. We examine some hypergroupoids of events, useful for a new algebraic-geometry perspective in the study and issues of probability applications. This paper considers some fundamental aspects of fuzzy classifications and their applications to problems of evaluation and decision in Architecture and Economics. Finally, we present hypergroups and join space associated with these classifications.   Iperstrutture algebriche e logica fuzzy nel trattamento dell’incertezza Si presentano alcuni aspetti fondamentali della relativamente recente teoria delle iperstrutture algebriche, importante strumento per una visione interdisciplinare di Geometria e Algebra. Si esaminano alcuni ipergruppoidi di eventi, utili per un nuovo punto di vista algebrico - geometrico nello studio e nelle applicazioni di alcune questioni di probabilità. Si considerano alcuni aspetti fondamentali delle classificazioni fuzzy e le loro applicazioni a problemi di valutazione e decisione in Architettura e in Economia. Si presentano infine ipergruppi e join space associati a tali classificazioni. Parole Chiave: Iperstrutture algebriche. Logica fuzzy. Applicazioni a Architettura e Economia.

Riemann surfaces and algebraic curves a first course in Hurwitz theory

CERN Document Server

Cavalieri, Renzo

2016-01-01

Hurwitz theory, the study of analytic functions among Riemann surfaces, is a classical field and active research area in algebraic geometry. The subject's interplay between algebra, geometry, topology and analysis is a beautiful example of the interconnectedness of mathematics. This book introduces students to this increasingly important field, covering key topics such as manifolds, monodromy representations and the Hurwitz potential. Designed for undergraduate study, this classroom-tested text includes over 100 exercises to provide motivation for the reader. Also included are short essays by guest writers on how they use Hurwitz theory in their work, which ranges from string theory to non-Archimedean geometry. Whether used in a course or as a self-contained reference for graduate students, this book will provide an exciting glimpse at mathematics beyond the standard university classes.

An algebraic approach to modeling in software engineering

International Nuclear Information System (INIS)

Loegel, C.J.; Ravishankar, C.V.

1993-09-01

Our work couples the formalism of universal algebras with the engineering techniques of mathematical modeling to develop a new approach to the software engineering process. Our purpose in using this combination is twofold. First, abstract data types and their specification using universal algebras can be considered a common point between the practical requirements of software engineering and the formal specification of software systems. Second, mathematical modeling principles provide us with a means for effectively analyzing real-world systems. We first use modeling techniques to analyze a system and then represent the analysis using universal algebras. The rest of the software engineering process exploits properties of universal algebras that preserve the structure of our original model. This paper describes our software engineering process and our experience using it on both research and commercial systems. We need a new approach because current software engineering practices often deliver software that is difficult to develop and maintain. Formal software engineering approaches use universal algebras to describe ''computer science'' objects like abstract data types, but in practice software errors are often caused because ''real-world'' objects are improperly modeled. There is a large semantic gap between the customer's objects and abstract data types. In contrast, mathematical modeling uses engineering techniques to construct valid models for real-world systems, but these models are often implemented in an ad hoc manner. A combination of the best features of both approaches would enable software engineering to formally specify and develop software systems that better model real systems. Software engineering, like mathematical modeling, should concern itself first and foremost with understanding a real system and its behavior under given circumstances, and then with expressing this knowledge in an executable form

Model theory and algebraic geometry an introduction to E. Hrushovski’s proof of the geometric Mordell-Lang conjecture

CERN Document Server

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.

Essays in the history of Lie groups and algebraic groups

CERN Document Server

Borel, Armand

2001-01-01

Lie groups and algebraic groups are important in many major areas of mathematics and mathematical physics. We find them in diverse roles, notably as groups of automorphisms of geometric structures, as symmetries of differential systems, or as basic tools in the theory of automorphic forms. The author looks at their development, highlighting the evolution from the almost purely local theory at the start to the global theory that we know today. Starting from Lie's theory of local analytic transformation groups and early work on Lie algebras, he follows the process of globalization in its two main frameworks: differential geometry and topology on one hand, algebraic geometry on the other. Chapters II to IV are devoted to the former, Chapters V to VIII, to the latter. The essays in the first part of the book survey various proofs of the full reducibility of linear representations of \\mathbf{SL}_2{(\\mathbb{C})}, the contributions of H. Weyl to representations and invariant theory for semisimple Lie groups, and con...

Numerical linear algebra with applications using Matlab

CERN Document Server

Ford, William

2014-01-01

Designed for those who want to gain a practical knowledge of modern computational techniques for the numerical solution of linear algebra problems, Numerical Linear Algebra with Applications contains all the material necessary for a first year graduate or advanced undergraduate course on numerical linear algebra with numerous applications to engineering and science. With a unified presentation of computation, basic algorithm analysis, and numerical methods to compute solutions, this book is ideal for solving real-world problems. It provides necessary mathematical background information for

Dynamical systems of algebraic origin

CERN Document Server

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...

Extremal rotating black holes in the near-horizon limit: Phase space and symmetry algebra

Directory of Open Access Journals (Sweden)

G. Compère

2015-10-01

Full Text Available We construct the NHEG phase space, the classical phase space of Near-Horizon Extremal Geometries with fixed angular momenta and entropy, and with the largest symmetry algebra. We focus on vacuum solutions to d dimensional Einstein gravity. Each element in the phase space is a geometry with SL(2,R×U(1d−3 isometries which has vanishing SL(2,R and constant U(1 charges. We construct an on-shell vanishing symplectic structure, which leads to an infinite set of symplectic symmetries. In four spacetime dimensions, the phase space is unique and the symmetry algebra consists of the familiar Virasoro algebra, while in d>4 dimensions the symmetry algebra, the NHEG algebra, contains infinitely many Virasoro subalgebras. The nontrivial central term of the algebra is proportional to the black hole entropy. The conserved charges are given by the Fourier decomposition of a Liouville-type stress-tensor which depends upon a single periodic function of d−3 angular variables associated with the U(1 isometries. This phase space and in particular its symmetries can serve as a basis for a semiclassical description of extremal rotating black hole microstates.

A Hierarchy of Proof Rules for Checking Differential Invariance of Algebraic Sets

Science.gov (United States)

2014-11-01

linear hybrid systems by linear algebraic methods. In SAS, volume 6337 of LNCS, pages 373–389. Springer, 2010. [19] E. W. Mayr. Membership in polynomial...383–394, 2009. [31] A. Tarski. A decision method for elementary algebra and geometry. Bull. Amer. Math. Soc., 59, 1951. [32] A. Tiwari. Abstractions...A Hierarchy of Proof Rules for Checking Differential Invariance of Algebraic Sets Khalil Ghorbal1 Andrew Sogokon2 André Platzer1 November 2014 CMU

Momentum-space cigar geometry in topological phases

Science.gov (United States)

Palumbo, Giandomenico

2018-01-01

In this paper, we stress the importance of momentum-space geometry in the understanding of two-dimensional topological phases of matter. We focus, for simplicity, on the gapped boundary of three-dimensional topological insulators in class AII, which are described by a massive Dirac Hamiltonian and characterized by an half-integer Chern number. The gap is induced by introducing a magnetic perturbation, such as an external Zeeman field or a ferromagnet on the surface. The quantum Bures metric acquires a central role in our discussion and identifies a cigar geometry. We first derive the Chern number from the cigar geometry and we then show that the quantum metric can be seen as a solution of two-dimensional non-Abelian BF theory in momentum space. The gauge connection for this model is associated to the Maxwell algebra, which takes into account the Lorentz symmetries related to the Dirac theory and the momentum-space magnetic translations connected to the magnetic perturbation. The Witten black-hole metric is a solution of this gauge theory and coincides with the Bures metric. This allows us to calculate the corresponding momentum-space entanglement entropy that surprisingly carries information about the real-space conformal field theory describing the defect lines that can be created on the gapped boundary.

Geometry of physical dispersion relations

International Nuclear Information System (INIS)

Raetzel, Dennis; Rivera, Sergio; Schuller, Frederic P.

2011-01-01

To serve as a dispersion relation, a cotangent bundle function must satisfy three simple algebraic properties. These conditions are derived from the inescapable physical requirements that local matter field dynamics must be predictive and allow for an observer-independent notion of positive energy. Possible modifications of the standard relativistic dispersion relation are thereby severely restricted. For instance, the dispersion relations associated with popular deformations of Maxwell theory by Gambini-Pullin or Myers-Pospelov are not admissible. Dispersion relations passing the simple algebraic checks derived here correspond to physically admissible Finslerian refinements of Lorentzian geometry.

Priority in Process Algebras

Science.gov (United States)

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.

On ''conformal spinor geometry'': An attempt to ''understand'' internal symmetry

International Nuclear Information System (INIS)

Budinich, P.

1981-09-01

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). C 4 and C 6 spinor geometry are analyzed but it seems that C 8 spinor geometry is necessary to construct Minkowski space Msup(3,1). C 6 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)

On ''conformal spinor geometry'': An attempt to ''understand'' internal symmetry

International Nuclear Information System (INIS)

Budinich, P.

1982-01-01

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 polynomials of the components of a pure spinor). C 4 and C 6 spinor geometry are analyzed, but it seems that C 8 spinor geometry is necessary to construct Minkowski space Msup(3,1). C 6 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 breaking spontaneously the original O(4,2) symmetry of the spinor equation. (author)

Towards relativistic quantum geometry

Energy Technology Data Exchange (ETDEWEB)

Ridao, Luis Santiago [Instituto de Investigaciones Físicas de Mar del Plata (IFIMAR), Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET), Mar del Plata (Argentina); Bellini, Mauricio, E-mail: mbellini@mdp.edu.ar [Departamento de Física, Facultad de Ciencias Exactas y Naturales, Universidad Nacional de Mar del Plata, Funes 3350, C.P. 7600, Mar del Plata (Argentina); Instituto de Investigaciones Físicas de Mar del Plata (IFIMAR), Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET), Mar del Plata (Argentina)

2015-12-17

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.

Multilinear Computing and Multilinear Algebraic Geometry

Science.gov (United States)

2016-08-10

Laplacians on graphs,” S. Mukherjee (Ed.), Geometry and Topology in Statistical Inference, Proc. Sympos. Appl. Math ., 73, AMS, Providence, RI, 2015...8–12, 2015. • “Fast(est) algorithms for structured matrices via tensor decompositions,” Applied Mathe - matics and Analysis Seminar, Duke University...Durham, NC, April 13, 2015. • “Fast(est) algorithms for structured matrices via tensor decompositions,” Applied Mathe - matics Seminar, Stanford

Generalized space-time supersymmetries, division algebras and octonionic M-theory

International Nuclear Information System (INIS)

Lukierski, Jerzy; Toppan, Francesco

2002-03-01

We describe the set of generalized Poincare and conformal superalgebras in D= 4,5 and 7 dimensions as two sequences of superalgebraic structures, taking values in the division algebras R, C and H. The generalized conformal superalgebras are described for D = 4 by OSp(1;8|R), for D = 5 by SU(4,4;1) and for D = 7 by U α U (8;1|H). The relation with other schemes, in particular the framework of conformal spin (super) algebras and Jordan (super) algebras is discussed. By extending the division-algebra-valued super-algebras to octonions we get in D= 11 an octonionic generalized Poincare superalgebra, which we call octonionic M-algebra, describing the octonionic M-theory. It contains 32 real supercharges but, due to the octonionic structure only 52 real bosonic generators remain independent in place of the 528 bosonic charges of standard M-algebra. In octonionic M-theory there is a sort of equivalence between the octonionic M2 (supermembrane) and the octonionic M5 (super-5-brane) sectors. We also define the octonionic generalized conformal M-superalgebra with 239 bosonic generators. (author)

On Fock Space Representations of quantized Enveloping Algebras related to Non-Commutative Differential Geometry

CERN Document Server

Jurco, B; Jurco, B; Schlieker, M

1995-01-01

In this paper we construct explicitly natural (from the geometrical point of view) Fock space representations (contragradient Verma modules) of the quantized enveloping algebras. In order to do so, we start from the Gauss decomposition of the quantum group and introduce the differential operators on the corresponding q-deformed flag manifold (asuumed as a left comodule for the quantum group) by a projection to it of the right action of the quantized enveloping algebra on the quantum group. Finally, we express the representatives of the elements of the quantized enveloping algebra corresponding to the left-invariant vector fields on the quantum group as first-order differential operators on the q-deformed flag manifold.

Numerical algebra, matrix theory, differential-algebraic equations and control theory festschrift in honor of Volker Mehrmann

CERN Document Server

Bollhöfer, Matthias; Kressner, Daniel; Mehl, Christian; Stykel, Tatjana

2015-01-01

This edited volume highlights the scientific contributions of Volker Mehrmann, a leading expert in the area of numerical (linear) algebra, matrix theory, differential-algebraic equations and control theory. These mathematical research areas are strongly related and often occur in the same real-world applications. The main areas where such applications emerge are computational engineering and sciences, but increasingly also social sciences and economics. This book also reflects some of Volker Mehrmann's major career stages. Starting out working in the areas of numerical linear algebra (his first full professorship at TU Chemnitz was in "Numerical Algebra," hence the title of the book) and matrix theory, Volker Mehrmann has made significant contributions to these areas ever since. The highlights of these are discussed in Parts I and II of the present book. Often the development of new algorithms in numerical linear algebra is motivated by problems in system and control theory. These and his later major work on ...

Unifying Ancient and Modern Geometries Through Octonions

International Nuclear Information System (INIS)

Catto, Sultan; Gürcan, Yasemin; Kurt, Levent; Khalfan, Amish

2016-01-01

We show the first unified description of some of the oldest known geometries such as the Pappus’ theorem with more modern ones like Desargues' theorem, Monge's theorem and Ceva's theorem, through octonions, the highest normed division algebra in eight dimensions. We also show important applications in hadronic physics, giving a full description of the algebra of color applicable to quark physics, and comment on further applications. (paper)

The Euclidean distance degree of an algebraic variety

NARCIS (Netherlands)

Draisma, J.; Horobet, E.; Ottaviani, G.; Sturmfels, B.; Thomas, R.R.

