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  • 1. Aas, Erik
    et al.
    Ayyer, Arvind
    Linusson, Svante
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Potka, Samu
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Limiting directions for random walks in classical affine Weyl groupsManuscript (preprint) (Other academic)
    Abstract [en]

    Let be a finite Weyl group and the corresponding affine Weyl group. A random element of can be obtained as a reduced random walk on the alcoves of . By a theorem of Lam (Ann. Probab. 2015), such a walk almost surely approaches one of many directions. We compute these directions when is , and and the random walk is weighted by Kac and dual Kac labels. This settles Lam's questions for types and in the affirmative and for type in the negative. The main tool is a combinatorial two row model for a totally asymmetric simple exclusion process called the -TASEP, with four parameters. By specializing the parameters in different ways, we obtain TASEPs for each of the Weyl groups mentioned above. Computing certain correlations in these TASEPs gives the desired limiting directions.

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  • 2.
    Abdallah, Nancy
    et al.
    Univ Borås, Borås, Sweden..
    Altafi, Nasrin
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.). KTH Royal Inst Technol, Dept Math, S-10044 Stockholm, Sweden.;Queens Univ, Dept Math & Stat, Kingston, ON K7L 3N6, Canada..
    Iarrobino, Anthony
    Northeastern Univ, Dept Math, Boston, MA 02115 USA..
    Seceleanu, Alexandra
    Univ Nebraska Lincoln, Dept Math, Lincoln, NE 68588 USA..
    Yameogo, Joachim
    Univ Cote dAzur, CNRS, LJAD, Nice, France..
    Lefschetz properties of some codimension three Artinian Gorenstein algebras2023In: Journal of Algebra, ISSN 0021-8693, E-ISSN 1090-266X, Vol. 625, p. 28-45Article in journal (Refereed)
    Abstract [en]

    Codimension two Artinian algebras have the strong and weak Lefschetz properties provided the characteristic is zero or greater than the socle degree. It is open to what extent such results might extend to codimension three Artinian Gorenstein algebras. Despite much work, the strong Lefschetz property for codimension three Artinian Gorenstein algebra has remained largely mysterious; our results build on and strengthen some of the previous results. We here show that every standard-graded codimension three Artinian Gorenstein algebra A having maximum value of the Hilbert function at most six has the strong Lefschetz property, provided that the characteristic is zero. When the characteristic is greater than the socle degree of A, we show that A is almost strong Lefschetz, they are strong Lefschetz except in the extremal pair of degrees.

  • 3.
    Abdallah, Nancy
    et al.
    University of Borås, Faculty of Textiles, Engineering and Business.
    Altafi, Nasrin
    Department of Mathematics, KTH Royal Institute of Technology, S-100 44 Stockholm, Sweden; Department of Mathematics and Statistics, Queen's University, Kingston, Ontario, K7L 3N6, Canada.
    Iarrobino, Anthony
    Department of Mathematics, Northeastern University, Boston, MA 02115, USA.
    Seceleanu, Alexandra
    Department of Mathematics, University of Nebraska-Lincoln, Lincoln, NE 68588, USA.
    Yaméogo, Joachim
    Université Côte d'Azur, CNRS, LJAD, France.
    Lefschetz properties of some codimension three Artinian Gorenstein algebras2023In: Journal of Algebra, ISSN 0021-8693, E-ISSN 1090-266X, Vol. 625, p. 28-45Article in journal (Refereed)
    Abstract [en]

    Codimension two Artinian algebras have the strong and weak Lefschetz propertiesprovided the characteristic is zero or greater than the socle degree. It is open to whatextent such results might extend to codimension three Artinian Gorenstein algebras. De-spite much work, the strong Lefschetz property for codimension three Artinian Gorensteinalgebra has remained largely mysterious; our results build on and strengthen some of theprevious results. We here show that every standard-graded codimension three ArtinianGorenstein algebra A having maximum value of the Hilbert function at most six has thestrong Lefschetz property, provided that the characteristic is zero. When the characteris-tic is greater than the socle degree of A, we show that A is almost strong Lefschetz, theyare strong Lefschetz except in the extremal pair of degrees.

  • 4.
    Abdallah, Nancy
    et al.
    University of Borås, Faculty of Textiles, Engineering and Business.
    Emsalem, Jacques
    Northeastern Univ, Dept Math, Boston.
    Iarrobino, Anthony
    Northeastern Univ, Dept Math, Boston.
    Yaméogo, Joachim
    Univ Cote Azur, CNRS, LJAD, Nice, France.
    Limits of graded Gorenstein algebras of Hilbert function $$(1,3^k,1)$$2024In: European Journal of Mathematics, ISSN 2199-675X, E-ISSN 2199-6768, Vol. 10, no 1, article id 9Article in journal (Refereed)
    Abstract [en]

    Let R= k [x, y, z], the polynomial ring over a field k. Several of the authors previously classified nets of ternary conics and their specializations over an algebraically closed field, Abdallah et al. (Eur J Math 9(2), Art. No. 22, 2023). We here show that when k is algebraically closed, and considering the Hilbert function sequence T =(1,3(k),1), k >= 2 (i.e. T = (1, 3, 3, ... , 3, 1) where k is the multiplicity of 3), then the family GT parametrizing graded Artinian algebra quotients A = R/I of R having Hilbert function T is irreducible, and G(T) is the closure of the family Gor(T) of Artinian Gorenstein algebras of Hilbert function T. We then classify up to isomorphism the elements of these families Gor(T) and of G(T). Finally, we give examples of codimension 3 Gorenstein sequences, such as (1, 3, 5, 3, 1), for which G(T) has several irreducible components, one being the Zariski closure of Gor(T).

