Mostrando recursos 1 - 20 de 3.027

  1. Midpoints for Thompson's metric on symmetric cones

    Lemmens, Bas; Roelands, Mark
    We characterise the affine span of the midpoints sets, $\mathcal{M}(x,y)$, for Thompson's metric on symmetric cones in terms of a translation of the zero-component of the Peirce decomposition of an idempotent. As a consequence we derive an explicit formula for the dimension of the affine span of $\mathcal{M}(x,y)$ in case the associated Euclidean Jordan algebra is simple. In particular, we find for $A$ and $B$ in the cone positive definite Hermitian matrices that \[ \dim({\rm aff}\, \mathcal{M}(A,B))=q^2, \] where $q$ is the number of eigenvalues $\mu$ of $A^{-1}B$, counting multiplicities, such that \[ \mu\neq \max\{\lambda_+(A^{-1}B),\lambda_-(A^{-1}B)^{-1}\}, \] where $\lambda_+(A^{-1}B):=\max \{\lambda\colon \lambda\in\sigma(A^{-1}B)\}$...

  2. L'anneau de cohomologie des variétés de Seifert non-orientables

    Bauval, Anne; Hayat, Claude
    If $p$ is a prime number, the cohomology ring with coefficients in $\mathbb{Z}/p\mathbb{Z}$ of an orientable or non-orientable Seifert manifold $M$ is obtained using a $\Delta$-simplicial decomposition of $M$. Several choices must be made before applying the Alexander-Whitney formula. The answers are given in terms of the classical cellular generators.

  3. Quadratic approximation in $\mathbb{F}_q(\!(T^{-1})\!)$

    Ooto, Tomohiro
    In this paper, we study Diophantine exponents $w_n$ and $w_n ^{*}$ for Laurent series over a finite field. Especially, we deal with the case $n=2$, that is, quadratic approximation. We first show that the range of the function $w_2-w_2 ^{*}$ is exactly the closed interval $[0,1]$. Next, we estimate an upper bound of the exponent $w_2$ of continued fractions with low complexity partial quotients.

  4. Complex structures and non-degenerate closed 2-forms of compact real parallelizable nilmanifolds

    Yamada, Takumi
    In this paper, we consider a relation of non-degenerate closed $2$-forms and complex structures on compact real parallelizable nilmanifolds.

  5. Rigidity of manifolds with boundary under a lower Ricci curvature bound

    Sakurai, Yohei
    We study Riemannian manifolds with boundary under a lower Ricci curvature bound, and a lower mean curvature bound for the boundary. We prove a volume comparison theorem of Bishop-Gromov type concerning the volumes of the metric neighborhoods of the boundaries. We conclude several rigidity theorems. As one of them, we obtain a volume growth rigidity theorem. We also show a splitting theorem of Cheeger-Gromoll type under the assumption of the existence of a single ray.

  6. Willmore-like functionals for surfaces in 3-dimensional Thurston geometries

    Berdinsky, Dmitry; Vyatkin, Yuri
    We find analogues of the Willmore functional for each of the Thurston geometries with $4$--dimensional isometry group such that the CMC--spheres in these geometries are critical points of these functionals.

  7. Polylogarithmic analogue of the Coleman-Ihara formula, I

    Nakamura, Hiroaki; Sakugawa, Kenji; Wojtkowiak, Zdzisław
    The Coleman-Ihara formula expresses Soule's $p$-adic characters restricted to $p$-local Galois group as the Coates-Wiles homomorphism multiplied by $p$-adic $L$-values at positive integers. In this paper, we show an analogous formula that $\ell$-adic polylogarithmic characters for $\ell=p$ restrict to the Coates-Wiles homomorphism multiplied by Coleman's $p$-adic polylogarithms at any roots of unity of order prime to $p$.

  8. Effects of randomization on asymptotic periodicity of nonsingular transformations

    Ishitani, Hiroshi; Ishitani, Kensuke
    It is known that the Perron--Frobenius operators of piecewise expanding $\mathcal{C}^2$ transformations possess an asymptotic periodicity of densities. On the other hand, external noise or measurement errors are unavoidable in practical systems; therefore, all realistic mathematical models should be regarded as random iterations of transformations. This paper aims to discuss the effects of randomization on the asymptotic periodicity of densities.

  9. The normal holonomy of $CR$-submanifolds

    Di Scala, Antonio J.; Vittone, Francisco
    We study the normal holonomy group, i.e. the holonomy group of the normal connection, of a $CR$-submanifold of a complex space form. We show that the normal holonomy group of a coisotropic submanifold acts as the holonomy representation of a Riemannian symmetric space. In case of a totally real submanifold we give two results about reduction of codimension. We describe explicitly the action of the normal holonomy in the case in which the totally real submanifold is contained in a totally real totally geodesic submanifold. In such a case we prove the compactness of the normal holonomy group.

  10. Salem numbers and automorphisms of abelian surfaces

    Reschke, Paul
    We classify two-dimensional complex tori admitting automorphisms with positive entropy in terms of the entropies they exhibit. For each possible positive value of entropy, we describe the set of two-dimensional complex tori admitting automorphisms with that entropy.

