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  • 301.
    Khrennikov, Andrei
    Linnéuniversitetet, Fakulteten för teknik (FTK), Institutionen för matematik (MA).
    Foundations of analysis on superspace — 1: Differential calculus2015Inngår i: P-Adic Numbers, Ultrametric Analysis, and Applications, ISSN 2070-0466, E-ISSN 2070-0474, Vol. 7, nr 2, s. 96-110Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    The recent experimental confirmation of the existence of Higgs boson stimulates theoretical research on supersymmetric models; in particular, mathematics of such modeling. Therefore we plan to present essentials of one special approach to “super-mathematics”, so called functional superanalysis (in the spirit of De Witt, Rogers, Vladimirov and Volovich, and the author of this review) in compact and clear form in series of review-type papers. This first paper is a review on super-differential calculus for the concrete model of the superspace (invented by Vladimirov and Volovich). In the next review we plan to present the integral supercalculus. The main distinguishing feature of functional superanalysis is that this is a real super-extension of analysis of Newton and Leibniz, opposite to algebraic models of Martin and Berezin. Here functions of commuting and anticommuting variables are no simply algebraic elements belonging to Grassmann algebras, but point-wise maps, from superspace into superspace. Finally, we remark that the first non-Archimedean physical model was based on invention by Vladimirov and Volovich of superspaces based on supercommutative Banach superalgebras over non-Archimedean (in particular, p-adic) fields. This model plays the basic role in theory of p-adic superstrings.

  • 302.
    Khrennikov, Andrei
    Växjö universitet, Fakulteten för matematik/naturvetenskap/teknik, Matematiska och systemtekniska institutionen. Matematik.
    Logical approach to p-adic probabilities2006Inngår i: Bulletin of the Section of Logic, ISSN 0138-0680, Vol. 35/1, s. 49-57Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    We study coupling between logic an p-adic probability

  • 303.
    Khrennikov, Andrei
    et al.
    Linnéuniversitetet, Fakulteten för teknik (FTK), Institutionen för matematik (MA).
    Kotovich, Nikolay
    Institute of System Analysis of Russian Academy of Science, Russia.
    Towards ultrametric modeling of unconscious creativity2015Inngår i: International Journal of Cognitive Informatics and Natural Intelligence, ISSN 1557-3958, E-ISSN 1557-3966, Vol. 8, nr 4, s. 98-109Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    Information processed by complex cognitive systems is characterized by the presence of various closely connected hierarchic structures. The most natural geometry for the representation of such structures is geometry of trees and the corresponding topology is the ultrametric topology of on these trees. And the p-adic trees provide the simplest model for representation of mental hierarchies. Moreover, p-adic trees can be endowed with the natural arithmetic reminding the usual arithmetic of real numbers. Therefore it is natural to start from the p-adic models of brain's functioning. In this note the authors apply this model to demonstrate the ability of the “p-adic brain” to process adequately the objects of the physical Euclidean space in the p-adic tree representation. This study also leads to p-adic modeling of brain's creativity and its ability to create abstract images. The authors' model is about unconscious processing of information by the brain. Therefore the authors can say about elements of coming theory of unconscious creativity.

  • 304.
    Khrennikov, Andrei
    et al.
    Linnéuniversitetet, Fakulteten för teknik (FTK), Institutionen för matematik (MA).
    Oleshko, Klaudia
    Univ. Nacional Autónoma de Mexico (UNAM), Mexico.
    de Jesús Correa López, Maria
    Caracterizac Yacimientos Act Prod Maloob Zaap, Mexico.
    Application of p-Adic Wavelets to Model Reaction–Diffusion Dynamics in Random Porous Media2016Inngår i: Journal of Fourier Analysis and Applications, ISSN 1069-5869, E-ISSN 1531-5851, Vol. 22, nr 4, s. 809-822Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    Fourier and more generally wavelet analysis over the fields of p-adic numbers are widely used in physics, biology and cognitive science, and recently in geophysics. In this note we present a model of the reaction–diffusion dynamics in random porous media, e.g., flow of fluid (oil, water or emulsion) in a a complex network of pores with known topology. Anomalous diffusion in the model is represented by the system of two equations of reaction–diffusion type, for the part of fluid not bound to solid’s interface (e.g., free oil) and for the part bound to solid’s interface (e.g., solids–bound oil). Our model is based on the p-adic (treelike) representation of pore-networks. We present the system of two p-adic reaction–diffusion equations describing propagation of fluid in networks of pores in random media and find its stationary solutions by using theory of p-adic wavelets. The use of p-adic wavelets (generalizing classical wavelet theory) gives a possibility to find the stationary solution in the analytic form which is typically impossible for anomalous diffusion in the standard representation based on the real numbers.