2013-01-01

The nearest point map of a real algebraic variety with respect to Euclidean distance is an algebraic function. For instance, for varieties of low rank matrices, the Eckart-Young Theorem states that this map is given by the singular value decomposition. This article develops a theory of such nearest

The Euclidean distance degree of an algebraic variety

NARCIS (Netherlands)

Draisma, J.; Horobet, E.; Ottaviani, G.; Sturmfels, B.; Thomas, R.R.

The nearest point map of a real algebraic variety with respect to Euclidean distance is an algebraic function. For instance, for varieties of low-rank matrices, the Eckart–Young Theorem states that this map is given by the singular value decomposition. This article develops a theory of such nearest

Advances in geometry and Lie algebras from supergravity

CERN Document Server

Frè, Pietro Giuseppe

2018-01-01

This book aims to provide an overview of several topics in advanced Differential Geometry and Lie Group Theory, all of them stemming from mathematical problems in supersymmetric physical theories. It presents a mathematical illustration of the main development in geometry and symmetry theory that occurred under the fertilizing influence of supersymmetry/supergravity. The contents are mainly of mathematical nature, but each topic is introduced by historical information and enriched with motivations from high energy physics, which help the reader in getting a deeper comprehension of the subject. .

Algebraic theory of locally nilpotent derivations

CERN Document Server

Freudenburg, Gene

2017-01-01

This book explores the theory and application of locally nilpotent derivations, a subject motivated by questions in affine algebraic geometry and having fundamental connections to areas such as commutative algebra, representation theory, Lie algebras and differential equations. The author provides a unified treatment of the subject, beginning with 16 First Principles on which the theory is based. These are used to establish classical results, such as Rentschler's Theorem for the plane and the Cancellation Theorem for Curves. More recent results, such as Makar-Limanov's theorem for locally nilpotent derivations of polynomial rings, are also discussed. Topics of special interest include progress in classifying additive actions on three-dimensional affine space, finiteness questions (Hilbert's 14th Problem), algorithms, the Makar-Limanov invariant, and connections to the Cancellation Problem and the Embedding Problem. A lot of new material is included in this expanded second edition, such as canonical factoriza...

Fiber-wise linear Poisson structures related to W∗-algebras

Science.gov (United States)

Odzijewicz, Anatol; Jakimowicz, Grzegorz; Sliżewska, Aneta

2018-01-01

In the framework of Banach differential geometry we investigate the fiber-wise linear Poisson structures as well as the Lie groupoid and Lie algebroid structures which are defined in the canonical way by the structure of a W∗-algebra (von Neumann algebra) M. The main role in this theory is played by the complex Banach-Lie groupoid G(M) ⇉ L(M) of partially invertible elements of M over the lattice L(M) of orthogonal projections of M. The Atiyah sequence and the predual Atiyah sequence corresponding to this groupoid are investigated from the point of view of Banach Poisson geometry. In particular we show that the predual Atiyah sequence fits in a short exact sequence of complex Banach sub-Poisson V B-groupoids with G(M) ⇉ L(M) as the side groupoid.

Coherent states and classical limit of algebraic quantum models

International Nuclear Information System (INIS)

Scutaru, H.

1983-01-01

The algebraic models for collective motion in nuclear physics belong to a class of theories the basic observables of which generate selfadjoint representations of finite dimensional, real Lie algebras, or of the enveloping algebras of these Lie algebras. The simplest and most used for illustrations model of this kind is the Lipkin model, which is associated with the Lie algebra of the three dimensional rotations group, and which presents all characteristic features of an algebraic model. The Lipkin Hamiltonian is the image, of an element of the enveloping algebra of the algebra SO under a representation. In order to understand the structure of the algebraic models the author remarks that in both classical and quantum mechanics the dynamics is associated to a typical algebraic structure which we shall call a dynamical algebra. In this paper he shows how the constructions can be made in the case of the algebraic quantum systems. The construction of the symplectic manifold M can be made in this case using a quantum analog of the momentum map which he defines

Numerical linear algebra theory and applications

CERN Document Server

Beilina, Larisa; Karchevskii, Mikhail

2017-01-01

This book combines a solid theoretical background in linear algebra with practical algorithms for numerical solution of linear algebra problems. Developed from a number of courses taught repeatedly by the authors, the material covers topics like matrix algebra, theory for linear systems of equations, spectral theory, vector and matrix norms combined with main direct and iterative numerical methods, least squares problems, and eigen problems. Numerical algorithms illustrated by computer programs written in MATLAB® are also provided as supplementary material on SpringerLink to give the reader a better understanding of professional numerical software for the solution of real-life problems. Perfect for a one- or two-semester course on numerical linear algebra, matrix computation, and large sparse matrices, this text will interest students at the advanced undergraduate or graduate level.

Causal structure and algebraic classification of non-dissipative linear optical media

International Nuclear Information System (INIS)

Schuller, Frederic P.; Witte, Christof; Wohlfarth, Mattias N.R.

2010-01-01

In crystal optics and quantum electrodynamics in gravitational vacua, the propagation of light is not described by a metric, but an area metric geometry. In this article, this prompts us to study conditions for linear electrodynamics on area metric manifolds to be well-posed. This includes an identification of the timelike future cones and their duals associated to an area metric geometry, and thus paves the ground for a discussion of the related local and global causal structures in standard fashion. In order to provide simple algebraic criteria for an area metric manifold to present a consistent spacetime structure, we develop a complete algebraic classification of area metric tensors up to general transformations of frame. This classification, valuable in its own right, is then employed to prove a theorem excluding the majority of algebraic classes of area metrics as viable spacetimes. Physically, these results classify and drastically restrict the viable constitutive tensors of non-dissipative linear optical media.

An introduction to central simple algebras and their applications to wireless communication

CERN Document Server

Berhuy, Gre�gory

2013-01-01

Central simple algebras arise naturally in many areas of mathematics. They are closely connected with ring theory, but are also important in representation theory, algebraic geometry and number theory. Recently, surprising applications of the theory of central simple algebras have arisen in the context of coding for wireless communication. The exposition in the book takes advantage of this serendipity, presenting an introduction to the theory of central simple algebras intertwined with its applications to coding theory. Many results or constructions from the standard theory are presented in classical form, but with a focus on explicit techniques and examples, often from coding theory. Topics covered include quaternion algebras, splitting fields, the Skolem-Noether Theorem, the Brauer group, crossed products, cyclic algebras and algebras with a unitary involution. Code constructions give the opportunity for many examples and explicit computations. This book provides an introduction to the theory of central alg...

Four-manifolds, geometries and knots

CERN Document Server

Hillman, Jonathan A

2007-01-01

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 14-18). In many cases the Euler characteristic, fundamental group and Stiefel-Whitney classes together form a complete system of invariants for the homotopy type of such manifolds, and the possible values of the invariants can be described explicitly. The strongest results are characterizations of manifolds which fibre homotopically over S^1 or an aspherical surface (up to homotopy equivalence) and infrasolvmanifolds (up to homeomorphism). As a consequence 2-knots whose groups are poly-Z are determined up to Gluck reconstruc...

Analytical SN solutions in heterogeneous slabs using symbolic algebra computer programs

International Nuclear Information System (INIS)

Warsa, J.S.

2002-01-01

A modern symbolic algebra computer program, MAPLE, is used to compute solutions to the well-known analytical discrete ordinates, or S N , solutions in one-dimensional, slab geometry. Symbolic algebra programs compute the solutions with arbitrary precision and are free of spatial discretization error so they can be used to investigate new discretizations for one-dimensional slab, geometry S N methods. Pointwise scalar flux solutions are computed for several sample calculations of interest. Sample MAPLE command scripts are provided to illustrate how easily the theory can be translated into a working solution and serve as a complete tool capable of computing analytical S N solutions for mono-energetic, one-dimensional transport problems

Non-commutative geometry and supersymmetry 2

International Nuclear Information System (INIS)

Hussain, F.; Thompson, G.

1991-05-01

Following the general construction of supersymmetric models, the model based on the idea of non-commutative geometry is formulated as a Yang-Mills theory of the graded Lie algebra U(2/1) over a graded space-time manifold. 4 refs

Linear Algebra and Smarandache Linear Algebra

OpenAIRE

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 vector spaces over finite p...

The geometry of special relativity

International Nuclear Information System (INIS)

Parizet, Jean

2008-01-01

This book for students in mathematics or physics shows the interest of geometry to understand special relativity as a consequence of invariance of Maxwell equations and of constancy of the speed of light. Space-time is actually provided with a geometrical structure and a physical interpretation: at each observer are associated his own time and his own physical space in which occur events he is concerned with. This leads to a natural approach to special relativity. The Lorentz group and its algebra are then studied by using matrices and the Pauli algebra. Quaternions are also addressed

Gauss Elimination: Workhorse of Linear Algebra.

Science.gov (United States)

1995-08-05

linear algebra computation for solving systems, computing determinants and determining the rank of matrix. All of these are discussed in varying contexts. These include different arithmetic or algebraic setting such as integer arithmetic or polynomial rings as well as conventional real (floating-point) arithmetic. These have effects on both accuracy and complexity analyses of the algorithm. These, too, are covered here. The impact of modern parallel computer architecture on GE is also

Introduction to vertex algebras, Borcherds algebras and the Monster Lie algebras

International Nuclear Information System (INIS)

Gebert, R.W.

1993-09-01

The theory of vertex algebras constitutes a mathematically rigorous axiomatic formulation of the algebraic origins of conformal field theory. In this context Borcherds algebras arise as certain ''physical'' subspaces of vertex algebras. The aim of this review is to give a pedagogical introduction into this rapidly-developing area of mathematics. Based on the machinery of formal calculus we present the axiomatic definition of vertex algebras. We discuss the connection with conformal field theory by deriving important implications of these axioms. In particular, many explicit calculations are presented to stress the eminent role of the Jacobi identity axiom for vertex algebras. As a class of concrete examples the vertex algebras associated with even lattices are constructed and it is shown in detail how affine Lie algebras and the fake Monster Lie algebra naturally appear. This leads us to the abstract definition of Borcherds algebras as generalized Kac-Moody algebras and their basic properties. Finally, the results about the simplest generic Borcherds algebras are analysed from the point of view of symmetry in quantum theory and the construction of the Monster Lie algebra is sketched. (orig.)

Differential geometry curves, surfaces, manifolds

CERN Document Server

Kohnel, Wolfgang

2002-01-01

This carefully written book is an introduction to the beautiful ideas and results of differential geometry. The first half covers the geometry of curves and surfaces, which provide much of the motivation and intuition for the general theory. Special topics that are explored include Frenet frames, ruled surfaces, minimal surfaces and the Gauss-Bonnet theorem. The second part is an introduction to the geometry of general manifolds, with particular emphasis on connections and curvature. The final two chapters are insightful examinations of the special cases of spaces of constant curvature and Einstein manifolds. The text is illustrated with many figures and examples. The prerequisites are undergraduate analysis and linear algebra.

Origami, Geometry and Art

Science.gov (United States)

Wares, Arsalan; Elstak, Iwan

2017-01-01

The purpose of this paper is to describe the mathematics that emanates from the construction of an origami box. We first construct a simple origami box from a rectangular sheet and then discuss some of the mathematical questions that arise in the context of geometry and algebra. The activity can be used as a context for illustrating how algebra…

Three-dimensional digital tomosynthesis iterative reconstruction, artifact reduction and alternative acquisition geometry

CERN Document Server

Levakhina, Yulia

2014-01-01

Yulia Levakhina gives an introduction to the major challenges of image reconstruction in Digital Tomosynthesis (DT), particularly to the connection of the reconstruction problem with the incompleteness of the DT dataset. The author discusses the factors which cause the formation of limited angle artifacts and proposes how to account for them in order to improve image quality and axial resolution of modern DT. The addressed methods include a weighted non-linear back projection scheme for algebraic reconstruction and?novel dual-axis acquisition geometry. All discussed algorithms and methods are supplemented by detailed illustrations, hints for practical implementation, pseudo-code, simulation results and real patient case examples.

Operator algebras and topology

International Nuclear Information System (INIS)

Schick, T.

2002-01-01

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 L 2 -cohomology, L 2 -Betti numbers and other L 2 -invariants. These invariants, their basic properties, and the central questions about them, are introduced in the third chapter. (author)

Group C∗-algebras without the completely bounded approximation property

DEFF Research Database (Denmark)

Haagerup, U.

2016-01-01

It is proved that: (1) The Fourier algebra A(G) of a simple Lie group G of real rank at least 2 with finite center does not have a multiplier bounded approximate unit. (2) The reduced C∗-algebra C∗ r of any lattice in a non-compact simple Lie group of real rank at least 2 with finite center does...... not have the completely bounded approximation property. Hence, the results obtained by de Canniere and the author for SOe (n, 1), n ≥ 2, and by Cowling for SU(n, 1) do not generalize to simple Lie groups of real rank at least 2. © 2016 Heldermann Verlag....

Cluster algebras in mathematical physics

International Nuclear Information System (INIS)

Francesco, Philippe Di; Gekhtman, Michael; Kuniba, Atsuo; Yamazaki, Masahito

2014-01-01

This special issue of Journal of Physics A: Mathematical and Theoretical contains reviews and original research articles on cluster algebras and their applications to mathematical physics. Cluster algebras were introduced by S Fomin and A Zelevinsky around 2000 as a tool for studying total positivity and dual canonical bases in Lie theory. Since then the theory has found diverse applications in mathematics and mathematical physics. Cluster algebras are axiomatically defined commutative rings equipped with a distinguished set of generators (cluster variables) subdivided into overlapping subsets (clusters) of the same cardinality subject to certain polynomial relations. A cluster algebra of rank n can be viewed as a subring of the field of rational functions in n variables. Rather than being presented, at the outset, by a complete set of generators and relations, it is constructed from the initial seed via an iterative procedure called mutation producing new seeds successively to generate the whole algebra. A seed consists of an n-tuple of rational functions called cluster variables and an exchange matrix controlling the mutation. Relations of cluster algebra type can be observed in many areas of mathematics (Plücker and Ptolemy relations, Stokes curves and wall-crossing phenomena, Feynman integrals, Somos sequences and Hirota equations to name just a few examples). The cluster variables enjoy a remarkable combinatorial pattern; in particular, they exhibit the Laurent phenomenon: they are expressed as Laurent polynomials rather than more general rational functions in terms of the cluster variables in any seed. These characteristic features are often referred to as the cluster algebra structure. In the last decade, it became apparent that cluster structures are ubiquitous in mathematical physics. Examples include supersymmetric gauge theories, Poisson geometry, integrable systems, statistical mechanics, fusion products in infinite dimensional algebras, dilogarithm

Algebra in Cuneiform

DEFF Research Database (Denmark)

Høyrup, Jens

with basic Assyriology but otherwise philological details are avoided. All of these texts are from the second half of the Old Babylonian period, that is, 1800–1600 BCE. It is indeed during this period that the “algebraic” discipline, and Babylonian mathematics in general, culminates. Even though a few texts...... particular culture. Finally, it describes the origin of the discipline and its impact in later mathematics, not least Euclid’s geometry and genuine algebra as created in medieval Islam and taken over in European medieval and Renaissance mathematics....

