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  • 151.
    Dieterich, Ernst
    Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Algebra och geometri.
    Lattices over group rings of cyclic p-groups and generalized factorspace categories1985Ingår i: Journal of the London Mathematical Society, ISSN 0024-6107, E-ISSN 1469-7750, Vol. s2-31, nr 3, s. 407-424Artikel i tidskrift (Refereegranskat)
  • 152.
    Dieterich, Ernst
    Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Algebra och geometri.
    Power-associative real division algebras1998Ingår i: Canadian Mathematical Society Conference Proceedings, Vol. 24, s. 139-144Artikel i tidskrift (Refereegranskat)
  • 153.
    Dieterich, Ernst
    Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Algebra och geometri.
    Quadratic division algebras revisited. (Remarks on an article by J. M. Osborn.)2000Ingår i: Proceedings of the American Mathematical Society, Vol. 128, s. 3159-3166Artikel i tidskrift (Refereegranskat)
  • 154.
    Dieterich, Ernst
    Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Algebra och geometri.
    Reduction of isolated singularities1987Ingår i: Commentarii Mathematici Helvetici, ISSN 0010-2571, E-ISSN 1420-8946, Vol. 62, nr 1, s. 654-676Artikel i tidskrift (Refereegranskat)
  • 155.
    Dieterich, Ernst
    Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Algebra och geometri.
    Representation types of group rings over complete discrete valuation rings1981Ingår i: Lecture notes in mathematics, ISSN 0075-8434, E-ISSN 1617-9692, Vol. 882, s. 369-389Artikel i tidskrift (Refereegranskat)
  • 156.
    Dieterich, Ernst
    Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Algebra och geometri.
    Representation types of group rings over complete discrete valuation rings II1985Ingår i: Lecture notes in mathematics, ISSN 0075-8434, E-ISSN 1617-9692, Vol. 1142, s. 112-125Artikel i tidskrift (Refereegranskat)
  • 157.
    Dieterich, Ernst
    Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Algebra och geometri.
    Solution of a non-domestic tame classification problem from integral representation theory of finite groups1991Ingår i: Memoirs of the American Mathematical Society, ISSN 0065-9266, E-ISSN 1947-6221, Vol. 92, nr 450, s. 1-140Artikel i tidskrift (Refereegranskat)
  • 158.
    Dieterich, Ernst
    Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Algebra och geometri.
    Tame curve singularities with large conductor1991Ingår i: Progress in Mathematics, Vol. 95, s. 327-341Artikel i tidskrift (Refereegranskat)
  • 159.
    Dieterich, Ernst
    Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Algebra och geometri.
    Tame orders1990Ingår i: Banach Center Publications, ISSN 0137-6934, E-ISSN 1730-6299, Vol. 26, nr 1, s. 233-261Artikel i tidskrift (Refereegranskat)
  • 160.
    Dieterich, Ernst
    Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Algebra och geometri.
    The Auslander-Reiten quiver of an isolated singularity1987Ingår i: Lecture notes in mathematics, ISSN 0075-8434, E-ISSN 1617-9692, Vol. 1273, s. 244-264Artikel i tidskrift (Refereegranskat)
  • 161.
    Dieterich, Ernst
    Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Algebra och geometri.
    Trace Invariants of Finite-Dimensional Algebras2016Ingår i: Communications in Algebra, ISSN 0092-7872, E-ISSN 1532-4125, Vol. 44, nr 5, s. 1852-1881Artikel i tidskrift (Refereegranskat)
  • 162.
    Dieterich, Ernst
    Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Algebra och geometri.
    Zur Klassifikation vierdimensionaler reeller Divisionsalgebren1998Ingår i: Mathematische Nachrichten, ISSN 0025-584X, E-ISSN 1522-2616, Vol. 194, s. 13-22Artikel i tidskrift (Refereegranskat)
  • 163.
    Dieterich, Ernst
    et al.
    Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Algebra och geometri.
    Fieseler, Karl-Heinz
    Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Algebra och geometri.
    Lindberg, Lars
    Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen.
    Liftings of dissident maps2006Ingår i: Journal of Pure and Applied Algebra, ISSN 0022-4049, E-ISSN 1873-1376, Vol. 204, nr 1, s. 133-154Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We study dissident maps ηη on RmRm for m∈{3,7}m∈{3,7} by investigating liftings Φ:Rm→RmΦ:Rm→Rm of the selfbijection ηP:P(Rm)→P(Rm),ηP[v]=(η(v∧Rm))⊥ induced by ηη. Our main result (Theorem 2.4) asserts the existence and uniqueness, up to a non-zero scalar multiple, of a lifting ΦΦ whose component functions are homogeneous polynomials of degree dd, relatively prime and without non-trivial common zero. We prove that 1⩽d⩽m-21⩽d⩽m-2.

