Sample records for algebraic topology foundations

  1. Algebraic Topology

    Oliver, Bob; Pawałowski, Krzystof


    As part of the scientific activity in connection with the 70th birthday of the Adam Mickiewicz University in Poznan, an international conference on algebraic topology was held. In the resulting proceedings volume, the emphasis is on substantial survey papers, some presented at the conference, some written subsequently.

  2. Boundedly controlled topology foundations of algebraic topology and simple homotopy theory

    Anderson, Douglas R


    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...

  3. On uniform topological algebras

    Azhari, M. El


    The uniform norm on a uniform normed Q-algebra is the only uniform Q-algebra norm on it. The uniform norm on a regular uniform normed Q-algebra with unit is the only uniform norm on it. Let A be a uniform topological algebra whose spectrum M (A) is equicontinuous, then A is a uniform normed algebra. Let A be a regular semisimple commutative Banach algebra, then every algebra norm on A is a Q-algebra norm on A.

  4. Algebraic topology and concurrency

    Fajstrup, Lisbeth; Raussen, Martin; Goubault, Eric


    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...

  5. Fractal Topology Foundations

    Porchon, Helene


    In this paper, we introduce the foundation of a fractal topological space constructed via a family of nested topological spaces endowed with subspace topologies, where the number of topological spaces involved in this family is related to the appearance of new structures on it. The greater the number of topological spaces we use, the stronger the subspace topologies we obtain. The fractal manifold model is brought up as an illustration of space that is locally homeomorphic to the fractal topo...

  6. Algebraic Topology Foundations of Supersymmetry and Symmetry Breaking in Quantum Field Theory and Quantum Gravity: A Review

    Ion C. Baianu


    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.

  7. Operator algebras and topology

    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)

  8. Topological convolution algebras

    Alpay, Daniel


    In this paper we introduce a new family of topological convolution algebras of the form $\\bigcup_{p\\in\\mathbb N} L_2(S,\\mu_p)$, where $S$ is a Borel semi-group in a locally compact group $G$, which carries an inequality of the type $\\|f*g\\|_p\\le A_{p,q}\\|f\\|_q\\|g\\|_p$ for $p > q+d$ where $d$ pre-assigned, and $A_{p,q}$ is a constant. We give a sufficient condition on the measures $\\mu_p$ for such an inequality to hold. We study the functional calculus and the spectrum of the elements of these algebras, and present two examples, one in the setting of non commutative stochastic distributions, and the other related to Dirichlet series.

  9. Algebraic K-theory and algebraic topology

    This contribution treats the various topological constructions of Algebraic K-theory together with the underlying homotopy theory. Topics covered include the plus construction together with its various ramifications and applications, Topological Hochschild and Cyclic Homology as well as K-theory of the ring of integers

  10. Foundations of combinatorial topology

    Pontryagin, L S


    Hailed by The Mathematical Gazette as ""an extremely valuable addition to the literature of algebraic topology,"" this concise but rigorous introductory treatment focuses on applications to dimension theory and fixed-point theorems. The lucid text examines complexes and their Betti groups, including Euclidean space, application to dimension theory, and decomposition into components; invariance of the Betti groups, with consideration of the cone construction and barycentric subdivisions of a complex; and continuous mappings and fixed points. Proofs are presented in a complete, careful, and eleg

  11. A1-algebraic topology over a field

    Morel, Fabien


    This text deals with A1-homotopy theory over a base field, i.e., with the natural homotopy theory associated to the category of smooth varieties over a field in which the affine line is imposed to be contractible. It is a natural sequel to the foundational paper on A1-homotopy theory written together with V. Voevodsky. Inspired by classical results in algebraic topology, we present new techniques, new results and applications related to the properties and computations of A1-homotopy sheaves, A1-homotogy sheaves, and sheaves with generalized transfers, as well as to algebraic vector bundles over affine smooth varieties.

  12. Hopf algebras and topological recursion

    We consider a model for topological recursion based on the Hopf algebra of planar binary trees defined by Loday and Ronco (1998 Adv. Math. 139 293–309 We show that extending this Hopf algebra by identifying pairs of nearest neighbor leaves, and thus producing graphs with loops, we obtain the full recursion formula discovered by Eynard and Orantin (2007 Commun. Number Theory Phys. 1 347–452). (paper)

  13. Topological gravity with exchange algebra

    Aoyama, S.


    A topological gravity is obtained by twisting the effective $(2,0)$ super\\-gravity. We show that this topological gravity has an infinite number of BRST invariant quantities with conformal weight $0$. They are a tower of OSp$(2,2)$ multiplets and satisfy the classical exchange algebra of OSp$(2,2)$. We argue that these BRST invariant quantities become physical operators in the quantum theory and their correlation functions are braided according to the quantum OSp$(2,2)$ group. These propertie...

  14. International Conference on Algebraic Topology

    Cohen, Ralph; Miller, Haynes; Ravenel, Douglas


    These are proceedings of an International Conference on Algebraic Topology, held 28 July through 1 August, 1986, at Arcata, California. The conference served in part to mark the 25th anniversary of the journal Topology and 60th birthday of Edgar H. Brown. It preceded ICM 86 in Berkeley, and was conceived as a successor to the Aarhus conferences of 1978 and 1982. Some thirty papers are included in this volume, mostly at a research level. Subjects include cyclic homology, H-spaces, transformation groups, real and rational homotopy theory, acyclic manifolds, the homotopy theory of classifying spaces, instantons and loop spaces, and complex bordism.

  15. Algebraic topology a first course

    Fulton, William


    To the Teacher. This book is designed to introduce a student to some of the important ideas of algebraic topology by emphasizing the re­ lations of these ideas with other areas of mathematics. Rather than choosing one point of view of modem topology (homotopy theory, simplicial complexes, singular theory, axiomatic homology, differ­ ential topology, etc.), we concentrate our attention on concrete prob­ lems in low dimensions, introducing only as much algebraic machin­ ery as necessary for the problems we meet. This makes it possible to see a wider variety of important features of the subject than is usual in a beginning text. The book is designed for students of mathematics or science who are not aiming to become practicing algebraic topol­ ogists-without, we hope, discouraging budding topologists. We also feel that this approach is in better harmony with the historical devel­ opment of the subject. What would we like a student to know after a first course in to­ pology (assuming we reject the answer: ...

  16. Topological ∗-algebras with *-enveloping Algebras II

    S J Bhatt


    Universal *-algebras *() exist for certain topological ∗-algebras called algebras with a *-enveloping algebra. A Frechet ∗-algebra has a *-enveloping algebra if and only if every operator representation of maps into bounded operators. This is proved by showing that every unbounded operator representation , continuous in the uniform topology, of a topological ∗-algebra , which is an inverse limit of Banach ∗-algebras, is a direct sum of bounded operator representations, thereby factoring through the enveloping pro-* algebra () of . Given a *-dynamical system (, , ), any topological ∗-algebra containing (, ) as a dense ∗-subalgebra and contained in the crossed product *-algebra *(, , ) satisfies ()=*(, , ). If $G = \\mathbb{R}$, if is an -invariant dense Frechet ∗-subalgebra of such that () = , and if the action on is -tempered, smooth and by continuous ∗-automorphisms: then the smooth Schwartz crossed product $S(\\mathbb{R}, B, )$ satisfies $E(S(\\mathbb{R}, B, )) = C^*(\\mathbb{R}, A, )$. When is a Lie group, the ∞-elements ∞(), the analytic elements () as well as the entire analytic elements () carry natural topologies making them algebras with a *-enveloping algebra. Given a non-unital *-algebra , an inductive system of ideals is constructed satisfying $A = C^*-\\mathrm{ind} \\lim I_$; and the locally convex inductive limit $\\mathrm{ind}\\lim I_$ is an -convex algebra with the *-enveloping algebra and containing the Pedersen ideal of . Given generators with weakly Banach admissible relations , we construct universal topological ∗-algebra (, ) and show that it has a *-enveloping algebra if and only if (, ) is *-admissible.

  17. An introduction to algebraic topology

    Rotman, Joseph J


    There is a canard that every textbook of algebraic topology either ends with the definition of the Klein bottle or is a personal communication to J. H. C. Whitehead. Of course, this is false, as a glance at the books of Hilton and Wylie, Maunder, Munkres, and Schubert reveals. Still, the canard does reflect some truth. Too often one finds too much generality and too little attention to details. There are two types of obstacle for the student learning algebraic topology. The first is the formidable array of new techniques (e. g. , most students know very little homological algebra); the second obstacle is that the basic defini­ tions have been so abstracted that their geometric or analytic origins have been obscured. I have tried to overcome these barriers. In the first instance, new definitions are introduced only when needed (e. g. , homology with coeffi­ cients and cohomology are deferred until after the Eilenberg-Steenrod axioms have been verified for the three homology theories we treat-singular, sim­ ...

  18. Sketches of a platypus: persistent homology and its algebraic foundations

    Vejdemo-Johansson, Mikael


    The subject of persistent homology has vitalized applications of algebraic topology to point cloud data and to application fields far outside the realm of pure mathematics. The area has seen several fundamentally important results that are rooted in choosing a particular algebraic foundational theory to describe persistent homology, and applying results from that theory to prove useful and important results. In this survey paper, we shall examine the various choices in use, and what they allo...

  19. Coverings of topological semi-abelian algebras

    Mucuk, Osman; Demir, Serap


    In this work, we study on a category of topological semi-abelian algebras which are topological models of given an algebraic theory T whose category of models is semi-abelian; and investigate some results on the coverings of topological models of such theories yielding semi-abelian categories. We also consider the internal groupoid structure in the semi-abelian category of T-algebras, and give a criteria for the lifting of internal groupoid structure to the covering groupoids.

  20. Algebra and topology for applications to physics

    Rozhkov, S. S.


    The principal concepts of algebra and topology are examined with emphasis on applications to physics. In particular, attention is given to sets and mapping; topological spaces and continuous mapping; manifolds; and topological groups and Lie groups. The discussion also covers the tangential spaces of the differential manifolds, including Lie algebras, vector fields, and differential forms, properties of differential forms, mapping of tangential spaces, and integration of differential forms.

  1. Al- Khwarizmi and axiomatic foundation of algebra

    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)




    Based upon Tomonoga-Rowe's many body theory, we find that the algebraic models, including IBM and FDSM are simplest extension of Rowe-Rosensteel's sp(3R).Dynkin-Gruber's subalgebra embedding method is applied to find an appropriate algebra and it's reduction chains conforming to physical requirement. The separated cases sp(6) and so(8) now appear as two branches stemming from the same root D6-O(12). Transitional ease between sp(6) and so(8) is inherently include.

  3. Algebraic topology of finite topological spaces and applications

    Barmak, Jonathan A


    This volume deals with the theory of finite topological spaces and its relationship with the homotopy and simple homotopy theory of polyhedra. The interaction between their intrinsic combinatorial and topological structures makes finite spaces a useful tool for studying problems in Topology, Algebra and Geometry from a new perspective. In particular, the methods developed in this manuscript are used to study Quillen’s conjecture on the poset of p-subgroups of a finite group and the Andrews-Curtis conjecture on the 3-deformability of contractible two-dimensional complexes. This self-contained work constitutes the first detailed exposition on the algebraic topology of finite spaces. It is intended for topologists and combinatorialists, but it is also recommended for advanced undergraduate students and graduate students with a modest knowledge of Algebraic Topology.

  4. Orlik-Solomon algebras in algebra and topology

    This is a survey of Orlik-Solomon algebras of hyperplane arrangements. These algebras first appeared in theorems due to Arnol'd, Brieskorn, and Orlik and Solomon as the cohomology algebras of the complements of complex hyperplane arrangements. Numerous applications of these algebras have subsequently been found. This survey is confined to studying Orlik-Solomon algebras per se and some of their applications to topology and combinatorics. Most of the results are taken from recent papers and preprints, although for the reader's convenience we also include relevant definitions and basic facts from the book Arrangements of hyperplanes by Orlik and Terao. For some of these facts new and more straightforward or shorter proofs are given

  5. P-commutative topological *-algebras

    If P(A) denotes the set of all continuous positive functionals on a unital complete Imc *-algebra and S(A) the extreme points of P(A), and if the spectrum of an element χ Ε A coincides with the set {f(χ): f Ε S(A)}, then A is shown to be P-commutative. Moreover, if A is unital symmetric Frechet Q Imc *-algebra, then this spectral condition is, in fact, necessary. Also, an isomorphism theorem between symmetric Frechet P-commutative Imc *-algebras is established. (author). 12 refs

  6. Algebraic definition of topological W gravity

    In this paper, the authors propose a definition of the topological W gravity using some properties of the principal three-dimensional subalgebra of a simple Lie algebra due to Kostant. In the authors' definition, structures of the two-dimensional topological gravity are naturally embedded in the extended theories. In accordance with the definition, the authors will present some explicit calculations for the W3 gravity

  7. Topics in algebraic and topological K-theory

    Baum, Paul Frank; Meyer, Ralf; Sánchez-García, Rubén; Schlichting, Marco; Toën, Bertrand


    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.

  8. Topologically Left Invariant Means on Semigroup Algebras

    Ali Ghaffari


    Let $M(S)$ be the Banach algebra of all bounded regular Borel measures on a locally compact Hausdorff semitopological semigroup with variation norm and convolution as multiplication. We obtain necessary and sufficient conditions for $M(S)^∗$ to have a topologically left invariant mean.

  9. An ideal topology type convergent theorem on scale effect algebras

    WU JunDe; ZHOU XuanChang; Minhyung CHO


    The famous Antosik-Mikusinski convergent theorem on the Abel topological groups has very extensive applications in measure theory, summation theory and other analysis fields. In this paper, we establish the theorem on a class of effect algebras equipped with the ideal topology. This paper shows also that the ideal topology of effect algebras is a useful topology in studying the quantum logic theory.

  10. On Power Idealization Filter Topologies of Lattice Implication Algebras


    The aim of this paper is to introduce power idealization filter topologies with respect to filter topologies and power ideals of lattice implication algebras, and to investigate some properties of power idealization filter topological spaces and their quotient spaces.

  11. Quantum Algebraic Approach to Refined Topological Vertex

    Awata, H; Shiraishi, J


    We establish the equivalence between the refined topological vertex of Iqbal-Kozcaz-Vafa and a certain representation theory of the quantum algebra of type W_{1+infty} introduced by Miki. Our construction involves trivalent intertwining operators Phi and Phi^* associated with triples of the bosonic Fock modules. Resembling the topological vertex, a triple of vectors in Z^2 is attached to each intertwining operator, which satisfy the Calabi-Yau and smoothness conditions. It is shown that certain matrix elements of Phi and Phi^* give the refined topological vertex C_{lambda mu nu}(t,q) of Iqbal-Kozcaz-Vafa. With another choice of basis, we recover the refined topological vertex C_{lambda mu}^nu(q,t) of Awata-Kanno. The gluing factors appears correctly when we consider any compositions of Phi and Phi^*. The spectral parameters attached to Fock spaces play the role of the K"ahler parameters.

  12. Topological isomorphisms for some universal operator algebras

    Hartz, Michael


    Let $I$ be a radical homogeneous ideal of complex polynomials in $d$ variables, and let $\\mathcal A_I$ be the norm-closed non-selfadjoint algebra generated by the compressions of the $d$-shift on Drury-Arveson space $H^2_d$ to the co-invariant subspace $H^2_d \\ominus I$. Then $\\mathcal A_I$ is the universal operator algebra for commuting row contractions subject to the relations in $I$. In this note, we study the question, under which conditions there are topological isomorphisms between two such algebras $\\mathcal A_I$ and $\\mathcal A_J$. We provide a positive answer to a conjecture of Davdison, Ramsey and Shalit: that $\\mathcal A_I$ and $\\mathcal A_J$ are topologically isomorphic if and only if there is an invertible linear map $A$ on $\\mathbb C^d$ which maps the vanishing locus of $J$ isometrically onto the vanishing locus of $I$. Most of the proof is devoted to showing that finite algebraic sums of full Fock spaces over subspaces of $\\mathbb C^d$ are closed. This allows us to show that the map $A$ induces...

