Algebra and Number Theory An Integrated Approach
Dixon, Martyn; Subbotin, Igor
2011-01-01
Explore the main algebraic structures and number systems that play a central role across the field of mathematics Algebra and number theory are two powerful branches of modern mathematics at the forefront of current mathematical research, and each plays an increasingly significant role in different branches of mathematics, from geometry and topology to computing and communications. Based on the authors' extensive experience within the field, Algebra and Number Theory has an innovative approach that integrates three disciplines-linear algebra, abstract algebra, and number theory-into one compr
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
Jarvis, Frazer
2014-01-01
The technical difficulties of algebraic number theory often make this subject appear difficult to beginners. This undergraduate textbook provides a welcome solution to these problems as it provides an approachable and thorough introduction to the topic. Algebraic Number Theory takes the reader from unique factorisation in the integers through to the modern-day number field sieve. The first few chapters consider the importance of arithmetic in fields larger than the rational numbers. Whilst some results generalise well, the unique factorisation of the integers in these more general number fields often fail. Algebraic number theory aims to overcome this problem. Most examples are taken from quadratic fields, for which calculations are easy to perform. The middle section considers more general theory and results for number fields, and the book concludes with some topics which are more likely to be suitable for advanced students, namely, the analytic class number formula and the number field sieve. This is the fi...
Weiss, Edwin
1998-01-01
Careful organization and clear, detailed proofs characterize this methodical, self-contained exposition of basic results of classical algebraic number theory from a relatively modem point of view. This volume presents most of the number-theoretic prerequisites for a study of either class field theory (as formulated by Artin and Tate) or the contemporary treatment of analytical questions (as found, for example, in Tate's thesis).Although concerned exclusively with algebraic number fields, this treatment features axiomatic formulations with a considerable range of applications. Modem abstract te
The theory of algebraic numbers
Pollard, Harry
1998-01-01
An excellent introduction to the basics of algebraic number theory, this concise, well-written volume examines Gaussian primes; polynomials over a field; algebraic number fields; and algebraic integers and integral bases. After establishing a firm introductory foundation, the text explores the uses of arithmetic in algebraic number fields; the fundamental theorem of ideal theory and its consequences; ideal classes and class numbers; and the Fermat conjecture. 1975 edition. References. List of Symbols. Index.
Classical theory of algebraic numbers
Ribenboim, Paulo
2001-01-01
Gauss created the theory of binary quadratic forms in "Disquisitiones Arithmeticae" and Kummer invented ideals and the theory of cyclotomic fields in his attempt to prove Fermat's Last Theorem These were the starting points for the theory of algebraic numbers, developed in the classical papers of Dedekind, Dirichlet, Eisenstein, Hermite and many others This theory, enriched with more recent contributions, is of basic importance in the study of diophantine equations and arithmetic algebraic geometry, including methods in cryptography This book has a clear and thorough exposition of the classical theory of algebraic numbers, and contains a large number of exercises as well as worked out numerical examples The Introduction is a recapitulation of results about principal ideal domains, unique factorization domains and commutative fields Part One is devoted to residue classes and quadratic residues In Part Two one finds the study of algebraic integers, ideals, units, class numbers, the theory of decomposition, iner...
Distribution theory of algebraic numbers
Yang, Chung-Chun
2008-01-01
The book timely surveys new research results and related developments in Diophantine approximation, a division of number theory which deals with the approximation of real numbers by rational numbers. The book is appended with a list of challenging open problems and a comprehensive list of references. From the contents: Field extensions Algebraic numbers Algebraic geometry Height functions The abc-conjecture Roth''s theorem Subspace theorems Vojta''s conjectures L-functions.
Algebraic number theory and code design for Rayleigh fading channels
Oggier, F
2014-01-01
Algebraic number theory is having an increasing impact in code design for many different coding applications, such as single antenna fading channels and more recently, MIMO systems.Extended work has been done on single antenna fading channels, and algebraic lattice codes have been proven to be an effective tool. The general framework has been settled in the last ten years and many explicit code constructions based on algebraic number theory are now available. The aim of this work is to provide both an overview on algebraic lattice code designs for Rayleigh fading channels, as well as a tutoria
Cryptography by means of linear algebra and number theory
Elfadel, Ajaeb
2014-01-01
ABSTRACT: This thesis focuses on the techniques of cryptography in linear algebra and number theory. We first give the necessary review on modular arithmetic. Under Linear Algebra, Hill cipher cryptographic technique and its variations are studied. Under number theory, on the other hand, the definition of Euler function, and some important theorems in this regard are given. The cryptographic techniques such as the Caesar cipher, Exponential transformations and the Public key cryptographic tec...
Partial Fractions in Calculus, Number Theory, and Algebra
Yackel, C. A.; Denny, J. K.
2007-01-01
This paper explores the development of the method of partial fraction decomposition from elementary number theory through calculus to its abstraction in modern algebra. This unusual perspective makes the topic accessible and relevant to readers from high school through seasoned calculus instructors.
Notes on the Theory of Algebraic Numbers
Wright, Steve
2015-01-01
A series of lecture notes on the elementary theory of algebraic numbers, using only knowledge of a first-semester graduate course in algebra (primarily groups and rings). No prerequisite knowledge of fields is required. Based primarily on the texts of E. Hecke, Lectures on the Theory of Algebraic Numbers, Springer-Verlag, 1981 (English translation by G. Brauer and J. Goldman) and D. Marcus, Number Fields, Springer, 1977.
Real Algebraic Number Theory I: Diophantine Approximation Groups
Gendron, T. M.
2012-01-01
This is the first of three papers introducing a paradigm within which global algebraic number theory for the reals may be formulated so as to make possible the synthesis of algebraic and transcendental number theory into a coherent whole. We introduce diophantine approximation groups and their associated Kronecker foliations, using them to provide new algebraic and geometric characterizations of K-linear and algebraic dependence. As a consequence we find reformulations -- as algebraic and geo...
Baxter Algebras, Stirling Numbers and Partitions
Guo, Li
2004-01-01
Recent developments of Baxter algebras have lead to applications to combinatorics, number theory and mathematical physics. We relate Baxter algebras to Stirling numbers of the first kind and the second kind, partitions and multinomial coefficients. This allows us to apply congruences from number theory to obtain congruences in Baxter algebras.
Elementary number theory an algebraic approach
Bolker, Ethan D
2007-01-01
This text uses the concepts usually taught in the first semester of a modern abstract algebra course to illuminate classical number theory: theorems on primitive roots, quadratic Diophantine equations, and the Fermat conjecture for exponents three and four. The text contains abundant numerical examples and a particularly helpful collection of exercises, many of which are small research problems requiring substantial study or outside reading. Some problems call for new proofs for theorems already covered or for inductive explorations and proofs of theorems found in later chapters.Ethan D. Bolke
Real Algebraic Number Theory I: Diophantine Approximation Groups
Gendron, T M
2012-01-01
The is the first of three papers introducing a paradigm within which global algebraic number theory for the reals may be formulated so as to make possible the synthesis of algebraic and transcendental number theory into a coherent whole. We introduce diophantine approximation groups and their associated Kronecker foliations, using them to provide new algebraic and geometric characterizations of K-linear and algebraic dependence. As a consequence we find reformulations -- as algebraic and geometric (graph) rigidities -- of the Theorems of Baker and Lindemann-Weierstrass, the Logarithm Conjecture and the Schanuel Conjecture. There is an Appendix describing diophantine approximation groups as model theoretic types.
Quantum Algorithms for Problems in Number Theory, Algebraic Geometry, and Group Theory
van Dam, Wim; Sasaki, Yoshitaka
2013-09-01
Quantum computers can execute algorithms that sometimes dramatically outperform classical computation. Undoubtedly the best-known example of this is Shor's discovery of an efficient quantum algorithm for factoring integers, whereas the same problem appears to be intractable on classical computers. Understanding what other computational problems can be solved significantly faster using quantum algorithms is one of the major challenges in the theory of quantum computation, and such algorithms motivate the formidable task of building a large-scale quantum computer. This article will review the current state of quantum algorithms, focusing on algorithms for problems with an algebraic flavor that achieve an apparent superpolynomial speedup over classical computation.
Bair, Sherry L.; Rich, Beverly S.
2011-01-01
This article characterizes the development of a deep and connected body of mathematical knowledge categorized by Ball and Bass' (2003b) model of Mathematical Knowledge for Teaching (MKT), as Specialized Content Knowledge for Teaching (SCK) in algebraic reasoning and number sense. The research employed multiple cases across three years from two…
Algebraic theta functions and Eisenstein-Kronecker numbers
Bannai, Kenichi; Kobayashi, Shinichi
2007-01-01
In this paper, we give an overview of our previous paper concerning the investigation of the algebraic and $p$-adic properties of Eisenstein-Kronecker numbers using Mumford's theory of algebraic theta functions.
Universal Algebras of Hurwitz Numbers
A. Mironov; Morozov, A; Natanzon, S.
2009-01-01
Infinite-dimensional universal Cardy-Frobenius algebra is constructed, which unifies all particular algebras of closed and open Hurwitz numbers and is closely related to the algebra of differential operators, familiar from the theory of Generalized Kontsevich Model.
Division Algebras and Quantum Theory
Baez, John C
2011-01-01
Quantum theory may be formulated using Hilbert spaces over any of the three associative normed division algebras: the real numbers, the complex numbers and the quaternions. Indeed, these three choices appear naturally in a number of axiomatic approaches. However, there are internal problems with real or quaternionic quantum theory. Here we argue that these problems can be resolved if we treat real, complex and quaternionic quantum theory as part of a unified structure. Dyson called this structure the "three-fold way". It is perhaps easiest to see it in the study of irreducible unitary representations of groups on complex Hilbert spaces. These representations come in three kinds: those that are not isomorphic to their own dual (the truly "complex" representations), those that are self-dual thanks to a symmetric bilinear pairing (which are "real", in that they are the complexifications of representations on real Hilbert spaces), and those that are self-dual thanks to an antisymmetric bilinear pairing (which are...
On the Theory of Algebraic Numbers with Elements of Small Height
Göral, Haydar
2015-01-01
In this paper, we study the field of algebraic numbers with elements of small height.We first show that the nonstandard algebraic numbers whose logarithmic height isinfinitesimal has the Mann property, and exploiting this we obtain a diophantine approximation result in number theory.Then, we prove that the theory of algebraic numbers with elements of small height is not simple and has the independence property.We also relate the simplicity of a certain pair with Lehmer's conjecture and obtain...
Problems and proofs in numbers and algebra
Millman, Richard S; Kahn, Eric Brendan
2015-01-01
Designed to facilitate the transition from undergraduate calculus and differential equations to learning about proofs, this book helps students develop the rigorous mathematical reasoning needed for advanced courses in analysis, abstract algebra, and more. Students will focus on both how to prove theorems and solve problem sets in-depth; that is, where multiple steps are needed to prove or solve. This proof technique is developed by examining two specific content themes and their applications in-depth: number theory and algebra. This choice of content themes enables students to develop an understanding of proof technique in the context of topics with which they are already familiar, as well as reinforcing natural and conceptual understandings of mathematical methods and styles. The key to the text is its interesting and intriguing problems, exercises, theorems, and proofs, showing how students will transition from the usual, more routine calculus to abstraction while also learning how to “prove” or “sol...
K-theory for ring C*-algebras - the case of number fields with higher roots of unity
Li, Xin
2012-01-01
We compute K-theory for ring C*-algebras in the case of higher roots of unity and thereby completely determine the K-theory for ring C*-algebras attached to rings of integers in arbitrary number fields.
Algebraic and structural automata theory
Mikolajczak, B
1991-01-01
Automata Theory is part of computability theory which covers problems in computer systems, software, activity of nervous systems (neural networks), and processes of live organisms development.The result of over ten years of research, this book presents work in the following areas of Automata Theory: automata morphisms, time-varying automata, automata realizations and relationships between automata and semigroups.Aimed at those working in discrete mathematics and computer science, parts of the book are suitable for use in graduate courses in computer science, electronics, telecommunications, and control engineering. It is assumed that the reader is familiar with the basic concepts of algebra and graph theory.
Matrix algebra, computational methods, and number theory. [Mysore, India, September 6-9, 1976
Energy Technology Data Exchange (ETDEWEB)
Ranganathan, N.R. (ed.)
1977-01-01
Ten papers were given at the meeting on the following topics: permutations, partitions, and L matrices; a matrix approach to certain number theoretic problems; analogs to the Hardy--Ramanujan theorems; some computational methods in physics; some aspects of infinite matrices; old Indian theory of numbers and its applications in nuclear and solid state physics; point pathos graphs of a graph; iterative methods for solving systems of linear equations to save computer time; numerical computation of surface and Coulomb entries of an axially deformed nucleus; and positron wave functions in solids. Two of the papers were abstracted and indexed separately. (RWR)
Rational and algebraic approximations of algebraic numbers and their application
Institute of Scientific and Technical Information of China (English)
袁平之
1997-01-01
Effective rational and algebraic approximations of a large class of algebraic numbers are obtained by Thue-Siegel’s method.As an application of this result,it is proved that; if D>0 is not a square,and ε=x0 +y0 D denotes the fundamental solution of x2-Dy2=-1,then x2+1=Dy4 is solvable if and only if y0=A2 where A is an integer.Moreover,if ≥64,then x2+1=Dy4 has at most one positive integral solution (x,y).
Certain number-theoretic episodes in algebra
Sivaramakrishnan, R
2006-01-01
Many basic ideas of algebra and number theory intertwine, making it ideal to explore both at the same time. Certain Number-Theoretic Episodes in Algebra focuses on some important aspects of interconnections between number theory and commutative algebra. Using a pedagogical approach, the author presents the conceptual foundations of commutative algebra arising from number theory. Self-contained, the book examines situations where explicit algebraic analogues of theorems of number theory are available. Coverage is divided into four parts, beginning with elements of number theory and algebra such as theorems of Euler, Fermat, and Lagrange, Euclidean domains, and finite groups. In the second part, the book details ordered fields, fields with valuation, and other algebraic structures. This is followed by a review of fundamentals of algebraic number theory in the third part. The final part explores links with ring theory, finite dimensional algebras, and the Goldbach problem.
Transformation groups and algebraic K-theory
Lück, Wolfgang
1989-01-01
The book focuses on the relation between transformation groups and algebraic K-theory. The general pattern is to assign to a geometric problem an invariant in an algebraic K-group which determines the problem. The algebraic K-theory of modules over a category is studied extensively and appplied to the fundamental category of G-space. Basic details of the theory of transformation groups sometimes hard to find in the literature, are collected here (Chapter I) for the benefit of graduate students. Chapters II and III contain advanced new material of interest to researchers working in transformation groups, algebraic K-theory or related fields.
Lectures on algebraic quantum field theory and operator algebras
Energy Technology Data Exchange (ETDEWEB)
Schroer, Bert [Berlin Univ. (Germany). Institut fuer Theoretische Physik. E-mail: schroer@cbpf.br
2001-04-01
In this series of lectures directed towards a mainly mathematically oriented audience I try to motivate the use of operator algebra methods in quantum field theory. Therefore a title as why mathematicians are/should be interested in algebraic quantum field theory would be equally fitting. besides a presentation of the framework and the main results of local quantum physics these notes may serve as a guide to frontier research problems in mathematical. (author)
Ônishi, Yoshihiro
2004-01-01
Hurwitz numbers are the Laurent coefficients of an elliptic function $\\wp(u)$ of cyclotomic type, and they are natural generalization of the Bernoulli numbers. This paper gives new generalization of Bernoulli and Hurwitz numbers for higher genus cases. They satisfy completely von Staudt-Clausen type theorem, an extension of von Staudt second theorem, and Kummer type congruence relation. The present paper is revised and combined version of math.NT/0304377 and math.NT/0312178 containing many nu...
Algebraic K-theory, K-regularity, and -duality of -stable C ∗-algebras
Mahanta, Snigdhayan
2015-12-01
We develop an algebraic formalism for topological -duality. More precisely, we show that topological -duality actually induces an isomorphism between noncommutative motives that in turn implements the well-known isomorphism between twisted K-theories (up to a shift). In order to establish this result we model topological K-theory by algebraic K-theory. We also construct an E ∞ -operad starting from any strongly self-absorbing C ∗-algebra . Then we show that there is a functorial topological K-theory symmetric spectrum construction on the category of separable C ∗-algebras, such that is an algebra over this operad; moreover, is a module over this algebra. Along the way we obtain a new symmetric spectra valued functorial model for the (connective) topological K-theory of C ∗-algebras. We also show that -stable C ∗-algebras are K-regular providing evidence for a conjecture of Rosenberg. We conclude with an explicit description of the algebraic K-theory of a x+ b-semigroup C ∗-algebras coming from number theory and that of -stabilized noncommutative tori.
Algebraic Numbers, Hyperbolicity, and Density Modulo One
Gorodnik, Alexander
2011-01-01
We prove the density of the sets of the form ${{\\lambda}_1^m {\\mu}_1^n {\\xi}_1 +...+{\\lambda}_k^m {\\mu}_k^n {\\xi}_k : m,n \\in \\mathbb N}$ modulo one, where $\\lambda_i$ and $\\mu_i$ are multiplicatively independent algebraic numbers satisfying some additional assumptions. The proof is based on analysing dynamics of higher-rank actions on compact abelean groups.
Ore, Oystein
1988-01-01
A prominent mathematician presents the principal ideas and methods of number theory within a historical and cultural framework. Oystein Ore's fascinating, accessible treatment requires only a basic knowledge of algebra. Topics include prime numbers, the Aliquot parts, linear indeterminate problems, congruences, Euler's theorem, classical construction problems, and many other subjects.
Kleene Algebra with Products and Iteration Theories
Kozen, Dexter; Mamouras, Konstantinos
2013-01-01
We develop a typed equational system that subsumes both iteration theories and typed Kleene algebra in a common framework. Our approach is based on cartesian categories endowed with commutative strong monads to handle nondeterminism.
Fourier theory and C∗-algebras
Bédos, Erik; Conti, Roberto
2016-07-01
We discuss a number of results concerning the Fourier series of elements in reduced twisted group C∗-algebras of discrete groups, and, more generally, in reduced crossed products associated to twisted actions of discrete groups on unital C∗-algebras. A major part of the article gives a review of our previous work on this topic, but some new results are also included.
Representation Theory of Algebraic Groups and Quantum Groups
Gyoja, A; Shinoda, K-I; Shoji, T; Tanisaki, Toshiyuki
2010-01-01
Invited articles by top notch expertsFocus is on topics in representation theory of algebraic groups and quantum groupsOf interest to graduate students and researchers in representation theory, group theory, algebraic geometry, quantum theory and math physics
L∞-algebra models and higher Chern-Simons theories
Ritter, Patricia; Sämann, Christian
2016-10-01
We continue our study of zero-dimensional field theories in which the fields take values in a strong homotopy Lie algebra. In the first part, we review in detail how higher Chern-Simons theories arise in the AKSZ-formalism. These theories form a universal starting point for the construction of L∞-algebra models. We then show how to describe superconformal field theories and how to perform dimensional reductions in this context. In the second part, we demonstrate that Nambu-Poisson and multisymplectic manifolds are closely related via their Heisenberg algebras. As a byproduct of our discussion, we find central Lie p-algebra extensions of 𝔰𝔬(p + 2). Finally, we study a number of L∞-algebra models which are physically interesting and which exhibit quantized multisymplectic manifolds as vacuum solutions.
C*-algebras and operator theory
Murphy, Gerald J
1990-01-01
This book constitutes a first- or second-year graduate course in operator theory. It is a field that has great importance for other areas of mathematics and physics, such as algebraic topology, differential geometry, and quantum mechanics. It assumes a basic knowledge in functional analysis but no prior acquaintance with operator theory is required.
Excision in algebraic K-theory and Karoubi's conjecture.
Suslin, A A; Wodzicki, M
1990-12-15
We prove that the property of excision in algebraic K-theory is for a Q-algebra A equivalent to the H-unitality of the latter. Our excision theorem, in particular, implies Karoubi's conjecture on the equality of algebraic and topological K-theory groups of stable C*-algebras. It also allows us to identify the algebraic K-theory of the symbol map in the theory of pseudodifferential operators. PMID:11607130
Ternary numbers and algebras. Reflexive numbers and Berger graphs
Dubrovskiy, A
2007-01-01
The Calabi-Yau spaces with SU(n) holonomy can be studied by the algebraic way through the integer lattice where one can construct the Newton reflexive polyhedra or the Berger graphs. Our conjecture is that the Berger graphs can be directly related with the $n$-ary algebras. To find such algebras we study the n-ary generalization of the well-known binary norm division algebras, ${\\mathbb R}$, ${\\mathbb C}$, ${\\mathbb H}$, ${\\mathbb O}$, which helped to discover the most important "minimal" binary simple Lie groups, U(1), SU(2) and G(2). As the most important example, we consider the case $n=3$, which gives the ternary generalization of quaternions and octonions, $3^n$, $n=2,3$, respectively. The ternary generalization of quaternions is directly related to the new ternary algebra and group which are related to the natural extensions of the binary $su(3)$ algebra and SU(3) group. Using this ternary algebra we found the solution for the Berger graph: a tetrahedron.
Topics in algebraic and topological K-theory
Baum, Paul Frank; Meyer, Ralf; Sánchez-García, Rubén; Schlichting, Marco; Toën, Bertrand
2011-01-01
This volume is an introductory textbook to K-theory, both algebraic and topological, and to various current research topics within the field, including Kasparov's bivariant K-theory, the Baum-Connes conjecture, the comparison between algebraic and topological K-theory of topological algebras, the K-theory of schemes, and the theory of dg-categories.
Galois theory, motives and transcendental numbers
Andre, Yves
2008-01-01
From its early beginnings up to nowadays, algebraic number theory has evolved in symbiosis with Galois theory: indeed, one could hold that it consists in the very study of the absolute Galois group of the field of rational numbers. Nothing like that can be said of transcendental number theory. Nevertheless, couldn't one associate conjugates and a Galois group to transcendental numbers such as $\\pi$? Beyond, can't one envision an appropriate Galois theory in the field of transcendental number ...
Shifted genus expanded W ∞ algebra and shifted Hurwitz numbers
Zheng, Quan
2016-05-01
We construct the shifted genus expanded W ∞ algebra, which is isomorphic to the central subalgebra A ∞ of infinite symmetric group algebra and to the shifted Schur symmetrical function algebra Λ* defined by Okounkov and Olshanskii. As an application, we get some differential equations for the generating functions of the shifted Hurwitz numbers; thus, we can express the generating functions in terms of the shifted genus expanded cut-and-join operators.
Mixed motives and algebraic K-theory
Jannsen, Uwe
1990-01-01
The relations that could or should exist between algebraic cycles, algebraic K-theory, and the cohomology of - possibly singular - varieties, are the topic of investigation of this book. The author proceeds in an axiomatic way, combining the concepts of twisted Poincaré duality theories, weights, and tensor categories. One thus arrives at generalizations to arbitrary varieties of the Hodge and Tate conjectures to explicit conjectures on l-adic Chern characters for global fields and to certain counterexamples for more general fields. It is to be hoped that these relations ions will in due course be explained by a suitable tensor category of mixed motives. An approximation to this is constructed in the setting of absolute Hodge cycles, by extending this theory to arbitrary varieties. The book can serve both as a guide for the researcher, and as an introduction to these ideas for the non-expert, provided (s)he knows or is willing to learn about K-theory and the standard cohomology theories of algebraic varietie...
An interface between physics and number theory
Duchamp, Gérard Henry Edmond; Solomon, Allan I; Goodenough, Silvia
2010-01-01
We extend the Hopf algebra description of a simple quantum system given previously, to a more elaborate Hopf algebra, which is rich enough to encompass that related to a description of perturbative quantum field theory (pQFT). This provides a {\\em mathematical} route from an algebraic description of non-relativistic, non-field theoretic quantum statistical mechanics to one of relativistic quantum field theory. Such a description necessarily involves treating the algebra of polyzeta functions, extensions of the Riemann Zeta function, since these occur naturally in pQFT. This provides a link between physics, algebra and number theory. As a by-product of this approach, we are led to indicate {\\it inter alia} a basis for concluding that the Euler gamma constant $\\gamma$ may be rational.
Inverse Determinant Sums and Connections Between Fading Channel Information Theory and Algebra
Vehkalahti, Roope
2011-01-01
Since the invention of space-time coding numerous algebraic methods have been applied to code design. In particular algebraic number theory and central simple algebras have been at the forefront of the research. In the first part of the paper we will push this direction further and show how the error probability of algebraic codes is tied to some central aspects of algebraic number theory and central simple algebras. In particular we prove how the error probability of several algebraic codes is tied to the corresponding zeta functions and unit groups. In the second part of this paper we turn to study what information theory can say about algebra. We will first derive some corollaries from the diversity-multiplexing gain tradeoff (DMT) Zheng and Tse and later show how these results can be used to analyze the unit group of orders of certain division algebras.
Algebraic Theories and (Infinity,1)-Categories
Cranch, James
2010-11-01
We adapt the classical framework of algebraic theories to work in the setting of (infinity,1)-categories developed by Joyal and Lurie. This gives a suitable approach for describing highly structured objects from homotopy theory. A central example, treated at length, is the theory of E_infinity spaces: this has a tidy combinatorial description in terms of span diagrams of finite sets. We introduce a theory of distributive laws, allowing us to describe objects with two distributing E_infinity stuctures. From this we produce a theory of E_infinity ring spaces. We also study grouplike objects, and produce theories modelling infinite loop spaces (or connective spectra), and infinite loop spaces with coherent multiplicative structure (or connective ring spectra). We use this to construct the units of a grouplike E_infinity ring space in a natural manner. Lastly we provide a speculative pleasant description of the K-theory of monoidal quasicategories and quasicategories with ring-like structures.
Nonassociativity, Malcev algebras and string theory
International Nuclear Information System (INIS)
Nonassociative structures have appeared in the study of D-branes in curved backgrounds. In recent work, string theory backgrounds involving three-form fluxes, where such structures show up, have been studied in more detail. We point out that under certain assumptions these nonassociative structures coincide with nonassociative Malcev algebras which had appeared in the quantum mechanics of systems with non-vanishing three-cocycles, such as a point particle moving in the field of a magnetic charge. We generalize the corresponding Malcev algebras to include electric as well as magnetic charges. These structures find their classical counterpart in the theory of Poisson-Malcev algebras and their generalizations. We also study their connection to Stueckelberg's generalized Poisson brackets that do not obey the Jacobi identity and point out that nonassociative string theory with a fundamental length corresponds to a realization of his goal to find a non-linear extension of quantum mechanics with a fundamental length. Similar nonassociative structures are also known to appear in the cubic formulation of closed string field theory in terms of open string fields, leading us to conjecture a natural string-field theoretic generalization of the AdS/CFT-like (holographic) duality. (Copyright copyright 2013 WILEY-VCH Verlag GmbH and Co. KGaA, Weinheim)
A Workshop on Algebraic Design Theory and Hadamard Matrices
2015-01-01
This volume develops the depth and breadth of the mathematics underlying the construction and analysis of Hadamard matrices and their use in the construction of combinatorial designs. At the same time, it pursues current research in their numerous applications in security and cryptography, quantum information, and communications. Bridges among diverse mathematical threads and extensive applications make this an invaluable source for understanding both the current state of the art and future directions. The existence of Hadamard matrices remains one of the most challenging open questions in combinatorics. Substantial progress on their existence has resulted from advances in algebraic design theory using deep connections with linear algebra, abstract algebra, finite geometry, number theory, and combinatorics. Hadamard matrices arise in a very diverse set of applications. Starting with applications in experimental design theory and the theory of error-correcting codes, they have found unexpected and important ap...
Introduction to algebraic independence theory
Philippon, Patrice
2001-01-01
In the last five years there has been very significant progress in the development of transcendence theory. A new approach to the arithmetic properties of values of modular forms and theta-functions was found. The solution of the Mahler-Manin problem on values of modular function j(tau) and algebraic independence of numbers pi and e^(pi) are most impressive results of this breakthrough. The book presents these and other results on algebraic independence of numbers and further, a detailed exposition of methods created in last the 25 years, during which commutative algebra and algebraic geometry exerted strong catalytic influence on the development of the subject.
Rigidification of algebras over essentially algebraic theories
Rosicky, J
2012-01-01
Badzioch and Bergner proved a rigidification theorem saying that each homotopy simplicial algebra is weakly equivalent to a simplicial algebra. The question is whether this result can be extended from algebraic theories to finite limit theories and from simplicial sets to more general monoidal model categories. We will present some answers to this question.
Andrews, George E
1994-01-01
Although mathematics majors are usually conversant with number theory by the time they have completed a course in abstract algebra, other undergraduates, especially those in education and the liberal arts, often need a more basic introduction to the topic.In this book the author solves the problem of maintaining the interest of students at both levels by offering a combinatorial approach to elementary number theory. In studying number theory from such a perspective, mathematics majors are spared repetition and provided with new insights, while other students benefit from the consequent simpl
Dual number coefficient octonion algebra, field equations and conservation laws
Chanyal, B. C.; Chanyal, S. K.
2016-08-01
Starting with octonion algebra, we develop the dual number coefficient octonion (DNCO) algebra having sixteen components. DNCO forms of generalized potential, field and current equations are discussed in consistent manner. We have made an attempt to write the DNCO form of generalized Dirac-Maxwell's equations in presence of electric and magnetic charges (dyons). Accordingly, we demonstrate the work-energy theorem of classical mechanics reproducing the continuity equation for dyons in terms of DNCO algebra. Further, we discuss the DNCO form of linear momentum conservation law for dyons.
Algebraic Graph Theory Morphisms, Monoids and Matrices
Knauer, Ulrich
2011-01-01
This is a highly self-contained book about algebraic graph theory which iswritten with a view to keep the lively and unconventional atmosphere of a spoken text to communicate the enthusiasm the author feels about this subject. The focus is on homomorphisms and endomorphisms, matrices and eigenvalues. Graph models are extremely useful for almost all applications and applicators as they play an important role as structuring tools. They allow to model net structures -like roads, computers, telephones -instances of abstract data structures -likelists, stacks, trees -and functional or object orient
Duality and products in algebraic (co)homology theories
Kowalzig, N.; Kraehmer, U.
2008-01-01
The origin and interplay of products and dualities in algebraic (co)homology theories is ascribed to a ×A-Hopf algebra structure on the relevant universal enveloping algebra. This provides a unified treatment for example of results by Van den Bergh about Hochschild (co)homology and by Huebschmann about Lie–Rinehart (co)homology.
Krichever-Novikov type algebras theory and applications
Schlichenmaier, Martin
2014-01-01
Krichever and Novikov introduced certain classes of infinite dimensionalLie algebrasto extend the Virasoro algebra and its related algebras to Riemann surfaces of higher genus. The author of this book generalized and extended them toa more general setting needed by the applications. Examples of applications are Conformal Field Theory, Wess-Zumino-Novikov-Witten models, moduli space problems, integrable systems, Lax operator algebras, and deformation theory of Lie algebra. Furthermore they constitute an important class of infinite dimensional Lie algebras which due to their geometric origin are
Cuntz-Krieger algebras and a generalization of Catalan numbers
Matsumoto, Kengo
2006-01-01
We first observe that the relations of the canonical generating isometries of the Cuntz algebra ${\\cal O}_N$ are naturally related to the $N$-colored Catalan numbers. For a directed graph $G$, we generalize the Catalan numbers by using the canonical generating partial isometries of the Cuntz-Krieger algebra ${\\cal O}_{A^G}$ for the transition matrix $A^G$ of $G$. The generalized Catalan numbers $c_n^G, n=0,1,2,...$ enumerate the number of Dyck paths and oriented rooted trees for the graph $G$...
Quantum groups and algebraic geometry in conformal field theory
International Nuclear Information System (INIS)
The classification of two-dimensional conformal field theories is described with algebraic geometry and group theory. This classification is necessary in a consistent formulation of a string theory. (author). 130 refs.; 4 figs.; schemes
Theory of Generalized Bernoulli-Hurwitz Numbers in the Algebraic Functions of Cyclotomic Type
Ônishi, Yoshihiro
2003-01-01
In this paper we announce some results obtained for certain algebraic functions, which we call of cyclotomic type. The main results properly resemble von Staudt-Clausen's theorem and Kummer's congruence for the Bernoulli numbers, and such theorems for the Hurwitz numbers.
International Conference on Semigroups, Algebras and Operator Theory
Meakin, John; Rajan, A
2015-01-01
This book discusses recent developments in semigroup theory and its applications in areas such as operator algebras, operator approximations and category theory. All contributing authors are eminent researchers in their respective fields, from across the world. Their papers, presented at the 2014 International Conference on Semigroups, Algebras and Operator Theory in Cochin, India, focus on recent developments in semigroup theory and operator algebras. They highlight current research activities on the structure theory of semigroups as well as the role of semigroup theoretic approaches to other areas such as rings and algebras. The deliberations and discussions at the conference point to future research directions in these areas. This book presents 16 unpublished, high-quality and peer-reviewed research papers on areas such as structure theory of semigroups, decidability vs. undecidability of word problems, regular von Neumann algebras, operator theory and operator approximations. Interested researchers will f...
Algebraic structure and Poisson's theory of mechanico-electrical systems
Institute of Scientific and Technical Information of China (English)
Liu Hong-Ji; Tang Yi-Fa; Fu Jing-Li
2006-01-01
The algebraic structure and Poisson's integral theory of mechanico-electrical systems are studied.The Hamilton canonical equations and generalized Hamilton canonical equations and their the contravariant algebraic forms for mechanico-electrical systems are obtained.The Lie algebraic structure and the Poisson's integral theory of Lagrange mechanico-electrical systems are derived.The Lie algebraic structure admitted and Poisson's integral theory of the Lagrange-Maxwell mechanico-electrical systems are presented.Two examples are presented to illustrate these results.
Integrable maps from Galois differential algebras, Borel transforms and number sequences
Tempesta, Piergiulio
A new class of integrable maps, obtained as lattice versions of polynomial dynamical systems is introduced. These systems are obtained by means of a discretization procedure that preserves several analytic and algebraic properties of a given differential equation, in particular symmetries and integrability (see Tempesta, 2010 [40]). Our approach is based on the properties of a suitable Galois differential algebra, that we shall call a Rota algebra. A formulation of the procedure in terms of category theory is proposed. In order to render the lattice dynamics confined, a Borel regularization is also adopted. As a byproduct of the theory, a connection between number sequences and integrability is discussed.
Australian Curriculum Linked Lessons: Reasoning in Number and Algebra
Day, Lorraine
2014-01-01
The Reasoning Proficiency in number and algebra is about children making sense of the mathematics by explaining their thinking, giving reasons for their decisions and describing mathematical situations and concepts. Lorraine Day notes, children need to be able to speak, read and write the language of mathematics while investigating pattern and…
Decomposition numbers for the cyclotomic Brauer algebras in characteristic zero
Bowman, C; De Visscher, M
2012-01-01
We study the representation theory of the cyclotomic Brauer algebra via truncation to idempotent subalgebras which are isomorphic to a product of walled and classical Brauer algebras. In particular, we determine the block structure and decomposition numbers in characteristic zero.
Misconceptions in Rational Numbers, Probability, Algebra, and Geometry
Rakes, Christopher R.
2010-01-01
In this study, the author examined the relationship of probability misconceptions to algebra, geometry, and rational number misconceptions and investigated the potential of probability instruction as an intervention to address misconceptions in all 4 content areas. Through a review of literature, 5 fundamental concepts were identified that, if…
Exceptional algebraic relations for reciprocal sums of Fibonacci and Lucas numbers
Elsner, Carsten; Shimomura, Shun; Shiokawa, Iekata
2011-09-01
We discuss algebraic relations for reciprocal sums of Fibonacci and Lucas numbers. For a certain set of 12 such sums, we show that any two numbers are algebraically independent, and that any three are algebraically independent except for those in 22 exceptional triplets. We explicitly present algebraic relations for some of these exceptional cases.
Lectures on Iwahori-Hecke Algebras and their Representation Theory
Cherednik, Ivan; Howe, Roger; Lusztig, George
2002-01-01
Two basic problems of representation theory are to classify irreducible representations and decompose representations occuring naturally in some other context. Algebras of Iwahori-Hecke type are one of the tools and were, probably, first considered in the context of representation theory of finite groups of Lie type. This volume consists of notes of the courses on Iwahori-Hecke algebras and their representation theory, given during the CIME summer school which took place in 1999 in Martina Franca, Italy.
Algebraic quantum field theory
International Nuclear Information System (INIS)
The basic assumption that the complete information relevant for a relativistic, local quantum theory is contained in the net structure of the local observables of this theory results first of all in a concise formulation of the algebraic structure of the superselection theory and an intrinsic formulation of charge composition, charge conjugation and the statistics of an algebraic quantum field theory. In a next step, the locality of massive particles together with their spectral properties are wed for the formulation of a selection criterion which opens the access to the massive, non-abelian quantum gauge theories. The role of the electric charge as a superselection rule results in the introduction of charge classes which in term lead to a set of quantum states with optimum localization properties. Finally, the asymptotic observables of quantum electrodynamics are investigated within the framework of algebraic quantum field theory. (author)
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 ...
Chung, Won Sang
2014-07-01
In this paper a new q-deformed oscillator algebra with an integer number eigenvalue and a half odd integer number eigenvalue is proposed. For this algebra, the associated energy spectrum and thermodynamic behavior is discussed.
Algebraic Quantization, Good Operators and Fractional Quantum Numbers
Aldaya Valverde, Víctor; Calixto Molina, Manuel; Guerrero García, Julio
1995-01-01
The problems arising when quantizing systems with periodic boundary conditions are analysed, in an algebraic (group-) quantization scheme, and the “failure” of the Ehrenfest theorem is clarified in terms of the already defined notion of good (and bad) operators. The analysis of “constrained” Heisenberg-Weyl groups according to this quantization scheme reveals the possibility for new quantum (fractional) numbers extending those allowed for Chern classes in traditional Geometric ...
Derivations of the Moyal Algebra and Noncommutative Gauge Theories
Directory of Open Access Journals (Sweden)
Jean-Christophe Wallet
2009-01-01
Full Text Available The differential calculus based on the derivations of an associative algebra underlies most of the noncommutative field theories considered so far. We review the essential properties of this framework and the main features of noncommutative connections in the case of non graded associative unital algebras with involution. We extend this framework to the case of Z2-graded unital involutive algebras. We show, in the case of the Moyal algebra or some related Z2-graded version of it, that the derivation based differential calculus is a suitable framework to construct Yang-Mills-Higgs type models on Moyal (or related algebras, the covariant coordinates having in particular a natural interpretation as Higgs fields. We also exhibit, in one situation, a link between the renormalisable NC φ4-model with harmonic term and a gauge theory model. Some possible consequences of this are briefly discussed.