Tracing a planar algebraic curve

International Nuclear Information System (INIS)

Chen Falai; Kozak, J.

1994-09-01

In this paper, an algorithm that determines a real algebraic curve is outlined. Its basic step is to divide the plane into subdomains that include only simple branches of the algebraic curve without singular points. Each of the branches is then stably and efficiently traced in the particular subdomain. Except for the tracing, the algorithm requires only a couple of simple operations on polynomials that can be carried out exactly if the coefficients are rational, and the determination of zeros of several polynomials of one variable. (author). 5 refs, 4 figs

Multivariable calculus and differential geometry

CERN Document Server

Walschap, Gerard

2015-01-01

This text is a modern in-depth study of the subject that includes all the material needed from linear algebra. It then goes on to investigate topics in differential geometry, such as manifolds in Euclidean space, curvature, and the generalization of the fundamental theorem of calculus known as Stokes' theorem.

Flag varieties an interplay of geometry, combinatorics, and representation theory

CERN Document Server

Lakshmibai, V

2009-01-01

Flag varieties are important geometric objects and their study involves an interplay of geometry, combinatorics, and representation theory. This book is detailed account of this interplay. In the area of representation theory, the book presents a discussion of complex semisimple Lie algebras and of semisimple algebraic groups; in addition, the representation theory of symmetric groups is also discussed. In the area of algebraic geometry, the book gives a detailed account of the Grassmannian varieties, flag varieties, and their Schubert subvarieties. Because of the connections with root systems, many of the geometric results admit elegant combinatorial description, a typical example being the description of the singular locus of a Schubert variety. This is shown to be a consequence of standard monomial theory (abbreviated SMT). Thus the book includes SMT and some important applications - singular loci of Schubert varieties, toric degenerations of Schubert varieties, and the relationship between Schubert variet...

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.

On Some Algebraic and Combinatorial Properties of Dunkl Elements

Science.gov (United States)

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.

Quantum W-algebras and elliptic algebras

International Nuclear Information System (INIS)

Feigin, B.; Kyoto Univ.; Frenkel, E.

1996-01-01

We define a quantum W-algebra associated to sl N as an associative algebra depending on two parameters. For special values of the parameters, this algebra becomes the ordinary W-algebra of sl N , or the q-deformed classical W-algebra of sl N . We construct free field realizations of the quantum W-algebras and the screening currents. We also point out some interesting elliptic structures arising in these algebras. In particular, we show that the screening currents satisfy elliptic analogues of the Drinfeld relations in U q (n). (orig.)

Tensorial spacetime geometries carrying predictive, interpretable and quantizable matter dynamics

International Nuclear Information System (INIS)

Rivera Hernandez, Sergio

2012-01-01

Which tensor fields G on a smooth manifold M can serve as a spacetime structure? In the first part of this thesis, it is found that only a severely restricted class of tensor fields can provide classical spacetime geometries, namely those that can carry predictive, interpretable and quantizable matter dynamics. The obvious dependence of this characterization of admissible tensorial spacetime geometries on specific matter is not a weakness, but rather presents an insight: it was Maxwell theory that justified Einstein to promote Lorentzian manifolds to the status of a spacetime geometry. Any matter that does not mimick the structure of Maxwell theory, will force us to choose another geometry on which the matter dynamics of interest are predictive, interpretable and quantizable. These three physical conditions on matter impose three corresponding algebraic conditions on the totally symmetric contravariant coefficient tensor field P that determines the principal symbol of the matter field equations in terms of the geometric tensor G: the tensor field P must be hyperbolic, time-orientable and energy-distinguishing. Remarkably, these physically necessary conditions on the geometry are mathematically already sufficient to realize all kinematical constructions familiar from Lorentzian geometry, for precisely the same structural reasons. This we were able to show employing a subtle interplay of convex analysis, the theory of partial differential equations and real algebraic geometry. In the second part of this thesis, we then explore general properties of any hyperbolic, time-orientable and energy-distinguishing tensorial geometry. Physically most important are the construction of freely falling non-rotating laboratories, the appearance of admissible modified dispersion relations to particular observers, and the identification of a mechanism that explains why massive particles that are faster than some massless particles can radiate off energy until they are slower than all

Tensorial spacetime geometries carrying predictive, interpretable and quantizable matter dynamics

Energy Technology Data Exchange (ETDEWEB)

Rivera Hernandez, Sergio

2012-02-15

Which tensor fields G on a smooth manifold M can serve as a spacetime structure? In the first part of this thesis, it is found that only a severely restricted class of tensor fields can provide classical spacetime geometries, namely those that can carry predictive, interpretable and quantizable matter dynamics. The obvious dependence of this characterization of admissible tensorial spacetime geometries on specific matter is not a weakness, but rather presents an insight: it was Maxwell theory that justified Einstein to promote Lorentzian manifolds to the status of a spacetime geometry. Any matter that does not mimick the structure of Maxwell theory, will force us to choose another geometry on which the matter dynamics of interest are predictive, interpretable and quantizable. These three physical conditions on matter impose three corresponding algebraic conditions on the totally symmetric contravariant coefficient tensor field P that determines the principal symbol of the matter field equations in terms of the geometric tensor G: the tensor field P must be hyperbolic, time-orientable and energy-distinguishing. Remarkably, these physically necessary conditions on the geometry are mathematically already sufficient to realize all kinematical constructions familiar from Lorentzian geometry, for precisely the same structural reasons. This we were able to show employing a subtle interplay of convex analysis, the theory of partial differential equations and real algebraic geometry. In the second part of this thesis, we then explore general properties of any hyperbolic, time-orientable and energy-distinguishing tensorial geometry. Physically most important are the construction of freely falling non-rotating laboratories, the appearance of admissible modified dispersion relations to particular observers, and the identification of a mechanism that explains why massive particles that are faster than some massless particles can radiate off energy until they are slower than all

Topics of differential geometry in hamiltonian and lagrangian mechanics and relativity

International Nuclear Information System (INIS)

Rodrigues, P.R.

1982-01-01

A little introduction to the tensor and exterior algebra just as to the differential geometry is made. Such a geometry is used in order to study the hamiltonian and lagrangian mechanics stressing their geometrical aspects. Some applications are done in relativity theory. (L.C.) [pt

An introduction to incidence geometry

CERN Document Server

De Bruyn, Bart

2016-01-01

This book gives an introduction to the field of Incidence Geometry by discussing the basic families of point-line geometries and introducing some of the mathematical techniques that are essential for their study. The families of geometries covered in this book include among others the generalized polygons, near polygons, polar spaces, dual polar spaces and designs. Also the various relationships between these geometries are investigated. Ovals and ovoids of projective spaces are studied and some applications to particular geometries will be given. A separate chapter introduces the necessary mathematical tools and techniques from graph theory. This chapter itself can be regarded as a self-contained introduction to strongly regular and distance-regular graphs. This book is essentially self-contained, only assuming the knowledge of basic notions from (linear) algebra and projective and affine geometry. Almost all theorems are accompanied with proofs and a list of exercises with full solutions is given at the end...

Tropical geometry of statistical models.

Science.gov (United States)

Pachter, Lior; Sturmfels, Bernd

2004-11-16

This article presents a unified mathematical framework for inference in graphical models, building on the observation that graphical models are algebraic varieties. From this geometric viewpoint, observations generated from a model are coordinates of a point in the variety, and the sum-product algorithm is an efficient tool for evaluating specific coordinates. Here, we address the question of how the solutions to various inference problems depend on the model parameters. The proposed answer is expressed in terms of tropical algebraic geometry. The Newton polytope of a statistical model plays a key role. Our results are applied to the hidden Markov model and the general Markov model on a binary tree.

Recoupling Lie algebra and universal ω-algebra

International Nuclear Information System (INIS)

Joyce, William P.

2004-01-01

We formulate the algebraic version of recoupling theory suitable for commutation quantization over any gradation. This gives a generalization of graded Lie algebra. Underlying this is the new notion of an ω-algebra defined in this paper. ω-algebra is a generalization of algebra that goes beyond nonassociativity. We construct the universal enveloping ω-algebra of recoupling Lie algebras and prove a generalized Poincare-Birkhoff-Witt theorem. As an example we consider the algebras over an arbitrary recoupling of Z n graded Heisenberg Lie algebra. Finally we uncover the usual coalgebra structure of a universal envelope and substantiate its Hopf structure

Perspectives in Analysis, Geometry, and Topology

CERN Document Server

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.

São Carlos Workshop on Real and Complex Singularities

CERN Document Server

Ruas, Maria

2007-01-01

The São Carlos Workshop on Real and Complex Singularities is the longest running workshop in singularities. It is held every two years and is a key international event for people working in the field. This volume contains papers presented at the eighth workshop, held at the IML, Marseille, July 19–23, 2004. The workshop offers the opportunity to establish the state of the art and to present new trends, new ideas and new results in all of the branches of singularities. This is reflected by the contributions in this book. The main topics discussed are equisingularity of sets and mappings, geometry of singular complex analytic sets, singularities of mappings, characteristic classes, classification of singularities, interaction of singularity theory with some of the new ideas in algebraic geometry imported from theoretical physics, and applications of singularity theory to geometry of surfaces in low dimensional euclidean spaces, to differential equations and to bifurcation theory.

Spin structures on algebraic curves and their applications in string theories

International Nuclear Information System (INIS)

Ferrari, F.

1990-01-01

The free fields on a Riemann surface carrying spin structures live on an unramified r-covering of the surface itself. When the surface is represented as an algebraic curve related to the vanishing of the Weierstrass polynomial, its r-coverings are algebraic curves as well. We construct explicitly the Weierstrass polynomial associated to the r-coverings of an algebraic curve. Using standard techniques of algebraic geometry it is then possible to solve the inverse Jacobi problem for the odd spin structures. As an application we derive the partition functions of bosonic string theories in many examples, including two general curves of genus three and four. The partition functions are explicitly expressed in terms of branch points apart from a factor which is essentially a theta constant. 53 refs., 4 figs. (Author)

Alfred Tarski early work in Poland—geometry and teaching

CERN Document Server

McFarland, Joanna; Smith, James

2014-01-01

Alfred Tarski (1901–1983) was a renowned Polish/American mathematician, a giant of the twentieth century, who helped establish the foundations of geometry, set theory, model theory, algebraic logic, and universal algebra. Throughout his career, he taught mathematics and logic at universities and sometimes in secondary schools. Many of his writings before 1939 were in Polish and remained inaccessible to most mathematicians and historians until now. This self-contained book focuses on Tarski’s early contributions to geometry and mathematics education, including the famous Banach–Tarski paradoxical decomposition of a sphere as well as high-school mathematical topics and pedagogy. These themes are significant since Tarski’s later research on geometry and its foundations stemmed in part from his early employment as a high-school mathematics teacher and teacher-trainer. The book contains careful translations and much newly uncovered social background of these works written during Tarski’s years in Poland....

Differential Hopf algebra structures on the universal enveloping algebra ofa Lie algebra

NARCIS (Netherlands)

N.W. van den Hijligenberg; R. Martini

1995-01-01

textabstractWe 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

Mathematics in the Real World.

Science.gov (United States)

Borenstein, Matt

1997-01-01

The abstract nature of algebra causes difficulties for many students. Describes "Real-World Data," an algebra course designed for students with low grades in algebra and provides multidisciplinary experiments (linear functions and variations; quadratic, square-root, and inverse relations; and exponential and periodic variation)…

The Yoneda algebra of a K2 algebra need not be another K2 algebra

OpenAIRE

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.

Teaching Quantitative Reasoning: A Better Context for Algebra

Directory of Open Access Journals (Sweden)

Eric Gaze

2014-01-01

Full Text Available This editorial questions the preeminence of algebra in our mathematics curriculum. The GATC (Geometry, Algebra, Trigonometry, Calculus sequence abandons the fundamental middle school math topics necessary for quantitative literacy, while the standard super-abundance of algebra taught in the abstract fosters math phobia and supports a culturally acceptable stance that math is not relevant to everyday life. Although GATC is seen as a pipeline to STEM (Science, Technology, Engineering, Mathematics, it is a mistake to think that the objective of producing quantitatively literate citizens is at odds with creating more scientists and engineers. The goal must be to create a curriculum that addresses the quantitative reasoning needs of all students, providing meaningful engagement in mathematics that will simultaneously develop quantitative literacy and spark an interest in STEM fields. In my view, such a curriculum could be based on a foundation of proportional reasoning leading to higher-order quantitative reasoning via modeling (including algebraic reasoning and problem solving and statistical literacy (through the exploration and study of data.

Algebraically special solutions in AdS/CFT

International Nuclear Information System (INIS)

Freitas, Gabriel Bernardi de; Reall, Harvey S.

2014-01-01

We investigate the AdS/CFT interpretation of the class of algebraically special solutions of Einstein gravity with a negative cosmological constant. Such solutions describe a CFT living in a 2+1 dimensional time-dependent geometry that, generically, has no isometries. The algebraically special condition implies that the expectation value of the CFT energy-momentum tensor is a local function of the boundary metric. When such a spacetime is slowly varying, the fluid/gravity approximation is valid and one can read off the values of certain higher order transport coefficients. To do this, we introduce a formalism for studying conformal, relativistic fluids in 2+1 dimensions that reduces everything to the manipulation of scalar quantities.