    We achieve a complete description of all dissident maps of degree one and we solve their isomorphism problem (Theorems 4.8 and 4.13). As a consequence, we achieve a complete description of all real quadratic division algebras of degree one and we solve their isomorphism problem (Theorems 5.1 and 5.3). Moreover we present examples of eight-dimensional real quadratic division algebras of degree 3 and 5 (Proposition 6.3). This extends earlier results of Osborn [Trans. Amer. Math. Soc. 105 (1962) 202–221], Hefendehl [Geometriae Dedicata 9 (1980) 129–152], Hefendehl-Hebeker [Arch. Math. 40 (1983) 50–60], Cuenca Mira et al. [Lin. Alg. Appl. 290 (1999) 1–22], Dieterich [Proc. Amer. Math. Soc. 128 (2000) 3159–3166] and Dieterich and Lindberg [Colloq. Math. 97 (2003) 251–276] on the classification of real quadratic division algebras.

  • 164.
    Dieterich, Ernst
    et al.
    Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Algebra och geometri.
    Lindberg, Lars
    Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Algebra och geometri.
    Dissident maps on the seven-dimensional Euclidean space2003Ingår i: Colloquium Mathematicum, ISSN 0010-1354, E-ISSN 1730-6302, Vol. 97, s. 251-276Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    Our article contributes to the classification of dissident maps on R7, which in turn contributes to the classification of 8-dimensional real division algebras. We study two large classes of dissident maps on R7. The first class is formed by all composed dissident maps, obtained from a vector product on R7 by composition with a definite endomorphism. The second class is formed by all doubled dissident maps, obtained as the purely imaginary parts of the structures of those 8-dimensional real quadratic division algebras which arise from a 4-dimensional real quadratic division algebra by doubling. For each of these two classes we exhibit a complete (but redundant) classification, given by a 49-parameter family of composed dissident maps and a 9-parameter family of doubled dissident maps respectively. The intersection of these two classes forms one isoclass of dissident maps only, namely the isoclass consisting of all vector products on R7.

  • 165.
    Dieterich, Ernst
    et al.
    Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Algebra och geometri.
    Wiedemann, Alfred
    The Auslander-Reiten quiver of a simple curve singularity1986Ingår i: Transactions of the American Mathematical Society, ISSN 0002-9947, E-ISSN 1088-6850, Vol. 294, nr 2, s. 455-475Artikel i tidskrift (Refereegranskat)
  • 166.
    Dieterich, Ernst
    et al.
    Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen.
    Öhman, Johan
    Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen.
    On the classification of four-dimensional quadratic division algebras over square-ordered fields2002Ingår i: Journal of the London Mathematical Society, Vol. 65, s. 285-302Artikel i tidskrift (Refereegranskat)
  • 167.
    Djordjevic, Marko
    Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Matematisk logik.
    Finite satisfiability and N-0-categorical structures with trivial dependence2006Ingår i: Journal of Symbolic Logic (JSL), ISSN 0022-4812, E-ISSN 1943-5886, Vol. 71, nr 3, s. 810-830Artikel i tidskrift (Refereegranskat)
  • 168.
    Djordjevic, Marko
    Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Matematisk logik.
    Finite variable logic, stability and finite models2001Ingår i: Journal of Symbolic Logic (JSL), ISSN 0022-4812, E-ISSN 1943-5886, Vol. 66, nr 2, s. 837-858Artikel i tidskrift (Refereegranskat)
  • 169.
    Djordjevic, Marko
    Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Matematisk logik.
    The finite submodel property and ω-categorical expansions of pregeometries2006Ingår i: Annals of Pure and Applied Logic, ISSN 0168-0072, E-ISSN 1873-2461, Vol. 139, nr 1-3, s. 201-229Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We prove, by a probabilistic argument, that a class of ω-categorical structures, on which algebraic closure defines a pregeometry, has the finite submodel property. This class includes any expansion of a pure set or of a vector space, projective space or affine space over a finite field such that the new relations are sufficiently independent of each other and over the original structure. In particular, the random graph belongs to this class, since it is a sufficiently independent expansion of an infinite set, with no structure. The class also contains structures for which the pregeometry given by algebraic closure is non-trivial.

  • 170.
    Dmytryshyn, Andrii
    Department of Computing Science, Umeå University, Umeå, Sweden.
    Structure preserving stratification of skew-symmetric matrix polynomials2017Ingår i: Linear Algebra and its Applications, ISSN 0024-3795, E-ISSN 1873-1856, Vol. 532, s. 266-286Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We study how elementary divisors and minimal indices of a skew-symmetric matrix polynomial of odd degree may change under small perturbations of the matrix coefficients. We investigate these changes qualitatively by constructing the stratifications (closure hierarchy graphs) of orbits and bundles for skew-symmetric linearizations. We also derive the necessary and sufficient conditions for the existence of a skew-symmetric matrix polynomial with prescribed degree, elementary divisors, and minimal indices.

