In this thesis the relationship between Gröbner bases and algebraic coding theory is investigated, and especially applications towards linear codes, with Reed-Müller codes as an illustrative example. We prove that each linear code can be described as a binomial ideal of a polynomial ring, and that a systematic encoding algorithm for such codes is given by the remainder of the information word computed with respect to the reduced Gröbner basis. Finally we show how to apply the representation of a code by its corresponding polynomial ring ideal to construct a class of codes containing the so called primitive Reed-Müller codes, with a few examples of this result.
Ahmad, Ohlson, and von Rosen (2011a) present asymptotic distribution of a one-sample test statistic under non-normality, when the data are high dimensional, i.e., when the dimension of the vector, p, may exceed the sample size, n. This paper extends the case to a two-sample statistic to test the difference of mean vectors of two independent multivariate distributions, again under high-dimensional set up. Using the asymptotic theory of U-statistics, and under mild assumptions on the traces of the unknown covariance matrices, the statistic is shown to follow an approximate normal distribution when n and p are large. However, no relationship between n and p is assumed. An extension to the paired case is given, which, being essentially a one-sample statistic, supplements the asymptotic results obtained in Ahmad, Ohlson, and von Rosen (2011a).
We study the algebra of semigroups of sets (i.e. families of sets closed under finite unions) and its applications. For each n greater than 1 we produce two finite nested families of pairwise different semigroups of sets consisting of subsets of R" without the Baire property.
The focus of this thesis is to introduce the concept of Kähler-Poisson algebras as analogues of algebras of smooth functions on Kähler manifolds. We first give here a review of the geometry of Kähler manifolds and Lie-Rinehart algebras. After that we give the definition and basic properties of Kähler-Poisson algebras. It is then shown that the Kähler type condition has consequences that allow for an identification of geometric objects in the algebra which share several properties with their classical counterparts. Furthermore, we introduce a concept of morphism between Kähler-Poisson algebras and show its consequences. Detailed examples are provided in order to illustrate the novel concepts.
We construct C -algebras for a class of surfaces that are inverse images of certain polynomials of arbitrary degree. By using the directed graph associated with a matrix, the representation theory can be understood in terms of “loop” and “string” representations, which are closely related to the dynamics of an iterated map in the plane. As a particular class of algebras, we introduce the “Hénon algebras,” for which the dynamical map is a generalized Hénon map, and give an example where irreducible representations of all dimensions exist.
We introduce C-Algebras of compact Riemann surfaces Σ as non-commutative analogues of the Poisson algebra of smooth functions on Σ . Representations of these algebras give rise to sequences of matrix-algebras for which matrix-commutators converge to Poisson-brackets as N → ∞. For a particular class of surfaces, interpolating between spheres and tori, we completely characterize (even for the intermediate singular surface) all finite dimensional representations of the corresponding C-algebras
We introduce C-Algebras (quantum analogues of compact Riemann surfaces), defined by polynomial relations in non-commutative variables and containing a real parameter that, when taken to zero, provides a classical non-linear, Poisson-bracket, obtainable from a single polynomial C(onstraint) function. For a continuous class of quartic constraints, we explicitly work out finite dimensional representations of the corresponding C-Algebras.
As n-ary operations, generalizing Lie and Poisson algebras, arise in many different physical contexts, it is interesting to study general ways of constructing explicit realizations of such multilinear structures. Generically, they describe the dynamics of a physical system, and there is a need of understanding their quantization. Hom-Nambu-Lie algebras provide a framework that might be an appropriate setting in which n-Lie algebras (n-ary Nambu-Lie algebras) can be deformed, and their quantization studied. We present a procedure to construct (n + 1)-ary Hom-Nambu-Lie algebras from n-ary Hom-Nambu-Lie algebras equipped with a generalized trace function. It turns out that the implications of the compatibility conditions, that are necessary for this construction, can be understood in terms of the kernel of the trace function and the range of the twisting maps. Furthermore, we investigate the possibility of defining (n + k)-Lie algebras from n-Lie algebras and a k-form satisfying certain conditions.