  • 305.
    Kildetoft, Tobias
    Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Algebra och geometri.
    Decomposition of Tensor Products Involving a Steinberg Module2017Inngår i: Algebras and Representation Theory, ISSN 1386-923X, E-ISSN 1572-9079, Vol. 20, nr 4, s. 951-975Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    We study the decomposition of tensor products between a Steinberg module and a costandard module, both as a module for the algebraic group G and when restricted to either a Frobenius kernel G (r) or a finite Chevalley group . In all three cases, we give formulas reducing this to standard character data for G. Along the way, we use a bilinear form on the characters of finite dimensional G-modules to give formulas for the dimension of homomorphism spaces between certain G-modules when restricted to either G (r) or . Further, this form allows us to give a new proof of the reciprocity between tilting modules and simple modules for G which has slightly weaker assumptions than earlier such proofs. Finally, we prove that in a suitable formulation, this reciprocity is equivalent to Donkin's tilting conjecture.

  • 306.
    Kildetoft, Tobias
    et al.
    Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Algebra och geometri.
    Mazorchuk, Volodymyr
    Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Algebra och geometri.
    Special Modules Over Positively Based Algebras2016Inngår i: Documenta Mathematica, ISSN 1431-0635, E-ISSN 1431-0643, Vol. 21, s. 1171-1192Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    We use the Perron-Frobenius Theorem to define, study and, in some sense, classify special simple modules over arbitrary finite dimensional positively based algebras. For group algebras of finite Weyl groups with respect to the Kazhdan-Lusztig basis, this agrees with Lusztig's notion of a special module introduced in [Lu2].

  • 307.
    Kitouni, Abdenour
    et al.
    Univ Haute Alsace, France.
    Makhlouf, Abdenacer
    Univ Haute Alsace, France.
    Silvestrov, Sergei
    Mälardalens högskola, Akademin för utbildning, kultur och kommunikation, Utbildningsvetenskap och Matematik.
    On (n+1)-Hom-Lie algebras induced by n-Hom-Lie algebras2016Inngår i: Georgian Mathematical Journal, ISSN 1072-947X, E-ISSN 1572-9176, Vol. 23, nr 1, s. 75-95Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    The purpose of this paper is to study the relationships between an n-Hom-Lie algebra and its induced (n + 1)-Hom-Lie algebra. We provide an overview of the theory and explore structure properties such as ideals, centers, derived series, solvability, nilpotency, central extensions, and the cohomology.

  • 308.
    Koenig, Steffen
    et al.
    Institute of Algebra and Number Theory, University of Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany.
    Külshammer, Julian
    Institute of Algebra and Number Theory, University of Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany.
    Ovsienko, Sergiy
    Mechanics and Mathematics Faculty, Kiev National University, 01033 Kiev, Ukraine.
    Quasi-hereditary algebras, exact Borel subalgebras, A∞-categories and boxes2014Inngår i: Advances in Mathematics, ISSN 0001-8708, E-ISSN 1090-2082, Vol. 262, s. 546-592Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    Highest weight categories arising in Lie theory are known to be associated with finite dimensional quasi-hereditary algebras such as Schur algebras or blocks of category O. An analogue of the PBW theorem will be shown to hold for quasi-hereditary algebras: Up to Morita equivalence each such algebra has an exact Borel subalgebra. The category F(\Delta) of modules with standard (Verma, Weyl, …) filtration, which is exact, but rarely abelian, will be shown to be equivalent to the category of representations of a directed box. This box is constructed as a quotient of a dg algebra associated with the A-infinity-structure on Ext(\Delta,\Delta). Its underlying algebra is an exact Borel subalgebra.

  • 309.
    Koerfer, Filip
    Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Algebra och geometri.
    Classification of Root Systems2018Independent thesis Basic level (degree of Bachelor), 10 poäng / 15 hpOppgave
  • 310.
    Kolyada, Viktor
    Karlstads universitet, Fakulteten för hälsa, natur- och teknikvetenskap (from 2013), Institutionen för matematik och datavetenskap.
    My First Meetings with Konstantin Oskolkov2013Inngår i: Recent Advances in Harmonic Analysis and Applications: In honor of Konstantin Oskolkov / [ed] Dmitriy Bilyk, Laura De Carli, Alexander Petukhov, Alexander M. Stokolos, Brett D. Wick, Springer, 2013, 1, Vol. 25, s. 27-29Kapittel i bok, del av antologi (Fagfellevurdert)
    Abstract [en]

    This note tells about our first meetings with Konstatin Oskolkov. We discuss also optimal estimates of the rate of convergence of Fourier series obtained by Oskolkov in 1975.

  • 311.
    Konstantopoulos, Takis
    Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Analys och sannolikhetsteori.
    A multilinear algebra proof of the Cauchy-Binet formula and a multilinear version of Parseval's identity2013Inngår i: Linear Algebra and its Applications, ISSN 0024-3795, E-ISSN 1873-1856, Vol. 439, nr 9, s. 2651-2658Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    We prove the Cauchy-Binet determinantal formula using multilinear algebra by first generalizing it to an identity not involving determinants. By extending the formula to abstract Hilbert spaces we obtain, as a corollary, a generalization of the classical Parseval identity.

