Basic Algebraic Structures Research Articles

Metarouting with automatic tunneling in multilayer networks

Metarouting allows for the modeling of routing protocols using an algebraic structure called routing algebra. Routing protocols requiring design or validation can easily be modeled using this approach. To date, however, existing research on routing algebras has mainly focused on applying this approach to routing protocols that are generally used in networks which have a single addressing and forwarding protocol. The basic algebraic structures used in such contexts are semirings, Sobrinho’s algebras and algebras of endomorphisms. In this paper, we propose the modification of these existing routing algebras to deal with networks that contain multiple forwarding protocols where tunnels are omnipresent. To achieve this, we define new algebraic structures derived from the three aforementioned ones, in order to model the generalized routing problem with automatic tunneling entitled valid paths algebra. All of our routing algebras are defined as semi-direct products of two structures, the well-known shortest paths algebra and the proposed valid paths algebra. These new algebras are isotonic and non-monotonic with a partial order. We propose a fixed point for those new algebras and we prove the iterative convergence to the optimal solution of the valid shortest paths problem.

A gentle alert on the definitions of basic algebraic structures

We provide a gentle alert that the standard definitions of basic algebraic structures—such as that of a group—contain aspects that may be questionable to students, and discuss what instructors could do to minimise the questionability.

Construction of an Infinite Cyclic Group Formed by Artificial Differential Neurons

Infinite cyclic groups created by various objects belong to the class to the class basic algebraic structures. In this paper, we construct the infinite cyclic group of differential neurons which are modifications of artificial neurons in analogy to linear ordinary differential operators of the n-th order. We also describe some of their basic properties.

Rota-Baxter operators on Turaev's Hopf group (co)algebras I: Basic definitions and related algebraic structures

We find a natural compatible condition between the Rota-Baxter operator and Turaev's (Hopf) group-(co)algebras, which leads to the concept of Rota-Baxter Turaev's (Hopf) group-(co)algebra. Two characterizations of Rota-Baxter Turaev's group-algebras (abbr. T-algebras) are obtained: one by Atkinson factorization and the other by T-quasi-idempotent elements. The relations among some related Turaev's group algebraic structures (such as (tri)dendriform T-algebras, Zinbiel T-algebras, pre-Lie T-algebras, Lie T-algebras) are discussed, and some concrete examples from the algebras of dimensions 2,3 and 4 are given. At last we prove that Rota-Baxter Poisson T-algebras can produce pre-Poisson T-algebras and Poisson T-algebras can be obtained from pre-Poisson T-algebras.

Fuzziness on Hopf Algebraic Structures with Its Application

The objective of writing this manuscript is to apply the concept of fuzzy set on some basic Hopf algebraic structures. In this manuscript, the novel concepts of fuzzy Hopf subalgebra, fuzzy Hopf ideal, and fuzzy H-submodule are proposed. Some properties of these concepts are discussed, and some significant results are also proved in it. The advantages of the proposed work are also studied in it. The application of the proposed work is also discussed in it.

Cayley graphs of basic algebraic structures

We present simple graph-theoretic characterizations for the Cayley graphs of monoids, right-cancellative monoids, left-cancellative monoids, and groups.

Horn versus full first-order: Complexity dichotomies in algebraic constraint satisfaction

We study techniques for deciding the computational complexity of infinite-domain constraint satisfaction problems. For certain basic algebraic structures Δ, we prove definability theorems of the following form: for every first-order expansion Γ of Δ, either Γ has a quantifier-free Horn definition in Δ, or there is an element d of Γ such that all non-empty relations in Γ contain a tuple of the form (d,…,d), or all relations with a first-order definition in Δ have a primitive positive definition in Γ. The results imply that several families of constraint satisfaction problems exhibit a complexity dichotomy: the problems are either polynomial-time solvable or NP-hard depending on the choice of the allowed relations. As concrete examples, we investigate fundamental algebraic constraint satisfaction problems. The first class consists of all relational structures with a first-order definition in (ℚ; +) that contain the relation {(x, y, z) ∈ ℚ3 | x + y = z}. The second class is the affine variant of the first class. In both cases, we obtain full dichotomies by utilizing our general methods.