Jorgensen, PET
1987-01-01
Historically, operator theory and representation theory both originated with the advent of quantum mechanics. The interplay between the subjects has been and still is active in a variety of areas.This volume focuses on representations of the universal enveloping algebra, covariant representations in general, and infinite-dimensional Lie algebras in particular. It also provides new applications of recent results on integrability of finite-dimensional Lie algebras. As a central theme, it is shown that a number of recent developments in operator algebras may be handled in a particularly e
Rich, Albert D.; Stoutemyer, David R.
2013-01-01
Simplification of fractional powers of positive rational numbers and of sums, products and powers of such numbers is taught in beginning algebra. Such numbers can often be expressed in many ways, as this article discusses in some detail. Since they are such a restricted subset of algebraic numbers, it might seem that good simplification of them must already be implemented in all widely used computer algebra systems. However, the algorithm taught in beginning algebra uses integer factorization...
Developments and retrospectives in Lie theory algebraic methods
Penkov, Ivan; Wolf, Joseph
2014-01-01
This volume reviews and updates a prominent series of workshops in representation/Lie theory, and reflects the widespread influence of those workshops in such areas as harmonic analysis, representation theory, differential geometry, algebraic geometry, and mathematical physics. Many of the contributors have had leading roles in both the classical and modern developments of Lie theory and its applications. This Work, entitled Developments and Retrospectives in Lie Theory, and comprising 26 articles, is organized in two volumes: Algebraic Methods and Geometric and Analytic Methods. This is the Algebraic Methods volume. The Lie Theory Workshop series, founded by Joe Wolf and Ivan Penkov and joined shortly thereafter by Geoff Mason, has been running for over two decades. Travel to the workshops has usually been supported by the NSF, and local universities have provided hospitality. The workshop talks have been seminal in describing new perspectives in the field covering broad areas of current research. Mos...
Algebraic and combinatorial Brill-Noether theory
Caporaso, Lucia
2011-01-01
The interplay between algebro-geometric and combinatorial Brill-Noether theory is studied. The Brill-Noether variety of a graph shown to be non-empty if the Brill-Noether number is non-negative, as a consequence of the analogous fact for smooth projective curves. Similarly, the existence of a graph for which the Brill-Noether variety is empty implies the emptiness of the corresponding Brill-Noether variety for a general curve. The main tool is a refinement of Baker's Specialization Lemma.
Function algebras on finite sets basic course on many-valued logic and clone theory
Lau, Dietlinde
2006-01-01
Gives an introduction to the theory of function algebras. This book gives the general concepts of the Universal Algebra in order to familiarize the reader from the beginning on with the algebraic side of function algebras. It is a source on function algebras for students and researchers in mathematical logic and theoretical computer science.
Kac-Moody algebras in gravity and M-theories
Houart, Laurent
2005-01-01
The formulation of gravity and M-theories as very-extended Kac-Moody invariant theories is reviewed. Exact solutions describing intersecting extremal brane configurations smeared in all directions but one are presented. The intersection rules characterising these solutions are neatly encoded in the algebra. The existence of dualities for all G+++ and their group theoretical-origin are discussed.
Bicomplex holomorphic functions the algebra, geometry and analysis of bicomplex numbers
Luna-Elizarrarás, M Elena; Struppa, Daniele C; Vajiac, Adrian
2015-01-01
The purpose of this book is to develop the foundations of the theory of holomorphicity on the ring of bicomplex numbers. Accordingly, the main focus is on expressing the similarities with, and differences from, the classical theory of one complex variable. The result is an elementary yet comprehensive introduction to the algebra, geometry and analysis of bicomplex numbers. Around the middle of the nineteenth century, several mathematicians (the best known being Sir William Hamilton and Arthur Cayley) became interested in studying number systems that extended the field of complex numbers. Hamilton famously introduced the quaternions, a skew field in real-dimension four, while almost simultaneously James Cockle introduced a commutative four-dimensional real algebra, which was rediscovered in 1892 by Corrado Segre, who referred to his elements as bicomplex numbers. The advantages of commutativity were accompanied by the introduction of zero divisors, something that for a while dampened interest in this subject. ...
Wilson operator algebras and ground states for coupled BF theories
Tiwari, Apoorv; Chen, Xiao; Ryu, Shinsei
2016-01-01
The multi-flavor $BF$ theories in (3+1) dimensions with cubic or quartic coupling are the simplest topological quantum field theories that can describe fractional braiding statistics between loop-like topological excitations (three-loop or four-loop braiding statistics). In this paper, by canonically quantizing these theories, we study the algebra of Wilson loop and Wilson surface operators, and multiplets of ground states on three torus. In particular, by quantizing these coupled $BF$ theori...
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.
The algebra of bipartite graphs and Hurwitz numbers of seamed surfaces
Alekseevskii, A. V.; Natanzon, S. M.
2008-08-01
We extend the definition of Hurwitz numbers to the case of seamed surfaces, which arise in new models of mathematical physics, and prove that they form a system of correlators for a Klein topological field theory in the sense defined in [1]. We find the corresponding Cardy-Frobenius algebras, which yield a method for calculating the Hurwitz numbers. As a by-product, we prove that the vector space generated by the bipartite graphs with n edges possesses a natural binary operation that makes this space into a non-commutative Frobenius algebra isomorphic to the algebra of intertwining operators for a representation of the symmetric group S_n on the space generated by the set of all partitions of a set of n elements.
Operator algebra from fusion rules: The infinite number of Ising theories
Energy Technology Data Exchange (ETDEWEB)
Fuchs, J. (Nationaal Inst. voor Kernfysica en Hoge-Energiefysica (NIKHEF), Amsterdam (Netherlands). Sectie H)
1989-12-25
It is described how the fusion rules of a conformal field theory can be employed to derive differential equations for the four-point functions of the theory, and thus to determine eventually the operator product coefficients for primary fields. The results are applied to the Ising fusion rules. A set of theories possessing these fusion rules is found which is labelled by two discrete parameters. For a specific value of one of the parameters, these are the level one Spin(2m+1) Wess-Zumino-Witten theories; it is shown that they represent an infinite number of inequivalent theories. (orig.).
Classification of hypergeometric identities for pi and other logarithms of algebraic numbers.
Chudnovsky, D V; Chudnovsky, G V
1998-03-17
This paper provides transcendental and algebraic framework for the classification of identities expressing pi and other logarithms of algebraic numbers as rapidly convergent generalized hypergeometric series in rational parameters. Algebraic and arithmetic relations between values of p+1Fp hypergeometric functions and their values are analyzed. The existing identities are explained, and new exhaustive classes of new ones are presented.
The Bezout Number of Piecewise Algebraic Curves
Institute of Scientific and Technical Information of China (English)
Dian Xuan GONG; Ren Hong WANG
2012-01-01
Based on the discussion of the number of roots of univariate spline and the common zero points of two piecewise algebraic curves,a lower upbound of Bezout number of two piecewise algebraic curves on any given non-obtuse-angled triangulation is found.Bezout number of two piecewise algebraic curves on two different partitions is also discussed in this paper.
Topological insulators and C∗-algebras: Theory and numerical practice
Hastings, Matthew B.; Loring, Terry A.
2011-07-01
We apply ideas from C∗-algebra to the study of disordered topological insulators. We extract certain almost commuting matrices from the free Fermi Hamiltonian, describing band projected coordinate matrices. By considering topological obstructions to approximating these matrices by exactly commuting matrices, we are able to compute invariants quantifying different topological phases. We generalize previous two dimensional results to higher dimensions; we give a general expression for the topological invariants for arbitrary dimension and several symmetry classes, including chiral symmetry classes, and we present a detailed K-theory treatment of this expression for time reversal invariant three dimensional systems. We can use these results to show non-existence of localized Wannier functions for these systems. We use this approach to calculate the index for time-reversal invariant systems with spin-orbit scattering in three dimensions, on sizes up to 12 3, averaging over a large number of samples. The results show an interesting separation between the localization transition and the point at which the average index (which can be viewed as an "order parameter" for the topological insulator) begins to fluctuate from sample to sample, implying the existence of an unsuspected quantum phase transition separating two different delocalized phases in this system. One of the particular advantages of the C∗-algebraic technique that we present is that it is significantly faster in practice than other methods of computing the index, allowing the study of larger systems. In this paper, we present a detailed discussion of numerical implementation of our method.
Expansion of real numbers by algebraic numbers
Kaneko, Hajime
2008-01-01
In this paper we represent the fractional part of ξαn, where ξ is a nonzero real number and α is an algebraic number. By using this representation, we give new lower bounds for the distance from ξαn to the nearest integer.
Isomorphisms of Algebraic Number Fields
van Hoeij, Mark
2010-01-01
Let $\\mathbb{Q}(\\alpha)$ and $\\mathbb{Q}(\\beta)$ be algebraic number fields. We describe a new method to find (if they exist) all isomorphisms, $\\mathbb{Q}(\\beta) \\rightarrow \\mathbb{Q}(\\alpha)$. The algorithm is particularly efficient if the number of isomorphisms is one.
Polynomial factorization and nonrandomness of bits of algebraic and some transcendental numbers
R. Kannan; Lenstra, Arjen K.; Lovasz, L.
1988-01-01
It is shown that the binary expansions of algebraic numbers do not form secure pseudorandom sequences, given sufficiently many initial bits of an algebraic number, its minimal polynomial can be reconstructed, and therefore the further bits of the algebraic number can be computed. This also enables the authors to devise a simple algorithm to factorise polynomials with rational coefficients. All algorithms work in polynomial time
Algebraic formulation of quantum theory, particle identity and entanglement
Govindarajan, T. R.
2016-08-01
Quantum theory as formulated in conventional framework using statevectors in Hilbert spaces misses the statistical nature of the underlying quantum physics. Formulation using operators 𝒞∗ algebra and density matrices appropriately captures this feature in addition leading to the correct formulation of particle identity. In this framework, Hilbert space is an emergent concept. Problems related to anomalies and quantum epistemology are discussed.
LeVeque, William J
1996-01-01
This excellent textbook introduces the basics of number theory, incorporating the language of abstract algebra. A knowledge of such algebraic concepts as group, ring, field, and domain is not assumed, however; all terms are defined and examples are given - making the book self-contained in this respect.The author begins with an introductory chapter on number theory and its early history. Subsequent chapters deal with unique factorization and the GCD, quadratic residues, number-theoretic functions and the distribution of primes, sums of squares, quadratic equations and quadratic fields, diopha
Polytopes, Fibonacci numbers, Hopf algebras, and quasi-symmetric functions
Buchstaber, Viktor M.; Erokhovets, Nikolai Yu
2011-04-01
This survey is devoted to the classical problem of flag numbers of convex polytopes, and contains an exposition of results obtained on the basis of connections between the theory of convex polytopes and a number of modern directions of research. Bibliography: 62 titles.
Polytopes, Fibonacci numbers, Hopf algebras, and quasi-symmetric functions
Energy Technology Data Exchange (ETDEWEB)
Buchstaber, Viktor M; Erokhovets, Nikolai Yu
2011-04-30
This survey is devoted to the classical problem of flag numbers of convex polytopes, and contains an exposition of results obtained on the basis of connections between the theory of convex polytopes and a number of modern directions of research. Bibliography: 62 titles.
Decomposition numbers for Brauer algebras of type G(m,p,n) in characteristic zero
Bowman, C.; Cox, A.
2013-01-01
We introduce Brauer algebras associated to complex reflection groups of type $G(m,p,n)$, and study their representation theory via Clifford theory. In particular, we determine the decomposition numbers of these algebras in characteristic zero.
Computing Bits of Algebraic Numbers
Datta, Samir
2011-01-01
We initiate the complexity theoretic study of the problem of computing the bits of (real) algebraic numbers. This extends the work of Yap on computing the bits of transcendental numbers like \\pi, in Logspace. Our main result is that computing a bit of a fixed real algebraic number is in C=NC1\\subseteq Logspace when the bit position has a verbose (unary) representation and in the counting hierarchy when it has a succinct (binary) representation. Our tools are drawn from elementary analysis and numerical analysis, and include the Newton-Raphson method. The proof of our main result is entirely elementary, preferring to use the elementary Liouville's theorem over the much deeper Roth's theorem for algebraic numbers. We leave the possibility of proving non-trivial lower bounds for the problem of computing the bits of an algebraic number given the bit position in binary, as our main open question. In this direction we show very limited progress by proving a lower bound for rationals.
Many rational points coding theory and algebraic geometry
Hurt, Norman E
2003-01-01
This monograph presents a comprehensive treatment of recent results on algebraic geometry as they apply to coding theory and cryptography, with the goal the study of algebraic curves and varieties with many rational points. They book surveys recent developments on abelian varieties, in particular the classification of abelian surfaces, hyperelliptic curves, modular towers, Kloosterman curves and codes, Shimura curves and modular jacobian surfaces. Applications of abelian varieties to cryptography are presented including a discussion of hyperelliptic curve cryptosystems. The inter-relationship of codes and curves is developed building on Goppa's results on algebraic-geometry cods. The volume provides a source book of examples with relationships to advanced topics regarding Sato-Tate conjectures, Eichler-Selberg trace formula, Katz-Sarnak conjectures and Hecke operators.
Brane Topological Field Theories and Hurwitz numbers for CW-complexes
Natanzon, Sergey M.
2009-01-01
We expand Topological Field Theory on some special CW-complexes (brane complexes). This Brane Topological Field Theory one-to-one corresponds to infinite dimensional Frobenius Algebras, graduated by CW-complexes of lesser dimension. We define general and regular Hurwitz numbers of brane complexes and prove that they generate Brane Topological Field Theories. For general Hurwitz numbers corresponding algebra is an algebra of coverings of lesser dimension. For regular Hurwitz numbers the Froben...
Arithmetic Deformation Theory of Lie Algebras
Rastegar, Arash
2012-01-01
This paper is devoted to deformation theory of graded Lie algebras over $\\Z$ or $\\Z_l$ with finite dimensional graded pieces. Such deformation problems naturally appear in number theory. In the first part of the paper, we use Schlessinger criteria for functors on Artin local rings in order to obtain universal deformation rings for deformations of graded Lie algebras and their graded representations. In the second part, we use a version of Schlessinger criteria for functors on the Artinian cat...
Algebraic Graph Theory (a short course for postgraduate students and researchers)
Shokrollahi, Arman
2008-01-01
Algebraic graph theory is the branch of mathematics that studies graphs by using algebraic properties of associated matrices. More in particular, spectral graph theory studies the relation between graph properties and the spectrum of the adjacency matrix or Laplace matrix. And the theory of association schemes and coherent configurations studies the algebra generated by associated matrices.
Elements of a theory of algebraic theories
Hyland, Martin
2013-01-01
Kleisli bicategories are a natural environment in which the combinatorics involved in various notions of algebraic theory can be handled in a uniform way. The setting allows a clear account of comparisons between such notions. Algebraic theories, symmetric operads and nonsymmetric operads are treated as examples.
On the smallest number of generators and the probability of generating an algebra
Petrenko, Bogdan V.; Mazur, Marcin; Kravchenko, Rostyslav V.
2012-01-01
In this paper we study algebraic and asymptotic properties of generating sets of algebras over orders in number fields. Let $A$ be an associative algebra over an order $R$ in an algebraic number field. We assume that $A$ is a free $R$-module of finite rank. We develop a technique to compute the smallest number of generators of $A$. For example, we prove that the ring $M_3(\\mathbb{Z})^{k}$ admits two generators if and only if $k\\leq 768$. For a given positive integer $m$, we define the density...
Gravity, Gauge Theories and Geometric Algebra
Lasenby, A; Gull, S F; Lasenby, Anthony; Doran, Chris; Gull, Stephen
1998-01-01
A new gauge theory of gravity is presented. The theory is constructed in a flat background spacetime and employs gauge fields to ensure that all relations between physical quantities are independent of the positions and orientations of the matter fields. In this manner all properties of the background spacetime are removed from physics, and what remains are a set of `intrinsic' relations between physical fields. The properties of the gravitational gauge fields are derived from both classical and quantum viewpoints. Field equations are then derived from an action principle, and consistency with the minimal coupling procedure selects an action that is unique up to the possible inclusion of a cosmological constant. This in turn singles out a unique form of spin-torsion interaction. A new method for solving the field equations is outlined and applied to the case of a time-dependent, spherically-symmetric perfect fluid. A gauge is found which reduces the physics to a set of essentially Newtonian equations. These e...
ALGEBRAIC COMBINATORICS ON TRACE MONOIDS: EXTENDING NUMBER THEORY TO WALKS ON GRAPHS
Giscard, P.-L; Rochet, P
2016-01-01
Trace monoids provide a powerful tool to study graphs, viewing walks as words whose letters, the edges of the graph, obey a specific commutation rule. A particular class of traces emerges from this framework, the hikes, whose alphabet is the set of simple cycles on the graph. We show that hikes characterize undirected graphs uniquely, up to isomorphism, and satisfy remarkable algebraic properties such as the existence and unicity of a prime factorization. Because of this, the set of hikes par...
Amenability and vanishing of L^2-Betti numbers: an operator algebraic approach
Alekseev, Vadim
2011-01-01
We recast the Foelner condition in an operator algebraic setting and prove that it implies a certain dimension flatness property. Furthermore, it is proven that the Foelner condition generalizes the existing notions of amenability and that the enveloping von Neumann algebra arising from a Foelner algebra is automatically injective. As an application we show how our techniques unify the previously known results concerning vanishing of L^2-Betti numbers for amenable groups, groupoids and quantum groups and moreover provides a large class of new examples of algebras with vanishing L^2-Betti numbers.
Mathematics of the 19th century mathematical logic, algebra, number theory, probability theory
Yushkevich, A
1992-01-01
This multi-authored effort, Mathematics of the nineteenth century (to be fol lowed by Mathematics of the twentieth century), is a sequel to the History of mathematics fram antiquity to the early nineteenth century, published in three 1 volumes from 1970 to 1972. For reasons explained below, our discussion of twentieth-century mathematics ends with the 1930s. Our general objectives are identical with those stated in the preface to the three-volume edition, i. e. , we consider the development of mathematics not simply as the process of perfecting concepts and techniques for studying real-world spatial forms and quantitative relationships but as a social process as weIl. Mathematical structures, once established, are capable of a certain degree of autonomous development. In the final analysis, however, such immanent mathematical evolution is conditioned by practical activity and is either self-directed or, as is most often the case, is determined by the needs of society. Proceeding from this premise, we intend...
On logical, algebraic, and probabilistic aspects of fuzzy set theory
Mesiar, Radko
2016-01-01
The book is a collection of contributions by leading experts, developed around traditional themes discussed at the annual Linz Seminars on Fuzzy Set Theory. The different chapters have been written by former PhD students, colleagues, co-authors and friends of Peter Klement, a leading researcher and the organizer of the Linz Seminars on Fuzzy Set Theory. The book also includes advanced findings on topics inspired by Klement’s research activities, concerning copulas, measures and integrals, as well as aggregation problems. Some of the chapters reflect personal views and controversial aspects of traditional topics, while others deal with deep mathematical theories, such as the algebraic and logical foundations of fuzzy set theory and fuzzy logic. Originally thought as an homage to Peter Klement, the book also represents an advanced reference guide to the mathematical theories related to fuzzy logic and fuzzy set theory with the potential to stimulate important discussions on new research directions in the fiel...
Number Theory, Analysis and Geometry
Goldfeld, Dorian; Jones, Peter
2012-01-01
Serge Lang was an iconic figure in mathematics, both for his own important work and for the indelible impact he left on the field of mathematics, on his students, and on his colleagues. Over the course of his career, Lang traversed a tremendous amount of mathematical ground. As he moved from subject to subject, he found analogies that led to important questions in such areas as number theory, arithmetic geometry, and the theory of negatively curved spaces. Lang's conjectures will keep many mathematicians occupied far into the future. In the spirit of Lang's vast contribution to mathematics, th
Wilson operator algebras and ground states for coupled BF theories
Tiwari, Apoorv; Ryu, Shinsei
2016-01-01
The multi-flavor $BF$ theories in (3+1) dimensions with cubic or quartic coupling are the simplest topological quantum field theories that can describe fractional braiding statistics between loop-like topological excitations (three-loop or four-loop braiding statistics). In this paper, by canonically quantizing these theories, we study the algebra of Wilson loop and Wilson surface operators, and multiplets of ground states on three torus. In particular, by quantizing these coupled $BF$ theories on the three-torus, we explicitly calculate the $\\mathcal{S}$- and $\\mathcal{T}$-matrices, which encode fractional braiding statistics and topological spin of loop-like excitations, respectively. In the coupled $BF$ theories with cubic and quartic coupling, the Hopf link and Borromean ring of loop excitations, together with point-like excitations, form composite particles.
Polylogarithm identities, cluster algebras and the N=4 supersymmetric theory
Vergu, C
2015-01-01
Scattering amplitudes in N = 4 super-Yang Mills theory can be computed to higher perturbative orders than in any other four-dimensional quantum field theory. The results are interesting transcendental functions. By a hidden symmetry (dual conformal symmetry) the arguments of these functions have a geometric interpretation in terms of configurations of points in CP^3 and they turn out to be cluster coordinates. We briefly introduce cluster algebras and discuss their Poisson structure and the Sklyanin bracket. Finally, we present a 40-term trilogarithm identity which was discovered by accident while studying the physical results.
DeWolf, Melissa; Bassok, Miriam; Holyoak, Keith J
2015-05-01
To understand the development of mathematical cognition and to improve instructional practices, it is critical to identify early predictors of difficulty in learning complex mathematical topics such as algebra. Recent work has shown that performance with fractions on a number line estimation task predicts algebra performance, whereas performance with whole numbers on similar estimation tasks does not. We sought to distinguish more specific precursors to algebra by measuring multiple aspects of knowledge about rational numbers. Because fractions are the first numbers that are relational expressions to which students are exposed, we investigated how understanding the relational bipartite format (a/b) of fractions might connect to later algebra performance. We presented middle school students with a battery of tests designed to measure relational understanding of fractions, procedural knowledge of fractions, and placement of fractions, decimals, and whole numbers onto number lines as well as algebra performance. Multiple regression analyses revealed that the best predictors of algebra performance were measures of relational fraction knowledge and ability to place decimals (not fractions or whole numbers) onto number lines. These findings suggest that at least two specific components of knowledge about rational numbers--relational understanding (best captured by fractions) and grasp of unidimensional magnitude (best captured by decimals)--can be linked to early success with algebraic expressions.
Valued Graphs and the Representation Theory of Lie Algebras
Lemay, Joel
2011-01-01
Quivers (directed graphs) and 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 that do not appear in the literature. Specifically, we prove that two such species have isomorphic tensor rings if and only if they are isomorphic as "crushed" species...
Clifford algebra, geometric algebra, and applications
Lundholm, Douglas
2009-01-01
These are lecture notes for a course on the theory of Clifford algebras, with special emphasis on their wide range of applications in mathematics and physics. Clifford algebra is introduced both through a conventional tensor algebra construction (then called geometric algebra) with geometric applications in mind, as well as in an algebraically more general form which is well suited for combinatorics, and for defining and understanding the numerous products and operations of the algebra. The various applications presented include vector space and projective geometry, orthogonal maps and spinors, normed division algebras, as well as simplicial complexes and graph theory.
Huang, Yu-tin; Johansson, Henrik
2013-04-26
We show that three-dimensional supergravity amplitudes can be obtained as double copies of either three-algebra super-Chern-Simons matter theory or two-algebra super-Yang-Mills theory when either theory is organized to display the color-kinematics duality. We prove that only helicity-conserving four-dimensional gravity amplitudes have nonvanishing descendants when reduced to three dimensions, implying the vanishing of odd-multiplicity S-matrix elements, in agreement with Chern-Simons matter theory. We explicitly verify the double-copy correspondence at four and six points for N = 12,10,8 supergravity theories and discuss its validity for all multiplicity.
Cohn, Harvey
1980-01-01
""A very stimulating book ... in a class by itself."" - American Mathematical MonthlyAdvanced students, mathematicians and number theorists will welcome this stimulating treatment of advanced number theory, which approaches the complex topic of algebraic number theory from a historical standpoint, taking pains to show the reader how concepts, definitions and theories have evolved during the last two centuries. Moreover, the book abounds with numerical examples and more concrete, specific theorems than are found in most contemporary treatments of the subject.The book is divided into three parts
Algebraic K-theory and abstract homotopy theory
Blumberg, Andrew J
2007-01-01
We decompose the K-theory space of a Waldhausen category in terms of its Dwyer-Kan simplicial localization. This leads to a criterion for functors to induce equivalences of K-theory spectra that generalizes and explains many of the criteria appearing in the literature.
Algebraic structures and eigenstates for integrable collective field theories
International Nuclear Information System (INIS)
Conditions for the construction of polynomial eigen-operators for the Hamiltonian of collective string field theories are explored. Such eigen-operators arise for only one monomial potential v(x)=μx2 in the collective field theory. They form a w∞-algebra isomorphic to the algebra of vertex operators in 2d gravity. Polynomial potentials of orders only strictly larger or smaller than 2 have no non-zero-energy polynomial eigen-operators. This analysis leads us to consider a particular potential ν(x)=μx2+g/x2. A Lie algebra of polynomial eigen-operators is then constructed for this potential. It is a symmetric 2-index Lie algebra, also represented as a subalgebra of U(sl(2)). (orig.)
The role of difficulty and gender in numbers, algebra, geometry and mathematics achievement
Rabab'h, Belal Sadiq Hamed; Veloo, Arsaythamby; Perumal, Selvan
2015-05-01
This study aims to identify the role of difficulty and gender in numbers, algebra, geometry and mathematics achievement among secondary schools students in Jordan. The respondent of the study were 337 students from eight public secondary school in Alkoura district by using stratified random sampling. The study comprised of 179 (53%) males and 158 (47%) females students. The mathematics test comprises of 30 items which has eight items for numbers, 14 items for algebra and eight items for geometry. Based on difficulties among male and female students, the findings showed that item 4 (fractions - 0.34) was most difficult for male students and item 6 (square roots - 0.39) for females in numbers. For the algebra, item 11 (inequality - 0.23) was most difficult for male students and item 6 (algebraic expressions - 0.35) for female students. In geometry, item 3 (reflection - 0.34) was most difficult for male students and item 8 (volume - 0.33) for female students. Based on gender differences, female students showed higher achievement in numbers and algebra compare to male students. On the other hand, there was no differences between male and female students achievement in geometry test. This study suggest that teachers need to give more attention on numbers and algebra when teaching mathematics.
On the binary expansions of algebraic numbers
Energy Technology Data Exchange (ETDEWEB)
Bailey, David H.; Borwein, Jonathan M.; Crandall, Richard E.; Pomerance, Carl
2003-07-01
Employing concepts from additive number theory, together with results on binary evaluations and partial series, we establish bounds on the density of 1's in the binary expansions of real algebraic numbers. A central result is that if a real y has algebraic degree D > 1, then the number {number_sign}(|y|, N) of 1-bits in the expansion of |y| through bit position N satisfies {number_sign}(|y|, N) > CN{sup 1/D} for a positive number C (depending on y) and sufficiently large N. This in itself establishes the transcendency of a class of reals {summation}{sub n{ge}0} 1/2{sup f(n)} where the integer-valued function f grows sufficiently fast; say, faster than any fixed power of n. By these methods we re-establish the transcendency of the Kempner--Mahler number {summation}{sub n{ge}0}1/2{sup 2{sup n}}, yet we can also handle numbers with a substantially denser occurrence of 1's. Though the number z = {summation}{sub n{ge}0}1/2{sup n{sup 2}} has too high a 1's density for application of our central result, we are able to invoke some rather intricate number-theoretical analysis and extended computations to reveal aspects of the binary structure of z{sup 2}.
Cryptography and computational number theory
Shparlinski, Igor; Wang, Huaxiong; Xing, Chaoping; Workshop on Cryptography and Computational Number Theory, CCNT'99
2001-01-01
This volume contains the refereed proceedings of the Workshop on Cryptography and Computational Number Theory, CCNT'99, which has been held in Singapore during the week of November 22-26, 1999. The workshop was organized by the Centre for Systems Security of the Na tional University of Singapore. We gratefully acknowledge the financial support from the Singapore National Science and Technology Board under the grant num ber RP960668/M. The idea for this workshop grew out of the recognition of the recent, rapid development in various areas of cryptography and computational number the ory. The event followed the concept of the research programs at such well-known research institutions as the Newton Institute (UK), Oberwolfach and Dagstuhl (Germany), and Luminy (France). Accordingly, there were only invited lectures at the workshop with plenty of time for informal discussions. It was hoped and successfully achieved that the meeting would encourage and stimulate further research in information and computer s...
Higher algebraic K-theory an overview
Lluis-Puebla, Emilio; Gillet, Henri; Soulé, Christophe; Snaith, Victor
1992-01-01
This book is a general introduction to Higher Algebraic K-groups of rings and algebraic varieties, which were first defined by Quillen at the beginning of the 70's. These K-groups happen to be useful in many different fields, including topology, algebraic geometry, algebra and number theory. The goal of this volume is to provide graduate students, teachers and researchers with basic definitions, concepts and results, and to give a sampling of current directions of research. Written by five specialists of different parts of the subject, each set of lectures reflects the particular perspective ofits author. As such, this volume can serve as a primer (if not as a technical basic textbook) for mathematicians from many different fields of interest.
On some algebraic and combinatorial properties of the number of substructures
Buchwalder, Xavier
2010-01-01
A new algebraic structure is described, that is a useful framework in which reconstruction problems and results can be expressed. A conjecture is made which would, provided it is true, help to address the problem of symmetries. A consequence of the abstract language in which the theory is formulated is the expression of relations between the numbers of substructures of a structure (for example, the number of subgraphs of a given isomorphism class in a graph). Moreover, a generalisation similar to the one achieved by P.J.Cameron and V.B.Mnukhin of the results of edge reconstruction to invariant algebras apply in this new structure. Examples are provided to show that the result of L.Lovasz is best possible if one knows nothing about the underlying group, and that the result of V.Muller is best possible if one knows only the order of the group. Thus, reconstruction problems are set in a theory that generates relations to address them, and at the same time, provides examples establishing the sharpness of the theo...
Dudley, Underwood
2008-01-01
Ideal for a first course in number theory, this lively, engaging text requires only a familiarity with elementary algebra and the properties of real numbers. Author Underwood Dudley, who has written a series of popular mathematics books, maintains that the best way to learn mathematics is by solving problems. In keeping with this philosophy, the text includes nearly 1,000 exercises and problems-some computational and some classical, many original, and some with complete solutions. The opening chapters offer sound explanations of the basics of elementary number theory and develop the fundamenta
Super Virasoro algebra and solvable supersymmetric quantum field theories
International Nuclear Information System (INIS)
Interesting and deep relationships between super Virasoro algebras and super soliton systems (super KdV, super mKdV and super sine-Gordon equations) are investigated at both classical and quantum levels. An infinite set of conserved quantities responsible for solvability is characterized by super Virasoro algebras only. Several members of the infinite set of conserved quantities are derived explicitly. (author)
Matrix algebra and sampling theory : The case of the Horvitz-Thompson estimator
Dol, W.; Steerneman, A.G.M.; Wansbeek, T.J.
1996-01-01
Matrix algebra is a tool not commonly employed in sampling theory. The intention of this paper is to help change this situation by showing, in the context of the Horvitz-Thompson (HT) estimator, the convenience of the use of a number of matrix-algebra results. Sufficient conditions for the consisten
Splitting full matrix algebras over algebraic number fields
Ivanyos, Gábor; Schicho, Joseph
2011-01-01
Let K be an algebraic number field of degree d and discriminant D over Q. Let A be an associative algebra over K given by structure constants such that A is siomorphic to the algebra M_n(K) of n by n matrices over K for some positive integer n. Suppose that d, n and D are bounded. Then an isomorphism of A with M_n(K) can be constructed by a polynomial time ff-algorithm. (An ff-algorithm is a deterministic procedure which is allowed to call oracles for factoring integers and factoring univariate polynomials over finite fields.) As a consequence, we obtain a polynomial time ff-algorithm to compute isomorphisms of central simple algebras of bounded degree over K.
The Clifford algebra of physical space and Dirac theory
Vaz, Jayme, Jr.
2016-09-01
The claim found in many textbooks that the Dirac equation cannot be written solely in terms of Pauli matrices is shown to not be completely true. It is only true as long as the term β \\psi in the usual Dirac factorization of the Klein-Gordon equation is assumed to be the product of a square matrix β and a column matrix ψ. In this paper we show that there is another possibility besides this matrix product, in fact a possibility involving a matrix operation, and show that it leads to another possible expression for the Dirac equation. We show that, behind this other possible factorization is the formalism of the Clifford algebra of physical space. We exploit this fact, and discuss several different aspects of Dirac theory using this formalism. In particular, we show that there are four different possible sets of definitions for the parity, time reversal, and charge conjugation operations for the Dirac equation.
The Clifford algebra of physical space and Dirac theory
Vaz, Jayme, Jr.
2016-09-01
The claim found in many textbooks that the Dirac equation cannot be written solely in terms of Pauli matrices is shown to not be completely true. It is only true as long as the term β \\psi in the usual Dirac factorization of the Klein–Gordon equation is assumed to be the product of a square matrix β and a column matrix ψ. In this paper we show that there is another possibility besides this matrix product, in fact a possibility involving a matrix operation, and show that it leads to another possible expression for the Dirac equation. We show that, behind this other possible factorization is the formalism of the Clifford algebra of physical space. We exploit this fact, and discuss several different aspects of Dirac theory using this formalism. In particular, we show that there are four different possible sets of definitions for the parity, time reversal, and charge conjugation operations for the Dirac equation.
Control Systems and Number Theory
Directory of Open Access Journals (Sweden)
Fuhuo Li
2012-01-01
and PID-controllers are applied successfully in the EV control by J.-Y. Cao and B.-G. Cao 2006 and Cao et al. 2007, which we may unify in our framework. Finally, we mention some similarities between control theory and zeta-functions.
Geary, David C; Hoard, Mary K; Nugent, Lara; Rouder, Jeffrey N
2015-12-01
The relation between performance on measures of algebraic cognition and acuity of the approximate number system (ANS) and memory for addition facts was assessed for 171 ninth graders (92 girls) while controlling for parental education, sex, reading achievement, speed of numeral processing, fluency of symbolic number processing, intelligence, and the central executive component of working memory. The algebraic tasks assessed accuracy in placing x,y pairs in the coordinate plane, speed and accuracy of expression evaluation, and schema memory for algebra equations. ANS acuity was related to accuracy of placements in the coordinate plane and expression evaluation but not to schema memory. Frequency of fact retrieval errors was related to schema memory but not to coordinate plane or expression evaluation accuracy. The results suggest that the ANS may contribute to or be influenced by spatial-numerical and numerical-only quantity judgments in algebraic contexts, whereas difficulties in committing addition facts to long-term memory may presage slow formation of memories for the basic structure of algebra equations. More generally, the results suggest that different brain and cognitive systems are engaged during the learning of different components of algebraic competence while controlling for demographic and domain general abilities. PMID:26255604
Evolution algebras and their applications
Tian, Jianjun Paul
2008-01-01
Behind genetics and Markov chains, there is an intrinsic algebraic structure. It is defined as a type of new algebra: as evolution algebra. This concept lies between algebras and dynamical systems. Algebraically, evolution algebras are non-associative Banach algebras; dynamically, they represent discrete dynamical systems. Evolution algebras have many connections with other mathematical fields including graph theory, group theory, stochastic processes, dynamical systems, knot theory, 3-manifolds, and the study of the Ihara-Selberg zeta function. In this volume the foundation of evolution algebra theory and applications in non-Mendelian genetics and Markov chains is developed, with pointers to some further research topics.
Schaum's outline of theory and problems of linear algebra
Lipschutz, Seymour
2001-01-01
This third edition of the successful outline in linear algebra--which sold more than 400,000 copies in its past two editions--has been thoroughly updated to increase its applicability to the fields in which linear algebra is now essential: computer science, engineering, mathematics, physics, and quantitative analysis. Revised coverage includes new problems relevant to computer science and a revised chapter on linear equations.
Current algebra and conformal field theory on a figure eight
Balachandran, A P; Sen-Gupta, K; Marmo, G; Salomonson, P; Simoni, A; Stern, A
1993-01-01
We examine the dynamics of a free massless scalar field on a figure eight network. Upon requiring the scalar field to have a well defined value at the junction of the network, it is seen that the conserved currents of the theory satisfy Kirchhoff's law, that is that the current flowing into the junction equals the current flowing out. We obtain the corresponding current algebra and show that, unlike on a circle, the left- and right-moving currents on the figure eight do not in general commute in quantum theory. Since a free scalar field theory on a one dimensional spatial manifold exhibits conformal symmetry, it is natural to ask whether an analogous symmetry can be defined for the figure eight. We find that, unlike in the case of a manifold, the action plus boundary conditions for the network are not invariant under separate conformal transformations associated with left- and right-movers. Instead, the system is, at best, invariant under only a single set of transformations. Its conserved current is also fou...
Generalized Heisenberg algebras and k-generalized Fibonacci numbers
Schork, Matthias
2007-04-01
It is shown how some of the recent results of de Souza et al (2006 J. Phys. A: Math. Gen. 39 10415) can be generalized to describe Hamiltonians whose eigenvalues are given as k-generalized Fibonacci numbers. Here k is an arbitrary integer and the cases considered by de Souza et al correspond to k = 2.
Zatorsky, Roman
2011-01-01
A new algebraic object is introduced - recurrent fractions, which is an n-dimensional generalization of continued fractions. It is used to describe an algorithm for rational approximations of algebraic irrational numbers. Some parametrization for generalized Pell's equations is constructed.
The Hilbert polynomial and linear forms in the logarithms of algebraic numbers
Aleksentsev, Yu M.
2008-12-01
We prove a new estimate for homogeneous linear forms with integer coefficients in the logarithms of algebraic numbers. We obtain a qualitative improvement of the estimate depending on the coefficients of the linear form and the best value of the constant in the estimate in the case when the number of logarithms is not too large.
International Nuclear Information System (INIS)
Several two dimensional quantum field theory models have more than one vacuum state. An investigation of super selection sectors in two dimensions from an axiomatic point of view suggests that there should be also states, called soliton or kink states, which interpolate different vacua. Familiar quantum field theory models, for which the existence of kink states have been proven, are the Sine-Gordon and the φ42-model. In order to establish the existence of kink states for a larger class of models, we investigate the following question: Which are sufficient conditions a pair of vacuum states has to fulfill, such that an interpolating kink state can be constructed? We discuss the problem in the framework of algebraic quantum field theory which includes, for example, the P(φ)2-models. We identify a large class of vacuum states, including the vacua of the P(φ)2-models, the Yukawa2-like models and special types of Wess-Zumino models, for which there is a natural way to construct an interpolating kink state. In two space-time dimensions, massive particle states are kink states. We apply the Haag-Ruelle collision theory to kink sectors in order to analyze the asymptotic scattering states. We show that for special configurations of n kinks the scattering states describe n freely moving non interacting particles. (orig.)