  • 5.
    Abdallah, Nancy
    et al.
    University of Borås, Faculty of Textiles, Engineering and Business.
    Schenck, Hal
    Free resolutions and Lefschetz properties of some Artin Gorenstein rings of codimension four2024In: Journal of symbolic computation, ISSN 0747-7171, E-ISSN 1095-855X, Vol. 121, p. 102257-102257, article id 102257Article in journal (Refereed)
    Abstract [en]

    In (Stanley, 1978), Stanley constructs an example of an Artinian Gorenstein (AG) ring A with non-unimodal H-vector (1,13,12,13,1). Migliore-Zanello show in (Migliore and Zanello, 2017) that for regularity r=4, Stanley's example has the smallest possible codimension c for an AG ring with non-unimodal H-vector.

    The weak Lefschetz property (WLP) has been much studied for AG rings; it is easy to show that an AG ring with non-unimodal H-vector fails to have WLP. In codimension c=3 it is conjectured that all AG rings have WLP. For c=4, Gondim shows in (Gondim, 2017) that WLP always holds for r≤4 and gives a family where WLP fails for any r≥7, building on Ikeda's example (Ikeda, 1996) of failure for r=5. In this note we study the minimal free resolution of A and relation to Lefschetz properties (both weak and strong) and Jordan type for c=4 and r≤6.

  • 6.
    Abebaw, Tilahun
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics. Addis Ababa University, Ethiopia.
    Bøgvad, Rikard
    Stockholm University, Faculty of Science, Department of Mathematics.
    Decomposition factors of D-modules on hyperplane configurations in general position2012In: Proceedings of the American Mathematical Society, ISSN 0002-9939, E-ISSN 1088-6826, Vol. 140, no 8, p. 2699-2711Article in journal (Refereed)
    Abstract [en]

    Let alpha(1), ... , alpha(m) be linear functions on C-n and X = C-n \ V(alpha), where alpha = Pi(m)(i=1) alpha(i) and V(alpha) = {p is an element of C-n : alpha(p) = 0}. The coordinate ring O-X = C[x](alpha) of X is a holonomic A(n)-module, where A(n) is the n-th Weyl algebra, and since holonomic A(n)-modules have finite length, O-X has finite length. We consider a twisted variant of this A(n)-module which is also holonomic. Define M-alpha(beta) to be the free rank 1 C[x](alpha)-module on the generator alpha(beta) (thought of as a multivalued function), where alpha(beta) = alpha(beta 1)(1) ... alpha(beta m)(m) and the multi-index beta = (beta(1), ... , beta(m)) is an element of C-m. It is straightforward to describe the decomposition factors of M-alpha(beta), when the linear functions alpha(1), ... , alpha(m) define a normal crossing hyperplane configuration, and we use this to give a sufficient criterion on beta for the irreducibility of M-alpha(beta), in terms of numerical data for a resolution of the singularities of V(alpha).

  • 7.
    Abrahamsson, Olle
    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
    Abstract [en]

    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.

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  • 8. Abramov, V.
    et al.
    Paal, E.Tallinn University of Technology.Silvestrov, Sergei D.Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.Stolin, A.Chalmers University of Techology.
    Proceedings of the 3rd Baltic-Nordic Workshop “Algebra, Geometry, and Mathematical Physics”2008Conference proceedings (editor) (Refereed)
  • 9.
    Abramov, Viktor
    et al.
    University of Tartu, Estonia.
    Silvestrov, Sergei
    Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
    3-Hom-Lie Algebras Based on σ-Derivation and Involution2020In: Advances in Applied Clifford Algebras, ISSN 0188-7009, E-ISSN 1661-4909, Vol. 30, no 3, article id 45Article in journal (Refereed)
    Abstract [en]

    We show that, having a Hom-Lie algebra and an element of its dual vector space that satisfies certain conditions, one can construct a ternary totally skew-symmetric bracket and prove that this ternary bracket satisfies the Hom-Filippov-Jacobi identity, i.e. this ternary bracket determines the structure of 3-Hom-Lie algebra on the vector space of a Hom-Lie algebra. Then we apply this construction to two Hom-Lie algebras constructed on an associative, commutative algebra using σ-derivation and involution, and we obtain two 3-Hom-Lie algebras.

  • 10.
    Abramsson, Matilda
    Jönköping University, School of Education and Communication, HLK, School Based Research, Mathematics Education Research.
    Elevers förståelse av likhetstecknet: En studie i årskurs 32016Independent thesis Advanced level (professional degree), 10 credits / 15 HE creditsStudent thesis
    Abstract [en]

    The aim of the study is to explore the understanding of the equal sign and how the teaching about the equal sign among third grade students can be varied to be as effective as possible.  The aim will be answered trough the questions: what patterns of variation can the studied students meet and what critical aspects have the students identified. Patterns of variation means that what is critical in the teaching should be varied to become visible. Critical aspects is what students need to identify to understand what should be learned. The foundation of the study is the Variation Theory, where patterns of variation and critical aspects are central concepts.

    The observations were accomplished during a third grade lesson and six students were selected for interviews about the equal sign. The result of the study shows that the students met six critical aspects during the lesson. For every critical aspect there were one or several patterns of variation that was exposed to the students. The result also states that the students who were interviewed have a relational and instrumental understanding of the equal sign. The students also have understanding of a critical aspect that they did not meet in the observed lesson, namely that all numbers have to enter in a task. Four out of six students have understanding of the critical aspect that there should be equivalence in a chain of similarities. The result also show that the students understanding of the equal sign is not dependent of that they meet patterns of variation in the teaching, but that they meet the critical aspects somehow.

