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Document Server@UHasselt (59.068 recursos)

Repository of the University of Hasselt containing publications in the fields of statistics, computer science, information strategies and material from the Institute for behavioural sciences.

Pure Mathematics

Mostrando recursos 1 - 20 de 64

  1. PBW deformations of Koszul algebras over a nonsemisimple ring

    He, Jiwei; Van Oystaeyen, Fred; Zhang, Yinhuo
    Let B be a generalized Koszul algebra over a finite dimensional algebra S. We construct a bimodule Koszul resolution of B when the projective dimension of SB equals two. Using this we prove a Poincaré–Birkhoff–Witt (PBW) type theorem for a deformation of a generalized Koszul algebra. When the projective dimension of SB is greater than two, we construct bimodule Koszul resolutions for generalized smash product algebras obtained from braidings between finite dimensional algebras and Koszul algebras, and then prove the PBW type theorem. The results obtained can be applied to standard Koszul Artin–Schelter Gorenstein algebras in the sense of Minamoto and...

  2. Division algebras in Gr-categories

    Huang, Hualin; Van Oystaeyen, Fred; ZHANG, Yinhuo
    We study division algebras in an arbitrary linear Gr-category, i.e., a category of finite-dimensional vector spaces graded by a groupwith associativity constraint given by a 3-cocycle. When the 3-cocycle is non-coboundary, this provides some interesting classes of nonassociative division algebras. In particular, when we work on Gr-categories over the field of real numbers, some quasi-associative version of the quaternions and octonions appear.

  3. Cyclicity of a fake saddle inside the quadratic vector fields

    DE MAESSCHALCK, Peter; Rebollo-Perdomo, S.; Torregrosa, J.
    This paper concerns the study of small-amplitude limit cycles that appear in the phase portrait near an unfolded fake saddle singularity. This degenerate singularity is also known as an impassable grain. The canonical form of the unperturbed vector field is like a degenerate flow box. Near the singularity, the phase portrait consists of parallel fibers, all but one of which have no singular points, and at the singular fiber, there is one node. We demonstrate different techniques in order to show that the cyclicity is bigger than or equal to two when the canonical form is quadratic. (C) 2014 Elsevier...

  4. Slow divergence integrals in generalized Li??nard equations near centers

    Huzak, Renato; De Maesschalck, Peter
    Using techniques from singular perturbations we show that for any n???6 and m???2 there are Li??nard equations {x??=y???F(x), y??=G(x)}, with F a polynomial of degree n and G a polynomial of degree m, having at least 2[n???2/2]+[m/2] hyperbolic limit cycles, where [???] denotes "the greatest integer equal or below".

  5. Noncommutative versions of some classical birational transformations

    Presotto, Dennis; Van den Bergh, Michel
    In this paper we generalize some classical birational transformations to the non-commutative case. In particular we show that 3-dimensional quadratic Sklyanin algebras (non-commutative projective planes) and 3-dimensional cubic Sklyanin algebras (non-commutative quadrics) have the same function field. In the same vein we construct and analogue of the Cremona transform for non-commutative projective planes.

  6. Green Rings of Pointed Rank One Hopf Algebras of Nilpotent Type

    WANG, Zhihua; Li, Libin; ZHANG, Yinhuo
    Let H be a finite dimensional pointed rank one Hopf algebra of nilpotent type. We first determine all finite dimensional indecomposable H-modules up to isomorphism, and then establish the Clebsch-Gordan formulas for the decompositions of the tensor products of indecomposable H-modules by virtue of almost split sequences. The Green ring r(H) of H will be presented in terms of generators and relations. It turns out that the Green ring r(H) is commutative and is generated by one variable over the Grothendieck ring G0(H) of H modulo one relation. Moreover, r(H) is Frobenius and symmetric with dual bases associated to almost...