  • 312.
    Koponen, Vera
    Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen.
    A limit law of almost l-partite graphs2013Inngår i: Journal of Symbolic Logic (JSL), ISSN 0022-4812, E-ISSN 1943-5886, Vol. 78, nr 3, s. 911-936Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    For integers l >= 1, d >= 0 we study (undirected) graphs with vertices 1,..., n such that the vertices can be partitioned into l parts such that every vertex has at most d neighbours in its own part. The set of all such graphs is denoted P-n (l, d). We prove a labelled first-order limitlaw, i.e., for every first-order sentence phi, the proportion of graphs in P-n (l, d) that satisfy phi converges as n -> infinity. By combining this result with a result of Hundack, Promel and Steger [12] we also prove that if 1 <= s(1) <=...<= s(1) are integers, then Forb(A(I),s(1),...,s(l)) has alabelled first-order limit law, where Forb (A(I),s(1),...,s(l)) denotes the set of all graphs with vertices 1,..., n, for some n, in which there is no subgraph isomorphic to the complete (l + 1)-partite graph with parts of sizes 1, S-1,..., S-l. In the course of doing this we also prove that there exists a first-order formula depending only on l and d, such that the proportion of g e P (I, d) with the following property approaches 1 as n ->infinity: there is a unique partition of {1,..., n} into l parts such that every vertex has at most d neighbours in its own part, and this partition, viewed as an equivalence relation, is defined by xi.

  • 313.
    Koponen, Vera
    Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen.
    Binary primitive homogeneous simple structures2017Inngår i: Journal of Symbolic Logic (JSL), ISSN 0022-4812, E-ISSN 1943-5886, Vol. 82, nr 1, s. 183-207Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    Suppose that M is countable, binary, primitive, homogeneous, and simple. We prove that the SU-rank of the complete theory of M is 1 and hence 1-based. It follows that M is a random structure. The conclusion that M is a random structure does not hold if the binarity condition is removed, as witnessed by the generic tetrahedron-free 3-hypergraph. However, to show that the generic tetrahedron-free 3-hypergraph is 1-based requires some work (it is known that it has the other properties) since this notion is defined in terms of imaginary elements. This is partly why we also characterize equivalence relations which are definable without parameters in the context of omega-categorical structures with degenerate algebraic closure. Another reason is that such characterizations may be useful in future research about simple (nonbinary) homogeneous structures.

  • 314.
    Koponen, Vera
    Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Algebra och geometri.
    Binary simple homogeneous structures are supersimple with finite rank2016Inngår i: Proceedings of the American Mathematical Society, ISSN 0002-9939, E-ISSN 1088-6826, Vol. 144, nr 4, s. 1745-1759Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    Suppose that M is an infinite structure with finite relational vocabulary such that every relation symbol has arity at most 2. If M is simple and homogeneous, then its complete theory is supersimple with finite SU-rank which cannot exceed the number of complete 2-types over the empty set.

  • 315.
    Koponen, Vera
    Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen.
    Homogeneous 1-based structures and interpretability in random structures2017Inngår i: Mathematical logic quarterly, ISSN 0942-5616, E-ISSN 1521-3870, Vol. 63, nr 1-2, s. 6-18Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    Let V be a finite relational vocabulary in which no symbol has arity greater than 2. Let math formula be countable V-structure which is homogeneous, simple and 1-based. The first main result says that if math formula is, in addition, primitive, then it is strongly interpretable in a random structure. The second main result, which generalizes the first, implies (without the assumption on primitivity) that if math formula is “coordinatized” by a set with SU-rank 1 and there is no definable (without parameters) nontrivial equivalence relation on M with only finite classes, then math formula is strongly interpretable in a random structure.

  • 316.
    Koponen, Vera
    Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Algebra och geometri.
    Random graphs with bounded maximum degree: asymptotic structure and a logical limit law2012Inngår i: Discrete mathematics and theoretical computer science (Online), ISSN 1462-7264, E-ISSN 1365-8050, Vol. 14, nr 2, s. 229-254Artikkel i tidsskrift (Fagfellevurdert)
  • 317.
    Koponen, Vera
    Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Algebra och geometri.
    Some connections between finite and infinite model theory2011Inngår i: Finite and Algorithmic Model Theory / [ed] Javier Esparza, Christian Michaux, Charles Steinhorn, Cambridge University Press, 2011, s. 109-139Kapittel i bok, del av antologi (Annet vitenskapelig)
  • 318.
    Koponen, Vera
    Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Algebra och geometri.
    Typical automorphism groups of finite nonrigid structures2015Inngår i: Archive for mathematical logic, ISSN 0933-5846, E-ISSN 1432-0665, Vol. 54, nr 5-6, s. 571-586Artikkel i tidsskrift (Fagfellevurdert)
  • 319.
    Koponen, Vera
    et al.
    Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Algebra och geometri.
    Hyttinen, Tapani
    Helsingfors Universitet.
    On compactness of logics that can express properties of symmetry or connectivity2015Inngår i: Studia Logica: An International Journal for Symbolic Logic, ISSN 0039-3215, E-ISSN 1572-8730, Vol. 103, nr 1, s. 1-20Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    A condition, in two variants, is given such that if a property P satisfies this condition, then every logic which is at least as strong as first-order logic and can express P fails to have the compactness property. The result is used to prove that for a number of natural properties P speaking about automorphism groups or connectivity, every logic which is at least as strong as first-order logic and can express P fails to have the compactness property. The basic idea underlying the results and examples presented here is that it is possible to construct a countable first-order theory T such that every model of T has a very rich automorphism group, but every finite subset T' of T has a model which is rigid.