  13. Barcelona Conference on Algebraic Topology

    Castellet, Manuel; Cohen, Frederick


    The papers in this collection, all fully refereed, original papers, reflect many aspects of recent significant advances in homotopy theory and group cohomology. From the Contents: A. Adem: On the geometry and cohomology of finite simple groups.- D.J. Benson: Resolutions and Poincar duality for finite groups.- C. Broto and S. Zarati: On sub-A*-algebras of H*V.- M.J. Hopkins, N.J. Kuhn, D.C. Ravenel: Morava K-theories of classifying spaces and generalized characters for finite groups.- K. Ishiguro: Classifying spaces of compact simple lie groups and p-tori.- A.T. Lundell: Concise tables of James numbers and some homotopyof classical Lie groups and associated homogeneous spaces.- J.R. Martino: Anexample of a stable splitting: the classifying space of the 4-dim unipotent group.- J.E. McClure, L. Smith: On the homotopy uniqueness of BU(2) at the prime 2.- G. Mislin: Cohomologically central elements and fusion in groups.

  14. C*-algebras over topological spaces

    Meyer, Ralf; Nest, Ryszard


    We define the filtrated K-theory of a C*-algebra over a finite topological space X and explain how to construct a spectral sequence that computes the bivariant Kasparov theory over X in terms of filtrated K-theory. For finite spaces with a totally ordered lattice of open subsets, this spectral...... sequence becomes an exact sequence as in the Universal Coefficient Theorem, with the same consequences for classification. We also exhibit an example where filtrated K-theory is not yet a complete invariant. We describe two C*-algebras over a space X with four points that have isomorphic filtrated K-theory...... without being KK(X)-equivalent. For this space X, we enrich filtrated K-theory by another K-theory functor to a complete invariant up to KK(X)-equivalence that satisfies a Universal Coefficient Theorem....

  15. Build an Early Foundation for Algebra Success

    Knuth, Eric; Stephens, Ana; Blanton, Maria; Gardiner, Angela


    Research tells us that success in algebra is a factor in many other important student outcomes. Emerging research also suggests that students who are started on an algebra curriculum in the earlier grades may have greater success in the subject in secondary school. What's needed is a consistent, algebra-infused mathematics curriculum all…

  16. Algebraic and Topological Aspects of Rough Set Theory

    Vlach, Milan


    The main purpose of this talk is to show how some widely known and well established algebraic and topological notions are closely related to notions and results introduced and rediscovered in the rough set literature.

  17. Birman-Wenzl-Murakami Algebra and Topological Basist

    周成城; 薛康; 王刚成; 孙春芳; 都桂娇


    In this paper, we use entangled states to construct 9 × 9-matrix representations of Temperley-Lieb algebra (TLA ), then a family of 9 × 9-matrix representations of Birman-Wenzl-Murakami algebra (t3 WMA ) have been presented. Based on which, three topological basis states have been found. And we apply topological basis states to recast ninedimensional BWMA into its three-dimensional counterpart. Finally, we find the topological basis states are spin singlet states in special ease.

  18. On compact multipliers of topological algebras

    It is shown that if the maximal ideal space Δ(A) of a semisimple commutative complete metrizable locally convex algebra contains no isolated points, then every compact multiplier is trivial. Particularly, compact multipliers on semisimple commutative Frechet algebras whose maximal ideal space has no isolated points are identically zero. (author). 5 refs

  19. Experimental and Theoretical Methods in Algebra, Geometry and Topology

    Veys, Willem; Bridging Algebra, Geometry, and Topology


    Algebra, geometry and topology cover a variety of different, but intimately related research fields in modern mathematics. This book focuses on specific aspects of this interaction. The present volume contains refereed papers which were presented at the International Conference “Experimental and Theoretical Methods in Algebra, Geometry and Topology”, held in Eforie Nord (near Constanta), Romania, during 20-25 June 2013. The conference was devoted to the 60th anniversary of the distinguished Romanian mathematicians Alexandru Dimca and Ştefan Papadima. The selected papers consist of original research work and a survey paper. They are intended for a large audience, including researchers and graduate students interested in algebraic geometry, combinatorics, topology, hyperplane arrangements and commutative algebra. The papers are written by well-known experts from different fields of mathematics, affiliated to universities from all over the word, they cover a broad range of topics and explore the research f...

  20. Exact Geometric-Topological Analysis of Algebraic Surfaces

    Berberich, Eric; Kerber, Michael; Sagraloff, Michael


    We present a method to compute the exact topology of a real algebraic surface $S$, implicitly given by a polynomial $f \\in \\mathbb{Q}[x,y,z]$ of arbitrary degree $N$. Additionally, our analysis provides geometric information as it supports the computation of arbitrary precise samples of $S$ including critical points. We use a projection approach, similar to Collins' cylindrical algebraic decomposition (cad). In comparison we reduce the number of output cells to $O(N^5...

  1. Ultrafilters: Where topological dynamics = algebra = combinatorics

    Blass, Andreas


    We survey some connections between topological dynamics, semigroups of ultrafilters, and combinatorics. As an application, we give a proof, based on ideas of Bergelson and Hindman, of the Hales-Jewett partition theorem.

  2. Topological Free Entropy Dimension for Approximately Divisible $C^*$-algebras

    LI, Weihua; Shen, Junhao


    Let $\\mathcal{A}$ be a unital separable approximately divisible C$^*$-algebra. We show that $\\mathcal{A}$ is generated by two self-adjoint elements and the topological free entropy dimension of any finite generating set of $\\mathcal{A}$ is less than or equal to 1.

  3. Determinant formula for the topological N = 2 superconformal algebra

    Doerrzapf, M


    The Kac determinant for the topological N = 2 superconformal algebra is presented as well as a detailed analysis of the singular vectors detected by the roots of the determinants. In addition we identify the standard Verma modules containing 'no-label' singular vectors (which are not detected directly by the roots of the determinants). We show that in standard Verma modules there are (at least) four different types of submodules, regarding size and shape. We also review the chiral determinant formula, for chiral Verma modules, adding new insights. Finally we transfer the results obtained to the Verma modules and singular vectors of the Ramond N = 2 algebra, which have been very poorly studied so far. This work clarifies several misconceptions and confusing claims appeared in the literature about the singular vectors, Verma modules and submodules of the topological N = 2 superconformal algebra.

  4. Determinant Formula for the Topological N=2 Superconformal Algebra

    Dörrzapf, M; Dörrzapf, Matthias; Gato-Rivera, Beatriz


    The Kac determinant for the Topological N=2 superconformal algebra is presented as well as a detailed analysis of the singular vectors detected by the roots of the determinants. In addition we identify the standard Verma modules containing `no-label' singular vectors (which are not detected directly by the roots of the determinants). We show that in standard Verma modules there are (at least) four different types of submodules, regarding size and shape. We also review the chiral determinant formula, for chiral Verma modules, adding new insights. Finally we transfer the results obtained to the Verma modules and singular vectors of the Ramond N=2 algebra, which have been very poorly studied so far. This work clarifies several misconceptions and confusing claims appeared in the literature about the singular vectors, Verma modules and submodules of the Topological N=2 superconformal algebra.

  5. An inner product for a topological algebra

    An inner product is defined on a commutative semisimple Frechet Q-algebra A. Several conditions concerning the completeness of the resultant inner product space A2 are established. Furthermore, a necessary condition is investigated under which A2 is shown to be incomplete. Also, it is proved that each maximal ideal M of A with M1 ≠ { 0 } is A2-closed and A = M + M1. (author). 5 refs

  6. Closed range multipliers on topological algebras

    We investigate several equivalent conditions under which a multiplier T on a semisimple commutative Frechet algebra A has closed range. It is shown here, for instance, that TA + kerT=A if an only if there is a commuting operator S (i.e., ST=TS) for which T=TST and S=STS, that this happens if and only if S may be taken to be a multiplier. Furthermore, it is proved that these conditions are also equivalent to the existence of a factorization T=PB, where P is an indempotent and B an invertible multiplier. (author). 6 refs

  7. Topological insulators and C∗-algebras: Theory and numerical practice

    Hastings, Matthew B.; Loring, Terry A.


    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.

  8. Topological conformal algebra and BRST algebra in non-critical string theories

    The operator algebra in non-critical string theories is studied by treating the cosmological term as a perturbation. The algebra of covariantly regularized BRST and related currents contains a twisted N = 2 superconformal algebra only at d = -2 in bosonic strings, and a twisted N = 3 superconformal algebra only at d = ±∞ in spinning strings. The bosonic string at d = -2 is examined by replacing the string coordinate by a fermionic matter with c = -2. The resulting bc-βγ system accommodates various forms of BRST cohomology, and the ghost number assignment and BRST cohomology are different in the c = -2 string theory and two-dimensional topological gravity. (author)

  9. Algebraic characterization of vector supersymmetry in topological field theories

    An algebraic cohomological characterization of a class of linearly broken Ward identities is provided. The examples of the topological vector supersymmetry and of the Landau ghost equation are discussed in detail. The existence of such a linearly broken Ward identities turns out to be related to BRST exact anti-field dependent cocycles with negative ghost number, according to the cohomological reformulation of the Noether theorem given by M. Henneaux et al. (author)

  10. Topological classification with additional symmetries from Clifford algebras

    Morimoto, Takahiro; Furusaki, Akira


    We classify topological insulators and superconductors in the presence of additional symmetries such as reflection or mirror symmetries. For each member of the 10 Altland-Zirnbauer symmetry classes, we have a Clifford algebra defined by operators of the generic (time-reversal, particle-hole, or chiral) symmetries and additional symmetries, together with gamma matrices in Dirac Hamiltonians representing topological insulators and superconductors. Following Kitaev's approach, we classify gapped phases of noninteracting fermions under additional symmetries by examining all possible distinct Dirac mass terms which can be added to the set of generators of the Clifford algebra. We find that imposing additional symmetries in effect changes symmetry classes and causes shifts in the periodic table of topological insulators and superconductors. Our results are in agreement with the classification under reflection symmetry recently reported by Chiu, Yao, and Ryu [Phys. Rev. B1098-012110.1103/PhysRevB.88.075142 88, 075142 (2013)]. Several examples are discussed including a topological crystalline insulator with mirror Chern numbers and mirror superconductors.

  11. Topological basis realization for BMW algebra and Heisenberg XXZ spin chain model

    Liu, Bo; Xue, Kang; Wang, Gangcheng; Liu, Ying; Sun, Chunfang


    In this paper, we study three-dimensional (3D) reduced Birman-Murakami-Wenzl (BMW) algebra based on topological basis theory. Several examples of BMW algebra representations are reviewed. We also discuss a special solution of BMW algebra, which can be used to construct Heisenberg XXZ model. The theory of topological basis provides a useful method to solve quantum spin chain models. It is also shown that the ground state of XXZ spin chain is superposition state of topological basis.

  12. Quantum field theory on toroidal topology: Algebraic structure and applications

    Khanna, F.C., E-mail: [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: [Centro Brasileiro de Pesquisas Físicas/MCT, 22290-180, Rio de Janeiro, RJ (Brazil); Malbouisson, J.M.C., E-mail: [Instituto de Física, Universidade Federal da Bahia, 40210-340, Salvador, BA (Brazil); Santana, A.E., E-mail: [International Center for Condensed Matter Physics, Instituto de Física, Universidade de Brasília, 70910-900, Brasília, DF (Brazil)


    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

  13. Algebraic Topology : New Trends in Localization and Periodicity : Barcelona Conference

    Casacuberta, Carles; Mislin, Guido


    Central to this collection of papers are new developments in the general theory of localization of spaces. This field has undergone tremendous change of late and is yielding new insight into the mysteries of classical homotopy theory. The present volume comprises the refereed articles submitted at the Conference on Algebraic Topology held in Sant Feliu de Guíxols, Spain, in June 1994. Several comprehensive articles on general localization clarify the basic tools and give a report on the state of the art in the subject matter. The text is therefore accessible not only to the professional mathematician but also to the advanced student.

  14. Construction Formulae for Singular Vectors of the Topological and of the Ramond N=2 Superconformal Algebras

    Gato-Rivera, Beatriz


    We write down one-to-one mappings between the singular vectors of the Neveu-Schwarz N=2 superconformal algebra and $16 + 16$ types of singular vectors of the Topological and of the Ramond N=2 superconformal algebras. As a result one obtains construction formulae for the latter using the construction formulae for the Neveu-Schwarz singular vectors due to D\\"orrzapf. The indecomposable singular vectors of the Topological and of the Ramond N=2 algebras (`no-label' and `no-helicity' singular vectors) cannot be mapped to singular vectors of the Neveu-Schwarz N=2 algebra, but to {\\it subsingular} vectors, for which no construction formulae exist.

  15. Algebraic K- and L-theory and applications to the topology of manifolds

    The development of geometric topology has led to the identification of specific algebraic structures of great richness and usefulness. A common theme in this area is the study of algebraic invariants of discrete groups or rings by topological methods. The resulting subject is now called algebraic K-theory. The purpose of these lecture notes is to survey some of the main constructions and techniques in algebraic K-theory, together with an indication of the topological background and applications. More details about proofs can be found in the references. The material is organized into some introductory sections, concerning linear and unitary K-theory, followed by descriptions of four important geometric problems and their related algebraic methods

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

    Delucchi, Emanuele; Moci, Luca


    Combinatorics plays a prominent role in contemporary mathematics, due to the vibrant development it has experienced in the last two decades and its many interactions with other subjects. This book arises from the INdAM conference "CoMeTA 2013 - Combinatorial Methods in Topology and Algebra,'' which was held in Cortona in September 2013. The event brought together emerging and leading researchers at the crossroads of Combinatorics, Topology and Algebra, with a particular focus on new trends in subjects such as: hyperplane arrangements; discrete geometry and combinatorial topology; polytope theory and triangulations of manifolds; combinatorial algebraic geometry and commutative algebra; algebraic combinatorics; and combinatorial representation theory. The book is divided into two parts. The first expands on the topics discussed at the conference by providing additional background and explanations, while the second presents original contributions on new trends in the topics addressed by the conference.

  17. On the topology of real algebraic plane curves

    Cheng, Jinsan; Lazard, Sylvain; Peñaranda, Luis; Pouget, Marc; Roullier, Fabrice; Tsigaridas, Elias


    coordinate system even if the curve is not in generic position. Previous methods based on the cylindrical algebraic decomposition use sub-resultant sequences and computations with polynomials with algebraic coefficients. A novelty of our approach is to replace these tools by Gröbner basis computations and...

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

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

  19. Open and Closed String field theory interpreted in classical Algebraic Topology

    Sullivan, Dennis


    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.

  20. Topology

    Hocking, John G


    ""As textbook and reference work, this is a valuable addition to the topological literature."" - Mathematical ReviewsDesigned as a text for a one-year first course in topology, this authoritative volume offers an excellent general treatment of the main ideas of topology. It includes a large number and variety of topics from classical topology as well as newer areas of research activity.There are four set-theoretic chapters, followed by four primarily algebraic chapters. Chapter I covers the fundamentals of topological and metrical spaces, mappings, compactness, product spaces, the Tychonoff t

  1. Topology

    Manetti, Marco


    This is an introductory textbook on general and algebraic topology, aimed at anyone with a basic knowledge of calculus and linear algebra. It provides full proofs and includes many examples and exercises. The covered topics include: set theory and cardinal arithmetic; axiom of choice and Zorn's lemma; topological spaces and continuous functions; connectedness and compactness; Alexandrov compactification; quotient topologies; countability and separation axioms; prebasis and Alexander's theorem; the Tychonoff theorem and paracompactness; complete metric spaces and function spaces; Baire spaces; homotopy of maps; the fundamental group; the van Kampen theorem; covering spaces; Brouwer and Borsuk's theorems; free groups and free product of groups; and basic category theory. While it is very concrete at the beginning, abstract concepts are gradually introduced. It is suitable for anyone needing a basic, comprehensive introduction to general and algebraic topology and its applications.