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  • 11. Aceto, Luca
    et al.
    Della Monica, Dario
    Goranko, Valentin
    Stockholm University, Faculty of Humanities, Department of Philosophy. University of Johannesburg, South Africa.
    Ingólfsdóttir, Anna
    Montanari, Angelo
    Sciavicco, Guido
    A complete classification of the expressiveness of interval logics of Allen’s relations: the general and the dense cases2016In: Acta Informatica, ISSN 0001-5903, E-ISSN 1432-0525, Vol. 53, no 3, p. 207-246Article in journal (Refereed)
    Abstract [en]

    Interval temporal logics take time intervals, instead of time points, as their primitive temporal entities. One of the most studied interval temporal logics is Halpern and Shoham’s modal logic of time intervals HS, which associates a modal operator with each binary relation between intervals over a linear order (the so-called Allen’s interval relations). In this paper, we compare and classify the expressiveness of all fragments of HS on the class of all linear orders and on the subclass of all dense linear orders. For each of these classes, we identify a complete set of definabilities between HS modalities, valid in that class, thus obtaining a complete classification of the family of all 4096 fragments of HS with respect to their expressiveness. We show that on the class of all linear orders there are exactly 1347 expressively different fragments of HS, while on the class of dense linear orders there are exactly 966 such expressively different fragments.

  • 12. Aczel, Peter
    et al.
    Crosilla, Laura
    Ishihara, Hajime
    Palmgren, Erik
    Uppsala University, Teknisk-naturvetenskapliga vetenskapsområdet, Mathematics and Computer Science, Department of Mathematics. Uppsala University, Teknisk-naturvetenskapliga vetenskapsområdet, Mathematics and Computer Science, Department of Mathematics, Mathematical Logic.
    Schuster, Peter
    Binary refinement implies discrete exponentiation2006In: Studia Logica, Vol. 84, p. 361-368Article in journal (Refereed)
  • 13. Afshari, Bahareh
    et al.
    Enqvist, Sebastian
    Stockholm University, Faculty of Humanities, Department of Philosophy.
    Leigh, Graham E.
    Cyclic proofs for the first-order µ-calculus2024In: Logic journal of the IGPL (Print), ISSN 1367-0751, E-ISSN 1368-9894, Vol. 32, no 1, p. 1-34Article in journal (Refereed)
    Abstract [en]

    We introduce a path-based cyclic proof system for first-order μ-calculus, the extension of first-order logic by second-order quantifiers for least and greatest fixed points of definable monotone functions. We prove soundness of the system and demonstrate it to be as expressive as the known trace-based cyclic systems of Dam and Sprenger. Furthermore, we establish cut-free completeness of our system for the fragment corresponding to the modal μ-calculus.

  • 14.
    Ageev, Dmitry S.
    et al.
    Russian Acad Sci, Dept Math Methods Quantum Technol, Steklov Math Inst, Gubkin Str 8, Moscow 119991, Russia..
    Bagrov, Andrey A.
    Uppsala University, Disciplinary Domain of Science and Technology, Physics, Department of Physics and Astronomy, Materials Theory.
    Iliasov, Askar A.
    Radboud Univ Nijmegen, Inst Mol & Mat, Heyendaalseweg 135, NL-6525 AJ Nijmegen, Netherlands..
    Coleman-Weinberg potential in p-adic field theory2020In: European Physical Journal C, ISSN 1434-6044, E-ISSN 1434-6052, Vol. 80, no 9, article id 859Article in journal (Refereed)
    Abstract [en]

    In this paper, we study lambda phi(4) scalar field theory defined on the unramified extension of p-adic numbers Q(pn). For different "space-time" dimensions n, we compute one-loop quantum corrections to the effective potential. Surprisingly, despite the unusual properties of non-Archimedean geometry, the Coleman-Weinberg potential of p-adic field theory has structure very similar to that of its real cousin. We also study two formal limits of the effective potential, p -> 1 and p -> infinity. We show that the p -> 1 limit allows to reconstruct the canonical result for real field theory from the p-adic effective potential and provide an explanation of this fact. On the other hand, in the p -> infinity limit, the theory exhibits very peculiar behavior with emerging logarithmic terms in the effective potential, which has no analogue in real theories.

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  • 15.
    Ahlman, Ove
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry.
    A zero-one law for l-colourable structures with a vectorspace pregeometry2012Independent thesis Advanced level (degree of Master (Two Years)), 20 credits / 30 HE creditsStudent thesis
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  • 16.
    Ahlman, Ove
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry.
    Homogenizable structures and model completeness2016In: Archive for mathematical logic, ISSN 0933-5846, E-ISSN 1432-0665, Vol. 55, no 7-8, p. 977-995Article in journal (Refereed)
    Abstract [en]

    A homogenizable structure M is a structure where we may add a finite amount of new relational symbols to represent some 0-definable relations in order to make the structure homogeneous. In this article we will divide the homogenizable structures into different classes which categorize many known examples and show what makes each class important. We will show that model completeness is vital for the relation between a structure and the amalgamation bases of its age and give a necessary and sufficient condition for an countably categorical model-complete structure to be homogenizable.

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  • 17.
    Ahlman, Ove
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
    >k-homogeneous infinite graphs2018In: Journal of combinatorial theory. Series B (Print), ISSN 0095-8956, E-ISSN 1096-0902, Vol. 128, p. 160-174Article in journal (Refereed)
    Abstract [en]

    In this article we give an explicit classification for the countably infinite graphs G which are, for some k, ≥k-homogeneous. It turns out that a ≥k  -homogeneous graph M is non-homogeneous if and only if it is either not 1-homogeneous or not 2-homogeneous, both cases which may be classified using ramsey theory.

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  • 18.
    Ahlman, Ove
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
    Limit Laws, Homogenizable Structures and Their Connections2018Doctoral thesis, comprehensive summary (Other academic)
    Abstract [en]

    This thesis is in the field of mathematical logic and especially model theory. The thesis contain six papers where the common theme is the Rado graph R. Some of the interesting abstract properties of R are that it is simple, homogeneous (and thus countably categorical), has SU-rank 1 and trivial dependence. The Rado graph is possible to generate in a probabilistic way. If we let K be the set of all finite graphs then we obtain R as the structure which satisfy all properties which hold with assymptotic probability 1 in K. On the other hand, since the Rado graph is homogeneous, it is also possible to generate it as a Fraïssé-limit of its age.

