Mostrando recursos 1 - 20 de 3.040

  1. An index formula for a bundle homomorphism of the tangent bundle into a vector bundle of the same rank, and its applications

    SAJI, Kentaro; UMEHARA, Masaaki; YAMADA, Kotaro
    In a previous work, the authors introduced the notion of ‘coherent tangent bundle’, which is useful for giving a treatment of singularities of smooth maps without ambient spaces. Two different types of Gauss–Bonnet formulas on coherent tangent bundles on $2$-dimensional manifolds were proven, and several applications to surface theory were given. ¶ Let $M^n$ ($n\ge 2$) be an oriented compact $n$-manifold without boundary and $TM^n$ its tangent bundle. Let $\mathcal{E}$ be a vector bundle of rank $n$ over $M^n$, and $\phi:TM^n\to \mathcal{E}$ an oriented vector bundle homomorphism. In this paper, we show that one of these two Gauss–Bonnet formulas can be generalized...

  2. Rotational beta expansion: ergodicity and soficness

    AKIYAMA, Shigeki; CAALIM, Jonathan
    We study a family of piecewise expanding maps on the plane, generated by composition of a rotation and an expansive similitude of expansion constant $\beta$. We give two constants $B_1$ and $B_2$ depending only on the fundamental domain that if $\beta$ > $B_1$ then the expanding map has a unique absolutely continuous invariant probability measure, and if $\beta$ > $B_2$ then it is equivalent to $2$-dimensional Lebesgue measure. Restricting to a rotation generated by $q$-th root of unity $\zeta$ with all parameters in $\mathbb{Q}(\zeta,\beta)$, the map gives rise to a sofic system when $\cos(2\pi/q) \in \mathbb{Q}(\beta)$ and $\beta$ is a...

  3. On stability of Leray's stationary solutions of the Navier–Stokes system in exterior domains

    KOBA, Hajime
    This paper studies the stability of a stationary solution of the Navier–Stokes system in $3$-D exterior domains. The stationary solution is called a Leray's stationary solution if the Dirichlet integral is finite. We apply an energy inequality and maximal $L^p$-in-time regularity for Hilbert space-valued functions to derive the decay rate with respect to time of energy solutions to a perturbed Navier–Stokes system governing a Leray's stationary solution.

  4. The analytic torsion of the finite metric cone over a compact manifold

    HARTMANN, Luiz; SPREAFICO, Mauro
    We give an explicit formula for the $L^2$ analytic torsion of the finite metric cone over an oriented compact connected Riemannian manifold. We provide an interpretation of the different factors appearing in this formula. We prove that the analytic torsion of the cone is the finite part of the limit obtained collapsing one of the boundaries, of the ratio of the analytic torsion of the frustum to a regularising factor. We show that the regularising factor comes from the set of the non square integrable eigenfunctions of the Laplace Beltrami operator on the cone.

  5. Sequentially Cohen–Macaulay Rees algebras

    TANIGUCHI, Naoki; PHUONG, Tran Thi; DUNG, Nguyen Thi; AN, Tran Nguyen
    This paper studies the question of when the Rees algebras associated to arbitrary filtration of ideals are sequentially Cohen–Macaulay. Although this problem has been already investigated by [CGT], their situation is quite a bit of restricted, so we are eager to try the generalization of their results.

  6. The Chabauty and the Thurston topologies on the hyperspace of closed subsets

    MATSUZAKI, Katsuhiko
    For a regularly locally compact topological space $X$ of $\rm T_0$ separation axiom but not necessarily Hausdorff, we consider a map $\sigma$ from $X$ to the hyperspace $C(X)$ of all closed subsets of $X$ by taking the closure of each point of $X$. By providing the Thurston topology for $C(X)$, we see that $\sigma$ is a topological embedding, and by taking the closure of $\sigma(X)$ with respect to the Chabauty topology, we have the Hausdorff compactification $\widehat X$ of $X$. In this paper, we investigate properties of $\widehat X$ and $C(\widehat X)$ equipped with different topologies. In particular, we consider...