  2. Construction Formulae for Singular Vectors of the Topological N=2 Superconformal Algebra

    Gato-Rivera, Beatriz


    The Topological N=2 Superconformal algebra has 29 different types of singular vectors (in complete Verma modules) distinguished by the relative U(1) charge and the BRST-invariance properties of the vector and of the primary on which it is built. Whereas one of these types only exists at level zero, the remaining 28 types exist for general levels and can be constructed already at level 1. In this paper we write down one-to-one mappings between 16 of these types of topological singular vectors and the singular vectors of the Antiperiodic NS algebra. As a result one obtains construction formulae for these 16 types of topological singular vectors using the construction formulae for the NS singular vectors due to Doerrzapf.

  3. Topological charge algebra of optical vortices in nonlinear interactions.

    Zhdanova, Alexandra A; Shutova, Mariia; Bahari, Aysan; Zhi, Miaochan; Sokolov, Alexei V


    We investigate the transfer of orbital angular momentum among multiple beams involved in a coherent Raman interaction. We use a liquid crystal light modulator to shape pump and Stokes beams into optical vortices with various integer values of topological charge, and cross them in a Raman-active crystal to produce multiple Stokes and anti-Stokes sidebands. We measure the resultant vortex charges using a tilted-lens technique. We verify that in every case the generated beams' topological charges obey a simple relationship, resulting from angular momentum conservation for created and annihilated photons, or equivalently, from phase-matching considerations for multiple interacting beams. PMID:26832066

  4. Topological basis associated with B–M–W algebra: Two-spin-1/2 realization

    In this letter, we study the two-spin-1/2 realization for the Birman–Murakami–Wenzl (B–M–W) algebra and the corresponding Yang–Baxter Ř(θ,ϕ) matrix. Based on the two-spin-1/2 realization for the B–M–W algebra, the three-dimensional topological space, which is spanned by topological basis, is investigated. By means of such topological basis realization, the four-dimensional Yang–Baxter Ř(θ,ϕ) can be reduced to Wigner DJ function with J=1. The entanglement and Berry phase in the spectral parameter space are also explored. The results show that one can obtain a set of entangled basis via Yang–Baxter Ř(θ,ϕ) matrix acting on the standard basis, and the entanglement degree is maximum when the Ři(θ,ϕ) turns to the braiding operator. - Highlights: • We study a two-spin-1/2 realization of Birman–Murakami–Wenzl algebra. • Topological basis for this model is constructed in this paper. • The reduced representation is related with Wigner D matrix

  5. Topological Membranes, Current Algebras and H-flux - R-flux Duality based on Courant Algebroids

    Bessho, Taiki; Ikeda, Noriaki; Watamura, Satoshi


    We construct a topological sigma model and a current algebra based on a Courant algebroid structure on a Poisson manifold. In order to construct models, we reformulate the Poisson Courant algebroid by supergeometric construction on a QP-manifold. A new duality of Courant algebroids which transforms H-flux and R-flux is proposed, where the transformation is interpreted as a canonical transformation of a graded symplectic manifold.

  6. Quantum field theory on toroidal topology: algebraic structure and applications

    Khanna, F C; Malbouisson, J M C; Santana, A E


    The development of quantum theory on a torus has a long history, and can be traced back to the 1920s, with the attempts by Nordstr\\"om, 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 particles 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 matted physics.

  7. Algebraic cohomology versus topological cohomology in constructing the Wess-Zumino term of the σ-model

    The construction of an invariant action of the σ-model based on relative Lie algebra cohomology is discussed and compared with the conventional topological interpretation. An example of the Wess-Zumino term for a topologically trivial target space is given. (orig.)

  8. Algebraic geometry

    Lefschetz, Solomon


    An introduction to algebraic geometry and a bridge between its analytical-topological and algebraical aspects, this text for advanced undergraduate students is particularly relevant to those more familiar with analysis than algebra. 1953 edition.

  9. Families of Singular and Subsingular Vectors of the Topological N=2 Superconformal Algebra

    Gato-Rivera, Beatriz; Gato-Rivera, Beatriz; Rosado, Jose Ignacio


    We analyze several issues concerning the singular vectors of the Topological N=2 Superconformal algebra. First we propose an algebraic mechanism to decide which types of singular vectors exist, regarding the relative U(1) charge and the BRST-invariance properties, finding four different types in chiral (incomplete) Verma modules and thirty-three different types in complete Verma modules. Then we investigate the family structure of the singular vectors, every member of a family being mapped to any other member by a chain of simple transformations involving the spectral flows. The families of singular vectors in chiral Verma modules follow a unique pattern (four vectors) and contain subsingular vectors. We write down these families until level 3, identifying the subsingular vectors. The families of singular vectors in complete Verma modules follow infinitely many different patterns, grouped roughly in six main kinds. We present a particularly interesting twelve-member family at levels 3 and 4, as well as the co...

  10. Calculating three loop ladder and V-topologies for massive operator matrix elements by computer algebra

    Three loop ladder and V-topology diagrams contributing to the massive operator matrix element AQg are calculated. The corresponding objects can all be expressed in terms of nested sums and recurrences depending on the Mellin variable N and the dimensional parameter ε. Given these representations, the desired Laurent series expansions in ε can be obtained with the help of our computer algebra toolbox. Here we rely on generalized hypergeometric functions and Mellin-Barnes representations, on difference ring algorithms for symbolic summation, on an optimized version of the multivariate Almkvist-Zeilberger algorithm for symbolic integration, and on new methods to calculate Laurent series solutions of coupled systems of differential equations. The solutions can be computed for general coefficient matrices directly for any basis also performing the expansion in the dimensional parameter in case it is expressible in terms of indefinite nested product-sum expressions. This structural result is based on new results of our difference ring theory. In the cases discussed we deal with iterative sum- and integral-solutions over general alphabets. The final results are expressed in terms of special sums, forming quasi-shuffle algebras, such as nested harmonic sums, generalized harmonic sums, and nested binomially weighted (cyclotomic) sums. Analytic continuations to complex values of N are possible through the recursion relations obeyed by these quantities and their analytic asymptotic expansions. The latter lead to a host of new constants beyond the multiple zeta values, the infinite generalized harmonic and cyclotomic sums in the case of V-topologies.

  11. Calculating three loop ladder and V-topologies for massive operator matrix elements by computer algebra

    Ablinger, J.; Behring, A.; Blümlein, J.; De Freitas, A.; von Manteuffel, A.; Schneider, C.


    Three loop ladder and V-topology diagrams contributing to the massive operator matrix element AQg are calculated. The corresponding objects can all be expressed in terms of nested sums and recurrences depending on the Mellin variable N and the dimensional parameter ε. Given these representations, the desired Laurent series expansions in ε can be obtained with the help of our computer algebra toolbox. Here we rely on generalized hypergeometric functions and Mellin-Barnes representations, on difference ring algorithms for symbolic summation, on an optimized version of the multivariate Almkvist-Zeilberger algorithm for symbolic integration, and on new methods to calculate Laurent series solutions of coupled systems of differential equations. The solutions can be computed for general coefficient matrices directly for any basis also performing the expansion in the dimensional parameter in case it is expressible in terms of indefinite nested product-sum expressions. This structural result is based on new results of our difference ring theory. In the cases discussed we deal with iterative sum- and integral-solutions over general alphabets. The final results are expressed in terms of special sums, forming quasi-shuffle algebras, such as nested harmonic sums, generalized harmonic sums, and nested binomially weighted (cyclotomic) sums. Analytic continuations to complex values of N are possible through the recursion relations obeyed by these quantities and their analytic asymptotic expansions. The latter lead to a host of new constants beyond the multiple zeta values, the infinite generalized harmonic and cyclotomic sums in the case of V-topologies.

  12. Topological centers of module actions and cohomological groups of Banach Algebras

    Azar, Kazem Azem Haghnejad


    In this paper, first we study some Arens regularity properties of module actions. Let $B$ be a Banach $A-bimodule$ and let ${Z}^\\ell_{B^{**}}(A^{**})$ and ${Z}^\\ell_{A^{**}}(B^{**})$ be the topological centers of the left module action $\\pi_\\ell:~A\\times B\\rightarrow B$ and the right module action $\\pi_r:~B\\times A\\rightarrow B$, respectively. We investigate some relationships between topological center of $A^{**}$, ${Z}_1({A^{**}})$ with respect to the first Arens product and topological centers of module actions ${Z}^\\ell_{B^{**}}(A^{**})$ and ${Z}^\\ell_{A^{**}}(B^{**})$. On the other hand, if $A$ has Mazure property and $B^{**}$ has the left $A^{**}-factorization$, then $Z^\\ell_{A^{**}}(B^{**})=B$, and so for a locally compact non-compact group $G$ with compact covering number $card(G)$, we have $Z^\\ell_{M(G)^{**}}{(L^1(G)^{**})}= {L^1(G)}$ and $Z^\\ell_{L^1(G)^{**}}{(M(G)^{**})}= {M(G)}$. By using the Arens regularity of module actions, we study some cohomological groups properties of Banach algebra and we...

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

    Eliashberg, Yakov; Maeda, Yoshiaki; Symplectic, Poisson, and Noncommutative geometry


    Symplectic geometry originated in physics, but it has flourished as an independent subject in mathematics, together with its offspring, symplectic topology. Symplectic methods have even been applied back to mathematical physics. Noncommutative geometry has developed an alternative mathematical quantization scheme based on a geometric approach to operator algebras. Deformation quantization, a blend of symplectic methods and noncommutative geometry, approaches quantum mechanics from a more algebraic viewpoint, as it addresses quantization as a deformation of Poisson structures. This volume contains seven chapters based on lectures given by invited speakers at two May 2010 workshops held at the Mathematical Sciences Research Institute: Symplectic and Poisson Geometry in Interaction with Analysis, Algebra and Topology (honoring Alan Weinstein, one of the key figures in the field) and Symplectic Geometry, Noncommutative Geometry and Physics. The chapters include presentations of previously unpublished results and ...

  14. Calculating Three Loop Ladder and V-Topologies for Massive Operator Matrix Elements by Computer Algebra

    Ablinger, J; Blümlein, J; De Freitas, A; von Manteuffel, A; Schneider, C


    Three loop ladder and $V$-topology diagrams contributing to the massive operator matrix element $A_{Qg}$ are calculated. The corresponding objects can all be expressed in terms of nested sums and recurrences depending on the Mellin variable $N$ and the dimensional parameter $\\varepsilon$. Given these representations, the desired Laurent series expansions in $\\varepsilon$ can be obtained with the help of our computer algebra toolbox. Here we rely on generalized hypergeometric functions and Mellin-Barnes representations, on difference ring algorithms for symbolic summation, on an optimized version of the multivariate Almkvist-Zeilberger algorithm for symbolic integration, and on new methods to calculate Laurent series solutions of coupled systems of differential equations. The solutions can be computed for general coefficient matrices directly for any basis also performing the expansion in the dimensional parameter in case it is expressible in terms of indefinite nested product-sum expressions. This structural ...

  15. Calculating three loop ladder and V-topologies for massive operator matrix elements by computer algebra

    Ablinger, J.; Schneider, C. [Johannes Kepler Univ., Linz (Austria). Research Inst. for Symbolic Computation; Behring, A.; Bluemlein, J.; Freitas, A. de [Deutsches Elektronen-Synchrotron (DESY), Zeuthen (Germany); Manteuffel, A. von [Mainz Univ. (Germany). Inst. fuer Physik


    Three loop ladder and V-topology diagrams contributing to the massive operator matrix element A{sub Qg} are calculated. The corresponding objects can all be expressed in terms of nested sums and recurrences depending on the Mellin variable N and the dimensional parameter ε. Given these representations, the desired Laurent series expansions in ε can be obtained with the help of our computer algebra toolbox. Here we rely on generalized hypergeometric functions and Mellin-Barnes representations, on difference ring algorithms for symbolic summation, on an optimized version of the multivariate Almkvist-Zeilberger algorithm for symbolic integration, and on new methods to calculate Laurent series solutions of coupled systems of differential equations. The solutions can be computed for general coefficient matrices directly for any basis also performing the expansion in the dimensional parameter in case it is expressible in terms of indefinite nested product-sum expressions. This structural result is based on new results of our difference ring theory. In the cases discussed we deal with iterative sum- and integral-solutions over general alphabets. The final results are expressed in terms of special sums, forming quasi-shuffle algebras, such as nested harmonic sums, generalized harmonic sums, and nested binomially weighted (cyclotomic) sums. Analytic continuations to complex values of N are possible through the recursion relations obeyed by these quantities and their analytic asymptotic expansions. The latter lead to a host of new constants beyond the multiple zeta values, the infinite generalized harmonic and cyclotomic sums in the case of V-topologies.

  16. The d≤1 union d≥25 and constrained KP hierarchy from BRST invariance in the c≠3 topological algebra

    The BRST invariance conditions in a highest-weight representation of the topological (≡twisted N=2) algebra captures the 'invariant' content of two-dimensional gravity coupled to matter. The topological algebra allows reductions to either the DDK-dressed matter or the 'Kontsevich-Miwa'-dressed matter related to Virasoro-constrained KP hierarchy. The two dressings of matter with the 'Liouville' differ also by their 'ghost numbers', which is similar to the existence of representatives of BRST cohomologies with different ghost numbers. The topological central charge c≠3 provides a two-fold covering of the allowed region d≤1 union d≥25 of the matter central charge d via d=(c+1)(c+6)/(c-3). The 'Liouville' field is identified as the ghost-free part of the topological U(1) current. The construction thus allows one to establish a direct relation (presumably an equivalence) between the Virasoro-constrained KP hierarchies, minimal models, and the BRST invariance condition for highest-weight states of the topological algebra

  17. Algebra

    Tabak, John


    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.

  18. Nonlinear functional analysis in Banach spaces and Banach algebras fixed point theory under weak topology for nonlinear operators and block operator matrices with applications

    Jeribi, Aref


    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

  19. Algebra


    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.

  20. Algebra

    Flanders, Harley


    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

  1. The topology of the Liouville foliation for the Kovalevskaya integrable case on the Lie algebra so(4)

    In this paper we study topological properties of an integrable case for Euler's equations on the Lie algebra so(4), which can be regarded as an analogue of the classical Kovalevskaya case in rigid body dynamics. In particular, for all values of the parameters of the system under consideration, the bifurcation diagrams of the momentum mapping are constructed, the types of critical points of rank 0 are determined, the bifurcations of Liouville tori are described, and the loop molecules are computed for all singular points of the bifurcation diagrams. It follows from the obtained results that some topological properties of the classical Kovalevskaya case can be obtained from the corresponding properties of the considered integrable case on the Lie algebra so(4) by taking a natural limit. Bibliography: 21 titles

  2. The topology of the Liouville foliation for the Kovalevskaya integrable case on the Lie algebra so(4)

    Kozlov, I K [M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics, Moscow (Russian Federation)


    In this paper we study topological properties of an integrable case for Euler's equations on the Lie algebra so(4), which can be regarded as an analogue of the classical Kovalevskaya case in rigid body dynamics. In particular, for all values of the parameters of the system under consideration, the bifurcation diagrams of the momentum mapping are constructed, the types of critical points of rank 0 are determined, the bifurcations of Liouville tori are described, and the loop molecules are computed for all singular points of the bifurcation diagrams. It follows from the obtained results that some topological properties of the classical Kovalevskaya case can be obtained from the corresponding properties of the considered integrable case on the Lie algebra so(4) by taking a natural limit. Bibliography: 21 titles.