    Paper I studies the binary structures which are simple, countably categorical, with SU-rank 1 and trivial algebraic closure. The main theorem shows that these structures are all possible to generate using a similar probabilistic method which is used to generate the Rado graph. Paper II looks at the simple homogeneous structures in general and give certain technical results on the subsets of SU-rank 1.

    Paper III considers the set K consisting of all colourable structures with a definable pregeometry and shows that there is a 0-1 law and almost surely a unique definable colouring. When generating the Rado graph we almost surely have only rigid structures in K. Paper IV studies what happens if the structures in K are only the non-rigid finite structures. We deduce that the limit structures essentially try to stay as rigid as possible, given the restriction, and that we in general get a limit law but not a 0-1 law.

    Paper V looks at the Rado graph's close cousin the random t-partite graph and notices that this structure is not homogeneous but almost homogeneous. Rather we may just add a definable binary predicate, which hold for any two elemenets which are in the same part, in order to make it homogeneous. This property is called being homogenizable and in Paper V we do a general study of homogenizable structures. Paper VI conducts a special case study of the homogenizable graphs which are the closest to being homogeneous, providing an explicit classification of these graphs.

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  • 19.
    Ahlman, Ove
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry.
    Simple structures axiomatized by almost sure theories2016In: Annals of Pure and Applied Logic, ISSN 0168-0072, E-ISSN 1873-2461, Vol. 167, no 5, p. 435-456Article in journal (Refereed)
    Abstract [en]

    In this article we give a classification of the binary, simple, ω-categorical structures with SU-rank 1 and trivial algebraic closure. This is done both by showing that they satisfy certain extension properties, but also by noting that they may be approximated by the almost sure theory of some sets of finite structures equipped with a probability measure. This study give results about general almost sure theories, but also considers certain attributes which, if they are almost surely true, generate almost sure theories with very specific properties such as ω-stability or strong minimality.

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  • 20.
    Ahlman, Ove
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry.
    To infinity and back: Logical limit laws and almost sure theories2014Licentiate thesis, comprehensive summary (Other academic)
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  • 21.
    Ahlman, Ove
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry.
    Koponen, Vera
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry.
    Limit laws and automorphism groups of random nonrigid structures2015In: Journal of Logic and Analysis, E-ISSN 1759-9008, Vol. 7, no 2, p. 1-53, article id 1Article in journal (Refereed)
    Abstract [en]

    A systematic study is made, for an arbitrary finite relational language with at least one symbol of arity at least 2, of classes of nonrigid finite structures. The well known results that almost all finite structures are rigid and that the class of finite structures has a zero-one law are, in the present context, the first layer in a hierarchy of classes of finite structures with increasingly more complex automorphism groups. Such a hierarchy can be defined in more than one way. For example, the kth level of the hierarchy can consist of all structures having at least k elements which are moved by some automorphism. Or we can consider, for any finite group G, all finite structures M such that G is a subgroup of the group of automorphisms of M; in this case the "hierarchy" is a partial order. In both cases, as well as variants of them, each "level" satisfies a logical limit law, but not a zero-one law (unless k = 0 or G is trivial). Moreover, the number of (labelled or unlabelled) n-element structures in one place of the hierarchy divided by the number of n-element structures in another place always converges to a rational number or to infinity as n -> infinity. All instances of the respective result are proved by an essentially uniform argument.

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  • 22.
    Ahlman, Ove
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry.
    Koponen, Vera
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry.
    On sets with rank one in simple homogeneous structures2015In: Fundamenta Mathematicae, ISSN 0016-2736, E-ISSN 1730-6329, Vol. 228, p. 223-250Article in journal (Refereed)
    Abstract [en]

    We study definable sets D of SU-rank 1 in Meq, where M is a countable homogeneous and simple structure in a language with finite relational vocabulary. Each such D can be seen as a 'canonically embedded structure', which inherits all relations on D which are definable in Meq, and has no other definable relations. Our results imply that if no relation symbol of the language of M has arity higher than 2, then there is a close relationship between triviality of dependence and D being a reduct of a binary random structure. Somewhat more precisely: (a) if for every n≥2, every n-type p(x1,...,xn) which is realized in D is determined by its sub-2-types q(xi,xj)⊆p, then the algebraic closure restricted to D is trivial; (b) if M has trivial dependence, then D is a reduct of a binary random structure.

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  • 23.
    Ahlman, Ove
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
    Koponen, Vera
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
    Random l-colourable structures with a pregeometry2017In: Mathematical logic quarterly, ISSN 0942-5616, E-ISSN 1521-3870, Vol. 63, no 1-2, p. 32-58Article in journal (Refereed)
    Abstract [en]

    We study finite -colourable structures with an underlying pregeometry. The probability measure that is usedcorresponds to a process of generating such structures by which colours are first randomly assigned to all1-dimensional subspaces and then relationships are assigned in such a way that the colouring conditions aresatisfied but apart from this in a random way. We can then ask what the probability is that the resulting structure,where we now forget the specific colouring of the generating process, has a given property. With this measurewe get the following results: (1) A zero-one law. (2) The set of sentences with asymptotic probability 1 has anexplicit axiomatisation which is presented. (3) There is a formula ξ (x, y) (not directly speaking about colours)such that, with asymptotic probability 1, the relation “there is an -colouring which assigns the same colourto x and y” is defined by ξ (x, y). (4) With asymptotic probability 1, an -colourable structure has a unique-colouring (up to permutation of the colours).

  • 24.
    Ahlsén, Daniel
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry.
    Classifying Categories: The Jordan-Hölder and Krull-Schmidt-Remak Theorems for Abelian Categories2018Independent thesis Advanced level (degree of Master (Two Years)), 20 credits / 30 HE creditsStudent thesis
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  • 25.
    Ahlsén, Daniel
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry.
    Limitless Analysis2014Independent thesis Basic level (degree of Bachelor), 10 credits / 15 HE creditsStudent thesis
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  • 26.
    Ahmad, M. Rauf
    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)
    Abstract [en]

    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).