Algebraic equations an introduction to the theories of Lagrange and Galois
Dehn, Edgar
2004-01-01
Meticulous and complete, this presentation of Galois' theory of algebraic equations is geared toward upper-level undergraduate and graduate students. The theories of both Lagrange and Galois are developed in logical rather than historical form. And they are given a more thorough exposition than is customary. For this reason, and also because the author concentrates on concrete applications of algebraic theory, Algebraic Equations is an excellent supplementary text, offering students a concrete introduction to the abstract principles of Galois theory. Of further value are the many numerical ex
Benschop, Nico F
2009-01-01
""Associative Digital Network Theory"" is intended for researchers at industrial laboratories, teachers and students at technical universities, in electrical engineering, computer science and applied mathematics departments, interested in new developments of modeling and designing digital networks (DN: state machines, sequential and combinational logic) in general, as a combined math/engineering discipline. As background an undergraduate level of modern applied algebra (Birkhoff-Bartee: ""Modern Applied Algebra"" - 1970, and Hartmanis-Stearns: ""Algebraic Structure of Sequential Machines"" - 1
Abstract algebra structure and application
Finston, David R
2014-01-01
This text seeks to generate interest in abstract algebra by introducing each new structure and topic via a real-world application. The down-to-earth presentation is accessible to a readership with no prior knowledge of abstract algebra. Students are led to algebraic concepts and questions in a natural way through their everyday experiences. Applications include: Identification numbers and modular arithmetic (linear) error-correcting codes, including cyclic codes ruler and compass constructions cryptography symmetry of patterns in the real plane Abstract Algebra: Structure and Application is suitable as a text for a first course on abstract algebra whose main purpose is to generate interest in the subject, or as a supplementary text for more advanced courses. The material paves the way to subsequent courses that further develop the theory of abstract algebra and will appeal to students of mathematics, mathematics education, computer science, and engineering interested in applications of algebraic concepts.
Solutions in Bosonic String Field Theory and Higher Spin Algebras in AdS
Polyakov, Dimitri
2015-01-01
We find a class of analytic solutions in open bosonic string field theory, parametrized by the chiral copy of higher spin algebra in $AdS_3$. The solutions are expressed in terms of the generating function for the products of Bell polynomials in derivatives of bosonic space-time coordinates $X^m(z)$ of the open string, which form is determined in this work. The products of these polynomials form a natural operator algebra realizations of $W_\\infty$ (area-preserving diffeomorphisms), enveloping algebra of SU(2) and higher spin algebra in $AdS_3$. The class of SFT solutions found can, in turn, be interpreted as the "enveloping of enveloping", or the enveloping of $AdS_3$ higher spin algebra. We also discuss the extensions of this class of solutions to superstring theory and their relations to higher spin algebras in higher space-time dimensions.
Homotopy Theory of Probability Spaces I: Classical independence and homotopy Lie algebras
Park, Jae-Suk
2015-01-01
This is the first installment of a series of papers whose aim is to lay a foundation for homotopy probability theory by establishing its basic principles and practices. The notion of a homotopy probability space is an enrichment of the notion of an algebraic probability space with ideas from algebraic homotopy theory. This enrichment uses a characterization of the laws of random variables in a probability space in terms of symmetries of the expectation. The laws of random variables are reinterpreted as invariants of the homotopy types of infinity morphisms between certain homotopy algebras. The relevant category of homotopy algebras is determined by the appropriate notion of independence for the underlying probability theory. This theory will be both a natural generalization and an effective computational tool for the study of classical algebraic probability spaces, while keeping the same central limit. This article is focused on the commutative case, where the laws of random variables are also described in t...
On the smallest number of generators and the probability of generating an algebra
Kravchenko, Rostyslav V; Petrenko, Bogdan V
2010-01-01
In this paper we study algebraic and asymptotic properties of generating sets of algebras over orders in number fields. Let $A$ be an associative algebra over an order $R$ in an algebraic number field. We assume that $A$ is a free $R$-module of finite rank. We develop a technique to compute the smallest number of generators of $A$. For example, we prove that the ring $\\M_3(\\mathbb{Z})^{k}$ admits two generators if and only if $k\\leq 768$. For a given positive integer $m$, we define the density of the set of all ordered $m$-tuples of elements of $A$ which generate it as an $R$-algebra. We express this density as a certain infinite product over the maximal ideals of $R$, and we interpret the resulting formula probabilistically. For example, we show that the probability that 2 random $3\\times 3$ matrices generate the ring $\\M_3(\\mathbb{Z})$ is equal to $\\dd (\\zeta(2)^2 \\zeta(3))^{-1}$, where $\\zeta$ is the Riemann zeta-function.
Experimental Number Theory, Part I : Tower Arithmetic
Gnang, Edinah K.
2011-01-01
We introduce in this section an Algebraic and Combinatorial approach to the theory of Numbers. The approach rests on the observation that numbers can be identified with familiar combinatorial objects namely rooted trees, which we shall here refer to as towers. The bijection between numbers and towers provides some insights into unexpected connexions between Number theory, combinatorics and discrete probability theory.
Mappings of Multimode Bose Algebra Preserving Number Operators
Doresic, Miroslav; Meljanac, Stjepan; Milekovic, Marijan
1994-01-01
We define a class of deformed multimode oscillator algebras which possess number operators and can be mapped to multimode Bose algebra.We construct and discuss the states in the Fock space and the corresponding number operators.
Higher theories of algebraic structures
Matsuoka, Takuo
2016-01-01
The notion of (symmetric) coloured operad or "multicategory" can be obtained from the notion of commutative algebra through a certain general process which we call "theorization" (where our term comes from an analogy with William Lawvere's notion of algebraic theory). By exploiting the inductivity in the structure of higher associativity, we obtain the notion of "$n$-theory" for every integer $n\\ge 0$, which inductively "theorizes" $n$ times, the notion of commutative algebra. As a result, (c...
Categories and Commutative Algebra
Salmon, P
2011-01-01
L. Badescu: Sur certaines singularites des varietes algebriques.- D.A. Buchsbaum: Homological and commutative algebra.- S. Greco: Anelli Henseliani.- C. Lair: Morphismes et structures algebriques.- B.A. Mitchell: Introduction to category theory and homological algebra.- R. Rivet: Anneaux de series formelles et anneaux henseliens.- P. Salmon: Applicazioni della K-teoria all'algebra commutativa.- M. Tierney: Axiomatic sheaf theory: some constructions and applications.- C.B. Winters: An elementary lecture on algebraic spaces.
Conferences on Combinatorial and Additive Number Theory
2014-01-01
This proceedings volume is based on papers presented at the Workshops on Combinatorial and Additive Number Theory (CANT), which were held at the Graduate Center of the City University of New York in 2011 and 2012. The goal of the workshops is to survey recent progress in combinatorial number theory and related parts of mathematics. The workshop attracts researchers and students who discuss the state-of-the-art, open problems, and future challenges in number theory.
Quantum field theory on toroidal topology: Algebraic structure and applications
Energy Technology Data Exchange (ETDEWEB)
Khanna, F.C., E-mail: khannaf@uvic.ca [Department of Physics and Astronomy, University of Victoria, Victoria, BC V8P 5C2 (Canada); TRIUMF, Vancouver, BC, V6T 2A3 (Canada); Malbouisson, A.P.C., E-mail: adolfo@cbpf.br [Centro Brasileiro de Pesquisas Físicas/MCT, 22290-180, Rio de Janeiro, RJ (Brazil); Malbouisson, J.M.C., E-mail: jmalboui@ufba.br [Instituto de Física, Universidade Federal da Bahia, 40210-340, Salvador, BA (Brazil); Santana, A.E., E-mail: asantana@unb.br [International Center for Condensed Matter Physics, Instituto de Física, Universidade de Brasília, 70910-900, Brasília, DF (Brazil)
2014-06-01
The development of quantum theory on a torus has a long history, and can be traced back to the 1920s, with the attempts by Nordström, Kaluza and Klein to define a fourth spatial dimension with a finite size, being curved in the form of a torus, such that Einstein and Maxwell equations would be unified. Many developments were carried out considering cosmological problems in association with particle physics, leading to methods that are useful for areas of physics, in which size effects play an important role. This interest in finite size effect systems has been increasing rapidly over the last decades, due principally to experimental improvements. In this review, the foundations of compactified quantum field theory on a torus are presented in a unified way, in order to consider applications in particle and condensed matter physics. The theory on a torus Γ{sub D}{sup d}=(S{sup 1}){sup d}×R{sup D−d} is developed from a Lie-group representation and c{sup ∗}-algebra formalisms. As a first application, the quantum field theory at finite temperature, in its real- and imaginary-time versions, is addressed by focusing on its topological structure, the torus Γ{sub 4}{sup 1}. The toroidal quantum-field theory provides the basis for a consistent approach of spontaneous symmetry breaking driven by both temperature and spatial boundaries. Then the superconductivity in films, wires and grains are analyzed, leading to some results that are comparable with experiments. The Casimir effect is studied taking the electromagnetic and Dirac fields on a torus. In this case, the method of analysis is based on a generalized Bogoliubov transformation, that separates the Green function into two parts: one is associated with the empty space–time, while the other describes the impact of compactification. This provides a natural procedure for calculating the renormalized energy–momentum tensor. Self interacting four-fermion systems, described by the Gross–Neveu and Nambu
Teaching of Real Numbers by Using the Archimedes-Cantor Approach and Computer Algebra Systems
Vorob'ev, Evgenii M.
2015-01-01
Computer technologies and especially computer algebra systems (CAS) allow students to overcome some of the difficulties they encounter in the study of real numbers. The teaching of calculus can be considerably more effective with the use of CAS provided the didactics of the discipline makes it possible to reveal the full computational potential of…
SU(1,1) and SU(2) Perelomov number coherent states: algebraic approach for general systems
Ojeda-Guillén, D; Granados, V D
2012-01-01
From the definition of the standard Perelomov coherent states (PCS) we introduce the Perelomov number coherent states (PNCS) of the su(1,1) Lie algebra for any realization of its generators. The displacement operator allows to apply a similarity transformation to the su(1,1) generators and construct a set of operators which close the su(1,1) Lie algebra, being the PNCS the basis for its unitary irreducible representation. We evaluate the Schr\\"odinger's uncertainty relationship (SUR) for a position and momentum-like operators (constructed from the Lie algebra generators) in the PNCS and show that it is minimized for the PCS. We obtain the time evolution of the PNCS and the algebra generators and prove that the SUR is minimized for t=0 and the PCS. Also, we obtain analogous results to these for the SU(2) PNCS. We emphasize that our treatment is strictly algebraic and does not depend on the explicit form of the PNCS, even we report those states.
The abc-conjecture for Algebraic Numbers
Institute of Scientific and Technical Information of China (English)
Jerzy BROWKIN
2006-01-01
The abc-conjecture for the ring of integers states that, for every ε＞ 0 and every triple of relatively prime nonzero integers (a, b, c) satisfying a + b = c, we have max(|a|, |b|, |c|) ≤ rad(abc)1+ε with a finite number of exceptions. Here the radical rad(m) is the product of all distinct prime factors of m.In the present paper we propose an abc-conjecture for the field of all algebraic numbers. It is based on the definition of the radical (in Section 1) and of the height (in Section 2) of an algebraic number.From this abc-conjecture we deduce some versions of Fermat's last theorem for the field of all algebraic numbers, and we discuss from this point of view known results on solutions of Fermat's equation in fields of small degrees over Q.
Indian Academy of Sciences (India)
Subhash J Bhatt
2006-05-01
Given an -tempered strongly continuous action of $\\mathbb{R}$ by continuous $∗$-automorphisms of a Frechet $∗$-algebra , it is shown that the enveloping -*-algebra $E(S(\\mathbb{R},A^∞,))$ of the smooth Schwartz crossed product $S(\\mathbb{R},A^∞,)$ of the Frechet algebra $A^∞$ of $C^∞$-elements of is isomorphic to the -*-crossed product $C^∗(\\mathbb{R}, E(A), )$ of the enveloping -*-algebra () of by the induced action. When is a hermitian $\\mathcal{Q}$-algebra, one gets -theory isomorphism $R K_∗(S(\\mathbb{R},A^∞,))=K_∗(C^∗(\\mathbb{R}, E(A),)$ for the representable -theory of Frechet algebras. An application to the differential structure of a *-algebra defined by densely defined differential seminorms is given.
Cohomological Donaldson-Thomas theory of a quiver with potential and quantum enveloping algebras
Davison, Ben
2016-01-01
This paper is a companion paper to 1512.08898, on the general definition of Donaldson--Thomas invariants for Jacobi algebras, or equivalently, the integrality conjecture for such algebras. In this paper we concentrate on the Hodge-theoretic aspects of the theory, and explore the structure of the Cohomological Hall algebra associated to a quiver and potential, introduced by Kontsevich and Soibelman. Via a study of the representation theory of these algebras, we introduce a perverse filtration on them, and prove that they are quantum enveloping algebras, for which the integrality theorem, and the wall crossing theorem relating DT invariants for different Bridgeland stability conditions, are a K-theoretic shadow of the existence of PBW bases.
Open and Closed String field theory interpreted in classical Algebraic Topology
Sullivan, Dennis
2003-01-01
There is an interpretation of open string field theory in algebraic topology. An interpretation of closed string field theory can be deduced from this open string theory to obtain as well the interpretation of open and closed string field theory combined.
Energy Technology Data Exchange (ETDEWEB)
Foussats, A.; Laura, R.; Zandron, O.
1986-09-01
Supergravity theories on a free differential algebra are examined and the factorisation condition is imposed leading to factorised solutions. H-gauge transformations for the pseudo-connections and pseudo-curvatures are also deduced.
International Nuclear Information System (INIS)
Supergravity theories on a free differential algebra are examined and the factorisation condition is imposed leading to factorised solutions. H-gauge transformations for the pseudo-connections and pseudo-curvatures are also deduced. (author)
Number theory arising from finite fields analytic and probabilistic theory
Knopfmacher, John
2001-01-01
""Number Theory Arising from Finite Fields: Analytic and Probabilistic Theory"" offers a discussion of the advances and developments in the field of number theory arising from finite fields. It emphasizes mean-value theorems of multiplicative functions, the theory of additive formulations, and the normal distribution of values from additive functions. The work explores calculations from classical stages to emerging discoveries in alternative abstract prime number theorems.
Singularity Theory for W-Algebra Potentials
J Gaite
1994-01-01
The Landau potentials of W3-algebra models are analyzed with algebraic-geometric methods. The number of ground states and the number of independent perturbations of every potential coincide and can be computed. This number agrees with the structure of ground states obtained in a previous paper, name
Evaluation of Tachiya's algebraic infinite products involving Fibonacci and Lucas numbers
Sondow, Jonathan
2011-09-01
In 2007, Tachiya gave necessary and sufficient conditions for the transcendence of certain infinite products involving Fibonacci numbers Fk and Lucas numbers Lk. In the present note, we explicitly evaluate two classes of his algebraic examples. Special cases are ∏ n = 1∞(1+1F2n+1) = 3/φ, ∏ n = 1/∞(1+1L2n+1) = 3-φ, where φ = (1+√5 )/2 is the golden ratio.
Characteristic Numbers of Matrix Lie Algebras
Zhang, Yu-Feng; Fan, En-Gui
2008-04-01
A notion of characteristic number of matrix Lie algebras is defined, which is devoted to distinguishing various Lie algebras that are used to generate integrable couplings of soliton equations. That is, the exact classification of the matrix Lie algebras by using computational formulas is given. Here the characteristic numbers also describe the relations between soliton solutions of the stationary zero curvature equations expressed by various Lie algebras.
A Short History of the Theory of Numbers
Alin Cristian Ioan
2014-01-01
The article treats some aspects of the history of the theory of algebraic numbers. The number theory is one of the oldest branches of mathematics which has its origins from second millennium BC, ancient documents dating from about 2000 BC being the Rhind papyrus and Golenischev papyrus, both of Egypt. Number theory is characterized by the simplicity of its fundamentals, its rigor and purity notions of its truths. One branch of number theory is the algebraic theory of numbers wh...
Algebraic K-theory and derived equivalences suggested by T-duality for torus orientifolds
Rosenberg, Jonathan
2016-01-01
We show that certain isomorphisms of (twisted) KR-groups that underlie T-dualities of torus orientifold string theories have purely algebraic analogues in terms of algebraic K-theory of real varieties and equivalences of derived categories of (twisted) coherent sheaves. The most interesting conclusion is a kind of Mukai duality in which the "dual abelian variety" to a smooth projective genus-1 curve over R with no real points is (mildly) noncommutative.
Cylindric-like algebras and algebraic logic
Ferenczi, Miklós; Németi, István
2013-01-01
Algebraic logic is a subject in the interface between logic, algebra and geometry, it has strong connections with category theory and combinatorics. Tarski’s quest for finding structure in logic leads to cylindric-like algebras as studied in this book, they are among the main players in Tarskian algebraic logic. Cylindric algebra theory can be viewed in many ways: as an algebraic form of definability theory, as a study of higher-dimensional relations, as an enrichment of Boolean Algebra theory, or, as logic in geometric form (“cylindric” in the name refers to geometric aspects). Cylindric-like algebras have a wide range of applications, in, e.g., natural language theory, data-base theory, stochastics, and even in relativity theory. The present volume, consisting of 18 survey papers, intends to give an overview of the main achievements and new research directions in the past 30 years, since the publication of the Henkin-Monk-Tarski monographs. It is dedicated to the memory of Leon Henkin.
Unified (p,q;α,γ,l)-deformation of oscillator algebra and two-dimensional conformal field theory
Energy Technology Data Exchange (ETDEWEB)
Burban, I.M., E-mail: burban@bitp.kiev.ua
2013-11-29
The unified (p,q;α,γ,l)-deformation of a number of well-known deformed oscillator algebras is introduced. The deformation is constructed by imputing new free parameters into the structure functions and by generalizing the defining relations of these algebras. The generalized Jordan–Schwinger and Holstein–Primakoff realizations of the U{sub pq}{sup αγl}(su(2)) algebra by the generalized (p,q;α,γ,l)-deformed operators are found. The generalized (p,q;α,γ,l)-deformation of the two-dimensional conformal field theory is established. By introducing the (p,q;α,γ,l)-operator product expansion (OPE) between the energy–momentum tensor and primary fields, we obtain the (p,q;α,γ,l)-deformed centerless Virasoro algebra. The two-point correlation function of the primary generalized (p,q;α,γ,l)-deformed fields is calculated.
Building Grassmann Numbers from PI-Algebras
Bentin, Ricardo M
2012-01-01
This works deals with the formal mathematical structure of so called Grassmann Numbers applied to Theoretical Physics, which is a basic concept on Berezin integration. To achieve this purpose we make use of some constructions from relative modern Polynomial Identity Algebras (PI-Algebras) applied to the special case of the Grassmann algebra.
Twisting theory for weak Hopf algebras
Institute of Scientific and Technical Information of China (English)
CHEN Ju-zhen; ZHANG Yan; WANG Shuan-hong
2008-01-01
The main aim of this paper is to study the twisting theory of weak Hopf algebras and give an equivalence between the (braided) monoidal categories of weak Hopf bimodules over the original and the twisted weak Hopf algebra to generalize the result from Oeckl (2000).
Function theory for a beltrami algebra
Directory of Open Access Journals (Sweden)
B. A. Case
1985-01-01
Full Text Available Complex functions are investigated which are solutions of an elliptic system of partial differential equations associated with a real parameter function. The functions f associated with a particualr parameter function g on a domain D form a Beltrami algebra denoted by the pair (D,g and a function theory is developed in this algebra. A strong conformality property holds for all functions in a (D,g algebra. For g≡|z|=r the algebra (D,r is that of the analytic functions.
St-Amant, Patrick
2010-01-01
We will see that key concepts of number theory can be defined for arbitrary operations. We give a generalized distributivity for hyperoperations (usual arithmetic operations and operations going beyond exponentiation) and a generalization of the fundamental theorem of arithmetic for hyperoperations. We also give a generalized definition of the prime numbers that are associated to an arbitrary n-ary operation and take a few steps toward the development of its modulo arithmetic by investigating a generalized form of Fermat's little theorem. Those constructions give an interesting way to interpret diophantine equations and we will see that the uniqueness of factorization under an arbitrary operation can be linked with the Riemann zeta function. This language of generalized primes and composites can be used to restate and extend certain problems such as the Goldbach conjecture.
Linear algebra meets Lie algebra: the Kostant-Wallach theory
Shomron, Noam; Parlett, Beresford N.
2008-01-01
In two languages, Linear Algebra and Lie Algebra, we describe the results of Kostant and Wallach on the fibre of matrices with prescribed eigenvalues of all leading principal submatrices. In addition, we present a brief introduction to basic notions in Algebraic Geometry, Integrable Systems, and Lie Algebra aimed at specialists in Linear Algebra.
Infinite number of conserved quantities and extended conformal algebra in the Thirring model
Energy Technology Data Exchange (ETDEWEB)
Inoue, K.; Odaka, K.; Omote, M.
1988-02-01
It is shown that the Thirring model has an infinite number of local conserved quantities, explicit forms of which are presented. These quantities are shown to be expressed in terms of scattering parameters. It will be shown that in this model there exists an extended symmetry algebra that includes the Virasoro algebra as its subalgebra.
Reconstruction of the number and positions of dipoles and quadrupoles using an algebraic method
Energy Technology Data Exchange (ETDEWEB)
Nara, Takaaki [University of Electro-Communications, 1-5-1, Chofugaoka, Chofu-city, Tokyo, 182-8585 (Japan)], E-mail: nara@mce.uec.ac.jp
2008-11-01
Localization of dipoles and quadrupoles is important in inverse potential analysis, since they can effectively express spatially extended sources with a small number of parmeters. This paper proposes an algebraic method for reconstruction of pole positions as well as the number of dipole-quadrupoles without providing an initial parameter guess or iterative computing forward solutions. It is also shown that a magnetoencephalography inverse problem with a source model of dipole-quadrupoles in 3D space is reduced into the same problem as in 2D space.
Teaching of real numbers by using the Archimedes-Cantor approach and computer algebra systems
Vorob'ev, Evgenii M.
2015-11-01
Computer technologies and especially computer algebra systems (CAS) allow students to overcome some of the difficulties they encounter in the study of real numbers. The teaching of calculus can be considerably more effective with the use of CAS provided the didactics of the discipline makes it possible to reveal the full computational potential of CAS. In the case of real numbers, the Archimedes-Cantor approach satisfies this requirement. The name of Archimedes brings back the exhaustion method. Cantor's name reminds us of the use of Cauchy rational sequences to represent real numbers. The usage of CAS with the Archimedes-Cantor approach enables the discussion of various representations of real numbers such as graphical, decimal, approximate decimal with precision estimates, and representation as points on a straight line. Exercises with numbers such as e, π, the golden ratio ϕ, and algebraic irrational numbers can help students better understand the real numbers. The Archimedes-Cantor approach also reveals a deep and close relationship between real numbers and continuity, in particular the continuity of functions.
Foliation theory in algebraic geometry
McKernan, James; Pereira, Jorge
2016-01-01
Featuring a blend of original research papers and comprehensive surveys from an international team of leading researchers in the thriving fields of foliation theory, holomorphic foliations, and birational geometry, this book presents the proceedings of the conference "Foliation Theory in Algebraic Geometry," hosted by the Simons Foundation in New York City in September 2013. Topics covered include: Fano and del Pezzo foliations; the cone theorem and rank one foliations; the structure of symmetric differentials on a smooth complex surface and a local structure theorem for closed symmetric differentials of rank two; an overview of lifting symmetric differentials from varieties with canonical singularities and the applications to the classification of AT bundles on singular varieties; an overview of the powerful theory of the variety of minimal rational tangents introduced by Hwang and Mok; recent examples of varieties which are hyperbolic and yet the Green-Griffiths locus is the whole of X; and a classificati...
Exploiting the Structure of Bipartite Graphs for Algebraic and Spectral Graph Theory Applications
Kunegis, Jérôme
2014-01-01
In this article, we extend several algebraic graph analysis methods to bipartite networks. In various areas of science, engineering and commerce, many types of information can be represented as networks, and thus the discipline of network analysis plays an important role in these domains. A powerful and widespread class of network analysis methods is based on algebraic graph theory, i.e., representing graphs as square adjacency matrices. However, many networks are of a very specific form that...
Geometric Algebras and Extensors
Fernandez, V. V.; Moya, A. M.; Rodrigues Jr., W. A.
2007-01-01
This is the first paper in a series (of four) designed to show how to use geometric algebras of multivectors and extensors to a novel presentation of some topics of differential geometry which are important for a deeper understanding of geometrical theories of the gravitational field. In this first paper we introduce the key algebraic tools for the development of our program, namely the euclidean geometrical algebra of multivectors Cl(V,G_{E}) and the theory of its deformations leading to met...
Hopf algebras and the combinatorics of connected graphs in quantum field theory
Mestre, Angela; Oeckl, Robert
2008-01-01
In this talk, we are concerned with the formulation and understanding of the combinatorics of time-ordered n-point functions in terms of the Hopf algebra of field operators. Mathematically, this problem can be formulated as one in combinatorics or graph theory. It consists in finding a recursive algorithm that generates all connected graphs in their Hopf algebraic representation. This representation can be used directly and efficiently in evaluating Feynman graphs as contributions to the n-po...
Kodera, Ryosuke
2016-01-01
We study quantized Coulomb branches of quiver gauge theories of Jordan type. We prove that the quantized Coulomb branch is isomorphic to the spherical graded Cherednik algebra in the unframed case, and is isomorphic to the spherical cyclotomic rational Cherednik algebra in the framed case. We also prove that the quantized Coulomb branch is a deformation of a subquotient of the Yangian of the affine $\\mathfrak{gl}(1)$.
Characteristic Numbers of Matrix Lie Algebras
Institute of Scientific and Technical Information of China (English)
QU Chang-Zheng; ZHANG Yu-Feng; LI Yan-Yan; FAN En-Gui
2008-01-01
A notion of characteristic number of matrix Lie algebras is defined, which is devoted to distinguishing various Lie Mgebras that ～re used to generate integrable couplings of soliton equations. That is, the exact classification of the matrix Lie algebras by using computational formulas is given. Here the characteristic numbers also describe the relations between soliton solutions of the stationary zero curvature equations expressed by various Lie algebras.
Operator algebras and topology
International Nuclear Information System (INIS)
These notes, based on three lectures on operator algebras and topology at the 'School on High Dimensional Manifold Theory' at the ICTP in Trieste, introduce a new set of tools to high dimensional manifold theory, namely techniques coming from the theory of operator algebras, in particular C*-algebras. These are extensively studied in their own right. We will focus on the basic definitions and properties, and on their relevance to the geometry and topology of manifolds. A central pillar of work in the theory of C*-algebras is the Baum-Connes conjecture. This is an isomorphism conjecture, as discussed in the talks of Luck, but with a certain special flavor. Nevertheless, it has important direct applications to the topology of manifolds, it implies e.g. the Novikov conjecture. In the first chapter, the Baum-Connes conjecture will be explained and put into our context. Another application of the Baum-Connes conjecture is to the positive scalar curvature question. This will be discussed by Stephan Stolz. It implies the so-called 'stable Gromov-Lawson-Rosenberg conjecture'. The unstable version of this conjecture said that, given a closed spin manifold M, a certain obstruction, living in a certain (topological) K-theory group, vanishes if and only M admits a Riemannian metric with positive scalar curvature. It turns out that this is wrong, and counterexamples will be presented in the second chapter. The third chapter introduces another set of invariants, also using operator algebra techniques, namely L2-cohomology, L2-Betti numbers and other L2-invariants. These invariants, their basic properties, and the central questions about them, are introduced in the third chapter. (author)
Derived Koszul Duality and Involutions in the Algebraic K-Theory of Spaces
Blumberg, Andrew J
2009-01-01
We interpret different constructions of algebraic $K$-theory of spaces as an instance of derived Koszul (or bar) duality and also as an instance of Morita equivalence. We relate the interplay between these two descriptions to the homotopy involution. We define a geometric analog of the Swan theory $G^{\\bZ}(\\bZ[\\pi])$ in terms of $\\Sigma^{\\infty}_{+} \\Omega X$ and show that it is the algebraic $K$-theory of the $E_{\\infty}$ ring spectrum $DX=S^{X_{+}}$.
Evaluation of Tachiya's algebraic infinite products involving Fibonacci and Lucas numbers
Sondow, Jonathan
2011-01-01
In 2007, Tachiya gave necessary and sufficient conditions for the transcendence of certain infinite products involving Fibonacci numbers $F_k$ and Lucas numbers $L_k$. In the present note, we explicitly evaluate two classes of his algebraic examples. Special cases are$$\\prod_{n=1}^{\\infty}(1+\\frac{1}{F_{2^n+1}})=\\frac{3}{\\phi}, \\qquad \\prod_{n=1}^{\\infty}(1+\\frac{1}{L_{2^n+1}})=3-\\phi\\, ,$$where $\\phi=(1+\\sqrt{5})/2$ is the golden ratio.
Algebraic K-theory of generalized schemes
DEFF Research Database (Denmark)
Anevski, Stella Victoria Desiree
Nikolai Durov has developed a generalization of conventional scheme theory in which commutative algebraic monads replace commutative unital rings as the basic algebraic objects. The resulting geometry is expressive enough to encompass conventional scheme theory, tropical algebraic geometry...... and geometry over the field with one element. It also permits the construction of important Arakelov theoretical objects, such as the completion \\Spec Z of Spec Z. In this thesis, we prove a projective bundle theorem for the eld with one element and compute the Chow rings of the generalized schemes Sp\\ec ZN......, appearing in the construction of \\Spec Z....
Higher Gauge Theories from Lie n-algebras and Off-Shell Covariantization
Carow-Watamura, Ursula; Ikeda, Noriaki; Kaneko, Yukio; Watamura, Satoshi
2016-01-01
We analyze higher gauge theories in various dimensions using a supergeometric method based on a differential graded symplectic manifold, called a QP-manifold, which is closely related to the BRST-BV formalism in gauge theories. Extensions of the Lie 2-algebra gauge structure are formulated within the Lie n-algebra induced by the QP-structure. We find that in 5 and 6 dimensions there are special extensions of the gauge algebra. In these cases, a restriction of the gauge symmetry by imposing constraints on the auxiliary gauge fields leads to a covariantized theory. As an example we show that we can obtain an off-shell covariantized higher gauge theory in 5 dimensions, which is similar to the one proposed in [1] (arxiv:1206.5643).
Equivariant K-theory and freeness of group actions on C*-algebras
Phillips, N Christopher
1987-01-01
Freeness of an action of a compact Lie group on a compact Hausdorff space is equivalent to a simple condition on the corresponding equivariant K-theory. This fact can be regarded as a theorem on actions on a commutative C*-algebra, namely the algebra of continuous complex-valued functions on the space. The successes of "noncommutative topology" suggest that one should try to generalize this result to actions on arbitrary C*-algebras. Lacking an appropriate definition of a free action on a C*-algebra, one is led instead to the study of actions satisfying conditions on equivariant K-theory - in the cases of spaces, simply freeness. The first third of this book is a detailed exposition of equivariant K-theory and KK-theory, assuming only a general knowledge of C*-algebras and some ordinary K-theory. It continues with the author's research on K-theoretic freeness of actions. It is shown that many properties of freeness generalize, while others do not, and that certain forms of K-theoretic freeness are related to ...
Algebra of Complex Vectors and Applications in Electromagnetic Theory and Quantum Mechanics
Directory of Open Access Journals (Sweden)
Kundeti Muralidhar
2015-08-01
Full Text Available A complex vector is a sum of a vector and a bivector and forms a natural extension of a vector. The complex vectors have certain special geometric properties and considered as algebraic entities. These represent rotations along with specified orientation and direction in space. It has been shown that the association of complex vector with its conjugate generates complex vector space and the corresponding basis elements defined from the complex vector and its conjugate form a closed complex four dimensional linear space. The complexification process in complex vector space allows the generation of higher n-dimensional geometric algebra from (n — 1-dimensional algebra by considering the unit pseudoscalar identification with square root of minus one. The spacetime algebra can be generated from the geometric algebra by considering a vector equal to square root of plus one. The applications of complex vector algebra are discussed mainly in the electromagnetic theory and in the dynamics of an elementary particle with extended structure. Complex vector formalism simplifies the expressions and elucidates geometrical understanding of the basic concepts. The analysis shows that the existence of spin transforms a classical oscillator into a quantum oscillator. In conclusion the classical mechanics combined with zeropoint field leads to quantum mechanics.
Singularity theory for W-algebra potentials
Gaite, J C
1993-01-01
The Landau potentials of $W_3$-algebra models are analyzed with algebraic-geometric methods. The number of ground states and the number of independent perturbations of every potential coincide and can be computed. This number agrees with the structure of ground states obtained in a previous paper, namely, as the phase structure of the IRF models of Jimbo et al. The singularities associated to these potentials are identified.
Characteristic numbers of algebraic varieties
Kotschick, D
2011-01-01
A rational linear combination of Chern numbers is an oriented diffeomorphism invariant of smooth complex projective varieties if and only if it is a linear combination of the Euler and Pontryagin numbers. In dimension at least three only multiples of the top Chern number, which is the Euler characteristic, are invariant under diffeomorphisms that are not necessarily orientation-preserving. In the space of Chern numbers there are two distinguished subspaces, one spanned by the Euler and Pontryagin numbers, the other spanned by the Hirzebruch--Todd numbers. Their intersection is the span of the Euler number and the signature.
Spin structures on algebraic curves and their applications in string theories
International Nuclear Information System (INIS)
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)
N Asili, Firouzabadi; M, K. Tavassoly; M, J. Faghihi
2015-06-01
Recently, nonlinear displaced number states (NDNSs) have been manually introduced, in which the deformation function f(n) has been artificially added to the previously well-known displaced number states (DNSs). Indeed, just a simple comparison has been performed between the standard coherent state and nonlinear coherent state for the formation of NDNSs. In the present paper, after expressing enough physical motivation of our procedure, four distinct classes of NDNSs are presented by applying algebraic and group treatments. To achieve this purpose, by considering the DNSs and recalling the nonlinear coherent states formalism, the NDNSs are logically defined through an algebraic consideration. In addition, by using a particular class of Gilmore-Perelomov-type of SU(1, 1) and a class of SU(2) coherent states, the NDNSs are introduced via group-theoretical approach. Then, in order to examine the nonclassical behavior of these states, sub-Poissonian statistics by evaluating Mandel parameter and Wigner quasi-probability distribution function associated with the obtained NDNSs are discussed, in detail.
Algebraic structures of sequences of numbers
Huang, I.-Chiau
2012-09-01
For certain sequences of numbers, commutative rings with a module structure over a non-commutative ring are constructed. Identities of these numbers are considered as realizations of algebraic relations.
Bitopological spaces theory, relations with generalized algebraic structures and applications
Dvalishvili, Badri
2005-01-01
This monograph is the first and an initial introduction to the theory of bitopological spaces and its applications. In particular, different families of subsets of bitopological spaces are introduced and various relations between two topologies are analyzed on one and the same set; the theory of dimension of bitopological spaces and the theory of Baire bitopological spaces are constructed, and various classes of mappings of bitopological spaces are studied. The previously known results as well the results obtained in this monograph are applied in analysis, potential theory, general topology, a
Proofs in Number Theory: History and Heresy.
Rowland, Tim
The domain of number theory lends itself particularly well to generic argument, presented with the intention of conveying the force and structure of a conventional generalized argument through the medium of a particular case. The potential of generic examples as a didactic tool is virtually unrecognized. Although the use of such examples has good…
Legendrian grid number one knots and augmentations of their differential algebras
Licata, Joan E
2010-01-01
In this article we study the differential graded algebra (DGA) invariant associated to Legendrian knots in tight lens spaces. Given a grid number one diagram for a knot in L(p, q), we show how to construct a special Lagrangian diagram suitable for computing the DGA invariant for the Legendrian knot specified by the diagram. We then specialize to L(p, p - 1) and show that for two families of knots, the existence of an augmentation of the DGA depends solely on the value of p.
On two notions of complexity of algebraic numbers
Bugeaud, Yann; Evertse, Jan-Hendrik
we derive new, improved lower bounds for the block complexity of an irrational algebraic number and for the number of digit changes in the b-ary expansion of an irrational algebraic number. To this end, we apply a quantitative version of the Subspace Theorem due to Evertse and Schlickewei (2002).
The $K$-theory of real graph $C*$-algebras
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...
Duality theories for Boolean algebras with operators
Givant, Steven
2014-01-01
In this new text, Steven Givant—the author of several acclaimed books, including works co-authored with Paul Halmos and Alfred Tarski—develops three theories of duality for Boolean algebras with operators. Givant addresses the two most recognized dualities (one algebraic and the other topological) and introduces a third duality, best understood as a hybrid of the first two. This text will be of interest to graduate students and researchers in the fields of mathematics, computer science, logic, and philosophy who are interested in exploring special or general classes of Boolean algebras with operators. Readers should be familiar with the basic arithmetic and theory of Boolean algebras, as well as the fundamentals of point-set topology.
Linear Algebra and Smarandache Linear Algebra
Vasantha, Kandasamy
2003-01-01
The present book, on Smarandache linear algebra, not only studies the Smarandache analogues of linear algebra and its applications, it also aims to bridge the need for new research topics pertaining to linear algebra, purely in the algebraic sense. We have introduced Smarandache semilinear algebra, Smarandache bilinear algebra and Smarandache anti-linear algebra and their fuzzy equivalents. Moreover, in this book, we have brought out the study of linear algebra and ve...
Gato-Rivera, Beatriz
2008-01-01
In 1998 the Adapted Ordering Method was developed for the study of the representation theory of the superconformal algebras in two dimensions. It allows: to determine the maximal dimension for a given type of space of singular vectors, to identify all singular vectors by only a few coefficients, to spot subsingular vectors and to set the basis for constructing embedding diagrams. In this talk I introduce the present version of the Adapted Ordering Method, published in J. Phys. A: Math. Theor. 41 (2008) 045201, which can be applied to general Lie algebras and superalgebras and their generalizations, provided they can be triangulated.
Institute of Scientific and Technical Information of China (English)
无
2004-01-01
Through most of Greek history, mathematicians concentrated on geometry, although Euclid considered the theory of numbers. The Greek mathematician Diophantus (3rd century),however, presented problems that had to be solved by what we would today call algebra. His book is thus the first algebra text.
KK-theory and Spectral Flow in von Neumann Algebras
DEFF Research Database (Denmark)
Kaad, Jens; Nest, Ryszard; Rennie, Adam
2007-01-01
We present a definition of spectral flow relative to any norm closed ideal J in any von Neumann algebra N. Given a path D(t) of selfadjoint operators in N which are invertible in N/J, the spectral flow produces a class in K_0(J). In the case when N is semifinite, the numerical spectral flow...