  11. The logarithms of Dehn twists on non-orientable surfaces

    Tsuji, Shunsuke
    We introduce a Lie algebra associated with a non-orientable surface, which is an analogue for the Goldman Lie algebra of an oriented surface. As an application, we deduce an explicit formula of the Dehn twist along an annulus simple closed curve on the surface as in Kawazumi--Kuno [4], [5] and Massuyeau--Turaev [7].

  12. Integrals on $p$-adic upper half planes and Hida families over totally real fields

    Ishikawa, Isao
    Bertolini--Darmon and Mok proved a formula of the second derivative of the two-variable $p$-adic $L$-function of a modular elliptic curve over a totally real field along the Hida family in terms of the image of a global point by some $p$-adic logarithm map. The theory of $p$-adic indefinite integrals and $p$-adic multiplicative integrals on $p$-adic upper half planes plays an important role in their work. In this paper, we generalize these integrals for $p$-adic measures which are not necessarily $\mathbb{Z}$-valued, and prove a formula of the second derivative of the two-variable $p$-adic $L$-function of an abelian variety of $\mathrm{GL}(2)$-type associated to a Hilbert modular form of weight 2.

  13. On certain 2-extensions of $\mathbb{Q}$ unramified at 2 and $\infty$

    Mizusawa, Yasushi
    Based on the method of Boston and Leedham-Green et al. for computing the Galois groups of tamely ramified $p$-extensions of number fields, this paper gives a large family of triples of odd prime numbers such that the maximal totally real $2$-extension of the rationals unramified outside the three prime numbers has the Galois group of order $512$ and derived length $3$. This family is characterized arithmetically, and the explicit presentation of the Galois group by generators and relations is also determined completely.

  14. Some exotic actions of finite groups on smooth 4-manifolds

    Sung, Chanyoung
    Using $G$-monopole invariants, we produce infinitely many exotic non-free actions of $\mathbb{Z}_{k}\oplus H$ on some connected sums of finite number of $S^{2}\times S^{2}$, $\mathbb{C}P_{2}$, $\overline{\mathbb{C}P}_{2}$, and $K3$ surfaces, where $k\geq 2$, and $H$ is any nontrivial finite group acting freely on $S^{3}$.

  15. Conjugacy class and discreteness in $\mathit{SL}(2, \mathbb{C})$

    Yang, Shihai; Zhao, Tiehong
    In this note we establish a new discreteness criterion for a non-elementary group $G$ in $\mathit{SL}(2, \mathbb{C})$. Namely, $G$ is discrete if all the two-generator subgroups are discrete, where one generator is a non-trivial element $f$ in $G$, and the other is in the conjugacy class of $f$.

  16. Some families of minimal elements for a partial ordering on prime knots

    Nagasato, Fumikazu; Tran, Anh T.
    We show that all twist knots and certain double twist knots are minimal elements for a partial ordering on the set of prime knots. The keys to these results are presentations of their character varieties using Chebyshev polynomials and a criterion for irreducibility of a polynomial of two variables. These give us an elementary method to discuss the number of irreducible components of the character varieties, which concludes the result essentially.

  17. Lie ideal enhancements of counting invariants

    Grindstaff, Gillian Roxanne; Nelson, Sam
    We define enhancements of the quandle counting invariant for knots and links with a finite labeling quandle $Q$ embedded in the quandle of units of a Lie algebra $\mathfrak{a}$ using Lie ideals. We provide examples demonstrating that the enhancement is stronger than the associated unenhanced counting invariant and image enhancement invariant.

  18. One-fixed-point actions on spheres and Smith sets

    Morimoto, Masaharu
    Let $G$ be a finite group. The Smith equivalence for real $G$-modules of finite dimension gives a subset of real representation ring, called the primary Smith set. Since the primary Smith set is not additively closed in general, it is an interesting problem to find a subset which is additively closed in the real representation ring and occupies a large portion of the primary Smith set. In this paper we introduce an additively closed subset of the primary Smith set by means of smooth one-fixed-point $G$-actions on spheres, and we give evidences that the subset occupies a large portion of the primary Smith set if $G$ is an Oliver group.

  19. Notes on quadratic integers and real quadratic number fields

    Park, Jeongho
    It is shown that when a real quadratic integer $\xi$ of fixed norm $\mu$ is considered, the fundamental unit $\varepsilon_{d}$ of the field $\mathbb{Q}(\xi) = \mathbb{Q}(\sqrt{d})$ satisfies $\log \varepsilon_{d} \gg (\log d)^{2}$ almost always. An easy construction of a more general set containing all the radicands $d$ of such fields is given via quadratic sequences, and the efficiency of this substitution is estimated explicitly. When $\mu = -1$, the construction gives all $d$'s for which the negative Pell's equation $X^{2} - d Y^{2} = -1$ (or more generally $X^{2} - D Y^{2} = -4$) is soluble. When $\mu$ is a prime, it gives all of the real quadratic fields in...

  20. The homotopy fixed point sets of spheres actions on rational complexes

    Hao, Yanlong; Liu, Xiugui; Sun, Qianwen
    In this paper, we describe the homotopy type of the homotopy fixed point sets of $S^{3}$-actions on rational spheres and complex projective spaces, and provide some properties of $S^{1}$-actions on a general rational complex.

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