Mostrando recursos 1 - 20 de 83

  1. A complete description of the antipodal set of most symmetric spaces of compact type

    Beyrer, Jonas
    It is known that the antipodal set of a Riemannian symmetric space of compact type $G/K$ consists of a union of $K$-orbits. We determine the dimensions of these $K$-orbits of most irreducible symmetric spaces of compact type. The symmetric spaces we are not going to deal with are those with restricted root system $\mathfrak{a}_r$ and a non-trivial fundamental group, which is not isomorphic to $\mathbb{Z}_2$ or $\mathbb{Z}_{r+1}$. For example, we show that the antipodal sets of the Lie groups $Spin(2r+1)\:\: r\geq 5$, $E_8$ and $G_2$ consist only of one orbit which is of dimension $2r$, 128 and 6, respectively; $SO(2r+1)$...

  2. Initial-boundary value problem for the degenerate hyperbolic equation of a hanging string

    Takayama, Masahiro
    We consider an initial-boundary value problem for the degenerate linear hyperbolic equation as a model of the motion of an inextensible string fixed at one end in the gravity field. We shall show the existence and the uniqueness of the solution and study the regularity of the solution.

  3. $L^2$-Burau maps and $L^2$-Alexander torsions

    Ben Aribi, Fathi; Conway, Anthony
    It is well known that the Burau representation of the braid group can be used to recover the Alexander polynomial of the closure of a braid. We define $L^2$-Burau maps and use them to compute some $L^2$-Alexander torsions of links. As an application, we prove that the $L^2$-Burau maps distinguish more braids than the Burau representation.

  4. Answer to a Question by Nakamura, Nakanishi, and Satoh involving crossing numbers of knots

    Ge, Jun; Jin, Xian'an; H. Kauffman, Louis; Lopes, Pedro; Zhang, Lianzhu
    In this paper we give a positive answer to a question raised by Nakamura, Nakanishi, and Satoh concerning an inequality involving crossing numbers of knots. We show it is an equality only for the trefoil and for the figure-eight knots.

  5. On Kohnen plus-space of Jacobi forms of half integral weight of matrix index

    Hayashida, Shuichi
    We introduce a plus-space of Jacobi forms, which is a certain subspace of Jacobi forms of half-integral weight of matrix index. This is an analogue to the Kohnen plus-space in the framework of Jacobi forms. We shall show a linear isomorphism between the plus-space of Jacobi forms and the space of Jacobi forms of integral weight of certain matrix index. Moreover, we shall show that this linear isomorphism is compatible with the action of Hecke operators of both spaces. This result is a kind of generalization of Eichler-Zagier-Ibukiyama correspondence, which is an isomorphism between the generalized plus-space of Siegel modular...

  6. Mazur manifolds and corks with small shadow complexities

    Naoe, Hironobu
    In this paper we show that there exist infinitely many Mazur type manifolds and corks with shadow complexity one among the 4-manifolds constructed from contractible special polyhedra having one true vertex by using the notion of Turaev's shadow. We also find such manifolds among 4-manifolds constructed from Bing's house. Our manifolds with shadow complexity one contain the Mazur manifolds $W^{\pm }(l,k)$ which were studied by Akbulut and Kirby.

  7. On-diagonal Heat Kernel Lower Bound for Strongly Local Symmetric Dirichlet Forms

    Lou, Shuwen
    This paper studies strongly local symmetric Dirichlet forms on general measure spaces. The underlying space is equipped with the intrinsic metric induced by the Dirichlet form, with respect to which the metric measure space does not necessarily satisfy volume-doubling property. Assuming Nash-type inequality, it is proved in this paper that outside a properly exceptional set, if a pointwise on-diagonal heat kernel upper bound in terms of the volume function is known a priori, then the comparable heat kernel lower bound also holds. The only assumption made on the volume growth rate is that it can be bounded by a continuous...

  8. Rank-one Perturbation of Weighted Shifts on a Directed Tree: Partial Normality and Weak Hyponormality

    R. Exner, George; Jung, Il Bong; Lee, Eun Young; Seo, Minjung
    A special rank-one perturbation $S_{t,n}$ of a weighted shift on a directed tree is constructed. Partial normality and weak hyponormality (including quasinormality, $p$-hyponormality, $p$-paranormality, absolute-$p$-paranormality and $A(p)$-class) of $S_{t,n}$ are characterized.

  9. Bloch's conjecture for Enriques varieties

    Laterveer, Robert
    Enriques varieties have been defined as higher-dimensional generalizations of Enriques surfaces. Bloch's conjecture implies that Enriques varieties should have trivial Chow group of zero-cycles. We prove this is the case for all known examples of irreducible Enriques varieties of index larger than $2$. The proof is based on results concerning the Chow motive of generalized Kummer varieties.

  10. On the flat geometry of the cuspidal edge

    Oset Sinha, Raúl; Tari, Farid
    We study the geometry of the cuspidal edge $M$ in $\mathbb{R}^3$ derived from its contact with planes and lines (referred to as flat geometry). The contact of $M$ with planes is measured by the singularities of the height functions on $M$. We classify submersions on a model of $M$ by diffeomorphisms and recover the contact of $M$ with planes from that classification. The contact of $M$ with lines is measured by the singularities of orthogonal projections of $M$. We list the generic singularities of the projections and obtain the generic deformations of the apparent contour (profile) when the direction of...

