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• 1.
Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
A Gröbner basis algorithm for fast encoding of Reed-Müller codes2016Independent thesis Basic level (degree of Bachelor), 10,5 credits / 16 HE creditsStudent thesis

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.

• 2.
Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, The Institute of Technology.
A two-sample test statistic for high-dimensional multivariate data under non-normality2011Report (Other academic)

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).

• 3.
Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering. National University of Rwanda, Rwanda.
THE ALGEBRA OF SEMIGROUPS OF SETS2015In: Mathematica Scandinavica, ISSN 0025-5521, E-ISSN 1903-1807, Vol. 116, no 2, p. 161-170Article in journal (Refereed)

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.

• 4. Buy this publication >>
Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
Kähler-Poisson Algebras2018Licentiate thesis, comprehensive summary (Other academic)

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.

• 5.
Department of Mathematics, Royal Institute of Technology, S-100 44 Stockholm, Sweden .
Representation theory of C  -algebras for a higher-order class of spheres and tori2008In: Journal of Mathematical Physics, ISSN 0022-2488, E-ISSN 1089-7658, Vol. 49, p. 053502-1-053502-13, article id 053502Article in journal (Refereed)

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.

• 6.
Institut des Hautes Études Scientifiques, Le Bois-Marie 35, route de Chartres, F-91440, Bures-sur-Yvette, France och Max Planck Institute for Gravitational Physics, Am Mühlenberg 1, D-14476, Golm, Germany .
Laboratoire de MIA, 4, rue des Frères Lumière, Université de Haute-Alsace, F-68093, Mulhouse, France . Université du Luxembourg, FSTC 162a, avenue de la Faïencerie, L-1511, Luxembourg City, Luxembourg . Department of Mathematics, KTH, S-10044, Stockholm, Sweden . Max Planck Institute for Gravitational Physics, Am Mühlenberg 1, D-14476, Golm, Germany .
Noncommutative Riemann Surfaces by Embeddings in R32009In: Communications in Mathematical Physics, ISSN 0010-3616, E-ISSN 1432-0916, Vol. 288, p. 403-429Article in journal (Refereed)

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

• 7.
Department of Mathematics, Royal Institute of Technology, Lindstedtsvägen 25 S-10044 Stockholm, Sweden.
Laboratoire de MIA, Université de Haute-Alsace, 4, rue des Frères Lumière, F-68093 Mulhouse, France. Department of Mathematics, Royal Institute of Technology, Lindstedtsvägen 25 S-10044 Stockholm, Sweden och Laboratoire de MIA, Université de Haute-Alsace, 4, rue des Frères Lumière, F-68093 Mulhouse, France. Department of Mathematics, Royal Institute of Technology, Lindstedtsvägen 25 S-10044 Stockholm, Sweden. Max Planck Institute for Gravitational Physics, Albert Einstein Institute, Am Mühlenberg 1 D-14476 Golm., Germany.
Fuzzy Riemann surfaces2009In: Journal of High Energy Physics (JHEP), ISSN 1126-6708, E-ISSN 1029-8479, Vol. 2009, no 024, p. 1-17Article in journal (Refereed)

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.

• 8.
Max Planck Institute for Gravitational Physics (AEI), Am Mühlenberg 1, D-14476 Golm, Germany.
Université de Haute Alsace, Laboratoire de Mathématiques, Informatique et Applications, 4, rue des Frères Lumière F-68093 Mulhouse, France . Mälardalen University, Division of Applied Mathematics, The School of Education, Culture and Communication, Box 883, 721 23 Västerås, Sweden och Centre for Mathematical Sciences, Lund University, Box 118, 221 00 Lund, Sweden .
Construction of n-Lie algebras and n-ary Hom-Nambu-Lie algebras2011In: Journal of Mathematical Physics, ISSN 0022-2488, E-ISSN 1089-7658, Vol. 52, article id 123502Article in journal (Refereed)

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.

• 9.
Max Planck Institute for Gravitational Physics (AEI), Am Mühlenberg 1, D-14476 Golm, Germany.
Laboratoire de Mathématiques, Informatique et Applications, Université de Haute Alsace, 4, rue des Frères Lumière, F-68093 Mulhouse, France . Centre for Mathematical Sciences, Lund University, Box 118, SE-221 00 Lund, Sweden .
Ternary Hom–Nambu–Lie algebras induced by Hom–Lie algebras2010In: Journal of Mathematical Physics, ISSN 0022-2488, E-ISSN 1089-7658, Vol. 51, p. 043515-1-043515-11, article id 43515Article in journal (Refereed)

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.