Algebraically special solutions in AdS/CFT

Energy Technology Data Exchange (ETDEWEB)

Freitas, Gabriel Bernardi de; Reall, Harvey S. [Department of Applied Mathematics and Theoretical Physics, University of Cambridge,Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WA (United Kingdom)

2014-06-26

We investigate the AdS/CFT interpretation of the class of algebraically special solutions of Einstein gravity with a negative cosmological constant. Such solutions describe a CFT living in a 2+1 dimensional time-dependent geometry that, generically, has no isometries. The algebraically special condition implies that the expectation value of the CFT energy-momentum tensor is a local function of the boundary metric. When such a spacetime is slowly varying, the fluid/gravity approximation is valid and one can read off the values of certain higher order transport coefficients. To do this, we introduce a formalism for studying conformal, relativistic fluids in 2+1 dimensions that reduces everything to the manipulation of scalar quantities.

Algebra Survival Guide A Conversational Handbook for the Thoroughly Befuddled

CERN Document Server

Rappaport, Josh

2011-01-01

If you think algebra has to be boring, confusing and unrelated to anything in the real world, think again! Written in a humorous, conversational style, this book gently nudges students toward success in pre-algebra and Algebra I. With its engaging question/answer format and helpful practice problems, glossary and index, it is ideal for homeschoolers, tutors and students striving for classroom excellence. It features funky icons and lively cartoons by award-winning Santa Fe artist Sally BlakemoreThe Algebra Survival Guide is the winner of a Paretns' Choice award, and it meets the Standards 2000

Nonlinear poisson brackets geometry and quantization

CERN Document Server

Karasev, M V

2012-01-01

This book deals with two old mathematical problems. The first is the problem of constructing an analog of a Lie group for general nonlinear Poisson brackets. The second is the quantization problem for such brackets in the semiclassical approximation (which is the problem of exact quantization for the simplest classes of brackets). These problems are progressively coming to the fore in the modern theory of differential equations and quantum theory, since the approach based on constructions of algebras and Lie groups seems, in a certain sense, to be exhausted. The authors' main goal is to describe in detail the new objects that appear in the solution of these problems. Many ideas of algebra, modern differential geometry, algebraic topology, and operator theory are synthesized here. The authors prove all statements in detail, thus making the book accessible to graduate students.

On Roots of Polynomials and Algebraically Closed Fields

Directory of Open Access Journals (Sweden)

Schwarzweller Christoph

2017-10-01

Full Text Available In this article we further extend the algebraic theory of polynomial rings in Mizar [1, 2, 3]. We deal with roots and multiple roots of polynomials and show that both the real numbers and finite domains are not algebraically closed [5, 7]. We also prove the identity theorem for polynomials and that the number of multiple roots is bounded by the polynomial’s degree [4, 6].

New WZW D-branes from the algebra of Wilson loop operators

International Nuclear Information System (INIS)

Monnier, Samuel

2009-01-01

We investigate the algebra generated by the topological Wilson loop operators in WZW models. Wilson loops describe the nontrivial fixed points of the boundary renormalization group flows triggered by Kondo perturbations. Their enveloping algebra therefore encodes all the fixed points which can be reached by sequences of Kondo flows. This algebra is easily described in the case of SU(2), but displays a very rich structure for higher rank groups. In the latter case, its action on known D-branes creates a profusion of new and generically non-rational D-branes. We describe their symmetries and the geometry of their worldvolumes. We briefly explain how to extend these results to coset models.

Differential Hopf algebra structures on the universal enveloping algebra of a Lie algebra

NARCIS (Netherlands)

van den Hijligenberg, N.W.; van den Hijligenberg, N.W.; Martini, Ruud

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

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.

Final Report: Geometry And Elementary Particle Physics

International Nuclear Information System (INIS)

Singer, Isadore M.

2008-01-01

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.

Thirty-three miniatures mathematical and algorithmic applications of linear algebra

CERN Document Server

Matousek, Jiří

2010-01-01

This volume contains a collection of clever mathematical applications of linear algebra, mainly in combinatorics, geometry, and algorithms. Each chapter covers a single main result with motivation and full proof in at most ten pages and can be read independently of all other chapters (with minor exceptions), assuming only a modest background in linear algebra. The topics include a number of well-known mathematical gems, such as Hamming codes, the matrix-tree theorem, the Lov�sz bound on the Shannon capacity, and a counterexample to Borsuk's conjecture, as well as other, perhaps less popular but similarly beautiful results, e.g., fast associativity testing, a lemma of Steinitz on ordering vectors, a monotonicity result for integer partitions, or a bound for set pairs via exterior products. The simpler results in the first part of the book provide ample material to liven up an undergraduate course of linear algebra. The more advanced parts can be used for a graduate course of linear-algebraic methods or for s...

Understanding geometric algebra Hamilton, Grassmann, and Clifford for computer vision and graphics

CERN Document Server

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

Introduction to relation algebras relation algebras

CERN Document Server

Givant, Steven

2017-01-01

The first volume of a pair that charts relation algebras from novice to expert level, this text offers a comprehensive grounding for readers new to the topic. Upon completing this introduction, mathematics students may delve into areas of active research by progressing to the second volume, Advanced Topics in Relation Algebras; computer scientists, philosophers, and beyond will be equipped to apply these tools in their own field. The careful presentation establishes first the arithmetic of relation algebras, providing ample motivation and examples, then proceeds primarily on the basis of algebraic constructions: subalgebras, homomorphisms, quotient algebras, and direct products. Each chapter ends with a historical section and a substantial number of exercises. The only formal prerequisite is a background in abstract algebra and some mathematical maturity, though the reader will also benefit from familiarity with Boolean algebra and naïve set theory. The measured pace and outstanding clarity are particularly ...

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...

Geometry-specified troposphere decorrelation for subcentimeter real-time kinematic solutions over long baselines

Science.gov (United States)

Li, Bofeng; Feng, Yanming; Shen, Yunzhong; Wang, Charles

2010-11-01

Real-time kinematic (RTK) GPS techniques have been extensively developed for applications including surveying, structural monitoring, and machine automation. Limitations of the existing RTK techniques that hinder their applications for geodynamics purposes are twofold: (1) the achievable RTK accuracy is on the level of a few centimeters and the uncertainty of vertical component is 1.5-2 times worse than those of horizontal components and (2) the RTK position uncertainty grows in proportional to the base-to-rover distances. The key limiting factor behind the problems is the significant effect of residual tropospheric errors on the positioning solutions, especially on the highly correlated height component. This paper develops the geometry-specified troposphere decorrelation strategy to achieve the subcentimeter kinematic positioning accuracy in all three components. The key is to set up a relative zenith tropospheric delay (RZTD) parameter to absorb the residual tropospheric effects and to solve the established model as an ill-posed problem using the regularization method. In order to compute a reasonable regularization parameter to obtain an optimal regularized solution, the covariance matrix of positional parameters estimated without the RZTD parameter, which is characterized by observation geometry, is used to replace the quadratic matrix of their "true" values. As a result, the regularization parameter is adaptively computed with variation of observation geometry. The experiment results show that new method can efficiently alleviate the model's ill condition and stabilize the solution from a single data epoch. Compared to the results from the conventional least squares method, the new method can improve the long-range RTK solution precision from several centimeters to the subcentimeter in all components. More significantly, the precision of the height component is even higher. Several geosciences applications that require subcentimeter real-time solutions can

Noncommutative spectral geometry, Bogoliubov transformations and neutrino oscillations

International Nuclear Information System (INIS)

Gargiulo, Maria Vittoria; Vitiello, Giuseppe; Sakellariadou, Mairi

2015-01-01

In this report we show that neutrino mixing is intrinsically contained in Connes’ noncommutatives pectral geometry construction, thanks to the introduction of the doubling of algebra, which is connected to the Bogoliubov transformation. It is known indeed that these transformations are responsible for the mixing, turning the mass vacuum state into the flavor vacuum state, in such a way that mass and flavor vacuum states are not unitary equivalent. There is thus a red thread that binds the doubling of algebra of Connes’ model to the neutrino mixing. (paper)

From Rota-Baxter algebras to pre-Lie algebras

International Nuclear Information System (INIS)

An Huihui; Ba, Chengming

2008-01-01

Rota-Baxter algebras were introduced to solve some analytic and combinatorial problems and have appeared in many fields in mathematics and mathematical physics. Rota-Baxter algebras provide a construction of pre-Lie algebras from associative algebras. In this paper, we give all Rota-Baxter operators of weight 1 on complex associative algebras in dimension ≤3 and their corresponding pre-Lie algebras

Algebra, Home Mortgages, and Recessions

Science.gov (United States)

Mariner, Jean A. Miller; Miller, Richard A.

2009-01-01

The current financial crisis and recession in the United States present an opportunity to discuss relevant applications of some topics in typical first-and second-year algebra and precalculus courses. Real-world applications of percent change, exponential functions, and sums of finite geometric sequences can help students understand the problems…

Quartic Poisson algebras and quartic associative algebras and realizations as deformed oscillator algebras

International Nuclear Information System (INIS)

Marquette, Ian

2013-01-01

We introduce the most general quartic Poisson algebra generated by a second and a fourth order integral of motion of a 2D superintegrable classical system. We obtain the corresponding quartic (associative) algebra for the quantum analog, extend Daskaloyannis construction obtained in context of quadratic algebras, and also obtain the realizations as deformed oscillator algebras for this quartic algebra. We obtain the Casimir operator and discuss how these realizations allow to obtain the finite-dimensional unitary irreducible representations of quartic algebras and obtain algebraically the degenerate energy spectrum of superintegrable systems. We apply the construction and the formula obtained for the structure function on a superintegrable system related to type I Laguerre exceptional orthogonal polynomials introduced recently

On Dimension Theory for a Certain Class of Simple AH Algebras

International Nuclear Information System (INIS)

Ho, Toan M.

2010-06-01

A class of unital diagonal AH algebras will be studied in this paper. The density property of the set of all elements which are nilpotent up to (left and right multiple) unitaries is presented. As a consequence, these algebras have stable rank one. Section 3 also shows that an algebra in this class has the property LP (i.e., the linear span of projections is dense) provided a certain condition. Finally, restricting our attention to a special subclass which includes Villadsen algebras of the first type, we give the necessary and sufficient condition of real rank zero. (author)

Subgroups of class groups of algebraic quadratic function fields

International Nuclear Information System (INIS)

Wang Kunpeng; Zhang Xianke

2001-09-01

Ideal class groups H(K) of algebraic quadratic function fields K are studied, by using mainly the theory of continued fractions of algebraic functions. Properties of such continued fractions are discussed first. Then a necessary and sufficient condition is given for the class group H(K) to contain a cyclic subgroup of any order n, this criterion condition holds true for both real and imaginary fields K. Furthermore, several series of function fields K, including real, inertia imaginary, as well as ramified imaginary quadratic function fields, are given, and their class groups H(K) are proved to contain cyclic subgroups of order n. (author)

The Schwinger Dyson equations and the algebra of constraints of random tensor models at all orders

International Nuclear Information System (INIS)

Gurau, Razvan

2012-01-01

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.

Algebrodynamics over complex space and phase extension of the Minkowski geometry

International Nuclear Information System (INIS)

Kassandrov, V. V.

2009-01-01

First principles should predetermine physical geometry and dynamics both together. In the 'algebrodynamics' they follow solely from the properties of biquaternion algebra B and the analysis over B. We briefly present the algebrodynamics over Minkowski background based on a nonlinear generalization to B of the Cauchi-Riemann analyticity conditions. Further, we consider the effective real geometry uniquely resulting from the structure of B multiplication and found it to be of the Minkowski type, with an additional phase invariant. Then we pass to study the primordial dynamics that takes place in the complex B space and brings into consideration a number of remarkable structures: an ensemble of identical correlated matter pre-elements ('duplicons'), caustic-like signals (interaction carriers), a concept of random complex time resulting in irreversibility of physical time at macrolevel, etc. In partucular, the concept of 'dimerous electron' naturally arises in the framework of complex algebrodynamics and, together with the above-mentioned phase invariant, allows for a novel approach to explanation of quantum interference phenomena alternative to recently accepted wave-particle dualism paradigm.

Yoneda algebras of almost Koszul algebras

Indian Academy of Sciences (India)

Abstract. Let k be an algebraically closed field, A a finite dimensional connected. (p,q)-Koszul self-injective algebra with p, q ≥ 2. In this paper, we prove that the. Yoneda algebra of A is isomorphic to a twisted polynomial algebra A![t; β] in one inde- terminate t of degree q +1 in which A! is the quadratic dual of A, β is an ...

Quantum cluster algebras and quantum nilpotent algebras

Science.gov (United States)

Goodearl, Kenneth R.; Yakimov, Milen T.

2014-01-01

A major direction in the theory of cluster algebras is to construct (quantum) cluster algebra structures on the (quantized) coordinate rings of various families of varieties arising in Lie theory. We prove that all algebras in a very large axiomatically defined class of noncommutative algebras possess canonical quantum cluster algebra structures. Furthermore, they coincide with the corresponding upper quantum cluster algebras. We also establish analogs of these results for a large class of Poisson nilpotent algebras. Many important families of coordinate rings are subsumed in the class we are covering, which leads to a broad range of applications of the general results to the above-mentioned types of problems. As a consequence, we prove the Berenstein–Zelevinsky conjecture [Berenstein A, Zelevinsky A (2005) Adv Math 195:405–455] for the quantized coordinate rings of double Bruhat cells and construct quantum cluster algebra structures on all quantum unipotent groups, extending the theorem of Geiß et al. [Geiß C, et al. (2013) Selecta Math 19:337–397] for the case of symmetric Kac–Moody groups. Moreover, we prove that the upper cluster algebras of Berenstein et al. [Berenstein A, et al. (2005) Duke Math J 126:1–52] associated with double Bruhat cells coincide with the corresponding cluster algebras. PMID:24982197

Geometric control theory and sub-Riemannian geometry

CERN Document Server

Boscain, Ugo; Gauthier, Jean-Paul; Sarychev, Andrey; Sigalotti, Mario

2014-01-01

This volume presents recent advances in the interaction between Geometric Control Theory and sub-Riemannian geometry. On the one hand, Geometric Control Theory used the differential geometric and Lie algebraic language for studying controllability, motion planning, stabilizability and optimality for control systems. The geometric approach turned out to be fruitful in applications to robotics, vision modeling, mathematical physics etc. On the other hand, Riemannian geometry and its generalizations, such as  sub-Riemannian, Finslerian  geometry etc., have been actively adopting methods developed in the scope of geometric control. Application of these methods  has led to important results regarding geometry of sub-Riemannian spaces, regularity of sub-Riemannian distances, properties of the group  of diffeomorphisms of sub-Riemannian manifolds, local geometry and equivalence of distributions and sub-Riemannian structures, regularity of the Hausdorff volume.