  7. On the fundamental groups of non-generic $\mathbb{R}$-join-type curves, II

    EYRAL, Christophe; OKA, Mutsuo
    We study the fundamental groups of (the complements of) plane complex curves defined by equations of the form $f(y)=g(x)$, where $f$ and $g$ are polynomials with real coefficients and real roots (so-called $\mathbb{R}$-join-type curves). For generic (respectively, semi-generic) such polynomials, the groups in question are already considered in [6] (respectively, in [3]). In the present paper, we compute the fundamental groups of $\mathbb{R}$-join-type curves under a simple arithmetic condition on the multiplicities of the roots of $f$ and $g$ without assuming any (semi-)genericity condition.

  8. Surface diffeomorphisms with connected but not path-connected minimal sets containing arcs

    NAKAYAMA, Hiromichi
    In 1955, Gottschalk and Hedlund introduced in their book that Jones constructed a minimal homeomorphism whose minimal set is connectd but not path-connected and contains infinitely many arcs. However the homeomorphism is defined only on this set. In 1991, Walker first constructed a homeomorphism of $S^1\times \mathbf{R}$ with such a minimal set. In this paper, we will show that Walker's homeomorphism cannot be a diffeomorphism (Theorem 2). Furthermore, we will construct a $C^\infty$ diffeomorphism of $S^1\times \mathbf{R}$ with a compact connected but not path-connected minimal set containing arcs (Theorem 1) by using the approximation by conjugation method.

  9. Classification of log del Pezzo surfaces of index three

    FUJITA, Kento; YASUTAKE, Kazunori
    A normal projective non-Gorenstein log-terminal surface $S$ is called a log del Pezzo surface of index three if the three-times of the anti-canonical divisor $-3K_S$ is an ample Cartier divisor. We classify all log del Pezzo surfaces of index three. The technique for the classification is based on the argument of Nakayama.

  10. Joint universality for Lerch zeta-functions

    LEE, Yoonbok; NAKAMURA, Takashi; PAŃKOWSKI, Łukasz
    For $0$ < $\alpha,$ $\lambda \leq 1$, the Lerch zeta-function is defined by $L(s;\alpha, \lambda) := \sum_{n=0}^\infty e^{2\pi i\lambda n} (n+\alpha)^{-s}$, where $\sigma$ > $1$. In this paper, we prove joint universality for Lerch zeta-functions with distinct $\lambda_1,\ldots,\lambda_m$ and transcendental $\alpha$.

  11. $L^{p}$ measure of growth and higher order Hardy–Sobolev–Morrey inequalities on $\Bbb{R}^{N}$

    RABIER, Patrick J.
    When the growth at infinity of a function $u$ on $\Bbb{R}^{N}$ is compared with the growth of $|x|^{s}$ for some $s\in \Bbb{R},$ this comparison is invariably made pointwise. This paper argues that the comparison can also be made in a suitably defined $L^{p}$ sense for every $1\leq p$ < $\infty$ and that, in this perspective, inequalities of Hardy, Sobolev or Morrey type account for the fact that sub $|x|^{-N/p}$ growth of $\nabla u$ in the $L^{p}$ sense implies sub $|x|^{1-N/p}$ growth of $u$ in the $L^{q}$ sense for well chosen values of $q.$ ¶ By investigating how sub $|x|^{s}$ growth of $\nabla...

  12. A construction of diffusion processes associated with sub-Laplacian on CR manifolds and its applications

    KONDO, Hiroki; TANIGUCHI, Setsuo
    A diffusion process associated with the real sub-Laplacian $\Delta_b$, the real part of the complex Kohn–Spencer Laplacian $\square_b$, on a strictly pseudoconvex CR manifold is constructed via the Eells–Elworthy–Malliavin method by taking advantage of the metric connection due to Tanaka and Webster. Using the diffusion process and the Malliavin calculus, the heat kernel and the Dirichlet problem for $\Delta_b$ are studied in a probabilistic manner. Moreover, distributions of stochastic line integrals along the diffusion process will be investigated.

  13. More on 2-chains with 1-shell boundaries in rosy theories

    KIM, SunYoung; LEE, Junguk
    In [4], B. Kim, and the authors classified 2-chains with 1-shell boundaries into either RN (renamable)-type or NR (non renamable)-type 2-chains up to renamability of support of subsummands of a 2-chain and introduced the notion of chain-walk, which was motivated from graph theory: a directed walk in a directed graph is a sequence of edges with compatible condition on initial and terminal vertices between sequential edges. We consider a directed graph whose vertices are 1-simplices whose supports contain $0$ and edges are plus/minus of $2$-simplices whose supports contain $0$. A chain-walk is a 2-chain induced from a directed walk in...