KK -theory and spectral flow in von Neumann algebras
DEFF Research Database (Denmark)
Kaad, Jens; Nest, Ryszard; Rennie, Adam
2012-01-01
We present a definition of spectral flow for any norm closed ideal J in any von Neumann algebra N. Given a path of selfadjoint operators in N which are invertible in N/J, the spectral flow produces a class in Ko (J). Given a semifinite spectral triple (A, H, D) relative to (N, t) with A separable...
An experimental investigation of the normality of irrational algebraic numbers
DEFF Research Database (Denmark)
Nielsen, Johan Sejr Brinch; Simonsen, Jakob Grue
2013-01-01
We investigate the distribution of digits of large prefixes of the expansion of irrational algebraic numbers to different bases. We compute 2.318 bits of the binary expansions (corresponding to 2.33.108 decimals) of the 39 least Pisot-Vijayaraghavan numbers, the 47 least known Salem numbers......, the least 20 square roots of positive integers that are not perfect squares, and 15 randomly generated algebraic irrationals. We employ these to compute the generalized serial statistics (roughly, the variant of the χ2-statistic apt for distribution of sequences of characters) of the distributions of digit......'s conjecture-that all irrational algebraic numbers are normal-is true, then it may have an empirical base: The distribution of digits in algebraic numbers appears close to equidistribution for large prefixes of their expansion. Of the 121 algebraic numbers studied, all numbers passed the maximum relative...
Threshold complexes and connections to number theory
Pakianathan, Jonathan; Winfree, Troy
2013-01-01
In this paper we study quota complexes (or equivalently in the case of scalar weights, threshold complexes) and how the topology of these quota complexes changes as the quota is changed. This problem is a simple ``linear\\" version of the general question in Morse Theory of how the topology of a space varies with a parameter. We give examples of natural and basic quota complexes where this problem frames questions about the distribution of primes, squares and divisors in number t...
The algebraic crossing number and the braid index of knots and links
Kawamuro, Keiko
2009-01-01
It has been conjectured that the algebraic crossing number of a link is uniquely determined in minimal braid representation. This conjecture is true for many classes of knots and links. The Morton-Franks-Williams inequality gives a lower bound for braid index. And sharpness of the inequality on a knot type implies the truth of the conjecture for the knot type. We prove that there are infinitely many examples of knots and links for which the inequality is not sharp but the conjecture is still true. We also show that if the conjecture is true for K and L, then it is also true for the (p,q)-cable of K and for the connect sum of K and L.
Properties of Quaternion Algebra over the Real Number Field and Zp
Institute of Scientific and Technical Information of China (English)
QIN Ying-bing
2010-01-01
The ring of quaternion over R, denoted by R[i,j,k], is a quaternion algebra. In this paper, the roots of quadratic equation with one variable in quaternion field are investigated and it is shown that it has infinitely many roots. Then the properties of quaternion algebra over Zp are discussed, and the order of its unit group is determined. Lastly, another ring isomorphism of M2(Zp) and the quaternion algebra over Zp when p satisfies some particular conditions are presented.
Izhakian, Zur; Rowen, Louis
2008-01-01
We develop the algebraic polynomial theory for "supertropical algebra," as initiated earlier over the real numbers by the first author. The main innovation there was the introduction of "ghost elements," which also play the key role in our structure theory. Here, we work somewhat more generally over an ordered monoid, and develop a theory which contains the analogs of several basic theorems of classical commutative algebra. This structure enables one to develop a Zariski-type algebraic geomet...
Virtual Betti numbers of real algebraic varieties
McCrory, Clint; Parusinski, Adam
2002-01-01
The weak factorization theorem for birational maps is used to prove that for all nonnegative i the ith mod 2 Betti number of compact nonsingular real algebraic varieties has a unique extension to a "virtual Betti number" beta_i defined for all real algebraic varieties, such that if Y is a closed subvariety of X, then beta_i(X) = beta_i(X\\Y) + beta_i(Y).
The number of simple modules of a cellular algebra
Institute of Scientific and Technical Information of China (English)
LI; Weixia; XI; Changchang
2005-01-01
Let n be a natural number, and let A be an indecomposable cellular algebra such that the spectrum of its Cartan matrix C is of theform {n, 1,..., 1}. In general, not every natural number could be the number of non-isomorphic simple modules over such a cellular algebra. Thus, two natural questions arise: (1) which numbers could be the number of non-isomorphic simple modules over such a cellular algebra A ? (2) Given such a number, is there a cellular algebra such that its Cartan matrix has the desired property ? In this paper, we shall completely answer the first question, and give a partial answer to the second question by constructing cellular algebras with the pre-described Cartan matrix.
Categorical Algebra and its Applications
1988-01-01
Categorical algebra and its applications contain several fundamental papers on general category theory, by the top specialists in the field, and many interesting papers on the applications of category theory in functional analysis, algebraic topology, algebraic geometry, general topology, ring theory, cohomology, differential geometry, group theory, mathematical logic and computer sciences. The volume contains 28 carefully selected and refereed papers, out of 96 talks delivered, and illustrates the usefulness of category theory today as a powerful tool of investigation in many other areas.
Algebraic independence results for reciprocal sums of Fibonacci and Lucas numbers
Stein, Martin
2011-09-01
Let Fn and Ln denote the Fibonacci and Lucas numbers, respectively. D. Duverney, Ke. Nishioka, Ku. Nishioka and I. Shiokawa proved that the values of the Fibonacci zeta function ζF(2s) = Σn = 1∞Fn-2s are transcendental for any s∈N using Nesterenko's theorem on Ramanujan functions P(q), Q(q), and R(q). They obtained similar results for the Lucas zeta function ζL(2s) = Σn = 1∞Ln-2s and some related series. Later, C. Elsner, S. Shimomura and I. Shiokawa found conditions for the algebraic independence of these series. In my PhD thesis I generalized their approach and treated the following problem: We investigate all subsets of { ∑ n = 1∞1/Fn2s1, ∑ n = 1∞(-1)n+1/Fn2s2, ∑ n = 1∞1/Ln2s3, ∑ n = 1∞(-1)n+1/Ln2s4:s1,s2,s3,s4∈N} and decide on their algebraic independence over Q. Actually this is a special case of a more general theorem for reciprocal sums of binary recurrent sequences.
Stoutemyer, D. R.
1977-01-01
The computer algebra language MACSYMA enables the programmer to include symbolic physical units in computer calculations, and features automatic detection of dimensionally-inhomogeneous formulas and conversion of inconsistent units in a dimensionally homogeneous formula. Some examples illustrate these features.
Spectral theory of linear operators and spectral systems in Banach algebras
Müller, Vladimir
2003-01-01
This book is dedicated to the spectral theory of linear operators on Banach spaces and of elements in Banach algebras. It presents a survey of results concerning various types of spectra, both of single and n-tuples of elements. Typical examples are the one-sided spectra, the approximate point, essential, local and Taylor spectrum, and their variants. The theory is presented in a unified, axiomatic and elementary way. Many results appear here for the first time in a monograph. The material is self-contained. Only a basic knowledge of functional analysis, topology, and complex analysis is assumed. The monograph should appeal both to students who would like to learn about spectral theory and to experts in the field. It can also serve as a reference book. The present second edition contains a number of new results, in particular, concerning orbits and their relations to the invariant subspace problem. This book is dedicated to the spectral theory of linear operators on Banach spaces and of elements in Banach alg...
Understanding geometric algebra for electromagnetic theory
Arthur, John W
2011-01-01
"This book aims to disseminate geometric algebra as a straightforward mathematical tool set for working with and understanding classical electromagnetic theory. It's target readership is anyone who has some knowledge of electromagnetic theory, predominantly ordinary scientists and engineers who use it in the course of their work, or postgraduate students and senior undergraduates who are seeking to broaden their knowledge and increase their understanding of the subject. It is assumed that the reader is not a mathematical specialist and is neither familiar with geometric algebra or its application to electromagnetic theory. The modern approach, geometric algebra, is the mathematical tool set we should all have started out with and once the reader has a grasp of the subject, he or she cannot fail to realize that traditional vector analysis is really awkward and even misleading by comparison"--Provided by publisher.
Classical and quantum Kummer shape algebras
Odzijewicz, A.; Wawreniuk, E.
2016-07-01
We study a family of integrable systems of nonlinearly coupled harmonic oscillators on the classical and quantum levels. We show that the integrability of these systems follows from their symmetry characterized by algebras, here called Kummer shape algebras. The resolution of identity for a wide class of reproducing kernels is found. A number of examples, illustrating this theory, are also presented.
Algebras with actions and automata
Directory of Open Access Journals (Sweden)
W. Kühnel
1982-01-01
Full Text Available In the present paper we want to give a common structure theory of left action, group operations, R-modules and automata of different types defined over various kinds of carrier objects: sets, graphs, presheaves, sheaves, topological spaces (in particular: compactly generated Hausdorff spaces. The first section gives an axiomatic approach to algebraic structures relative to a base category B, slightly more powerful than that of monadic (tripleable functors. In section 2 we generalize Lawveres functorial semantics to many-sorted algebras over cartesian closed categories. In section 3 we treat the structures mentioned in the beginning as many-sorted algebras with fixed scalar or input object and show that they still have an algebraic (or monadic forgetful functor (theorem 3.3 and hence the general theory of algebraic structures applies. These structures were usually treated as one-sorted in the Lawvere-setting, the action being expressed by a family of unary operations indexed over the scalars. But this approach cannot, as the one developed here, describe continuity of the action (more general: the action to be a B-morphism, which is essential for the structures mentioned above, e.g. modules for a sheaf of rings or topological automata. Finally we discuss consequences of theorem 3.3 for the structure theory of various types of automata. The particular case of algebras with fixed natural numbers object has been studied by the authors in [23].
Iachello, Francesco
2015-01-01
This course-based primer provides an introduction to Lie algebras and some of their applications to the spectroscopy of molecules, atoms, nuclei and hadrons. In the first part, it concisely presents the basic concepts of Lie algebras, their representations and their invariants. The second part includes a description of how Lie algebras are used in practice in the treatment of bosonic and fermionic systems. Physical applications considered include rotations and vibrations of molecules (vibron model), collective modes in nuclei (interacting boson model), the atomic shell model, the nuclear shell model, and the quark model of hadrons. One of the key concepts in the application of Lie algebraic methods in physics, that of spectrum generating algebras and their associated dynamic symmetries, is also discussed. The book highlights a number of examples that help to illustrate the abstract algebraic definitions and includes a summary of many formulas of practical interest, such as the eigenvalues of Casimir operators...
Elliptic Tales Curves, Counting, and Number Theory
Ash, Avner
2012-01-01
Elliptic Tales describes the latest developments in number theory by looking at one of the most exciting unsolved problems in contemporary mathematics--the Birch and Swinnerton-Dyer Conjecture. The Clay Mathematics Institute is offering a prize of 1 million to anyone who can discover a general solution to the problem. In this book, Avner Ash and Robert Gross guide readers through the mathematics they need to understand this captivating problem. The key to the conjecture lies in elliptic curves, which are cubic equations in two variables. These equations may appear simple, yet they arise from
Fields of algebraic numbers with bounded local degrees and their properties
Checcoli, Sara
2010-01-01
We provide a characterization of infinite algebraic Galois extensions of the rationals with uniformly bounded local degrees, giving a detailed proof of all the results announced in a paper by Checcoli and Zannier and obtaining relevant generalizations for them. In particular we show that that for an infinite Galois extension of the rationals the following three properties are equivalent: having uniformly bounded local degrees at every prime; having uniformly bounded local degrees at almost every prime; having Galois group of finite exponent. The proof of this result enlightens interesting connections with Zelmanov's work on the Restricted Burnside Problem. We give a formula to explicitly compute bounds for the local degrees of a infinite extension in some special cases. We relate the uniform boundedness of the local degrees to other properties: being a subfield of the compositum of all number fields of bounded degree; being generated by elements of bounded degree. We prove that the above properties are equiva...
Conference on Number Theory and Arithmetic Geometry
Silverman, Joseph; Stevens, Glenn; Modular forms and Fermat’s last theorem
1997-01-01
This volume contains expanded versions of lectures given at an instructional conference on number theory and arithmetic geometry held August 9 through 18, 1995 at Boston University. Contributor's includeThe purpose of the conference, and of this book, is to introduce and explain the many ideas and techniques used by Wiles in his proof that every (semi-stable) elliptic curve over Q is modular, and to explain how Wiles' result can be combined with Ribet's theorem and ideas of Frey and Serre to show, at long last, that Fermat's Last Theorem is true. The book begins with an overview of the complete proof, followed by several introductory chapters surveying the basic theory of elliptic curves, modular functions, modular curves, Galois cohomology, and finite group schemes. Representation theory, which lies at the core of Wiles' proof, is dealt with in a chapter on automorphic representations and the Langlands-Tunnell theorem, and this is followed by in-depth discussions of Serre's conjectures, Galois deformations, ...
Quantum exchange algebra and exact operator solution of A sub 2 -Toda field theory
Takimoto, Y; Kurokawa, H; Fujiwara, T
1999-01-01
Locality is analyzed for Toda field theories by noting novel chiral description in the conventional non-chiral formalism. It is shown that the canonicity of the interacting to free field mapping described by the classical solution is automatically guaranteed by the locality. Quantum Toda theories are investigated by applying the method of free field quantization. We give Toda exponential operators associated with fundamental weight vectors as bilinear forms of chiral fields satisfying characteristic quantum exchange algebra. It is shown that the locality leads to non-trivial relations among the R-matrix and the expansion coefficients of the exponential operators. The Toda exponentials are obtained for a A sub 2 -system by extending the algebraic method developed for the Liouville theory. The canonical commutation relations and the operatorial field equations are also examined.
Quantum Exchange Algebra and Exact Operator Solution of $A_{2}$-Toda Field Theory
Takimoto, Y; Kurokawa, H; Fujiwara, T
1999-01-01
Locality is analyzed for Toda field theories by noting novel chiral description in the conventional nonchiral formalism. It is shown that the canonicity of the interacting to free field mapping described by the classical solution is automatically guaranteed by the locality. Quantum Toda theories are investigated by applying the method of free field quantization. We give Toda exponential operators associated with fundamental weight vectors as bilinear forms of chiral fields satisfying characteristic quantum exchange algebra. It is shown that the locality leads to nontrivial relations among the ${\\cal R}$-matrix and the expansion coefficients of the exponential operators. The Toda exponentials are obtained for $A_2$-system by extending the algebraic method developed for Liouville theory. The canonical commutation relations and the operatorial field equations are also examined.
The Green formula and heredity of algebras
Institute of Scientific and Technical Information of China (English)
无
2005-01-01
[1]Green, J. A., Hall algebras, hereditary algebras and quantum groups, Invent. Math. 1995, 120: 361-377.[2]Ringel, C. M., Green's theorem on Hall algebras, in Representations of Algebras and Related Topics, CMS Conference Proceedings 19, Providence, 1996, 185-245.[3]Xiao J., Drinfeld double and Ringel-Green theory of Hall Algebras, J. Algebra, 1997, 190: 100-144.[4]Sevenhant, B., Van den Bergh, M., A relation between a conjecture of Kac and the structure of the Hall algebra,J. Pure Appl. Algebra, 2001, 160: 319-332.[5]Deng B., Xiao, J., On double Ringel-Hall algebras, J. Algebra, 2002, 251: 110-149.
Algebraic Independence of Certain Generalized Mahler Type Numbers
Institute of Scientific and Technical Information of China (English)
Yao Chen ZHU
2007-01-01
In this paper the generalized Mahler type number Mh(g;A,T) is defined, and in the case of multiplicatively dependent parameters gi, hi(1≤ I≤ s) the algebraic independence of the numbers Mh, (gi; A, T)(1≤I≤s) is proved, where A and T are certain infinite sequences of non-negative integers and of positive integers, respectively. Furthermore, the algebraic independence result on values of a certain function connected with the generalized Mahler type number and its derivatives at algebraic numbers is also given.
Quantum double actions on operator algebras and orbifold quantum field theories
International Nuclear Information System (INIS)
Starting from a local quantum field theory with an unbroken compact symmetry group G in 1+1 dimensional spacetime we construct disorder fields implementing gauge transformations on the fields (order variables) localized in a wedge region. Enlarging the local algebras by these disorder fields we obtain a nonlocal field theory, the fixpoint algebras of which under the appropriately extended action of the group G are shown to satisfy Haag duality in every simple sector. The specifically 1+1 dimensional phenomenon of violation of Haag duality of fixpoint nets is thereby clarified. In the case of a finite group G the extended theory is acted upon in a completely canonical way by the quantum double D(G) and satisfies R-matrix commutation relations as well as a Verlinde algebra. Furthermore, our methods are suitable for a concise and transparent approach to bosonization. The main technical ingredient is a strengthened version of the split property which should hold in all reasonable massive theories. In the appendices (part of) the results are extended to arbitary locally compact groups and our methods are adapted to chiral theories on the circle. (orig.)
Chudnovsky, Gregory; Cohn, Harvey; Nathanson, Melvyn
1989-01-01
The New York Number Theory Seminar was organized in 1982 to provide a forum for the presentation and discussion of recent advances in higher arithmetic and its applications. Papers included in this volume are based on the lectures presented by their authors at the Seminar at the Graduate Center of C.U.N.Y. in 1985-88. Papers in the volume cover a wide spectrum of number theoretic topics ranging from additive number theory and diophantine approximations to algebraic number theory and relations with algebraic geometry and topology.
Quantum field theories on algebraic curves. I. Additive bosons
International Nuclear Information System (INIS)
Using Serre's adelic interpretation of cohomology, we develop a 'differential and integral calculus' on an algebraic curve X over an algebraically closed field k of constants of characteristic zero, define algebraic analogues of additive multi-valued functions on X and prove the corresponding generalized residue theorem. Using the representation theory of the global Heisenberg algebra and lattice Lie algebra, we formulate quantum field theories of additive and charged bosons on an algebraic curve X. These theories are naturally connected with the algebraic de Rham theorem. We prove that an extension of global symmetries (Witten's additive Ward identities) from the k-vector space of rational functions on X to the vector space of additive multi-valued functions uniquely determines these quantum theories of additive and charged bosons.
Quantum field theories on algebraic curves. I. Additive bosons
Takhtajan, Leon A.
2013-04-01
Using Serre's adelic interpretation of cohomology, we develop a `differential and integral calculus' on an algebraic curve X over an algebraically closed field k of constants of characteristic zero, define algebraic analogues of additive multi-valued functions on X and prove the corresponding generalized residue theorem. Using the representation theory of the global Heisenberg algebra and lattice Lie algebra, we formulate quantum field theories of additive and charged bosons on an algebraic curve X. These theories are naturally connected with the algebraic de Rham theorem. We prove that an extension of global symmetries (Witten's additive Ward identities) from the k-vector space of rational functions on X to the vector space of additive multi-valued functions uniquely determines these quantum theories of additive and charged bosons.
Xu, Jing; Fan, Yi-Zheng; Tan, Ying-Ying
2014-01-01
In this paper we characterize the unique graph whose algebraic connectivity is minimum among all connected graphs with given order and fixed matching number or edge covering number, and present two lower bounds for the algebraic connectivity in terms of the matching number or edge covering number.
Symbolic Computation, Number Theory, Special Functions, Physics and Combinatorics
Ismail, Mourad
2001-01-01
These are the proceedings of the conference "Symbolic Computation, Number Theory, Special Functions, Physics and Combinatorics" held at the Department of Mathematics, University of Florida, Gainesville, from November 11 to 13, 1999. The main emphasis of the conference was Com puter Algebra (i. e. symbolic computation) and how it related to the fields of Number Theory, Special Functions, Physics and Combinatorics. A subject that is common to all of these fields is q-series. We brought together those who do symbolic computation with q-series and those who need q-series in cluding workers in Physics and Combinatorics. The goal of the conference was to inform mathematicians and physicists who use q-series of the latest developments in the field of q-series and especially how symbolic computa tion has aided these developments. Over 60 people were invited to participate in the conference. We ended up having 45 participants at the conference, including six one hour plenary speakers and 28 half hour speakers. T...
Finite automata, their algebras and grammars towards a theory of formal expressions
Büchi, J Richard
1989-01-01
The author, who died in 1984, is well-known both as a person and through his research in mathematical logic and theoretical computer science. In the first part of the book he presents the new classical theory of finite automata as unary algebras which he himself invented about 30 years ago. Many results, like his work on structure lattices or his characterization of regular sets by generalized regular rules, are unknown to a wider audience. In the second part of the book he extends the theory to general (non-unary, many-sorted) algebras, term rewriting systems, tree automata, and pushdown automata. Essentially Büchi worked independent of other rersearch, following a novel and stimulating approach. He aimed for a mathematical theory of terms, but could not finish the book. Many of the results are known by now, but to work further along this line presents a challenging research program on the borderline between universal algebra, term rewriting systems, and automata theory. For the whole book and aga...
Number theory and the periodicity of matter
Boeyens, Jan CA
2007-01-01
The main purpose of the book is to communicate a fundamental principle to the scientific world. The eventual impact of the subject matter is considered to be much wider than the readership of the preliminary accounts which have been published. The number principle at issue is known to be of wide general interest and the book has also been written to be accessible to nonspecialists. The potential readership should extend way beyond academic scientists. The discovery described in this book could be of seminal significance, also in other fields where the golden ratio is known to be of fundamental importance. The most obvious connection is with Fibonacci phylotaxis in the study of botanical growth and the number basis of DNA coding. In another context it may impinge on crystallographic periodicity and the structure of quasicrystals. These topics are beyond the scope of this book and hence it is all the more important that the power of number theory to describe physical systems be disseminated more widely.
Methods of algebraic geometry in control theory
Falb, Peter
1999-01-01
"Control theory represents an attempt to codify, in mathematical terms, the principles and techniques used in the analysis and design of control systems. Algebraic geometry may, in an elementary way, be viewed as the study of the structure and properties of the solutions of systems of algebraic equations. The aim of this book is to provide access to the methods of algebraic geometry for engineers and applied scientists through the motivated context of control theory" .* The development which culminated with this volume began over twenty-five years ago with a series of lectures at the control group of the Lund Institute of Technology in Sweden. I have sought throughout to strive for clarity, often using constructive methods and giving several proofs of a particular result as well as many examples. The first volume dealt with the simplest control systems (i.e., single input, single output linear time-invariant systems) and with the simplest algebraic geometry (i.e., affine algebraic geometry). While this is qui...
A Short History of the Theory of Numbers
Directory of Open Access Journals (Sweden)
Alin Cristian Ioan
2014-08-01
Full Text Available The article treats some aspects of the history of the theory of algebraic numbers. The number theory is one of the oldest branches of mathematics which has its origins from second millennium BC, ancient documents dating from about 2000 BC being the Rhind papyrus and Golenischev papyrus, both of Egypt. Number theory is characterized by the simplicity of its fundamentals, its rigor and purity notions of its truths. One branch of number theory is the algebraic theory of numbers which, after Hilbert, is a wonderful monument of beauty and incomparable harmony.
Representation Theory of Quantized Enveloping Algebras with Interpolating Real Structure
Directory of Open Access Journals (Sweden)
Kenny De Commer
2013-12-01
Full Text Available Let g be a compact simple Lie algebra. We modify the quantized enveloping ∗-algebra associated to g by a real-valued character on the positive part of the root lattice. We study the ensuing Verma module theory, and the associated quotients of these modified quantized enveloping ∗-algebras. Restricting to the locally finite part by means of a natural adjoint action, we obtain in particular examples of quantum homogeneous spaces in the operator algebraic setting.
Spectral numbers in Floer theories
Usher, Michael
2007-01-01
The chain complexes underlying Floer homology theories typically carry a real-valued filtration, allowing one to associate to each Floer homology class a spectral number defined as the infimum of the filtration levels of chains representing that class. These spectral numbers have been studied extensively in the case of Hamiltonian Floer homology by Oh, Schwarz, and others. We prove that the spectral number associated to any nonzero Floer homology class is always finite, and that the infimum in the definition of the spectral number is always attained. In the Hamiltonian case, this implies that what is known as the "nondegenerate spectrality" axiom holds on all closed symplectic manifolds. Our proofs are entirely algebraic and rather elementary, and apply to any Floer-type theory (including Novikov homology) satisfying certain standard formal properties provided that one works with coefficients in a Novikov ring whose degree-zero part \\Lambda_0 is a field. The key ingredient is a theorem about linear transforma...
Field algebra, Hilbert space and observables in two-dimensional higher-derivative field theories
International Nuclear Information System (INIS)
We discuss structural aspects related to the field algebra of two-dimensional higher-derivative quantum field theories. We present general selection criteria for the proper field subalgebra that generates Wightman functions satisfying the asymptotic factorization property and which define a semi-definite inner product Hilbert space. The positive definite inner product Hilbert space, which contains as a subspace of states the general Wightman functions of the corresponding standard canonical models, is a quotient space obtained by equivalence classes. For higher-derivative local gauge theories, besides the Lowenstein-Swieca condition, an additional condition must be imposed on the field algebra in order to obtain a physical subspace of states satisfying a reasonable set of physically meaningful axioms. (author)
Vazzana, Anthony; Garth, David
2007-01-01
One of the oldest branches of mathematics, number theory is a vast field devoted to studying the properties of whole numbers. Offering a flexible format for a one- or two-semester course, Introduction to Number Theory uses worked examples, numerous exercises, and two popular software packages to describe a diverse array of number theory topics.
Schneider, Hans
1989-01-01
Linear algebra is one of the central disciplines in mathematics. A student of pure mathematics must know linear algebra if he is to continue with modern algebra or functional analysis. Much of the mathematics now taught to engineers and physicists requires it.This well-known and highly regarded text makes the subject accessible to undergraduates with little mathematical experience. Written mainly for students in physics, engineering, economics, and other fields outside mathematics, the book gives the theory of matrices and applications to systems of linear equations, as well as many related t
Lee, Kerry; Khng, Kiat Hui; Ng, Swee Fong; Ng Lan Kong, Jeremy
2013-01-01
In Singapore, primary school students are taught to use bar diagrams to represent known and unknown values in algebraic word problems. However, little is known about students' understanding of these graphical representations. We investigated whether students use and think of the bar diagrams in a concrete or a more abstract fashion. We also…
Directory of Open Access Journals (Sweden)
Ion C. Baianu
2009-04-01
Full Text Available A novel algebraic topology approach to supersymmetry (SUSY and symmetry breaking in quantum field and quantum gravity theories is presented with a view to developing a wide range of physical applications. These include: controlled nuclear fusion and other nuclear reaction studies in quantum chromodynamics, nonlinear physics at high energy densities, dynamic Jahn-Teller effects, superfluidity, high temperature superconductors, multiple scattering by molecular systems, molecular or atomic paracrystal structures, nanomaterials, ferromagnetism in glassy materials, spin glasses, quantum phase transitions and supergravity. This approach requires a unified conceptual framework that utilizes extended symmetries and quantum groupoid, algebroid and functorial representations of non-Abelian higher dimensional structures pertinent to quantized spacetime topology and state space geometry of quantum operator algebras. Fourier transforms, generalized Fourier-Stieltjes transforms, and duality relations link, respectively, the quantum groups and quantum groupoids with their dual algebraic structures; quantum double constructions are also discussed in this context in relation to quasi-triangular, quasi-Hopf algebras, bialgebroids, Grassmann-Hopf algebras and higher dimensional algebra. On the one hand, this quantum algebraic approach is known to provide solutions to the quantum Yang-Baxter equation. On the other hand, our novel approach to extended quantum symmetries and their associated representations is shown to be relevant to locally covariant general relativity theories that are consistent with either nonlocal quantum field theories or local bosonic (spin models with the extended quantum symmetry of entangled, 'string-net condensed' (ground states.
GOLDMAN ALGEBRA, OPERS AND THE SWAPPING ALGEBRA
Labourie, François
2012-01-01
We define a Poisson Algebra called the {\\em swapping algebra} using the intersection of curves in the disk. We interpret a subalgebra of the fraction algebra of the swapping algebra -- called the {\\em algebra of multifractions} -- as an algebra of functions on the space of cross ratios and thus as an algebra of functions on the Hitchin component as well as on the space of $\\mathsf{SL}_n(\\mathbb R)$-opers with trivial holonomy. We relate this Poisson algebra to the Atiyah--Bott--Goldman symple...
Wassermann, Antony
1998-01-01
Fusion of positive energy representations is defined using Connes' tensor product for bimodules over a von Neumann algebra. Fusion is computed using the analytic theory of primary fields and explicit solutions of the Knizhnik-Zamolodchikov equation.
Wild Pfister forms over Henselian fields, K-theory, and conic division algebras
Garibaldi, Skip
2010-01-01
The epicenter of this paper concerns Pfister quadratic forms over a field $F$ with a Henselian discrete valuation. All characteristics are considered but we focus on the most complicated case where the residue field has characteristic 2 but $F$ does not. We also prove results about round quadratic forms, composition algebras, generalizations of composition algebras we call conic algebras, and central simple associative symbol algebras. Finally we give relationships between these objects and Kato's filtration on the Milnor $K$-groups of $F$.
The number of arrows in the quiver of tilting modules over a path algebra of type $A$ and $D$
Kase, Ryoichi
2011-01-01
Happel and Unger defined a partial order on the set of basic tilting modules. The tilting quiver is the Hasse diagram of the poset of basic tilting modules. We determine the number of arrows in the tilting quiver over a path algebra of type $A$ or $D$.
A pseudoexponentiation-like structure on the algebraic numbers
Mantova, Vincenzo
2012-01-01
Pseudoexponential fields are exponential fields similar to complex exponentiation satisfying the Schanuel Property, which is the abstract statement of Schanuel's Conjecture, and an adapted form of existential closure. Here we show that if we remove the Schanuel Property and just care about existential closure, it is possible to create several existentially closed exponential functions on the algebraic numbers that still have similarities with complex exponentiation. The main difficulties are related to the arithmetic of algebraic numbers, and can be overcome with known results about specialisations of multiplicatively independent functions on algebraic varieties.
Matsumoto, Kohji
2002-01-01
The book includes several survey articles on prime numbers, divisor problems, and Diophantine equations, as well as research papers on various aspects of analytic number theory such as additive problems, Diophantine approximations and the theory of zeta and L-function Audience Researchers and graduate students interested in recent development of number theory
Algebra and Theory of Order-Deterministic Pomsets
Rensink, A.
1996-01-01
This paper is about partially ordered multisets (pomsets for short). We investigate a particular class of pomsets that we call order-deterministic, properly including all partially ordered sets, which satisfies a number of interesting properties: among other things, it forms a distributive lattice u
Introduction to the theory of abstract algebras
Pierce, Richard S
2014-01-01
Intended for beginning graduate-level courses, this text introduces various aspects of the theory of abstract algebra. The book is also suitable as independent reading for interested students at that level as well as a primary source for a one-semester course that an instructor may supplement to expand to a full year. Author Richard S. Pierce, a Professor of Mathematics at Seattle's University of Washington, places considerable emphasis on applications of the theory and focuses particularly on lattice theory.After a preliminary review of set theory, the treatment presents the basic definitions
Saito, Asaki; Yasutomi, Shin-ichi; Tamura, Jun-ichi; Ito, Shunji
2015-06-01
We introduce a true orbit generation method enabling exact simulations of dynamical systems defined by arbitrary-dimensional piecewise linear fractional maps, including piecewise linear maps, with rational coefficients. This method can generate sufficiently long true orbits which reproduce typical behaviors (inherent behaviors) of these systems, by properly selecting algebraic numbers in accordance with the dimension of the target system, and involving only integer arithmetic. By applying our method to three dynamical systems—that is, the baker's transformation, the map associated with a modified Jacobi-Perron algorithm, and an open flow system—we demonstrate that it can reproduce their typical behaviors that have been very difficult to reproduce with conventional simulation methods. In particular, for the first two maps, we show that we can generate true orbits displaying the same statistical properties as typical orbits, by estimating the marginal densities of their invariant measures. For the open flow system, we show that an obtained true orbit correctly converges to the stable period-1 orbit, which is inherently possessed by the system.
Nekrasov and Argyres-Douglas theories in spherical Hecke algebra representation
Rim, Chaiho
2016-01-01
AGT conjecture connects Nekrasov instanton partition function of 4D quiver gauge theory with 2D Liouville conformal blocks. We re-investigate this connection using the central extension of spherical Hecke algebra in q-coordinate representation, q being the instanton expansion parameter. Based on AFLT basis together with Matsuo's interwiner we construct gauge conformal state and demonstrate its equivalence to the Liouville conformal state with careful attention to the proper scaling behavior of the state. Using the colliding limit of regular states, we obtain the formal expression of irregular conformal states corresponding to Argyres-Douglas theory which involves summation of functions over Young diagrams.
Algebraic independence of p-adic numbers
Nesterenko, Yu V.
2008-06-01
We prove lower bounds for the transcendence degree of fields generated by values of the p-adic exponential function. In particular, we estimate the transcendence degree of the field \\mathbb Q(e^{\\alpha_1},\\dots,e^{\\alpha_d}), where \\alpha_1,\\dots,\\alpha_d are algebraic (over the field of rational numbers) p-adic numbers that form a basis of a finite extension of \\mathbb Q.
A note on Quarks and numbers theory
Hage-Hassan, Mehdi
2013-01-01
We express the basis vectors of Cartan fundamental representations of unitary groups by binary numbers. We determine the expression of Gel'fand basis of SU (3) based on the usual subatomic quarks notations and we represent it by binary numbers. By analogy with the mesons and quarks we find a new property of prime numbers.
Representations of Conformal Nets, Universal C*-Algebras and K-Theory
Carpi, Sebastiano; Conti, Roberto; Hillier, Robin; Weiner, Mihály
2013-05-01
We study the representation theory of a conformal net {{A}} on S 1 from a K-theoretical point of view using its universal C*-algebra {C^*({A})}. We prove that if {{A}} satisfies the split property then, for every representation π of {{A}} with finite statistical dimension, {π(C^*({A}))} is weakly closed and hence a finite direct sum of type I∞ factors. We define the more manageable locally normal universal C*-algebra {C_ln^*({A})} as the quotient of {C^*({A})} by its largest ideal vanishing in all locally normal representations and we investigate its structure. In particular, if {{A}} is completely rational with n sectors, then {C_ln^*({A})} is a direct sum of n type I∞ factors. Its ideal {{K}_{A}} of compact operators has nontrivial K-theory, and we prove that the DHR endomorphisms of {C^*({A})} with finite statistical dimension act on {{K}_{A}}, giving rise to an action of the fusion semiring of DHR sectors on {K_0({K}_{A})}. Moreover, we show that this action corresponds to the regular representation of the associated fusion algebra.
Solvable Lie Algebras in Type IIA, Type IIB and M Theories
Andrianopoli, Laura; Ferrara, Sergio; Fré, P; Minasian, R; Trigiante, M
1997-01-01
We study some applications of solvable Lie algebras in type IIA, type IIB and M theories. RR and NS generators find a natural geometric interpretation in this framework. Special emphasis is given to the counting of the abelian nilpotent ideals (translational symmetries of the scalar manifolds) in arbitrary D dimensions. These are seen to be related, using Dynkin diagram techniques, to one-form counting in D+1 dimensions. A recipy for gauging isometries in this framework is also presented. In particular, we list the gauge groups both for compact and translational isometries. The former agree with some results already existing in gauged supergravity. The latter should be possibly related to the study of partial supersymmetry breaking, as suggested by a similar role played by solvable Lie algebras in N=2 gauged supergravity.
S-duality and the prepotential of N=2* theories (I): the ADE algebras
Billo', M; Fucito, F; Lerda, A; Morales, J F
2015-01-01
The prepotential of N=2* supersymmetric theories with unitary gauge groups in an Omega-background satisfies a modular anomaly equation that can be recursively solved order by order in an expansion for small mass. By requiring that S-duality acts on the prepotential as a Fourier transform we generalise this result to N=2* theories with gauge algebras of the D and E type and show that their prepotentials can be written in terms of quasi-modular forms of SL(2,Z). The results are checked against microscopic multi-instanton calculus based on localization for the A and D series and reproduce the known 1-instanton prepotential of the pure N=2 theories for any gauge group of ADE type. Our results can also be used to obtain the multi-instanton terms in the exceptional theories for which the microscopic instanton calculus and the ADHM construction are not available.
S-duality and the prepotential in N={2}^{star } theories (I): the ADE algebras
Billó, M.; Frau, M.; Fucito, F.; Lerda, A.; Morales, J. F.
2015-11-01
The prepotential of N={2}^{star } supersymmetric theories with unitary gauge groups in an Ω background satisfies a modular anomaly equation that can be recursively solved order by order in an expansion for small mass. By requiring that S-duality acts on the prepotential as a Fourier transform we generalise this result to N={2}^{star } theories with gauge algebras of the D and E type and show that their prepotentials can be written in terms of quasi-modular forms of SL(2, {Z}) . The results are checked against microscopic multi-instanton calculus based on localization for the A and D series and reproduce the known 1-instanton prepotential of the pure N=2 theories for any gauge group of ADE type. Our results can also be used to obtain the multi-instanton terms in the exceptional theories for which the microscopic instanton calculus and the ADHM construction are not available.
Dimer models and Calabi-Yau algebras
Broomhead, Nathan
2008-01-01
In this thesis we study dimer models, as introduced in string theory, which give a way of writing down a class of non-commutative `superpotential' algebras. Some examples are 3-dimensional Calabi-Yau algebras, as defined by Ginzburg, and some are not. We consider two types of `consistency' condition on dimer models, and show that a `geometrically consistent' model is `algebraically consistent'. Finally we prove that the algebras obtained from algebraically consistent dimer models are 3-dimensional Calabi-Yau algebras.
K-theory of Continuous Deformations of C*-algebras
Institute of Scientific and Technical Information of China (English)
Takahiro SUDO
2007-01-01
We study K-theory of continuous deformations of C*-algebras to obtain that their K-theory is the same as that of the fiber at zero. We also consider continuous or discontinuous deformations of Cuntz and Toeplitz algebras.
Semi-Hopf Algebra and Supersymmetry
Gunara, Bobby Eka
1999-01-01
We define a semi-Hopf algebra which is more general than a Hopf algebra. Then we construct the supersymmetry algebra via the adjoint action on this semi-Hopf algebra. As a result we have a supersymmetry theory with quantum gauge group, i.e., quantised enveloping algebra of a simple Lie algebra. For the example, we construct the Lagrangian N=1 and N=2 supersymmetry.