  • 320.
    kouri, emmi
    Stockholms universitet, Naturvetenskapliga fakulteten, Institutionen för matematikämnets och naturvetenskapsämnenas didaktik.
    ”Det blir lika med…”: En studie om hur elever i en årskurs 4 resonerar kring likhetstecknet2010Independent thesis Basic level (degree of Bachelor), 10 poäng / 15 hpOppgave
    Abstract [sv]

    I denna studie undersöktes hur några elever i en årskurs 4 resonerar kring likhetstecknet. Undersökningen gjordes med hjälp av kvalitativa intervjuer. Eleverna var valda och placerade på måfå i grupper där de sedan fick diskutera kring tre matematiska likheter.

    Studien visar på tendenser av relationell förståelse för likhetstecknet hos eleverna, men samtidigt problematiseras skillnaden mellan den relationella och den operationella förståelsen. Studien kommer fram till att eleverna diskuterar om uppgiftens konstiga utformning när operationen är på högra sidan av likhetstecknet och svaret är på den vänstra sidan. Eleverna hänvisar till kutym och läroböcker. Av studien framkommer det också att eleverna jämför likheterna ofta med balansvågen, både muntligt men också kroppsligt. Studien tenderar även att ge en bild av att elevers svar skiljer sig beroende på vilka uppgifterna är. 

  • 321.
    Kragh, Thomas
    Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Algebra och geometri.
    Homotopy equivalence of nearby Lagrangians and the Serre spectral sequence2017Inngår i: Mathematische Annalen, ISSN 0025-5831, E-ISSN 1432-1807, Vol. 368, nr 3-4, s. 945-970Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    We construct using relatively basic techniques a spectral sequence for exact Lagrangians in cotangent bundles similar to the one constructed by Fukaya, Seidel, and Smith. That spectral sequence was used to prove that exact relative spin Lagrangians in simply connected cotangent bundles with vanishing Maslov class are homology equivalent to the base (a similar result was also obtained by Nadler). The ideas in that paper were extended by Abouzaid who proved that vanishing Maslov class alone implies homotopy equivalence. In this paper we present a short proof of the fact that any exact Lagrangian with vanishing Maslov class is homology equivalent to the base and that the induced map on fundamental groups is an isomorphism. When the fundamental group of the base is pro-finite this implies homotopy equivalence.

  • 322.
    Krasnov, Yakov
    et al.
    Department of Mathematics, Bar-Ilan University, Ramat Gan, Israel.
    Tkachev, Vladimir
    Linköpings universitet, Matematiska institutionen, Matematik och tillämpad matematik. Linköpings universitet, Tekniska fakulteten.
    Idempotent Geometry in Generic Algebras2018Inngår i: Advances in Applied Clifford Algebras, ISSN 0188-7009, E-ISSN 1661-4909, Vol. 28, nr 5Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    Using the syzygy method, established in our earlier paper (Krasnov and Tkachev, 2018), we characterize the combinatorial stratification of the variety of two-dimensional real generic algebras. We show that there exist exactly three different homotopic types of such algebras and relate this result to potential applications and known facts from qualitative theory of quadratic ODEs. The genericity condition is crucial. For example, the idempotent geometry in Clifford algebras or Jordan algebras of Clifford type is very different: such algebras always contain nontrivial submanifolds of idempotents.

  • 323.
    Krohn, Johannes
    Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Algebra och geometri.
    Legendrian Conormals of 2-Knots2016Independent thesis Advanced level (degree of Master (Two Years)), 20 poäng / 30 hpOppgave
  • 324.
    Kronosjö, Anton
    Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Algebra och geometri.
    Hurwitz 1, 2, 4, 8-sats2013Independent thesis Basic level (degree of Bachelor), 10 poäng / 15 hpOppgave
  • 325.
    Kurujyibwami, Celestin
    Linköpings universitet, Matematiska institutionen, Matematik och tillämpad matematik. Linköpings universitet, Tekniska fakulteten.
    Admissible transformations and the group classification of Schrödinger equations2017Doktoravhandling, med artikler (Annet vitenskapelig)
    Abstract [en]

    We study admissible transformations and solve group classification problems for various classes of linear and nonlinear Schrödinger equations with an arbitrary number n of space variables.