A Workshop on Algebraic Design Theory and Hadamard Matrices

CERN Document Server

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...

Analysis and Geometry : MIMS-GGTM, in Honour of Mohammed Salah Baouendi

CERN Document Server

Kacimi, Aziz; Kallel, Sadok; Mir, Nordine

2015-01-01

This book includes selected papers presented at the MIMS (Mediterranean Institute for the Mathematical Sciences) - GGTM (Geometry and Topology Grouping for the Maghreb) conference, held in memory of Mohammed Salah Baouendi, a most renowned figure in the field of several complex variables, who passed away in 2011. All research articles were written by leading experts, some of whom are prize winners in the fields of complex geometry, algebraic geometry and analysis. The book offers a valuable resource for all researchers interested in recent developments in analysis and geometry.

Introduction to differential geometry for engineers

CERN Document Server

Doolin, Brian F

2013-01-01

This outstanding guide supplies important mathematical tools for diverse engineering applications, offering engineers the basic concepts and terminology of modern global differential geometry. Suitable for independent study as well as a supplementary text for advanced undergraduate and graduate courses, this volume also constitutes a valuable reference for control, systems, aeronautical, electrical, and mechanical engineers.The treatment's ideas are applied mainly as an introduction to the Lie theory of differential equations and to examine the role of Grassmannians in control systems analysis. Additional topics include the fundamental notions of manifolds, tangent spaces, vector fields, exterior algebra, and Lie algebras. An appendix reviews concepts related to vector calculus, including open and closed sets, compactness, continuity, and derivative.

Inequalities, Assessment and Computer Algebra

Science.gov (United States)

Sangwin, Christopher J.

2015-01-01

The goal of this paper is to examine single variable real inequalities that arise as tutorial problems and to examine the extent to which current computer algebra systems (CAS) can (1) automatically solve such problems and (2) determine whether students' own answers to such problems are correct. We review how inequalities arise in contemporary…

Differential Hopf algebra structures on the Universal Enveloping Algebra of a Lie Algebra

NARCIS (Netherlands)

van den Hijligenberg, N.W.; van den Hijligenberg, N.; Martini, Ruud

1995-01-01

We discuss a method to construct a De Rham complex (differential algebra) of Poincaré–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 algebrastructure of U(g).

Women in numbers Europe II contributions to number theory and arithmetic geometry

CERN Document Server

Ozman, Ekin; Johnson-Leung, Jennifer; Newton, Rachel

2018-01-01

Inspired by the September 2016 conference of the same name, this second volume highlights recent research in a wide range of topics in contemporary number theory and arithmetic geometry. Research reports from projects started at the conference, expository papers describing ongoing research, and contributed papers from women number theorists outside the conference make up this diverse volume. Topics cover a broad range of topics such as arithmetic dynamics, failure of local-global principles, geometry in positive characteristics, and heights of algebraic integers. The use of tools from algebra, analysis and geometry, as well as computational methods exemplifies the wealth of techniques available to modern researchers in number theory. Exploring connections between different branches of mathematics and combining different points of view, these papers continue the tradition of supporting and highlighting the contributions of women number theorists at a variety of career stages. Perfect for students and researche...

The relation between quantum W algebras and Lie algebras

International Nuclear Information System (INIS)

Boer, J. de; Tjin, T.

1994-01-01

By quantizing the generalized Drinfeld-Sokolov reduction scheme for arbitrary sl 2 embeddings we show that a large set W of quantum W algebras can be viewed as (BRST) cohomologies of affine Lie algebras. The set W contains many known W algebras such as W N and W 3 (2) . Our formalism yields a completely algorithmic method for calculating the W algebra generators and their operator product expansions, replacing the cumbersome construction of W algebras as commutants of screening operators. By generalizing and quantizing the Miura transformation we show that any W algebra in W can be embedded into the universal enveloping algebra of a semisimple affine Lie algebra which is, up to shifts in level, isomorphic to a subalgebra of the original affine algebra. Therefore any realization of this semisimple affine Lie algebra leads to a realization of the W algebra. In particular, one obtains in this way a general and explicit method for constructing the free field realizations and Fock resolutions for all algebras in W. Some examples are explicitly worked out. (orig.)

2-Local derivations on matrix algebras over semi-prime Banach algebras and on AW*-algebras

International Nuclear Information System (INIS)

Ayupov, Shavkat; Kudaybergenov, Karimbergen

2016-01-01

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 2 n (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. (paper)

Quantum cluster algebra structures on quantum nilpotent algebras

CERN Document Server

Goodearl, K R

2017-01-01

All algebras in a very large, axiomatically defined class of quantum nilpotent algebras are proved to possess quantum cluster algebra structures under mild conditions. Furthermore, it is shown that these quantum cluster algebras always equal the corresponding upper quantum cluster algebras. Previous approaches to these problems for the construction of (quantum) cluster algebra structures on (quantized) coordinate rings arising in Lie theory were done on a case by case basis relying on the combinatorics of each concrete family. The results of the paper have a broad range of applications to these problems, including the construction of quantum cluster algebra structures on quantum unipotent groups and quantum double Bruhat cells (the Berenstein-Zelevinsky conjecture), and treat these problems from a unified perspective. All such applications also establish equality between the constructed quantum cluster algebras and their upper counterparts.

Developments in special geometry

International Nuclear Information System (INIS)

Mohaupt, Thomas; Vaughan, Owen

2012-01-01

We review the special geometry of N = 2 supersymmetric vector and hypermultiplets with emphasis on recent developments and applications. A new formulation of the local c-map based on the Hesse potential and special real coordinates is presented. Other recent developments include the Euclidean version of special geometry, and generalizations of special geometry to non-supersymmetric theories. As applications we discuss the proof that the local r-map and c-map preserve geodesic completeness, and the construction of four- and five-dimensional static solutions through dimensional reduction over time. The shared features of the real, complex and quaternionic version of special geometry are stressed throughout.

Geometry of Yang-Mills fields

International Nuclear Information System (INIS)

Atiyah, M.F.

1978-01-01

In this talk I shall explain how information about classical solutions of Yang-Mills equations can be obtained, rather surprisingly, from algebraic geometry. Although direct physical interest is restricted to the case of four dimensions I shall begin by discussing the two-dimensional case. Besides preparing the ground for the four-dimensional problem this has independent mathematical (and possible physical) interest, and very complete results can be obtained. (orig.) [de

Homological mirror symmetry and tropical geometry

CERN Document Server

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...

The Standard Model in noncommutative geometry: fundamental fermions as internal forms

Science.gov (United States)

Dąbrowski, Ludwik; D'Andrea, Francesco; Sitarz, Andrzej

2018-05-01

Given the algebra, Hilbert space H, grading and real structure of the finite spectral triple of the Standard Model, we classify all possible Dirac operators such that H is a self-Morita equivalence bimodule for the associated Clifford algebra.

Differential and complex geometry origins, abstractions and embeddings

CERN Document Server

Wells, Jr , Raymond O

2017-01-01

Differential and complex geometry are two central areas of mathematics with a long and intertwined history. This book, the first to provide a unified historical perspective of both subjects, explores their origins and developments from the sixteenth to the twentieth century. Providing a detailed examination of the seminal contributions to differential and complex geometry up to the twentieth century embedding theorems, this monograph includes valuable excerpts from the original documents, including works of Descartes, Fermat, Newton, Euler, Huygens, Gauss, Riemann, Abel, and Nash. Suitable for beginning graduate students interested in differential, algebraic or complex geometry, this book will also appeal to more experienced readers.

Regularity of C*-algebras and central sequence algebras

DEFF Research Database (Denmark)

Christensen, Martin S.

The main topic of this thesis is regularity properties of C*-algebras and how these regularity properties are re ected in their associated central sequence algebras. The thesis consists of an introduction followed by four papers [A], [B], [C], [D]. In [A], we show that for the class of simple...... Villadsen algebra of either the rst type with seed space a nite dimensional CW complex, or the second type, tensorial absorption of the Jiang-Su algebra is characterized by the absence of characters on the central sequence algebra. Additionally, in a joint appendix with Joan Bosa, we show that the Villadsen...... algebra of the second type with innite stable rank fails the corona factorization property. In [B], we consider the class of separable C*-algebras which do not admit characters on their central sequence algebra, and show that it has nice permanence properties. We also introduce a new divisibility property...

Asymptotic identity in min-plus algebra: a report on CPNS.

Science.gov (United States)

Li, Ming; Zhao, Wei

2012-01-01

Network calculus is a theory initiated primarily in computer communication networks, especially in the aspect of real-time communications, where min-plus algebra plays a role. Cyber-physical networking systems (CPNSs) are recently developing fast and models in data flows as well as systems in CPNS are, accordingly, greatly desired. Though min-plus algebra may be a promising tool to linearize any node in CPNS as can be seen from its applications to the Internet computing, there are tough problems remaining unsolved in this regard. The identity in min-plus algebra is one problem we shall address. We shall point out the confusions about the conventional identity in the min-plus algebra and present an analytical expression of the asymptotic identity that may not cause confusions.

On the C*-algebras of foliations in the plane

CERN Document Server

Wang, Xiaolu

1987-01-01

The main result of this original research monograph is the classification of C*-algebras of ordinary foliations of the plane in terms of a class of -trees. It reveals a close connection between some most recent developments in modern analysis and low-dimensional topology. It introduces noncommutative CW-complexes (as the global fibred products of C*-algebras), among other things, which adds a new aspect to the fast-growing field of noncommutative topology and geometry. The reader is only required to know basic functional analysis. However, some knowledge of topology and dynamical systems will be helpful. The book addresses graduate students and experts in the area of analysis, dynamical systems and topology.

Hom-Novikov algebras

International Nuclear Information System (INIS)

Yau, Donald

2011-01-01

We study a twisted generalization of Novikov algebras, called Hom-Novikov algebras, in which the two defining identities are twisted by a linear map. It is shown that Hom-Novikov algebras can be obtained from Novikov algebras by twisting along any algebra endomorphism. All algebra endomorphisms on complex Novikov algebras of dimensions 2 or 3 are computed, and their associated Hom-Novikov algebras are described explicitly. Another class of Hom-Novikov algebras is constructed from Hom-commutative algebras together with a derivation, generalizing a construction due to Dorfman and Gel'fand. Two other classes of Hom-Novikov algebras are constructed from Hom-Lie algebras together with a suitable linear endomorphism, generalizing a construction due to Bai and Meng.

Matematica Para La Escuela Secundaria, Primer Curso de Algebra (Parte 2). Traduccion Preliminar de la Edicion Inglesa Revisada. (Mathematics for High School, First Course in Algebra, Part 2. Preliminary Translation of the Revised English Edition).

Science.gov (United States)

Allen, Frank B.; And Others

This is part two of a three-part SMSG algebra text for high school students. The principal objective of the text is to help the student develop an understanding and appreciation of some of the algebraic structure as a basis for the techniques of algebra. Chapter topics include addition and multiplication of real numbers, subtraction and division…

A note on the Akivis algebra of a smooth hyporeductive loop

International Nuclear Information System (INIS)

Issa, A.N.

2002-05-01

Using the fundamental tensors of a smooth loop and the differential geometric characterization of smooth hyporeductive loops, the Akivis operations of a local smooth hyporeductive loop are expressed through the two binary and the one ternary operations of the hyporeductive triple algebra (h.t.a.) associated with the given hyporeductive loop. Those Akivis operations are also given in terms of Lie brackets of a Lie algebra of vector fields with the hyporeductive decomposition which generalizes the reductive decomposition of Lie algebras. A nontrivial real two-dimensional h.t.a. is presented. (author)

College Algebra in Context: A Project Incorporating Social Issues

Directory of Open Access Journals (Sweden)

Michael T. Catalano

2010-01-01

Full Text Available This paper discusses the development of an innovative college algebra text designed for use in a data-driven, activity-oriented college algebra course, incorporating realistic problem situations emphasizing social and economic issues, including hunger and poverty, energy, and the environment. The course incorporates quantitative literacy themes, is informed by existing college algebra texts within the college algebra reform movement, and implements a collaborative pedagogical approach intended to provide future K-12 teachers an alternative model for the teaching of mathematics. The paper contains a short history of the project development phase, supported by an NSF grant (DUE #0442979, as well as the perceived role of the project in the college algebra reform and quantitative literacy movements. We make a short case for redefining the content of a college algebra course and acknowledging that for many students, it has become a terminal mathematics course. A description of the contents of the text, its relation to more traditional college algebra content, and four example student activities are included (on the topics of homelessness, the effects of airline deregulation, real estate versus savings as investment instruments, and the 2008 election. A summary of evaluation and assessment data from five years of pilot-testing, done primarily in conjuction with our NSF grant evaluation plan, is provided.

Algebra task sheets : grades pk-2

CERN Document Server

Reed, Nat

2009-01-01

For grades PK-2, our Common Core State Standards-based resource meets the algebraic concepts addressed by the NCTM standards and encourages the students to learn and review the concepts in unique ways. Each task sheet is organized around a central problem taken from real-life experiences of the students.

Pure Jauch-Piron states on von Neumann algebras

International Nuclear Information System (INIS)

Hamhalter, J.

1993-01-01

We study Jauch-Piron states and two-valued measures on von Neumann algebra. We prove as the main result that, under some set-theoretical assumption, a pure state of a von Neumann algebra A not containing a central abelian portion is Jauch-Piron if and only if it is σ-additive. Moreover, we show that this result holds for type I factor indenpendently on the set-theoretical axiomatics. As a consequence we obtain a lucid characterization of pure Jauch-Piron states on von Neumann algebras acting on a Hilbert space with real-nonmeasurable dimension (this can be viewed as a generalization of the paper). We also characterize the von Neumann algebras whose logic of projections is Jauch-Piron. Finally, we prove that every two-valued measure on the projection logic of A, where A contains no type I 2 central portion, has to be concentrated at an abelian direct summand of A. (orig.)