  • 27. Ahmed, Chwas
    et al.
    Fröberg, Ralf
    Stockholm University, Faculty of Science, Department of Mathematics.
    Rafiq Namiq, Mohammed
    The graded Betti numbers of truncation of ideals in polynomial rings2023In: Journal of Algebraic Combinatorics, ISSN 0925-9899, E-ISSN 1572-9192, Vol. 57, no 4, p. 1303-1312Article in journal (Refereed)
    Abstract [en]

    Let R=K[x1,…,xn], a graded algebra S=R/I satisfies Nk,p if I is generated in degree k, and the graded minimal resolution is linear the first p steps, and the k-index of S is the largest p such that S satisfies Nk,p. Eisenbud and Goto have shown that for any graded ring R/I, then R/I≥k, where I≥k=I∩Mk and M=(x1,…,xn), has a k-linear resolution (satisfies Nk,p for all p) if k≫0. For a squarefree monomial ideal I, we are here interested in the ideal Ik which is the squarefree part of I≥k. The ideal I is, via Stanley–Reisner correspondence, associated to a simplicial complex ΔI. In this case, all Betti numbers of R/Ik for k>min{deg(u)∣u∈I}, which of course are a much finer invariant than the index, can be determined from the Betti diagram of R/I and the f-vector of ΔI. We compare our results with the corresponding statements for I≥k. (Here I is an arbitrary graded ideal.) In this case, we show that the Betti numbers of R/I≥k can be determined from the Betti numbers of R/I and the Hilbert series of R/I≥k.

  • 28.
    Ahmed, Chwas
    et al.
    Univ Sulaimani, Coll Sci, Dept Math, Sulaymaniyah, Kurdistan, Iraq.;Univ Leeds, Sch Math, Leeds, England..
    Martin, Paul
    Univ Leeds, Sch Math, Leeds, England..
    Mazorchuk, Volodymyr
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra, Logic and Representation Theory.
    Tonal partition algebras: fundamental and geometrical aspects of representation theory2024In: Communications in Algebra, ISSN 0092-7872, E-ISSN 1532-4125, Vol. 52, no 1, p. 233-271Article in journal (Refereed)
    Abstract [en]

    For l, n is an element of N we define tonal partition algebra P-l (n) over Z[delta]. We construct modules {Delta mu} mu for P-l (n) over Z[delta], and hence over any integral domain containing Z[delta] (such as C[delta]), that pass to a complete set of irreducible modules over the field of fractions. We show that P-l (n) is semisimple there. That is, we construct for the tonal partition algebras a modular system in the sense of Brauer. Using a "geometrical" index set for the Delta-modules, we give an order with respect to which the decomposition matrix over C (with d. C-x) is upper-unitriangular. We establish several crucial properties of the Delta-modules. These include a tower property, with respect to n, in the sense of Green and Cox-Martin-Parker-Xi; contravariant forms with respect to a natural involutive antiautomorphism; a highest weight category property; and branching rules.

  • 29. Ahrens, Benedikt
    et al.
    Huber, Simon
    Mörtberg, Anders
    Stockholm University, Faculty of Science, Department of Mathematics.
    Preface to the MSCS Issue 31.1 (2021) Homotopy Type Theory and Univalent Foundations2021In: Mathematical Structures in Computer Science, ISSN 0960-1295, E-ISSN 1469-8072, Vol. 31, no 1, p. 1-2Article in journal (Other academic)
  • 30. Ahrens, Benedikt
    et al.
    Lumsdaine, Peter Lefanu
    Stockholm University, Faculty of Science, Department of Mathematics.
    Displayed Categories2019In: Logical Methods in Computer Science, E-ISSN 1860-5974, Vol. 15, no 1, article id 20Article in journal (Refereed)
    Abstract [en]

    We introduce and develop the notion of displayed categories. A displayed category over a category C is equivalent to 'a category D and functor F : D -> C', but instead of having a single collection of 'objects of D' with a map to the objects of C, the objects are given as a family indexed by objects of C, and similarly for the morphisms. This encapsulates a common way of building categories in practice, by starting with an existing category and adding extra data/properties to the objects and morphisms. The interest of this seemingly trivial reformulation is that various properties of functors are more naturally defined as properties of the corresponding displayed categories. Grothendieck fibrations, for example, when defined as certain functors, use equality on objects in their definition. When defined instead as certain displayed categories, no reference to equality on objects is required. Moreover, almost all examples of fibrations in nature are, in fact, categories whose standard construction can be seen as going via displayed categories. We therefore propose displayed categories as a basis for the development of fibrations in the type-theoretic setting, and similarly for various other notions whose classical definitions involve equality on objects. Besides giving a conceptual clarification of such issues, displayed categories also provide a powerful tool in computer formalisation, unifying and abstracting common constructions and proof techniques of category theory, and enabling modular reasoning about categories of multi-component structures. As such, most of the material of this article has been formalised in Coq over the UniMath library, with the aim of providing a practical library for use in further developments.

  • 31.
    Aigner, Mats
    et al.
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
    Tjatyrko, Vitalij
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
    Nyagahakwa, Venuste
    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)
    Abstract [en]

    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.