  • 171.
    Dmytryshyn, Andrii
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för datavetenskap.
    Structure preserving stratification of skew-symmetric matrix polynomials2017Ingår i: Linear Algebra and its Applications, ISSN 0024-3795, E-ISSN 1873-1856, Vol. 532, s. 266-286Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We study how elementary divisors and minimal indices of a skew-symmetric matrix polynomial of odd degree may change under small perturbations of the matrix coefficients. We investigate these changes qualitatively by constructing the stratifications (closure hierarchy graphs) of orbits and bundles for skew-symmetric linearizations. We also derive the necessary and sufficient conditions for the existence of a skew-symmetric matrix polynomial with prescribed degree, elementary divisors, and minimal indices.

  • 172.
    Dmytryshyn, Andrii
    et al.
    Umeå universitet, Teknisk-naturvetenskapliga fakulteten, Institutionen för datavetenskap.
    Dopico, Froilán
    Universidad Carlos III de Madrid.
    Generic skew-symmetric matrix polynomials with fixed rank and fixed odd grade2018Ingår i: Linear Algebra and its Applications, ISSN 0024-3795, E-ISSN 1873-1856, Vol. 536, s. 1-18Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We show that the set of m×m complex skew-symmetric matrix polynomials of odd grade d, i.e., of degree at most d, and (normal) rank at most 2r is the closure of the single set of matrix polynomials with the certain, explicitly described, complete eigenstructure. This complete eigenstructure corresponds to the most generic m×m complex skew-symmetric matrix polynomials of odd grade d and rank at most 2r. In particular, this result includes the case of skew-symmetric matrix pencils (d=1).

  • 173.
    Dmytryshyn, Andrii
    et al.
    Department of Computing Science, Umeå University, Umeå, Sweden.
    Dopico, Froilán M.
    Departamento de Matemáticas, Universidad Carlos III de Madrid, Leganés, Spain.
    Generic complete eigenstructures for sets of matrix polynomials with bounded rank and degree2017Ingår i: Linear Algebra and its Applications, ISSN 0024-3795, E-ISSN 1873-1856, Vol. 535, s. 213-230Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    The set POLd,rm×n of m×n complex matrix polynomials of grade d and (normal) rank at most r in a complex (d+1)mn dimensional space is studied. For r=1,...,min{m,n}−1, we show that POLd,rm×n is the union of the closures of the rd+1 sets of matrix polynomials with rank r, degree exactly d, and explicitly described complete eigenstructures. In addition, for the full-rank rectangular polynomials, i.e. r=min{m,n} and mn, we show that POLd,rm×n coincides with the closure of a single set of the polynomials with rank r, degree exactly d, and the described complete eigenstructure. These complete eigenstructures correspond to generic m×n matrix polynomials of grade d and rank at most r.

  • 174.
    Dmytryshyn, Andrii
    et al.
    Department of Computing Science, Umeå University, Umeå, Sweden.
    Dopico, Froilán M.
    Departamento de Matemáticas, Universidad Carlos III de Madrid, Leganés, Spain.
    Generic skew-symmetric matrix polynomials with fixed rank and fixed odd grade2018Ingår i: Linear Algebra and its Applications, ISSN 0024-3795, E-ISSN 1873-1856, Vol. 536, s. 1-18Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We show that the set of m×m complex skew-symmetric matrix polynomials of odd grade d, i.e., of degree at most d, and (normal) rank at most 2r is the closure of the single set of matrix polynomials with the certain, explicitly described, complete eigenstructure. This complete eigenstructure corresponds to the most generic m×m complex skew-symmetric matrix polynomials of odd grade d and rank at most 2r. In particular, this result includes the case of skew-symmetric matrix pencils (d=1).

  • 175.
    Dmytryshyn, Andrii
    et al.
    Department of Computing Science and HPC2N, Umeå University, Umeå, Sweden.
    Kågström, Bo
    Department of Computing Science and HPC2N, Umeå University, Umeå, Sweden.
    Sergeichuk, Vladimir V.
    Institute of Mathematics, Kiev, Ukraine.
    Skew-symmetric matrix pencils: codimension counts and the solution of a pair of matrix equations2013Ingår i: Linear Algebra and its Applications, ISSN 0024-3795, E-ISSN 1873-1856, Vol. 438, nr 8, s. 3375-3396Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    The homogeneous system of matrix equations (X(T)A + AX, (XB)-B-T + BX) = (0, 0), where (A, B) is a pair of skew-symmetric matrices of the same size is considered: we establish the general solution and calculate the codimension of the orbit of (A, B) under congruence. These results will be useful in the development of the stratification theory for orbits of skew-symmetric matrix pencils.