Niederreiter, Harald
2015-01-01
This textbook effectively builds a bridge from basic number theory to recent advances in applied number theory. It presents the first unified account of the four major areas of application where number theory plays a fundamental role, namely cryptography, coding theory, quasi-Monte Carlo methods, and pseudorandom number generation, allowing the authors to delineate the manifold links and interrelations between these areas. Number theory, which Carl-Friedrich Gauss famously dubbed the queen of mathematics, has always been considered a very beautiful field of mathematics, producing lovely results and elegant proofs. While only very few real-life applications were known in the past, today number theory can be found in everyday life: in supermarket bar code scanners, in our cars’ GPS systems, in online banking, etc. Starting with a brief introductory course on number theory in Chapter 1, which makes the book more accessible for undergraduates, the authors describe the four main application areas in Chapters...
Introduction to the representation theory of algebras
Barot, Michael
2015-01-01
This book gives a general introduction to the theory of representations of algebras. It starts with examples of classification problems of matrices under linear transformations and explains the three common setups: representation of quivers, modules over algebras and additive functors over certain categories. The main part is devoted to (i) module categories, presenting the unicity of the decomposition into indecomposable modules, the Auslander–Reiten theory and the technique of knitting; (ii) the use of combinatorial tools such as dimension vectors and integral quadratic forms; and (iii) deeper theorems such as Gabriel‘s Theorem, the trichotomy and the Theorem of Kac – all accompanied by further examples. Each section includes exercises to facilitate understanding. By keeping the proofs as basic and comprehensible as possible and introducing the three languages at the beginning, this book is suitable for readers from the advanced undergraduate level onwards and enables them to consult related, specifi...
On the algebraic numbers computable by some generalized Ehrenfest urns
Albenque, Marie
2011-01-01
This article deals with some stochastic population protocols, motivated by theoretical aspects of distributed computing. We modelize the problem by a large urn of black and white balls from which at every time unit a fixed number of balls are drawn and their colors are changed according to the number of black balls among them. When the time and the number of balls both tend to infinity the proportion of black balls converges to an algebraic number. We prove that, surprisingly enough, not every algebraic number can be "computed" this way.
An adventurer's guide to number theory
Friedberg, Richard
1995-01-01
In this delightful guide, a noted mathematician and teacher offers a witty, historically oriented introduction to number theory, dealing with properties of numbers and with numbers as abstract concepts. Written for readers with an understanding of arithmetic and beginning algebra, the book presents the classical discoveries of number theory, including the work of Pythagoras, Euclid, Diophantus, Fermat, Euler, Lagrange and Gauss.Unlike many authors, however, Mr. Friedberg encourages students to think about the imaginative, playful qualities of numbers as they consider such subjects as primes
On the number of finite algebraic structures
Aichinger, Erhard; McKenzie, Ralph
2011-01-01
We prove that every clone of operations on a finite set A, if it contains a Malcev operation, is finitely related -- i.e., identical with the clone of all operations respecting R for some finitary relation R over A. It follows that for a fixed finite set A, the set of all such Malcev clones is countable. This completes the solution of a problem that was first formulated in 1980, or earlier: how many Malcev clones can finite sets support? More generally, we prove that every finite algebra with few subpowers has a finitely related clone of term operations. Hence modulo term equivalence and a renaming of the elements, there are only countably many finite algebras with few subpowers, and thus only countably many finite algebras with a Malcev term.
Decomposition Theory in the Teaching of Elementary Linear Algebra.
London, R. R.; Rogosinski, H. P.
1990-01-01
Described is a decomposition theory from which the Cayley-Hamilton theorem, the diagonalizability of complex square matrices, and functional calculus can be developed. The theory and its applications are based on elementary polynomial algebra. (KR)
Computer algebra in quantum field theory integration, summation and special functions
Schneider, Carsten
2013-01-01
The book focuses on advanced computer algebra methods and special functions that have striking applications in the context of quantum field theory. It presents the state of the art and new methods for (infinite) multiple sums, multiple integrals, in particular Feynman integrals, difference and differential equations in the format of survey articles. The presented techniques emerge from interdisciplinary fields: mathematics, computer science and theoretical physics; the articles are written by mathematicians and physicists with the goal that both groups can learn from the other field, including
Algebraic K-theory of strict ring spectra
Rognes, John
2014-01-01
We view strict ring spectra as generalized rings. The study of their algebraic K-theory is motivated by its applications to the automorphism groups of compact manifolds. Partial calculations of algebraic K-theory for the sphere spectrum are available at regular primes, but we seek more conceptual answers in terms of localization and descent properties. Calculations for ring spectra related to topological K-theory suggest the existence of a motivic cohomology theory for strictly commutative ri...
Realization of $W_{1+\\infty}$ and Virasoro Algebras in Supersymmetric Theories on Four Manifolds
Johansen, Andrei
1994-01-01
We demonstrate that a supersymmetric theory twisted on a K\\"ahler four manifold $M=\\Sigma_1 \\times \\Sigma_2 ,$ where $\\Sigma_{1,2}$ are 2D Riemann surfaces, possesses a "left-moving" conformal stress tensor on $\\Sigma_1$ ($\\Sigma_2$) in the BRST cohomology. The central charge of the Virasoro algebra has a purely geometric origin and is proportional to the Euler characteristic of the $\\Sigma_2$ ($\\Sigma_1$) surface. This structure is shown to be invariant under renormalization group. We also g...
Algebraic K-theory of crystallographic groups the three-dimensional splitting case
Farley, Daniel Scott
2014-01-01
The Farrell-Jones isomorphism conjecture in algebraic K-theory offers a description of the algebraic K-theory of a group using a generalized homology theory. In cases where the conjecture is known to be a theorem, it gives a powerful method for computing the lower algebraic K-theory of a group. This book contains a computation of the lower algebraic K-theory of the split three-dimensional crystallographic groups, a geometrically important class of three-dimensional crystallographic group, representing a third of the total number. The book leads the reader through all aspects of the calculation. The first chapters describe the split crystallographic groups and their classifying spaces. Later chapters assemble the techniques that are needed to apply the isomorphism theorem. The result is a useful starting point for researchers who are interested in the computational side of the Farrell-Jones isomorphism conjecture, and a contribution to the growing literature in the field.
Won, Chang-Hee; Michel, Anthony N
2008-01-01
This volume - dedicated to Michael K. Sain on the occasion of his seventieth birthday - is a collection of chapters covering recent advances in stochastic optimal control theory and algebraic systems theory. Written by experts in their respective fields, the chapters are thematically organized into four parts: Part I focuses on statistical control theory, where the cost function is viewed as a random variable and performance is shaped through cost cumulants. In this respect, statistical control generalizes linear-quadratic-Gaussian and H-infinity control. Part II addresses algebraic systems th
Turner doubles and generalized Schur algebras
Evseev, Anton; Kleshchev, Alexander
2016-01-01
Turner's Conjecture describes all blocks of symmetric groups and Hecke algebras up to derived equivalence in terms of certain double algebras. With a view towards a proof of this conjecture, we develop a general theory of Turner doubles. In particular, we describe doubles as explicit maximal symmetric subalgebras of certain generalized Schur algebras and establish a Schur-Weyl duality with wreath product algebras.
Reduction of filtered k-theory and a characterization of Cuntz-Krieger algebras
DEFF Research Database (Denmark)
Arklint, Sara E.; Bentmann, Rasmus Moritz; Katsura, Takeshi
2014-01-01
We show that filtered K-theory is equivalent to a substantially smaller invariant for all real-rank-zero C*-algebras with certain primitive ideal spaces—including the infinitely many so-called accordion spaces for which filtered K-theory is known to be a complete invariant. As a consequence, we...
Algebraic model theory for languages without equality
Elgueta Montó, Raimon
1994-01-01
In our opinion, it is fair to distinguish two separate branches in the origins of model theory. The first one, the model theory of first-order logic, can be traced back to the pioneering work of L. Lowenheim, T. Skolem, K. Gödel, A. Tarski and A.I. MaI 'cev, published before the mid 30's. This branch was put forward during the 40s' and 50s’ by several authors, including A. Tarski, L. Henkin, A. Robinson, J. Los. Their contribution, however, was rather influenced by modern algebra, a disciplin...
Higher AGT Correspondences, W-algebras, and Higher Quantum Geometric Langlands Duality from M-Theory
Tan, Meng-Chwan
2016-01-01
We further explore the implications of our framework in [arXiv:1301.1977, arXiv:1309.4775], and physically derive, from the principle that the spacetime BPS spectra of string-dual M-theory compactifications ought to be equivalent, (i) a 5d AGT correspondence for any compact Lie group, (ii) a 5d and 6d AGT correspondence on ALE space of type ADE, and (iii) identities between the ordinary, q-deformed and elliptic affine W-algebras associated with the 4d, 5d and 6d AGT correspondence, respectively, which also define a quantum geometric Langlands duality and its higher analogs formulated by Feigin-Frenkel-Reshetikhin in [3,4]. As an offshoot, we are led to the sought-after connection between the gauge-theoretic realization of the geometric Langlands correspondence by Kapustin-Witten [5,6] and its algebraic CFT formulation by Beilinson-Drinfeld [7], where one can also understand Wilson and 't Hooft-Hecke line operators in 4d gauge theory as monodromy loop operators in 2d CFT, for example. In turn, this will allow ...
Boundedly controlled topology foundations of algebraic topology and simple homotopy theory
Anderson, Douglas R
1988-01-01
Several recent investigations have focused attention on spaces and manifolds which are non-compact but where the problems studied have some kind of "control near infinity". This monograph introduces the category of spaces that are "boundedly controlled" over the (usually non-compact) metric space Z. It sets out to develop the algebraic and geometric tools needed to formulate and to prove boundedly controlled analogues of many of the standard results of algebraic topology and simple homotopy theory. One of the themes of the book is to show that in many cases the proof of a standard result can be easily adapted to prove the boundedly controlled analogue and to provide the details, often omitted in other treatments, of this adaptation. For this reason, the book does not require of the reader an extensive background. In the last chapter it is shown that special cases of the boundedly controlled Whitehead group are strongly related to lower K-theoretic groups, and the boundedly controlled theory is compared to Sie...
Residue number systems theory and applications
Mohan, P V Ananda
2016-01-01
This new and expanded monograph improves upon Mohan's earlier book, Residue Number Systems (Springer, 2002) with a state of the art treatment of the subject. Replete with detailed illustrations and helpful examples, this book covers a host of cutting edge topics such as the core function, the quotient function, new Chinese Remainder theorems, and large integer operations. It also features many significant applications to practical communication systems and cryptography such as FIR filters and elliptic curve cryptography. Starting with a comprehensive introduction to the basics and leading up to current research trends that are not yet widely distributed in other publications, this book will be of interest to both researchers and students alike.
On algebraic structure of the set of prime numbers
Zahedi, Ramin
2012-01-01
The set of prime numbers has been analyzed, based on their algebraic and arithmetical structure. Here by obtaining a sort of linear formula for the set of prime numbers, they are redefined and identified; under a systematic procedure it has been shown that the set of prime numbers is combinations (unions and intersections) of some subsets of natural numbers, with more primary structures. In fact generally, the logical essence of obtained formula for prime numbers is similar to formula 2n - 1 ...
Estimation of the Bezout number for piecewise algebraic curve
Institute of Scientific and Technical Information of China (English)
WANG; Renhong(王仁宏); XU; Zhiqiang(许志强)
2003-01-01
A piecewise algebraic curve is a curve determined by the zero set of a bivariate spline function.In this paper, a conjecture on triangulation is confirmed. The relation between the piecewise linear algebraiccurve and four-color conjecture is also presented. By Morgan-Scott triangulation, we will show the instabilityof Bezout number of piecewise algebraic curves. By using the combinatorial optimization method, an upperbound of the Bezout number defined as the maximum finite number of intersection points of two piecewisealgebraic curves is presented.
Semigroups and computer algebra in algebraic structures
Bijev, G.
2012-11-01
Some concepts in semigroup theory can be interpreted in several algebraic structures. A generalization fA,B,fA,B(X) = A(X')B of the complement operator (') on Boolean matrices is made, where A and B denote any rectangular Boolean matrices. While (') is an isomorphism between Boolean semilattices, the generalized complement operator is homomorphism in the general case. The map fA,B and its general inverse (fA,B)+ have quite similar properties to those in the linear algebra and are useful for solving linear equations in Boolean matrix algebras. For binary relations on a finite set, necessary and sufficient conditions for the equation αξβ = γ to have a solution ξ are proved. A generalization of Green's equivalence relations in semigroups for rectangular matrices is proposed. Relationships between them and the Moore-Penrose inverses are investigated. It is shown how any generalized Green's H-class could be constructed by given its corresponding linear subspaces and converted into a group isomorphic to a linear group. Some information about using computer algebra methods concerning this paper is given.
Young, Matthew B
2016-01-01
We introduce a new class of representations of the cohomological Hall algebras of Kontsevich and Soibelman which we call cohomological Hall modules, or CoHM for short. These representations are constructed from self-dual representations of a quiver with contravariant involution $\\sigma$ and provide a mathematical model for the space of BPS states in orientifold string theory. We use the CoHM to define a generalization of cohomological Donaldson-Thomas theory of quivers which allows the quiver representations to have orthogonal and symplectic structure groups. The associated invariants are called orientifold Donaldson-Thomas invariants. We prove the integrality conjecture for orientifold Donaldson-Thomas invariants of $\\sigma$-symmetric quivers. We also formulate precise conjectures regarding the geometric meaning of these invariants and the freeness of the CoHM of a $\\sigma$-symmetric quiver. We prove the freeness conjecture for disjoint union quivers, loop quivers and the affine Dynkin quiver of type $\\widet...
On the Theory of Generalized Algebraic Transformations
Strecka, Jozef
2010-01-01
This book deals with the theory of generalized algebraic transformations, which is elaborated with the aim to provide a relatively simple theoretical tool that enables an exact treatment of diverse more complex lattice-statistical models. In addition to a brief historical account on the developments of this exact mapping method, the versatility of generalized algebraic transformations will be convincingly evidenced when providing exact results for two different families of exactly solvable models. The family of exactly solved Ising models brings a deeper insight into various aspects closely associated especially with phase transitions and critical phenomena. The second class of exactly solved Ising-Heisenberg models sheds light on striking quantum manifestations of spontaneously long-range ordered systems, which are closely connected with a mutual interplay between quantum and cooperative phenomena.
Higher order Fourier analysis as an algebraic theory I
Szegedy, Balazs
2009-01-01
Ergodic theory, Higher order Fourier analysis and the hyper graph regularity method are three possible approaches to Szemer\\'edi type theorems in abelian groups. In this paper we develop an algebraic theory that creates a connection between these approaches. Our main method is to take the ultra product of abelian groups and to develop a precise algebraic theory of higher order characters on it. These results then can be turned back into approximative statements about finite Abelian groups.
Twisted vertex algebras, bicharacter construction and boson-fermion correspondences
International Nuclear Information System (INIS)
The boson-fermion correspondences are an important phenomena on the intersection of several areas in mathematical physics: representation theory, vertex algebras and conformal field theory, integrable systems, number theory, cohomology. Two such correspondences are well known: the types A and B (and their super extensions). As a main result of this paper we present a new boson-fermion correspondence of type D-A. Further, we define a new concept of twisted vertex algebra of order N, which generalizes super vertex algebra. We develop the bicharacter construction which we use for constructing classes of examples of twisted vertex algebras, as well as for deriving formulas for the operator product expansions, analytic continuations, and normal ordered products. By using the underlying Hopf algebra structure we prove general bicharacter formulas for the vacuum expectation values for two important groups of examples. We show that the correspondences of types B, C, and D-A are isomorphisms of twisted vertex algebras
WEAKLY ALGEBRAIC REFLEXIVITY AND STRONGLY ALGEBRAIC REFLEXIVITY
Institute of Scientific and Technical Information of China (English)
TaoChangli; LuShijie; ChenPeixin
2002-01-01
Algebraic reflexivity introduced by Hadwin is related to linear interpolation. In this paper, the concepts of weakly algebraic reflexivity and strongly algebraic reflexivity which are also related to linear interpolation are introduced. Some properties of them are obtained and some relations between them revealed.
Exceptional Vertex Operator Algebras and the Virasoro Algebra
Tuite, Michael P.
2008-01-01
We consider exceptional vertex operator algebras for which particular Casimir vectors constructed from the primary vectors of lowest conformal weight are Virasoro descendants of the vacuum. We discuss constraints on these theories that follow from an analysis of appropriate genus zero and genus one two point correlation functions. We find explicit differential equations for the partition function in the cases where the lowest weight primary vectors form a Lie algebra or a Griess algebra. Exam...
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
Number theory an introduction via the density of primes
Fine, Benjamin
2016-01-01
Now in its second edition, this textbook provides an introduction and overview of number theory based on the density and properties of the prime numbers. This unique approach offers both a firm background in the standard material of number theory, as well as an overview of the entire discipline. All of the essential topics are covered, such as the fundamental theorem of arithmetic, theory of congruences, quadratic reciprocity, arithmetic functions, and the distribution of primes. New in this edition are coverage of p-adic numbers, Hensel's lemma, multiple zeta-values, and elliptic curve methods in primality testing. Key topics and features include: A solid introduction to analytic number theory, including full proofs of Dirichlet's Theorem and the Prime Number Theorem Concise treatment of algebraic number theory, including a complete presentation of primes, prime factorizations in algebraic number fields, and unique factorization of ideals Discussion of the AKS algorithm, which shows that primality testing is...
Number theory and modular forms papers in memory of Robert A Rankin
Ono, Ken
2003-01-01
Robert A. Rankin, one of the world's foremost authorities on modular forms and a founding editor of The Ramanujan Journal, died on January 27, 2001, at the age of 85. Rankin had broad interests and contributed fundamental papers in a wide variety of areas within number theory, geometry, analysis, and algebra. To commemorate Rankin's life and work, the editors have collected together 25 papers by several eminent mathematicians reflecting Rankin's extensive range of interests within number theory. Many of these papers reflect Rankin's primary focus in modular forms. It is the editors' fervent hope that mathematicians will be stimulated by these papers and gain a greater appreciation for Rankin's contributions to mathematics. This volume would be an inspiration to students and researchers in the areas of number theory and modular forms.
A universal characterization of higher algebraic K-theory
Blumberg, Andrew J; Tabuada, Goncalo
2010-01-01
We establish a universal characterization of the higher algebraic $K$-theory of stable infinity categories. Specifically, we prove that the (connective) algebraic $K$-theory spectrum construction is the universal functor, with values in a stable presentable infinity category, which inverts Morita equivalences, preserves filtered colimits, and satisfies Waldhausen's additivity theorem. In order to prove these results, we construct and study a suitable localization of the category of presheaves of spectra on small stable infinity categories. In this category, Waldhausen's S.-construction corresponds to the suspension functor and the algebraic K-theory spectrum becomes co-representable. This latter result leads to a complete classification of all natural transformations from algebraic K-theory to topological Hochschild homology (THH) and topological cyclic homology (TC). In particular, we obtain a canonical universal description of the cyclotomic trace map.
NATO Advanced Study Institute on Structural Theory of Automata, Semigroups and Universal Algebra
Rosenberg, Ivo; Goldstein, Martin
2005-01-01
Several of the contributions to this volume bring forward many mutually beneficial interactions and connections between the three domains of the title. Developing them was the main purpose of the NATO ASI summerschool held in Montreal in 2003. Although some connections, for example between semigroups and automata, were known for a long time, developing them and surveying them in one volume is novel and hopefully stimulating for the future. Another aspect is the emphasis on the structural theory of automata that studies ways to contstruct big automata from small ones. The volume also has contributions on top current research or surveys in the three domains. One contribution even links clones of universal algebra with the computational complexity of computer science. Three contributions introduce the reader to research in the former East block.
Promoting Number Theory in High Schools or Birthday Problem and Number Theory
Srinivasan, V. K.
2010-01-01
The author introduces the birthday problem in this article. This can amuse willing members of any birthday party. This problem can also be used as the motivational first day lecture in number theory for the gifted students in high schools or in community colleges or in undergraduate classes in colleges.
Gromov-Witten theory, Hurwitz numbers, and Matrix models, I
Okounkov, Andrei; Pandharipande, Rahul
2001-01-01
The main goal of the paper is to present a new approach via Hurwitz numbers to Kontsevich's combinatorial/matrix model for the intersection theory of the moduli space of curves. A secondary goal is to present an exposition of the circle of ideas involved: Hurwitz numbers, Gromov-Witten theory of the projective line, matrix integrals, and the theory of random trees. Further topics will be treated in a sequel.
Wild kernels for higher K-theory of division and semi-simple algebras
Quo Xue Jun
2003-01-01
Let SIGMA be a semi-simple algebra over a number field F. In this paper, we prove that for all n >= 0, the wild kernel WK sub n (SIGMA):Ker(K sub n (SIGMA) -> PI sub f sub i sub n sub i sub t sub e subupsilon K sub n (SIGMA subupsilon)) is contained in the torsion part of the image of the natural homomorphism K sub n (LAMBDA) -> K sub n (SIGMA), where LAMBDA is a maximal order in SIGMA. In particular, WK sub n (SIGMA) is finite. In the process, we prove that if LAMBDA is a maximal order in a central division algebra D over F, then the kernel of the reduction map K sub 2 sub n sub - sub 1 (LAMBDA) -> suppi sup subupsilon PI sub f sub i sub n sub i sub t sub e subupsilon K sub 2 sub n sub - sub 1 (d subupsilon) is finite. In paragraph 3 we investigate the connections between WK sub n (D) and div(K sub n (D)) and prove that divK sub 2 (SIGMA) is a subset of WK sub 2 (SIGMA); if the index of D is square free, then div(K sub 2 (D)) approx = div(K sub 2 (F)), WK sub 2 (F) approx = WK sub 2 (D) and vertical bar WK s...
Tsue, Yasuhiko; Providência, Constança; Providência, João da; Yamamura, Masatoshi
2016-08-01
The minimum weight states of the Lipkin model consisting of n single-particle levels and obeying the SU(n) algebra are investigated systematically. The basic idea is to use the SU(2) algebra, which is independent of the SU(n) algebra. This idea has already been presented by the present authors in the case of the conventional Lipkin model consisting of two single-particle levels and obeying the SU(2) algebra. If this idea is followed, the minimum weight states are determined for any fermion number appropriately occupying n single-particle levels. Naturally, the conventional minimum weight state is included: all fermions occupy energetically the lowest single-particle level in the absence of interaction. The cases n=2, 3, 4, and 5 are discussed in some detail.
Jeribi, Aref
2015-01-01
Uncover the Useful Interactions of Fixed Point Theory with Topological StructuresNonlinear Functional Analysis in Banach Spaces and Banach Algebras: Fixed Point Theory under Weak Topology for Nonlinear Operators and Block Operator Matrices with Applications is the first book to tackle the topological fixed point theory for block operator matrices with nonlinear entries in Banach spaces and Banach algebras. The book provides researchers and graduate students with a unified survey of the fundamental principles of fixed point theory in Banach spaces and algebras. The authors present several exten
Three Hopf algebras and their common simplicial and categorical background
Gálvez-Carrillo, Imma; Tonks, Andrew
2016-01-01
We consider three a priori totally different setups for Hopf algebras from number theory, mathematical physics and algebraic topology. These are the Hopf algebras of Goncharov for multiple zeta values, that of Connes--Kreimer for renormalization, and a Hopf algebra constructed by Baues to study double loop spaces. We show that these examples can be successively unified by considering simplicial objects, cooperads with multiplication and Feynman categories at the ultimate level. These considerations open the door to new constructions and reinterpretation of known constructions in a large common framework.
The concrete theory of numbers: initial numbers and wonderful properties of numbers repunit
Tarasov, Boris V
2007-01-01
In this work initial numbers and repunit numbers have been studied. All numbers have been considered in a decimal notation. The problem of simplicity of initial numbers has been studied. Interesting properties of numbers repunit are proved: $gcd(R_a, R_b) = R_{gcd(a,b)}$; $R_{ab}/(R_aR_b)$ is an integer only if $gcd(a,b) = 1$, where $a\\geq1$, $b\\geq1$ are integers. Dividers of numbers repunit, are researched by a degree of prime number.
Number Theory and Public-Key Cryptography.
Lefton, Phyllis
1991-01-01
Described are activities in the study of techniques used to conceal the meanings of messages and data. Some background information and two BASIC programs that illustrate the algorithms used in a new cryptographic system called "public-key cryptography" are included. (CW)
Algebraic differential calculus for gauge theories
International Nuclear Information System (INIS)
The guiding idea in this paper is that, from the point of view of physics, functions and fields are more important than the (space time) manifold over which they are defined. The line pursued in these notes belongs to the general framework of ideas that replaces the space M by the ring of functions on it. Our essential observation, underlying this work, is that much of mathematical physics requires only a few differential operators (Lie derivative, d, δ) operating on modules of sections of suitable bundles. A connection (=gauge potential) can be described by a lift of vector fields from the base to the total space of a principal bundle. Much of the information can be encoded in the lift without reference to the bundle structures. In this manner, one arrives at an 'algebraic differential calculus' and its graded generalization that we are going to discuss. We are going to give an exposition of 'algebraic gauge theory' in both ungraded and graded versions. We show how to deal with the essential features of electromagnetism, Dirac, Kaluza-Klein and 't Hooft-Polyakov monopoles. We also show how to break the symmetry from SU(2) to U(1) without Higgs field. We briefly show how to deal with tests particles in external fields and with the Lagrangian formulation of field theories. (orig./HSI)
Soft Drinks, Mind Reading, and Number Theory
Schultz, Kyle T.
2009-01-01
Proof is a central component of mathematicians' work, used for verification, explanation, discovery, and communication. Unfortunately, high school students' experiences with proof are often limited to verifying mathematical statements or relationships that are already known to be true. As a result, students often fail to grasp the true nature of…
Diagrammatic Kazhdan-Lusztig theory for the (walled) Brauer algebra
Cox, Anton
2010-01-01
We determine the decomposition numbers for the Brauer and walled Brauer algebra in characteristic zero in terms of certain polynomials associated to cap and curl diagrams (recovering a result of Martin in the Brauer case). We consider a second family of polynomials associated to such diagrams, and use these to determine projective resolutions of the standard modules. We then relate these two families of polynomials to Kazhdan-Lusztig theory via the work of Lascoux-Sch\\"utzenberger and Boe, inspired by work of Brundan and Stroppel in the cap diagram case.
Cluster algebras and Poisson geometry
Gekhtman, M.; Shapiro, M.; Vainshtein, A.
2002-01-01
We introduce a Poisson variety compatible with a cluster algebra structure and a compatible toric action on this variety. We study Poisson and topological properties of the union of generic orbits of this toric action. In particular, we compute the number of connected components of the union of generic toric orbits for cluster algebras over real numbers. As a corollary we compute the number of connected components of refined open Bruhat cells in Grassmanians G(k,n) over real numbers.
The algebraic numbers definable in various exponential fields
Kirby, Jonathan; Onshuus, Alf
2011-01-01
We prove the following theorems: Theorem 1: For any E-field with cyclic kernel, in particular $\\mathbb C$ or the Zilber fields, all real abelian algebraic numbers are pointwise definable. Theorem 2: For the Zilber fields, the only pointwise definable algebraic numbers are the real abelian numbers.
On the number of connected components of random algebraic hypersurfaces
Fyodorov, Yan V.; Lerario, Antonio; Lundberg, Erik
2015-09-01
We study the expectation of the number of components b0(X) of a random algebraic hypersurface X defined by the zero set in projective space RPn of a random homogeneous polynomial f of degree d. Specifically, we consider invariant ensembles, that is Gaussian ensembles of polynomials that are invariant under an orthogonal change of variables. Fixing n, under some rescaling assumptions on the family of ensembles (as d → ∞), we prove that Eb0(X) has the same order of growth as [ Eb0(X ∩ RP1) ] n. This relates the average number of components of X to the classical problem of M. Kac (1943) on the number of zeros of the random univariate polynomial f|RP1. The proof requires an upper bound for Eb0(X), which we obtain by counting extrema using Random Matrix Theory methods from Fyodorov (2013), and it also requires a lower bound, which we obtain by a modification of the barrier method from Lerario and Lundberg (2015) and Nazarov and Sodin (2009). We also provide quantitative upper bounds on implied constants; for the real Fubini-Study model these estimates provide super-exponential decay (as n → ∞) of the leading coefficient (in d) of Eb0(X) .
Sierpinski, Waclaw
1988-01-01
Since the publication of the first edition of this work, considerable progress has been made in many of the questions examined. This edition has been updated and enlarged, and the bibliography has been revised.The variety of topics covered here includes divisibility, diophantine equations, prime numbers (especially Mersenne and Fermat primes), the basic arithmetic functions, congruences, the quadratic reciprocity law, expansion of real numbers into decimal fractions, decomposition of integers into sums of powers, some other problems of the additive theory of numbers and the theory of Gaussian
Beta-conjugates of real algebraic numbers as Puiseux expansions
Verger-Gaugry, Jean-Louis
2011-01-01
The beta-conjugates of a base of numeration $\\beta > 1$, $\\beta$ being a Parry number, were introduced by Boyd, in the context of the R\\'enyi-Parry dynamics of numeration system and the beta-transformation. These beta-conjugates are canonically associated with $\\beta$. Let $\\beta > 1$ be a real algebraic number. A more general definition of the beta-conjugates of $\\beta$ is introduced in terms of the Parry Upper function $f_{\\beta}(z)$ of the beta-transformation. We introduce the concept of a germ of curve at $(0,1/\\beta) \\in \\mathbb{C}^{2}$ associated with $f_{\\beta}(z)$ and the reciprocal of the minimal polynomial of $\\beta$. This germ is decomposed into irreducible elements according to the theory of Puiseux, gathered into conjugacy classes. The beta-conjugates of $\\beta$, in terms of the Puiseux expansions, are given a new equivalent definition in this new context. If $\\beta$ is a Parry number the (Artin-Mazur) dynamical zeta function $\\zeta_{\\beta}(z)$ of the beta-transformation, simply related to $f_{\\b...
Vertex Algebras, Kac-Moody Algebras, and the Monster
Borcherds, Richard E.
1986-05-01
It is known that the adjoint representation of any Kac-Moody algebra A can be identified with a subquotient of a certain Fock space representation constructed from the root lattice of A. I define a product on the whole of the Fock space that restricts to the Lie algebra product on this subquotient. This product (together with a infinite number of other products) is constructed using a generalization of vertex operators. I also construct an integral form for the universal enveloping algebra of any Kac-Moody algebra that can be used to define Kac-Moody groups over finite fields, some new irreducible integrable representations, and a sort of affinization of any Kac-Moody algebra. The ``Moonshine'' representation of the Monster constructed by Frenkel and others also has products like the ones constructed for Kac-Moody algebras, one of which extends the Griess product on the 196884-dimensional piece to the whole representation.
Theory of loop algebra on multi-loop kinematic chains and its application
Institute of Scientific and Technical Information of China (English)
HUANG Zhen; DING HuaFeng
2007-01-01
Based on the mathematic representation of loops of kinematic chains, this paper proposes the "" operation of loops and its basic laws and establishes the basic theorem system of the loop algebra of kinematic chains. Then the basis loop set and its determination conditions, and the ways to obtain the crucial perimeter topological graph are presented. Furthermore, the characteristic perimeter topological graph and the characteristic adjacency matrix are also developed. The most important characteristic of this theory is that for a topological graph which is drawn or labeled in any way, both the resulting characteristic perimeter topological graph and the characteristic adjacency matrix obtained through this theory are unique, and each has one-to-one correspondence with its kinematic chain. This characteristic dramatically simplifies the isomorphism identification and establishes a theoretical basis for the numeralization of topological graphs, and paves the way for numeralization and computerization of the structural synthesis and mechanism design further. Finally, this paper also proposes a concise isomorphism identification method of kinematic chains based on the concept of characteristic adjacency matrix.
Theory of loop algebra on multi-loop kinematic chains and its application
Institute of Scientific and Technical Information of China (English)
2007-01-01
Based on the mathematic representation of loops of kinematic chains, this paper proposes the " ⊕ " operation of loops and its basic laws and establishes the basic theorem system of the loop algebra of kinematic chains. Then the basis loop set and its determination conditions, and the ways to obtain the crucial perimeter topological graph are presented. Furthermore, the characteristic perimeter topo-logical graph and the characteristic adjacency matrix are also developed. The most important characteristic of this theory is that for a topological graph which is drawn or labeled in any way, both the resulting characteristic perimeter topological graph and the characteristic adjacency matrix obtained through this theory are unique, and each has one-to-one correspondence with its kinematic chain. This character-istic dramatically simplifies the isomorphism identification and establishes a theoretical basis for the numeralization of topological graphs, and paves the way for numeralization and computerization of the structural synthesis and mechanism design further. Finally, this paper also proposes a concise isomorphism identifica-tion method of kinematic chains based on the concept of characteristic adjacency matrix.
Putnam, Ian F.
2010-03-01
We investigate the C*-algebras associated to aperiodic structures called model sets obtained by the cut-and-project method. These C*-algebras are Morita equivalent to crossed product C*-algebras obtained from dynamics on a disconnected version of the internal space. This construction may be made from more general data, which we call a hyperplane system. From a hyperplane system, others may be constructed by a process of reduction and we show how the C*-algebras involved are related to each other. In particular, there are natural elements in the Kasparov KK-groups for the C*-algebra of a hyperplane system and that of its reduction. The induced map on K-theory fits in a six-term exact sequence. This provides a new method of the computation of the K-theory of such C*-algebras which is done completely in the setting of non-commutative geometry.
Number theory III Diophantine geometry
1991-01-01
From the reviews of the first printing of this book, published as Volume 60 of the Encyclopaedia of Mathematical Sciences: "Between number theory and geometry there have been several stimulating influences, and this book records of these enterprises. This author, who has been at the centre of such research for many years, is one of the best guides a reader can hope for. The book is full of beautiful results, open questions, stimulating conjectures and suggestions where to look for future developments. This volume bears witness of the broad scope of knowledge of the author, and the influence of several people who have commented on the manuscript before publication ... Although in the series of number theory, this volume is on diophantine geometry, and the reader will notice that algebraic geometry is present in every chapter. ... The style of the book is clear. Ideas are well explained, and the author helps the reader to pass by several technicalities. Reading and rereading this book I noticed that the topics ...
Number Worlds: Visual and Experimental Access to Elementary Number Theory Concepts
Sinclair, Nathalie; Zazkis, Rina; Liljedahl, Peter
2004-01-01
Recent research demonstrates that many issues related to the structure of natural numbers and the relationship among numbers are not well grasped by students. In this article, we describe a computer-based learning environment called "Number Worlds" that was designed to support the exploration of elementary number theory concepts by making the…
Vidal, Juan Climent
2012-01-01
For single-sorted algebras, Fujiwara defined, through the concept of family of basic mapping-formulas, a notion of morphism which generalizes the ordinary notion of homomorphism between algebras and an equivalence relation, the conjugation, on the families of basic mapping-formulas, which corresponds to the relation of inner isomorphism for algebras. In this paper we extend the theory of Fujiwara about morphisms to the many-sorted algebras, by defining the concept of polyderivor between many-sorted signatures, which assigns to basic sorts, words and to formal operations, families of derived terms, and under which the standard signature morphisms, the basic mapping-formulas of Fujiwara, and the derivors of Goguen-Thatcher-Wagner are subsumed. Then, by means of the homomorphisms between B\\'enabou algebras, which are the algebraic counterpart of the finitary many-sorted algebraic theories of B\\'enabou, we define the composition of polyderivors from which we get a corresponding category, isomorphic to the categor...
Harmonic functions on groups and Fourier algebras
Chu, Cho-Ho
2002-01-01
This research monograph introduces some new aspects to the theory of harmonic functions and related topics. The authors study the analytic algebraic structures of the space of bounded harmonic functions on locally compact groups and its non-commutative analogue, the space of harmonic functionals on Fourier algebras. Both spaces are shown to be the range of a contractive projection on a von Neumann algebra and therefore admit Jordan algebraic structures. This provides a natural setting to apply recent results from non-associative analysis, semigroups and Fourier algebras. Topics discussed include Poisson representations, Poisson spaces, quotients of Fourier algebras and the Murray-von Neumann classification of harmonic functionals.
Supersymmetry Algebra in Super Yang-Mills Theories
Yokoyama, Shuichi
2015-01-01
We compute supersymmetry algebra (superalgebra) in supersymmetric Yang-Mills theories (SYM) consisting of a vector multiplet including fermionic contribution in six dimensions. We show that the contribution of fermion is given by boundary terms. From six dimensional results we determine superalgebras of five and four dimensional SYM by dimensional reduction. In five dimensional superalgebra the Kaluza-Klein momentum and the instanton particle charge are not the same but algebraically indistinguishable. We also extend this calculation including a hyper multiplet and for maximally SYM. We derive extended supersymmetry algebras in those four dimensional SYM with the holomorphic coupling constant given in hep-th/9408099.
Path operator algebras in conformal quantum field theories
International Nuclear Information System (INIS)
Two different kinds of path algebras and methods from noncommutative geometry are applied to conformal field theory: Fusion rings and modular invariants of extended chiral algebras are analyzed in terms of essential paths which are a path description of intertwiners. As an example, the ADE classification of modular invariants for minimal models is reproduced. The analysis of two-step extensions is included. Path algebras based on a path space interpretation of character identities can be applied to the analysis of fusion rings as well. In particular, factorization properties of character identities and therefore of the corresponding path spaces are - by means of K-theory - related to the factorization of the fusion ring of Virasoro- and W-algebras. Examples from nonsupersymmetric as well as N=2 supersymmetric minimal models are discussed. (orig.)
Algebraic Formulation of the Operatorial Perturbation Theory; 1
Müller, A H; Müller, Ary W. Espinosa; Vásquez, Adelio R. Matamala
1996-01-01
A new totally algebraic formalism based on general, abstract ladder operators has been proposed. This approach heavily grounds in the superoperator formalism of Primas. However it is necessary to introduce many improvements in his formalism. In this regard, it has been introduced a new set of superoperators featured by their algebraic structure. Also, two lemmas and one theorem have been developed in order to algebraically reformulate the theory on more rigorous grounds. Finally, we have been able to build a coherent and self-contained formalism independent on any matricial representation , removing in this way the degeneracy problem .
Hom-alternative algebras and Hom-Jordan algebras
Makhlouf, Abdenacer
2009-01-01
The purpose of this paper is to introduce Hom-alternative algebras and Hom-Jordan algebras. We discuss some of their properties and provide construction procedures using ordinary alternative algebras or Jordan algebras. Also, we show that a polarization of Hom-associative algebra leads to Hom-Jordan algebra.
Approximation to real numbers by cubic algebraic integers I
Roy, Damien
2002-01-01
In 1969, H. Davenport and W. M. Schmidt studied the problem of approximation to a real number \\xi by algebraic integers of degree at most three. They did so, using geometry of numbers, by resorting to the dual problem of finding simultaneous approximations to \\xi and \\xi^2 by rational numbers with the same denominator. In this paper, we show that their measure of approximation for the dual problem is optimal and that it is realized for a countable set of real numbers \\xi. We give several prop...