  3. An Analysis of Finite Volume, Finite Element, and Finite Difference Methods Using Some Concepts from Algebraic Topology

    Mattiussi, Claudio


    In this paper we apply the ideas of algebraic topology to the analysis of the finite volume and finite element methods, illuminating the similarity between the discretization strategies adopted by the two methods, in the light of a geometric interpretation proposed for the role played by the weighting functions in finite elements. We discuss the intrinsic discrete nature of some of the factors appearing in the field equations, underlining the exception represented by the constitutive term, th...

  4. An Algebraic Topological Construct of Classical Loop Gravity and the prospect of Higher Dimensions

    Venkatesh, Madhavan


    In this paper, classical gravity is reformulated in terms of loops, via an algebraic topological approach. The main component is the loop group, whose elements consist of pairs of cobordant loops. A Chas-Sullivan product is described on the cobordism, and three other products, namely the 'vertical', 'horizontal' and 'total' products are re-introduced. (They have already been defined in an earlier paper by the author). A loop calculus is introduced on the space of loops, consisting of the loop variation functional,the loop derivative, Mandelstam derivative, and what the author wishes to call, the Gambini-Pullin contact functional. The loop derivative happens to be a generator of the group of loops, and the Gambini-Pullin functional is an infinitesimal generator of diffeomorphisms. A toy model of gravity is formulated in terms of the above, and it is proven that the total product provides for the Maurer-Cartan structure of the space. Further, new quantities labeled as 'momenta', 'velocity' and 'energy' are intr...

  5. Mikusi\\'nski's Operational Calculus with Algebraic Foundations and Applications to Bessel Functions

    Bengochea, Gabriel; G, Gabriel López


    We construct an operational calculus supported on the algebraic operational calculus introduced by Bengochea and Verde. With this operational calculus we study the solution of certain Bessel type equations.

  6. Real Algebraic Geometry

    Mahé, Louis; Roy, Marie-Françoise


    Ten years after the first Rennes international meeting on real algebraic geometry, the second one looked at the developments in the subject during the intervening decade - see the 6 survey papers listed below. Further contributions from the participants on recent research covered real algebra and geometry, topology of real algebraic varieties and 16thHilbert problem, classical algebraic geometry, techniques in real algebraic geometry, algorithms in real algebraic geometry, semialgebraic geometry, real analytic geometry. CONTENTS: Survey papers: M. Knebusch: Semialgebraic topology in the last ten years.- R. Parimala: Algebraic and topological invariants of real algebraic varieties.- Polotovskii, G.M.: On the classification of decomposing plane algebraic curves.- Scheiderer, C.: Real algebra and its applications to geometry in the last ten years: some major developments and results.- Shustin, E.L.: Topology of real plane algebraic curves.- Silhol, R.: Moduli problems in real algebraic geometry. Further contribu...

  7. Noncommutative spectral synthesis for the involutive Banach algebra associated with a topological dynamical system

    de Jeu, Marcel


    If X is a compact Hausdorff space, supplied with a homeomorphism, then a crossed product involutive Banach algebra is naturally associated with these data. If X consists of one point, then this algebra is the group algebra of the integers. In this paper, we study spectral synthesis for the closed ideals of this associated algebra in two versions, one modeled after C(X), and one modeled after the group algebra of the integers. We identify the closed ideals which are equal to (what is the analogue of) the kernel of their hull, and determine when this holds for all closed ideals, i.e., when spectral synthesis holds. In both models, this is the case precisely when the homeomorphism has no periodic points.

  8. Geometrical and topological foundations of theoretical physics: from gauge theories to string program

    Luciano Boi


    Full Text Available We study the role of geometrical and topological concepts in the recent developments of theoretical physics, notably in non-Abelian gauge theories and superstring theory, and further we show the great significance of these concepts for a deeper understanding of the dynamical laws of physics. This work aims to demonstrate that the global topological properties of the manifold's model of spacetime play a major role in quantum field theory and that, therefore, several physical quantum effects arise from the nonlocal metrical and topological structure of this manifold. We mathematically argue the need for building new structures of space with different topology. This means, in particular, that the “hidden” symmetries of fundamental physics can be related to the phenomenon of topological change of certain classes of (presumably nonsmooth manifolds.

  9. On the algebraic and topological structure of the set of Tur\\'an densities

    Grosu, Codrut


    The present paper is concerned with the various algebraic structures supported by the set of Tur\\'an densities. We prove that the set of Tur\\'an densities of finite families of r-graphs is a non-trivial commutative semigroup, and as a consequence we construct explicit irrational densities for any r >= 3. The proof relies on a technique recently developed by Pikhurko. We also show that the set of all Tur\\'an densities forms a graded ring, and from this we obtain a short proof of a theorem of P...

  10. Topological theory of dynamical systems recent advances

    Aoki, N


    This monograph aims to provide an advanced account of some aspects of dynamical systems in the framework of general topology, and is intended for use by interested graduate students and working mathematicians. Although some of the topics discussed are relatively new, others are not: this book is not a collection of research papers, but a textbook to present recent developments of the theory that could be the foundations for future developments. This book contains a new theory developed by the authors to deal with problems occurring in diffentiable dynamics that are within the scope of general topology. To follow it, the book provides an adequate foundation for topological theory of dynamical systems, and contains tools which are sufficiently powerful throughout the book. Graduate students (and some undergraduates) with sufficient knowledge of basic general topology, basic topological dynamics, and basic algebraic topology will find little difficulty in reading this book.

  11. Introduction to topology

    Gamelin, Theodore W


    A fresh approach to introductory topology, this volume explains nontrivial applications of metric space topology to analysis, clearly establishing their relationship. Also, topics from elementary algebraic topology focus on concrete results with minimal algebraic formalism. The first two chapters consider metric space and point-set topology; the second two, algebraic topological material. 1983 edition. Solutions to Selected Exercises. List of Notations. Index. 51 illustrations.

  12. Foundations of relational realism a topological approach to quantum mechanics and the philosophy of nature

    Epperson, Michael


    This book presents an intuitive interpretation of quantum mechanics, based on a revised decoherent histories interpretation, structured within a category theoretic topological formalism. More broadly, as a philosophical enterprise, the authors propose this conceptual framework as a speculative ontological program that includes a rigorous mathematical formalism, providing a coherent and intuitive ontological scheme that is both novel and applicable practically to the physical sciences.

  13. Foundations

    Harteveld, Casper

    A building will more likely collapse if it does not have any proper foundations. Similarly, the design philosophy of Triadic Game Design (TGD) needs to reside on solid building blocks, otherwise the concept will collapse as well. In this level I will elaborate on these building blocks. First I will explain what the general idea of TGD is. It is a design philosophy, for sure, but one which stresses that an “optimum” needs to be found in a design space constituted by three different worlds: Reality, Meaning, and Play. Additionally, these worlds need to be considered simultaneously and be treated equally. The latter requires balancing the worlds which may result in different tensions, within and between two or three of the worlds. I continue by discussing each of the worlds and showing their perspective on the field of games with a meaningful purpose. From this, we clearly see that it is feasible to think of each world and that the idea makes sense. I substantiate this further by relating the notion of player and similar approaches to this framework. This level is quite a tough pill to swallow yet essential for finishing the other levels. Do not cheat or simply skip this level, but just take a big cup of coffee or tea and start reading it.

  14. Conceptual Foundations of Soliton Versus Particle Dualities Toward a Topological Model for Matter

    Kouneiher, Joseph


    The idea that fermions could be solitons was actually confirmed in theoretical models in 1975 in the case when the space-time is two-dimensional and with the sine-Gordon model. More precisely S. Coleman showed that two different classical models end up describing the same fermions particle, when the quantum theory is constructed. But in one model the fermion is a quantum excitation of the field and in the other model the particle is a soliton. Hence both points of view can be reconciliated.The principal aim in this paper is to exhibit a solutions of topological type for the fermions in the wave zone, where the equations of motion are non-linear field equations, i.e. using a model generalizing sine- Gordon model to four dimensions, and describe the solutions for linear and circular polarized waves. In other words, the paper treat fermions as topological excitations of a bosonic field.

  15. Relative Homological Algebra Volume 1


    This is the second revised edition of an introduction to contemporary relative homological algebra. It supplies important material essential to understand topics in algebra, algebraic geometry and algebraic topology. Each section comes with exercises providing practice problems for students as well as additional important results for specialists. The book is also suitable for an introductory course in commutative and ordinary homological algebra.

  16. Coarse categories I: foundations

    Luu, Viêt-Trung


    Following Roe and others (see, e.g., [MR1451755]), we (re)develop coarse geometry from the foundations, taking a categorical point of view. In this paper, we concentrate on the discrete case in which topology plays no role. Our theory is particularly suited to the development of the Roe (C*-)algebras C*(X) and their K-theory on the analytic side; we also hope that it will be of use in the strictly geometric/algebraic setting of controlled topology and algebra. We leave these topics to future papers. Crucial to our approach are nonunital coarse spaces, for which we introduce locally proper maps. Our coarse category Crs generalizes the usual one: its objects are nonunital coarse spaces and its morphisms (locally proper) coarse maps modulo closeness. Crs is much richer than the usual unital coarse category. As such, it has all nonzero limits and all colimits. We examine various other categorical issues. E.g., Crs does not have a terminal object, so we substitute a termination functor which will be important in t...

  17. Topological fields

    Warner, S


    Aimed at those acquainted with basic point-set topology and algebra, this text goes up to the frontiers of current research in topological fields (more precisely, topological rings that algebraically are fields).The reader is given enough background to tackle the current literature without undue additional preparation. Many results not in the text (and many illustrations by example of theorems in the text) are included among the exercises. Sufficient hints for the solution of the exercises are offered so that solving them does not become a major research effort for the reader. A comprehensive bibliography completes the volume.

  18. Evolution algebras and their applications

    Tian, Jianjun Paul


    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.

  19. Homotopy Invariant Commutative Algebra over fields

    Greenlees, J. P. C.


    These notes illustrates the power of formulating ideas of commutative algebra in a homotopy invariant form. They can then be applied to derived categories of rings or ring spectra. These ideas are powerful in classical algebra, in representation theory of groups, in classical algebraic topology and elsewhere. The notes grew out of a series of lectures given during the `Interactions between Representation Theory, Algebraic Topology and Commutative Algebra' (IRTATCA) at the CRM (Barcelona) in S...

  20. Algebraic renormalization perturbative twisted considerations on topological Yang-Mills theory and on N=2 supersymmetric gauge theories

    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)

  1. On Hadamard algebras

    Carlos C. Peña


    Full Text Available Topological algebras of sequences of complex numbers are introduced, endowed with a Hadamard product type. The complex homomorphisms on these algebras are characterized, and units, prime cyclic ideals, prime closed ideals, and prime minimal ideals, discussed. Existence of closed and maximal ideals are investigated, and it is shown that the Jacobson and nilradicals are both trivial.

  2. Differential topology

    Mukherjee, Amiya


    This book presents a systematic and comprehensive account of the theory of differentiable manifolds and provides the necessary background for the use of fundamental differential topology tools. The text includes, in particular, the earlier works of Stephen Smale, for which he was awarded the Fields Medal. Explicitly, the topics covered are Thom transversality, Morse theory, theory of handle presentation, h-cobordism theorem, and the generalised Poincaré conjecture. The material is the outcome of lectures and seminars on various aspects of differentiable manifolds and differential topology given over the years at the Indian Statistical Institute in Calcutta, and at other universities throughout India. The book will appeal to graduate students and researchers interested in these topics. An elementary knowledge of linear algebra, general topology, multivariate calculus, analysis, and algebraic topology is recommended.

  3. Categorical Algebra and its Applications


    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.

  4. On various avatars of the Pasquier algebra

    A Pasquier algebra is a commutative associative algebra of normal matrices attached to a graph. I review various appearances of such algebras in different contexts: operator product algebras and structure constants in conformal theories and lattice models, integrable N = 2 supersymmetric models and their topological partners. (author)

  5. Continuous Fields of Kirchberg C*-algebras

    Dadarlat, Marius; Pasnicu, Cornel


    In this paper we study the C*-algebras associated to continuous fields over locally compact metrisable zero dimensional spaces whose fibers are Kirchberg C*-algebras satisfying the UCT. We show that these algebras are inductive limits of finite direct sums of Kirchberg algebras and they are classified up to isomorphism by topological invariants.

  6. Algebraic extensions of fields

    McCarthy, Paul J


    ""...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

  7. Two's company, three (or more) is a simplex : Algebraic-topological tools for understanding higher-order structure in neural data.

    Giusti, Chad; Ghrist, Robert; Bassett, Danielle S


    The language of graph theory, or network science, has proven to be an exceptional tool for addressing myriad problems in neuroscience. Yet, the use of networks is predicated on a critical simplifying assumption: that the quintessential unit of interest in a brain is a dyad - two nodes (neurons or brain regions) connected by an edge. While rarely mentioned, this fundamental assumption inherently limits the types of neural structure and function that graphs can be used to model. Here, we describe a generalization of graphs that overcomes these limitations, thereby offering a broad range of new possibilities in terms of modeling and measuring neural phenomena. Specifically, we explore the use of simplicial complexes: a structure developed in the field of mathematics known as algebraic topology, of increasing applicability to real data due to a rapidly growing computational toolset. We review the underlying mathematical formalism as well as the budding literature applying simplicial complexes to neural data, from electrophysiological recordings in animal models to hemodynamic fluctuations in humans. Based on the exceptional flexibility of the tools and recent ground-breaking insights into neural function, we posit that this framework has the potential to eclipse graph theory in unraveling the fundamental mysteries of cognition. PMID:27287487

  8. Topology an introduction

    Waldmann, Stefan


    This book provides a concise introduction to topology and is necessary for courses in differential geometry, functional analysis, algebraic topology, etc. Topology is a fundamental tool in most branches of pure mathematics and is also omnipresent in more applied parts of mathematics. Therefore students will need fundamental topological notions already at an early stage in their bachelor programs. While there are already many excellent monographs on general topology, most of them are too large for a first bachelor course. Topology fills this gap and can be either used for self-study or as the basis of a topology course.

  9. An Algebra of Reversible Computation

    Wang, Yong


    We design an axiomatization for reversible computation called reversible ACP (RACP). It has four extendible modules, basic reversible processes algebra (BRPA), algebra of reversible communicating processes (ARCP), recursion and abstraction. Just like process algebra ACP in classical computing, RACP can be treated as an axiomatization foundation for reversible computation.

  10. Algebra II workbook for dummies

    Sterling, Mary Jane


    To succeed in Algebra II, start practicing now Algebra II builds on your Algebra I skills to prepare you for trigonometry, calculus, and a of myriad STEM topics. Working through practice problems helps students better ingest and retain lesson content, creating a solid foundation to build on for future success. Algebra II Workbook For Dummies, 2nd Edition helps you learn Algebra II by doing Algebra II. Author and math professor Mary Jane Sterling walks you through the entire course, showing you how to approach and solve the problems you encounter in class. You'll begin by refreshing your Algebr

  11. The theory of algebraic numbers

    Pollard, Harry


    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.

  12. The foundations of statistics

    Savage, Leonard J


    Classic analysis of the foundations of statistics and development of personal probability, one of the greatest controversies in modern statistical thought. Revised edition. Calculus, probability, statistics, and Boolean algebra are recommended.