  • 176.
    Dmytryshyn, Andrii
    et al.
    Department of Computing Science and HPC2N, Umeå University, Umeå, Sweden.
    Kågström, Bo
    Department of Computing Science and HPC2N, Umeå University, Umeå, Sweden.
    Sergeichuk, Vladimir V.
    Ukrainian Acad Sci, Kiev, Ukraine.
    Symmetric matrix pencils: codimension counts and the solution of a pair of matrix equations2014Ingår i: The Electronic Journal of Linear Algebra, ISSN 1537-9582, E-ISSN 1081-3810, Vol. 27, s. 1-18Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    The set of all solutions to the homogeneous system of matrix equations (X-T A + AX, X-T B + BX) = (0, 0), where (A, B) is a pair of symmetric matrices of the same size, is characterized. In addition, the codimension of the orbit of (A, B) under congruence is calculated. This paper is a natural continuation of the article [A. Dmytryshyn, B. Kagstrom, and V. V. Sergeichuk. Skew-symmetric matrix pencils: Codimension counts and the solution of a pair of matrix equations. Linear Algebra Appl., 438:3375-3396, 2013.], where the corresponding problems for skew-symmetric matrix pencils are solved. The new results will be useful in the development of the stratification theory for orbits of symmetric matrix pencils.

  • 177.
    Dubsky, Brendan
    Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen.
    Koszulity of some path categories2017Ingår i: Communications in Algebra, ISSN 0092-7872, E-ISSN 1532-4125, Vol. 45, nr 9, s. 4084-4092Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We prove Koszulity of certain linear path categories obtained from connected graphs with some infinite directed walk. These categories can be viewed as locally quadratic dual to preprojective algebras.

  • 178.
    Dubsky, Brendan Frisk
    et al.
    Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen.
    Guo, Xiangqian
    Zhengzhou Univ, Sch Math & Stat, Zhengzhou 450001, Henan, Peoples R China.
    Yao, Yufeng
    Shanghai Maritime Univ, Dept Math, Shanghai 201306, Peoples R China.
    Zhao, Kaiming
    Hebei Normal Teachers Univ, Coll Math & Informat Sci, Shijiazhuang 050016, Hebei, Peoples R China;Wilfrid Laurier Univ, Dept Math, Waterloo, ON N2L 3C5, Canada.
    Simple modules over the Lie algebras of divergence zero vector fields on a torus2019Ingår i: Forum mathematicum, ISSN 0933-7741, E-ISSN 1435-5337, Vol. 31, nr 3, s. 727-741Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    Let n >= 2 be an integer, S-n the Lie algebra of divergence zero vector fields on an n-dimensional torus, and K-n the Weyl algebra over the Laurent polynomial algebra A(n) = C[x(1)(+/- 1), x(2)(+/- 1), . . . , x(n)(+/- 1)]. For any sln-module V and any module P over K-n, we define an S-n-module structure on the tensor product P circle times V. In this paper, necessary and sufficient conditions for the S-n-modules P circle times V to be simple are given, and an isomorphism criterion for nonminuscule S-n-modules is provided. More precisely, all nonminuscule S-n-modules are simple, and pairwise nonisomorphic. For minuscule S-n-modules, minimal and maximal submodules are concretely determined.

  • 179.
    Ekdahl, Anna-Lena
    Stockholms universitet, Naturvetenskapliga fakulteten, Institutionen för matematikämnets och naturvetenskapsämnenas didaktik.
    Elevers skilda sätt att erfara talmönster - en studie av elever i årskurs 3 och 42012Självständigt arbete på avancerad nivå (masterexamen), 20 poäng / 30 hpStudentuppsats (Examensarbete)
    Abstract [sv]

    Matematiken handlar i mångt och mycket om att lösa problem och se mönster. Talmönster är en viktig del inom algebran och aritmetiken och är det fenomen som jag i denna studie vill undersöka elevers uppfattningar av. Syftet med föreliggande kvalitativa studie är att skapa kunskap om elevers skilda sätt att erfara talmönster, såväl talföljder som visuella talmönster. Därutöver syftar studien till att identifiera kritiska aspekter utifrån de skilda sätt som talmönstren erfars av eleverna.

    Nio elever i årskurs 3 och 4 har intervjuats utifrån ett antal talmönster. Fenomenografin och variationsteorin utgör studiens teoretiska utgångspunkter och har använts för att analysera materialet. I analysen har förutom likheter och skillnader mellan sätten att erfara, innehållet i elevutsagorna analyserats utifrån erfarandets referentiella och strukturella aspekt.

    Resultatet av den fenomenografiska analysen har utmynnat i följande sex beskrivningskategorier:

    Jämn förflyttning, Konstant eller icke-konstant skillnad, Kombination av delar, Relation mellan vissa delar, Olika del- och helhetsstrukturer och Utöver angiven helhet. I analysen har de aspekter som eleverna fokuserat på varit vägledande för att skilja kategorierna åt och identifiera sex kritiska aspekter. En av dessa kritiska aspekter handlar om att urskilja att förhållandet mellan delarna i mönstret kan se olika ut. En annan kritisk aspekt är fråga om att kunna urskilja delarnas inbördes relation, relationernas förhållande till helheten och den icke angivna helheten. En tredje innebär att delarna behöver urskiljas samtidigt som helheten. Inte nödvändigtvis samtliga delar, men tillräckligt många för att se en regelbundenhet.