On tea, donuts and non-commutative geometry

Directory of Open Access Journals (Sweden)

Igor V. Nikolaev

2018-03-01

Full Text Available As many will agree, it feels good to complement a cup of tea by a donut or two. This sweet relationship is also a guiding principle of non-commutative geometry known as Serre Theorem. We explain the algebra behind this theorem and prove that elliptic curves are complementary to the so-called non-commutative tori.

Algebra a teaching and source book

CERN Document Server

Shult, Ernest

2015-01-01

This book presents a graduate-level course on modern algebra. It can be used as a teaching book – owing to the copious exercises – and as a source book for those who wish to use the major theorems of algebra. The course begins with the basic combinatorial principles of algebra: posets, chain conditions, Galois connections, and dependence theories. Here, the general Jordan–Holder Theorem becomes a theorem on interval measures of certain lower semilattices. This is followed by basic courses on groups, rings and modules; the arithmetic of integral domains; fields; the categorical point of view; and tensor products. Beginning with introductory concepts and examples, each chapter proceeds gradually towards its more complex theorems. Proofs progress step-by-step from first principles. Many interesting results reside in the exercises, for example, the proof that ideals in a Dedekind domain are generated by at most two elements. The emphasis throughout is on real understanding as opposed to memorizing a catech...

Topics in noncommutative algebra the theorem of Campbell, Baker, Hausdorff and Dynkin

CERN Document Server

Bonfiglioli, Andrea

2012-01-01

Motivated by the importance of the Campbell, Baker, Hausdorff, Dynkin Theorem in many different branches of Mathematics and Physics (Lie group-Lie algebra theory, linear PDEs, Quantum and Statistical Mechanics, Numerical Analysis, Theoretical Physics, Control Theory, sub-Riemannian Geometry), this monograph is intended to: 1) fully enable readers (graduates or specialists, mathematicians, physicists or applied scientists, acquainted with Algebra or not) to understand and apply the statements and numerous corollaries of the main result; 2) provide a wide spectrum of proofs from the modern literature, comparing different techniques and furnishing a unifying point of view and notation; 3) provide a thorough historical background of the results, together with unknown facts about the effective early contributions by Schur, Poincaré, Pascal, Campbell, Baker, Hausdorff and Dynkin; 4) give an outlook on the applications, especially in Differential Geometry (Lie group theory) and Analysis (PDEs of subelliptic type); ...

The geometry of lie algebras and broken SO(6) symmetries

International Nuclear Information System (INIS)

Lawrence, T.R.

2001-10-01

Non-linear realisations of the groups SU(2), SO(1,4) and SO(2,4) are analysed, described by the coset spaces SU(2)/U(1), SO(1,4)/SO(1,3) and SO(2,4)/SO(1,3) x SO(1,1). The Lie algebras of certain special unitary and special orthogonal groups are studied and their projection operators are determined in order to facilitate the above analyses, in particular that of SO(2,4)/SO(l,3) x SO(1,1). The analysis consists of determining the transformation properties of the Goldstone bosons, constructing the most general possible Lagrangian for the realisations and finding the metric of the coset space. (author)

Geometry of the Universe

International Nuclear Information System (INIS)

Gurevich, L.Eh.; Gliner, Eh.B.

1978-01-01

Problems of investigating the Universe space-time geometry are described on a popular level. Immediate space-time geometries, corresponding to three cosmologic models are considered. Space-time geometry of a closed model is the spherical Riemann geonetry, of an open model - is the Lobachevskij geometry; and of a plane model - is the Euclidean geometry. The Universe real geometry in the contemporary epoch of development is based on the data testifying to the fact that the Universe is infinitely expanding

Multivariate calculus and geometry

CERN Document Server

Dineen, Seán

2014-01-01

Multivariate calculus can be understood best by combining geometric insight, intuitive arguments, detailed explanations and mathematical reasoning. This textbook has successfully followed this programme. It additionally provides a solid description of the basic concepts, via familiar examples, which are then tested in technically demanding situations. In this new edition the introductory chapter and two of the chapters on the geometry of surfaces have been revised. Some exercises have been replaced and others provided with expanded solutions. Familiarity with partial derivatives and a course in linear algebra are essential prerequisites for readers of this book. Multivariate Calculus and Geometry is aimed primarily at higher level undergraduates in the mathematical sciences. The inclusion of many practical examples involving problems of several variables will appeal to mathematics, science and engineering students.

Extended Virasoro algebra and algebra of area preserving diffeomorphisms

International Nuclear Information System (INIS)

Arakelyan, T.A.

1990-01-01

The algebra of area preserving diffeomorphism plays an important role in the theory of relativistic membranes. It is pointed out that the relation between this algebra and the extended Virasoro algebra associated with the generalized Kac-Moody algebras G(T 2 ). The highest weight representation of these infinite-dimensional algebras as well as of their subalgebras is studied. 5 refs

Leibniz algebroids, twistings and exceptional generalized geometry

Science.gov (United States)

Baraglia, D.

2012-05-01

We investigate a class of Leibniz algebroids which are invariant under diffeomorphisms and symmetries involving collections of closed forms. Under appropriate assumptions we arrive at a classification which in particular gives a construction starting from graded Lie algebras. In this case the Leibniz bracket is a derived bracket and there are higher derived brackets resulting in an L∞-structure. The algebroids can be twisted by a non-abelian cohomology class and we prove that the twisting class is described by a Maurer-Cartan equation. For compact manifolds we construct a Kuranishi moduli space of this equation which is shown to be affine algebraic. We explain how these results are related to exceptional generalized geometry.

Asymptotic Identity in Min-Plus Algebra: A Report on CPNS

Science.gov (United States)

Li, Ming; Zhao, Wei

2012-01-01

Network calculus is a theory initiated primarily in computer communication networks, especially in the aspect of real-time communications, where min-plus algebra plays a role. Cyber-physical networking systems (CPNSs) are recently developing fast and models in data flows as well as systems in CPNS are, accordingly, greatly desired. Though min-plus algebra may be a promising tool to linearize any node in CPNS as can be seen from its applications to the Internet computing, there are tough problems remaining unsolved in this regard. The identity in min-plus algebra is one problem we shall address. We shall point out the confusions about the conventional identity in the min-plus algebra and present an analytical expression of the asymptotic identity that may not cause confusions. PMID:21822446

An application of the division algebras, Jordan algebras and split composition algebras

International Nuclear Information System (INIS)

Foot, R.; Joshi, G.C.

1992-01-01

It has been established that the covering group of the Lorentz group in D = 3, 4, 6, 10 can be expressed in a unified way, based on the four composition division algebras R, C, Q and O. In this paper, the authors discuss, in this framework, the role of the complex numbers of quantum mechanics. A unified treatment of quantum-mechanical spinors is given. The authors provide an explicit demonstration that the vector and spinor transformations recently constructed from a subgroup of the reduced structure group of the Jordan algebras M n 3 are indeed the Lorentz transformations. The authors also show that if the division algebras in the construction of the covering groups of the Lorentz groups in D = 3, 4, 6, 10 are replaced by the split composition algebras, then the sequence of groups SO(2, 2), SO(3, 3) and SO(5, 5) result. The analysis is presumed to be self-contained as the relevant aspects of the division algebras and Jordan algebras are reviewed. Some applications to physical theory are indicated

Interactions Between Representation Ttheory, Algebraic Topology and Commutative Algebra

CERN Document Server

Pitsch, Wolfgang; Zarzuela, Santiago

2016-01-01

This book includes 33 expanded abstracts of selected talks given at the two workshops "Homological Bonds Between Commutative Algebra and Representation Theory" and "Brave New Algebra: Opening Perspectives," and the conference "Opening Perspectives in Algebra, Representations, and Topology," held at the Centre de Recerca Matemàtica (CRM) in Barcelona between January and June 2015. These activities were part of the one-semester intensive research program "Interactions Between Representation Theory, Algebraic Topology and Commutative Algebra (IRTATCA)." Most of the abstracts present preliminary versions of not-yet published results and cover a large number of topics (including commutative and non commutative algebra, algebraic topology, singularity theory, triangulated categories, representation theory) overlapping with homological methods. This comprehensive book is a valuable resource for the community of researchers interested in homological algebra in a broad sense, and those curious to learn the latest dev...

Geometric Algebra Computing

CERN Document Server

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

Monomial algebras

CERN Document Server

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.

Iwahori-Hecke algebras and Schur algebras of the symmetric group

CERN Document Server

Mathas, Andrew

1999-01-01

This volume presents a fully self-contained introduction to the modular representation theory of the Iwahori-Hecke algebras of the symmetric groups and of the q-Schur algebras. The study of these algebras was pioneered by Dipper and James in a series of landmark papers. The primary goal of the book is to classify the blocks and the simple modules of both algebras. The final chapter contains a survey of recent advances and open problems. The main results are proved by showing that the Iwahori-Hecke algebras and q-Schur algebras are cellular algebras (in the sense of Graham and Lehrer). This is proved by exhibiting natural bases of both algebras which are indexed by pairs of standard and semistandard tableaux respectively. Using the machinery of cellular algebras, which is developed in Chapter 2, this results in a clean and elegant classification of the irreducible representations of both algebras. The block theory is approached by first proving an analogue of the Jantzen sum formula for the q-Schur algebras. T...

Cubic and quartic planar differential systems with exact algebraic limit cycles

Directory of Open Access Journals (Sweden)

Ahmed Bendjeddou

2011-01-01

Full Text Available We construct cubic and quartic polynomial planar differential systems with exact limit cycles that are ovals of algebraic real curves of degree four. The result obtained for the cubic case generalizes a proposition of [9]. For the quartic case, we deduce for the first time a class of systems with four algebraic limit cycles and another for which nested configurations of limit cycles occur.

Jordan algebras versus C*- algebras

International Nuclear Information System (INIS)

Stormer, E.

1976-01-01

The axiomatic formulation of quantum mechanics and the problem of whether the observables form self-adjoint operators on a Hilbert space, are discussed. The relation between C*- algebras and Jordan algebras is studied using spectral theory. (P.D.)

Geometric model of topological insulators from the Maxwell algebra

Science.gov (United States)

Palumbo, Giandomenico

2017-11-01

We propose a novel geometric model of time-reversal-invariant topological insulators in three dimensions in presence of an external electromagnetic field. Their gapped boundary supports relativistic quantum Hall states and is described by a Chern-Simons theory, where the gauge connection takes values in the Maxwell algebra. This represents a non-central extension of the Poincaré algebra and takes into account both the Lorentz and magnetic-translation symmetries of the surface states. In this way, we derive a relativistic version of the Wen-Zee term and we show that the non-minimal coupling between the background geometry and the electromagnetic field in the model is in agreement with the main properties of the relativistic quantum Hall states in the flat space.

Implicative Algebras

African Journals Online (AJOL)

Tadesse

In this paper we introduce the concept of implicative algebras which is an equivalent definition of lattice implication algebra of Xu (1993) and further we prove that it is a regular Autometrized. Algebra. Further we remark that the binary operation → on lattice implicative algebra can never be associative. Key words: Implicative ...

Gleason-kahane-Żelazko theorem for spectrally bounded algebra

Directory of Open Access Journals (Sweden)

S. H. Kulkarni

2005-01-01

Full Text Available We prove by elementary methods the following generalization of a theorem due to Gleason, Kahane, and Żelazko. Let A be a real algebra with unit 1 such that the spectrum of every element in A is bounded and let φ:A→ℂ be a linear map such that φ(1=1 and (φ(a2+(φ(b2≠0 for all a, b in A satisfying ab=ba and a2+b2 is invertible. Then φ(ab=φ(aφ(b for all a, b in A. Similar results are proved for real and complex algebras using Ransford's concept of generalized spectrum. With these ideas, a sufficient condition for a linear transformation to be multiplicative is established in terms of generalized spectrum.

C*-algebras of holonomy-diffeomorphisms and quantum gravity: I

International Nuclear Information System (INIS)

Aastrup, Johannes; Grimstrup, Jesper Møller

2013-01-01

A new approach to a unified theory of quantum gravity based on noncommutative geometry and canonical quantum gravity is presented. The approach is built around a *-algebra generated by local holonomy-diffeomorphisms on a 3-manifold and a quantized Dirac-type operator, the two capturing the kinematics of quantum gravity formulated in terms of Ashtekar variables. We prove that the separable part of the spectrum of the algebra is contained in the space of measurable connections modulo gauge transformations and we give limitations to the non-separable part. The construction of the Dirac-type operator—and thus the application of noncommutative geometry—is motivated by the requirement of diffeomorphism invariance. We conjecture that a semi-finite spectral triple, which is invariant under volume-preserving diffeomorphisms, arises from a GNS construction of a semi-classical state. Key elements of quantum field theory emerge from the construction in a semi-classical limit, as does an almost commutative algebra. Finally, we note that the spectrum of loop quantum gravity emerges from a discretization of our construction. Certain convergence issues are left unresolved. This paper is the first of two where the second paper [1] is concerned with mathematical details and proofs concerning the spectrum of the holonomy-diffeomorphism algebra. (paper)

Number theory III Diophantine geometry

CERN Document Server

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 ...

Real-time surgery simulation of intracranial aneurysm clipping with patient-specific geometries and haptic feedback

Science.gov (United States)

Fenz, Wolfgang; Dirnberger, Johannes

2015-03-01

Providing suitable training for aspiring neurosurgeons is becoming more and more problematic. The increasing popularity of the endovascular treatment of intracranial aneurysms leads to a lack of simple surgical situations for clipping operations, leaving mainly the complex cases, which present even experienced surgeons with a challenge. To alleviate this situation, we have developed a training simulator with haptic interaction allowing trainees to practice virtual clipping surgeries on real patient-specific vessel geometries. By using specialized finite element (FEM) algorithms (fast finite element method, matrix condensation) combined with GPU acceleration, we can achieve the necessary frame rate for smooth real-time interaction with the detailed models needed for a realistic simulation of the vessel wall deformation caused by the clamping with surgical clips. Vessel wall geometries for typical training scenarios were obtained from 3D-reconstructed medical image data, while for the instruments (clipping forceps, various types of clips, suction tubes) we use models provided by manufacturer Aesculap AG. Collisions between vessel and instruments have to be continuously detected and transformed into corresponding boundary conditions and feedback forces, calculated using a contact plane method. After a training, the achieved result can be assessed based on various criteria, including a simulation of the residual blood flow into the aneurysm. Rigid models of the surgical access and surrounding brain tissue, plus coupling a real forceps to the haptic input device further increase the realism of the simulation.

