Geometric properties of a certain class of compact dynamical horizons in locally rotationally symmetric class II spacetimes

In this paper, we study the geometry of a certain class of compact dynamical horizons with a time-dependent induced metric in locally rotationally symmetric class II spacetimes. We first obtain a compactness condition for embedded [Formula: see text]-manifolds in these spacetimes, satisfying the weak energy condition, with non-negative isotropic pressure [Formula: see text]. General conditions for a [Formula: see text]-manifold to be a dynamical horizon are imposed, as well as certain genericity conditions, which in the case of locally rotationally symmetric class II spacetimes reduces to the statement that “the weak energy condition is strictly satisfied or otherwise violated”. The compactness condition is presented as a spatial first-order partial differential equation in the sheet expansion [Formula: see text], in the form [Formula: see text], where [Formula: see text] is the Gaussian curvature of [Formula: see text]-surfaces in the spacetime and [Formula: see text] is a real number parametrizing the differential equation, where [Formula: see text] can take on only two values, [Formula: see text] and [Formula: see text]. Using geometric arguments, it is shown that the case [Formula: see text] can be ruled out and the [Formula: see text] ([Formula: see text]-dimensional sphere) geometry of compact dynamical horizons for the case [Formula: see text] is established. Finally, an invariant characterization of this class of compact dynamical horizons is also presented.

Coupled fixed points and α-dense curves

We present a new iterative method, based on the so called α-dense curves, to approximate coupled fixed points of nonexpansive mappings. Compactness condition on the mapping or its domain of definition is necessary. As application, we construct a sequence which converges to a solution of certain system of integral equations of Volterra type.

Controllability of Fractional Neutral Stochastic Integro-Differential Systems with Infinite Delay

This paper is concerned with the controllability of a class of fractional neutral stochastic integro-differential systems with infinite delay in an abstract space. By employing fractional calculus and Sadovskii's fixed point principle without assuming severe compactness condition on the semigroup, a set of sufficient conditions are derived for achieving the controllability result.

Estimate of Number of Periodic Solutions of Second-Order Asymptotically Linear Difference System

We investigate the number of periodic solutions of second-order asymptotically linear difference system. The main tools are Morse theory and twist number, and the discussion in this paper is divided into three cases. As the system is resonant at infinity, we use perturbation method to study the compactness condition of functional. We obtain some new results concerning the lower bounds of the nonconstant periodic solutions for discrete system.

A general strong Nyman-Beurling criterion for the Riemann hypothesis

For each [FORMULA] formally consider its Miintz transform [FORMULA]. For certain ?'s with both [FORMULA] it is true that the Riemann hypothesis holds if and only if ? is in the L2 closure of the vector space generated by the dilations [FORMULA]. Such is the case for example when ? = X(0,1) where the above statement reduces to the strong Nyman criterion already established by the author. In this note we show that the necessity implication holds for any continuously differentiable function ? vanishing at infinity and satisfying [FORMULA]. If in addition ? is of compact support, then the sufficiency implication also holds true. It would be convenient to remove this compactness condition .