The need to consider n -ary algebraic structures, generalizing Lie and Poisson algebras, has become increasingly important in physics, and it should therefore be of interest to study the mathematical concepts related to n -ary algebras. The purpose of this paper is to investigate ternary multiplications (as deformations of n -Lie structures) constructed from the binary multiplication of a Hom–Lie algebra, a linear twisting map, and a trace function satisfying certain compatibility conditions. We show that the relation between the kernels of the twisting maps and the trace function plays an important role in this context and provide examples of Hom–Nambu–Lie algebras obtained using this construction.
Several classes of *-algebras associated to teh action of an affine transformation are considered, and an investigation of the interplay between the different classes is initiated. Connections are established that relate representations of *-algebras, geometry of algebraic surfaces, dynamics of affine transformations, graphs and algebras coming from a quantization procedure of Poisson structures. In particular, algebras related to surgaced being inverse images of fourth order polynomials (in ) are studied in detail, and a close link between representation theory and geometric properties is established for compact as well as non-compact surfaces.
In order to investigate to what extent the calculus of classical (pseudo-) Riemannian manifolds can be extended to a noncommutative setting, we introduce pseudo-Riemannian calculi of modules over noncommutative algebras. In this framework, it is possible to prove an analogue of Levi-Civitas theorem, which states that there exists at most one torsion-free and metric connection for a given (metric) module, satisfying the requirements of a real metric calculus. Furthermore, the corresponding curvature operator has the same symmetry properties as the classical Riemannian curvature. As our main motivating example, we consider a pseudo-Riemannian calculus over the noncommutative 3-sphere and explicitly determine the torsion-free and metric connection, as well as the curvature operator together with its scalar curvature.
Let A be a principal ideal domain (PID) or more generally a Dedekind domain and let F be a coherent functor from the category of finitely generated A-modules to itself. We classify the half-exact coherent functors F. In particular, we show that if F is a half-exact coherent functor over a Dedekind domain A, then F is a direct sum of functors of the form Hom(A) (P,-), Hom(A) (A/p(s),-) and A/p(s) circle times -, where P is a finitely generated projective A-module, p a nonzero prime ideal in A and s amp;gt;= 1.
Asymptotic properties of high powers of an ideal related to a coherent functor F are investigated. It is shown that when N is an artinian module the sets of attached prime ideals Att(A) F(0 :(N) a(n)) are the same for n large enough. Also it is shown that for an artinian module N if the modules F(0 :(N) a(n)) have finite length and for a finitely generated module M if the modules F(M/a(n) M) have finite length, their lengths are given by polynomials in n, for large n. When A is local it is shown that, the Betti numbers beta(i)(F(M /a(n) M)) and the Bass numbers mu(i)(F(M / a(n) M)) are given by polynomials in n for large n. (C) 2018 Elsevier Inc. All rights reserved.
Let (L; C) be the (up to isomorphism unique) countable homogeneous structure carrying a binary branching C-relation. We study the reducts of (L; C), i.e., the structures with domain L that are first-order definable in (L; C). We show that up to existential interdefinability, there are finitely many such reducts. This implies that there are finitely many reducts up to first-order interdefinability, thus confirming a conjecture of Simon Thomas for the special case of (L; C). We also study the endomorphism monoids of such reducts and show that they fall into four categories.
We elaborate on the existence and construction of the so-called combinatorial configurations. The main result is that for fixed degrees the existence of such configurations is given by a numerical semigroup. The proof is constructive giving a method to obtain combinatorial configurations with parameters large enough.
Genom att analysera domstolsmaterialet från rättegången mot fildelningssiten The Pirat Bay, i relation till en idéhistorisk diskussion om äganderätt, har uppsatsen funnit att den liberala tanketraditionen och dess juridiska institutioner står inför en betydelseglidning vad gället begreppsparet ”Egendom” och ”Stöld”. Det har visat sig att Lockes naturtillstånd, varseblivningen av ”det oändliga” på jorden, har skiftat plats; från ”naturen” ut till ”cyberspace”, vilket har resulterat i att fildelningstekniken skapat en ny matematik som omöjliggör tidigare egendomsdefinition.
This note verifies a conjecture of Armitage and Goldstein that annular domains may be characterized as quadrature domains for harmonic functions with respect to a uniformly distributed measure on a sphere.