  7. On compact generation of deformed schemes

    Lowen, Wendy; Van den Bergh, Michel
    We obtain a theorem which allows to prove compact generation of derived categories of Grothendieck categories, based upon certain coverings by localizations. This theorem follows from an application of Rouquier's cocovering theorem in the triangulated context, and it implies Neeman's result on compact generation of quasi-compact separated schemes. We prove an application of our theorem to non-commutative deformations of such schemes, based upon a change from Koszul complexes to Chevalley-Eilenberg complexes. (C) 2013 Elsevier Inc. All rights reserved.

  8. Geometric classification of 4-dimensional superalgebras

    Armour, Aaron; Zhang, Yinhuo
    In this paper, we give a geometric classification of 4-dimensional superalgebras over an algebraic closed field. It turns out that the number of irreducible components of the variety of 4-dimensional superalgebras Salg$_4$ under the Zariski topology is between 20 and 22. One of the significant differences between the variety Alg$_n$ and the variety Salg$_n$ is that Salg$_n$ is disconnected while Alg$_n$ is connected. Under certain conditions on $n$, one can show that the variety Salg$_n$ is the disjoint union of $n$ connected subvrieties. We shall present the degeneration diagrams of the 4 disjoint connected subvarieties Salg$^i_4$ of Salg$_4$.

  9. On (co)homology of Frobenius Poisson algebras

    Zhu, Can; Van Oystaeyen, Fred; ZHANG, Yinhuo
    In this paper, we study Poisson (co)homology of a Frobenius Poisson algebra. More precisely, we show that there exists a duality between Poisson homology and Poisson cohomology of Frobenius Poisson algebras, similar to that between Hochschild homology and Hochschild cohomology of Frobenius algebras. Then we use the non-degenerate bilinear form on a unimodular Frobenius Poisson algebra to construct a Batalin-Vilkovisky structure on the Poisson cohomology ring making it into a Batalin-Vilkovisky algebra.

  10. Primary birth of canard cycles in slow-fast codimension 3 elliptic bifurcations

    HUZAK, Renato; DE MAESSCHALCK, Peter; DUMORTIER, Freddy
    In this paper we continue the study of ``large" small-amplitude limit cycles in slow-fast codimension 3 elliptic bifurcations which is initiated in [8]. Our treatment is based on blow-up and good normal forms.

  11. Slow Divergence Integrals in Classical Li??nard Equations Near Centers

    DE MAESSCHALCK, Peter; HUZAK, Renato
    We significantly improve lower bounds for the number of limit cycles for polynomial classical Li??nard equations, aiming at stating lower bounds that are reasonable enough to be optimal. The techniques used are the notion of slow divergence integral from the geometric theory of planar slow-fast systems.

  12. Canard cycle transition at a slow-fast passage through a jump point

    DE MAESSCHALCK, Peter; DUMORTIER, Freddy; Roussarie, Robert
    We introduce transitory canard cycles for slow???fast vector fields in the plane. Such cycles separate ???canards without head??? and ???canardswithhead???,like for example in the Van der Pol equation. We obtain optimal upper bounds on the number of periodic orbits that can appear near the cycle underwhatever condition on the related slow divergence integralI,including the challenging caseI=0.

  13. Gevrey asymptotics of series in Mourtada-type compensators used for linearization of an analytic 1:-1 resonant saddle

    Bonckaert, Patrick
    Given a 1:-1 resonant saddle singularity of a planar analytic vector field, we provide a linearization procedure using a series expansion in compensators of Mourtada-type, and show that this series has Gevrey-1 asymptotics. In case of an analytic Poincar\'e-Dulac normal form we show that this transformation is analytic as a function of the compensators

  14. Three Time-Scales In An Extended Bonhoeffer???Van Der Pol Oscillator

    DE MAESSCHALCK, Peter; Popovic, Nikola; KUTAFINA, Ekaterina
    We consider an extended three-dimensional Bonhoeffer-van der Pol oscillator which generalises the planar FitzHugh-Nagumo model from mathematical neuroscience, and which was recently studied by Sekikawa et al. and by Freire and Gallas. Focussing on a parameter regime which has hitherto been neglected, and in which the governing equations evolve on three distinct time-scales, we propose a reduction to a model problem that was formulated by Krupa et al. as a canonical form for such systems. Based on previously obtained results in, we characterise completely the mixed-mode dynamics of the resulting three time-scale extended Bonhoeffer-van der Pol oscillator from the point of...