  13. Foundations of factor analysis

    Mulaik, Stanley A


    Introduction Factor Analysis and Structural Theories Brief History of Factor Analysis as a Linear Model Example of Factor AnalysisMathematical Foundations for Factor Analysis Introduction Scalar AlgebraVectorsMatrix AlgebraDeterminants Treatment of Variables as Vectors Maxima and Minima of FunctionsComposite Variables and Linear Transformations Introduction Composite Variables Unweighted Composite VariablesDifferentially Weighted Composites Matrix EquationsMulti

  14. Representations of fundamental groups of algebraic varieties

    Zuo, Kang


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

  15. Noncommutative topological dynamics

    Ramos, C. Correia [Department of Mathematics, Universidade de Evora, Rua Roma-tilde o Ramalho, 59, 7000-671 Evora (Portugal)] e-mail:; Martins, Nuno [Department of Mathematics, Instituto Superior Tecnico, Av. Rovisco Pais, 1, 1049-001 Lisbon (Portugal)] e-mail:; Severino, Ricardo [Department of Mathematics, Universidade do Minho, Campus de Gualtar, 4710 Braga (Portugal)] e-mail:; Ramos, J. Sousa [Department of Mathematics, Instituto Superior Tecnico, Av. Rovisco Pais, 1, 1049-001 Lisbon (Portugal)] e-mail:


    We study noncommutative dynamical systems associated to unimodal and bimodal maps of the interval. To these maps we associate subshifts and the correspondent AF-algebras and Cuntz-Krieger algebras. As an example we consider systems having equal topological entropy log(1 + {phi}), where {phi} is the golden number, but distinct chaotic behavior and we show how a new numerical invariant allows to distinguish that complexity. Finally, we give a statistical interpretation to the topological numerical invariants associated to bimodal maps.

  16. Lattice Operators and Topologies

    Eva Cogan


    Working within a complete (not necessarily atomic) Boolean algebra, we use a sublattice to define a topology on that algebra. Our operators generalize complement on a lattice which in turn abstracts the set theoretic operator. Less restricted than those of Banaschewski and Samuel, the operators exhibit some surprising behaviors. We consider properties of such lattices and their interrelations. Many of these properties are abstractions and generalizations of topological spaces. The approach is...

  17. Left Artinian Algebraic Algebras

    S. Akbari; M. Arian-Nejad


    Let R be a left artinian central F-algebra, T(R) = J(R) + [R, R],and U(R) the group of units of R. As one of our results, we show that, if R is algebraic and char F = 0, then the number of simple components of -R = R/J(R)is greater than or equal to dimF R/T(R). We show that, when char F = 0 or F is uncountable, R is algebraic over F if and only if [R, R] is algebraic over F. As another approach, we prove that R is algebraic over F if and only if the derived subgroup of U(R) is algebraic over F. Also, we present an elementary proof for a special case of an old question due to Jacobson.

  18. Topological rings

    Warner, S


    This text brings the reader to the frontiers of current research in topological rings. The exercises illustrate many results and theorems while a comprehensive bibliography is also included. The book is aimed at those readers acquainted with some very basic point-set topology and algebra, as normally presented in semester courses at the beginning graduate level or even at the advanced undergraduate level. Familiarity with Hausdorff, metric, compact and locally compact spaces and basic properties of continuous functions, also with groups, rings, fields, vector spaces and modules, and with Zorn''s Lemma, is also expected.

  19. Geometric Algebra for Physicists

    Doran, Chris; Lasenby, Anthony


    Preface; Notation; 1. Introduction; 2. Geometric algebra in two and three dimensions; 3. Classical mechanics; 4. Foundations of geometric algebra; 5. Relativity and spacetime; 6. Geometric calculus; 7. Classical electrodynamics; 8. Quantum theory and spinors; 9. Multiparticle states and quantum entanglement; 10. Geometry; 11. Further topics in calculus and group theory; 12. Lagrangian and Hamiltonian techniques; 13. Symmetry and gauge theory; 14. Gravitation; Bibliography; Index.

  20. Finitary Algebraic Superspace

    Zapatrin, R R


    An algebraic scheme is suggested in which discretized spacetime turns out to be a quantum observable. As an example, a toy model producing spacetimes of four points with different topologies is presented. The possibility of incorporating this scheme into the framework of non-commutative differential geometry is discussed.

  1. Twin TQFTs and Frobenius Algebras

    Carmen Caprau


    Full Text Available We introduce the category of singular 2-dimensional cobordisms and show that it admits a completely algebraic description as the free symmetric monoidal category on a twin Frobenius algebra, by providing a description of this category in terms of generators and relations. A twin Frobenius algebra (C,W,z,z∗ consists of a commutative Frobenius algebra C, a symmetric Frobenius algebra W, and an algebra homomorphism z:C→W with dual z∗:W→C, satisfying some extra conditions. We also introduce a generalized 2-dimensional Topological Quantum Field Theory defined on singular 2-dimensional cobordisms and show that it is equivalent to a twin Frobenius algebra in a symmetric monoidal category.

  2. Hochschild homology of structured algebras

    Wahl, Nathalie; Westerland, Craig Christopher


    We give a general method for constructing explicit and natural operations on the Hochschild complex of algebras over any prop with A∞-multiplication—we think of such algebras as A∞-algebras “with extra structure”. As applications, we obtain an integral version of the Costello......–Kontsevich–Soibelman moduli space action on the Hochschild complex of open TCFTs, the Tradler–Zeinalian and Kaufmann actions of Sullivan diagrams on the Hochschild complex of strict Frobenius algebras, and give applications to string topology in characteristic zero. Our main tool is a generalization of the Hochschild complex....

  3. Topological and geometrical quantum computation in cohesive Khovanov homotopy type theory

    Ospina, Juan


    The recently proposed Cohesive Homotopy Type Theory is exploited as a formal foundation for central concepts in Topological and Geometrical Quantum Computation. Specifically the Cohesive Homotopy Type Theory provides a formal, logical approach to concepts like smoothness, cohomology and Khovanov homology; and such approach permits to clarify the quantum algorithms in the context of Topological and Geometrical Quantum Computation. In particular we consider the so-called "open-closed stringy topological quantum computer" which is a theoretical topological quantum computer that employs a system of open-closed strings whose worldsheets are open-closed cobordisms. The open-closed stringy topological computer is able to compute the Khovanov homology for tangles and for hence it is a universal quantum computer given than any quantum computation is reduced to an instance of computation of the Khovanov homology for tangles. The universal algebra in this case is the Frobenius Algebra and the possible open-closed stringy topological quantum computers are forming a symmetric monoidal category which is equivalent to the category of knowledgeable Frobenius algebras. Then the mathematical design of an open-closed stringy topological quantum computer is involved with computations and theorem proving for generalized Frobenius algebras. Such computations and theorem proving can be performed automatically using the Automated Theorem Provers with the TPTP language and the SMT-solver Z3 with the SMT-LIB language. Some examples of application of ATPs and SMT-solvers in the mathematical setup of an open-closed stringy topological quantum computer will be provided.

  4. Quantum-mechanical topological entropy

    We introduce a notion of topological entropy for automorphisms of arbitrary (non-commutative, but unital) nuclear C*-algebras A, generalizing the 'classical' topological entropy of a homeomorphism T: X→X of an arbitrary (compact, but not necessarily metric) space. In the sense that the topological dynamical system (X,T) is equivalently viewed as the C*-dynamical system given by the induced automorphism of the Abelian C*-algebra A=C(x) of all (complex valued) continuous functions on X. We calculate our 'quantum topological' entropy for shift automorphisms on AF-algebras associated with topological Markov chains (i.e. 'quantum topological' Markov chains); and also a natural physical interpretation is added

  5. Quantum computation using geometric algebra

    Matzke, Douglas James

    This dissertation reports that arbitrary Boolean logic equations and operators can be represented in geometric algebra as linear equations composed entirely of orthonormal vectors using only addition and multiplication Geometric algebra is a topologically based algebraic system that naturally incorporates the inner and anticommutative outer products into a real valued geometric product, yet does not rely on complex numbers or matrices. A series of custom tools was designed and built to simplify geometric algebra expressions into a standard sum of products form, and automate the anticommutative geometric product and operations. Using this infrastructure, quantum bits (qubits), quantum registers and EPR-bits (ebits) are expressed symmetrically as geometric algebra expressions. Many known quantum computing gates, measurement operators, and especially the Bell/magic operators are also expressed as geometric products. These results demonstrate that geometric algebra can naturally and faithfully represent the central concepts, objects, and operators necessary for quantum computing, and can facilitate the design and construction of quantum computing tools.

  6. Completely Positive Definite Maps on σ-C*-algebras

    许天周; 段培超; 郑庆琳


    @@ Recently, there has been increased interest [1-8] in topological *-algebras that are inverselimits of C*-algebras, called Pro-C*-algebras. These algebras were introduced in [5] as a gene-ralization of C*-algebras were called locally C*-algebras. The same objects have been studiedvarious term, in [1-8], it is shown in [6-7] that they arise naturally in certain aspects of C*-algebraslike the tangent algebras of C*-algebras, multipliers of Pedersen's ideal, non-commutative ana-logues of classical Lie groups and K-theory.

  7. Algebraic K-theory, K-regularity, and -duality of -stable C ∗-algebras

    Mahanta, Snigdhayan


    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.

  8. Monomial algebras

    Villarreal, Rafael


    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.

  9. Topics in general topology

    Morita, K


    Being an advanced account of certain aspects of general topology, the primary purpose of this volume is to provide the reader with an overview of recent developments.The papers cover basic fields such as metrization and extension of maps, as well as newly-developed fields like categorical topology and topological dynamics. Each chapter may be read independently of the others, with a few exceptions. It is assumed that the reader has some knowledge of set theory, algebra, analysis and basic general topology.

  10. Classification theory of topological insulators and superconductors and its application to topological crystalline insulators

    We review the classification theory of topological insulators and superconductors based on the K-theory and the Clifford algebras. We also review an application of the classification theory to the topological crystalline insulators with reflection symmetry. (author)

  11. Topological Strings and $D$-Branes

    Vancea, Ion V.


    In this talk we give a brief review of the algebraic structure behind the open and closed topological strings and $D$-branes and emphasize the role of tensor category and the Frobenius algebra. Also, we speculate on the possibility of generalizing the topological strings and the $D$-branes through the subfactor theory.

  12. On a certain class of operator algebras and their derivations

    Given a von Neumann algebra M with a faithful normal finite trace, we introduce the so-called finite tracial algebra Mf as the intersection of Lp-spaces Lp(M, μ) over all p ≥ and over all faithful normal finite traces μ on M. Basic algebraic and topological properties of finite tracial algebras are studied. We prove that all derivations on these algebras are inner. (author)

  13. Excision in algebraic K-theory and Karoubi's conjecture.

    Suslin, A A; Wodzicki, M


    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

  14. Cluster algebras and Poisson geometry

    Gekhtman, M.; Shapiro, M.; Vainshtein, A.


    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.

  15. Certain number-theoretic episodes in algebra

    Sivaramakrishnan, R


    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.

  16. Algebra and Number Theory An Integrated Approach

    Dixon, Martyn; Subbotin, Igor


    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

  17. Phase boundaries in algebraic conformal QFT

    We describe the structure of local algebras in relativistic conformal QFT with phase boundaries (topological defects) in two space-time dimensions. The phase boundary conditions are produced by the irreducible components of a certain universal construction.

  18. Fall Foliage Topology Seminars


    This book demonstrates the lively interaction between algebraic topology, very low dimensional topology and combinatorial group theory. Many of the ideas presented are still in their infancy, and it is hoped that the work here will spur others to new and exciting developments. Among the many techniques disussed are the use of obstruction groups to distinguish certain exact sequences and several graph theoretic techniques with applications to the theory of groups.

  19. Supertropical algebra

    Izhakian, Zur; Rowen, Louis


    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...

  20. Operator algebras for multivariable dynamics

    Davidson, Kenneth R.; Katsoulis, Elias G.


    Let $X$ be a locally compact Hausdorff space with $n$ proper continuous self maps $\\tau_i:X \\to X$ for $1 \\le i \\le n$. To this we associate two topological conjugacy algebras which emerge as the natural candidates for the universal algebra of the system, the tensor algebra $\\A(X, \\tau)$ and the semicrossed product $\\rC_0(X)\\times_\\tau\\Fn$. We introduce a concept of conjugacy for multidimensional systems, which we coin piecewise conjugacy. We prove that the piecewise conjugacy class of the sy...

  1. Differential topology an introduction

    Gauld, David B


    Offering classroom-proven results, Differential Topology presents an introduction to point set topology via a naive version of nearness space. Its treatment encompasses a general study of surgery, laying a solid foundation for further study and greatly simplifying the classification of surfaces.This self-contained treatment features 88 helpful illustrations. Its subjects include topological spaces and properties, some advanced calculus, differentiable manifolds, orientability, submanifolds and an embedding theorem, and tangent spaces. Additional topics comprise vector fields and integral curv

  2. On an inner product over Imc *-algebras

    An inner product is defined on a commutative *-semisimple separable Frechet Q Imc *-algebra A and the resultant space Am is shown to be a topological *-algebra. It is also proved that every maximal ideal M of A is Am -closed and thus A can be decomposed into A = M + Mperpendicular. Further, it is shown that a norm can be defined on A which makes it into a pre C* -algebra. A completeness criterion for Am is established. (author). 10 refs

  3. Arrangement Computation for Planar Algebraic Curves

    Berberich, Eric; Emeliyanenko, Pavel; Kobel, Alexander; Sagraloff, Michael


    We present a new certified and complete algorithm to compute arrangements of real planar algebraic curves. Our algorithm provides a geometric-topological analysis of the decomposition of the plane induced by a finite number of algebraic curves in terms of a cylindrical algebraic decomposition of the plane. Compared to previous approaches, we improve in two main aspects: Firstly, we significantly reduce the amount of exact operations, that is, our algorithms only uses resultant and gcd as pure...

  4. Noncommutative algebras associated to complexes and graphs

    Gelfand, Israel; Gelfand, Sergei; Retakh, Vladimir


    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.

  5. Common idempotents in compact left topological left semirings

    Saveliev, Denis I


    A classical result of topological algebra states that any compact left topological semigroup has an idempotent. We refine this by showing that any compact left topological left semiring has a common, i.e. additive and multiplicative simultaneously, idempotent. As an application, we partially answer a question related to algebraic properties of ultrafilters over natural numbers. Finally, we observe that similar arguments establish the existence of common idempotents in much more general, non-associative universal algebras.

  6. Ultrafilters and topologies on groups

    Zelenyuk, Yevhen


    This book presents the relationship between ultrafilters and topologies on groups. It shows how ultrafilters are used in constructing topologies on groups with extremal properties and how topologies on groups serve in deriving algebraic results aboutultrafilters. Topics covered include: topological and left topological groups, ultrafilter semigroups, local homomorphisms and automorphisms, subgroups and ideal structure of ßG, almost maximal spaces and projectives of finite semigroups, resolvability of groups. This is a self-contained book aimed at graduate students and researchers working in to

  7. On Soft Topology

    ENGİNOĞLU, Serdar; Çağman, Naim; KARATAŞ, Serkan; Aydin, Tuğçe


    Soft set theory was introduced by Molodtsov in 1999. Until now many versions of it have been developed and applied to a lot of areas from algebra to decision making problems. One of these areas is Soft Topology [Çağman, N., Karataş, S., Enginoglu, S., Soft topology, Computers and Mathematics with Applications, 62, 351-358, 2011]. However, it has some difficulties and mistakes. In this paper, for further study on the soft topology, we have made fit this concept which is important for developme...

  8. An algebraic result on the topological closure of the set of rational points on a sphere whose center is non-rational

    Matsushita, Jun-ichi


    Let $S$ be a sphere in $\\mathbf{R}^n$ whose center is not in $\\mathbf{Q}^n$. We pose the following problem on $S$. \\[ \\text{``What is the closure of $S \\cap \\mathbf{Q}^n$ with respect to the Euclidean topology?''} \\] In this paper we give a simple solution for this problem in the special case that the center $a = (a_i) \\in \\mathbf{R}^n$ of $S$ satisfies \\[ \\left\\{ \\sum_{i=1}^n r_i (a_i - b_i); \\ r_1, \\dots, r_n \\in \\mathbf{Q} \\right\\} = K \\] for some $b = (b_i) \\in S \\cap \\m...