    Studiens resultat har gett didaktiska implikationer om vad eleverna i en undervisningssituation behöver ges möjlighet att urskilja för att utveckla ett mer innehållsrikt och differentierat sätt att erfara talmönster.

    Resultatet diskuteras utifrån tidigare internationella undersökningar. Det förs även en diskussion om vad studiens resultat kan tillföra och de didaktiska implikationer resultatet ger.

  • 180.
    Ekholm, Tomas
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Kovarik, Hynek
    Laptev, Ari
    Hardy inequalities for p-Laplacians with Robin boundary conditions2015Ingår i: Nonlinear Analysis, ISSN 0362-546X, E-ISSN 1873-5215, Vol. 128, s. 365-379Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    In this paper we study the best constant in a Hardy inequality for the p-Laplace operator on convex domains with Robin boundary conditions. We show, in particular, that the best constant equals ((p-1)/p)(p) whenever Dirichlet boundary conditions are imposed on a subset of the boundary of non-zero measure. We also discuss some generalizations to non-convex domains.

  • 181.
    Ekström, Fredrik
    et al.
    Lund University.
    Silvestrov, Sergei
    Mälardalens högskola, Akademin för utbildning, kultur och kommunikation, Utbildningsvetenskap och Matematik.
    Commutants and Centers in a 6-Parameter Family of Quadratically Linked Quantum Plane Algebras2014Ingår i: Springer Proceedings in Mathematics and Statistics, Springer, 2014, Vol. 85, s. 37-59Konferensbidrag (Refereegranskat)
    Abstract [en]

    We consider a family of associative algebras, defined as the quotient of a free algebra with the ideal generated by a set of multi-parameter deformed commutation relations between four generators consisting of five quantum plane relations between pairs of generators and one sub-quadratic relation inter-linking all four generators. For generic parameter vectors, the center and the commutants of the two of the generators are described and conditions on the parameters for these commutants to be itself commutative or non-commutative are obtained.

  • 182.
    Elchinger, O.
    et al.
    Laboratoire de Mathématiques, Informatique et Applications, Université de Haute Alsace, Mulhouse, France.
    Lundengård, Karl
    Mälardalens högskola, Akademin för utbildning, kultur och kommunikation, Utbildningsvetenskap och Matematik.
    Makhlouf, A.
    Laboratoire de Mathématiques Informatique et Applications, Université de Haute Alsace, Mulhouse, France.
    Silvestrov, Sergei
    Mälardalens högskola, Akademin för utbildning, kultur och kommunikation, Utbildningsvetenskap och Matematik.
    Brackets with (τ,σ)-derivations and (p,q)-deformations of Witt and Virasoro algebras2016Ingår i: Forum mathematicum, ISSN 0933-7741, E-ISSN 1435-5337, Vol. 28, nr 4, s. 657-673Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    The aim of this paper is to study some brackets defined on (τ,σ)-derivations satisfying quasi-Lie identities. Moreover, we provide examples of (p, q)-deformations of Witt and Virasoro algebras as well as sl(2) algebra. These constructions generalize the results obtained by Hartwig, Larsson and Silvestrov on σ-derivations, arising in connection with discretizations and deformations of algebras of vector fields.

  • 183.
    Eliasson, Jonas
    Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen. Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Matematisk logik.
    Ultrapowers as sheaves on a category of ultrafilters2001Licentiatavhandling, monografi (Övrigt vetenskapligt)
  • 184.
    Eliasson, Jonas
    Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen.
    Ultrapowers as sheaves on a category of ultrafilters2004Ingår i: Archive for mathematical logic, ISSN 0933-5846, E-ISSN 1432-0665, Vol. 43, nr 7, s. 825-843Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    In the paper we investigate the topos of sheaves on a category of ultrafilters. The category is described with the help of the Rudin-Keisler ordering of ultrafilters. It is shown that the topos is Boolean and two-valued and that the axiom of choice does not hold in it. We prove that the internal logic in the topos does not coincide with that in any of the ultrapowers. We also show that internal set theory, an axiomatic nonstandard set theory, can be modeled in the topos.

  • 185.
    Eliasson, Jonas
    Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen.
    Ultrasheaves2003Doktorsavhandling, sammanläggning (Övrigt vetenskapligt)
    Abstract [en]

    This thesis treats ultrasheaves, sheaves on the category of ultrafilters.

    In the classical theory of ultrapowers, you start with an ultrafilter and, given a structure, you construct the ultrapower of the structure over the ultrafilter. The fundamental result is Los's theorem for ultrapowers giving the connection between what formulas are satisfied in the ultrapower and in the original structure. In this thesis we instead start with the category of ultrafilters (denoted U). On this category U we build the topos of sheaves on U (the ultrasheaves), which we think of as generalized ultrapowers.