Separable algebras

CERN Document Server

Ford, Timothy J

2017-01-01

This book presents a comprehensive introduction to the theory of separable algebras over commutative rings. After a thorough introduction to the general theory, the fundamental roles played by separable algebras are explored. For example, Azumaya algebras, the henselization of local rings, and Galois theory are rigorously introduced and treated. Interwoven throughout these applications is the important notion of étale algebras. Essential connections are drawn between the theory of separable algebras and Morita theory, the theory of faithfully flat descent, cohomology, derivations, differentials, reflexive lattices, maximal orders, and class groups. The text is accessible to graduate students who have finished a first course in algebra, and it includes necessary foundational material, useful exercises, and many nontrivial examples.

Pocket book of integrals and mathematical formulas

CERN Document Server

Tallarida, Ronald J

1999-01-01

Elementary Algebra and GeometryFundamental Properties (Real Numbers)ExponentsFractional ExponentsIrrational ExponentsLogarithmsFactorialsBinomial TheoremFactors and ExpansionProgressionsComplex NumbersPolar FormPermutationsCombinationsAlgebraic EquationsGeometryDeterminants, Matrices, and Systems of EquationsDeterminantsEvaluation by CofactorsProperties of DeterminantsMatricesOperationsPropertiesTransposeIdentity MatrixAdjointInverse MatrixSystems of Linear EquationsMatrix Solution

Generalized EMV-Effect Algebras

Science.gov (United States)

Borzooei, R. A.; Dvurečenskij, A.; Sharafi, A. H.

2018-04-01

Recently in Dvurečenskij and Zahiri (2017), new algebraic structures, called EMV-algebras which generalize both MV-algebras and generalized Boolean algebras, were introduced. We present equivalent conditions for EMV-algebras. In addition, we define a partial algebraic structure, called a generalized EMV-effect algebra, which is close to generalized MV-effect algebras. Finally, we show that every generalized EMV-effect algebra is either an MV-effect algebra or can be embedded into an MV-effect algebra as a maximal ideal.

Topological vertex, string amplitudes and spectral functions of hyperbolic geometry

Energy Technology Data Exchange (ETDEWEB)

Guimaraes, M.E.X.; Rosa, T.O. [Universidade Federal Fluminense, Instituto de Fisica, Av. Gal. Milton Tavares de Souza, s/n, CEP 24210-346, Niteroi, RJ (Brazil); Luna, R.M. [Universidade Estadual de Londrina, Departamento de Fisica, Caixa Postal 6001, Londrina, Parana (Brazil)

2014-05-15

We discuss the homological aspects of the connection between quantum string generating function and the formal power series associated to the dimensions of chains and homologies of suitable Lie algebras. Our analysis can be considered as a new straightforward application of the machinery of modular forms and spectral functions (with values in the congruence subgroup of SL(2,Z)) to the partition functions of Lagrangian branes, refined vertex and open string partition functions, represented by means of formal power series that encode Lie algebra properties. The common feature in our examples lies in the modular properties of the characters of certain representations of the pertinent affine Lie algebras and in the role of Selberg-type spectral functions of a hyperbolic three-geometry associated with q-series in the computation of the string amplitudes. (orig.)

Special set linear algebra and special set fuzzy linear algebra

OpenAIRE

Kandasamy, W. B. Vasantha; Smarandache, Florentin; Ilanthenral, K.

2009-01-01

The authors in this book introduce the notion of special set linear algebra and special set fuzzy Linear algebra, which is an extension of the notion set linear algebra and set fuzzy linear algebra. These concepts are best suited in the application of multi expert models and cryptology. This book has five chapters. In chapter one the basic concepts about set linear algebra is given in order to make this book a self contained one. The notion of special set linear algebra and their fuzzy analog...

Constant curvature algebras and higher spin action generating functions

International Nuclear Information System (INIS)

Hallowell, K.; Waldron, A.

2005-01-01

The algebra of differential geometry operations on symmetric tensors over constant curvature manifolds forms a novel deformation of the sl(2,R)-bar R 2 Lie algebra. We present a simple calculus for calculations in its universal enveloping algebra. As an application, we derive generating functions for the actions and gauge invariances of massive, partially massless and massless (for both Bose and Fermi statistics) higher spins on constant curvature backgrounds. These are formulated in terms of a minimal set of covariant, unconstrained, fields rather than towers of auxiliary fields. Partially massless gauge transformations are shown to arise as degeneracies of the flat, massless gauge transformation in one dimension higher. Moreover, our results and calculus offer a considerable simplification over existing techniques for handling higher spins. In particular, we show how theories of arbitrary spin in dimension d can be rewritten in terms of a single scalar field in dimension 2d where the d additional dimensions correspond to coordinate differentials. We also develop an analogous framework for spinor-tensor fields in terms of the corresponding superalgebra

Unitary W-algebras and three-dimensional higher spin gravities with spin one symmetry

International Nuclear Information System (INIS)

Afshar, Hamid; Creutzig, Thomas; Grumiller, Daniel; Hikida, Yasuaki; Rønne, Peter B.

2014-01-01

We investigate whether there are unitary families of W-algebras with spin one fields in the natural example of the Feigin-Semikhatov W_n"("2")-algebra. This algebra is conjecturally a quantum Hamiltonian reduction corresponding to a non-principal nilpotent element. We conjecture that this algebra admits a unitary real form for even n. Our main result is that this conjecture is consistent with the known part of the operator product algebra, and especially it is true for n=2 and n=4. Moreover, we find certain ranges of allowed levels where a positive definite inner product is possible. We also find a unitary conformal field theory for every even n at the special level k+n=(n+1)/(n−1). At these points, the W_n"("2")-algebra is nothing but a compactified free boson. This family of W-algebras admits an ’t Hooft limit. Further, in the case of n=4, we reproduce the algebra from the higher spin gravity point of view. In general, gravity computations allow us to reproduce some leading coefficients of the operator product.

Banach Synaptic Algebras

Science.gov (United States)

Foulis, David J.; Pulmannov, Sylvia

2018-04-01

Using a representation theorem of Erik Alfsen, Frederic Schultz, and Erling Størmer for special JB-algebras, we prove that a synaptic algebra is norm complete (i.e., Banach) if and only if it is isomorphic to the self-adjoint part of a Rickart C∗-algebra. Also, we give conditions on a Banach synaptic algebra that are equivalent to the condition that it is isomorphic to the self-adjoint part of an AW∗-algebra. Moreover, we study some relationships between synaptic algebras and so-called generalized Hermitian algebras.

Configuration spaces geometry, topology and representation theory

CERN Document Server

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.

Grassmann algebras

International Nuclear Information System (INIS)

Garcia, R.L.

1983-11-01

The Grassmann algebra is presented briefly. Exponential and logarithm of matrices functions, whose elements belong to this algebra, are studied with the help of the SCHOONSCHIP and REDUCE 2 algebraic manipulators. (Author) [pt

Geometric calculus: a new computational tool for Riemannian geometry

International Nuclear Information System (INIS)

Moussiaux, A.; Tombal, P.

1988-01-01

We compare geometric calculus applied to Riemannian geometry with Cartan's exterior calculus method. The correspondence between the two methods is clearly established. The results obtained by a package written in an algebraic language and doing general manipulations on multivectors are compared. We see that the geometric calculus is as powerful as exterior calculus

Converting nested algebra expressions into flat algebra expressions

NARCIS (Netherlands)

Paredaens, J.; Van Gucht, D.

1992-01-01

Nested relations generalize ordinary flat relations by allowing tuple values to be either atomic or set valued. The nested algebra is a generalization of the flat relational algebra to manipulate nested relations. In this paper we study the expressive power of the nested algebra relative to its

Geometry and dynamics of integrable systems

CERN Document Server

Matveev, Vladimir

2016-01-01

Based on lectures given at an advanced course on integrable systems at the Centre de Recerca Matemàtica in Barcelona, these lecture notes address three major aspects of integrable systems: obstructions to integrability from differential Galois theory; the description of singularities of integrable systems on the basis of their relation to bi-Hamiltonian systems; and the generalization of integrable systems to the non-Hamiltonian settings. All three sections were written by top experts in their respective fields. Native to actual problem-solving challenges in mechanics, the topic of integrable systems is currently at the crossroads of several disciplines in pure and applied mathematics, and also has important interactions with physics. The study of integrable systems also actively employs methods from differential geometry. Moreover, it is extremely important in symplectic geometry and Hamiltonian dynamics, and has strong correlations with mathematical physics, Lie theory and algebraic geometry (including mir...

The SUSY oscillator from local geometry: Dynamics and coherent states

International Nuclear Information System (INIS)

Thienel, H.P.

1994-01-01

The choice of a coordinate chart on an analytical R n (R a n ) provides a representation of the n-dimensional SUSY oscillator. The corresponding Hilbert space is Cartan's exterior algebra endowed with a suitable scalar product. The exterior derivative gives rise to the algebra of the n-dimensional SUSY oscillator. Its euclidean dynamics is an inherent consequence of the geometry imposed by the Lie derivative generating the dilations, i.e. evolution of the quantum system corresponds to parametrization of a sequence of charts by euclidean time. Coherent states emerge as a natural structure related to the Lie derivative generating the translations. (orig.)

Fibered F-Algebra

OpenAIRE

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.

The bubble algebra: structure of a two-colour Temperley-Lieb Algebra

International Nuclear Information System (INIS)

Grimm, Uwe; Martin, Paul P

2003-01-01

We define new diagram algebras providing a sequence of multiparameter generalizations of the Temperley-Lieb algebra, suitable for the modelling of dilute lattice systems of two-dimensional statistical mechanics. These algebras give a rigorous foundation to the various 'multi-colour algebras' of Grimm, Pearce and others. We determine the generic representation theory of the simplest of these algebras, and locate the nongeneric cases (at roots of unity of the corresponding parameters). We show by this example how the method used (Martin's general procedure for diagram algebras) may be applied to a wide variety of such algebras occurring in statistical mechanics. We demonstrate how these algebras may be used to solve the Yang-Baxter equations

Three lectures on quantum groups: Representations, duality, real forms

International Nuclear Information System (INIS)

Dobrev, V.K.

1992-07-01

Quantum groups appeared first as quantum algebra, i.e. as one parameter deformations of the numerical enveloping algebras of complex Lie algebras, in the study of the algebraic aspects of quantum integrable systems. Then quantum algebras related to triparametric solutions of the quantum Yang-Baxter equation were axiomatically introduced as (pseudo) quasi-triangular Hopf algebras. Later, a theory of formal deformations has been developed and the notion of quasi-Hopf algebra has been introduced. In other approaches to quantum groups the objects are called quantum matrix groups and are Hopf algebras in chirality to the quantum algebras. The representations of U q (G), the chirality and the real forms associated to these approaches are discussed here. Refs

An Algebraic Criterion of Zero Solutions of Some Dynamic Systems

Directory of Open Access Journals (Sweden)

Ying Wang

2012-01-01

Full Text Available We establish some algebraic results on the zeros of some exponential polynomials and a real coefficient polynomial. Based on the basic theorem, we develop a decomposition technique to investigate the stability of two coupled systems and their discrete versions, that is, to find conditions under which all zeros of the exponential polynomials have negative real parts and the moduli of all roots of a real coefficient polynomial are less than 1.

Algebraic monoids, group embeddings, and algebraic combinatorics

CERN Document Server

Li, Zhenheng; Steinberg, Benjamin; Wang, Qiang

2014-01-01

This book contains a collection of fifteen articles and is dedicated to the sixtieth birthdays of Lex Renner and Mohan Putcha, the pioneers of the field of algebraic monoids.   Topics presented include:   v  structure and representation theory of reductive algebraic monoids v  monoid schemes and applications of monoids v  monoids related to Lie theory v  equivariant embeddings of algebraic groups v  constructions and properties of monoids from algebraic combinatorics v  endomorphism monoids induced from vector bundles v  Hodge–Newton decompositions of reductive monoids   A portion of these articles are designed to serve as a self-contained introduction to these topics, while the remaining contributions are research articles containing previously unpublished results, which are sure to become very influential for future work. Among these, for example, the important recent work of Michel Brion and Lex Renner showing that the algebraic semigroups are strongly π-regular.   Graduate students as well a...

Weinberg-Salam theory based on a Z2-graded algebra

International Nuclear Information System (INIS)

Iizuka, Jugoro; Morita, Katsusada; Kase, Hiromi; Okumura, Yositaka; Tanaka-Yamawaki, Mieko.

1994-01-01

Generalized differential calculus on discrete space M 4 xZ 2 which is an underlying space-time in the non-commutative geometry for the standard model is reformulated in terms of a Z 2 -graded algebra, even and odd elements of which being pairs of complex matrices defined over Minkowski space-time with different properties of product and involution. It is shown that the Z 2 -grading is equivalent to that of Coquereaux et al. if the pair is represented by 2x2 matrices, although our formalism has closer contact with the differential calculus on the discrete space. A graded differential algebra is then defined, in which the exterior derivative with respect to the pair is assumed to determine the pattern of symmetry breaking of the theory. On the basis of it the Weinberg-Salam theory in both bosonic and fermionic sectors is constructed. It is pointed out that, in contrast to usual assertion in non-commutative geometry, the Weinberg angle and the Higgs mass in the tree level are not fixed separately but related through m H = 2√2εm W sinθ W . Connes' prescription of constructing gauge-invariant Lagrangian, which is based on the assumption that there arise only logarithmic divergences from one-loop diagrams, corresponds to the case ε = 1. In principle, however, the parameter ε is arbitrary due to possible presence of Sitarz' linear term so that noncommutative geometry alone says nothing about the Higgs mass. (author)

Approximation of complex algebraic numbers by algebraic numbers of bounded degree

OpenAIRE

Bugeaud, Yann; Evertse, Jan-Hendrik

2007-01-01

We investigate how well complex algebraic numbers can be approximated by algebraic numbers of degree at most n. We also investigate how well complex algebraic numbers can be approximated by algebraic integers of degree at most n+1. It follows from our investigations that for every positive integer n there are complex algebraic numbers of degree larger than n that are better approximable by algebraic numbers of degree at most n than almost all complex numbers. As it turns out, these numbers ar...