Computer algebra and operators
Fateman, Richard; Grossman, Robert
1989-01-01
The symbolic computation of operator expansions is discussed. Some of the capabilities that prove useful when performing computer algebra computations involving operators are considered. These capabilities may be broadly divided into three areas: the algebraic manipulation of expressions from the algebra generated by operators; the algebraic manipulation of the actions of the operators upon other mathematical objects; and the development of appropriate normal forms and simplification algorithms for operators and their actions. Brief descriptions are given of the computer algebra computations that arise when working with various operators and their actions.
An Integrated Theory of Whole Number and Fractions Development
Siegler, Robert S.; Thompson, Clarissa A.; Schneider, Michael
2011-01-01
This article proposes an integrated theory of acquisition of knowledge about whole numbers and fractions. Although whole numbers and fractions differ in many ways that influence their development, an important commonality is the centrality of knowledge of numerical magnitudes in overall understanding. The present findings with 11- and 13-year-olds…
Supersymmetry algebra in super Yang-Mills theories
Yokoyama, Shuichi
2015-01-01
We compute supersymmetry algebra (superalgebra) in supersymmetric Yang-Mills theories (SYM) consisting of a vector multiplet including fermionic contribution in six dimensions. We show that the contribution of fermion is given by boundary terms. From six dimensional results we determine superalgebras of five and four dimensional SYM by dimensional reduction. In five dimensional superalgebra the Kaluza-Klein momentum and the instanton particle charge are not the same but algebraically indistin...
Monotone complete C*-algebras and generic dynamics
Saitô, Kazuyuki
2015-01-01
This monograph is about monotone complete C*-algebras, their properties and the new classification theory. A self-contained introduction to generic dynamics is also included because of its important connections to these algebras. Our knowledge and understanding of monotone complete C*-algebras has been transformed in recent years. This is a very exciting stage in their development, with much discovered but with many mysteries to unravel. This book is intended to encourage graduate students and working mathematicians to attack some of these difficult questions. Each bounded, upward directed net of real numbers has a limit. Monotone complete algebras of operators have a similar property. In particular, every von Neumann algebra is monotone complete but the converse is false. Written by major contributors to this field, Monotone Complete C*-algebras and Generic Dynamics takes readers from the basics to recent advances. The prerequisites are a grounding in functional analysis, some point set topology and an eleme...
Introduction to algebra and trigonometry
Kolman, Bernard
1981-01-01
Introduction to Algebra and Trigonometry provides a complete and self-contained presentation of the fundamentals of algebra and trigonometry.This book describes an axiomatic development of the foundations of algebra, defining complex numbers that are used to find the roots of any quadratic equation. Advanced concepts involving complex numbers are also elaborated, including the roots of polynomials, functions and function notation, and computations with logarithms. This text also discusses trigonometry from a functional standpoint. The angles, triangles, and applications involving triangles are
Alladi, Krishnaswami
2008-01-01
Contains chapters on number theory and related topics. This title covers topics that focus on multipartitions, congruences and identities, the formulas of Koshliakov and Guinand in Ramanujan's "Lost Notebook", alternating sign matrices and the Weyl character formulas, theta functions in complex analysis, and elliptic functions
Representations of quivers over the algebra of dual numbers
Ringel, Claus Michael
2011-01-01
The representations of a quiver Q over a field k have been studied for a long time. It seems to be worthwhile to consider also representations of Q over arbitrary finite-dimensional k-algebras A. Here we draw the attention to the case when A = k[\\epsilon] is the algebra of dual numbers, thus to the \\Lambda-modules with \\Lambda = kQ[\\epsilon]. The algebra \\Lambda{} is a 1-Gorenstein algebra, therefore the torsionless \\Lambda-modules are known to be of special interest (as the Gorenstein-projective or maximal Cohen-Macaulay modules). They form a Frobenius category \\Cal L, thus the corresponding stable category is a triangulated category. As we show, the category \\Cal L is the category of perfect differential kQ-modules and the stable category of \\Cal L is triangle equivalent to the orbit category of the derived category D^b(mod kQ) modulo the shift. The homology functor H from mod \\Lambda{} to \\mod kQ yields a bijection between the indecomposables in the stable category of \\Cal L and those in mod kQ. Our main i...
Contributions to the structure theory of non-simple C*-algebras
DEFF Research Database (Denmark)
Bentmann, Rasmus Moritz
This thesis is mainly concerned with classification results for non-simple purely ininite C*-algebras, specifically Cuntz-Krieger algebras and graph C*-algebras, and continuous fields of Kirchberg algebras. In Article A, we perform some computations concerning projective dimension in filtrated K-theory....... In joint work with Sara Arklint and Takeshi Katsura, we provide a range result complementing Gunnar Restor's classification theorem for Cuntz-Kieger algebras (Article B) and we investigate reduction of filtrated K-theory for C*-algebras of real rank zero, thereby obtaining a characterization of Cuntz...
Permutation Centralizer Algebras and Multi-Matrix Invariants
Mattioli, Paolo
2016-01-01
We introduce a class of permutation centralizer algebras which underly the combinatorics of multi-matrix gauge invariant observables. One family of such non-commutative algebras is parametrised by two integers. Its Wedderburn-Artin decomposition explains the counting of restricted Schur operators, which were introduced in the physics literature to describe open strings attached to giant gravitons and were subsequently used to diagonalize the Gaussian inner product for gauge invariants of 2-matrix models. The structure of the algebra, notably its dimension, its centre and its maximally commuting sub-algebra, is related to Littlewood-Richardson numbers for composing Young diagrams. It gives a precise characterization of the minimal set of charges needed to distinguish arbitrary matrix gauge invariants, which are related to enhanced symmetries in gauge theory. The algebra also gives a star product for matrix invariants. The centre of the algebra allows efficient computation of a sector of multi-matrix correlator...
Prime numbers, quantum field theory and the Goldbach conjecture
Sanchis-Lozano, Miguel-Angel; Navarro-Salas, Jose
2012-01-01
Motivated by the Goldbach and Polignac conjectures in Number Theory, we propose the factorization of a classical non-interacting real scalar field (on a two-cylindrical spacetime) as a product of either two or three (so-called primer) fields whose Fourier expansion exclusively contains prime modes. We undertake the canonical quantization of such primer fields and construct the corresponding Fock space by introducing creation operators $a_p^{\\dag}$ (labeled by prime numbers $p$) acting on the vacuum. The analysis of our model, based on the standard rules of quantum field theory, suggests intriguing connections between different topics in Number Theory, notably the Riemann hypothesis and the Goldbach and Polignac conjectures. Our analysis also suggests that the (non) renormalizability properties of the proposed model could be linked to the possible validity or breakdown of the Goldbach conjecture for large integer numbers.
Optical potentials in algebraic scattering theory
Energy Technology Data Exchange (ETDEWEB)
Levay, Peter [Institute of Theoretical Physics, Technical University of Budapest, Budapest (Hungary)
1999-02-12
Using the theory of induced representations new realizations for the Lie algebras of the groups SO(2, 1), SO(2, 2), SO(3, 2) are found. The eigenvalue problem of the Casimir operators yield Schroedinger equations with non-Hermitian interaction terms (i.e. optical potentials). For the group SO(2, 2) we have a two-parameter family of (matrix-valued) potentials containing terms of Poeschl-Teller and Gendenshtein type. We calculate the S-matrices for special values of this two-parameter family. In particular we also include a derivation of the S-matrix for the two-dimensional scattering problem on a complex Gendenshtein potential. The canonically transformed realization results in a non-local optical potential. (author)
C*-algebras of Toeplitz type associated with algebraic number fields
Cuntz, Joachim; Laca, Marcelo
2011-01-01
We associate with the ring $R$ of algebraic integers in a number field a C*-algebra $\\mathfrak T[R]$. It is an extension of the C*-algebra $\\mathfrak A[R]$ studied previously in \\cite{Cun}, \\cite{CuLi}, \\cite{CuLi2} and, in contrast to $\\mathfrak A[R]$, it is functorial under homomorphisms of rings. It can also be defined using the left regular representation of the $ax+b$-semigroup $R\\rtimes R^\\times$ on $\\ell^2 (R\\rtimes R^\\times)$. The algebra $\\mathfrak T[R]$ carries a natural one-parameter automorphism group $(\\sigma_t)_{t\\in\\mathbb R}$. We determine its KMS-structure. The technical difficulties that we encounter are due to the presence of the class group in the case where $R$ is not a principal ideal domain. In that case, for a fixed large inverse temperature, the simplex of KMS-states splits over the class group. The "partition functions" are partial Dedekind $\\zeta$-functions. We prove a result characterizing the asymptotic behavior of quotients of such partial $\\zeta$-functions, which we then use to ...
The number of complete exceptional sequences for a Dynkin algebra
Obaid, Mustafa A. A.; Nauman, S. Khalid; Shammakh, Wafa S. Al; Fakieh, Wafaa M.; Ringel, Claus Michael
2013-01-01
We consider Dynkin algebras, these are the hereditary artin algebras of finite representation type. The indecomposable modules for a Dynkin algebra correspond bijectively to the positive roots of a Dynkin diagram. Given a Dynkin algebra with n simple modules, a complete exceptional sequence is a sequence M_1,..., M_n of indecomposable modules such that Hom(M_i,M_j) = 0 = Ext(M_i,M_j) for i > j. The aim of this paper is to determine the number of complete exceptional sequences for any Dynkin a...
Quaternionen and Geometric Algebra (Quaternionen und Geometrische Algebra)
Horn, Martin Erik
2007-01-01
In the last one and a half centuries, the analysis of quaternions has not only led to further developments in mathematics but has also been and remains an important catalyst for the further development of theories in physics. At the same time, Hestenes geometric algebra provides a didactically promising instrument to model phenomena in physics mathematically and in a tangible manner. Quaternions particularly have a catchy interpretation in the context of geometric algebra which can be used didactically. The relation between quaternions and geometric algebra is presented with a view to analysing its didactical possibilities.
Arithmetic gravity and Yang-Mills theory: An approach to adelic physics via algebraic spaces
Schmidt, Rene
2008-01-01
This work is a dissertation thesis written at the WWU Muenster (Germany), supervised by Prof. Dr. Raimar Wulkenhaar. We present an approach to adelic physics based on the language of algebraic spaces. Relative algebraic spaces X over a base S are considered as fundamental objects which describe space-time. This yields a formulation of general relativity which is covariant with respect to changes of the chosen domain of numbers S. With regard to adelic physics the choice of S as an excellent Dedekind scheme is of interest (because this way also the finite prime spots, i.e. the p-adic degrees of freedom are taken into account). In this arithmetic case, it turns out that X is a Neron model. This enables us to make concrete statements concerning the structure of the space-time described by X. Furthermore, some solutions of the arithmetic Einstein equations are presented. In a next step, Yang-Mills gauge fields are incorporated.
Rota-Baxter algebras and the Hopf algebra of renormalization
Energy Technology Data Exchange (ETDEWEB)
Ebrahimi-Fard, K.
2006-06-15
Recently, the theory of renormalization in perturbative quantum field theory underwent some exciting new developments. Kreimer discovered an organization of Feynman graphs into combinatorial Hopf algebras. The process of renormalization is captured by a factorization theorem for regularized Hopf algebra characters. Hereby the notion of Rota-Baxter algebras enters the scene. In this work we develop in detail several mathematical aspects of Rota-Baxter algebras as they appear also in other sectors closely related to perturbative renormalization, to wit, for instance multiple-zeta-values and matrix differential equations. The Rota-Baxter picture enables us to present the algebraic underpinning for the Connes-Kreimer Birkhoff decomposition in a concise way. This is achieved by establishing a general factorization theorem for filtered algebras. Which in turn follows from a new recursion formula based on the Baker-Campbell-Hausdorff formula. This allows us to generalize a classical result due to Spitzer to non-commutative Rota-Baxter algebras. The Baker-Campbell-Hausdorff based recursion turns out to be a generalization of Magnus' expansion in numerical analysis to generalized integration operators. We will exemplify these general results by establishing a simple representation of the combinatorics of renormalization in terms of triangular matrices. We thereby recover in the presence of a Rota-Baxter operator the matrix representation of the Birkhoff decomposition of Connes and Kreimer. (orig.)
Rota-Baxter algebras and the Hopf algebra of renormalization
International Nuclear Information System (INIS)
Recently, the theory of renormalization in perturbative quantum field theory underwent some exciting new developments. Kreimer discovered an organization of Feynman graphs into combinatorial Hopf algebras. The process of renormalization is captured by a factorization theorem for regularized Hopf algebra characters. Hereby the notion of Rota-Baxter algebras enters the scene. In this work we develop in detail several mathematical aspects of Rota-Baxter algebras as they appear also in other sectors closely related to perturbative renormalization, to wit, for instance multiple-zeta-values and matrix differential equations. The Rota-Baxter picture enables us to present the algebraic underpinning for the Connes-Kreimer Birkhoff decomposition in a concise way. This is achieved by establishing a general factorization theorem for filtered algebras. Which in turn follows from a new recursion formula based on the Baker-Campbell-Hausdorff formula. This allows us to generalize a classical result due to Spitzer to non-commutative Rota-Baxter algebras. The Baker-Campbell-Hausdorff based recursion turns out to be a generalization of Magnus' expansion in numerical analysis to generalized integration operators. We will exemplify these general results by establishing a simple representation of the combinatorics of renormalization in terms of triangular matrices. We thereby recover in the presence of a Rota-Baxter operator the matrix representation of the Birkhoff decomposition of Connes and Kreimer. (orig.)
Amos, K.; Fraser, P.; Karataglidis, S.; van der Knijff, D.; Svenne, J. P.; Canton, L.; Pisent, G.
2005-01-01
A multi-channel algebraic scattering theory, to find solutions of coupled-channel scattering problems with interactions determined by collective models, has been structured to ensure that the Pauli principle is not violated. Positive (scattering) and negative (sub-threshold) solutions can be found to predict both the compound nucleus sub-threshold spectrum and all resonances due to coupled channel effects that occur on a smooth energy varying background.
Regular algebra and finite machines
Conway, John Horton
2012-01-01
World-famous mathematician John H. Conway based this classic text on a 1966 course he taught at Cambridge University. Geared toward graduate students of mathematics, it will also prove a valuable guide to researchers and professional mathematicians.His topics cover Moore's theory of experiments, Kleene's theory of regular events and expressions, Kleene algebras, the differential calculus of events, factors and the factor matrix, and the theory of operators. Additional subjects include event classes and operator classes, some regulator algebras, context-free languages, communicative regular alg
Baianu, Ion C; Brown, Ronald; 10.3842/SIGMA.2009.051
2009-01-01
A novel algebraic topology approach to supersymmetry (SUSY) and symmetry breaking in quantum field and quantum gravity theories is presented with a view to developing a wide range of physical applications. These include: controlled nuclear fusion and other nuclear reaction studies in quantum chromodynamics, nonlinear physics at high energy densities, dynamic Jahn-Teller effects, superfluidity, high temperature superconductors, multiple scattering by molecular systems, molecular or atomic paracrystal structures, nanomaterials, ferromagnetism in glassy materials, spin glasses, quantum phase transitions and supergravity. This approach requires a unified conceptual framework that utilizes extended symmetries and quantum groupoid, algebroid and functorial representations of non-Abelian higher dimensional structures pertinent to quantized spacetime topology and state space geometry of quantum operator algebras. Fourier transforms, generalized Fourier-Stieltjes transforms, and duality relations link, respectively, t...
Prime Numbers, Quantum Field Theory and the Goldbach Conjecture
Sanchis-Lozano, Miguel-Angel; Barbero G., J. Fernando; Navarro-Salas, José
2012-09-01
Motivated by the Goldbach conjecture in number theory and the Abelian bosonization mechanism on a cylindrical two-dimensional space-time, we study the reconstruction of a real scalar field as a product of two real fermion (so-called prime) fields whose Fourier expansion exclusively contains prime modes. We undertake the canonical quantization of such prime fields and construct the corresponding Fock space by introducing creation operators bp\\dag — labeled by prime numbers p — acting on the vacuum. The analysis of our model, based on the standard rules of quantum field theory and the assumption of the Riemann hypothesis, allows us to prove that the theory is not renormalizable. We also comment on the potential consequences of this result concerning the validity or breakdown of the Goldbach conjecture for large integer numbers.
Algebra: A Challenge at the Crossroads of Policy and Practice
Stein, Mary Kay; Kaufman, Julia Heath; Sherman, Milan; Hillen, Amy F.
2011-01-01
The authors review what is known about early and universal algebra, including who is getting access to algebra and student outcomes associated with algebra course taking in general and specifically with universal algebra policies. The findings indicate that increasing numbers of students, some of whom are underprepared, are taking algebra earlier.…
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.
Cohen, Greg
2010-01-01
In Graph Theory a number of results were devoted to studying the computational complexity of the number modulo 2 of a graph's edge set decompositions of various kinds, first of all including its Hamiltonian decompositions, as well as the number modulo 2 of, say, Hamiltonian cycles/paths etc. While the problems of finding a Hamiltonian decomposition and Hamiltonian cycle are NP-complete, counting these objects modulo 2 in polynomial time is yet possible for certain types of regular undirected graphs. Some of the most known examples are the theorems about the existence of an even number of Hamiltonian decompositions in a 4-regular graph and an even number of such decompositions where two given edges e and g belong to different cycles (Thomason, 1978), as well as an even number of Hamiltonian cycles passing through any given edge in a regular odd-degreed graph (Smith's theorem). The present article introduces a new algebraic technique which generalizes the notion of counting modulo 2 via applying fields of Chara...
Prime Numbers, quantum field theory and the Goldbach conjecture
Sanchis Lozano, Miguel Ángel; Barbero González, J. Fernando; Navarro Salas, José
2012-01-01
Motivated by the Goldbach conjecture in Number Theory and the abelian bosonization mechanism on a cylindrical two-dimensional spacetime we study the reconstruction of a real scalar field as a product of two real fermion (so-called \\textit{prime}) fields whose Fourier expansion exclusively contains prime modes. We undertake the canonical quantization of such prime fields and construct the corresponding Fock space by introducing creation operators $b_{p}^{\\dag}$ --labeled by prime numbers $p$--...
Structure and applications of real C*-algebras
Rosenberg, Jonathan
2015-01-01
For a long time, practitioners of the art of operator algebras always worked over the complex numbers, and nobody paid much attention to real C*-algebras. Over the last thirty years, that situation has changed, and it's become apparent that real C*-algebras have a lot of extra structure not evident from their complexifications. At the same time, interest in real C*-algebras has been driven by a number of compelling applications, for example in the classification of manifolds of positive scalar curvature, in representation theory, and in the study of orientifold string theories. We will discuss a number of interesting examples of these, and how the real Baum-Connes conjecture plays an important role.
Elementary number theory with programming
Lewinter, Marty
2015-01-01
A successful presentation of the fundamental concepts of number theory and computer programming Bridging an existing gap between mathematics and programming, Elementary Number Theory with Programming provides a unique introduction to elementary number theory with fundamental coverage of computer programming. Written by highly-qualified experts in the fields of computer science and mathematics, the book features accessible coverage for readers with various levels of experience and explores number theory in the context of programming without relying on advanced prerequisite knowledge and con
Computational linear and commutative algebra
Kreuzer, Martin
2016-01-01
This book combines, in a novel and general way, an extensive development of the theory of families of commuting matrices with applications to zero-dimensional commutative rings, primary decompositions and polynomial system solving. It integrates the Linear Algebra of the Third Millennium, developed exclusively here, with classical algorithmic and algebraic techniques. Even the experienced reader will be pleasantly surprised to discover new and unexpected aspects in a variety of subjects including eigenvalues and eigenspaces of linear maps, joint eigenspaces of commuting families of endomorphisms, multiplication maps of zero-dimensional affine algebras, computation of primary decompositions and maximal ideals, and solution of polynomial systems. This book completes a trilogy initiated by the uncharacteristically witty books Computational Commutative Algebra 1 and 2 by the same authors. The material treated here is not available in book form, and much of it is not available at all. The authors continue to prese...
Finite dimensional semigroup quadratic algebras with minimal number of relations
Iyudu, Natalia
2011-01-01
A quadratic semigroup algebra is an algebra over a field given by the generators $x_1,...,x_n$ and a finite set of quadratic relations each of which either has the shape $x_jx_k=0$ or the shape $x_jx_k=x_lx_m$. We prove that a quadratic semigroup algebra given by $n$ generators and $d\\leq \\frac{n^2+n}{4}$ relations is always infinite dimensional. This strengthens the Golod--Shafarevich estimate for the above class of algebras. Our main result however is that for every $n$, there is a finite dimensional quadratic semigroup algebra with $n$ generators and $\\delta_n$ generators, where $\\delta_n$ is the first integer greater than $\\frac{n^2+n}{4}$. This shows that the above Golod-Shafarevich type estimate for semigroup algebras is sharp.
Omni-Lie Color Algebras and Lie Color 2-Algebras
Zhang, Tao
2013-01-01
Omni-Lie color algebras over an abelian group with a bicharacter are studied. The notions of 2-term color $L_{\\infty}$-algebras and Lie color 2-algebras are introduced. It is proved that there is a one-to-one correspondence between Lie color 2-algebras and 2-term color $L_{\\infty}$-algebras.
Field Theory on Noncommutative Space-Time and the Deformed Virasoro Algebra
Chaichian, M.; Demichev, A.; Presnajder, P.
2000-01-01
We consider a field theoretical model on the noncommutative cylinder which leads to a discrete-time evolution. Its Euclidean version is shown to be equivalent to a model on the complex $q$-plane. We reveal a direct link between the model on a noncommutative cylinder and the deformed Virasoro algebra constructed earlier on an abstract mathematical background. As it was shown, the deformed Virasoro generators necessarily carry a second index (in addition to the usual one), whose meaning, howeve...
An invitation to general algebra and universal constructions
Bergman, George M
2015-01-01
Rich in examples and intuitive discussions, this book presents General Algebra using the unifying viewpoint of categories and functors. Starting with a survey, in non-category-theoretic terms, of many familiar and not-so-familiar constructions in algebra (plus two from topology for perspective), the reader is guided to an understanding and appreciation of the general concepts and tools unifying these constructions. Topics include: set theory, lattices, category theory, the formulation of universal constructions in category-theoretic terms, varieties of algebras, and adjunctions. A large number of exercises, from the routine to the challenging, interspersed through the text, develop the reader's grasp of the material, exhibit applications of the general theory to diverse areas of algebra, and in some cases point to outstanding open questions. Graduate students and researchers wishing to gain fluency in important mathematical constructions will welcome this carefully motivated book.
On the complexity of the binary expansions of algebraic irrational numbers (survey)
Kaneko, Hajime
2010-07-01
Borel conjectured that all irrational numbers are normal in any integral base α. For each positive number ξ and integer α greater than 1, ξ is normal in base α if and only if the sequence ξαn (n = 0,1,…) is uniformly distributed modulo 1. In this paper we survey not only the digit of algebraic irrational numbers in integral base but also the fractional parts of geometric progressions whose common ratios are algebraic numbers greater than 1. In our main results, we give new lower bounds for the number of digit changes in the binary expansions of algebraic irrational numbers.
Indian Academy of Sciences (India)
Cătălin Ciupală
2005-02-01
In this paper we introduce non-commutative fields and forms on a new kind of non-commutative algebras: -algebras. We also define the Frölicher–Nijenhuis bracket in the non-commutative geometry on -algebras.
Universal Algebra and Mathematical Logic
Luo, Zhaohua
2011-01-01
In this paper, first-order logic is interpreted in the framework of universal algebra, using the clone theory developed in three previous papers. We first define the free clone T(L, C) of terms of a first order language L over a set C of parameters in a standard way. The free right algebra F(L, C) of formulas over T(L, C) is then generated by atomic formulas. Structures for L over C are represented as perfect valuations of F(L, C), and theories of L are represented as filters of F(L). Finally...
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.
Mattson Solomon transform and algebra codes
DEFF Research Database (Denmark)
Martínez-Moro, E.; Benito, Diego Ruano
2009-01-01
In this note we review some results of the first author on the structure of codes defined as subalgebras of a commutative semisimple algebra over a finite field (see Martínez-Moro in Algebra Discrete Math. 3:99-112, 2007). Generator theory and those aspects related to the theory of Gröbner bases...
Quantum fields, periods and algebraic geometry
Kreimer, Dirk
2014-01-01
We discuss how basic notions of graph theory and associated graph polynomials define questions for algebraic geometry, with an emphasis given to an analysis of the structure of Feynman rules as determined by those graph polynomials as well as algebraic structures of graphs. In particular, we discuss the appearance of renormalization scheme independent periods in quantum field theory.
International Conference on Automorphic Forms and Number Theory
Al-Baali, Mehiddin; Ibukiyama, Tomoyoshi; Rupp, Florian
2014-01-01
This edited volume presents a collection of carefully refereed articles covering the latest advances in Automorphic Forms and Number Theory, that were primarily developed from presentations given at the 2012 “International Conference on Automorphic Forms and Number Theory,” held in Muscat, Sultanate of Oman. The present volume includes original research as well as some surveys and outlines of research altogether providing a contemporary snapshot on the latest activities in the field and covering the topics of: Borcherds products Congruences and Codes Jacobi forms Siegel and Hermitian modular forms Special values of L-series Recently, the Sultanate of Oman became a member of the International Mathematical Society. In view of this development, the conference provided the platform for scientific exchange and collaboration between scientists of different countries from all over the world. In particular, an opportunity was established for a close exchange between scientists and students of Germany, Oman, and J...
S-numbers of elementary operators on C*-algebras
Anoussis, M; Todorov, I G
2008-01-01
We study the s-numbers of elementary operators acting on C*-algebras. The main results are the following: If $\\tau$ is any tensor norm and $a,b\\in B(H)$ are such that the sequences $s(a),s(b)$ of their singular numbers belong to a stable Calkin space $J$ then the sequence of approximation numbers of $a\\otimes_{\\tau} b$ belongs to $J$. If $A$ is a C*-algebra, $J$ is a stable Calkin space, $s$ is an s-number function, and $a_i, b_i \\in A,$ $i=1,...,m$ are such that $s(\\pi(a_i)), s(\\pi(b_i)) \\in J$, $i=1,...,m$ for some faithful representation $\\pi$ of $A$ then $s(\\sum_{i=1}^{m} M_{a_i,b_i})\\in J$. The converse implication holds if and only if the ideal of compact elements of $A$ has finite spectrum. We also prove a quantitative version of a result of Ylinen.
Energy Technology Data Exchange (ETDEWEB)
Fucito, F.; Tanzini, A. [Rome Univ. 2 (Italy). Dipt. di Fisica; Vilar, L.C.Q.; Ventura, O.S.; Sasaki, C.A.G. [Centro Brasileiro de Pesquisas Fisicas (CBPF), Rio de Janeiro, RJ (Brazil); Sorella, S.P. [Universidade do Estado (UERJ), Rio de Janeiro, RJ (Brazil). Inst. de Fisica
1997-07-01
The aim of these notes is to provide a simple and pedagogical (as much as possible) introduction to what is nowadays commonly called Algebraic Renormalization. As the same itself let it understand, the Algebraic Renormalization gives a systematic set up in order to analyse the quantum extension of a given set of classical symmetries. The framework is purely algebraic, yielding a complete characterization of all possible anomalies and invariant counterterms without making use of any explicit computation of the Feynman diagrams. This goal is achieved by collecting, with the introduction of suitable ghost fields, all the symmetries into a unique operation summarized by a generalized Slavnov-Taylor (or master equation) identity which is the starting point for the quantum analysis. The Slavnov-Taylor identity allows to define a nilpotent operator whose cohomology classes in the space of the integrated local polynomials in the fields and their derivatives with dimensions bounded by power counting give all nontrivial anomalies and counterterms. I other words, the proof of the renormalizability is reduced to the computation of some cohomology classes. (author) 28 refs., 2 figs.
Lower bounds on the class number of algebraic function fields defined over any finite field
Ballet, Stéphane
2011-01-01
We give lower bounds on the number of effective divisors of degree $\\leq g-1$ with respect to the number of places of certain degrees of an algebraic function field of genus $g$ defined over a finite field. We deduce lower bounds and asymptotics for the class number, depending mainly on the number of places of a certain degree. We give examples of towers of algebraic function fields having a large class number.
Algebraic Systems and Pushdown Automata
Petre, Ion; Salomaa, Arto
We concentrate in this chapter on the core aspects of algebraic series, pushdown automata, and their relation to formal languages. We choose to follow here a presentation of their theory based on the concept of properness. We introduce in Sect. 2 some auxiliary notions and results needed throughout the chapter, in particular the notions of discrete convergence in semirings and C-cycle free infinite matrices. In Sect. 3 we introduce the algebraic power series in terms of algebraic systems of equations. We focus on interconnections with context-free grammars and on normal forms. We then conclude the section with a presentation of the theorems of Shamir and Chomsky-Schützenberger. We discuss in Sect. 4 the algebraic and the regulated rational transductions, as well as some representation results related to them. Section 5 is dedicated to pushdown automata and focuses on the interconnections with classical (non-weighted) pushdown automata and on the interconnections with algebraic systems. We then conclude the chapter with a brief discussion of some of the other topics related to algebraic systems and pushdown automata.
Antieigenvalue analysis for continuum mechanics, economics, and number theory
Directory of Open Access Journals (Sweden)
Gustafson Karl
2016-01-01
Full Text Available My recent book Antieigenvalue Analysis, World-Scientific, 2012, presented the theory of antieigenvalues from its inception in 1966 up to 2010, and its applications within those forty-five years to Numerical Analysis, Wavelets, Statistics, Quantum Mechanics, Finance, and Optimization. Here I am able to offer three further areas of application: Continuum Mechanics, Economics, and Number Theory. In particular, the critical angle of repose in a continuum model of granular materials is shown to be exactly my matrix maximum turning angle of the stress tensor of the material. The important Sharpe ratio of the Capital Asset Pricing Model is now seen in terms of my antieigenvalue theory. Euclid’s Formula for Pythagorean triples becomes a special case of my operator trigonometry.
Geometric representation of interval exchange maps over algebraic number fields
Poggiaspalla, G.; Lowenstein, J. H.; Vivaldi, F.
2008-01-01
This paper is concerned with the restriction of interval exchange transformations (IETs) to algebraic number fields, which leads to maps on lattices. We characterize renormalizability arithmetically and study its relationships with a geometrical quantity that we call the drift vector. We exhibit some examples of renormalizable IETs with zero and non-zero drift vectors and carry out some investigations of their properties. In particular we look for evidence of the finite decomposition property on a family of IETs extending the example studied in Lowenstein et al (2007 Dyn. Syst. 22 73-106).
LeVeque, William J
2002-01-01
Classic two-part work now available in a single volume assumes no prior theoretical knowledge on reader's part and develops the subject fully. Volume I is a suitable first course text for advanced undergraduate and beginning graduate students. Volume II requires a much higher level of mathematical maturity, including a working knowledge of the theory of analytic functions. Contents range from chapters on binary quadratic forms to the Thue-Siegel-Roth Theorem and the Prime Number Theorem. Includes numerous problems and hints for their solutions. 1956 edition. Supplementary Reading. List of Symb
Classical algebra its nature, origins, and uses
Cooke, Roger L
2008-01-01
This insightful book combines the history, pedagogy, and popularization of algebra to present a unified discussion of the subject. Classical Algebra provides a complete and contemporary perspective on classical polynomial algebra through the exploration of how it was developed and how it exists today. With a focus on prominent areas such as the numerical solutions of equations, the systematic study of equations, and Galois theory, this book facilitates a thorough understanding of algebra and illustrates how the concepts of modern algebra originally developed from classical algebraic precursors. This book successfully ties together the disconnect between classical and modern algebraand provides readers with answers to many fascinating questions that typically go unexamined, including: What is algebra about? How did it arise? What uses does it have? How did it develop? What problems and issues have occurred in its history? How were these problems and issues resolved? The author answers these questions and more,...
Relations Between BZMVdM-Algebra and Other Algebras
Institute of Scientific and Technical Information of China (English)
高淑萍; 邓方安; 刘三阳
2003-01-01
Some properties of BZMVdM-algebra are proved, and a new operator is introduced. It is shown that the substructure of BZMVdM-algebra can produce a quasi-lattice implication algebra. The relations between BZMVdM-algebra and other algebras are discussed in detail. A pseudo-distance function is defined in linear BZMVdM-algebra, and its properties are derived.
Algebraic K-theory of discrete subgroups of Lie groups.
Farrell, F T; Jones, L E
1987-05-01
Let G be a Lie group (with finitely many connected components) and Gamma be a discrete, cocompact, torsion-free subgroup of G. We rationally calculate the algebraic K-theory of the integral group ring ZGamma in terms of the homology of Gamma with trivial rational coefficients. PMID:16593834
Exact Ramsey Theory: Green-Tao numbers and SAT
Kullmann, Oliver
2010-01-01
We consider the links between Ramsey theory in the integers, based on van der Waerden's theorem, and (boolean, CNF) SAT solving. We aim at using the problems from exact Ramsey theory, concerned with computing Ramsey-type numbers, as a rich source of test problems, where especially methods for solving hard problems can be developed. In order to control the growth of the problem instances, we introduce "transversal extensions" as a natural way of constructing mixed parameter tuples (k_1, ..., k_m) for van-der-Waerden-like numbers N(k_1, ..., k_m), such that the growth of these numbers is guaranteed to be linear. Based on Green-Tao's theorem we introduce the "Green-Tao numbers" grt(k_1, ..., k_m), which in a sense combine the strict structure of van der Waerden problems with the (pseudo-)randomness of the distribution of prime numbers. Using standard SAT solvers (look-ahead, conflict-driven, and local search) we determine the basic values. It turns out that already for this single form of Ramsey-type problems, w...
Spacetime algebra and electron physics
Doran, C J L; Gull, S F; Somaroo, S; Challinor, A D
1996-01-01
This paper surveys the application of geometric algebra to the physics of electrons. It first appeared in 1996 and is reproduced here with only minor modifications. Subjects covered include non-relativistic and relativistic spinors, the Dirac equation, operators and monogenics, the Hydrogen atom, propagators and scattering theory, spin precession, tunnelling times, spin measurement, multiparticle quantum mechanics, relativistic multiparticle wave equations, and semiclassical mechanics.
Renormalization and Hopf Algebraic Structure of the 5-Dimensional Quartic Tensor Field Theory
Avohou, Remi Cocou; Tanasa, Adrian
2015-01-01
This paper is devoted to the study of renormalization of the quartic melonic tensor model in dimension (=rank) five. We review the perturbative renormalization and the computation of the one loop beta function, confirming the asymptotic freedom of the model. We then define the Connes-Kreimer-like Hopf algebra describing the combinatorics of the renormalization of this model and we analyze in detail, at one- and two-loop levels, the Hochschild cohomology allowing to write the combinatorial Dyson-Schwinger equations. Feynman tensor graph Hopf subalgebras are also exhibited.
Tur\\'an numbers for $K_{s,t}$-free graphs: topological obstructions and algebraic constructions
Blagojević, Pavle; Karasev, Roman
2011-01-01
We show that every hypersurface in $\\R^s\\times \\R^s$ contains a large grid, i.e., the set of the form $S\\times T$, with $S,T\\subset \\R^s$. We use this to deduce that the known constructions of extremal $K_{2,2}$-free and $K_{3,3}$-free graphs cannot be generalized to a similar construction of $K_{s,s}$-free graphs for any $s\\geq 4$. We also give new constructions of extremal $K_{s,t}$-free graphs for large $t$.
Tubular algebras and affine Kac-Moody algebras
Institute of Scientific and Technical Information of China (English)
Zheng-xin CHEN; Ya-nan LIN
2007-01-01
The purpose of this paper is to construct quotient algebras L(A)C1/I(A) of complex degenerate composition Lie algebras L(A)C1 by some ideals, where L(A)C1 is defined via Hall algebras of tubular algebras A, and to prove that the quotient algebras L(A)C1/I(A) are isomorphic to the corresponding affine Kac-Moody algebras. Moreover, it is shown that the Lie algebra Lre(A)C1 generated by A-modules with a real root coincides with the degenerate composition Lie algebra L(A)C1 generated by simple A-modules.
Tubular algebras and affine Kac-Moody algebras
Institute of Scientific and Technical Information of China (English)
2007-01-01
The purpose of this paper is to construct quotient algebras L(A)1C/I(A) of complex degenerate composition Lie algebras L(A)1C by some ideals, where L(A)1C is defined via Hall algebras of tubular algebras A, and to prove that the quotient algebras L(A)1C/I(A) are isomorphic to the corresponding affine Kac-Moody algebras. Moreover, it is shown that the Lie algebra Lre(A)1C generated by A-modules with a real root coincides with the degenerate composition Lie algebra L(A)1C generated by simple A-modules.
IS-代数的特征数%Characteristic numbers of IS-algebras
Institute of Scientific and Technical Information of China (English)
杨闻起
2011-01-01
The concept of characteristic numbers of IS-algebras is introduced and its properties are discussed. The rela tionship between characteristic numbers of IS-algebras and the characteristic numbers of rings is given, and three kinds of IS-algebras are characterized by the characteristic numbers.%引入了IS-代数的特征数的概念,讨论了它的性质；给出了IS-代数的特征数与环的特征数之间的关系,并用特征数刻画了无零因子、BCK和拟结合的IS-代数.
Algebraic topology and concurrency
DEFF Research Database (Denmark)
Fajstrup, Lisbeth; Raussen, Martin; Goubault, Eric
2006-01-01
We show in this article that some concepts from homotopy theory, in algebraic topology,are relevant for studying concurrent programs. We exhibit a natural semantics of semaphore programs, based on partially ordered topological spaces, which are studied up to “elastic deformation” or homotopy, giv...
关于代数整数与代数数的一个注记%A note about algebric integer and algebra number
Institute of Scientific and Technical Information of China (English)
徐丽媛; 陈良云
2011-01-01
In this paper,it is proved that an algebraic number can be seen as an eigenvalue of a matrix over rational field, and an algebraic integer can be seen as an eigenvalue of a matrix over integral ring. Then the important conclusion in mathematics that all algebraic integers form a ring and the field of its fractions is an algebraic number field is directly and clearly proved, in history the prove of this conclusion is much difficult for people to understand.%证明了代数数是有理数系数方阵的特征值,代数整数是整数系数方阵的特征值.由此出发,完全用线性代数与矩阵计算的方法简洁地证明了代数整数对加减法和乘法封闭,从而构成一个环(代数整数环)；所有代数数对加减乘除封闭,从而构成一个域(代数数域).