    The aim of the thesis is twofold. The first is the construction of the new theory of uniform seminormalized classes of differential equations and its application to solving group classification problems for these classes. Point transformations connecting two equations (source and target) from the class under study may have special properties of semi-normalization. This makes the group classification of that class using the algebraic method more involved. To extend this method we introduce the new notion of uniformly semi-normalized classes. Various types of uniform semi-normalization are studied: with respect to the corresponding equivalence group, with respect to a proper subgroup of the equivalence group as well as the corresponding types of weak uniform semi-normalization. An important kind of uniform semi-normalization is given by classes of homogeneous linear differential equations, which we call uniform semi-normalization with respect to linear superposition of solutions.

    The class of linear Schrödinger equations with complex potentials is of this type and its group classification can be effectively carried out within the framework of the uniform semi-normalization. Computing the equivalence groupoid and the equivalence group of this class, we show that it is uniformly seminormalized with respect to linear superposition of solutions. This allow us to apply the version of the algebraic method for uniformly semi-normalized classes and to reduce the group classification of this class to the classification of appropriate subalgebras of its equivalence algebra. To single out the classification cases, integers that are invariant under equivalence transformations are introduced. The complete group classification of linear Schrödinger equations is carried out for the cases n = 1 and n = 2.

    The second aim is to study group classification problem for classes of generalized nonlinear Schrödinger equations which are not uniformly semi-normalized. We find their equivalence groupoids and their equivalence groups and then conclude whether these classes are normalized or not. The most appealing classes are the class of nonlinear Schrödinger equations with potentials and modular nonlinearities and the class of generalized Schrödinger equations with complex-valued and, in general, coefficients of Laplacian term. Both these classes are not normalized. The first is partitioned into an infinite number of disjoint normalized subclasses of three kinds: logarithmic nonlinearity, power nonlinearity and general modular nonlinearity. The properties of the Lie invariance algebras of equations from each subclass are studied for arbitrary space dimension n, and the complete group classification is carried out for each subclass in dimension (1+2). The second class is successively reduced into subclasses until we reach the subclass of (1+1)-dimensional linear Schrödinger equations with variable mass, which also turns out to be non-normalized. We prove that this class is mapped by a family of point transformations to the class of (1+1)-dimensional linear Schrödinger equations with unique constant mass.

  • 326.
    Källström, Rolf
    Högskolan i Gävle, Akademin för teknik och miljö, Avdelningen för elektronik, matematik och naturvetenskap, Matematik.
    Purity of Branch and Critical locus2013Inngår i: Journal of Algebra, ISSN 0021-8693, E-ISSN 1090-266X, Vol. 379, s. 156-178Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    To a dominant morphism $X/S \to Y/S$ of N\oe therian integral $S$-schemes one has the inclusion $C_{X/Y}\subset B_{X/Y}$ of the critical locus in the branch locus of $X/Y$. Starting from the notion of locally complete intersection morphisms, we give conditions on the modules of relative differentials $\Omega_{X/Y}$, $\Omega_{X/S}$, and $\Omega_{Y/S}$ that imply bounds on the codimensions of $ C_{X/Y}$ and $ B_{X/Y}$. These bounds generalise to a wider class of morphisms the classical purity results for finite morphisms by Zariski-Nagata-Auslander, and Faltings and Grothendieck, and van der Waerden's purity for birational morphisms.

  • 327.
    Källström, Rolf
    et al.
    University of Gävle, Department of Mathematics.
    Tadesse, Yohannes
    Stockholms universitet, Naturvetenskapliga fakulteten, Matematiska institutionen.
    Hilbert series of modules over Lie algebroidsArtikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    We consider modules M over Lie algebroids g_A which are of  finite type over a local noetherian ring A.  Using ideals  J\subseteq A such that g_A . J \subseteq J  and the length  l_{g_A}(M/JM)< \infty we can define in a natural way the  Hilbert series of M with respect to the defining ideal J.  This  notion is in particular studied for modules over the Lie algebroid  of k-linear derivations g_A=T_A(I) that preserve an ideal  I\subseteq A, for example when A is the ring of convergent  power series.

  • 328.
    Källström, Rolf
    et al.
    Högskolan i Gävle, Akademin för teknik och miljö, Avdelningen för elektronik, matematik och naturvetenskap, Matematik.
    Tadesse, Yohannes
    Stockholms universitet.
    Hilbert series of modules over Lie algebroids2015Inngår i: Journal of Algebra, ISSN 0021-8693, E-ISSN 1090-266X, Vol. 432, s. 129-184Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    We consider modules M over Lie algebroids gA which are of finite type over a local noetherian ring A. Using ideals J ⊂ A such that gA ·J ⊂ J and the length ℓgA (M/JM) < ∞ we can define in a natural way the Hilbert series of M with respect to the defining ideal J. This notion is in particular studied for modules over the Lie algebroid of k-linear derivations gA = TA(I) that preserve an ideal I ⊂ A, for example when A = On, the ring of convergent power series. Hilbert series over Stanley-Reisner rings are also considered.