We introduce a notion of resultant of two meromorphic functions on a compact Riemann surface and demonstrate its usefulness in several respects. For example, we exhibit several integral formulas for the resultant, relate it to potential theory and give explicit formulas for the algebraic dependence between two meromorphic functions on a compact Riemann surface. As a particular application, the exponential transform of a quadrature domain in the complex plane is expressed in terms of the resultant of two meromorphic functions on the Schottky double of the domain.
It is well known that every closed Riemann surface S of genus g amp;gt;= 2, admitting a group G of conformal automorphisms so that S/G has triangular signature, can be defined over a finite extension of Q. It is interesting to know, in terms of the algebraic structure of G, if S can in fact be defined over Q. This is the situation if G is either abelian or isomorphic to A x Z(2), where A is an abelian group. On the other hand, as shown by Streit and Wolfart, if G congruent to Z(p) x Z(q), where p, q amp;gt; 3 are prime integers, then S is not necessarily definable over Q. In this paper, we observe that if G congruent to Z(2)(2) x Z(m) with m amp;gt;= 3, then S can be defined over Q. Moreover, we describe explicit models for S, the corresponding groups of automorphisms, and an isogenous decomposition of their Jacobian varieties as product of Jacobians of hyperelliptic Riemann surfaces.
'Groups St Andrews 2005' was held in the University of St Andrews in August 2005 and this first volume of a two-volume book contains selected papers from the international conference. Four main lecture courses were given at the conference, and articles based on their lectures form a substantial part of the Proceedings. This volume contains the contributions by Peter Cameron (Queen Mary, London) and Rostislav Grogorchuk (Texas A&M, USA). Apart from the main speakers, refereed survey and research articles were contributed by other conference participants. Arranged in alphabetical order, these articles cover a wide spectrum of modern group theory. The regular Proceedings of Groups St Andrews conferences have provided snapshots of the state of research in group theory throughout the past 25 years. Earlier volumes have had a major impact on the development of group theory and it is anticipated that this volume will be equally important.
Given a compact Riemann surface X with an action of a finite group G, the group algebra provides an isogenous decomposition of its Jacobian variety JX, known as the group algebra decomposition of JX. We obtain a method to concretely build a decomposition of this kind. Our method allows us to study the geometry of the decomposition. For instance, we build several decompositions in order to determine which one has kernel of smallest order. We apply this method to families of trigonal curves up to genus 10.
Syftet med denna litteraturstudie var att granska och analysera vad forskning visar beträffande elevers svårigheter för området bråk, samt hur lärare kan underlätta elevers förståelse för bråk i årskurs 4-6. I litteraturstudien har databassökningar gjorts via UniSearch, ERIC, MathEduc och Google Scholar. Även manuella sökningar har använts. Bråk tonas ned i dagens undervisning och resultat från TIMSS och PISA visar att bråk är ett, för elever såväl som lärare, problematiskt område i matematiken. Forskare framhåller bland annat elevers svårigheter för täljaren och nämnarens innebörd, samt jämförelse och beräkning av bråkuttryck. Vidare framhäver forskare att lärare med bland annat diskussioner i klassrummet, praktiskt material, samt en verklighetsanknuten och elevcentrerad bråkundervisning främjar elevers inlärning. Resultaten visar dock att forskare inte är eniga vad gäller de svårigheter elever har på området bråk.
Using the syzygy method, established in our earlier paper (Krasnov and Tkachev, 2018), we characterize the combinatorial stratification of the variety of two-dimensional real generic algebras. We show that there exist exactly three different homotopic types of such algebras and relate this result to potential applications and known facts from qualitative theory of quadratic ODEs. The genericity condition is crucial. For example, the idempotent geometry in Clifford algebras or Jordan algebras of Clifford type is very different: such algebras always contain nontrivial submanifolds of idempotents.
In this paper, we study the variety of all nonassociative (NA) algebras from the idempotent point of view. We are interested, in particular, in the spectral properties of idempotents when algebra is generic, i.e. idempotents are in general position. Our main result states that in this case, there exist at least $n−1$ nontrivial obstructions (syzygies) on the Peirce spectrum of a generic NA algebra of dimension $n$. We also discuss the exceptionality of the eigenvalue $\lambda=\frac12$ which appears in the spectrum of idempotents in many classical examples of NA algebras and characterize its extremal properties in metrised algebras. solution.