Infinite Conformal Algebras in Supersymmetric Theories on Four Manifolds
Johansen, Andrei
1994-01-01
We study a supersymmetric theory twisted on a K\\"ahler four manifold $M=\\Sigma_1 \\times \\Sigma_2 ,$ where $\\Sigma_{1,2}$ are 2D Riemann surfaces. We demonstrate that it possesses a "left-moving" conformal stress tensor on $\\Sigma_1$ ($\\Sigma_2$) in a BRST cohomology, which generates the Virasoro algebra with the conventional commutation relations. The central charge of the Virasoro algebra has a purely geometric origin and is proportional to the Euler characteristic $\\c$ of the $\\Sigma_2$ ($\\...
The distribution of prime numbers and associated problems in number theory
International Nuclear Information System (INIS)
Some problems in number theory, namely the gaps between consecutive primes, the distribution of primes in arithmetic progressions, Brun-Titchmarsh theorem, Fermat's last theorem, The Thue equation, the gaps between square-free numbers are discussed
Cosmological Applications of Algebraic Quantum Field Theory in Curved Spacetimes
Hack, Thomas-Paul
2015-01-01
This monograph provides a largely self--contained and broadly accessible exposition of two cosmological applications of algebraic quantum field theory (QFT) in curved spacetime: a fundamental analysis of the cosmological evolution according to the Standard Model of Cosmology and a fundamental study of the perturbations in Inflation. The two central sections of the book dealing with these applications are preceded by sections containing a pedagogical introduction to the subject as well as introductory material on the construction of linear QFTs on general curved spacetimes with and without gauge symmetry in the algebraic approach, physically meaningful quantum states on general curved spacetimes, and the backreaction of quantum fields in curved spacetimes via the semiclassical Einstein equation. The target reader should have a basic understanding of General Relativity and QFT on Minkowski spacetime, but does not need to have a background in QFT on curved spacetimes or the algebraic approach to QFT. In particul...
Cosmological applications of algebraic quantum field theory in curved spacetimes
Hack, Thomas-Paul
2016-01-01
This book provides a largely self-contained and broadly accessible exposition on two cosmological applications of algebraic quantum field theory (QFT) in curved spacetime: a fundamental analysis of the cosmological evolution according to the Standard Model of Cosmology; and a fundamental study of the perturbations in inflation. The two central sections of the book dealing with these applications are preceded by sections providing a pedagogical introduction to the subject. Introductory material on the construction of linear QFTs on general curved spacetimes with and without gauge symmetry in the algebraic approach, physically meaningful quantum states on general curved spacetimes, and the backreaction of quantum fields in curved spacetimes via the semiclassical Einstein equation is also given. The reader should have a basic understanding of General Relativity and QFT on Minkowski spacetime, but no background in QFT on curved spacetimes or the algebraic approach to QFT is required.
Schmidt number of pure states in bipartite quantum systems as an algebraic-geometric invariant
Chen, H
2001-01-01
Our previous work about algebraic-geometric invariants of the mixed states are extended and a stronger separability criterion is given. We also show that the Schmidt number of pure states in bipartite quantum systems, a classical concept, is actually an algebraic-geometric invariant.
Algebraic formulation of the operatorial perturbation theory; 2, applications
Espinosa-Müller, A W
1996-01-01
The algebraic approach to operator perturbation method has been applied to two quantum--mechanical systems ``The Stark Effect in the Harmonic Oscillator'' and ``The Generalized Zeeman Effect''. To that end, two realizations of the superoperators involved in the formalism have been carried out. The first of them has been based on the Heisenberg--Dirac algebra of \\hat{a}^\\dagger, \\hat{a}, \\hat{1} operators, the second one has been based in the angular momemtum algebra of \\hat{L}_+, \\hat{L}_- and \\hat{L}_0 operators. The successful results achieved in predicting the discrete spectra of both systems have put in evidence the reliability and accuracy of the theory.
Algebraic extensions of fields
McCarthy, Paul J
1991-01-01
""...clear, unsophisticated and direct..."" - MathThis textbook is intended to prepare graduate students for the further study of fields, especially algebraic number theory and class field theory. It presumes some familiarity with topology and a solid background in abstract algebra. Chapter 1 contains the basic results concerning algebraic extensions. In addition to separable and inseparable extensions and normal extensions, there are sections on finite fields, algebraically closed fields, primitive elements, and norms and traces. Chapter 2 is devoted to Galois theory. Besides the fundamenta
Noncommutative algebras associated to complexes and graphs
Gelfand, Israel; Gelfand, Sergei; Retakh, Vladimir
2000-01-01
This is a first of our papers devoted to "noncommutative topology and graph theory". Its origin is the paper math.QA/0002238 by I. Gelfand, V. Retakh, and R.L. Wilson where a new class of noncommutative algebras $Q_n$ was introduced. The algebra $Q_n$ is closely related to factorizations of a generic polynomial of degree $n$ over a division algebra into linear factors.
Tsue, Y.; Providência, C.; Providência, J. d.; Yamamura, M.
2012-10-01
The relation between two approaches to the su(2)-algebraic many-fermion model is discussed: (1) the BCS-Bogoliubov approach in terms of the use of the quasiparticles representing all the degrees of freedom except those related to the Cooper-pairs and (2) the conventional algebraic approach in terms of the use of the minimum weight states, from which the Cooper-pairs are excluded. In order to arrive at the goal, the idea of the quasiparticles is brought up in the conservation of the fermion number. Under the c-number replacement for the three su(2)-generators, the quasiparticles suggested in this paper are reduced to those in the BCS-Bogoliubov approach. It is also shown that the two approaches are equivalent through the c-number replacement. Further, a certain modification of the BCS-Bogoliubov approach is discussed.
Energy Technology Data Exchange (ETDEWEB)
Connes, A.; Kreimer, D. [Institut des Hautes Etudes Sci., Bures sur Yvette (France)
2000-03-01
This paper gives a complete selfcontained proof of our result (1999) showing that renormalization in quantum field theory is a special instance of a general mathematical procedure of extraction of finite values based on the Riemann-Hilbert problem. We shall first show that for any quantum field theory, the combinatorics of Feynman graphs gives rise to a Hopf algebra H which is commutative asan algebra. It is the dual Hopf algebra of the enveloping algebra of a Lie algebra G whose basis is labelled by the one particle irreducible Feynman graphs. The Lie bracket of two such graphs is computed from insertions of one graph in the other and vice versa. The corresponding Lie group G is the group of characters of H. We show then that, using dimensional regularization, the bare (unrenormalized) theory gives rise to a loop {gamma}(z) element of G, z element of C, where C is a small circle of complex dimensions around the integer dimension D of space-time. Our main result is that the renormalized theory is just the evaluation at z=D of the holomorphic part {gamma}{sub +} of the Birkhoff decomposition of {gamma}. We begin to analyse the group G and show that it is a semi-direct product of an easily understood abelian group by a highly non-trivial group closely tied up with groups of diffeomorphisms. (orig.)
Algebras, dialgebras, and polynomial identities
Bremner, Murray R
2012-01-01
This is a survey of some recent developments in the theory of associative and nonassociative dialgebras, with an emphasis on polynomial identities and multilinear operations. We discuss associative, Lie, Jordan, and alternative algebras, and the corresponding dialgebras; the KP algorithm for converting identities for algebras into identities for dialgebras; the BSO algorithm for converting operations in algebras into operations in dialgebras; Lie and Jordan triple systems, and the corresponding disystems; and a noncommutative version of Lie triple systems based on the trilinear operation abc-bca. The paper concludes with a conjecture relating the KP and BSO algorithms, and some suggestions for further research. Most of the original results are joint work with Raul Felipe, Luiz A. Peresi, and Juana Sanchez-Ortega.
Boolean functions of an odd number of variables with maximum algebraic immunity
Institute of Scientific and Technical Information of China (English)
LI Na; QI WenFeng
2007-01-01
In this paper, we study Boolean functions of an odd number of variables with maximum algebraic immunity, We identify three classes of such functions, and give some necessary conditions of such functions, which help to examine whether a Boolean function of an odd number of variables has the maximum algebraic immunity. Further, some necessary conditions for such functions to have also higher nonlinearity are proposed, and a class of these functions are also obtained. Finally,we present a sufficient and necessary condition for Boolean functions of an odd number of variables to achieve maximum algebraic immunity and to be also 1-resilient.
Shafarevich, I
1994-01-01
This book, the first printing of which was published as volume 38 of the Encyclopaedia of Mathematical Sciences, presents a modern approach to homological algebra, based on the systematic use of the terminology and ideas of derived categories and derived functors. The book contains applications of homological algebra to the theory of sheaves on topological spaces, to Hodge theory, and to the theory of modules over rings of algebraic differential operators (algebraic D-modules). The authors Gelfand and Manin explain all the main ideas of the theory of derived categories. Both authors are well-known researchers and the second, Manin, is famous for his work in algebraic geometry and mathematical physics. The book is an excellent reference for graduate students and researchers in mathematics and also for physicists who use methods from algebraic geometry and algebraic topology.
The Planar Algebra of a Semisimple and Cosemisimple Hopf Algebra
Indian Academy of Sciences (India)
Vijay Kodiyalam; V S Sunder
2006-11-01
To a semisimple and cosemisimple Hopf algebra over an algebraically closed field, we associate a planar algebra defined by generators and relations and show that it is a connected, irreducible, spherical, non-degenerate planar algebra with non-zero modulus and of depth two. This association is shown to yield a bijection between (the isomorphism classes, on both sides, of) such objects.
Fermi-Dirac statistics and the number theory
Kubasiak, A; Zakrzewski, J; Lewenstein, M
2005-01-01
We relate the Fermi-Dirac statistics of an ideal Fermi gas in a harmonic trap to partitions of given integers into distinct parts, studied in number theory. Using methods of quantum statistical physics we derive analytic expressions for cumulants of the probability distribution of the number of different partitions.
On graded decomposition numbers for cyclotomic Hecke algebras in quantum characteristic 2
Evseev, Anton
2013-01-01
Brundan and Kleshchev introduced graded decomposition numbers for representations of cyclotomic Hecke algebras of type $A$, which include group algebras of symmetric groups. Graded decomposition numbers are certain Laurent polynomials, whose values at 1 are the usual decomposition numbers. We show that in quantum characteristic 2 every such polynomial has non-zero coefficients either only in odd or only in even degrees. As a consequence, we find the first examples of graded decomposition numb...
Algebraic discrete Morse theory for the hull resolution
Norén, Patrik
2015-01-01
We study how powerful algebraic discrete Morse theory is when applied to hull resolutions. The main result describes all cases when the hull resolution of the edge ideal of the complement of a triangle-free graph can be made minimal using algebraic discrete Morse theory.
APPLICATION OF INFORMATION THEORY AND A.S.C. ANALYSIS FOR EXPERIMENTAL RESEARCH IN NUMBER THEORY
Directory of Open Access Journals (Sweden)
Lutsenko Y. V.
2014-03-01
Full Text Available Is it possible to automate the study of the properties of numbers and their relationship so that the results of this study can be formulated in the form of statements, indicating the specific quantity of information stored in them? To answer this question it is offered to apply the same method that is widely tested and proved in studies of real objects and their relations in various fields to study the properties of numbers in the theory of numbers namely - the automated system-cognitive analysis (A.S.C. analysis, based on information theory
K-theory of locally finite graph C∗-algebras
Iyudu, Natalia
2013-09-01
We calculate the K-theory of the Cuntz-Krieger algebra OE associated with an infinite, locally finite graph, via the Bass-Hashimoto operator. The formulae we get express the Grothendieck group and the Whitehead group in purely graph theoretic terms. We consider the category of finite (black-and-white, bi-directed) subgraphs with certain graph homomorphisms and construct a continuous functor to abelian groups. In this category K0 is an inductive limit of K-groups of finite graphs, which were calculated in Cornelissen et al. (2008) [3]. In the case of an infinite graph with the finite Betti number we obtain the formula for the Grothendieck group K0(OE)=Z, where β(E) is the first Betti number and γ(E) is the valency number of the graph E. We note that in the infinite case the torsion part of K0, which is present in the case of a finite graph, vanishes. The Whitehead group depends only on the first Betti number: K1(OE)=Z. These allow us to provide a counterexample to the fact, which holds for finite graphs, that K1(OE) is the torsion free part of K0(OE).
An introduction to Clifford algebras and spinors
Vaz, Jayme
2016-01-01
This text explores how Clifford algebras and spinors have been sparking a collaboration and bridging a gap between Physics and Mathematics. This collaboration has been the consequence of a growing awareness of the importance of algebraic and geometric properties in many physical phenomena, and of the discovery of common ground through various touch points: relating Clifford algebras and the arising geometry to so-called spinors, and to their three definitions (both from the mathematical and physical viewpoint). The main point of contact are the representations of Clifford algebras and the periodicity theorems. Clifford algebras also constitute a highly intuitive formalism, having an intimate relationship to quantum field theory. The text strives to seamlessly combine these various viewpoints and is devoted to a wider audience of both physicists and mathematicians. Among the existing approaches to Clifford algebras and spinors this book is unique in that it provides a didactical presentation of the topic and i...
Hopf algebras of rooted forests, cocyles, and free Rota-Baxter algebras
Zhang, Tianjie; Gao, Xing; Guo, Li
2016-10-01
The Hopf algebra and the Rota-Baxter algebra are the two algebraic structures underlying the algebraic approach of Connes and Kreimer to renormalization of perturbative quantum field theory. In particular, the Hopf algebra of rooted trees serves as the "baby model" of Feynman graphs in their approach and can be characterized by certain universal properties involving a Hochschild 1-cocycle. Decorated rooted trees have also been applied to study Feynman graphs. We will continue the study of universal properties of various spaces of decorated rooted trees with such a 1-cocycle, leading to the concept of a cocycle Hopf algebra. We further apply the universal properties to equip a free Rota-Baxter algebra with the structure of a cocycle Hopf algebra.
Factorizations of invertible operators and $K$-theory of $C^*$-algebras
Zhang, Shuang
1992-01-01
Let $\\Scr A$ be a unital C*-algebra. We describe \\it K-skeleton factorizations \\rm of all invertible operators on a Hilbert C*-module $\\Scr H_{\\Scr A}$, in particular on $\\Scr H=l^2$, with the Fredholm index as an invariant. We then outline the isomorphisms $K_0(\\Scr A) \\cong \\pi _{2k}([p]_0)\\cong \\pi _{2k} ({GL}^p_r(\\Scr A))$ and $K_1(\\Scr A)\\cong \\pi _{2k+1}([p]_0)\\cong \\pi _{2k+1}(GL^p_r(\\Scr A))$ for $k\\ge 0 $, where $[p]_0$ denotes the class of all compact perturbations of a projection $...
Basic matrix algebra and transistor circuits
Zelinger, G
1963-01-01
Basic Matrix Algebra and Transistor Circuits deals with mastering the techniques of matrix algebra for application in transistors. This book attempts to unify fundamental subjects, such as matrix algebra, four-terminal network theory, transistor equivalent circuits, and pertinent design matters. Part I of this book focuses on basic matrix algebra of four-terminal networks, with descriptions of the different systems of matrices. This part also discusses both simple and complex network configurations and their associated transmission. This discussion is followed by the alternative methods of de
On Lagrange＇s Algebraic Equation Theory and Its Influence%拉格朗日的代数方程求解理论及其影响
Institute of Scientific and Technical Information of China (English)
赵增逊
2012-01-01
Lagrange＇s theory of algebraic equations is an integral part of the resolution history of algebraic equations,and the theory has an important impact on later algebraist.To reveal the content of Lagrange＇s algebraic equations theory,express its far-reaching implications,the article,based on original literature, concisely describes the content of Lagrange＇s algebraic equations theory and its influence.Therefore, understanding Lagrange＇s theory of algebraic equations is not only conducive to the understanding of the auxiliary equation theory,the connotation of permutation theory,but also is more helpful to find out the entire resolution history of algebraic equation.%拉格朗日的代数方程求解理论是整个代数方程求解史中不可或缺的一部分,并且该理论对以后的代数学家产生了重要的影响。为展示拉格朗日代数方程求解理论的内容,说明该理论产生的深远影响,从原始文献出发,叙述了拉格朗日的代数方程求解理论的内容,重点阐述了该理论产生的重要影响。因此,清楚拉格朗日的代数方程求解理论不仅有利于了解辅助方程理论、置换思想的内涵,更有利于清楚整个代数方程的求解历史。
Algebra model and security analysis for cryptographic protocols
Institute of Scientific and Technical Information of China (English)
HUAI Jinpeng; LI Xianxian
2004-01-01
More and more cryptographic protocols have been used to achieve various security requirements of distributed systems in the open network environment. However cryptographic protocols are very difficult to design and analyze due to the complexity of the cryptographic protocol execution, and a large number of problems are unsolved that range from the theory framework to the concrete analysis technique. In this paper, we build a new algebra called cryptographic protocol algebra (CPA) for describing the message operations with many cryptographic primitives, and proposed a new algebra model for cryptographic protocols based on the CPA. In the model, expanding processes of the participant's knowledge on the protocol runs are characterized with some algebraic notions such as subalgebra, free generator and polynomial algebra, and attack processes are modeled with a new notion similar to that of the exact sequence used in homological algebra. Then we develope a mathematical approach to the cryptographic protocol security analysis. By using algebraic techniques, we have shown that for those cryptographic protocols with some symmetric properties, the execution space generated by an arbitrary number of participants may boil down to a smaller space generated by several honest participants and attackers. Furthermore we discuss the composability problem of cryptographic protocols and give a sufficient condition under which the protocol composed of two correct cryptographic protocols is still correct, and we finally offer a counterexample to show that the statement may not be true when the condition is not met.
Basic algebraic topology and its applications
Adhikari, Mahima Ranjan
2016-01-01
This book provides an accessible introduction to algebraic topology, a ﬁeld at the intersection of topology, geometry and algebra, together with its applications. Moreover, it covers several related topics that are in fact important in the overall scheme of algebraic topology. Comprising eighteen chapters and two appendices, the book integrates various concepts of algebraic topology, supported by examples, exercises, applications and historical notes. Primarily intended as a textbook, the book oﬀers a valuable resource for undergraduate, postgraduate and advanced mathematics students alike. Focusing more on the geometric than on algebraic aspects of the subject, as well as its natural development, the book conveys the basic language of modern algebraic topology by exploring homotopy, homology and cohomology theories, and examines a variety of spaces: spheres, projective spaces, classical groups and their quotient spaces, function spaces, polyhedra, topological groups, Lie groups and cell complexes, etc. T...
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.
K-theory of the chair tiling via AF-algebras
Julien, Antoine; Savinien, Jean
2016-08-01
We compute the K-theory groups of the groupoid C∗-algebra of the chair tiling, using a new method. We use exact sequences of Putnam to compute these groups from the K-theory groups of the AF-algebras of the substitution and the induced lower dimensional substitutions on edges and vertices.
DEFF Research Database (Denmark)
Schmidt, Thomas Lundsgaard
2016-01-01
such a map, generalising the transformation groupoid of a local homeomorphism first introduced by Renault in \\cite{re}. We conduct a detailed study of the relationship between the dynamics of $\\phi$, the properties of these groupoids, the structure of their corresponding reduced groupoid $C^*$-algebras......, and, for certain classes of maps, the K-theory of these algebras. When the map $\\phi$ is transitive, we show that the algebras $C^*_r(\\Gamma_\\phi)$ and $C^*_r(\\Gamma_\\phi^+)$ are purely infinite and satisfy the Universal Coefficient Theorem. Furthermore, we find necessary and sufficient conditions for...... simplicity of these algebras in terms of dynamical properties of $\\phi$. We proceed to consider the situation when the algebras are non-simple, and describe the primitive ideal spectrum in this case. We prove that any irreducible representation factors through the $C^*$-algebra of the reduction of the...
Fundamental number theory with applications
Mollin, Richard A
2008-01-01
An update of the most accessible introductory number theory text available, Fundamental Number Theory with Applications, Second Edition presents a mathematically rigorous yet easy-to-follow treatment of the fundamentals and applications of the subject. The substantial amount of reorganizing makes this edition clearer and more elementary in its coverage. New to the Second Edition Removal of all advanced material to be even more accessible in scope New fundamental material, including partition theory, generating functions, and combinatorial number theory Expa
Boundary quotients of the Toeplitz algebra of the affine semigroup over the natural numbers
Brownlowe, Nathan; Laca, Marcelo; Raeburn, Iain
2010-01-01
We study the Toeplitz algebra $\\TT(\\N\\rtimes\\N^\\times)$ and three quotients of this algebra: the $C^*$-algebra $\\qn$ recntly introduced by Cuntz, and two new ones, which we call the additive and multiplicative boundary quotients. These quotients are universal for Nica-covariant representations of $\\N\\rtimes\\N^\\times$ satisfying extra relations, and can be realised as partial crossed products. We use the structure theory for partial crossed products to prove a uniqueness theorem for the additive boundary quotient, and use the recent analysis of KMS states on $\\TT(\
Masoero, Davide; Raimondo, Andrea; Valeri, Daniele
2016-09-01
We assess the ODE/IM correspondence for the quantum g -KdV model, for a non-simply laced Lie algebra g. This is done by studying a meromorphic connection with values in the Langlands dual algebra of the affine Lie algebra g^{(1)} , and constructing the relevant {Ψ} -system among subdominant solutions. We then use the {Ψ} -system to prove that the generalized spectral determinants satisfy the Bethe Ansatz equations of the quantum g -KdV model. We also consider generalized Airy functions for twisted Kac-Moody algebras and we construct new explicit solutions to the Bethe Ansatz equations. The paper is a continuation of our previous work on the ODE/IM correspondence for simply-laced Lie algebras.
Masoero, Davide; Valeri, Daniele
2015-01-01
We assess the ODE/IM correspondence for the quantum $\\mathfrak{g}$-KdV model, for a non-simply laced Lie algebra $\\mathfrak{g}$. This is done by studying a meromorphic connection with values in the Langlands dual algebra of the affine Lie algebra ${\\mathfrak{g}}^{(1)}$, and constructing the relevant $\\Psi$-system among subdominant solutions. We then use the $\\Psi$-system to prove that the generalized spectral determinants satisfy the Bethe Ansatz equations of the quantum $\\mathfrak{g}$-KdV model. We also consider generalized Airy functions for twisted Kac--Moody algebras and we construct new explicit solutions to the Bethe Ansatz equations. The paper is a continuation of our previous work on the ODE/IM correspondence for simply-laced Lie algebras.
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
The Number Crunch game: a simple vehicle for building algebraic reasoning skills
Sugden, Steve
2012-03-01
A newspaper numbers game based on simple arithmetic relationships is discussed. Its potential to give students of elementary algebra practice in semi-ad hoc reasoning and to build general arithmetic reasoning skills is explored.
Unsolved problems in number theory
Guy, Richard K
1994-01-01
Unsolved Problems in Number Theory contains discussions of hundreds of open questions, organized into 185 different topics. They represent numerous aspects of number theory and are organized into six categories: prime numbers, divisibility, additive number theory, Diophantine equations, sequences of integers, and miscellaneous. To prevent repetition of earlier efforts or duplication of previously known results, an extensive and up-to-date collection of references follows each problem. In the second edition, not only extensive new material has been added, but corrections and additions have been included throughout the book.
Distribution of real algebraic numbers of arbitrary degree in short intervals
Bernik, V. I.; Götze, F.
2015-02-01
We consider real algebraic numbers α of degree \\operatorname{deg}α=n and height H=H(α). There are intervals I\\subset{R} of length \\vert I\\vert whose interiors contain no real algebraic numbers α of any degree with H(α)\\lt\\frac12\\vert I\\vert-1. We prove that one can always find a constant c_1=c_1(n) such that if Q is a positive integer and Q \\gt c_1\\vert I\\vert-1, then the interior of I contains at least c_2(n)Qn+1\\vert I\\vert real algebraic numbers α with \\operatorname{deg}α=n and H(α)≤ Q. We use this result to solve a problem of Bugeaud on the regularity of the set of real algebraic numbers in short intervals.
A precise result on the arithmetic of non-principal orders in algebraic number fields
Philipp, Andreas
2011-01-01
Let $R$ be an order in an algebraic number field. If $R$ is a principal order, then many explicit results on its arithmetic are available. Among others, $R$ is half-factorial if and only if the class group of $R$ has at most two elements. Much less is known for non-principal orders. Using a new semigroup theoretical approach, we study half-factoriality and further arithmetical properties for non-principal orders in algebraic number fields.
Higher order Fourier analysis as an algebraic theory II
Szegedy, Balazs
2009-01-01
Our approach to higher order Fourier analysis is to study the ultra product of finite (or compact) Abelian groups on which a new algebraic theory appears. This theory has consequences on finite (or compact) groups usually in the form of approximative statements. The present paper is the second part of a paper in which higher order characters and decompositions were introduced. We generalize the concept of the Pontrjagin dual group and introduce higher order versions of it. We study the algebraic structure of the higher order dual groups. We prove a simple formula for the Gowers uniformity norms in terms of higher order decompositions. We present a simple spectral algorithm to produce higher order decompositions. We briefly study a multi linear version of Fourier analysis. Along these lines we obtain new inverse theorems for Gowers's norms.
Garrett, Paul B
2007-01-01
Designed for an advanced undergraduate- or graduate-level course, Abstract Algebra provides an example-oriented, less heavily symbolic approach to abstract algebra. The text emphasizes specifics such as basic number theory, polynomials, finite fields, as well as linear and multilinear algebra. This classroom-tested, how-to manual takes a more narrative approach than the stiff formalism of many other textbooks, presenting coherent storylines to convey crucial ideas in a student-friendly, accessible manner. An unusual feature of the text is the systematic characterization of objects by universal
Index maps in the K-theory of graph algebras
DEFF Research Database (Denmark)
Meier Carlsen, Toke; Eilers, Søren; Tomforde, Mark
2012-01-01
Let C*(E) be the graph C*-algebra associated to a graph E and let J be a gauge-invariant ideal in C*(E). We compute the cyclic six-term exact sequence in K-theory associated to the extension in terms of the adjacency matrix associated to E. The ordered six-term exact sequence is a complete stable...
Geometry, algebra and applications from mechanics to cryptography
Encinas, Luis; Gadea, Pedro; María, Mª
2016-01-01
This volume collects contributions written by different experts in honor of Prof. Jaime Muñoz Masqué. It covers a wide variety of research topics, from differential geometry to algebra, but particularly focuses on the geometric formulation of variational calculus; geometric mechanics and field theories; symmetries and conservation laws of differential equations, and pseudo-Riemannian geometry of homogeneous spaces. It also discusses algebraic applications to cryptography and number theory. It offers state-of-the-art contributions in the context of current research trends. The final result is a challenging panoramic view of connecting problems that initially appear distant.
Relative integral basis for algebraic number fields
Mohmood Haghighi
1986-01-01
At first conditions are given for existence of a relative integral basis for OK≅Okn−1⊕I with [K;k]=n. Then the constrtiction of the ideal I in OK≅Okn−1⊕I is given for proof of existence of a relative integral basis for OK4(m1,m2)/Ok(m3). Finally existence and construction of the relative integral basis for OK6(n3,−3)/Ok3(n3),OK6(n3,−3)/Ok2(−3) for some values of n are given.
The whole truth about whole numbers an elementary introduction to number theory
Forman, Sylvia
2015-01-01
The Whole Truth About Whole Numbers is an introduction to the field of Number Theory for students in non-math and non-science majors who have studied at least two years of high school algebra. Rather than giving brief introductions to a wide variety of topics, this book provides an in-depth introduction to the field of Number Theory. The topics covered are many of those included in an introductory Number Theory course for mathematics majors, but the presentation is carefully tailored to meet the needs of elementary education, liberal arts, and other non-mathematical majors. The text covers logic and proofs, as well as major concepts in Number Theory, and contains an abundance of worked examples and exercises to both clearly illustrate concepts and evaluate the students’ mastery of the material.
Nonmonotonic logics and algebras
Institute of Scientific and Technical Information of China (English)
CHAKRABORTY Mihir Kr; GHOSH Sujata
2008-01-01
Several nonmonotonie logic systems together with their algebraic semantics are discussed. NM-algebra is defined.An elegant construction of an NM-algebra starting from a Boolean algebra is described which gives rise to a few interesting algebraic issues.
Pinsky, Ross G
2014-01-01
The primary intent of the book is to introduce an array of beautiful problems in a variety of subjects quickly, pithily and completely rigorously to graduate students and advanced undergraduates. The book takes a number of specific problems and solves them, the needed tools developed along the way in the context of the particular problems. It treats a mélange of topics from combinatorial probability theory, number theory, random graph theory and combinatorics. The problems in this book involve the asymptotic analysis of a discrete construct as some natural parameter of the system tends to infinity. Besides bridging discrete mathematics and mathematical analysis, the book makes a modest attempt at bridging disciplines. The problems were selected with an eye toward accessibility to a wide audience, including advanced undergraduate students. The book could be used for a seminar course in which students present the lectures.
Feynman-diagram evaluation in the electroweak theory with computer algebra
Weiglein, G.
2001-01-01
The evaluation of quantum corrections in the theory of the electroweak and strong interactions via higher-order Feynman diagrams requires complicated and laborious calculations, which however can be structured in a strictly algorithmic way. These calculations are ideally suited for the application of computer algebra systems, and computer algebra has proven to be a very valuable tool in this field already over several decades. It is sketched how computer algebra is presently applied in evalua...
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.
APPLICATION OF INFORMATION THEORY AND A.S.C. ANALYSIS FOR EXPERIMENTAL RESEARCH IN NUMBER THEORY
Lutsenko Y. V.
2014-01-01
Is it possible to automate the study of the properties of numbers and their relationship so that the results of this study can be formulated in the form of statements, indicating the specific quantity of information stored in them? To answer this question it is offered to apply the same method that is widely tested and proved in studies of real objects and their relations in various fields to study the properties of numbers in the theory of numbers namely - the automated system-cognitive anal...
Hopf Algebra Structure of a Model Quantum Field Theory
Solomon, A I; Blasiak, P; Horzela, A; Penson, K A
2006-01-01
Recent elegant work on the structure of Perturbative Quantum Field Theory (PQFT) has revealed an astonishing interplay between analysis(Riemann Zeta functions), topology (Knot theory), combinatorial graph theory (Feynman Diagrams) and algebra (Hopf structure). The difficulty inherent in the complexities of a fully-fledged field theory such as PQFT means that the essential beauty of the relationships between these areas can be somewhat obscured. Our intention is to display some, although not all, of these structures in the context of a simple zero-dimensional field theory; i.e. a quantum theory of non-commuting operators which do not depend on spacetime. The combinatorial properties of these boson creation and annihilation operators, which is our chosen example, may be described by graphs, analogous to the Feynman diagrams of PQFT, which we show possess a Hopf algebra structure. Our approach is based on the partition function for a boson gas. In a subsequent note in these Proceedings we sketch the relationship...
Finite and Infinite W Algebras and their Applications
Tjin, T
1993-01-01
In this paper we present a systematic study of $W$ algebras from the Hamiltonian reduction point of view. The Drinfeld-Sokolov (DS) reduction scheme is generalized to arbitrary $sl_2$ embeddings thus showing that a large class of W algebras can be viewed as reductions of affine Lie algebras. The hierarchies of integrable evolution equations associated to these classical W algebras are constructed as well as the generalized Toda field theories which have them as Noether symmetry algebras. The problem of quantising the DS reductions is solved for arbitrary $sl_2$ embeddings and it is shown that any W algebra can be embedded into an affine Lie algebra. This also provides us with an algorithmic method to write down free field realizations for arbitrary W algebras. Just like affine Lie algebras W algebras have finite underlying structures called `finite W algebras'. We study the classical and quantum theory of these algebras, which play an important role in the theory of ordinary W algebras, in detail as well as s...
Short-distance analysis for algebraic euclidean field theory
Schlingemann, D
1999-01-01
Recently D. Buchholz and R. Verch have proposed a method for implementing in algebraic quantum field theory ideas from renormalization group analysis of short-distance (high energy) behavior by passing to certain scaling limit theories. Buchholz and Verch distinguish between different types of theories where the limit is unique, degenerate, or classical, and the method allows in principle to extract the `ultraparticle' content of a given model, i.e. to identify particles (like quarks and gluons) that are not visible at finite distances due to `confinement'. It is therefore of great importance for the physical interpretation of the theory. The method has been illustrated in a simple model in with some rather surprising results. This paper will focus on the question how the short distance behavior of models defined by euclidean means is reflected in the corresponding behavior of their Minkowski counterparts. More specifically, we shall prove that if a euclidean theory has some short distance limit, then it is p...
On the algebraic K-theory of the complex K-theory spectrum
Ausoni, Christian
2010-03-01
Let p>3 be a prime, let ku be the connective complex K-theory spectrum, and let K(ku) be the algebraic K-theory spectrum of ku. We study the p-primary homotopy type of the spectrum K(ku) by computing its mod (p,v_1) homotopy groups. We show that up to a finite summand, these groups form a finitely generated free module over a polynomial algebra F_p[b], where b is a class of degree 2p+2 defined as a higher Bott element.
On Fusion Algebras and Modular Matrices
Gannon, T
1997-01-01
We consider the fusion algebras arising in e.g. Wess-Zumino-Witten conformal field theories, affine Kac-Moody algebras at positive integer level, and quantum groups at roots of unity. Using properties of the modular matrix $S$, we find small sets of primary fields (equivalently, sets of highest weights) which can be identified with the variables of a polynomial realization of the $A_r$ fusion algebra at level $k$. We prove that for many choices of rank $r$ and level $k$, the number of these variables is the minimum possible, and we conjecture that it is in fact minimal for most $r$ and $k$. We also find new, systematic sources of zeros in the modular matrix $S$. In addition, we obtain a formula relating the entries of $S$ at fixed points, to entries of $S$ at smaller ranks and levels. Finally, we identify the number fields generated over the rationals by the entries of $S$, and by the fusion (Verlinde) eigenvalues.
Current algebra and the local nature of symmetries in local quantum theory
International Nuclear Information System (INIS)
In this report we mainly discuss the problem of finding local observables which measure the charges in a volume smaller than their localization region, in particular providing the existence of local observables with a specific physical interpretation. In the same way we can also establish the existence of a version of the current algebra structure. Similar local observables can be constructed for the energy-momentum; we also comment on the local implementation of supersymmetries. (orig./HSI)
Directory of Open Access Journals (Sweden)
M. Sivasubramanian
2009-01-01
Full Text Available Problem statement: After formulating the special theory of relativity in 1905, Albert Einstein politely remarked: for velocities that are greater than light our deliberations become meaningless. In 1962, Sudarshan and his co-researchers proposed a hypothesis that particles/objects whose rest mass is imaginary can travel by birth faster than light. After the publication of Sudarshans research, many scholars began to probe into faster than light phenomena. In extended relativity, many properties of tachyons have been found. But still this micro second, the velocity of a free tachyon with respect to us is unknown. In this research the researchers found tachyon velocity. Approach: In this research, Einsteins variation of mass with velocity equation was transformed into quadratic equation. We introduced a new hypothesis to find the roots of the quadratic equation. Results: By introducing a new hypothesis in tachyon algebra, the researchers found that the velocity of superluminal objects with respect to us is v = c√3 where c is the velocity of the light. Conclusion/Recommendations: But the road to tachyon is too long. Hereafter it is up to experimental physicists to establish the existence/generation of tachyons.
Analytic number theory, approximation theory, and special functions in honor of Hari M. Srivastava
Rassias, Michael
2014-01-01
This book, in honor of Hari M. Srivastava, discusses essential developments in mathematical research in a variety of problems. It contains thirty-five articles, written by eminent scientists from the international mathematical community, including both research and survey works. Subjects covered include analytic number theory, combinatorics, special sequences of numbers and polynomials, analytic inequalities and applications, approximation of functions and quadratures, orthogonality, and special and complex functions. The mathematical results and open problems discussed in this book are presented in a simple and self-contained manner. The book contains an overview of old and new results, methods, and theories toward the solution of longstanding problems in a wide scientific field, as well as new results in rapidly progressing areas of research. The book will be useful for researchers and graduate students in the fields of mathematics, physics, and other computational and applied sciences.
A Mathematical Theory of Origami Numbers and Constructions
Alperin, Roger
1999-01-01
We give a hierarchial set of axioms for mathematical origami. The hierachy gives the fields of Pythagorean numbers, first discussed by Hilbert, the field of Euclidean constructible numbers which are obtained by the usual constructions of straightedge and compass, and the Origami numbers, which is also the field generated from the intersections of conics or equivalently the marked ruler.
On the Isomorphism Conjecture in algebraic K-theory
Bartels, Arthur; Farrell, Tom; Jones, Lowell; Reich, Holger
2001-01-01
The Isomorphism Conjecture is a conceptional approach towards a calculation of the algebraic K-theory of a group ring RG, where G is an infinite group. In this paper we prove the conjecture in dimensions n
Combinatorial Algebra for second-quantized Quantum Theory
Blasiak, P; Solomon, A I; Horzela, A; Penson, K A
2010-01-01
We describe an algebra G of diagrams which faithfully gives a diagrammatic representation of the structures of both the Heisenberg-Weyl algebra H - the associative algebra of the creation and annihilation operators of quantum mechanics - and U(L_H), the enveloping algebra of the Heisenberg Lie algebra L_H. We show explicitly how G may be endowed with the structure of a Hopf algebra, which is also mirrored in the structure of U(L_H). While both H and U(L_H) are images of G, the algebra G has a richer structure and therefore embodies a finer combinatorial realization of the creation-annihilation system, of which it provides a concrete model.
Multidimensional analysis algebras and systems for science and engineering
Hart, George W
1995-01-01
This book deals with the mathematical properties of dimensioned quantities, such as length, mass, voltage, and viscosity. Beginning with a careful examination of how one expresses the numerical results of a measurement and uses these results in subsequent manipulations, the author rigorously constructs the notion of dimensioned numbers and discusses their algebraic structure. The result is a unification of linear algebra and traditional dimensional analysis that can be extended from the scalars to which the traditional analysis is perforce restricted to multidimensional vectors of the sort frequently encountered in engineering, systems theory, economics, and other applications.
Rings of quotients of incidence algebras and path algebras
DEFF Research Database (Denmark)
Esparza, Eduardo Ortega
2006-01-01
We compute the maximal right/left/symmetric rings of quotients of finite dimensional incidence and graph algebras. We show that these rings of quotients are Morita equivalent to incidence algebras and path algebras respectively, with respect to simpler, well determined partially ordered sets...
Algebraic Independence and Mahler's method
Zorin, Evgeniy
2011-01-01
We give some new results on algebraic independence within Mahler's method, including algebraic independence of values at transcendental points. We also give some new measures of algebraic independence for infinite series of numbers. In particular, our results furnishes, for $n\\geq 1$ arbitrarily large, new examples of sets $(\\theta_1,...,\\theta_n)\\in\\mrr^n$ normal in the sense of definition formulated by Grigory Chudnovsky (1980).