Fuzzy modal-like approximation operators based on double residuated lattices

In many applications we have a set of objects together with their properties. Since the available information is usually incomplete and/or imprecise, the true knowledge about subsets of objects can be determined approximately only. In this paper, we discuss a fuzzy generalisation of two pairs of relation-based operators suitable for fuzzy set approximations, which have been recently investigated by Düntsch and Gediga. Double residuated lattices, introduced by Orlowska and Radzikowska, are taken as basic algebraic structures. Main properties of these operators are presented.

Real analysis: an introduction to the theory of real functions and integration

PART I. AN INTRODUCTION TO GENERAL TOPOLOGY SET-THEORETIC AND ALGEBRAIC PRELIMINARIES Sets and Basic Notation Functions Set Operations under Maps Relations and Well-Ordering Principle Cartesian Product Cardinality Basic Algebraic Structures ANALYSIS OF METRIC SPACES Definitions and Notations The Structure of Metric Spaces5 Convergence in Metric Spaces Continuous Mappings in Metric Spaces Complete Metric Spaces Compactness Linear and Normed Linear Spaces ELEMENTS OF POINT SET TOPOLOGY Topological Spaces Bases and Subbases for Topological Spaces Convergence of Sequences in Topological Spaces and Countability Continuity in Topological Spaces Product Topology Notes on Subspaces and Compactnes Function Spaces and Ascoli's Theorem Stone-Weierstrass Approximation Theorem Filter and Net Convergence Separation Functions on Locally Compact Spaces PART II. BASICS OF MEASURE AND INTEGRATION MEASURABLE SPACES AND MEASURABLE FUNCTIONS Systems of Sets System's Generators Measurable Functions MEASURES Set Functions Extension of Set Functions to a Measure Lebesgue and Lebesgue-Stieltjes Measures Image Measures Extended Real-Valued Measurable Functions Simple Functions ELEMENTS OF INTEGRATION Integration on C -1(W,S) Main Convergence Theorems Lebesgue and Riemann Integrals on R Integration with Respect to Image Measures Measures Generated by Integrals. Absolute Continuity. Orthogonality Product Measures of Finitely Many Measurable Spaces and Fubini's Theorem Applications of Fubini's Theorem CALCULUS IN EUCLIDEAN SPACES Differentiation Change of Variables PART III. FURTHER TOPICS IN INTEGRATION ANALYSIS IN ABSTRACT SPACES Signed and Complex Measures Absolute Continuity Singularity Lp Spaces Modes of Convergence Uniform Integrability Radon Measures on Locally Compact Hausdorff Spaces Measure Derivatives CALCULUS ON THE REAL LINE Monotone Functions Functions of Bounded Variation Absolute Continuous Functions Singular Functions BIBLIOGRAPHY

State sum invariants of 3-manifolds and quantum 6 j-symbols

IN THE 1980s the topology of low dimensional manifolds has experienced the most remarkable intervention of ideas developed in rather distant areas of mathematics. In the 4dimensional topology this process was initiated by S. Donaldson. He applied the theory of the Yang-Mills equation and instantons to study 4-manifolds. In dimension 3 a similar breakthrough was made by V. Jones. He discovered his famous polynomial of links in 3-sphere S3 via an astonishing use of von Neumann algebras. It has been soon understood that deep notions of statistical mechanics and quantum field theory stay behind the Jones polynomial (see [8], [16], [18]). The relevant basic algebraic structures turn out to be the Yang-Baxter equation, the R-matrices, and the quantum groups (see [S], [6], [7]). This viewpoint, in particular, enables one to generalize the Jones polynomial to links in arbitrary compact oriented 3-manifolds (see [ 131). In this paper we present a new approach to constructing “quantum” invariants of 3-manifolds. Our approach is intrinsic and purely combinatorial. The invariant of a manifold is defined as a certain state sum computed on an arbitrary triangulation of the manifold. The state sum in question is based on the so-called quantum 6j-symbols associated with the quantized universal enveloping algebra U,&(C)) where CJ is a complex root of 1 of a certain degree z > 2 (see [9]). The state sum on a triangulation X of a compact 3-manifold M is defined, roughly speaking, as follows. Assume for simplicity that M is closed, i.e. 8M = @. We consider “colorings” of X which associate with edges of X elements of the set of colors (0, l/2, 1, . . . , (I 2)/2). H avm a coloring of X we associate . g with each 3-simplex of X the q-6j-symbol

