We sketch a proof of the fact that the Gromov boundary of a hyperbolic group does not have a global cut point if it is connected. This implies, by a theorem of Bestvina and Mess, that the boundary is locally connected if it is connected. The object of this note is to sketch a proof of the following theorem: Theorem 1. If ? is a one ended hyperbolic group, then the Gromov boundary @? of ? does n...

Keywords: Upper and lower solutions p-Laplacian operator Fractional differential equation Integral boundary condition Eigenvalue a b s t r a c t In this paper, we are concerned with the eigenvalue problem of a class of singular p-Lapla-cian fractional differential equations involving the Riemann–Stieltjes integral boundary condition. The conditions for the existence of at least one positive sol...

Let Σg,r be a compact oriented surface of genus g with r boundary components. We determine the abelianization of the symmetric mapping class group M̂(g,r)(p2) of a double unbranched cover p2 : Σ2g−1,2r → Σg,r using the Riemann constant, Schottky theta constant, and the theta multiplier. We also give lower bounds of the abelianizations of some finite index subgroups of the mapping class group.

The purpose of this paper is to study the higher order asymptotic distributions of the eigenvalues associated with a class of Sturm-Liouville problem with equation of the form w??=(?2f(x)?R(x)) (1), on [a,b, where ? is a real parameter and f(x) is a real valued function in C2(a,b which has a single zero (so called turning point) at point 0x=x and R(x) is a continuously differentiable function. ...

We prove tight lower bounds for the coefficients of the generalized h-vector of a rational polytope with a symmetry of prime order that is fixed–point–free on the boundary. These bounds generalize results of R. Stanley and R. Adin for the h–vector of a simplicial rational polytope with a central symmetry or a symmetry of prime order respectively.

We are concerned with the higher-order nonlinear three-point boundary value problems: x n f t, x, x′, . . . , x n−1 , n ≥ 3, with the three point boundary conditions g x a , x′ a , . . . , x n−1 a 0; x i b μi, i 0, 1, . . . , n − 3;h x c , x′ c , . . . , x n−1 c 0, where a < b < c, f : a, c × R → R −∞, ∞ is continuous, g, h : R → R are continuous, andμi ∈ R, i 0, 1, . . . , n − 3 are arbitrary ...