  14. Deformations of Killing spinors on Sasakian and 3-Sasakian manifolds

    van COEVERING, Craig
    We consider some natural infinitesimal Einstein deformations on Sasakian and 3-Sasakian manifolds. Some of these are infinitesimal deformations of Killing spinors and further some integrate to actual Killing spinor deformations. In particular, on 3-Sasakian 7 manifolds these yield infinitesimal Einstein deformations preserving 2, 1, or none of the 3 independent Killing spinors. Toric 3-Sasakian manifolds provide non-trivial examples with integrable deformation preserving precisely 2 Killing spinors. Thus in contrast to the case of parallel spinors the dimension of Killing spinors is not preserved under Einstein deformations but is only upper semi-continuous.

  15. Automorphicity and mean-periodicity

    OLIVER, Thomas
    If $C$ is a smooth projective curve over a number field $k$, then, under fair hypotheses, its $L$-function admits meromorphic continuation and satisfies the anticipated functional equation if and only if a related function is $\mathfrak{X}$-mean-periodic for some appropriate functional space $\mathfrak{X}$. Building on the work of Masatoshi Suzuki for modular elliptic curves, we will explore the dual relationship of this result to the widely believed conjecture that such $L$-functions should be automorphic. More precisely, we will directly show the orthogonality of the matrix coefficients of $GL_{2g}$-automorphic representations to the vector spaces $\mathcal{T}(h(\mathcal{S},\{k_i\},s))$, which are constructed from the Mellin transforms...

  16. Darboux curves on surfaces I

    GARCIA, Ronaldo; LANGEVIN, Rémi; WALCZAK, Paweł
    In 1872, G. Darboux defined a family of curves on surfaces of $\mathbb{R}^3$ which are preserved by the action of the Möbius group and share many properties with geodesics. Here, we characterize these curves under the view point of Lorentz geometry and prove that they are geodesics in a 3-dimensional sub-variety of a quadric $\Lambda^4$ contained in the 5-dimensional Lorentz space $\mathbb{R}^5_1$ naturally associated to the surface. We construct a new conformal object: the Darboux plane-field $\mathcal{D}$ and give a condition depending on the conformal principal curvatures of the surface which guarantees its integrability. We show that $\mathcal{D}$ is integrable...

  17. Equivariant weight filtration for real algebraic varieties with action

    PRIZIAC, Fabien
    We show the existence of a weight filtration on the equivariant homology of real algebraic varieties equipped with a finite group action, by applying group homology to the weight complex of McCrory and Parusiński. If the group is of even order, we can not extract additive invariants directly from the induced spectral sequence. ¶ Nevertheless, we construct finite additive invariants in terms of bounded long exact sequences, recovering Fichou's equivariant virtual Betti numbers in some cases. In the case of the two-elements group, we recover these additive invariants by using globally invariant chains and the equivariant version of Guillén and Navarro...

  18. A uniqueness of periodic maps on surfaces

    HIROSE, Susumu; KASAHARA, Yasushi
    Kulkarni showed that, if $g$ is greater than $3$, a periodic map on an oriented surface $\Sigma_g$ of genus $g$ with order not smaller than $4g$ is uniquely determined by its order, up to conjugation and power. In this paper, we show that, if $g$ is greater than $30$, the same phenomenon happens for periodic maps on the surfaces with orders more than $8g/3$, and, for any integer $N$, there is $g > N$ such that there are periodic maps of $\Sigma_g$ of order $8g/3$ which are not conjugate up to power each other. Moreover, as a byproduct of our...

  19. Realizing compactly generated pseudo-groups of dimension one

    MEIGNIEZ, Gaël
    Many compactly generated pseudo-groups of local transformations on 1-manifolds are realizable as the transverse dynamic of a foliation of codimension 1 on a compact manifold of dimension 3 or 4.

  20. Locally standard torus actions and $h'$-numbers of simplicial posets

    AYZENBERG, Anton
    We consider the orbit type filtration on a manifold with a locally standard torus action and study the corresponding spectral sequence in homology. When all proper faces of the orbit space are acyclic and the free part of the action is trivial, this spectral sequence can be described in full. The ranks of diagonal terms of its second page are equal to $h'$-numbers of a simplicial poset dual to the orbit space. Betti numbers of a manifold with a locally standard torus action are computed: they depend on the combinatorics and topology of the orbit space but not on the...

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