  • 32.
    Akbari, Saieed
    et al.
    Sharif Univ Technol, Dept Math Sci, Tehran, Iran.
    Aryapoor, Masood
    Inst Res Fundamental Sci IPM, Sch Math, Tehran, Iran.
    Jamaali, M.
    Sharif Univ Technol, Dept Math Sci, Tehran, Iran.
    Chromatic number and clique number of subgraphs of regular graph of matrix algebras2012In: Linear Algebra and its Applications, ISSN 0024-3795, E-ISSN 1873-1856, Vol. 436, no 7, p. 2419-2424Article in journal (Refereed)
    Abstract [en]

    Let R be a ring and X subset of R be a non-empty set. The regular graph of X, Gamma(X), is defined to be the graph with regular elements of X (non-zero divisors of X) as the set of vertices and two vertices are adjacent if their sum is a zero divisor. There is an interesting question posed in BCC22. For a field F, is the chromatic number of Gamma(GL(n)(F)) finite? In this paper, we show that if G is a soluble sub-group of GL(n)(F), then x (Gamma(G)) < infinity. Also, we show that for every field F, chi (Gamma(M-n(F))) = chi (Gamma(M-n(F(x)))), where x is an indeterminate. Finally, for every algebraically closed field F, we determine the maximum value of the clique number of Gamma(< A >), where < A > denotes the subgroup generated by A is an element of GL(n)(F). (C) 2011 Elsevier Inc. All rights reserved.

  • 33.
    Al Homsi, Rania
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry.
    Equation Solving in Indian Mathematics2018Independent thesis Basic level (degree of Bachelor), 10 credits / 15 HE creditsStudent thesis
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  • 34.
    Al-Ayyoub, Ibrahim
    et al.
    Sultan Qaboos University; Jordan University of Science and Technology.
    Nasernejad, Mehrdad
    Ferdowsi University of Mashhad.
    Khashyarmanesh, Kazem
    Ferdowsi University of Mashhad.
    Roberts, Leslie G.
    Queen's University.
    Crispin Quiñonez, Veronica
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry.
    Results on the normality of square-free monomial ideals and cover ideals under some graph operations2021In: Mathematica Scandinavica, ISSN 0025-5521, E-ISSN 1903-1807, Vol. 127, no 3, p. 441-457Article in journal (Refereed)
    Abstract [en]

    In this paper, we introduce techniques for producing normal square-free monomial ideals fromold such ideals. These techniques are then used to investigate the normality of cover ideals undersome graph operations. Square-free monomial ideals that come out as linear combinations of twonormal ideals are shown to be not necessarily normal; under such a case we investigate the integralclosedness of all powers of these ideals.

  • 35.
    Alekseev, Aleksandr
    et al.
    Independent University of Moscow (IUM), Bolshoy Vlasyevskiy Pereulok 11, Moscow, 119002, Russian Federation.
    Arutyunov, Andronick
    V. A. Trapeznikov Institute of Control Sciences of Russian Academy of Sciences, 65 Profsoyuznaya Street, Moscow, 117997, Russian Federation.
    Silvestrov, Sergei
    Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
    On (σ,τ)-Derivations of Group Algebra as Category Characters2023In: Non-commutative and Non-associative Algebra and Analysis Structures: SPAS 2019, Västerås, Sweden, September 30 - October 2 / [ed] Sergei Silvestrov, Anatoliy Malyarenko, Springer , 2023, p. 81-99Conference paper (Refereed)
    Abstract [en]

    For the space of (σ,τ)-derivations of the group algebra C[G] of a discrete countable group G, the decomposition theorem for the space of (σ,τ)-derivations, generalising the corresponding theorem on ordinary derivations on group algebras, is established in an algebraic context using groupoids and characters. Several corollaries and examples describing when all (σ,τ)-derivations are inner are obtained. Considered in details are cases of (σ,τ)-nilpotent groups and (σ,τ)-FC groups.

  • 36.
    Alexandersson, Per
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Hopkins, Sam
    Zaimi, Gjergji
    Restricted Birkhoff Polytopes and Ehrhart Period Collapse2023In: Discrete & Computational Geometry, ISSN 0179-5376, E-ISSN 1432-0444Article in journal (Refereed)
    Abstract [en]

    We show that the polytopes obtained from the Birkhoff polytope by imposing additional inequalities restricting the “longest increasing subsequence” have Ehrhart quasi-polynomials which are honest polynomials, even though they are just rational polytopes in general. We do this by defining a continuous, piecewise-linear bijection to a certain Gelfand–Tsetlin polytope. This bijection is not an integral equivalence but it respects lattice points in the appropriate way to imply that the two polytopes have the same Ehrhart (quasi-)polynomials. In fact, the bijection is essentially the Robinson–Schensted–Knuth correspondence.

  • 37.
    Alexandersson, Per
    et al.
    Stockholm Univ, Dept Math, SE-10691 Stockholm, Sweden..
    Oguz, Ezgi Kantarci
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Linusson, Svante
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Promotion and cyclic sieving on families of SSYT2021In: Arkiv för matematik, ISSN 0004-2080, E-ISSN 1871-2487, Vol. 59, no 2, p. 247-274Article in journal (Refereed)
    Abstract [en]

    We examine a few families of semistandard Young tableaux, for which we observe the cyclic sieving phenomenon under promotion. The first family we consider consists of stretched hook shapes, where we use the cocharge generating polynomial as CSP-polynomial. The second family contains skew shapes, consisting of disjoint rectangles. Again, the charge generating polynomial together with promotion exhibits the cyclic sieving phenomenon. This generalizes earlier results by B. Rhoades and later B. Fontaine and J. Kamnitzer. Finally, we consider certain skew ribbons, where promotion behaves in a predictable manner. This result is stated in the form of a bicyclic sieving phenomenon. One of the tools we use is a novel method for computing charge of skew semistandard tableaux, in the case when every number in the tableau occurs with the same frequency.