  • 329.
    Külshammer, Julian
    Univ Stuttgart, Inst Algebra & Number Theory, D-70569 Stuttgart, Germany.
    Auslander-Reiten theory of Frobenius-Lusztig kernels2013Inngår i: Bulletin of the London Mathematical Society, ISSN 0024-6093, E-ISSN 1469-2120, Vol. 45, nr 6, s. 1285-1300Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    In this paper, we show that the tree class of a component of the stable Auslander–Reiten quiver of a Frobenius–Lusztig kernel is one of the three infinite Dynkin diagrams. For the special case of the small quantum group, we show that the periodic components are homogeneous tubes and that the non‐periodic components have shape ℤ[A] if the component contains a module for the infinite‐dimensional quantum group.

  • 330.
    Külshammer, Julian
    Christian-Albrechts-Universität zu Kiel, Ludewig-Meyn-Str. 4, 24098 Kiel, Germany .
    Auslander-Reiten theory of small half quantum groups2014Inngår i: Proceedings of the American Mathematical Society, ISSN 0002-9939, E-ISSN 1088-6826, Vol. 142, s. 1109-1119Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    For the small half quantum groups u_\zeta(b) and u_\zeta(n) we show that the components of the stable Auslander-Reiten quiver containing gradable modules are of the form Z[A_\infty].

  • 331.
    Külshammer, Julian
    Univ Kiel, D-24098 Kiel, Germany.
    Biserial algebras via subalgebras and the path algebra of D42011Inngår i: Journal of Algebra, ISSN 0021-8693, E-ISSN 1090-266X, Vol. 331, nr 1, s. 58-67Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    We give two new criteria for a basic algebra to be biserial. The first one states that an algebra is biserial iff all subalgebras of the form eAe where e is supported by at most 4 vertices are biserial. The second one gives some condition on modules that must not exist for a biserial algebra. These modules have properties similar to the module with dimension vector (1,1,1,1) for the path algebra of the quiver D4.

    Both criteria generalize criteria for an algebra to be Nakayama. They rely on the description of a basic biserial algebra in terms of quiver and relations given by R. Vila-Freyer and W. Crawley-Boevey.

  • 332. Külshammer, Julian
    In the bocs seat: quasi-hereditary algebras and representation type2017Inngår i: Representation theory—current trends and perspectives / [ed] Henning Krause, Peter Littelmann, Gunter Malle, Karl-Hermann Neeb, Christoph Schweigert, European Mathematical Society Publishing House, 2017, s. 375-426Konferansepaper (Fagfellevurdert)
    Abstract [en]

    This paper surveys bocses, quasi-hereditary algebras and their relationship which was established in a recent result by Koenig, Ovsienko, and the author. Particular emphasis is placed on applications of this result to the representation type of the category filtered by standard modules for a quasi-hereditary algebra. In this direction, joint work with Thiel is presented showing that the subcategory of modules filtered by Weyl modules for tame Schur algebras is of finite representation type. The paper also includes a new proof for the classication of quasi-hereditary algebras with two simple modules, a result originally obtained by Membrillo–Hernández.

  • 333.
    Külshammer, Julian
    Univ Stuttgart, Inst Algebra & Number Theory, Pfaffenwaldring 57, D-70569 Stuttgart, Germany.
    Pro-species of algebras I: Basic properties2017Inngår i: Algebras and Representation Theory, ISSN 1386-923X, E-ISSN 1572-9079, Vol. 20, nr 5, s. 1215-1238Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    In this paper, we generalise part of the theory of hereditary algebras to the context of pro-species of algebras. Here, a pro-species is a generalisation of Gabriel’s concept of species gluing algebras via projective bimodules along a quiver to obtain a new algebra. This provides a categorical perspective on a recent paper by Geiß et al. (2016). In particular, we construct a corresponding preprojective algebra, and establish a theory of a separated pro-species yielding a stable equivalence between certain functorially finite subcategories.

  • 334.
    Külshammer, Julian
    Christian-Albrechts-Universität zu Kiel, Ludewig-Meyn-Str. 4, 24098 Kiel, Germany.
    Representation type of Frobenius-Lusztig kernels2013Inngår i: Quarterly Journal of Mathematics, ISSN 0033-5606, E-ISSN 1464-3847, Vol. 64, nr 2, s. 471-488Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    In this article, we show that almost all blocks of all Frobenius–Lusztig kernels are of wild representation type extending results of Feldvoss and Witherspoon, who proved this result for the principal block of the zeroth Frobenius–Lusztig kernel. Furthermore, we verify the conjecture that there are infinitely many Auslander–Reiten components for a finite-dimensional algebra of infinite representation type for selfinjective algebras whose cohomology satisfies certain finiteness conditions.

