Mostrando recursos 1 - 20 de 21

  1. On the most expected number of components for random links

    Ichihara, Kazuhiro; Yoshida, Ken-ichi
    We consider a random link, which is defined as the closure of a braid obtained from a random walk on the braid group. For such a random link, the expected value for the number of components was calculated by Jiming Ma. In this paper, we determine the most expected number of components for a random link, and further, consider the most expected partition of the number of strings for a random braid.

  2. Minimal timelike surfaces in a certain homogeneous Lorentzian 3-manifold

    Lee, Sungwook
    The 2-parameter family of certain homogeneous Lorentzian 3-manifolds which includes Minkowski 3-space, de Sitter 3-space, and Minkowski motion group is considered. Each homogeneous Lorentzian 3-manifold in the 2-parameter family has a solvable Lie group structure with left invariant metric. A generalized integral representation formula which is the unification of representation formulas for minimal timelike surfaces in those homogeneous Lorentzian 3-manifolds is obtained. The normal Gauß map of minimal timelike surfaces in those homogeneous Lorentzian 3-manifolds and its harmonicity are discussed.

  3. Schottky via the punctual Hilbert scheme

    Gulbrandsen, Martin G.; Lahoz, Martí
    We show that a smooth projective curve of genus $g$ can be reconstructed from its polarized Jacobian $(X, \Theta)$ as a certain locus in the Hilbert scheme $\mathrm{Hilb}^d(X)$, for $d=3$ and for $d=g+2$, defined by geometric conditions in terms of the polarization $\Theta$. The result is an application of the Gunning-Welters trisecant criterion and the Castelnuovo-Schottky theorem by Pareschi-Popa and Grushevsky, and its scheme theoretic extension by the authors.

  4. Spectral zeta functions of graphs and the Riemann zeta function in the critical strip

    Friedli, Fabien; Karlsson, Anders
    We initiate the study of spectral zeta functions $\zeta_X$ for finite and infinite graphs $X$, instead of the Ihara zeta function, with a perspective towards zeta functions from number theory and connections to hypergeometric functions. The Riemann hypothesis is shown to be equivalent to an approximate functional equation of graph zeta functions. The latter holds at all points where Riemann's zeta function $\zeta(s)$ is non-zero. This connection arises via a detailed study of the asymptotics of the spectral zeta functions of finite torus graphs in the critcal strip and estimates on the real part of the logarithmic derivative of $\zeta(s)$....

  5. A polynomial defined by the $SL(2;\mathbb{C})$-Reidemeister torsion for a homology 3-sphere obtained by a Dehn surgery along a $(2p,q)$-torus knot

    Kitano, Teruaki
    Let $K$ be a $(2p,q)$-torus knot. Here $p$ and $q$ are coprime odd positive integers. Let $M_n$ be a 3-manifold obtained by a $1/n$-Dehn surgery along $K$. We consider a polynomial $\sigma_{(2p,q,n)}(t)$ whose zeros are the inverses of the Reidemeister torsion of $M_n$ for $\mathit{SL}(2;\mathbb{C})$-irreducible representations under some normalization. Johnson gave a formula for the case of the $(2,3)$-torus knot under another normalization. We generalize this formula for the case of $(2p,q)$-torus knots by using Tchebychev polynomials.

  6. Monodromy representations of hypergeometric systems with respect to fundamental series solutions

    Matsumoto, Keiji
    We study the monodromy representation of the generalized hypergeometric differential equation and that of Lauricella's $F_C$ system of hypergeometric differential equations. We use fundamental systems of solutions expressed by the hypergeometric series. We express non-diagonal circuit matrices as reflections with respect to root vectors with all entries 1. We present a simple way to obtain circuit matrices.

  7. Holomorphic isometric embeddings of the projective line into quadrics

    Macia, Oscar; Nagatomo, Yasuyuki; Takahashi, Masaro
    We discuss holomorphic isometric embeddings of the projective line into quadrics using the generalisation of the theorem of do Carmo-Wallach in [14] to provide a description of their moduli spaces up to image and gauge equivalence. Moreover, we show rigidity of the real standard map from the projective line into quadrics.

  8. Bounds on the Tamagawa numbers of a crystalline representation over towers of cyclotomic extensions

    Lei, Antonio
    In this paper, we study the Tamagawa numbers of a crystalline representation over a tower of cyclotomic extensions under certain technical conditions on the representation. In particular, we show that we may improve the asymptotic bounds given in the thesis of Arthur Laurent in certain cases.

  9. Boundedness of the maximal operator on Musielak-Orlicz-Morrey spaces

    Maeda, Fumi-Yuki; Ohno, Takao; Shimomura, Tetsu
    We give the boundedness of the maximal operator on Musielak-Orlicz-Morrey spaces, which is an improvement of [7, Theorem 4.1]. We also discuss the sharpness of our conditions.