Simplicities and Automorphisms of a Sp ecial Infinite Dimensional Lie Algebra
Institute of Scientific and Technical Information of China (English)
YU De-min; LI Ai-hua
2013-01-01
In this paper, a special infinite dimensional Lie algebra is studied. The infinite dimensional Lie algebra appears in the fields of conformal theory, mathematical physics, statistic mechanics and Hamilton operator. The infinite dimensional Lie algebras is pop-ularized Virasoro-like Lie algebra. Isomorphisms, homomorphisms, ideals of the infinite dimensional Lie algebra are studied.
Perturbative algebraic quantum field theory at finite temperature
Energy Technology Data Exchange (ETDEWEB)
Lindner, Falk
2013-08-15
We present the algebraic approach to perturbative quantum field theory for the real scalar field in Minkowski spacetime. In this work we put a special emphasis on the inherent state-independence of the framework and provide a detailed analysis of the state space. The dynamics of the interacting system is constructed in a novel way by virtue of the time-slice axiom in causal perturbation theory. This method sheds new light in the connection between quantum statistical dynamics and perturbative quantum field theory. In particular it allows the explicit construction of the KMS and vacuum state for the interacting, massive Klein-Gordon field which implies the absence of infrared divergences of the interacting theory at finite temperature, in particular for the interacting Wightman and time-ordered functions.
C∗-completions and the DFR-algebra
Forger, Michael; Paulino, Daniel V.
2016-02-01
The aim of this paper is to present the construction of a general family of C∗-algebras which includes, as a special case, the "quantum spacetime algebra" introduced by Doplicher, Fredenhagen, and Roberts. It is based on an extension of the notion of C∗-completion from algebras to bundles of algebras, compatible with the usual C∗-completion of the appropriate algebras of sections, combined with a novel definition for the algebra of the canonical commutation relations using Rieffel's theory of strict deformation quantization. Taking the C∗-algebra of continuous sections vanishing at infinity, we arrive at a functor associating a C∗-algebra to any Poisson vector bundle and recover the original DFR-algebra as a particular example.
Graph theory, irreducibility, and structural analysis of differential-algebraic equation systems
Pryce, John D.; Nedialkov, Nedialko S.; Tan, Guangning
2014-01-01
The $\\Sigma$-method for structural analysis of a differential-algebraic equation (DAE) system produces offset vectors from which the sparsity pattern of a system Jacobian is derived. This pattern implies a block-triangular form (BTF) of the DAE that can be exploited to speed up numerical solution. The paper compares this fine BTF with the usually coarser BTF derived from the sparsity pattern of the \\sigmx. It defines a Fine-Block Graph with weighted edges, which gives insight into the relatio...
Lattice-ordered matrix algebras over real algebraic numbers UFD rings
Li, Fei; Qiu, Derong
2010-01-01
Let $ R \\subset \\R \\cap \\bar{\\Q} $ be a unique factorization domain. In this paper, we obtain a sufficient and necessary condition such that the Weinberg's conjecture holds for the $ n \\times n $ matrix ring $ M_{n}(R) \\ (n \\geq 2). $ Moreover, we give all the lattice orders (up to isomorphisms) on a full $ 2 \\times 2 $ matrix algebra over $ R. $
The geometry of blueprints. Part I: Algebraic background and scheme theory
Lorscheid, Oliver
2011-01-01
A blueprint generalizes both commutative (semi-)rings and commutative monoids. This generalization allows a simultaneous treatment of ideals resp. congruences for rings and monoids and leads to a common scheme theory. In particular, it bridges the gap between usual schemes and $\\mathbb{F}_1$-schemes (after Kato, Deitmar and Connes-Consani). Beside this unification, the category of blueprints contains new interesting objects as "improved" cyclotomic field extensions $\\mathbb{F}_{1^n}$ of $\\mathbb{F}_1$ and "archimedean valuation rings". It also yields a notion of semi-ring schemes. This first paper lays the foundation for subsequent projects, which are devoted to the following problems: Tits' idea of Chevalley groups over $\\mathbb{F}_1$, congruence schemes, sheaf cohomology and $K$-theory and a unified view on analytic geometry over $\\mathbb{F}_1$, adic spaces (after Huber), analytic spaces (after Berkovich) and tropical geometry.
Nourani, Cyrus F
2014-01-01
IntroductionCategorical PreliminariesCategories and FunctorsMorphismsFunctorsCategorical ProductsNatural TransformationsProducts on Models Preservation of LimitsModel Theory and Topoi More on Universal ConstructionsChapter ExercisesInfinite Language CategoriesBasicsLimits and Infinitary Languages Generic Functors and Language String ModelsFunctorial Morphic Ordered Structure ModelsChapter ExercisesFunctorial Morphic Ordered Structure ModelsFunctorial Fragment M
The Logical Syntax of Number Words: Theory, Acquisition and Processing
Musolino, Julien
2009-01-01
Recent work on the acquisition of number words has emphasized the importance of integrating linguistic and developmental perspectives [Musolino, J. (2004). The semantics and acquisition of number words: Integrating linguistic and developmental perspectives. "Cognition 93", 1-41; Papafragou, A., Musolino, J. (2003). Scalar implicatures: Scalar…
DEFF Research Database (Denmark)
Schmidt, Thomas Lundsgaard; Thomsen, Klaus
2015-01-01
We consider a construction of $C^*$-algebras from continuous piecewise monotone maps on the circle which generalizes the crossed product construction for homeomorphisms and more generally the construction of Renault, Deaconu and Anantharaman-Delaroche for local homeomorphisms. Assuming that the map...... is surjective and not locally injective we give necessary and sufficient conditions for the simplicity of the $C^*$-algebra and show that it is then a Kirchberg algebra. We provide tools for the calculation of the K-theory groups and turn them into an algorithmic method for Markov maps....
Homomorphisms between JC*-algebras and Lie C*-algebras
Institute of Scientific and Technical Information of China (English)
Chun Gil PARK; Jin Chuan HOU; Sei Qwon OH
2005-01-01
It is shown that every almost *-homomorphism h: A → B of a unital JC*-algebra A to a unital JC*-algebra B is a *-homomorphism when h(rx) = rh(x) (r ＞ 1) for all x ∈ A, and that every almost linear mapping h: A → B is a *-homomorphism when h(2nu o y) = h(2nu) o h(y),h(3nu o y) = h(3nu) o h(y) or h(qnu o y) = h(qnu) o h(y) for all unitaries u ∈ A, all y ∈ A, and n = 0, 1, Here the numbers 2, 3, q depend on the functional equations given in the almost linear mappings.We prove that every almost *-homomorphism h: A → B of a unital Lie C*-algebra A to a unital Lie C*-algebra B is a *-homomorphism when h(rx) = rh(x) (r ＞ 1) for all x ∈ A.
Algebraic independence of real numbers with low density of nonzero digits (survey)
Kaneko, Hajime
2010-07-01
We study transcendence and algebraic independence of the value f(z;w(n)) = ∑n = 1∞zw(n), where z is an algebraic number with 0<|z|<1 and w(n) (n = 1,2,…) is a strictly increasing sequence of nonnegative integers. First we survey the case where w(n) is lacunary. In our main results, we give criteria for algebraic independence of the values f(z;w(n)) in the case where z = 1/α for some integer α greater than 1 and w(n) is not lacunary. In particular, using our criteria, we deduce that the uncountable set {ηl = ∑n = 1∞α-[fl(n)]|l≥1} is algebraically independent, where fl(n) = exp((log n)1+l).
Algebra and geometry of Hamilton's quaternions
Krishnaswami, Govind S
2016-01-01
Inspired by the relation between the algebra of complex numbers and plane geometry, William Rowan Hamilton sought an algebra of triples for application to three dimensional geometry. Unable to multiply and divide triples, he invented a non-commutative division algebra of quadruples, in what he considered his most significant work, generalizing the real and complex number systems. We give a motivated introduction to quaternions and discuss how they are related to Pauli matrices, rotations in three dimensions, the three sphere, the group SU(2) and the celebrated Hopf fibrations.
Algebraic connectivity and graph robustness.
Energy Technology Data Exchange (ETDEWEB)
Feddema, John Todd; Byrne, Raymond Harry; Abdallah, Chaouki T. (University of New Mexico)
2009-07-01
Recent papers have used Fiedler's definition of algebraic connectivity to show that network robustness, as measured by node-connectivity and edge-connectivity, can be increased by increasing the algebraic connectivity of the network. By the definition of algebraic connectivity, the second smallest eigenvalue of the graph Laplacian is a lower bound on the node-connectivity. In this paper we show that for circular random lattice graphs and mesh graphs algebraic connectivity is a conservative lower bound, and that increases in algebraic connectivity actually correspond to a decrease in node-connectivity. This means that the networks are actually less robust with respect to node-connectivity as the algebraic connectivity increases. However, an increase in algebraic connectivity seems to correlate well with a decrease in the characteristic path length of these networks - which would result in quicker communication through the network. Applications of these results are then discussed for perimeter security.
Peng, Jie; Kan, Haibin
It is well known that Boolean functions used in stream and block ciphers should have high algebraic immunity to resist algebraic attacks. Up to now, there have been many constructions of Boolean functions achieving the maximum algebraic immunity. In this paper, we present several constructions of rotation symmetric Boolean functions with maximum algebraic immunity on an odd number of variables which are not symmetric, via a study of invertible cyclic matrices over the binary field. In particular, we generalize the existing results and introduce a new method to construct all the rotation symmetric Boolean functions that differ from the majority function on two orbits. Moreover, we prove that their nonlinearities are upper bounded by 2^{n-1}-\\binom{n-1}{\\lfloor\\frac{n}{2}\\rfloor}+2(n-6).
Algebraic structure and Poisson method for a weakly nonholonomic system
Institute of Scientific and Technical Information of China (English)
无
2011-01-01
The algebraic structure and the Poisson method for a weakly nonholonomic system are studied.The differential equations of motion of the system can be written in a contravariant algebra form and its algebraic structure is discussed.The Poisson theory for the systems which possess Lie algebra structure is generalized to the weakly nonholonomic system.An example is given to illustrate the application of the result.
Asghar, HJ; Steinfeld, R.; Li, S.; Kaafar, MA; Pieprzyk, J
2015-01-01
Human identification protocols are challenge-response protocols that rely on human computational ability to reply to random challenges from the server based on a public function of a shared secret and the challenge to authenticate the human user. One security criterion for a human identification protocol is the number of challenge-response pairs the adversary needs to observe before it can deduce the secret. In order to increase this number, protocol designers have tried to construct protocol...
Algebraic and geometric structures of Special Relativity
Giulini, Domenico
2006-01-01
I review, some of the algebraic and geometric structures that underlie the theory of Special Relativity. This includes a discussion of relativity as a symmetry principle, derivations of the Lorentz group, its composition law, its Lie algebra, comparison with the Galilei group, Einstein synchronization, the lattice of causally and chronologically complete regions in Minkowski space, rigid motion (the Noether-Herglotz theorem), and the geometry of rotating reference frames. Representation-theor...
Vague Congruences and Quotient Lattice Implication Algebras
Directory of Open Access Journals (Sweden)
Xiaoyan Qin
2014-01-01
Full Text Available The aim of this paper is to further develop the congruence theory on lattice implication algebras. Firstly, we introduce the notions of vague similarity relations based on vague relations and vague congruence relations. Secondly, the equivalent characterizations of vague congruence relations are investigated. Thirdly, the relation between the set of vague filters and the set of vague congruences is studied. Finally, we construct a new lattice implication algebra induced by a vague congruence, and the homomorphism theorem is given.
$K_0$ of locally finite graph $C^*$-algebras via Betti numbers
Iyudu, Natalia
2010-01-01
We calculate the K-theory of Cuntz-Krieger algebras associated to locally finite infinite graphs via Bass-Hashimoto operator. This calculation gives a result which is expressed in purely graph theoretic terms, that is we get a formula via the first Betti number of the graph. One of our main tools is the consideration of embeddings of monoids of the Murray von Neumann equivalence classes of projections. We calculate inductive limits in the associated matrix category.
International Nuclear Information System (INIS)
Expository notes on Clifford algebras and spinors with a detailed discussion of Majorana, Weyl, and Dirac spinors. The paper is meant as a review of background material, needed, in particular, in now fashionable theoretical speculations on neutrino masses. It has a more mathematical flavour than the over twenty-six-year-old Introduction to Majorana masses [M84] and includes historical notes and biographical data on past participants in the story. (author)
Algebra, Arithmetic, and Geometry
Tschinkel, Yuri
2009-01-01
The two volumes of "Algebra, Arithmetic, and Geometry: In Honor of Y.I. Manin" are composed of invited expository articles and extensions detailing Manin's contributions to the subjects, and are in celebration of his 70th birthday. The well-respected and distinguished contributors include: Behrend, Berkovich, Bost, Bressler, Calaque, Carlson, Chambert-Loir, Colombo, Connes, Consani, Dabrowski, Deninger, Dolgachev, Donaldson, Ekedahl, Elsenhans, Enriques, Etingof, Fock, Friedlander, Geemen, Getzler, Goncharov, Harris, Iskovskikh, Jahnel, Kaledin, Kapranov, Katz, Kaufmann, Kollar, Kont
Hagerty, Gary; Smith, Stanley; Goodwin, Danielle
2010-01-01
In 2001, Black Hills State University (BHSU) redesigned college algebra to use the computer-based mastery learning program, Assessment and Learning in Knowledge Spaces [1], historical development of concepts modules, whole class discussions, cooperative activities, relevant applications problems, and many fewer lectures. This resulted in a 21%…
Picard-Fuchs equations, Integrable Systems, and higher Algebraic K-theory
del Angel, Pedro L.; Müller-Stach, Stefan
2002-01-01
This paper continues our previous work done in math.AG/0008207 and is an attempt to establish a conceptual framework which generalizes the work of Manin on the relation between non-linear second order ODEs of type Painleve VI and integrable systems. The principle behind everything is a strong interaction between K-theory and Picard-Fuchs type differential equations via Abel-Jacobi maps. Our main result is an extension of a theorem of Donagi and Markman to our setup.
Combinatorics and commutative algebra
Stanley, Richard P
1996-01-01
Some remarkable connections between commutative algebra and combinatorics have been discovered in recent years. This book provides an overview of two of the main topics in this area. The first concerns the solutions of linear equations in nonnegative integers. Applications are given to the enumeration of integer stochastic matrices (or magic squares), the volume of polytopes, combinatorial reciprocity theorems, and related results. The second topic deals with the face ring of a simplicial complex, and includes a proof of the Upper Bound Conjecture for Spheres. An introductory chapter giving background information in algebra, combinatorics and topology broadens access to this material for non-specialists. New to this edition is a chapter surveying more recent work related to face rings, focusing on applications to f-vectors. Included in this chapter is an outline of the proof of McMullen's g-conjecture for simplicial polytopes based on toric varieties, as well as a discussion of the face rings of such special ...
On numbers and endgames: Combinatorial game theory in chess endgames
Elkies, Noam D.
1999-01-01
In an investigation of the applications of Combinatorial Game Theory to chess, we construct novel mutual Zugzwang positions, explain an otherwise mysterious pawn endgame from "A Guide to Chess Endings" (Euwe and Hooper), show positions containing non-integer values (fractions, switches, tinies, and loopy games), and pose open problems concerning the values that may be realized by positions on either standard or nonstandard chessboards.
Uniform Algebras Over Complete Valued Fields
Mason, Jonathan W
2012-01-01
UNIFORM algebras have been extensively investigated because of their importance in the theory of uniform approximation and as examples of complex Banach algebras. An interesting question is whether analogous algebras exist when a complete valued field other than the complex numbers is used as the underlying field of the algebra. In the Archimedean setting, this generalisation is given by the theory of real function algebras introduced by S. H. Kulkarni and B. V. Limaye in the 1980s. This thesis establishes a broader theory accommodating any complete valued field as the underlying field by involving Galois automorphisms and using non-Archimedean analysis. The approach taken keeps close to the original definitions from the Archimedean setting. Basic function algebras are defined and generalise real function algebras to all complete valued fields. Several examples are provided. Each basic function algebra is shown to have a lattice of basic extensions related to the field structure. In the non-Archimedean settin...
A Representation of Real and Complex Numbers in Quantum Theory
Benioff, Paul
2005-01-01
A quantum theoretic representation of real and complex numbers is described here as equivalence classes of Cauchy sequences of quantum states of finite strings of qubits. There are 4 types of qubits each with associated single qubit annihilation creation (a-c) operators that give the state and location of each qubit type on a 2 dimensional integer lattice. The string states, defined as finite products of creation operators acting on the qubit vacuum state correspond to complex rational number...
Forward error correction based on algebraic-geometric theory
A Alzubi, Jafar; M Chen, Thomas
2014-01-01
This book covers the design, construction, and implementation of algebraic-geometric codes from Hermitian curves. Matlab simulations of algebraic-geometric codes and Reed-Solomon codes compare their bit error rate using different modulation schemes over additive white Gaussian noise channel model. Simulation results of Algebraic-geometric codes bit error rate performance using quadrature amplitude modulation (16QAM and 64QAM) are presented for the first time and shown to outperform Reed-Solomon codes at various code rates and channel models. The book proposes algebraic-geometric block turbo codes. It also presents simulation results that show an improved bit error rate performance at the cost of high system complexity due to using algebraic-geometric codes and Chase-Pyndiah’s algorithm simultaneously. The book proposes algebraic-geometric irregular block turbo codes (AG-IBTC) to reduce system complexity. Simulation results for AG-IBTCs are presented for the first time.
Lattice W-algebras and logarithmic CFTs
Gainutdinov, A. M.; Saleur, H.; Tipunin, I. Yu.
2012-01-01
This paper is part of an effort to gain further understanding of 2D Logarithmic Conformal Field Theories (LCFTs) by exploring their lattice regularizations. While all work so far has dealt with the Virasoro algebra (or the product of left and right Virasoro), the best known (although maybe not the most relevant physically) LCFTs in the continuum are characterized by a W-algebra symmetry, whose presence is powerful, but difficult to understand physically. We explore here the origin of this sym...
Realisation of a Lorentz algebra in Lorentz violating theory
Energy Technology Data Exchange (ETDEWEB)
Ganguly, Oindrila [S. N. Bose National Centre for Basic Sciences, Kolkata (India)
2012-11-15
A Lorentz non-invariant higher derivative effective action in flat spacetime, characterised by a constant vector, can be made invariant under infinitesimal Lorentz transformations by restricting the allowed field configurations. These restricted fields are defined as functions of the background vector in such a way that background dependence of the dynamics of the physical system is no longer manifest. We show here that they also provide a field basis for the realisation of a Lorentz algebra and allow the construction of a Poincare invariant symplectic two-form on the covariant phase space of the theory. (orig.)
Central simple Poisson algebras
Institute of Scientific and Technical Information of China (English)
SU; Yucai; XU; Xiaoping
2004-01-01
Poisson algebras are fundamental algebraic structures in physics and symplectic geometry. However, the structure theory of Poisson algebras has not been well developed. In this paper, we determine the structure of the central simple Poisson algebras related to locally finite derivations, over an algebraically closed field of characteristic zero.The Lie algebra structures of these Poisson algebras are in general not finitely-graded.
Certain associative algebras similar to $U(sl_{2})$ and Zhu's algebra $A(V_{L})$
Dong, Chongying; Li, Haisheng; Mason, Geoffrey
1996-01-01
It is proved that Zhu's algebra for vertex operator algebra associated to a positive-definite even lattice of rank one is a finite-dimensional semiprimitive quotient algebra of certain associative algebra introduced by Smith. Zhu's algebra for vertex operator algebra associated to any positive-definite even lattice is also calculated and is related to a generalization of Smith's algebra.
Homogeneous spaces, multi-moment maps and (2,3)-trivial algebras
DEFF Research Database (Denmark)
Madsen, T. B.; Swann, A.
2011-01-01
For geometries with a closed three-form we briefly overview the notion of multi-moment maps. We then give concrete examples of multi-moment maps for homogeneous hypercomplex and nearly Kahler manifolds. A special role in the theory is played by Lie algebras with second and third Betti numbers equal...... to zero. These we call (2,3)-trivial. We provide a number of examples of such algebras including a complete list in dimensions up to and including five....
Perturbative quantization of Yang-Mills theory with classical double as gauge algebra
Energy Technology Data Exchange (ETDEWEB)
Ruiz Ruiz, F. [Universidad Complutense de Madrid, Departamento de Fisica Teorica I, Madrid (Spain)
2016-02-15
Perturbative quantization of Yang-Mills theory with a gauge algebra given by the classical double of a semisimple Lie algebra is considered. The classical double of a real Lie algebra is a nonsemisimple real Lie algebra that admits a nonpositive definite invariant metric, the indefiniteness of the metric suggesting an apparent lack of unitarity. It is shown that the theory is UV divergent at one loop and that there are no radiative corrections at higher loops. One-loop UV divergences are removed through renormalization of the coupling constant, thus introducing a renormalization scale. The terms in the classical action that would spoil unitarity are proved to be cohomologically trivial with respect to the Slavnov-Taylor operator that controls gauge invariance for the quantum theory. Hence they do not contribute gauge invariant radiative corrections to the quantum effective action and the theory is unitary. (orig.)
Algebraicity and Asymptotics: An explosion of BPS indices from algebraic generating series
Mainiero, Tom
2016-01-01
It is an observation of Kontsevich and Soibelman that generating series that produce certain (generalized) Donaldson Thomas invariants are secretly algebraic functions over the rationals. From a physical perspective this observation arises naturally for DT invariants that appear as BPS indices in theories of class S[A]: explicit algebraic equations (that completely determine these series) can be derived using (degenerate) spectral networks. In this paper, we conjecture an algebraic equation associated to DT invariants for the Kronecker 3-quiver with dimension vectors (3n,2n), n>0 in the non-trivial region of its stability parameter space. Using a functional equation due to Reineke, we show algebraicity of generating series for Euler characteristics of stable moduli for the Kronecker m-quiver assuming algebraicity of generating series for DT invariants. In the latter part of the paper we deduce very explicit results on the asymptotics of DT invariants/Euler characteristics under the assumption of algebraicity ...
Fundamental structures of algebra and discrete mathematics
Foldes, Stephan
2011-01-01
Introduces and clarifies the basic theories of 12 structural concepts, offering a fundamental theory of groups, rings and other algebraic structures. Identifies essentials and describes interrelationships between particular theories. Selected classical theorems and results relevant to current research are proved rigorously within the theory of each structure. Throughout the text the reader is frequently prompted to perform integrated exercises of verification and to explore examples.
Blanchard, Philippe; Hellmich, Mario; Ługiewicz, Piotr; Olkiewicz, Robert
Quantum mechanics is the greatest revision of our conception of the character of the physical world since Newton. Consequently, David Hilbert was very interested in quantum mechanics. He and John von Neumann discussed it frequently during von Neumann's residence in Göttingen. He published in 1932 his book Mathematical Foundations of Quantum Mechanics. In Hilbert's opinion it was the first exposition of quantum mechanics in a mathematically rigorous way. The pioneers of quantum mechanics, Heisenberg and Dirac, neither had use for rigorous mathematics nor much interest in it. Conceptually, quantum theory as developed by Bohr and Heisenberg is based on the positivism of Mach as it describes only observable quantities. It first emerged as a result of experimental data in the form of statistical observations of quantum noise, the basic concept of quantum probability.
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.
Kappa-deformation of Heisenberg algebra and coalgebra; generalized Poincare algebras and R-matrix
Meljanac, Stjepan; Strajn, Rina
2012-01-01
We deform Heisenberg algebra and corresponding coalgebra by twist. We present undeformed and deformed tensor identities. Coalgebras for the generalized Poincar\\'{e} algebras have been constructed. The exact universal $R$-matrix for the deformed Heisenberg (co)algebra is found. We show, up to the third order in the deformation parameter, that in the case of $\\kappa$-Poincar\\'{e} Hopf algebra this $R$-matrix can be expressed in terms of Poincar\\'{e} generators only. This implies that the states of any number of identical particles can be defined in a $\\kappa$-covariant way.
Degeneration, Rigidity and Irreducible Components of Hopf Algebras
Institute of Scientific and Technical Information of China (English)
Abdenacer Makhlouf
2005-01-01
The aim of this work is to discuss the concepts of degeneration, deformation and rigidity of Hopf algebras and to apply them to the geometric study of the varieties of Hopf algebras. The main result is the description of the n-dimensional rigid Hopf algebras and the irreducible components for n ＜ 14 and n = p2 with p a prime number.
Lie groups, lie algebras, and representations an elementary introduction
Hall, Brian
2015-01-01
This textbook treats Lie groups, Lie algebras and their representations in an elementary but fully rigorous fashion requiring minimal prerequisites. In particular, the theory of matrix Lie groups and their Lie algebras is developed using only linear algebra, and more motivation and intuition for proofs is provided than in most classic texts on the subject. In addition to its accessible treatment of the basic theory of Lie groups and Lie algebras, the book is also noteworthy for including: a treatment of the Baker–Campbell–Hausdorff formula and its use in place of the Frobenius theorem to establish deeper results about the relationship between Lie groups and Lie algebras motivation for the machinery of roots, weights and the Weyl group via a concrete and detailed exposition of the representation theory of sl(3;C) an unconventional definition of semisimplicity that allows for a rapid development of the structure theory of semisimple Lie algebras a self-contained construction of the representations of compac...
Vortex lattice theory: A linear algebra approach
Chamoun, George C.
Vortex lattices are prevalent in a large class of physical settings that are characterized by different mathematical models. We present a coherent and generalized Hamiltonian fluid mechanics-based formulation that reduces all vortex lattices into a classic problem in linear algebra for a non-normal matrix A. Via Singular Value Decomposition (SVD), the solution lies in the null space of the matrix (i.e., we require nullity( A) > 0) as well as the distribution of its singular values. We demonstrate that this approach provides a good model for various types of vortex lattices, and makes it possible to extract a rich amount of information on them. The contributions of this thesis can be classified into four main points. The first is asymmetric equilibria. A 'Brownian ratchet' construct was used which converged to asymmetric equilibria via a random walk scheme that utilized the smallest singular value of A. Distances between configurations and equilibria were measured using the Frobenius norm ||·||F and 2-norm ||·||2, and conclusions were made on the density of equilibria within the general configuration space. The second contribution used Shannon Entropy, which we interpret as a scalar measure of the robustness, or likelihood of lattices to occur in a physical setting. Third, an analytic model was produced for vortex street patterns on the sphere by using SVD in conjunction with expressions for the center of vorticity vector and angular velocity. Equilibrium curves within the configuration space were presented as a function of the geometry, and pole vortices were shown to have a critical role in the formation and destruction of vortex streets. The fourth contribution entailed a more complete perspective of the streamline topology of vortex streets, linking the bifurcations to critical points on the equilibrium curves.
Lie algebras and linear differential equations.
Brockett, R. W.; Rahimi, A.
1972-01-01
Certain symmetry properties possessed by the solutions of linear differential equations are examined. For this purpose, some basic ideas from the theory of finite dimensional linear systems are used together with the work of Wei and Norman on the use of Lie algebraic methods in differential equation theory.
A note on the "logarithmic-W_3" octuplet algebra and its Nichols algebra
Semikhatov, A M
2013-01-01
We describe a Nichols-algebra-motivated construction of an octuplet chiral algebra that is a "W_3-counterpart" of the triplet algebra of (p,1) logarithmic models of two-dimensional conformal field theory.
$A\\mathcal{T}$-Algebras and Extensions of $AT$-Algebras
Indian Academy of Sciences (India)
Hongliang Yao
2010-04-01
Lin and Su classified $A\\mathcal{T}$-algebras of real rank zero. This class includes all $A\\mathbb{T}$-algebras of real rank zero as well as many *-algebras which are not stably finite. An $A\\mathcal{T}$-algebra often becomes an extension of an $A\\mathbb{T}$-algebra by an -algebra. In this paper, we show that there is an essential extension of an $A\\mathbb{T}$-algebra by an -algebra which is not an $A\\mathcal{T}$-algebra. We describe a characterization of an extension of an $A\\mathbb{T}$-algebra by an -algebra if is an $A\\mathcal{T}$-algebra.
Affine and degenerate affine BMW algebras: The center
Daugherty, Zajj; Virk, Rahbar
2011-01-01
The degenerate affine and affine BMW algebras arise naturally in the context of Schur-Weyl duality for orthogonal and symplectic Lie algebras and quantum groups, respectively. Cyclotomic BMW algebras, affine Hecke algebras, cyclotomic Hecke algebras, and their degenerate versions are quotients. In this paper the theory is unified by treating the orthogonal and symplectic cases simultaneously; we make an exact parallel between the degenerate affine and affine cases via a new algebra which takes the role of the affine braid group for the degenerate setting. A main result of this paper is an identification of the centers of the affine and degenerate affine BMW algebras in terms of rings of symmetric functions which satisfy a "cancellation property" or "wheel condition" (in the degenerate case, a reformulation of a result of Nazarov). Miraculously, these same rings also arise in Schubert calculus, as the cohomology and K-theory of isotropic Grassmanians and symplectic loop Grassmanians. We also establish new inte...
The logical syntax of number words: theory, acquisition and processing.
Musolino, Julien
2009-04-01
Recent work on the acquisition of number words has emphasized the importance of integrating linguistic and developmental perspectives [Musolino, J. (2004). The semantics and acquisition of number words: Integrating linguistic and developmental perspectives. Cognition93, 1-41; Papafragou, A., Musolino, J. (2003). Scalar implicatures: Scalar implicatures: Experiments at the semantics-pragmatics interface. Cognition, 86, 253-282; Hurewitz, F., Papafragou, A., Gleitman, L., Gelman, R. (2006). Asymmetries in the acquisition of numbers and quantifiers. Language Learning and Development, 2, 76-97; Huang, Y. T., Snedeker, J., Spelke, L. (submitted for publication). What exactly do numbers mean?]. Specifically, these studies have shown that data from experimental investigations of child language can be used to illuminate core theoretical issues in the semantic and pragmatic analysis of number terms. In this article, I extend this approach to the logico-syntactic properties of number words, focusing on the way numerals interact with each other (e.g. Three boys are holding two balloons) as well as with other quantified expressions (e.g. Three boys are holding each balloon). On the basis of their intuitions, linguists have claimed that such sentences give rise to at least four different interpretations, reflecting the complexity of the linguistic structure and syntactic operations involved. Using psycholinguistic experimentation with preschoolers (n=32) and adult speakers of English (n=32), I show that (a) for adults, the intuitions of linguists can be verified experimentally, (b) by the age of 5, children have knowledge of the core aspects of the logical syntax of number words, (c) in spite of this knowledge, children nevertheless differ from adults in systematic ways, (d) the differences observed between children and adults can be accounted for on the basis of an independently motivated, linguistically-based processing model [Geurts, B. (2003). Quantifying kids. Language
Duplex numbers, diffusion systems, and generalized quantum mechanics
KOCIK, Jerzy
2012-01-01
We show that the relation between the Schr\\"odinger equation and diffusion processes has an algebraic nature and can be revealed via the structure of "duplex numbers." This helps one to clarify that quantum mechanics cannot be reduced to diffusion theory. Also, a generalized version of quantum mechanics where $\\C$ is replaced by a normed algebra with a unit is proposed.
Knots, BPS States, and Algebraic Curves
Garoufalidis, Stavros; Kucharski, Piotr; Sułkowski, Piotr
2016-08-01
We analyze relations between BPS degeneracies related to Labastida-Mariño-Ooguri-Vafa (LMOV) invariants and algebraic curves associated to knots. We introduce a new class of such curves, which we call extremal A-polynomials, discuss their special properties, and determine exact and asymptotic formulas for the corresponding (extremal) BPS degeneracies. These formulas lead to nontrivial integrality statements in number theory, as well as to an improved integrality conjecture, which is stronger than the known M-theory integrality predictions. Furthermore, we determine the BPS degeneracies encoded in augmentation polynomials and show their consistency with known colored HOMFLY polynomials. Finally, we consider refined BPS degeneracies for knots, determine them from the knowledge of super-A-polynomials, and verify their integrality. We illustrate our results with twist knots, torus knots, and various other knots with up to 10 crossings.
Linear algebra and projective geometry
Baer, Reinhold
2005-01-01
Geared toward upper-level undergraduates and graduate students, this text establishes that projective geometry and linear algebra are essentially identical. The supporting evidence consists of theorems offering an algebraic demonstration of certain geometric concepts. These focus on the representation of projective geometries by linear manifolds, of projectivities by semilinear transformations, of collineations by linear transformations, and of dualities by semilinear forms. These theorems lead to a reconstruction of the geometry that constituted the discussion's starting point, within algebra
Perturbative algebraic quantum field theory an introduction for mathematicians
Rejzner, Kasia
2016-01-01
Perturbative Algebraic Quantum Field Theory (pAQFT), the subject of this book, is a complete and mathematically rigorous treatment of perturbative quantum field theory (pQFT) that doesn’t require the use of divergent quantities. We discuss in detail the examples of scalar fields and gauge theories and generalize them to QFT on curved spacetimes. pQFT models describe a wide range of physical phenomena and have remarkable agreement with experimental results. Despite this success, the theory suffers from many conceptual problems. pAQFT is a good candidate to solve many, if not all of these conceptual problems. Chapters 1-3 provide some background in mathematics and physics. Chapter 4 concerns classical theory of the scalar field, which is subsequently quantized in chapters 5 and 6. Chapter 7 covers gauge theory and chapter 8 discusses QFT on curved spacetimes and effective quantum gravity. The book aims to be accessible researchers and graduate students interested in the mathematical foundations of pQFT are th...
Al- Khwarizmi and axiomatic foundation of algebra
International Nuclear Information System (INIS)
This paper intends to investigate the axiomatic foundations of algebra, as they were presented in the book of algebra of al-Khwarizmi (9 th century), and as they were developed in many subsequent Arabic works. The paper gives also a description of algebra evolution towards a discipline independent ofgeometry and arithmetic: the two disciplines whosemarriage had led to its birth.By an in depth reading of some details in the text of al Khwarizmi , we concluded that this mathematician intended to lay down the axiomatic foundations of that new discipline. His resort to arithmetical and geometrical means was a way of making his theory more accessible. He used them to justify the axioms: those that were not explicitly introduced per se, and those that were remained implicit. The paper also relies on some unedited writingsof al-Khwarizmi's successors, which could shedlight on the ways they used to consolidate the foundations of algebra and improve its methods. (author)
L^2-Betti numbers, isomorphism conjectures and noncommutative localization
Reich, Holger
2003-01-01
In this paper we discuss how the question about the rationality of L^2-Betti numbers is related to the Isomorphism Conjecture in algebraic K-theory and why in this context noncommutative localization appears as an important tool.
On Euclid’s algorithm and elementary number theory
Backhouse, Roland; Ferreira, João F.
2011-01-01
Algorithms can be used to prove and to discover new theorems. This paper shows how algorithmic skills in general, and the notion of invariance in particular, can be used to derive many results from Euclid’s algorithm. We illustrate how to use the algorithm as a verification interface (i.e., how to verify theorems) and as a construction interface (i.e., how to investigate and derive new theorems). The theorems that we verify are well-known and most of them are included in standard number-t...
Introduction to abstract algebra
Nicholson, W Keith
2012-01-01
Praise for the Third Edition ". . . an expository masterpiece of the highest didactic value that has gained additional attractivity through the various improvements . . ."-Zentralblatt MATH The Fourth Edition of Introduction to Abstract Algebra continues to provide an accessible approach to the basic structures of abstract algebra: groups, rings, and fields. The book's unique presentation helps readers advance to abstract theory by presenting concrete examples of induction, number theory, integers modulo n, and permutations before the abstract structures are defined. Readers can immediately be
Application of Fuzzy Algebra in Automata theory
Directory of Open Access Journals (Sweden)
Kharatti Lal
2016-06-01
Full Text Available In our first application we consider strings of fuzzy singletons as input to a fuzzy finite state machine. The notion of fuzzy automata was introduced in [58]. There has been considerable growth in the area [18]. In this section present a theory of free fuzzy monoids and apply the results to the area of fuzzy automata. In fuzzy automata, the set of strings of input symbols can be considered to be a free monoid. We introduced the motion of fuzzy strings of input symbols, where the fuzzy strings from free fuzzy submonoids of the free monoids of input strings. We show that fuzzy automata with fuzzy input are equivalent to fuzzy automata with crisp input. Hence the result of fuzzy automata theory can be immediately applied to those of fuzzy automata theory with fuzzy input. The result are taken from [7] and [34].
An algebraic approach towards the classification of 2 dimensional conformal field theories
International Nuclear Information System (INIS)
This thesis treats an algebraic method for the construction of 2-dimensional conformal field theories. The method consists of the study of the representation theory of the Virasoro algebra and suitable extensions of this. The classification of 2-dimensional conformal field theories is translated into the classification of combinations of representations which satisfy certain consistence conditions (unitarity and modular invariance). For a certain class of 2-dimensional field theories, namely the one with central charge c = 1 from the theory of Kac-Moody algebra's. there exist indications, but as yet mainly hope, that this construction will finally lead to a classification of 2-dimensional conformal field theories. 182 refs.; 2 figs.; 26 tabs
Monakhov, Vadim V
2016-01-01
We introduced fermionic variables in complex modules over real Clifford algebras of even dimension which are analog of the Witt basis. We built primitive idempotents which are a set of equivalent Clifford vacuums. It is shown that the modules are decomposed into direct sum of minimal left ideals generated by these idempotents and that the fermionic variables can be considered as more fundamental mathematical objects than spinors.
Dechant, Pierre-Philippe
2016-01-01
In this paper, we make the case that Clifford algebra is the natural framework for root systems and reflection groups, as well as related groups such as the conformal and modular groups: The metric that exists on these spaces can always be used to construct the corresponding Clifford algebra. Via the Cartan-Dieudonn\\'e theorem all the transformations of interest can be written as products of reflections and thus via `sandwiching' with Clifford algebra multivectors. These multivector groups can be used to perform concrete calculations in different groups, e.g. the various types of polyhedral groups, and we treat the example of the tetrahedral group $A_3$ in detail. As an aside, this gives a constructive result that induces from every 3D root system a root system in dimension four, which hinges on the facts that the group of spinors provides a double cover of the rotations, the space of 3D spinors has a 4D euclidean inner product, and with respect to this inner product the group of spinors can be shown to be cl...
LOCAL AUTOMORPHISMS OF SEMISIMPLE ALGEBRAS AND GROUP ALGEBRAS
Institute of Scientific and Technical Information of China (English)
Wang Dengyin; Guan Qi; Zhan9 Dongju
2011-01-01
Let F be a field of characteristic not 2,and let A be a finite-dimensional semisimple F-algebra.All local automorphisms of A are characterized when all the degrees of A are larger than 1.If F is further assumed to be an algebraically closed field of characteristic zero,K a finite group,FK the group algebra of K over F,then all local automorphisms of FK are also characterized.