Innehållet i studien handlar om att kategorisera olika typer av fel som elever i åk 1 på gymnasiet gör i algebra. Data utgörs av 80 elevprov skrivna av elever på samhällsvetenskapsprogrammet och VVS- och fastighetsprogrammet läsåret 2017/2018 och 2018/2019. Uppgifterna som eleverna har fått göra är lösa ekvationer, förenkla uttryck, räkna värdet av ett uttryck samt problemlösning. Elevernas svar har analyserats och kategoriserats i sex feltyper: 1. Förståelsefel, 2. Procedurfel, 3. Modelleringsfel eller problemlösningsfel, 4. Resonemangsfel, 5. Redovisningsfel eller kommunikationsfel, 6. Övriga fel.
I resultatet preseneteras varje feltyp illustrerad med elevexempel. Med tidigre forskning som utgångspunkt identifieras och diskuteras vilka missuppfattningar och svårigheter som kan vara den bakomliggande orsaken till att eleverna gjort dessa fel.
Några exempel på orsaker är att eleverna inte uppfattar variabelns (x) symboliska värde, förstår inte variablers generella beteckning (a och b), att variabeln kan representera en siffra, eleverna övergeneraliserar, förstår inte räkning med negativa tal, kan inte hantera aritmetik, förstår inte likhetstecknets betydelse, har oeffektiv resonemang (gissar, testar sig fram), samt skriver av uppgiften fel.
We study admissible transformations and solve group classification problems for various classes of linear and nonlinear Schrödinger equations with an arbitrary number n of space variables.
The aim of the thesis is twofold. The first is the construction of the new theory of uniform seminormalized classes of differential equations and its application to solving group classification problems for these classes. Point transformations connecting two equations (source and target) from the class under study may have special properties of semi-normalization. This makes the group classification of that class using the algebraic method more involved. To extend this method we introduce the new notion of uniformly semi-normalized classes. Various types of uniform semi-normalization are studied: with respect to the corresponding equivalence group, with respect to a proper subgroup of the equivalence group as well as the corresponding types of weak uniform semi-normalization. An important kind of uniform semi-normalization is given by classes of homogeneous linear differential equations, which we call uniform semi-normalization with respect to linear superposition of solutions.
The class of linear Schrödinger equations with complex potentials is of this type and its group classification can be effectively carried out within the framework of the uniform semi-normalization. Computing the equivalence groupoid and the equivalence group of this class, we show that it is uniformly seminormalized with respect to linear superposition of solutions. This allow us to apply the version of the algebraic method for uniformly semi-normalized classes and to reduce the group classification of this class to the classification of appropriate subalgebras of its equivalence algebra. To single out the classification cases, integers that are invariant under equivalence transformations are introduced. The complete group classification of linear Schrödinger equations is carried out for the cases n = 1 and n = 2.
The second aim is to study group classification problem for classes of generalized nonlinear Schrödinger equations which are not uniformly semi-normalized. We find their equivalence groupoids and their equivalence groups and then conclude whether these classes are normalized or not. The most appealing classes are the class of nonlinear Schrödinger equations with potentials and modular nonlinearities and the class of generalized Schrödinger equations with complex-valued and, in general, coefficients of Laplacian term. Both these classes are not normalized. The first is partitioned into an infinite number of disjoint normalized subclasses of three kinds: logarithmic nonlinearity, power nonlinearity and general modular nonlinearity. The properties of the Lie invariance algebras of equations from each subclass are studied for arbitrary space dimension n, and the complete group classification is carried out for each subclass in dimension (1+2). The second class is successively reduced into subclasses until we reach the subclass of (1+1)-dimensional linear Schrödinger equations with variable mass, which also turns out to be non-normalized. We prove that this class is mapped by a family of point transformations to the class of (1+1)-dimensional linear Schrödinger equations with unique constant mass.
We describe admissible point transformations in the class of (1+2)-dimensional linear Schrodinger equations with complex potentials. We prove that any point transformation connecting two equations from this class is the composition of a linear superposition transformation of the corresponding initial equation and an equivalence transformation of the class. This shows that the class under study is semi-normalized.