    The theorem for ultrapowers corresponding to Los's theorem is Moerdijk's theorem, first proved by Moerdijk for the topos Sh(F) of sheaves on filters. In the thesis we prove that Los's theorem follows from Moerdijk's theorem. We also investigate the exact relation between the topos of ultrasheaves and Moerdijk's topos Sh(F) and prove that Sh(U) is the double negation subtopos of Sh(F).

    The connection between ultrapowers and ultrasheaves is investigated in detail. We also prove some model theoretic results for ultrasheaves, for instance we prove that they are saturated models. The Rudin-Keisler ordering is a tool used in set theory to study ultrafilters. It has a strong relationship to the category U. Blass has given a model theoretic characterization of this ordering and in the thesis we give a new proof of his result.

    One common use of ultrapowers is to give non-standard models. In the thesis we prove that you can model internal set theory (IST) in the ultrasheaves. IST, introduced by Nelson, is a non-standard set theory, an axiomatic approach to non-standard mathematics.

  • 186.
    Emmenegger, Jacopo
    Stockholms universitet, Naturvetenskapliga fakulteten, Matematiska institutionen.
    A category-theoretic version of the identity type weak factorization systemManuskript (preprint) (Övrigt vetenskapligt)
    Abstract [en]

    Gambino and Garner proved that the syntactic category of a dependent type theory with identity types can be endowed with a weak factorization system structure, called identity type weak factorization system. In this paper we consider an enrichment of Joyal's notion of tribe which allow us to prove a purely category-theoretic version of the identity type weak factorization system, thus generalizing Gambino and Garner's result. We investigate then how it relates with other well-known weak factorization systems, namely those arising from Quillen model structures on the category of topological spaces and on the category of small groupoids.

  • 187.
    Emmenegger, Jacopo
    Stockholms universitet, Naturvetenskapliga fakulteten, Matematiska institutionen.
    Exact completion and type-theoretic structures2019Doktorsavhandling, sammanläggning (Övrigt vetenskapligt)
    Abstract [en]

    This thesis consists of four papers and is a contribution to the study of representations of extensional properties in intensional type theories using, mainly, the language and tools from category theory. Our main focus is on exact completions of categories with weak finite limits as a category-theoretic description of the setoid construction in Martin-Löf's intensional type theory.

    Paper I, which is joint work with Erik Palmgren, provides sufficient conditions for such an exact completion to produce a model of the system CETCS (Constructive Elementary Theory of the Category of Sets), a finite axiomatisation of the theory of well-pointed locally cartesian closed pretoposes with a natural numbers object and enough projectives. In particular, we use a condition inspired by Aczel's set-theoretic Fullness Axiom to obtain the local cartesian closure of an exact completion. As an application, we obtain a simple  uniform proof that the category of setoids is a model of CETCS.

    Paper II was prompted by the discovery of an overlooked issue in the characterisationof local cartesian closure for exact completions due to Carboni and Rosolini. In this paper we clarify the problem, show that their characterisation is still valid when the base category has finite limits, and provide a complete solution in the general case of a category with weak finite limits.

    In paper III we generalise the approach used in paper I to obtain the local cartesian closure of an exact completion to arbitrary categories with finite limits. We then show how this condition inspired by the Fullness Axiom naturally arises in several homotopy categories and apply this result to obtain the local cartesian closure of the exact completion of the homotopy category of spaces, thus answering a question left open by Marino Gran and Enrico Vitale.

    Finally, in paper IV we abandon the pure category-theoretic approach and instead present a type-theoretic construction, formalised in Coq, of W-types in the category of setoids from dependent W-types in the underlying intensional theory. In particular, contrary to previous approaches, this construction does not require the assumption of Uniqueness of Identity Proofs nor recursion into a type universe.

  • 188.
    Emmenegger, Jacopo
    Stockholms universitet, Naturvetenskapliga fakulteten, Matematiska institutionen.
    On the local cartesian closure of exact completionsManuskript (preprint) (Övrigt vetenskapligt)
    Abstract [en]

    A characterisation of cartesian closure of exact completions as a property of the projective objects was given by Carboni and Rosolini. We show that the argument used to prove that characterisation is equivalent to the projectives being closed under binary products (equivalently, being internally projective). The property in question is the existence of weak simple products (a slight strengthening of weak exponentials) and the argument used relies on two claims: that weak simple products endow the internal logic with universal quantification, and that an exponential is the quotient of a weak exponential. We show that either these claims hold if and only if the projectives are internally projectives, which entails that Carboni and Rosolini's characterisation only applies to ex/lex completions. We then argue that this limitation depends on the universal property of weak simple products, and derive from this observation an alternative notion, which we call generalised weak simple product. We conclude by showing that existence of generalised weak simple products in the subcategory of projectives is equivalent to the cartesian closure of the exact category, thus obtaining a complete characterisation of (local) cartesian closure for exact completions of categories with weak finite limits.