  15. Mixed mode oscillations in the Bonhoeffer-van der Pol oscillator with weak periodic perturbation

    KUTAFINA, Ekaterina
    Following the paper of Shimizu et al. (Phys Lett A 375:1566, 2011), we consider the Bonhoeffer-van der Pol oscillator with non-autonomous periodic perturbation. We show that the presence of mixed mode oscillations reported in that paper can be explained using the geometric singular perturbation theory. The considered model can be re-written as a four-dimensional (locally three-dimensional) autonomous system, which under certain conditions has a folded saddle-node singularity and additionally can be treated as a three time scale one.

  16. The Green rings of Taft algebras

    Chen, Huixiang; Van Oystaeyen, Fred; ZHANG, Yinhuo
    We compute the Green ring of the Taft algebra $H_n(q)$, where $n$ is a positive integer greater than 1, and $q$ is an $n$-th root of unity. It turns out that the Green ring $r(H_n(q))$ of the Taft algebra $H_n(q)$ is a commutative ring generated by two elements subject to certain relations defined recursively. Concrete examples for $n=2,3, ... , 8$ are given.

  17. Noncommutative quadric surfaces

    Smith, S. Paul; VAN DEN BERGH, Michel
    The 4-dimensional Sklyanin algebra is the homogeneous coordinate ring of a noncommutative analogue of projective 3-space. The degree-two component of the algebra contains a 2-dimensional subspace of central elements. The zero loci of those central elements, except 0, form a pencil of noncommutative quadric surfaces. We show that the behavior of this pencil is similar to that of a generic pencil of quadrics in the commutative projective 3-space. There are exactly four singular quadrics in the pencil. The singular and non-singular quadrics are characterized by whether they have one or two rulings by noncommutative lines. The Picard groups of the...

  18. Calabi-Yau pointed Hopf algebras of finite Cartan type

    YU, Xiaolan; ZHANG, Yinhuo
    We study the Calabi???Yau property of pointed Hopf algebra U(D,??) of finite Cartan type. It turns out that this class of pointed Hopf algebras constructed by N. Andruskiewitsch and H.-J. Schneider contains many Calabi???Yau Hopf algebras. To give concrete examples of new Calab???Yau Hopf algebras, we classify the Calabi???Yau pointed Hopf algebras U(D,??) of dimension less than 5.

  19. DUALIZING COMPLEXES OF NOETHERIAN COMPLETE ALGEBRAS VIA COALGEBRAS

    He, J. -W.; Torrecillas, B.; Van Oystaeyen, F.; ZHANG, Yinhuo
    Let A be a noetherian complete basic semiperfect algebra over an algebraically closed field, and C=A degrees be its dual coalgebra. If A is Artin-Schelter regular, then the local cohomology of A is isomorphic to a shift of twisted bimodule C-1(sigma*) with sigma a coalgebra automorphism. This yields that the balanced dualinzing complex of A is a shift of the twisted bimodule (sigma*)A(1). If sigma is an inner automorphism, then A is Calabi-Yau. An appendix is included to prove a duality theorem of the bounded derived category of quasi-finite comodules over an artinian coalgebra.

  20. Some New Examples of Nondegenerate Quiver Potentials

    de Volcsey, Louis de Thanhoffer; van den Bergh, Michel
    We prove a technical result which allows us to establish the nondegeneracy of potentials on quivers in some previously unknown or nonobvious cases. Our result applies to certain McKay quivers and also to potentials derived from geometric helices on Del Pezzo surfaces.

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