  • 38.
    Alexandersson, Per
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Uhlin, Joakim
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Cyclic sieving, skew Macdonald polynomials and Schur positivity2020In: Algebraic Combinatorics, ISSN 2589-5486, Vol. 3, no 4, p. 913-939Article in journal (Refereed)
    Abstract [en]

    When λ is a partition, the specialized non-symmetric Macdonald polynomial Eλ(x; q; 0) is symmetric and related to a modified Hall–Littlewood polynomial. We show that whenever all parts of the integer partition λ are multiples of n, the underlying set of fillings exhibit the cyclic sieving phenomenon (CSP) under an n-fold cyclic shift of the columns. The corresponding CSP polynomial is given by Eλ(x; q; 0). In addition, we prove a refined cyclic sieving phenomenon where the content of the fillings is fixed. This refinement is closely related to an earlier result by B. Rhoades. We also introduce a skew version of Eλ(x; q; 0). We show that these are symmetric and Schur positive via a variant of the Robinson–Schenstedt–Knuth correspondence and we also describe crystal raising and lowering operators for the underlying fillings. Moreover, we show that the skew specialized non-symmetric Macdonald polynomials are in some cases vertical-strip LLT polynomials. As a consequence, we get a combinatorial Schur expansion of a new family of LLT polynomials.

  • 39.
    Ali, Hassan
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra, Logic and Representation Theory.
    Exceptional Groups and their Generators2022Independent thesis Advanced level (degree of Master (Two Years)), 20 credits / 30 HE creditsStudent thesis
    Abstract [en]

    Exceptional algebraic groups are divided into five types, namely G2, F4,E6, E7 and E8. In this thesis we discuss G2, F4 and E6. We discuss the exceptionalalgebraic groups via octonion algebras and Jordan algebras. We firstconsider the groups of type G2. Groups of type G2 are automorphism groupsof octonion algebras, a form of composition algebras. We take the algebra of Zorn vector matrices and find the possible values of automorphisms of thisalgebra with the help of U-operators. We also discuss the product of two andthree U-operators. Then we discuss Albert algebras, since groups of type E6and F4 are related to these algebras. The Albert algebras are a form of Jordanalgebras. We also study the U-operators in Albert algebras. In this thesis wework over algebraically closed fields of characteristic zero.

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  • 40.
    Alkass, David
    Mälardalen University, School of Education, Culture and Communication.
    Finite Posets as Prime Spectra of Commutative Noetherian Rings2024Independent thesis Basic level (degree of Bachelor), 10 credits / 15 HE creditsStudent thesis
    Abstract [en]

    We study partially ordered sets of prime ideals as found in commutative Noetherian rings. These structures, commonly known as prime spectra, have long been a popular topic in the field of commutative algebra. As a consequence, there are many related questions that remain unanswered. Among them is the question of what partially ordered sets appear as Spec(A) of some Noetherian ring A, asked by Kaplansky during the 1950's. As a partial case of Kaplansky's question, we consider finite posets that are ring spectra of commutative Noetherian rings. Specifically, we show that finite spectra of such rings are always order-isomorphic to a bipartite graph. However, the most significant undertaking of this study is that of devising a constructive methodology for finding a ring with prime spectrum that is order-isomorphic to an arbitrary bipartite graph. As a result, we prove that any complete bipartite graph is order-isomorphic to the prime spectrum of some ring of essentially finite type over the field of rational numbers. Moreover, a series of potential generalizations and extensions are proposed to further enhance the constructive methodology. Ultimately, the results of this study constitute an original contribution and perspective on questions related to commutative ring spectra.

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  • 41.
    Al-khafaji, Aea
    et al.
    Jönköping University, School of Education and Communication.
    Mutmain, Sonia
    Jönköping University, School of Education and Communication.
    En litteraturstudie om vad forskning pekar ut som möjliga orsaker till elevers missuppfattningar inom algebra: – bokstavssymboler, operationssymboler och prioriteringsregler.2023Independent thesis Basic level (professional degree), 10 credits / 15 HE creditsStudent thesis
    Abstract [sv]

    Aritmetik handlar om de fyra räknesätten och innefattar beräkningar och operationer på tal. Algebra är däremot matematikområdet där elever räknar med bokstäver istället för med bara tal. Forskning har visat att när elever arbetar med algebra uppstår vissa missuppfattningar. Syftet med studien var att undersöka vad den matematiska forskningen pekar på som kan vara orsak till att elever har problem med områden av algebra. Denna litteraturstudie baserar sig på matematikdidaktisk forskning. Materialet har analyserats och bearbetats både enskilt och gemensamt utifrån förbestämda urvalskriterier. Det analyserade materialet består av åtta vetenskapliga artiklar, en review-artikel och ett konferensbidrag.

    I den forskning som har analyserats i studien beskrivs det olika anledningar till missuppfattningar med bokstavssymboler, operationssymboler och prioriteringsregler inom algebra. De två oftast nämnda anledningar är att elever saknar väsentliga kunskaper inom aritmetik och saknar en konceptuell kunskap. Några enstaka texter pekade ut läromedel och lärarens förklaringar som möjliga orsaker till elevers missuppfattningar.

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  • 42.
    Alpan, Gökalp
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Partial Differential Equations.
    Kozhan, Rostyslav
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Partial Differential Equations.
    Hermitian and non-Hermitian perturbations of chiral Gaussian beta-ensembles2022In: Journal of Mathematical Physics, ISSN 0022-2488, E-ISSN 1089-7658, Vol. 63, no 4, article id 043505Article in journal (Refereed)
    Abstract [en]

    We compute the joint eigenvalue distribution for the rank one Hermitian and non-Hermitian perturbations of chiral Gaussian beta-ensembles (beta > 0) of random matrices.

  • 43.
    Alper, Jarod
    et al.
    Department of Mathematics, University of Washington, Box 354350, Seattle, WA 98195-4350, USA, Box 354350.
    Hall, Jack
    School of Mathematics & Statistics, The University of Melbourne, Parkville, VIC 3010, Australia.
    Halpern-Leistner, Daniel
    Department of Mathematics, Cornell University, 310 Mallot Hall, Ithaca, NY 14853, USA, 310 Mallot Hall.
    Rydh, David
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Artin algebraization for pairs with applications to the local structure of stacks and Ferrand pushouts2024In: Forum of Mathematics, Sigma, E-ISSN 2050-5094, Vol. 12, article id e20Article in journal (Refereed)
    Abstract [en]

    We give a variant of Artin algebraization along closed subschemes and closed substacks. Our main application is the existence of étale, smooth or syntomic neighborhoods of closed subschemes and closed substacks. In particular, we prove local structure theorems for stacks and their derived counterparts and the existence of henselizations along linearly fundamental closed substacks. These results establish the existence of Ferrand pushouts, which answers positively a question of Temkin-Tyomkin.

