Applications of shuffle product to restricted decomposition formulas for multiple zeta values

In this paper we obtain a recursive formula for the shuffle product and apply it to derive two restricted decomposition formulas for multiple zeta values (MZVs). The first formula generalizes the decomposition formula of Euler and is similar to the restricted formula of Eie and Wei for MZVs with one strings of 1's. The second formula generalizes the previous results to the product of two MZVs with one and two strings of 1's respectively.Comment: 11 page

Curve shortening flow on Riemann surfaces with possible ambient conic singularities

In this paper, we study the curve shortening flow (CSF) on Riemann surfaces. We generalize Huisken's comparison function to Riemann surfaces and surfaces with conic singularities. We reprove the Gage-Hamilton-Grayson theorem on surfaces. We also prove that for embedded simple closed curves, CSF can not touch conic singularities with cone angles $\leq \pi$.Comment: 28 pages, 4 figure

Five-dimensional metric f(R)f(R) gravity and the accelerated universe

The metric $f(R)$ theories of gravity are generalized to five-dimensional spacetimes. By assuming a hypersurface-orthogonal Killing vector field representing the compact fifth dimension, the five-dimensional theories are reduced to their four-dimensional formalism. Then we study the cosmology of a special class of $f(R)=\alpha R^m$ models in a spatially flat FRW spacetime. It is shown that the parameter $m$ can be constrained to a certain range by the current observed deceleration parameter, and its lower bound corresponds to the Kaluza-Klein theory. It turns out that both expansion and contraction of the extra dimension may prescribe the smooth transition from the deceleration era to the acceleration era in the recent past as well as an accelerated scenario for the present universe. Hence five-dimensional $f(R)$ gravity can naturally account for the present accelerated expansion of the universe. Moreover, the models predict a transition from acceleration to deceleration in the future, followed by a cosmic recollapse within finite time. This differs from the prediction of the five-dimensional Brans-Dicke theory but is in consistent with a recent prediction based on loop quantum cosmology.Comment: 14 pages, 9 figures; Version published in PR