  • 189.
    Emmenegger, Jacopo
    Stockholms universitet, Naturvetenskapliga fakulteten, Matematiska institutionen.
    The Fullness Axiom and exact completions of homotopy categoriesManuskript (preprint) (Övrigt vetenskapligt)
    Abstract [en]

    We use a category-theoretic formulation of Aczel's Fullness Axiom from Constructive Set Theory to derive the local cartesian closure of an exact completion. As an application, we prove that such a formulation is valid in the homotopy category of any model category satisfying mild requirements, thus obtaining in particular the local cartesian closure of the exact completion of topological spaces and homotopy classes of maps. Under a type-theoretic reading, these results provide a general motivation for the local cartesian closure of the category of setoids. However, results and proofs are formulated solely in the language of categories, and no knowledge of type theory or constructive set theory is required on the reader's part.

  • 190.
    Emmenegger, Jacopo
    Stockholms universitet, Naturvetenskapliga fakulteten, Matematiska institutionen.
    W-types in setoidsManuskript (preprint) (Övrigt vetenskapligt)
    Abstract [en]

    W-types and their categorical analogue, initial algebras for polynomial endofunctors, are an important tool in predicative systems to replace transfinite recursion on well-orderings. Current arguments to obtain W-types in quotient completions rely on assumptions, like Uniqueness of Identity Proofs, or on constructions that involve recursion into a universe, that limit their applicability to a specific setting. We present an argument, verified in Coq, that instead uses dependent W-types in the underlying type theory to construct W-types in the setoid model. The immediate advantage is to have a proof more type-theoretic in flavour, which directly uses recursion on the underlying W-type to prove initiality. Furthermore, taking place in intensional type theory and not requiring any recursion into a universe, it may be generalised to various categorical quotient completions, with the aim of finding a uniform construction of extensional W-types.

  • 191.
    Emmenegger, Jacopo
    et al.
    Stockholms universitet, Naturvetenskapliga fakulteten, Matematiska institutionen.
    Palmgren, Erik
    Stockholms universitet, Naturvetenskapliga fakulteten, Matematiska institutionen.
    Exact completion and constructive theories of setsManuskript (preprint) (Övrigt vetenskapligt)
    Abstract [en]

    In the present paper we use the theory of exact completions to study categorical properties of small setoids in Martin-Loef type theory and, more generally, of models of the Constructive Elementary Theory of the Category of Sets, in terms of properties of their subcategories of choice objects (i.e. objects satisfying the axiom of choice). Because of these intended applications, we deal with categories that lack equalisers and just have weak ones, but whose objects can be regarded as collections of global elements. In this context, we study the internal logic of the categories involved, and employ this analysis to give a sufficient condition for the local cartesian closure of an exact completion. Finally, we apply this result to show when an exact completion produces a model of CETCS.

  • 192.
    Epstein, Melker
    Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Algebra och geometri.
    Powers and Products of Monomial Ideals2016Självständigt arbete på grundnivå (kandidatexamen), 10 poäng / 15 hpStudentuppsats (Examensarbete)
  • 193.
    Eriksson, Anders
    Stockholms universitet.
    Differential operators on some classes of rings2000Doktorsavhandling, sammanläggning (Övrigt vetenskapligt)
  • 194.
    Ernst, Thomas
    Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen.
    On several q-special matrices, including the q-Bernoulli and q-Euler matrices2018Ingår i: Linear Algebra and its Applications, ISSN 0024-3795, E-ISSN 1873-1856, Vol. 542, s. 422-440Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    In the spirit of our earlier articles [12,14,11], and our work [13], we define two dual q-Bernoulli polynomials, with corresponding vector and matrix forms. Following Aceto Trigiante [1], the q-L matrix, the indefinite q-integral of the q-Pascal matrix is the link between the q-Cauchy and the q-Bernoulli matrix. The q-analogue of the Bernoulli complementary argument theorem can be expressed in matrix form through the diagonal An matrix. For the q-Euler polynomials corresponding results are obtained. The umbral calculus for generating functions of q-Appell polynomials is shown to be equivalent to a transform method, which maps polynomials to matrices, a true q-analogue of Arponen [6]. This is manifested by the Vein [21] matrix, which occurs as the transform of the q-difference operator. The Aceto Trigiante shifted q-Bernoulli matrix has a simple connection to the q-Bernoulli Arponen matrix through the q-Pascal matrix. We reintroduce certain q-Stirling numbers is an element of 7L(q) from [12], which will be needed for the polynomial matrix definitions.

  • 195.
    Ernst, Thomas
    Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen.
    On the q-Lie group of q-Appell polynomial matrices and related factorizations2018Ingår i: Special Matrices, ISSN 2300-7451, Vol. 6, nr 1, s. 93-109Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    In the spirit of our earlier paper [10] and Zhang andWang [16], we introduce the matrix of multiplicative q-Appell polynomials of order M is an element of Z. This is the representation of the respective q-Appell polynomials in ke-ke basis. Based on the fact that the q-Appell polynomials form a commutative ring [11], we prove that this set constitutes a q-Lie group with two dual q-multiplications in the sense of [9]. A comparison with earlier results on q-Pascal matrices gives factorizations according to [7], which are specialized to q-Bernoulli and q-Euler polynomials. We also show that the corresponding q-Bernoulli and q-Euler matrices form q-Lie subgroups. In the limit q -> 1 we obtain corresponding formulas for Appell polynomial matrices. We conclude by presenting the commutative ring of generalized q-Pascal functional matrices, which operates on all functions f is an element of C-q(infinity).