  • 44.
    Alsaody, Seidon
    Department of Mathematical and Statistical Sciences, University of Alberta, AB, Canada.
    Albert algebras over rings and related torsors2020In: Canadian Journal of Mathematics - Journal Canadien de Mathematiques, ISSN 0008-414X, E-ISSN 1496-4279, Vol. 73, no 3, p. 875-898Article in journal (Refereed)
    Abstract [en]

    We study exceptional Jordan algebras and related exceptional group schemes over commutative rings from a geometric point of view, using appropriate torsors to parametrize and explain classical and new constructions, and proving that over rings, they give rise to nonisomorphic structures.

    We begin by showing that isotopes of Albert algebras are obtained as twists by a certain  F4-torsor with total space a group of type  E6  and, using this, that Albert algebras over rings in general admit nonisomorphic isotopes even in the split case, as opposed to the situation over fields. We then consider certain  D4 -torsors constructed from reduced Albert algebras, and show how these give rise to a class of generalised reduced Albert algebras constructed from compositions of quadratic forms. Showing that this torsor is nontrivial, we conclude that the Albert algebra does not uniquely determine the underlying composition, even in the split case. In a similar vein, we show that a given reduced Albert algebra can admit two coordinate algebras which are nonisomorphic and have nonisometric quadratic forms, contrary, in a strong sense, to the case over fields, established by Albert and Jacobson.

  • 45.
    Alsaody, Seidon
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry.
    Corestricted Group Actions and Eight-Dimensional Absolute Valued Algebras2012Report (Other academic)
    Abstract [en]

    A condition for when two eight-dimensional absolute valued algebras are isomorphic was given in [4]. We use this condition to deduce a description (in the sense of Dieterich, [9]) of the category of such algebras, and show how previous descriptions of some full subcategories fit in this description. Led by the structure of these examples, we aim at systematically constructing new subcategories whose classification is manageable. To this end we propose, in greater generality, the definition of sharp stabilizers for group actions, and use these to obtain conditions for when certain subcategories of groupoids are full. This we apply to the category of eight-dimensional absolute valued algebras and obtain a class of subcategories, for which we simplify, and partially solve, the classification problem.

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  • 46.
    Alsaody, Seidon
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics.
    Groups of type E6 and E7 over rings via Brown algebras and related torsors2024In: Journal of Algebra, ISSN 0021-8693, E-ISSN 1090-266X, Vol. 656, p. 24-46Article in journal (Refereed)
    Abstract [en]

    We study structurable algebras and their associated Freudenthal triple systems over commutative rings. The automorphism groups of these triple systems are exceptional groups of type E 7 , and we realize groups of type E 6 as centralizers. When 6 is invertible, we further give a geometric description of homogeneous spaces of type E 7 / E 6 , and show that they parametrize principal isotopes of Brown algebras. As opposed to the situation over fields, we show that such isotopes may be non-isomorphic. (c) 2023 The Author. Published by Elsevier Inc. This is an open access article under the CC BY license (http:// creativecommons .org /licenses /by /4 .0/).

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  • 47.
    Alsaody, Seidon
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry.
    Morphisms in the Category of Finite Dimensional Absolute Valued Algebras2011Report (Other academic)
    Abstract [en]

    This is a study of morphisms in the category of nite dimensional absolute valued algebras, whose codomains have dimension four. We begin by citing and transferring a classication of an equivalent category. Thereafter, we give a complete description of morphisms from one-dimensional algebras, partly via solutions of real polynomials, and a complete, explicit description of morphisms from two-dimensional algebras. We then give an account of the reducibility of the morphisms, and for the morphisms from two-dimensional algebras we describe the orbits under the actions of the automorphism groups involved. Parts of these descriptions rely on a suitable choice of a cross-section of four-dimensional absolute valued algebras, and we thus end by providing an explicit means of transferring these results to algebras outside this crosssection.

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  • 48.
    Alsaody, Seidon
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry.
    On Finite-Dimensional Absolute Valued Algebras2012Licentiate thesis, comprehensive summary (Other academic)
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  • 49.
    Alsaody, Seidon
    et al.
    Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry.
    Chernousov, Vladimir
    Univ Alberta, Dept Math & Stat Sci, Edmonton, AB T6G 2G1, Canada..
    Pianzola, Arturo
    Univ Alberta, Dept Math & Stat Sci, Edmonton, AB T6G 2G1, Canada.;Ctr Altos Estudios Ciencias Exactas, Ave Mayo 866, RA-1084 Buenos Aires, DF, Argentina..
    On the Tits-Weiss conjecture and the Kneser-Tits conjecture for E-7,1(78) and E-8,2(78) (With an Appendix by R. M. Weiss)2021In: Forum of Mathematics, Sigma, ISSN 2050-5094, Vol. 9, article id e75Article in journal (Refereed)
    Abstract [en]

    We prove that the structure group of any Albert algebra over an arbitrary field is R-trivial. This implies the TitsWeiss conjecture for Albert algebras and the Kneser-Tits conjecture for isotropic groups of type E-7,1(78), E-8,2(78). As a further corollary, we show that some standard conjectures on the groups of R-equivalence classes in algebraic groups and the norm principle are true for strongly inner forms of type E-1(6).

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  • 50.
    Al-Shujary, Ahmed
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
    Kähler-Poisson Algebras2018Licentiate thesis, comprehensive summary (Other academic)
    Abstract [en]

    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.

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    Kähler-Poisson Algebras
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