  • 335. Külshammer, Julian
    et al.
    Bodzenta, Agnieszka
    Ringel duality as an instance of Koszul duality2018Inngår i: Journal of Algebra, ISSN 0021-8693, E-ISSN 1090-266X, Vol. 506, s. 129-187Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    S. Koenig, S. Ovsienko and the second author showed that every quasi-hereditary algebra is Morita equivalent to the right algebra, i.e. the opposite algebra of the left dual, of a coring. Let A be an associative algebra and V an A-coring whose right algebra R is quasi-hereditary. In this paper, we give a combinatorial description of an associative algebra B and a B-coring W whose right algebra is the Ringel dual of R. We apply our results in small examples to obtain restrictions on the A-infinity-structure of the Ext-algebra of standard modules over a class of quasi-hereditary algebras related to birational morphisms of smooth surfaces.

  • 336. Külshammer, Julian
    et al.
    Jasso, Gustavo
    Nakayama-type phenomena in higher Auslander-Reiten theory2018Inngår i: Representations of algebras / [ed] Graham J. Leuschke, Frauke Bleher, Ralf Schiffler, Dan Zacharia, American Mathematical Society (AMS), 2018, Vol. 705Konferansepaper (Fagfellevurdert)
  • 337.
    Lacoin, Hubert
    et al.
    Univ Paris 09, CEREMADE, UMR CNRS 7534, F-75775 Paris 16, France..
    Tykesson, Johan
    Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen.
    On the easiest way to connect k points in the Random Interlacements process2013Inngår i: Latin American Journal of Probability and Mathematical Statistics, ISSN 1980-0436, E-ISSN 1980-0436, Vol. 10, nr 1, s. 505-524Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    We consider the random interlacements process with intensity u on Z(d), d >= 5 (call it I-u), built from a Poisson point process on the space of doubly infinite nearest neighbor trajectories on Z(d). For k >= 3 we want to determine the minimal number of trajectories from the point process that is needed to link together k points in I-u. Let n(k, d) := [d/2(k -1)] - (k - 2). We prove that almost surely given any k points x(1),..., x(k) is an element of I-u, there is a sequence of n(k, d) trajectories gamma(1),..., gamma(n(k,d)) from the underlying Poisson point process such that the union of their traces U-i =1(n(k,d)) Tr(gamma(i)) is a connected set containing x(1),..., x(k). Moreover we show that this result is sharp, i.e. that a.s. one can find x(1),..., x(k) is an element of I-u that cannot be linked together by n(k, d) - 1 trajectories.

  • 338.
    Lakso, Johan
    Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Algebra och geometri.
    Det allmänna måttproblemet & Hausdorffs paradox2015Independent thesis Basic level (degree of Bachelor), 10 poäng / 15 hpOppgave
  • 339.
    Larsson, Daniel
    et al.
    Uppsala University.
    Sigurdsson, Gunnar
    Royal Institute of Technology (KTH).
    Silvestrov, Sergei D.
    Mälardalens högskola, Akademin för utbildning, kultur och kommunikation.
    Quasi-Lie deformations on the algebra F[t]/(tn)2008Inngår i: Journal of Generalized Lie Theory and Applications, ISSN 1736-5279, E-ISSN 1736-4337, Vol. 2, nr 3, s. 201-205Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    This paper explores the quasi-deformation scheme devised by Hartwig, Larsson and Silvestrov as applied to the simple Lie algebra sl2(F). One of the main points of this method is that the quasi-deformed algebra comes endowed with a canonical twisted Jacobi identity. We show in the present article that when the quasi-deformation method is applied to sl2(F) via representations by twisted derivations on the algebra F[t]/(tN) one obtains interesting new multi-parameter families of almost quadratic algebras.

  • 340.
    Larsson, Daniel
    et al.
    Uppsala University.
    Silvestrov, Sergei D.
    Lund University.
    On generalized N-complexes coming from twisted derivations2009Inngår i: Generalized Lie theory in mathematics, physics and beyond / [ed] Silvestrov, S.D.; Paal, E.; Abramov, V.; Stolin, A., Berlin, Heidelberg: Springer , 2009, s. 81-88Kapittel i bok, del av antologi (Fagfellevurdert)
    Abstract [en]

    Inspired by a result of V. Abramov on q-differential graded algebras, we prove a theorem, analogous to Abramov's result but in a slightly different set-up, using a σ- (twisted) derivation as the differential-like map. As an application, we construct a generalized N-complex based on the ring of Laurent polynomials.

