Viscous Matter in FRW Cosmology

We investigate the dynamics of dust matter with bulk viscosity effects. We explored the analogy dynamical problem to Chaplygin gas. Due to this analogy we give exact solutions for the FRW cosmology with viscosity coefficient parameterized by the Belinskii–Khalatnikov power law dependence with respect to energy density. These exact solutions are given in the form of hypergeometrical functions. We proved simple theorem which illustrated as viscosity effects can solved the initial singularity problem present in standard cosmological model.

A Note on the Topological Transversality Theorem for Weakly Upper Semicontinuous, Weakly Compact Maps on Locally Convex Topological Vector Spaces

A simple theorem is presented that automatically generates the topological transversality theorem and Leray–Schauder alternatives for weakly upper semicontinuous, weakly compact maps. An application is given to illustrate our results.

FACTORS OF CARMICHAEL NUMBERS AND AN EVEN WEAKER -TUPLES CONJECTURE

One of the open questions in the study of Carmichael numbers is whether, for a given $R\geq 3$ , there exist infinitely many Carmichael numbers with exactly $R$ prime factors. Chernick [‘On Fermat’s simple theorem’, Bull. Amer. Math. Soc. 45 (1935), 269–274] proved that Dickson’s $k$ -tuple conjecture would imply a positive result for all such $R$ . Wright [‘Factors of Carmichael numbers and a weak $k$ -tuples conjecture’, J. Aust. Math. Soc. 100(3) (2016), 421–429] showed that a weakened version of Dickson’s conjecture would imply that there are an infinitude of $R$ for which there are infinitely many such Carmichael numbers. In this paper, we improve on our 2016 result by weakening the required conjecture even further.

A remark on positive topological entropy ofN-buffer switched flow networks

We present a simpler elementary proof on positive topological entropy of theN-buffer switched flow networks based on a new simple theorem on positive topological entropy of continuous map on compact metric space.

NONEXISTENCE THEOREMS FOR TRAVERSABLE WORMHOLES

Gauss–Bonnet formula is used to derive a new and simple theorem of nonexistence of vacuum static nonsingular Lorentzian wormholes. We also derive simple proofs for the nonexistence of Lorentzian wormhole solutions for some classes of static matter such as, for instance, real scalar fields with a generic potential obeying ϕV′(ϕ)≥0 and massless fermion fields.