  • 196.
    Eshagh, Mehdi
    KTH, Skolan för arkitektur och samhällsbyggnad (ABE), Samhällsplanering och miljö, Geoinformatik och Geodesi.
    Generalized Least Squares Adjustment of Gauss-Helmert Model2005Konferensbidrag (Refereegranskat)
  • 197.
    Espíndola, Christian
    Stockholms universitet, Naturvetenskapliga fakulteten, Matematiska institutionen.
    A short proof of Glivenko theorems for intermediate predicate logics2013Ingår i: Archive for mathematical logic, ISSN 0933-5846, E-ISSN 1432-0665, Vol. 52, nr 7-8, s. 823-826Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We give a simple proof-theoretic argument showing that Glivenko’s theorem for propositional logic and its version for predicate logic follow as an easy consequence of the deduction theorem, which also proves some Glivenko type theorems relating intermediate predicate logics between intuitionistic and classical logic. We consider two schemata, the double negation shift (DNS) and the one consisting of instances of the principle of excluded middle for sentences (REM). We prove that both schemata combined derive classical logic, while each one of them provides a strictly weaker intermediate logic, and neither of them is derivable from the other. We show that over every intermediate logic there exists a maximal intermediate logic for which Glivenko’s theorem holds. We deduce as well a characterization of DNS, as the weakest (with respect to derivability) scheme that added to REM derives classical logic.

  • 198.
    Espíndola, Christian
    Stockholms universitet, Naturvetenskapliga fakulteten, Matematiska institutionen.
    Achieving completeness: from constructive set theory to large cardinals2016Doktorsavhandling, monografi (Övrigt vetenskapligt)
    Abstract [en]

    This thesis is an exploration of several completeness phenomena, both in the constructive and the classical settings. After some introductory chapters in the first part of the thesis where we outline the background used later on, the constructive part contains a categorical formulation of several constructive completeness theorems available in the literature, but presented here in an unified framework. We develop them within a constructive reverse mathematical viewpoint, highlighting the metatheory used in each case and the strength of the corresponding completeness theorems.

    The classical part of the thesis focuses on infinitary intuitionistic propositional and predicate logic. We consider a propositional axiomatic system with a special distributivity rule that is enough to prove a completeness theorem, and we introduce weakly compact cardinals as the adequate metatheoretical assumption for this development. Finally, we return to the categorical formulation focusing this time on infinitary first-order intuitionistic logic. We propose a first-order system with a special rule, transfinite transitivity, that embodies both distributivity as well as a form of dependent choice, and study the extent to which completeness theorems can be established. We prove completeness using a weakly compact cardinal, and, like in the constructive part, we study disjunction-free fragments as well. The assumption of weak compactness is shown to be essential for the completeness theorems to hold.

  • 199.
    Espíndola, Christian
    Stockholms universitet, Naturvetenskapliga fakulteten, Matematiska institutionen.
    Semantic Completeness of First-Order Theories in Constructive Reverse Mathematics2016Ingår i: Notre Dame Journal of Formal Logic, ISSN 0029-4527, E-ISSN 1939-0726, Vol. 57, nr 2, s. 281-286Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We introduce a general notion of semantic structure for first-order theories, covering a variety of constructions such as Tarski and Kripke semantics, and prove that, over Zermelo–Fraenkel set theory (ZF), the completeness of such semantics is equivalent to the Boolean prime ideal theorem (BPI). Using a result of McCarty (2008), we conclude that the completeness of Kripke semantics is equivalent, over intuitionistic Zermelo–Fraenkel set theory (IZF), to the Law of Excluded Middle plus BPI. Along the way, we also prove the equivalence, over ZF, between BPI and the completeness theorem for Kripke semantics for both first-order and propositional theories.

  • 200.
    Everitt, Tom
    et al.
    Stockholms universitet, Naturvetenskapliga fakulteten, Matematiska institutionen.
    Lattimore, Tor
    Hutter, Marcus
    Free Lunch for Optimisation under the Universal Distribution2014Ingår i: 2014 IEEE Congress on Evolutionary Computation (CEC), New York: IEEE Computer Society, 2014, s. 167-174Konferensbidrag (Refereegranskat)
    Abstract [en]

    Function optimisation is a major challenge in computer science. The No Free Lunch theorems state that if all functions with the same histogram are assumed to be equally probable then no algorithm outperforms any other in expectation. We argue against the uniform assumption and suggest a universal prior exists for which there is a free lunch, but where no particular class of functions is favoured over another. We also prove upper and lower boundson the size of the free lunch.

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