  9. Introduction to algebra and trigonometry

    Kolman, Bernard


    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

  10. Algebraic Groups


    The workshop continued a series of Oberwolfach meetings on algebraic groups, started in 1971 by Tonny Springer and Jacques Tits who both attended the present conference. This time, the organizers were Michel Brion, Jens Carsten Jantzen, and Raphaël Rouquier. During the last years, the subject of...... algebraic groups (in a broad sense) has seen important developments in several directions, also related to representation theory and algebraic geometry. The workshop aimed at presenting some of these developments in order to make them accessible to a "general audience" of algebraic group-theorists, and to...

  11. An inner product for a Banach-algebra

    An inner product is defined on a commutative Banach algebra with an essential involution and the resultant inner product space is shown to be a topological algebra. Several conditions for its completeness are established and moreover, a decomposition theorem is proved. It is shown that every commutative Banach algebra with an essential involution has an auxiliary norm which turns it into an A*-algebra. (author). 6 refs

  12. Linear Algebra and Smarandache Linear Algebra

    Vasantha, Kandasamy


    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...

  13. Combinatorics and commutative algebra

    Stanley, Richard P


    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 ...

  14. Focus on Fractions to Scaffold Algebra

    Ooten, Cheryl Thomas


    Beginning algebra is a gatekeeper course into the pipeline to higher mathematics courses required for respected professions in engineering, science, statistics, mathematics, education, and technology. Beginning algebra can also be a perfect storm if the necessary foundational skills are not within a student's grasp. What skills ensure beginning…

  15. Segal algebras in commutative Banach algebras

    INOUE, Jyunji; TAKAHASI, Sin-Ei


    The notion of Reiter's Segal algebra in commutative group algebras is generalized to a notion of Segal algebra in more general classes of commutative Banach algebras. Then we introduce a family of Segal algebras in commutative Banach algebras under considerations and study some properties of them.

  16. Algebra-Geometry of Piecewise Algebraic Varieties

    Chun Gang ZHU; Ren Hong WANG


    Algebraic variety is the most important subject in classical algebraic geometry.As the zero set of multivariate splines,the piecewise algebraic variety is a kind generalization of the classical algebraic variety.This paper studies the correspondence between spline ideals and piecewise algebraic varieties based on the knowledge of algebraic geometry and multivariate splines.

  17. The algebraic structure of the Onsager algebra

    DATE, ETSURO; Roan, Shi-shyr


    We study the Lie algebra structure of the Onsager algebra from the ideal theoretic point of view. A structure theorem of ideals in the Onsager algebra is obtained with the connection to the finite-dimensional representations. We also discuss the solvable algebra aspect of the Onsager algebra through the formal Lie algebra theory.

  18. Foundations of predictive analytics

    Wu, James


    Drawing on the authors' two decades of experience in applied modeling and data mining, Foundations of Predictive Analytics presents the fundamental background required for analyzing data and building models for many practical applications, such as consumer behavior modeling, risk and marketing analytics, and other areas. It also discusses a variety of practical topics that are frequently missing from similar texts. The book begins with the statistical and linear algebra/matrix foundation of modeling methods, from distributions to cumulant and copula functions to Cornish--Fisher expansion and o

  19. Quantum field theory in topology changing spacetimes

    The goal of this diploma thesis is to present an overview of how to reduce the problem of topology change of general spacetimes to the investigation of elementary cobordisms. In the following we investigate the possibility to construct quantum fields on elementary cobordisms, in particular we discuss the trousers topology. Trying to avoid the problems occuring at spacetimes with instant topology change we use a model for simulating topology change. We construct the algebra of observables for a free scalar field with the algebraic approach to quantum field theory. Therefore we determine a fundamental solution of the eld equation. (orig.)

  20. Spectral flows and twisted topological theories

    Gato-Rivera, Beatriz; Gato-Rivera, Beatriz; Rosado, Jose Ignacio


    We analyze the action of the spectral flows on N=2 twisted topological theories. We show that they provide a useful mapping between the two twisted topological theories associated to a given N=2 superconformal theory. This mapping can also be viewed as a topological algebra automorphism. In particular null vectors are mapped into null vectors, considerably simplifying their computation. We give the level 2 results. Finally we discuss the spectral flow mapping in the case of the DDK and KM realizations of the topological algebra.

  1. College algebra

    Kolman, Bernard


    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

  2. Zonotopal algebra

    Holtz, Olga; Ron, Amos


    A wealth of geometric and combinatorial properties of a given linear endomorphism $X$ of $\\R^N$ is captured in the study of its associated zonotope $Z(X)$, and, by duality, its associated hyperplane arrangement ${\\cal H}(X)$. This well-known line of study is particularly interesting in case $n\\eqbd\\rank X \\ll N$. We enhance this study to an algebraic level, and associate $X$ with three algebraic structures, referred herein as {\\it external, central, and internal.} Each algebraic structure is ...

  3. Hom-Akivis algebras

    Issa, A. Nourou


    Hom-Akivis algebras are introduced. The commutator-Hom-associator algebra of a non-Hom-associative algebra (i.e. a Hom-nonassociative algebra) is a Hom-Akivis algebra. It is shown that non-Hom-associative algebras can be obtained from nonassociative algebras by twisting along algebra automorphisms while Hom-Akivis algebras can be obtained from Akivis algebras by twisting along algebra endomorphisms. It is pointed out that a Hom-Akivis algebra associated to a Hom-alternative algebra is a Hom-M...

  4. The Cyclotomic Birman-Murakami-Wenzl Algebras

    Yu, Shona


    ----- Please see the pdf file for the actual abstract and important remarks, which could not be put here due to the arXiv length restrictions. ----- This thesis presents a study of the cyclotomic BMW (Birman-Murakami-Wenzl) algebras, introduced by Haring-Oldenburg as a generalization of the BMW algebras associated with the cyclotomic Hecke algebras of type G(k,1,n) (also known as Ariki-Koike algebras) and type B knot theory involving affine/cylindrical tangles. They are shown to be free of rank k^n (2n-1)!! and to have a topological realization as a certain cylindrical analogue of the Kauffman Tangle algebra. Furthermore, the cyclotomic BMW algebras are proven to be cellular, in the sense of Graham and Lehrer. This Ph.D. thesis, completed at the University of Sydney, was submitted September 2007 and passed December 2007.

  5. Topology Explains Why Automobile Sunshades Fold Oddly

    Feist, Curtis; Naimi, Ramin


    Automobile sunshades always fold into an "odd" number of loops. The explanation why involves elementary topology (braid theory and linking number, both explained in detail here with definitions and examples), and an elementary fact from algebra about symmetric group.

  6. Double-Framed Soft Sets with Applications in BCK/BCI-Algebras

    Young Bae Jun; Sun Shin Ahn


    In order to lay a foundation for providing a soft algebraic tool in considering many problems that contain uncertainties, the notion of double-framed soft sets is introduced, and applications in BCK/BCI-algebras are discussed. The notions of double-framed soft algebras in BCK/BCI-algebras are introduced, and related properties are investigated. Characterizations of double-framed soft algebras are considered. Product and int-uni structure of double-framed soft algebras are discussed, and sever...

  7. Brans–Dicke gravity theory from topological gravity

    We consider a model that suggests a mechanism by which the four dimensional Brans–Dicke gravity theory may emerge from the topological gravity action. To achieve this goal, both the Lie algebra and the symmetric invariant tensor that define the topological gravity Lagrangian are constructed by means of the Lie algebra S-expansion procedure with an appropriate abelian semigroup S

  8. Enveloping algebras

    Since the works of Gelfand, Harish-Chandra, Kostant and Duflo, a new theory has earned its place in the field of mathematics, due to the abundance of its results and the coherence of its methods: the theory of enveloping algebras. This study is the first to present the whole subject in textbook form. The most recent results are included, as well as complete proofs, starting from the elementary theory of Lie algebras. (Auth.)

  9. Topological methods in Galois representation theory

    Snaith, Victor P


    An advanced monograph on Galois representation theory by one of the world's leading algebraists, this volume is directed at mathematics students who have completed a graduate course in introductory algebraic topology. Topics include abelian and nonabelian cohomology of groups, characteristic classes of forms and algebras, explicit Brauer induction theory, and much more. 1989 edition.

  10. Which multiplier algebras are $W^*$-algebras?

    Akemann, Charles A.; Amini, Massoud; Asadi, Mohammad B.


    We consider the question of when the multiplier algebra $M(\\mathcal{A})$ of a $C^*$-algebra $\\mathcal{A}$ is a $ W^*$-algebra, and show that it holds for a stable $C^*$-algebra exactly when it is a $C^*$-algebra of compact operators. This implies that if for every Hilbert $C^*$-module $E$ over a $C^*$-algebra $\\mathcal{A}$, the algebra $B(E)$ of adjointable operators on $E$ is a $ W^*$-algebra, then $\\mathcal{A}$ is a $C^*$-algebra of compact operators. Also we show that a unital $C^*$-algebr...

  11. Alternative algebraic approaches in quantum chemistry

    Mezey, Paul G., E-mail: [Canada Research Chair in Scientific Modeling and Simulation, Department of Chemistry and Department of Physics and Physical Oceanography, Memorial University of Newfoundland, 283 Prince Philip Drive, St. John' s, NL A1B 3X7 (Canada)


    Various algebraic approaches of quantum chemistry all follow a common principle: the fundamental properties and interrelations providing the most essential features of a quantum chemical representation of a molecule or a chemical process, such as a reaction, can always be described by algebraic methods. Whereas such algebraic methods often provide precise, even numerical answers, nevertheless their main role is to give a framework that can be elaborated and converted into computational methods by involving alternative mathematical techniques, subject to the constraints and directions provided by algebra. In general, algebra describes sets of interrelations, often phrased in terms of algebraic operations, without much concern with the actual entities exhibiting these interrelations. However, in many instances, the very realizations of two, seemingly unrelated algebraic structures by actual quantum chemical entities or properties play additional roles, and unexpected connections between different algebraic structures are often giving new insight. Here we shall be concerned with two alternative algebraic structures: the fundamental group of reaction mechanisms, based on the energy-dependent topology of potential energy surfaces, and the interrelations among point symmetry groups for various distorted nuclear arrangements of molecules. These two, distinct algebraic structures provide interesting interrelations, which can be exploited in actual studies of molecular conformational and reaction processes. Two relevant theorems will be discussed.

  12. Alternative algebraic approaches in quantum chemistry

    Various algebraic approaches of quantum chemistry all follow a common principle: the fundamental properties and interrelations providing the most essential features of a quantum chemical representation of a molecule or a chemical process, such as a reaction, can always be described by algebraic methods. Whereas such algebraic methods often provide precise, even numerical answers, nevertheless their main role is to give a framework that can be elaborated and converted into computational methods by involving alternative mathematical techniques, subject to the constraints and directions provided by algebra. In general, algebra describes sets of interrelations, often phrased in terms of algebraic operations, without much concern with the actual entities exhibiting these interrelations. However, in many instances, the very realizations of two, seemingly unrelated algebraic structures by actual quantum chemical entities or properties play additional roles, and unexpected connections between different algebraic structures are often giving new insight. Here we shall be concerned with two alternative algebraic structures: the fundamental group of reaction mechanisms, based on the energy-dependent topology of potential energy surfaces, and the interrelations among point symmetry groups for various distorted nuclear arrangements of molecules. These two, distinct algebraic structures provide interesting interrelations, which can be exploited in actual studies of molecular conformational and reaction processes. Two relevant theorems will be discussed

  13. Algebraic entropy for algebraic maps

    We propose an extension of the concept of algebraic entropy, as introduced by Bellon and Viallet for rational maps, to algebraic maps (or correspondences) of a certain kind. The corresponding entropy is an index of the complexity of the map. The definition inherits the basic properties from the definition of entropy for rational maps. We give an example with positive entropy, as well as two examples taken from the theory of Bäcklund transformations. (letter)

  14. Homotopy DG algebras induce homotopy BV algebras

    Terilla, John; Tradler, Thomas; Wilson, Scott O.


    Let TA denote the space underlying the tensor algebra of a vector space A. In this short note, we show that if A is a differential graded algebra, then TA is a differential Batalin-Vilkovisky algebra. Moreover, if A is an A-infinity algebra, then TA is a commutative BV-infinity algebra.

  15. Symmetries of topological field theories in the BV-framework

    Gieres, F.; Grimstrup, J. M.; Nieder, H.; Pisar, T.; Schweda, M.


    Topological field theories of Schwarz-type generally admit symmetries whose algebra does not close off-shell, e.g. the basic symmetries of BF models or vector supersymmetry of the gauge-fixed action for Chern-Simons theory (this symmetry being at the origin of the perturbative finiteness of the theory). We present a detailed discussion of all these symmetries within the algebraic approach to the Batalin-Vilkovisky formalism. Moreover, we discuss the general algebraic construction of topologic...

  16. Homology for higher-rank graphs and twisted C*-algebras

    Kumjian, Alex; Pask, David; Sims, Aidan


    We introduce a homology theory for k-graphs and explore its fundamental properties. We establish connections with algebraic topology by showing that the homology of a k-graph coincides with the homology of its topological realisation as described by Kaliszewski et al. We exhibit combinatorial versions of a number of standard topological constructions, and show that they are compatible, from a homological point of view, with their topological counterparts. We show how to twist the C*-algebra o...

  17. Duality theories for Boolean algebras with operators

    Givant, Steven


    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.

  18. Advanced linear algebra

    Cooperstein, Bruce


    Advanced Linear Algebra, Second Edition takes a gentle approach that starts with familiar concepts and then gradually builds to deeper results. Each section begins with an outline of previously introduced concepts and results necessary for mastering the new material. By reviewing what students need to know before moving forward, the text builds a solid foundation upon which to progress. The new edition of this successful text focuses on vector spaces and the maps between them that preserve their structure (linear transformations). Designed for advanced undergraduate and beginning graduate stud

  19. Spectral synthesis for Banach Algebras II

    Feinstein, J. F.; Somerset, D. W. B.


    This paper continues the study of spectral synthesis and the topologies $\\tau_{\\infty}$ and $\\tau_r$ on the ideal space of a Banach algebra, concentrating particularly on the class of Haagerup tensor products of C$^*$-algebras. For this class, it is shown that spectral synthesis is equivalent to the Hausdorffness of $\\tau_{\\infty}$. Under a weak extra condition, spectral synthesis is shown to be equivalent to the Hausdorffness of $\\tau_r$.

  20. Piecewise algebraic varieties

    WANG Renhong; ZHU Chungang


    The piecewise algebraic variety is a generalization of the classical algebraic variety. This paper discusses some properties of piecewise algebraic varieties and their coordinate rings based on the knowledge of algebraic geometry.

  1. Linear algebra

    Edwards, Harold M


    In his new undergraduate textbook, Harold M Edwards proposes a radically new and thoroughly algorithmic approach to linear algebra Originally inspired by the constructive philosophy of mathematics championed in the 19th century by Leopold Kronecker, the approach is well suited to students in the computer-dominated late 20th century Each proof is an algorithm described in English that can be translated into the computer language the class is using and put to work solving problems and generating new examples, making the study of linear algebra a truly interactive experience Designed for a one-semester course, this text adopts an algorithmic approach to linear algebra giving the student many examples to work through and copious exercises to test their skills and extend their knowledge of the subject Students at all levels will find much interactive instruction in this text while teachers will find stimulating examples and methods of approach to the subject

  2. Linear algebra

    Liesen, Jörg


    This self-contained textbook takes a matrix-oriented approach to linear algebra and presents a complete theory, including all details and proofs, culminating in the Jordan canonical form and its proof. Throughout the development, the applicability of the results is highlighted. Additionally, the book presents special topics from applied linear algebra including matrix functions, the singular value decomposition, the Kronecker product and linear matrix equations. The matrix-oriented approach to linear algebra leads to a better intuition and a deeper understanding of the abstract concepts, and therefore simplifies their use in real world applications. Some of these applications are presented in detailed examples. In several ‘MATLAB-Minutes’ students can comprehend the concepts and results using computational experiments. Necessary basics for the use of MATLAB are presented in a short introduction. Students can also actively work with the material and practice their mathematical skills in more than 300 exerc...