• 10.
Albert Einstein Institute, Golm, Germany..
Centre for Mathematical Sciences, Lund University, Box 118, 221 00 Lund, Sweden .
Affine transformation crossed product type algebras and noncommutative surfaces2009In: Operator structures and dynamical systems :: July 21-25 2008, Lorentz Center, Leiden, the Netherlands, satellite conference of the fifth European Congress of Mathematics, American Mathematical Society (AMS), 2009, 503, p. 1-25Chapter in book (Refereed)

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 $\mathbb{R}^3$) are studied in detail, and a close link between representation theory and geometric properties is established for compact as well as non-compact surfaces.

• 11.
Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
Riemannian curvature of the noncommutative 3-sphere2017In: Journal of Noncommutative Geometry, ISSN 1661-6952, E-ISSN 1661-6960, Vol. 11, no 2, p. 507-536Article in journal (Refereed)

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.

• 12.
Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering. Univ Zambia, Zambia.
Half-exact coherent functors over Dedekind domains2019In: Journal of Algebra and its Applications, ISSN 0219-4988, E-ISSN 1793-6829, Vol. 18, no 5, article id 1950099Article in journal (Refereed)

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.

• 13.
Univ Zambia, Zambia.
Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
Coherent functors and asymptotic stability2019In: Journal of Algebra, ISSN 0021-8693, E-ISSN 1090-266X, Vol. 522Article in journal (Refereed)

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.

• 14.
Technical University of Dresden, Germany.
Linköping University, Department of Computer and Information Science, Software and Systems. Linköping University, Faculty of Science & Engineering. Vienna University of Technology, Austria.
THE REDUCTS OF THE HOMOGENEOUS BINARY BRANCHING C-RELATION2016In: Journal of Symbolic Logic (JSL), ISSN 0022-4812, E-ISSN 1943-5886, Vol. 81, no 4, p. 1255-1297Article in journal (Refereed)

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.

• 15.
Universitat Rovira i Virgili, Tarragona, Catalonia, Spain .
Universitat Rovira i Virgili, Tarragona, Catalonia, Spain .
The semigroup of combinatorial configurations2012In: Semigroup Forum, ISSN 0037-1912, E-ISSN 1432-2137, Vol. 84, no 1, p. 91-96Article in journal (Refereed)

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.

• 16.
Linköping University, Department of Social and Welfare Studies. Linköping University, Department of Social and Welfare Studies.
Egendom och Stöld: Den juridiska hegemonins svårigheter med teknikens nya matematik2009Independent thesis Basic level (degree of Bachelor), 10 credits / 15 HE creditsStudent thesis

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.

• 17.
Univ Coll Dublin, Ireland.
Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
A characterization of annular domains by quadrature identities2019In: Bulletin of the London Mathematical Society, ISSN 0024-6093, E-ISSN 1469-2120, Vol. 51, no 3, p. 436-442Article in journal (Refereed)

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.

• 18.
KTH, Matemiska institutionen.
KTH, Matemiska institutionen.
The resultant on compact Riemann surfaces2009In: Communications in Mathematical Physics, ISSN 0010-3616, E-ISSN 1432-0916, Vol. 286, no 1, p. 313-358Article in journal (Refereed)

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.

• 19.
Univ Claude Bernard Lyon 1, France; Univ Cote Azur, France.
Linköping University, Department of Computer and Information Science, Software and Systems. Linköping University, Faculty of Science & Engineering. Linköping University, Department of Computer and Information Science, Software and Systems. Linköping University, Faculty of Science & Engineering.
Leveraging access mode declarations in a model for memory consistency in heterogeneous systems2020In: The Journal of logical and algebraic methods in programming, ISSN 2352-2208, E-ISSN 2352-2216, Vol. 110, article id UNSP 100498Article in journal (Refereed)

On a system that exposes disjoint memory spaces to the software, a program has to address memory consistency issues and perform data transfers so that it always accesses valid data. Several approaches exist to ensure the consistency of the memory accessed. Here we are interested in the verification of a declarative approach where each component of a computation is annotated with an access mode declaring which part of the memory is read or written by the component. The programming framework uses the component annotations to guarantee the validity of the memory accesses. This is the mechanism used in VectorPU, a C++ library for programming CPU-GPU heterogeneous systems. This article proves the correctness of the software cache-coherence mechanism used in VectorPU. Beyond the scope of VectorPU, this article provides a simple and effective formalization of memory consistency mechanisms based on the explicit declaration of the effect of each component on each memory space. The formalism we propose also takes into account arrays for which a single validity status is stored for the whole array; additional mechanisms for dealing with overlapping arrays are also studied. (C) 2019 Elsevier Inc. All rights reserved.

