On the Coincidence of the Classes of Finite Groups $ E_{\pi_{x}} $ and $ D_{\pi_{x}} $

Let \( \pi_{x} \) be the set of primes greater than \( x \). We prove that for all \( x\in{??} \) the classes of finite groups \( D_{\pi_{x}} \) and \( E_{\pi_{x}} \) coincide; i.e., a finite group \( G \) possesses a \( \pi_{x} \)-Hall subgroup if and only if \( G \) satisfies the complete analog of the Sylow Theorems for a \( \pi_{x} \)-subgroup.

     

Ramanujan''s $\,_1\psi_1$ 公式的新证明

In this paper, we give a short proof of the celebrated Ramanujan＇s1ψ1 Formula.     

On the deleted product criterion for embeddability of manifolds in $ \Bbb {R}^m $

Commentarii Mathematici Helvetici - For a space N let $ \tilde N = \{(x,y) \in N \times N|x \neq y \} $ . Let $\Bbb {Z}_2 $ act on Ñ and on $ S^{m-1} $ by exchanging factors and antipodes,...     

L-Factor of Irreducible x1×x2(×)σ

The L-factor of irreducible x1×x2(×)σ defined by Piatetski-Shapiro is computed by using non-split Bessel functional.     

赋$\beta$-范空间中的最佳逼近问题

本文讨论赋$\beta$-范空间中的最佳逼近问题.以[1]引进的共轭锥为工具,借助[2]中关于$\beta$-次半范的Hahn-Banach延拓定理,第二节给出赋$\beta$-范空间的闭子空间中最佳逼近元的特征,第三节得到赋$\beta$-范空间中任何凸子集或子空间均为半Chebyshev集的充要条件是空间本身严格凸,文章最后证明了严格凸的赋$\beta$-范空间中任何有限维子空间都是Chebyshev集.     

Generalized ($\kappa,\mu $)-contact metric manifolds with $\|{grad}\kappa \| =$ constant

We locally classify the 3-dimensional generalized ($\kappa,\mu$)-contact metric manifolds, which satisfy the condition $\Vert grad\kappa \Vert =$const. ($\not=0$). This class of manifolds is determined by two arbitrary functions of one variable.     

Higher generation subgroup sets and the $ \Sigma $-invariants of graph groups

We present a general condition, based on the idea of n-generating subgroup sets, which implies that a given character represents a point in the homotopical or homological -invariants of the group G. Let be a finite simplicial graph, the flag complex induced by , and the graph group, or 'right angled Artin group', defined by . We use our result on n-generating subgroup sets to describe the homotopical and homological -invariants of in terms of the topology of subcomplexes of . In particular, this work determines the finiteness properties of kernels of maps from graph groups to abelian groups. This is the first complete computation of the -invariants for a family of groups whose higher invariants are not determined - either implicitly or explicitly - by ¹. Received: October 18, 1996     

Hierarchy of graph matchbox manifolds

We study a class of graph foliated spaces, or graph matchbox manifolds, initially constructed by Kenyon and Ghys. For graph foliated spaces we introduce a quantifier of dynamical complexity which we call its level. We develop the fusion construction, which allows us to associate to every two graph foliated spaces a third one which contains the former two in its closure. Although the underlying idea of the fusion is simple, it gives us a powerful tool to study graph foliated spaces. Using fusion, we prove that there is a hierarchy of graph foliated spaces at infinite levels. We also construct examples of graph foliated spaces with various dynamical and geometric properties.     

Profinite properties of graph manifolds

Let M be a closed, orientable, irreducible, geometrizable 3-manifold. We prove that the profinite topology on the fundamental group of π ₁(M) is efficient with respect to the JSJ decomposition of M. We go on to prove that π ₁(M) is good, in the sense of Serre, if all the pieces of the JSJ decomposition are. We also prove that if M is a graph manifold then π ₁(M) is conjugacy separable.     

Closed graph and open mapping theorems for topological $ \tilde \lc $‐modules and applications

We present closed graph and open mapping theorems for ‐linear maps acting between suitable classes of topological and locally convex topological ‐modules. This is done by adaptation of De Wilde's theory of webbed spaces and Adasch–Ernst–Keim's theory of barrelled spaces to the context of locally convex and topological ‐modules, respectively. We give applications of the previous theorems to Colombeau theory as well to the theory of Banach ‐modules. In particular we obtain a necessary condition for ??^∞‐hypoellipticity on the symbol of a partial differential operator with generalized constant coefficients (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Non-separable surfaces in cubed manifolds

We show that there are 3-manifolds with cubings of non-positive curvature such that their fundamental groups are not subgroup separable (LERF). We also give explicit examples of non-separable surfaces in certain cubed manifolds.

     

Constant angle surfaces in $ {\Bbb S}^2\times {\Bbb R} $

In this article we study surfaces in for which the unit normal makes a constant angle with the -direction. We give a complete classification for surfaces satisfying this simple geometric condition.     

Almost $ \omega $ -Categorical Weakly $ o $ -Minimal Theories of Convexity Rank 1

Siberian Mathematical Journal - Studying the properties of almost $ \omega $ -categorical weakly $ o $ -minimal theories of convexity rank 1, we prove the orthogonality of every family of...     

Pfaff $\tau$-functions

Consider the two-dimensional Toda lattice, with certain skew-symmetric initial condition, which is preserved along the locus of the space of time variables. Restricting the solution to , we obtain another hierarchy called Pfaff lattice, which has its own tau function, being equal to the square root of the restriction of 2D-Toda tau function. We study its bilinear and Fay identities, W and Virasoro symmetries, relation to symmetric and symplectic matrix integrals and quasiperiodic solutions. Received: 20 September 1999 / Published online: 1 February 2002     

关于同余式$\sigma(n)\equiv 1\pmod n$

Let k ≥ 2 be an integer, and let σ(n) denote the sum of the positive divisors of an integer n. We call n a quasi-multiperfect number if σ(n) = kn + 1. In this paper, we give some necessary properties of them.