  3. Clifford algebra, geometric algebra, and applications

    Lundholm, Douglas; Svensson, Lars


    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 v...

  4. Cofree Hopf algebras on Hopf bimodule algebras

    Fang, Xin; Jian, Run-Qiang


    We investigate a Hopf algebra structure on the cotensor coalgebra associated to a Hopf bimodule algebra which contains universal version of Clifford algebras and quantum groups as examples. It is shown to be the bosonization of the quantum quasi-shuffle algebra built on the space of its right coinvariants. The universal property and a Rota-Baxter algebra structure are established on this new algebra.

  5. Generalized exterior algebras

    Marchuk, Nikolay


    Exterior algebras and differential forms are widely used in many fields of modern mathematics and theoretical physics. In this paper we define a notion of $N$-metric exterior algebra, which depends on $N$ matrices of structure constants. The usual exterior algebra (Grassmann algebra) can be considered as 0-metric exterior algebra. Clifford algebra can be considered as 1-metric exterior algebra. $N$-metric exterior algebras for $N\\geq2$ can be considered as generalizations of the Grassmann alg...

  6. Word Hopf algebras

    Hazewinkel, Michiel


    Two important generalizations of the Hopf algebra of symmetric functions are the Hopf algebra of noncommutative symmetric functions and its graded dual the Hopf algebra of quasisymmetric functions. A common generalization of the latter is the selfdual Hopf algebra of permutations (MPR Hopf algebra). This latter Hopf algebra can be seen as a Hopf algebra of endomorphisms of a Hopf algebra. That turns out to be a fruitful way of looking at things and gives rise to wide ranging further generaliz...

  7. Contemporary developments in algebraic K-theory

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

  8. Linear algebra

    Allenby, Reg


    As the basis of equations (and therefore problem-solving), linear algebra is the most widely taught sub-division of pure mathematics. Dr Allenby has used his experience of teaching linear algebra to write a lively book on the subject that includes historical information about the founders of the subject as well as giving a basic introduction to the mathematics undergraduate. The whole text has been written in a connected way with ideas introduced as they occur naturally. As with the other books in the series, there are many worked examples.Solutions to the exercises are available onlin

  9. Lie algebras

    Jacobson, Nathan


    Lie group theory, developed by M. Sophus Lie in the 19th century, ranks among the more important developments in modern mathematics. Lie algebras comprise a significant part of Lie group theory and are being actively studied today. This book, by Professor Nathan Jacobson of Yale, is the definitive treatment of the subject and can be used as a textbook for graduate courses.Chapter I introduces basic concepts that are necessary for an understanding of structure theory, while the following three chapters present the theory itself: solvable and nilpotent Lie algebras, Carlan's criterion and its

  10. Linear algebra

    Stoll, R R


    Linear Algebra is intended to be used as a text for a one-semester course in linear algebra at the undergraduate level. The treatment of the subject will be both useful to students of mathematics and those interested primarily in applications of the theory. The major prerequisite for mastering the material is the readiness of the student to reason abstractly. Specifically, this calls for an understanding of the fact that axioms are assumptions and that theorems are logical consequences of one or more axioms. Familiarity with calculus and linear differential equations is required for understand

  11. Abstract algebra

    Deskins, W E


    This excellent textbook provides undergraduates with an accessible introduction to the basic concepts of abstract algebra and to the analysis of abstract algebraic systems. These systems, which consist of sets of elements, operations, and relations among the elements, and prescriptive axioms, are abstractions and generalizations of various models which evolved from efforts to explain or discuss physical phenomena.In Chapter 1, the author discusses the essential ingredients of a mathematical system, and in the next four chapters covers the basic number systems, decompositions of integers, diop

  12. Analysing hybrid drive system topologies

    Jonasson, Karin


    In this thesis a simulation model is presented that enables a comparison of different hybrid topologies, with respect to fuel consumption, emissions and performance. The obtained results stress the properties of the different topologies and form a foundation for the choice of hybrid topology. The simulation models included in this thesis are the result of collaboration with Petter Strandh at the Division of Combustion Engines, Department of Heat and Power Engineering, Lund University. The stu...


    Labourie, François


    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...

  14. Smarandache Jordan Algebras - abstract

    Vasantha Kandasamy, W. B.; Christopher, S.; A. Victor Devadoss


    We prove a S-commutative Jordan Algebra is a S-weakly commutative Jordan algebra. We define a S-Jordan algebra to be S-simple Jordan algebras if the S-Jordan algebra has no S-Jordan ideals. We obtain several other interesting notions and results on S-Jordan algebras.

  15. Ordered Algebraic Structures : the 1991 Conrad Conference

    Holland, C


    This volume contains a selection of papers presented at the 1991 Conrad Conference, held in Gainesville, Florida, USA, in December, 1991. Together, these give an overview of some recent advances in the area of ordered algebraic structures. The first part of the book is devoted to ordered permutation groups and universal, as well as model-theoretic, aspects. The second part deals with material variously connected to general topology and functional analysis. Collectively, the contents of the book demonstrate the wide applicability of order-theoretic methods, and how ordered algebraic structures have connections with many research disciplines. For researchers and graduate students whose work involves ordered algebraic structures.

  16. Adinkras for Clifford Algebras, and Worldline Supermultiplets

    Doran, C F; Gates, S J; Hübsch, T; Iga, K M; Landweber, G D; Miller, R L


    Adinkras are a graphical depiction of representations of the N-extended supersymmetry algebra in one dimension, on the worldline. These diagrams represent the component fields in a supermultiplet as vertices, and the action of the supersymmetry generators as edges. In a previous work, we showed that the chromotopology (topology with colors) of an Adinkra must come from a doubly even binary linear code. Herein, we relate Adinkras to Clifford algebras, and use this to construct, for every such code, a supermultiplet corresponding to that code. In this way, we correlate the well-known classification of representations of Clifford algebras to the classification of Adinkra chromotopologies.

  17. Mathematical foundations of thermodynamics

    Giles, R; Stark, M; Ulam, S


    Mathematical Foundations of Thermodynamics details the core concepts of the mathematical principles employed in thermodynamics. The book discusses the topics in a way that physical meanings are assigned to the theoretical terms. The coverage of the text includes the mechanical systems and adiabatic processes; topological considerations; and equilibrium states and potentials. The book also covers Galilean thermodynamics; symmetry in thermodynamics; and special relativistic thermodynamics. The book will be of great interest to practitioners and researchers of disciplines that deal with thermodyn

  18. Exact WKB analysis and cluster algebras

    We develop the mutation theory in the exact WKB analysis using the framework of cluster algebras. Under a continuous deformation of the potential of the Schrödinger equation on a compact Riemann surface, the Stokes graph may change the topology. We call this phenomenon the mutation of Stokes graphs. Along the mutation of Stokes graphs, the Voros symbols, which are monodromy data of the equation, also mutate due to the Stokes phenomenon. We show that the Voros symbols mutate as variables of a cluster algebra with surface realization. As an application, we obtain the identities of Stokes automorphisms associated with periods of cluster algebras. The paper also includes an extensive introduction of the exact WKB analysis and the surface realization of cluster algebras for nonexperts. This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ‘Cluster algebras in mathematical physics’. (paper)

  19. Towards a classification of rational Hopf algebras

    Rational Hopf algebras, i.e. certain quasitriangular weak quasi-Hopf *-algebras, are expected to describe the quantum symmetry of rational field theories. In this paper methods are developed which allow for a classification of all rational Hopf algebras that are compatible with some prescribed set of fusion rules. The algebras are parametrized by the solutions of the square, pentagon and hexagon identities. As examples, we classify all solutions for fusion rules with not more than three sectors, as well as for the level three affine A1(1) fusion rules. We also establish several general properties of rational Hopf algebras and present a graphical description of the coassociator in terms of labelled tetrahedra. The latter construction allows to make contact with conformal field theory fusing matrices and with invariants of three-manifolds and topological lattice field theory. (orig.)

  20. Algebraic Stacks

    Tomás L Gómez


    This is an expository article on the theory of algebraic stacks. After introducing the general theory, we concentrate in the example of the moduli stack of vector bundles, giving a detailed comparison with the moduli scheme obtained via geometric invariant theory.

  1. Brane Topological Field Theories and Hurwitz numbers for CW-complexes

    Natanzon, Sergey M.


    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...

  2. Bose condensation in topologically ordered quantum liquids

    Neupert, Titus; He, Huan; von Keyserlingk, Curt; Sierra, German; Bernevig, Andrei

    The condensation of bosons can induce transitions between topological quantum field theories (TQFTs). This as been previously investigated through the formalism of Frobenius algebras and with the use of Vertex lifting coefficients. We discuss an alternative, algebraic approach to boson condensation in TQFTs that is physically motivated and computationally efficient. With a minimal set of assumptions, such as commutativity of the condensation with the fusion of anyons, we can prove a number of theorems linking boson condensation in TQFTs with algebra extensions in conformal field theories and with the problem of factorization of completely positive matrices over the positive integers. We propose an algorithm for obtaining a condensed theory fusion algebra and its modular matrices. For example, this formalism can be used to build multi-layer TQFTs which could be a starting point to build three-dimensional topologically ordered phases. Using this formalism, we also give examples of bosons that cannot undergo a condensation transition due to topological obstructions.

  3. Topologies on Abelian Groups

    Zelenyuk, E. G.; Protasov, I. V.


    A filter phi on an abelian group G is called a T-filter if there exists a Hausdorff group topology under which phi converges to zero. G{phi} will denote the group G with the largest topology among those making phi converge to zero. This method of defining a group topology is completely equivalent to the definition of an abstract group by defining relations. We shall obtain characterizations of T-filters and of T-sequences; among these, we shall pay particular attention to T-sequences on the integers. The method of T-sequences will be used to construct a series of counterexamples for several open problems in topological algebra. For instance there exists, on every infinite abelian group, a topology distinguishing between sequentiality and the Fréchet-Urysohn property (this solves a problem posed by V.I. Malykhin) we also find a topology on the group of integers admitting no nontrivial continuous character, thus solving a problem of Nienhuys. We show also that on every infinite abelian group there exists a free ultrafilter which is not a T-ultrafilter.

  4. Topological Features In Cancer Gene Expression Data

    Lockwood, Svetlana; Krishnamoorthy, Bala


    We present a new method for exploring cancer gene expression data based on tools from algebraic topology. Our method selects a small relevant subset from tens of thousands of genes while simultaneously identifying nontrivial higher order topological features, i.e., holes, in the data. We first circumvent the problem of high dimensionality by dualizing the data, i.e., by studying genes as points in the sample space. Then we select a small subset of the genes as landmarks to construct topologic...

  5. The Universal C*-Algebra of the Electromagnetic Field

    Buchholz, Detlev; Ciolli, Fabio; Ruzzi, Giuseppe; Vasselli, Ezio


    A universal C*-algebra of the electromagnetic field is constructed. It is represented in any quantum field theory which incorporates electromagnetism and expresses basic features of the field such as Maxwell's equations, Poincaré covariance and Einstein causality. Moreover, topological properties of the field resulting from Maxwell's equations are encoded in the algebra, leading to commutation relations with values in its center. The representation theory of the algebra is discussed with focus on vacuum representations, fixing the dynamics of the field.

  6. Exact Symbolic-Numeric Computation of Planar Algebraic Curves

    Berberich, Eric; Emeliyanenko, Pavel; Kobel, Alexander; Sagraloff, Michael


    We present a novel certified and complete algorithm to compute arrangements of real planar algebraic curves. It provides a geometric-topological analysis of the decomposition of the plane induced by a finite number of algebraic curves in terms of a cylindrical algebraic decomposition. From a high-level perspective, the overall method splits into two main subroutines, namely an algorithm denoted Bisolve to isolate the real solutions of a zero-dimensional bivariate system, and an algorithm deno...

  7. The Calkin representation for a certain class of algebras of unbounded operators

    Let D=Dsup(infinity)(T), T=T* >= I selfadjoint. It is proved that the closure of the finite dimensional operators on D with respect to the uniform topology tausub(D) is the only twosided tausub(D) - closed * - ideal C in the maximal operator * - algebra α+(D) on D. Moreover the quotient algebra α+(h)/C equipped with the factor topology induced by tausub(D) is algebraically and topologically isomorphic to an appropriate Op*-algebra A(D)[tausub(D-circumflex)]. The isomorphism is constructed explicitly. This generalizes the classical Calkin result to the unbounded case

  8. PKD Foundation

    ... Up for Advocacy Alerts Donate Ways to Give Foundation Partners Estate Gift Planning Tribute Giving News About ... know Learn More About PKD and the PKD Foundation Make a Gift I Have PKD I Know ...

  9. Scleroderma Foundation

    ... Ways to Add Sun Protection This Summer Scleroderma Foundation National Director of Programs and Services Named Chair ... Get more news headlines >> Buzzworthy Reads>> Facebook Scleroderma Foundation E-Newsletter Signup Get the latest news. Please ...

  10. Vasculitis Foundation

    ... fund the most promising studies. Support the Vasculitis Foundation Donate Now Join VF Team Brandon and Victory ... map feature to find a doctor or Vasculitis Foundation chapter near you. Participate in research Help find ...


    Jipu Ma


    @@ Let B(Rn) be the set of all,n×n real matrices,Sr the set of all matrices with rank r,0≤r≤n,and S#r the number of arcwise connected components of Sr.It is well-known that Sn=GL (Rn) is a Lie group and also a smooth hypersurface in B(Rn) with the dimension n×n.

  12. Two-dimensional topological gravity and equivariant cohomology

    Getzler, E. (Dept. of Mathematics, MIT, Cambridge, MA (United States))


    The analogy between topological string theory and equivariant cohomology for differentiable actions of the circle group on manifolds has been widely remarked on. One of our aims in this paper is to make this analogy precise. We show that topological string theory is the ''derived functor'' of semi-relative cohomology, just as equivariant cohomology is the derived functor of basic cohomology. That homological algebra finds a place in the study of topological string theory should not surprise the reader, granted that topological string theory is the conformal field theorist's algebraic topology. In an earlier paper (1994), we have shown that the cohomology of a topological conformal field theory carries the structure of a Batalin-Vilkovisky algebra (actually, two commuting such structures, corresponding to the two chiral sectors of the theory). In the second part of this paper, we describe the analogous algebraic structure on the equivariant cohomology of a topological conformal field theory: we call this structure a gravity algebra. This algebraic structure is a certain generalization of a Lie algebra, and is distinguished by the fact that it has an infinite sequence of independent operations [l brace]a[sub 1],..,a[sub k][r brace], k [>=] 2, satisfying quadratic relations generalizing the Jacobi rule. (orig.)

  13. The Dyer-Lashof algebra and the Steenrod algebra for generalized homology and cohomology

    An analogue R of the Dyer-Lashof algebra R and an analogue A of the Steenrod algebra A are defined for generalized homology and cohomology theories. It is shown that if there is an E∞-multiplicative structure on a spectrum H, then on the corresponding generalized cohomology H*(X) of a topological space X there is an action AxH*(X)→H*(X) of the Steenrod algebra, while if the space X is an E∞-space, then on the generalized homology H*(X) there is an action RxH*(X)→H*(X) of the Dyer-Lashof algebra. These actions are computed for cobordism of topological spaces. A connection is established between the Steenrod operations and the Landweber-Novikov operations

  14. Wavelets and quantum algebras

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

  15. Central simple Poisson algebras

    SU; Yucai; XU; Xiaoping


    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.