Operadic formulation of topological vertex algebras and gerstenhaber or Batalin-Vilkovisky algebras

International Nuclear Information System (INIS)

Huang Yizhi

1994-01-01

We give the operadic formulation of (weak, strong) topological vertex algebras, which are variants of topological vertex operator algebras studied recently by Lian and Zuckerman. As an application, we obtain a conceptual and geometric construction of the Batalin-Vilkovisky algebraic structure (or the Gerstenhaber algebra structure) on the cohomology of a topological vertex algebra (or of a weak topological vertex algebra) by combining this operadic formulation with a theorem of Getzler (or of Cohen) which formulates Batalin-Vilkovisky algebras (or Gerstenhaber algebras) in terms of the homology of the framed little disk operad (or of the little disk operad). (orig.)

Vladimir Igorevich Arnold (1937–2010)

Indian Academy of Sciences (India)

IAS Admin

History is full of examples of scientists who made early brilliant contributions ... equations, symplectic geometry, real algebraic geometry, the calculus of variations, hydrodynamics, and magneto-hydrodynamics, often discovering links between.

Refined algebraic quantisation in a system with nonconstant gauge invariant structure functions

International Nuclear Information System (INIS)

Martínez-Pascual, Eric

2013-01-01

In a previous work [J. Louko and E. Martínez-Pascual, “Constraint rescaling in refined algebraic quantisation: Momentum constraint,” J. Math. Phys. 52, 123504 (2011)], refined algebraic quantisation (RAQ) within a family of classically equivalent constrained Hamiltonian systems that are related to each other by rescaling one momentum-type constraint was investigated. In the present work, the first steps to generalise this analysis to cases where more constraints occur are developed. The system under consideration contains two momentum-type constraints, originally abelian, where rescalings of these constraints by a non-vanishing function of the coordinates are allowed. These rescalings induce structure functions at the level of the gauge algebra. Providing a specific parametrised family of real-valued scaling functions, the implementation of the corresponding rescaled quantum momentum-type constraints is performed using RAQ when the gauge algebra: (i) remains abelian and (ii) undergoes into an algebra of a nonunimodular group with nonconstant gauge invariant structure functions. Case (ii) becomes the first example known to the author where an open algebra is handled in refined algebraic quantisation. Challenging issues that arise in the presence of non-gauge invariant structure functions are also addressed

Groups, matrices, and vector spaces a group theoretic approach to linear algebra

CERN Document Server

Carrell, James B

2017-01-01

This unique text provides a geometric approach to group theory and linear algebra, bringing to light the interesting ways in which these subjects interact. Requiring few prerequisites beyond understanding the notion of a proof, the text aims to give students a strong foundation in both geometry and algebra. Starting with preliminaries (relations, elementary combinatorics, and induction), the book then proceeds to the core topics: the elements of the theory of groups and fields (Lagrange's Theorem, cosets, the complex numbers and the prime fields), matrix theory and matrix groups, determinants, vector spaces, linear mappings, eigentheory and diagonalization, Jordan decomposition and normal form, normal matrices, and quadratic forms. The final two chapters consist of a more intensive look at group theory, emphasizing orbit stabilizer methods, and an introduction to linear algebraic groups, which enriches the notion of a matrix group. Applications involving symm etry groups, determinants, linear coding theory ...

Wn(2) algebras

International Nuclear Information System (INIS)

Feigin, B.L.; Semikhatov, A.M.

2004-01-01

We construct W-algebra generalizations of the sl-circumflex(2) algebra-W algebras W n (2) generated by two currents E and F with the highest pole of order n in their OPE. The n=3 term in this series is the Bershadsky-Polyakov W 3 (2) algebra. We define these algebras as a centralizer (commutant) of the Uqs-bar (n vertical bar 1) quantum supergroup and explicitly find the generators in a factored, 'Miura-like' form. Another construction of the W n (2) algebras is in terms of the coset sl-circumflex(n vertical bar 1)/sl-circumflex(n). The relation between the two constructions involves the 'duality' (k+n-1)(k'+n-1)=1 between levels k and k' of two sl-circumflex(n) algebras

Architectural geometry

KAUST Repository

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.

Architectural geometry

KAUST Repository

Pottmann, Helmut; Eigensatz, Michael; Vaxman, Amir; Wallner, Johannes

2014-01-01

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 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.

A foundation for props, algebras, and modules

CERN Document Server

Yau, Donald

2015-01-01

PROPs and their variants are extremely general and powerful machines that encode operations with multiple inputs and multiple outputs. In this respect PROPs can be viewed as generalizations of operads that would allow only a single output. Variants of PROPs are important in several mathematical fields, including string topology, topological conformal field theory, homotopical algebra, deformation theory, Poisson geometry, and graph cohomology. The purpose of this monograph is to develop, in full technical detail, a unifying object called a generalized PROP. Then with an appropriate choice of p

The vacuum preserving Lie algebra of a classical W-algebra

International Nuclear Information System (INIS)

Feher, L.; Tsutsui, I.

1993-07-01

We simplify and generalize an argument due to Bowcock and Watts showing that one can associate a finite Lie algebra (the 'classical vacuum preserving algebra') containing the Moebius sl(2) subalgebra to any classical W-algebra. Our construction is based on a kinematical analysis of the Poisson brackets of quasi-fields. In the case of the W S G -subalgebra S of a simple Lie algebra G, we exhibit a natural isomorphism between this finite Lie algebra and G whereby the Moebius sl(2) is identified with S. (orig.)

Algorithmic Algebraic Combinatorics and Gröbner Bases

CERN Document Server

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

Nonflexible Lie-admissible algebras

International Nuclear Information System (INIS)

Myung, H.C.

1978-01-01

We discuss the structure of Lie-admissible algebras which are defined by nonflexible identities. These algebras largely arise from the antiflexible algebras, 2-varieties and associator dependent algebras. The nonflexible Lie-admissible algebras in our discussion are in essence byproducts of the study of nonassociative algebras defined by identities of degree 3. The main purpose is to discuss the classification of simple Lie-admissible algebras of nonflexible type

On 2-Banach algebras

International Nuclear Information System (INIS)

Mohammad, N.; Siddiqui, A.H.

1987-11-01

The notion of a 2-Banach algebra is introduced and its structure is studied. After a short discussion of some fundamental properties of bivectors and tensor product, several classical results of Banach algebras are extended to the 2-Banach algebra case. A condition under which a 2-Banach algebra becomes a Banach algebra is obtained and the relation between algebra of bivectors and 2-normed algebra is discussed. 11 refs

The Minnesota notes on Jordan algebras and their applications

CERN Document Server

Walcher, Sebastian

1999-01-01

This volume contains a re-edition of Max Koecher's famous Minnesota Notes. The main objects are homogeneous, but not necessarily convex, cones. They are described in terms of Jordan algebras. The central point is a correspondence between semisimple real Jordan algebras and so-called omega-domains. This leads to a construction of half-spaces which give an essential part of all bounded symmetric domains. The theory is presented in a concise manner, with only elementary prerequisites. The editors have added notes on each chapter containing an account of the relevant developments of the theory since these notes were first written.

Algebra

CERN Document Server

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

Low Dimensional Vessiot-Guldberg-Lie Algebras of Second-Order Ordinary Differential Equations

Directory of Open Access Journals (Sweden)

Rutwig Campoamor-Stursberg

2016-03-01

Full Text Available A direct approach to non-linear second-order ordinary differential equations admitting a superposition principle is developed by means of Vessiot-Guldberg-Lie algebras of a dimension not exceeding three. This procedure allows us to describe generic types of second-order ordinary differential equations subjected to some constraints and admitting a given Lie algebra as Vessiot-Guldberg-Lie algebra. In particular, well-known types, such as the Milne-Pinney or Kummer-Schwarz equations, are recovered as special cases of this classification. The analogous problem for systems of second-order differential equations in the real plane is considered for a special case that enlarges the generalized Ermakov systems.

Spectral dimension of quantum geometries

International Nuclear Information System (INIS)

Calcagni, Gianluca; Oriti, Daniele; Thürigen, Johannes

2014-01-01

The spectral dimension is an indicator of geometry and topology of spacetime and a tool to compare the description of quantum geometry in various approaches to quantum gravity. This is possible because it can be defined not only on smooth geometries but also on discrete (e.g., simplicial) ones. In this paper, we consider the spectral dimension of quantum states of spatial geometry defined on combinatorial complexes endowed with additional algebraic data: the kinematical quantum states of loop quantum gravity (LQG). Preliminarily, the effects of topology and discreteness of classical discrete geometries are studied in a systematic manner. We look for states reproducing the spectral dimension of a classical space in the appropriate regime. We also test the hypothesis that in LQG, as in other approaches, there is a scale dependence of the spectral dimension, which runs from the topological dimension at large scales to a smaller one at short distances. While our results do not give any strong support to this hypothesis, we can however pinpoint when the topological dimension is reproduced by LQG quantum states. Overall, by exploring the interplay of combinatorial, topological and geometrical effects, and by considering various kinds of quantum states such as coherent states and their superpositions, we find that the spectral dimension of discrete quantum geometries is more sensitive to the underlying combinatorial structures than to the details of the additional data associated with them. (paper)

Lukasiewicz-Moisil algebras

CERN Document Server

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.

Abstract Algebra for Algebra Teaching: Influencing School Mathematics Instruction

Science.gov (United States)

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…

It's a Wonderful Life: Using Public Domain Cinema Clips To Teach Affective Objectives and Illustrate Real-World Algebra Applications.

Science.gov (United States)

Palmer, Loretta

A basic algebra unit was developed at Utah Valley State College to emphasize applications of mathematical concepts in the work world, using video and computer-generated graphics to integrate textual material. The course was implemented in three introductory algebra sections involving 80 students and taught algebraic concepts using such areas as…

An invitation to noncommutative geometry

CERN Document Server

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

Geometrical treatment of Clifford algebras and spinors with applications to the dynamics of spin particles

International Nuclear Information System (INIS)

Dimakis, A.

1983-01-01

The algebraic structure of the real Clifford algebras (CA) of vector spaces with non-degenerated scalar product of arbitrary signature is studied, and a classification formula for this is obtained. The latter is based on three sequences of integer numbers from which one is the Radon-Harwitz sequence. A new representation method of real CA is constructed. This leads to a geometrical representation of real Clifford algebras in which the representation spaces are subspaces of the CA itself (''spinor spaces''). One of these spinor spaces is a subalgebra of the original CA. The relation between CA and external algebras is studied. Each external algebra with a scalar product possesses the structure of a CA. From the geometric representation developed here then follows that spinors are inhomogeneous external forms. The transformation behaviour of spinors under the orthogonal, as well as under the general linear group is studied. By means of these algebraic results the spinor connexion and the covariant Dirac operator on a differential manifold are introduced. In the geometrical representation a further in ternal SL(2,R) symmetry of the Dirac equation (DE) is shown. Furthermore other equivalent formulations of the DE can be obtained. Of special interest is the tetrade formulation of the DE. A generalization of the DE is introduced. The equations of motion of the classical relativistic spin particle are derived by means of spinors and CA from a variational principle. From this interesting formal analogies with the supersymmetric theories of the spin particle result. Finally the DE in the curved space-time is established and studied in the tetrade formulation. Using the methods developed here a new exact solution of the coupled Einstein-Curtan-Dirac theory was found (massice ''Ghost-Dirac fields''). (orig.) [de

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.

Wavelets and quantum algebras

International Nuclear Information System (INIS)

Ludu, A.; Greiner, M.

1995-09-01

A non-linear associative algebra is realized in terms of translation and dilation operators, and a wavelet structure generating algebra is obtained. We show that this algebra is a q-deformation of the Fourier series generating algebra, and reduces to this for certain value of the deformation parameter. This algebra is also homeomorphic with the q-deformed su q (2) algebra and some of its extensions. Through this algebraic approach new methods for obtaining the wavelets are introduced. (author). 20 refs

Lie-algebra expansions, Chern-Simons theories and the Einstein-Hilbert Lagrangian

International Nuclear Information System (INIS)

Edelstein, Jose D.; Hassaine, Mokhtar; Troncoso, Ricardo; Zanelli, Jorge

2006-01-01

Starting from gravity as a Chern-Simons action for the AdS algebra in five dimensions, it is possible to modify the theory through an expansion of the Lie algebra that leads to a system consisting of the Einstein-Hilbert action plus non-minimally coupled matter. The modified system is gauge invariant under the Poincare group enlarged by an Abelian ideal. Although the resulting action naively looks like general relativity plus corrections due to matter sources, it is shown that the non-minimal couplings produce a radical departure from GR. Indeed, the dynamics is not continuously connected to the one obtained from Einstein-Hilbert action. In a matter-free configuration and in the torsionless sector, the field equations are too strong a restriction on the geometry as the metric must satisfy both the Einstein and pure Gauss-Bonnet equations. In particular, the five-dimensional Schwarzschild geometry fails to be a solution; however, configurations corresponding to a brane-world with positive cosmological constant on the worldsheet are admissible when one of the matter fields is switched on. These results can be extended to higher odd dimensions

Novikov-Jordan algebras

OpenAIRE

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.

Using Linear Algebra to Introduce Computer Algebra, Numerical Analysis, Data Structures and Algorithms (and To Teach Linear Algebra, Too).

Science.gov (United States)

Gonzalez-Vega, Laureano

1999-01-01

Using a Computer Algebra System (CAS) to help with the teaching of an elementary course in linear algebra can be one way to introduce computer algebra, numerical analysis, data structures, and algorithms. Highlights the advantages and disadvantages of this approach to the teaching of linear algebra. (Author/MM)

(Quasi-)Poisson enveloping algebras

OpenAIRE

Yang, Yan-Hong; Yao, Yuan; 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.

Iterated Leavitt Path Algebras

International Nuclear Information System (INIS)

Hazrat, R.

2009-11-01

Leavitt path algebras associate to directed graphs a Z-graded algebra and in their simplest form recover the Leavitt algebras L(1,k). In this note, we introduce iterated Leavitt path algebras associated to directed weighted graphs which have natural ± Z grading and in their simplest form recover the Leavitt algebras L(n,k). We also characterize Leavitt path algebras which are strongly graded. (author)