The longest matching consecutive subsequence between two N-ary expansions found applications in information theory and molecular biology. In this paper, we establish an almost sure limit theorem for the longest matching consecutive subsequence, and investigate the size of the set of points violating the limit in terms of Hausdorff dimension.
Let X be a set, τ1, τ2 topologies on X and Bp(X, τi) the family of all subsets of X possessing the Baire property in (X, τi), i = 1, 2. In this paper we study conditions on τ1 and τ2 that imply a relationship (for example, inclusion or equality) between the families Bp(X, τ1) and Bp(X, τ2). We are mostly interested in the case where the topology τ2 is formed with the help of a local function defined by the topology τ1 and an ideal of sets I on X. We also consider several applications of the local function defined by the usual topology on the reals and the ideal of all meager sets there, for proving some known facts.
In this thesis the last two digits of m^k, for different cases of the positive integers m and k, in the base of 10 has been determined. Moreover, using fundamental theory from elementary number theory and abstract algebra, results most helpful in finding the last two digits in any base b has been regarded and developed, such as how to reduce large m and k to more manageable numbers.
It is proved that a numerical semigroup can be associated to the triangle-free -configurations, and some results on existence are deduced. For example it is proved that for any there exists infinitely many -configurations. Most proofs are given from a graph theoretical point of view, in the sense that the configurations are represented by their incidence graphs. An application to private information retrieval is described.
It is proved that the numerical semigroups associated to the combinatorial configurations satisfy a family of non-linear symmetric patterns. Also, these numerical semigroups are studied for two particular classes of combinatorial configurations.
In their paper of 1993, Meyer and Neutsch established the existence of a 48-dimensional associative subalgebra in the Griess algebra G. By exhibiting an explicit counter example, the present paper shows a gap in the proof one of the key results in Meyer and Neutsch’s paper, which states that an idempotent a in the Griess algebra is indecomposable if and only its Peirce 1-eigenspace (i.e. the 1-eigenspace of the linear transformation L : x → ax) is one-dimensional. The present paper fixes this gap, and shows a more general result: let V be a real commutative nonassociative algebra with an associative inner product, and let c be a nonzero idempotent of V such that its Peirce 1-eigenspace is a subalgebra; then, c is indecomposable if and only if its Peirce 1-eigenspace is one-dimensional. The proof of this result is based on a general variational argument for real commutative metrised algebras with inner product.
We establish a natural correspondence between (the equivalence classes of) cubic solutions of a cubic eiconal equation and (the isomorphy classes of) cubic Jordan algebras.
By using a non-associative algebra argument, we prove that any cubic homogeneous polynomial solution to the p-Laplace equation in R^n is identically zero for any n>2 and any p distinct from 1 and 2.
In this paper, we address the following question: Why certain nonassociative algebra structures emerge in the regularity theory of elliptic type PDEs and also in constructing nonclassical and singular solutions? The aim of the paper is twofold. Firstly, to give a survey of diverse examples on nonregular solutions to elliptic PDEs with emphasis on recent results on nonclassical solutions to fully nonlinear equations. Secondly, to define an appropriate algebraic formalism which makes the analytic part of the construction of nonclassical solutions more transparent.
In this short survey we give a background and explain some recent developments in algebraic minimal cones and nonassociative algebras. A part of this paper is recollections of my collaboration with my teacher, PhD supervisor and a colleague, Vladimir Miklyukov on minimal surface theory that motivated the present research.
Enterprises need patent transfer strategies to improve their technology management. This paper proposes a combinatorial optimization model that is based on intelligent computing to support enterprises’ decision making in developing patent transfer strategy. The model adopts the Black–Scholes Option Pricing Model and Arbitrage Pricing Theory to estimate a patent’s value. Based on the estimation, a hybrid genetic algorithm is applied that combines genetic algorithms and greedy strategy for the optimization purpose. Encode repairing and a single-point crossover are applied as well. To validate this proposed model, a case study is conducted. The results indicate that the proposed model is effective for achieving optimal solutions. The combinatorial optimization model can help enterprise promote their benefits from patent sale and support the decision making process when enterprises develop patent transfer strategies.