• 20.
University of La Frontera, Chile.
Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering. University of La Frontera, Chile. University of La Frontera, Chile.
Quasiplatonic curves with symmetry group Z(2)(2) x Z(m) are definable over Q2017In: Bulletin of the London Mathematical Society, ISSN 0024-6093, E-ISSN 1469-2120, Vol. 49, no 1, p. 165-183Article in journal (Refereed)

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.

• 21.
Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
On the space of cyclic trigonal Riemann surfaces of genus 42007In: Groups St Andrews 2005 Volume 2 / [ed] C. M. Campbell, M. R. Quick, E. F. Robertson, G. C. Smith, Cambridge, UK: Cambridge University Press, 2007, 1, p. 504-516Chapter in book (Other academic)

'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.

• 22.
Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering. University of Chile, Chile.
On the group algebra decomposition of a Jacobian variety2016In: REVISTA DE LA REAL ACADEMIA DE CIENCIAS EXACTAS FISICAS Y NATURALES SERIE A-MATEMATICAS, ISSN 1578-7303, Vol. 110, no 1, p. 185-199Article in journal (Refereed)

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.

• 23.
Linköping University, Department of Behavioural Sciences and Learning.
Linköping University, Department of Behavioural Sciences and Learning.
Olika aspekter av bråk: En litteraturstudie om elevers svårigheter och hur lärare kan underlätta elevers förståelse för bråk i årskurs 4-6.2017Independent thesis Basic level (professional degree), 10 credits / 15 HE creditsStudent thesis

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.

• 24.
Department of Mathematics, Bar-Ilan University, Ramat Gan, Israel.
Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
Idempotent Geometry in Generic Algebras2018In: Advances in Applied Clifford Algebras, ISSN 0188-7009, E-ISSN 1661-4909, Vol. 28, no 5, article id UNSP 84Article in journal (Refereed)

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.

• 25.
Department of Mathematics, Bar-Ilan University, Israel.
Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
Varieties of idempotents in nonassociative algebras2019In: Topics in Clifford Analysis / [ed] Bernstein, Swanhild, Springer International Publishing , 2019, p. 80-112Conference paper (Refereed)

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.

• 26.
Linköping University, Department of Behavioural Sciences and Learning. Linköping University, Faculty of Educational Sciences.
Linköping University, Department of Behavioural Sciences and Learning. Linköping University, Faculty of Educational Sciences.
Algebra på gymnasiet = Svårt?!: Förekomst av felsvar och feltyper vid åk 1-gymnasieelevers beräkningar inom algebra2019Independent thesis Advanced level (professional degree), 10 credits / 15 HE creditsStudent thesis

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.

• 27. Buy this publication >>
Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
Admissible transformations and the group classification of Schrödinger equations2017Doctoral thesis, comprehensive summary (Other academic)

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.

1. Equivalence groupoid for (1+2)-dimensional linear Schrodinger equations with complex potentials
Open this publication in new window or tab >>Equivalence groupoid for (1+2)-dimensional linear Schrodinger equations with complex potentials
2015 (English)In: SEVENTH INTERNATIONAL WORKSHOP: GROUP ANALYSIS OF DIFFERENTIAL EQUATIONS AND INTEGRABLE SYSTEMS (GADEISVII), IOP Publishing: Conference Series / Institute of Physics (IoP) , 2015, Vol. 621, no UNSP 012008, p. UNSP 012008-Conference paper, Published paper (Refereed)
Abstract [en]

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.

Place, publisher, year, edition, pages
IOP Publishing: Conference Series / Institute of Physics (IoP), 2015
Series
Journal of Physics Conference Series, ISSN 1742-6588 ; 621
Mathematics
Identifiers
urn:nbn:se:liu:diva-120668 (URN)10.1088/1742-6596/621/1/012008 (DOI)000357939100008 ()
Conference
7th International Workshop on Group Analysis of Differential Equations and Integrable Systems (GADEIS)
Available from: 2015-08-20 Created: 2015-08-20 Last updated: 2017-05-15
• 28.
Guangzhou Univ, Peoples R China.
Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, Faculty of Science & Engineering.
On longest matching consecutive subsequence2019In: International Journal of Number Theory, ISSN 1793-0421, Vol. 15, no 8, p. 1745-1758Article in journal (Refereed)

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.