  • 341.
    Larsson, Erik
    Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Algebra och geometri.
    λ-Calculus and Decidability2017Independent thesis Basic level (degree of Bachelor), 10 poäng / 15 hpOppgave
  • 342. Laustsen, Niels Jakob
    et al.
    Silvestrov, S. D.
    Lund University.
    Heisenberg-Lie commutation relations in Banach algebras2009Inngår i: Mathematical Proceedings of the Royal Irish Academy, ISSN 1393-7197, E-ISSN 2009-0021, Vol. 109, nr 2, s. 163-186Artikkel i tidsskrift (Fagfellevurdert)
  • 343.
    Lenells, Jonatan
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.).
    Matrix Riemann-Hilbert problems with jumps across Carleson contours2018Inngår i: Monatshefte für Mathematik (Print), ISSN 0026-9255, E-ISSN 1436-5081, Vol. 186, nr 1, s. 111-152Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    We develop a theory of n x n-matrix Riemann-Hilbert problems for a class of jump contours and jump matrices of low regularity. Our basic assumption is that the contour Gamma is a finite union of simple closed Carleson curves in the Riemann sphere. In particular, unbounded contours with cusps, corners, and nontransversal intersections are allowed. We introduce a notion of L-p-Riemann-Hilbert problem and establish basic uniqueness results and Fredholm properties. We also investigate the implications of Fredholmness for the unique solvability and prove a theorem on contour deformation.

  • 344.
    Lenngren, Nils
    Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Algebra, geometri och logik.
    k-uniform tilings by regular polygons2009Independent thesis Basic level (degree of Bachelor), 10 poäng / 15 hpOppgave
  • 345.
    Leward, Oscar
    Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Algebra och geometri.
    Graph TheoryThe Four Color Theorem2014Independent thesis Basic level (degree of Bachelor), 10 poäng / 15 hpOppgave
  • 346.
    Li, Jing
    et al.
    School of Mathematics and Statistics, Southwest University, Chongqing, 40071, China ; School of Mathematics and Information, Langfang Teachers’ College, Langfang, 065000, China.
    Peng, Aihui
    Southwest University, Chongqing, China.
    Song, Naiqing
    School of Mathematics and Statistics, Southwest University, Chongqing, 40071, China.
    Teaching algebraic equations with variation in Chinese classroom2011Inngår i: Early Algebraization: A Global Dialogue from Multiple Perspectives / [ed] Jinfa Cai & Eric Knuth, Springer, 2011, s. 529-556Kapittel i bok, del av antologi (Fagfellevurdert)
    Abstract [en]

    This chapter gives a detailed analysis of how teaching with variation is helpful for students’ learning of algebraic equations by using typical teaching episodes in grade seven in China. Also, it provides a demonstration showing how variation is used as an effective way of teaching through the discussion after the analysis.

  • 347.
    Lidberg, Petter
    Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Algebra och geometri.
    Barycentric and harmonic coordinates2012Independent thesis Advanced level (degree of Master (Two Years)), 20 poäng / 30 hpOppgave
  • 348.
    Lilja, Katarina
    Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Algebra och geometri.
    Kvaternioner – algebraiska och geometriska aspekter2013Independent thesis Basic level (degree of Bachelor), 10 poäng / 15 hpOppgave
  • 349.
    Lindeberg, Ludvig
    et al.
    Högskolan i Jönköping, Högskolan för lärande och kommunikation, HLK, Skolnära forskning, Matematikdidaktik.
    Johansson, Tom
    Högskolan i Jönköping, Högskolan för lärande och kommunikation, HLK, Skolnära forskning, Matematikdidaktik.
    Undervisning med växande geometriska mönster: En litteraturstudie om arbetssätt och metoder.2018Independent thesis Basic level (university diploma), 10 poäng / 15 hpOppgave
    Abstract [sv]

    Växande geometriska mönster ses som ett redskap för att få elever från ett aritmetiskt till ett algebraiskt tänk om mönster successivt ersätts med tal och bokstavsbeteckningar. TIMSS- rapporten från 2015 redovisar att svenska elever i årskurs 8 har dåliga resultat inom algebra och geometri men samtidigt är aritmetik ett av de starkaste områdena. Syftet är därför att undersöka vilka teorier om lärande av växande geometriska mönster som finns samt vilka arbetssätt och metoder som används i undervisning om växande geometriska mönster. Studien är en litteraturstudie, där vetenskapligt material analyseras utifrån forskningsfrågor. I denna studie består materialet av tidskriftsartiklar, konferensbidrag och en avhandling.

     

    Resultatet visar på två skilda teorier angående hur undervisning om växande geometriska mönster ska bedrivas, hierarkisk och dynamisk undervisning. Den hierarkiska teorins ståndpunkt är att kunskap byggs från grunden och att gå från det enkla till det komplexa för att elever ska lära sig generalisera. Tvärtemot anser förespråkare för en dynamisk undervisning att elever bör tvingas generalisera redan från början. Resultatet visar också att elever har lättare att generalisera och se samband mellan formel och mönster om de får lära sig analysera det växande geometriska mönstrets delar och samband mellan figurer redan från början.

  • 350.
    Lindqvist, Anders
    Uppsala universitet, Teknisk-naturvetenskapliga vetenskapsområdet, Matematisk-datavetenskapliga sektionen, Matematiska institutionen, Algebra och geometri.
    On four-dimensional unital division algebras over finite fields2015Independent thesis Basic level (degree of Bachelor), 10 poäng / 15 hpOppgave
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