  10. A remark on almost sure global well-posedness of the energy-critical defocusing nonlinear wave equations in the periodic setting

    Oh, Tadahiro; Pocovnicu, Oana
    In this note, we prove almost sure global well-posedness of the energy-critical defocusing nonlinear wave equation on $\mathbb{T}^d$, $d = 3, 4,$ and $5$, with random initial data below the energy space.

  11. A remark on almost sure global well-posedness of the energy-critical defocusing nonlinear wave equations in the periodic setting

    Oh, Tadahiro; Pocovnicu, Oana
    In this note, we prove almost sure global well-posedness of the energy-critical defocusing nonlinear wave equation on $\mathbb{T}^d$, $d = 3, 4,$ and $5$, with random initial data below the energy space.

  12. Gauss maps of toric varieties

    Furukawa, Katsuhisa; Ito, Atsushi
    We investigate Gauss maps of (not necessarily normal) projective toric varieties over an algebraically closed field of arbitrary characteristic. The main results are as follows: (1) The structure of the Gauss map of a toric variety is described in terms of combinatorics in any characteristic. (2) We give a developability criterion in the toric case. In particular, we show that any toric variety whose Gauss map is degenerate must be the join of some toric varieties in characteristic zero. (3) As applications, we provide two constructions of toric varieties whose Gauss maps have some given data (e.g., fibers, images) in positive characteristic.

  13. Gauss maps of toric varieties

    Furukawa, Katsuhisa; Ito, Atsushi
    We investigate Gauss maps of (not necessarily normal) projective toric varieties over an algebraically closed field of arbitrary characteristic. The main results are as follows: (1) The structure of the Gauss map of a toric variety is described in terms of combinatorics in any characteristic. (2) We give a developability criterion in the toric case. In particular, we show that any toric variety whose Gauss map is degenerate must be the join of some toric varieties in characteristic zero. (3) As applications, we provide two constructions of toric varieties whose Gauss maps have some given data (e.g., fibers, images) in positive characteristic.

  14. The maximal ideal cycles over normal surface singularities with ${\Bbb C}^*$-action

    Tomari, Masataka; Tomaru, Tadashi
    The maximal ideal cycles and the fundamental cycles are defined on the exceptional sets of resolution spaces of normal complex surface singularities. The former (resp. later) is determined by the analytic (resp. topological) structure of the singularities. We study such cycles for normal surface singularities with ${\Bbb C}^*$-action. Assuming the existence of a reduced homogeneous function of the minimal degree, we prove that these two cycles coincide if the coefficients on the central curve of the exceptional set of the minimal good resolution coincide.

  15. The maximal ideal cycles over normal surface singularities with ${\Bbb C}^*$-action

    Tomari, Masataka; Tomaru, Tadashi
    The maximal ideal cycles and the fundamental cycles are defined on the exceptional sets of resolution spaces of normal complex surface singularities. The former (resp. later) is determined by the analytic (resp. topological) structure of the singularities. We study such cycles for normal surface singularities with ${\Bbb C}^*$-action. Assuming the existence of a reduced homogeneous function of the minimal degree, we prove that these two cycles coincide if the coefficients on the central curve of the exceptional set of the minimal good resolution coincide.

  16. Atomic decompositions of weighted Hardy spaces with variable exponents

    Ho, Kwok-Pun
    We establish the atomic decompositions for the weighted Hardy spaces with variable exponents. These atomic decompositions also reveal some intrinsic structures of atomic decomposition for Hardy type spaces.

  17. Atomic decompositions of weighted Hardy spaces with variable exponents

    Ho, Kwok-Pun
    We establish the atomic decompositions for the weighted Hardy spaces with variable exponents. These atomic decompositions also reveal some intrinsic structures of atomic decomposition for Hardy type spaces.

  18. A note on stable sheaves on Enriques surfaces

    Yoshioka, Kōta
    We shall give a necessary and sufficient condition for the existence of stable sheaves on Enriques surfaces based on results of Kim, Yoshioka, Hauzer and Nuer. For unnodal Enriques surfaces, we also study the relation of virtual Hodge “polynomial” of the moduli stacks.

  19. A note on stable sheaves on Enriques surfaces

    Yoshioka, Kōta
    We shall give a necessary and sufficient condition for the existence of stable sheaves on Enriques surfaces based on results of Kim, Yoshioka, Hauzer and Nuer. For unnodal Enriques surfaces, we also study the relation of virtual Hodge “polynomial” of the moduli stacks.

  20. Seidel elements and potential functions of holomorphic disc counting

    González, Eduardo; Iritani, Hiroshi
    Let $M$ be a symplectic manifold equipped with a Hamiltonian circle action and let $L$ be an invariant Lagrangian submanifold of $M$. We study the problem of counting holomorphic disc sections of the trivial $M$-bundle over a disc with boundary in $L$ through degeneration. We obtain a conjectural relationship between the potential function of $L$ and the Seidel element associated to the circle action. When applied to a Lagrangian torus fibre of a semi-positive toric manifold, this degeneration argument reproduces a conjecture (now a theorem) of Chan-Lau-Leung-Tseng [8, 9] relating certain correction terms appearing in the Seidel elements with the potential function.

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