• 29.
Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
Sets with the Baire property in topologies formed from a given topology and ideals of sets2017In: Questions and answers in General Topology, ISSN 0918-4732, Vol. 35, no 1, p. 59-76Article in journal (Refereed)

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.

• 30.
The last two digits of mk2012Independent thesis Basic level (degree of Bachelor), 10,5 credits / 16 HE creditsStudent thesis

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.

• 31.
Universitat Rovira i Virgili, Tarragona, Catalonia, Spain .
Numerical semigroups from combinatorial configurations2012Conference paper (Other academic)
• 32.
Universitat Rovira i Virgili, Tarragona, Catalonia, Spain.
Universitat Rovira i Virgili, Tarragona, Catalonia, Spain.
Associating a numerical semigroup to the triangle-free configurations2011In: Advances in Mathematics of Communications, ISSN 1930-5346, E-ISSN 1930-5338, Vol. 5, no 2, p. 351-371Article in journal (Refereed)

It is proved that a numerical semigroup can be associated to the triangle-free $\displaystyle{\left({r},{k}\right)}$-configurations, and some results on existence are deduced. For example it is proved that for any $\displaystyle{r},{k}\geq{2}$ there exists infinitely many $\displaystyle{\left({r},{k}\right)}$-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.

• 33.
Linköping University, Department of Computer and Information Science, Database and information techniques. Linköping University, The Institute of Technology.
Universitat Rovira i Virgili, Spain.
Linear non-homogenous patterns and prime power generators in numerical semigroups associated to combinatorial configurations2014In: Semigroup Forum, ISSN 0037-1912, E-ISSN 1432-2137, Vol. 44, no 1, p. 11-20Article in journal (Refereed)

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.

• 34.
Universitat Rovira i Virgili, Tarragona, Catalonia, Spain .
Universitat Rovira i Virgili, Tarragona, Catalonia, Spain .
The semigroup of combinatorial configurations2010Conference paper (Other academic)
• 35.
Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
A correction of the decomposability result in a paper by Meyer-Neutsch2018In: Journal of Algebra, ISSN 0021-8693, E-ISSN 1090-266X, Vol. 504, p. 432-439Article in journal (Refereed)

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.

• 36.
Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology.
A Jordan algebra approach to the cubic eiconal equation2014In: Journal of Algebra, ISSN 0021-8693, E-ISSN 1090-266X, Vol. 419, p. 34-51Article in journal (Refereed)

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.

• 37.
Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
On an extremal property of Jordan algebras of Clifford type2019In: Communications in Algebra, ISSN 0092-7872, E-ISSN 1532-4125, Vol. 47, no 3, p. 1057-1066Article in journal (Refereed)
• 38.
Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
On the non-vanishing property for real analytic solutions of the p-Laplace equation2016In: Proceedings of the American Mathematical Society, ISSN 0002-9939, E-ISSN 1088-6826, Vol. 144Article in journal (Refereed)

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.

• 39.
Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
Spectral properties of nonassociative algebras and breaking regularity for nonlinear elliptic type PDEs2019In: St. Petersburg Mathematical Journal, ISSN 1061-0022, E-ISSN 1547-7371, Vol. 31, no 2, p. 255-278Article in journal (Refereed)

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.

• 40.
Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
V.M. Miklyukov: from Dimension 8 to Nonassociative Algebras2019In: Mathematical Physics and Computer Simulation, ISSN 2222-8896, Vol. 22, no 2, p. 33-50Article in journal (Refereed)

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.

• 41.
Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
SPECTRAL PROPERTIES OF NONASSOCIATIVE ALGEBRAS AND BREAKING REGULARITY FOR NONLINEAR ELLIPTIC TYPE PDEs2020In: St. Petersburg Mathematical Journal, ISSN 1061-0022, E-ISSN 1547-7371, Vol. 31, no 2, p. 223-240Article in journal (Refereed)

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? Our aim in the paper is twofold. First, 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. Second, to define an appropriate algebraic formalism, which makes the analytic part of the construction of nonclassical solutions more transparent.

• 42.
Wuhan Engineering Consulting Bureau, Wuhan, China.
Faculty of TechnologyUniversity of Vaasa, Vaasa, Finland. Faculty of TechnologyUniversity of Vaasa, Vaasa, Finland. Old Dominion University, Norfolk, USA.
A combinatorial optimization model for enterprise patent transfer2015In: Journal of Special Topics in Information Technology and Management, ISSN 1385-951X, E-ISSN 1573-7667, Vol. 16, no 4, p. 327-337Article in journal (Refereed)

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.

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