Mostrando recursos 1 - 20 de 28

  1. A coupling of Brownian motions in the $\mathcal{L}_0$-geometry

    Amaba, Takafumi; Kuwada, Kazumasa
    Under a complete Ricci flow, we construct a coupling of two Brownian motions such that their $\mathcal{L}_0$-distance is a supermartingale. This recovers a result of Lott [J. Lott, Optimal transport and Perelman's reduced volume, Calc. Var. Partial Differential Equations 36 (2009), no. 1, 49–84.] on the monotonicity of $\mathcal{L}_0$-distance between heat distributions.

  2. Sharp $L^p$-bounds for the martingale maximal function

    Osȩkowski, Adam
    The paper studies sharp weighted $L^p$ inequalities for the martingale maximal function. Proofs exploit properties of certain special functions of four variables and self-improving properties of $A_p$ weights.

  3. Stochastic calculus for Markov processes associated with semi-Dirichlet forms

    Chen, Chuan-Zhong; Ma, Li; Sun, Wei
    We present a new Fukushima type decomposition in the framework of semi-Dirichlet forms. This generalizes the result of Ma, Sun and Wang [17, Theorem 1.4] by removing the condition (S). We also extend Nakao's integral to semi-Dirichlet forms and derive Itô's formula related to it.

  4. The rates of the $L^p$-convergence of the Euler-Maruyama and Wong-Zakai approximations of path-dependent stochastic differential equations under the Lipschitz condition

    Aida, Shigeki; Kikuchi, Takanori; Kusuoka, Seiichiro
    We consider the rates of the $L^p$-convergence of the Euler-Maruyama and Wong-Zakai approximations of path-dependent stochastic differential equations under the Lipschitz condition on the coefficients. By a transformation, the stochastic differential equations of Markovian type with reflecting boundary condition on sufficiently good domains are to be associated with the equations concerned in the present paper. The obtained rates of the $L^p$-convergence are the same as those in the case of the stochastic differential equations of Markovian type without boundaries.

  5. Worpitzky partitions for root systems and characteristic quasi-polynomials

    Yoshinaga, Masahiko
    For a given irreducible root system, we introduce a partition of (coweight) lattice points inside the dilated fundamental parallelepiped into those of partially closed simplices. This partition can be considered as a generalization and a lattice points interpretation of the classical formula of Worpitzky. ¶ This partition, and the generalized Eulerian polynomial, recently introduced by Lam and Postnikov, can be used to describe the characteristic (quasi)polynomials of Shi and Linial arrangements. As an application, we prove that the characteristic quasi-polynomial of the Shi arrangement turns out to be a polynomial. We also present several results on the location of zeros of...

  6. On Frobenius manifolds from Gromov–Witten theory of orbifold projective lines with $r$ orbifold points

    Shiraishi, Yuuki
    We prove that the Frobenius structure constructed from the Gromov–Witten theory for an orbifold projective line with at most $r$ orbifold points is uniquely determined by the WDVV equations with certain natural initial conditions.

  7. Exponentially weighted Polynomial approximation for absolutely continuous functions

    Itoh, Kentaro; Sakai, Ryozi; Suzuki, Noriaki
    We discuss a polynomial approximation on $\mathbb{R}$ with a weight $w$ in $\mathcal{F}(C^{2} +)$ (see Section 2). The de la Vallée Poussin mean $v_n(f)$ of an absolutely continuous function $f$ is not only a good approximation polynomial of $f$, but also its derivatives give an approximation for the derivative $f'$. More precisely, for $1 \leq p \leq \infty$, we have $\lim_{n \rightarrow \infty}\|(f - v_{n}(f))w\|_{L^{p}(\mathbb{R})} =0$ and $\lim_{n \rightarrow \infty}\|(f' - v_{n}(f)')w\|_{L^{p}(\mathbb{R})} =0$ whenever $f''w \in L^{p}(\mathbb{R})$.

  8. 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.

  9. 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.

  10. 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.

  11. 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 $\zeta(s)$....

  12. 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.

  13. 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.

  14. 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.

  15. 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.

  16. 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.

  17. 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.

  18. 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.

  19. 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.

  20. 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.

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