  16. The Onsager Algebra

    El-Chaar, Caroline


    In this thesis, four realizations of the Onsager algebra are explored. We begin with its original definition as introduced by Lars Onsager. We then examine how the Onsager algebra can be presented as a Lie algebra with two generators and two relations. The third realization of the Onsager algebra consists of viewing it as an equivariant map algebra which then gives us the tools to classify its closed ideals. Finally, we examine the Onsager algebra as a subalgebra of the tetrahedron algebra. U...

  17. Linear algebra tools for data mining

    Simovici, Dan A


    This comprehensive volume presents the foundations of linear algebra ideas and techniques applied to data mining and related fields. Linear algebra has gained increasing importance in data mining and pattern recognition, as shown by the many current data mining publications, and has a strong impact in other disciplines like psychology, chemistry, and biology. The basic material is accompanied by more than 550 exercises and supplements, many accompanied with complete solutions and MATLAB applications.

  18. States of MV-algebras

    Flaminio, T.; Kroupa, Tomáš

    Vol.3. London: College Publications, 2015 - (Cintula, P.; Fermüller, C.; Noguera, C.), s. 1183-1236. (Studies in Logic - Mathematical Logic and Foundations. 58). ISBN 978-1-84890-193-3 R&D Projects: GA ČR GAP402/12/1309 Institutional support: RVO:67985556 Keywords : state * MV- algebra * Lukasiewicz logic Subject RIV: BA - General Mathematics

  19. First steps in brave new commutative algebra

    Greenlees, J. P. C.


    This is an expository account of completion and local cohomology in brave new commutative algebra, especially as it applies to completion theorems and their duals in equivariant topology. The first part is fairly direct, but the second considers Morita theory and Gorenstein ring spectra. This draws on work of the author with Benson, Dwyer, Iyengar and May, amongst others.

  20. Quasi-algebras and general Weyl quantization

    In this paper we show how the systematic use of the topological properties of the quasi-sup(*)-algebra L(S,S') leads to a systematization of the quantization procedure. With that as background, the multiplication of certain classes of pairs of operators of L(S,S') and the corresponding twisted product of their sybmols are defined. (orig./HSI)

  1. C*-algebras and operator theory

    Murphy, Gerald J


    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.

  2. Exotic Elliptic Algebras

    Chirvasitu, Alex; Smith, S. Paul


    This paper examines a general method for producing twists of a comodule algebra by tensoring it with a torsor then taking co-invariants. We examine the properties that pass from the original algebra to the twisted algebra and vice versa. We then examine the special case where the algebra is a 4-dimensional Sklyanin algebra viewed as a comodule algebra over the Hopf algebra of functions on the non-cyclic group of order 4 with the torsor being the 2x2 matrix algebra. The twisted algebra is an "...

  3. C*-algebras of tilings with infinite rotational symmetry

    Whittaker, Michael F


    A tiling with infinite rotational symmetry, such as the Conway-Radin Pinwheel Tiling, gives rise to a topological dynamical system to which an \\'etale equivalence relation is associated. A groupoid C*-algebra for a tiling is produced and a separating dense set is exhibited in the C*-algebra which encodes the structure of the topological dynamical system. In the case of a substitution tiling, natural subsets of this separating dense set are used to define an AT-subalgebra of the C*-algebra. Finally our results are applied to the Pinwheel Tiling.

  4. Skew category algebras associated with partially defined dynamical systems

    Lundström, Patrik; Öinert, Per Johan


    We introduce partially defined dynamical systems defined on a topological space. To each such system we associate a functor s from a category G to Topop and show that it defines what we call a skew category algebra A ⋊σ G. We study the connection between topological freeness of s and, on the one ...

  5. Nonmonotonic logics and algebras

    CHAKRABORTY Mihir Kr; GHOSH Sujata


    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.

  6. Fibered F-Algebra

    Kleyn, Aleks


    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.

  7. Topological and geometric measurements of force chain structure

    Giusti, Chad; Owens, Eli T; Daniels, Karen E; Bassett, Danielle S


    Developing quantitative methods for characterizing structural properties of force chains in densely packed granular media is an important step toward understanding or predicting large-scale physical properties of a packing. A promising framework in which to develop such methods is network science, which can be used to translate particle locations and force contacts to a graph in which particles are represented by nodes and forces between particles are represented by weighted edges. Applying network-based community-detection techniques to extract force chains opens the door to developing statistics of force chain structure, with the goal of identifying shape differences across packings, and providing a foundation on which to build predictions of bulk material properties from mesoscale network features. Here, we discuss a trio of related but fundamentally distinct measurements of mesoscale structure of force chains in arbitrary 2D packings, including a novel statistic derived using tools from algebraic topology...

  8. Nijenhuis Operators on n-Lie Algebras

    Jie-Feng, Liu; Yun-He, Sheng; Yan-Qiu, Zhou; Cheng-Ming, Bai


    In this paper, we study (n ‑ 1)-order deformations of an n-Lie algebra and introduce the notion of a Nijenhuis operator on an n-Lie algebra, which could give rise to trivial deformations. We prove that a polynomial of a Nijenhuis operator is still a Nijenhuis operator. Finally, we give various constructions of Nijenhuis operators and some examples. Supported by National Natural Science Foundation of China under Grant Nos. 11471139, 11271202, 11221091, 11425104, Specialized Research Fund for the Doctoral Program of Higher Education under Grant No. 20120031110022, and National Natural Science Foundation of Jilin Province under Grant No. 20140520054JH

  9. Multi-Matrix Models and Noncommutative Frobenius Algebras Obtained from Symmetric Groups and Brauer Algebras

    Kimura, Yusuke


    It has been understood that correlation functions of multi-trace operators in SYM can be neatly computed using the group algebra of symmetric groups or walled Brauer algebras. On the other hand, such algebras have been known to construct 2D topological field theories (TFTs). After reviewing the construction of 2D TFTs based on symmetric groups, we construct 2D TFTs based on walled Brauer algebras. In the construction, the introduction of a dual basis manifests a similarity between the two theories. We next construct a class of 2D field theories whose physical operators have the same symmetry as multi-trace operators constructed from some matrices. Such field theories correspond to non-commutative Frobenius algebras. A matrix structure arises as a consequence of the noncommutativity. Correlation functions of the Gaussian complex multi-matrix models can be translated into correlation functions of the two-dimensional field theories.

  10. Corporate Foundations

    Herlin, Heidi; Thusgaard Pedersen, Janni


    This paper aims to explore the potential of Danish corporate foundations as boundary organizations facilitating relationships between their founding companies and non-governmental organizations (NGOs). Hitherto, research has been silent about the role of corporate foundations in relation to cross......-sector partnerships. The results of this paper are based on interviews, participant observations, and organizational documents from a 19-month empirical study of a Danish corporate foundation. Findings suggest that corporate foundations have potential to act as boundary organizations and facilitate collaborative...... action between business and NGOs through convening, translation, collaboration, and mediation. Our study provides valuable insights into the tri-part relationship of company foundation NGO by discussing the implications of corporate foundations taking an active role in the realm of corporate social...

  11. Rigid current Lie algebras

    Goze, Michel; Remm, Elisabeth


    A current Lie algebra is contructed from a tensor product of a Lie algebra and a commutative associative algebra of dimension greater than 2. In this work we are interested in deformations of such algebras and in the problem of rigidity. In particular we prove that a current Lie algebra is rigid if it is isomorphic to a direct product gxg...xg where g is a rigid Lie algebra.

  12. Solvable quadratic Lie algebras


    A Lie algebra endowed with a nondegenerate, symmetric, invariant bilinear form is called a quadratic Lie algebra. In this paper, the author investigates the structure of solvable quadratic Lie algebras, in particular, the solvable quadratic Lie algebras whose Cartan subalgebras consist of semi-simple elements, the author presents a procedure to construct a class of quadratic Lie algebras from the point of view of cohomology and shows that all solvable quadratic Lie algebras can be obtained in this way.

  13. Graded cluster algebras

    Grabowski, Jan


    In the cluster algebra literature, the notion of a graded cluster algebra has been implicit since the origin of the subject. In this work, we wish to bring this aspect of cluster algebra theory to the foreground and promote its study. We transfer a definition of Gekhtman, Shapiro and Vainshtein to the algebraic setting, yielding the notion of a multi-graded cluster algebra. We then study gradings for finite type cluster algebras without coefficients, giving a full classification. Translating ...

  14. Three Hopf algebras and their common simplicial and categorical background

    Gálvez-Carrillo, Imma; Tonks, Andrew


    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.

  15. Higher algebraic K-theory an overview

    Lluis-Puebla, Emilio; Gillet, Henri; Soulé, Christophe; Snaith, Victor


    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.

  16. Algebraic Soft-Decision Decoding of Hermitian Codes

    Lee, Kwankyu; O'Sullivan, Michael E.


    An algebraic soft-decision decoder for Hermitian codes is presented. We apply Koetter and Vardy's soft-decision decoding framework, now well established for Reed-Solomon codes, to Hermitian codes. First we provide an algebraic foundation for soft-decision decoding. Then we present an interpolation algorithm finding the Q-polynomial that plays a key role in the decoding. With some simulation results, we compare performances of the algebraic soft-decision decoders for Hermitian codes and Reed-S...

  17. Piecewise-Koszul algebras


    It is a small step toward the Koszul-type algebras. The piecewise-Koszul algebras are,in general, a new class of quadratic algebras but not the classical Koszul ones, simultaneously they agree with both the classical Koszul and higher Koszul algebras in special cases. We give a criteria theorem for a graded algebra A to be piecewise-Koszul in terms of its Yoneda-Ext algebra E(A), and show an A∞-structure on E(A). Relations between Koszul algebras and piecewise-Koszul algebras are discussed. In particular, our results are related to the third question of Green-Marcos.

  18. On vertex Leibniz algebras

    Li, Haisheng; Tan, Shaobin; Wang, Qing


    In this paper, we study a notion of what we call vertex Leibniz algebra. This notion naturally extends that of vertex algebra without vacuum, which was previously introduced by Huang and Lepowsky. We show that every vertex algebra without vacuum can be naturally extended to a vertex algebra. On the other hand, we show that a vertex Leibniz algebra can be embedded into a vertex algebra if and only if it admits a faithful module. To each vertex Leibniz algebra we associate a vertex algebra with...

  19. Universal algebra

    Grätzer, George


    Universal Algebra, heralded as ". . . the standard reference in a field notorious for the lack of standardization . . .," has become the most authoritative, consistently relied on text in a field with applications in other branches of algebra and other fields such as combinatorics, geometry, and computer science. Each chapter is followed by an extensive list of exercises and problems. The "state of the art" account also includes new appendices (with contributions from B. Jónsson, R. Quackenbush, W. Taylor, and G. Wenzel) and a well-selected additional bibliography of over 1250 papers and books which makes this a fine work for students, instructors, and researchers in the field. "This book will certainly be, in the years to come, the basic reference to the subject." --- The American Mathematical Monthly (First Edition) "In this reviewer's opinion [the author] has more than succeeded in his aim. The problems at the end of each chapter are well-chosen; there are more than 650 of them. The book is especially sui...

  20. Putting Algebra Progress Monitoring into Practice: Insights from the Field

    Foegen, Anne; Morrison, Candee


    Algebra progress monitoring is a research-based practice that extends a long history of research in curriculum-based measurement (CBM). This article describes the theoretical foundations and research evidence for algebra progress monitoring, along with critical features of the practice. A detailed description of one practitioner's implementation…

  1. A class of simple $C^*$-algebras arising from certain nonsofic subshifts

    Matsumoto, Kengo


    We present a class of subshifts $Z_N, N = 1,2,...$ whose associated $C^*$-algebras ${\\cal O}_{Z_N}$ are simple, purely infinite and not stably isomorphic to any Cuntz-Krieger algebra nor to Cuntz algebra. The class of the subshifts is the first examples whose associated $C^*$-algebras are not stably isomorphic to any Cuntz-Krieger algebra nor to Cuntz algebra. The subshifts $Z_N$ are coded systems whose languages are context free. We compute the topological entropy for the subshifts and show ...

  2. Yoneda algebras of almost Koszul algebras

    Zheng Lijing


    Let be an algebraically closed field, a finite dimensional connected (, )-Koszul self-injective algebra with , ≥ 2. In this paper, we prove that the Yoneda algebra of is isomorphic to a twisted polynomial algebra $A^!$ [ ; ] in one indeterminate of degree +1 in which $A^!$ is the quadratic dual of , is an automorphism of $A^!$, and = () for each $t \\in A^!$. As a corollary, we recover Theorem 5.3 of [2].

  3. Bucket foundations

    Foglia, Aligi; Ibsen, Lars Bo

    In this report, bearing behaviour and installation of bucket foundations are reviewed. Different methods and standards are compared with the experimental data presented in Foglia and Ibsen (2014a). The most important studies on these topics are suggested. The review is focused on the response of...... monopod bucket foundations supporting offshore wind turbines....

  4. Algebra cohomology over a commutative algebra revisited

    Pirashvili, Teimuraz


    The aim of this paper is to give a relatively easy bicomplex which computes the Shukla, or Quillen cohomology in the category of associative algebras over a commutative algebra $A$, in the case when $A$ is an algebra over a field.


    TaoChangli; LuShijie; ChenPeixin


    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.

  6. Topological library, pt.3 spectral sequences in topology

    Novikov, S P; Golubyatnikov, V P


    The final volume of the three-volume edition, this book features classical papers on algebraic and differential topology published in the 1950s-1960s. The partition of these papers among the volumes is rather conditional. The original methods and constructions from these works are properly documented for the first time in this book. No existing book covers the beautiful ensemble of methods created in topology starting from approximately 1950. That is, from Serre's celebrated "singular homologies of fiber spaces." Sample Chapter(s)Chapter 1: Singular homology of fiber spaces - Intr

  7. Enveloping algebras of some quantum Lie algebras

    Pourkia, Arash


    We define a family of Hopf algebra objects, $H$, in the braided category of $\\mathbb{Z}_n$-modules (known as anyonic vector spaces), for which the property $\\psi^2_{H\\otimes H}=id_{H\\otimes H}$ holds. We will show that these anyonic Hopf algebras are, in fact, the enveloping (Hopf) algebras of particular quantum Lie algebras, also with the property $\\psi^2=id$. Then we compute the braided periodic Hopf cyclic cohomology of these Hopf algebras. For that, we will show the following fact: analog...

  8. Foundation Structure


    Method of installing a bucket foundation structure comprising one, two, three or more skirts, into soils in a controlled manner. The method comprises two stages: a first stage being a design phase and the second stage being an installation phase. In the first stage, design parameters are determined...... relating to the loads on the finished foundation structure; soil profile on the location; allowable installation tolerances, which parameters are used to estimate the minimum diameter and length of the skirts of the bucket. The bucket size is used to simulate load situations and penetration into foundation...

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

    Cassidy, T.; Phan, C.; Shelton, B.


    The Yoneda algebra of a Koszul algebra or a D-Koszul algebra is Koszul. K2 algebras are a natural generalization of Koszul algebras, and one would hope that the Yoneda algebra of a K2 algebra would be another K2 algebra. We show that this is not necessarily the case by constructing a monomial K2 algebra for which the corresponding Yoneda algebra is not K2.

  10. Topology-controlled volume rendering

    Weber, Gunther H.; Dillard, Scott E.; Carr, Hamish; Pascucci, Valerio; Hamann, Bernd


    Topology provides a foundation for the development of mathematically sound tools for processing and exploration of scalar fields. Existing topology-based methods can be used to identify interesting features in volumetric data sets, to find seed sets for accelerated isosurface extraction, or to treat individual connected components as distinct entities for isosurfacing or interval volume rendering. We describe a framework for direct volume rendering based on segmenting ...