  11. A remark on conditions that a diffusion in the natural scale is a martingale

    Shimizu, Yuuki; Nakano, Fumihiko
    We consider a diffusion processes $\{ X_t \}$ on an interval in the natural scale. Some results are known under which $\{ X_t \}$ is a martingale, and we give simple and analytic proofs for them.

  12. Type numbers of quaternion hermitian forms and supersingular abelian varieties

    Ibukiyama, Tomoyoshi
    The word \textit{type number} of an algebra means classically the number of isomorphism classes of maximal orders in the algebra, but here we consider quaternion hermitian lattices in a fixed genus and their right orders. Instead of inner isomorphism classes of right orders, we consider isomorphism classes realized by similitudes of the quaternion hermitian forms.The number $T$ of such isomorphism classes are called \textit{type number} or \textit{$G$-type number}, where $G$ is the group of quaternion hermitian similitudes. We express $T$ in terms of traces of some special Hecke operators. This is a generalization of the result announced in [5] (I)...

  13. Comparison theorems in pseudo-Hermitian geometry and applications

    Dong, Yuxin; Zhang, Wei
    In this paper, we study the theory of geodesics with respect to the Tanaka-Webster connection in a pseudo-Hermitian manifold, aiming to generalize some comparison results in Riemannian geometry to the case of pseudo-Hermitian geometry. Some Hopf-Rinow type, Cartan-Hadamard type and Bonnet-Myers type results are established.

  14. Biharmonic submanifolds in a Riemannian manifold

    Koiso, Norihito; Urakawa, Hajime
    In this paper, we solve affirmatively B.-Y. Chen's conjecture for hypersurfaces in the Euclidean space, under a generic condition. More precisely, every biharmonic hypersurface of the Euclidean space must be minimal if their principal curvatures are simple, and the associated frame field is irreducible.

  15. On the digital representation of integers with bounded prime factors

    Bugeaud, Yann
    Let $b \ge 2$ be an integer. Not much is known on the representation in base $b$ of prime numbers or of numbers whose prime factors belong to a given, finite set. Among other results, we establish that any sufficiently large integer which is not a multiple of $b$ and has only small (in a suitable sense) prime factors has at least four nonzero digits in its representation in base $b$.

  16. On calculations of the twisted Alexander ideals for spatial graphs, handlebody-knots and surface-links

    Ishii, Atsushi; Nikkuni, Ryo; Oshiro, Kanako
    There are many studies about twisted Alexander invariants for knots and links, but calculations of twisted Alexander invariants for spatial graphs, handlebody-knots, and surface-links have not been demonstrated well. In this paper, we give some remarks to calculate the twisted Alexander ideals for spatial graphs, handlebody-knots and surface-links, and observe their behaviors. For spatial graphs, we calculate the invariants of Suzuki's theta-curves and show that the invariants are nontrivial for Suzuki's theta-curves whose Alexander ideals are trivial. For handlebody-knots, we give a remark on abelianizations and calculate the invariant of the handlebody-knots up to six crossings. For surface-links, we correct...

  17. Galois covers of type $(p,\cdots,p)$, vanishing cycles formula, and the existence of torsor structures

    Saïdi, Mohamed; Williams, Nicholas
    In this article we prove a local Riemman-Hurwitz formula which compares the dimensions of the spaces of vanishing cycles in a finite Galois cover of type $(p,p,\cdots,p)$ between formal germs of $p$-adic curves and which generalises the formula proven in [6] in the case of Galois covers of degree $p$. We also investigate the problem of the existence of a torsor structure for a finite Galois cover of type $(p,p,\cdots,p)$ between $p$-adic schemes.

  18. Étale endomorphisms of 3-folds. I

    Fujimoto, Yoshio
    This paper is the first part of our project towards classifications of smooth projective $3$-folds $X$ with $\kappa(X) = -\infty$ admitting a non-isomorphic étale endomorphism. We can prove that for any extremal ray $R$ of divisorial type, the contraction morphism $\pi_R\colon X\to X'$ associated to $R$ is the blowing-up of a smooth $3$-fold $X'$ along an elliptic curve. The difficulty is that there may exist infinitely many extremal rays on $X$. Thus we introduce the notion of an `ESP' which is an infinite sequence of non-isomorphic finite étale coverings of $3$-folds with constant Picard number. We can run the minimal...

  19. $\delta$-homogeneity in Finsler geometry and the positive curvature problem

    Xu, Ming; Zhang*, Lei
    In this paper, we explore the similarity between normal homogeneity and $\delta$-homogeneity in Finsler geometry. They are both non-negatively curved Finsler spaces. We show that any connected $\delta$-homogeneous Finsler space is $G$-$\delta$-homogeneous, for some suitably chosen connected quasi-compact $G$. So $\delta$-homogeneous Finsler metrics can be defined by a bi-invariant singular metric on $G$ and submersion, just as normal homogeneous metrics, using a bi-invariant Finsler metric on $G$ instead. More careful analysis shows, in the space of all Finsler metrics on $G/H$, the subset of all $G$-$\delta$-homogeneous ones is in fact the closure for the subset of all $G$-normal ones, in...

  20. A rigidity of equivariant holomorphic maps into a complex Grassmannian induced from orthogonal direct sums of holomorphic line bundles

    Koga, Isami
    In the present paper, we study holomorphic maps induced from orthogonal direct sums of holomorphic line bundles over a compact simply connected homogeneous Kähler manifold into a complex Grassmannian. Then we show if such maps are equivariant, then they are unique up to complex isometry.

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