Write a book!

Five years ago, I discovered the computer for myself. This was certainly no accident, as some people thought not having read carefully enough the books I wrote. I always included one or the other algorithm in them for potential use as a calculation device. Most of these algorithms were very naive I must admit. Anyway, five years ago, I started using the computer. Since I had to learn the art right from the beginning, I began teaching the subject to my students. This seems absurd, but it is the way professors learn. I realized immediately several things. First of all, there existed no book presenting algebra in such a way that one could compute in the end the objects one is dealing with; I say "existed," since the book of Lipson [2] appeared in the meantime. Also, Buchberger, Collins and Loos [1] is a very valuable publication indeed in this context. Secondly, our students - I am speaking of Western Germany and of students of mathematics - do not got enough computing experience. The situation seems improving, but still a lot has to be done in my opinion. Thirdly, applied mathematics or rather applicable mathematics is still considered to be numerical mathematics by many collegues. Of course, one has to integrate numerically differential equations to get a probe near Saturn, but the quality of the pictures we get from there depends heavily on algebra and algebraic computing. Hence algebra has to be taught in a new way. All the beautiful achievements of the last two or three decades in computational algebra should become part of our teaching. For this purpose books must be written. Lipson [2] is a good example for what I mean, but it does not cover enough ground. I am actually writing one planned to have two or three volumes. It will be concerned with the basic algebraic structures including programs ready to use. A preliminary draft of some of them has been published [3]. Unfortunately, this preprint is out of print, but a friend of yours may happen to have got a copy as well as more material about the book to inform you.

The basic algebraic structures in categories of derivations

The basic algebraic structures within the categories of derivations determined by rewriting systems are presented. The similarity congruence relation in categories of derivations is given in three versions. The syntax category is formed by taking derivations modulo similarity. This category is a free strict monoidal category, a simple form of a 2-category. The syntax category is central to the study of rewriting systems, morphisms in the category generalizing the notion of “derivation tree,” so a detailed development is given. Griffith's interchange operators on derivations form a 2-category over a category of derivations. Representability of a similarity class is defined and shown to imply the existence of group of operators on the class, induced by interchanges. Uniform representability of rewriting systems is defined and shown to imply that the set of left divisors of each derivation in the syntax category is a distributive lattice.

Convolution and correlation algebras

An algebraic characterization of convolution and correlation is outlined. The basic algebraic structures generated on a suitable vector space by the two operations are described. The convolution induces an associative Abelian algebra over the real field; the correlation induces a not-associative, not-commutative — but Lieadmissible algebra — with a left unity. The algebraic connection between the two algebras is found to coincide with the relation of isotopy, an extension of the concept of equivalence. The interest of these algebraic structures with respect to information processing is discussed.

Inequalities for structure functions of deep inelastic lepton-nucleon scattering giving tests of basic algebraic structures

Inequalities for the structure functions of deep inelastic electro-and-neutrino-production on nucleons are derived (a) from the quark parton model or equivalently from a hypothesis of Fritzsch and Gell-Mann on the leading light cone singularity of current commutators. (b) from the quark model with neutral gluons using current algebra techniques to the same extent as for the Callan-Gross relation. Present data satisfy these inequalities, and give stringent predictions for future neutrino-production experiments. On the other hand, three models with integrally charged partons, the SUB-model, the Drell-Levy-Yan model and the Bjorken-Glashow model are found to be in trouble by comparison with experiment.