The issue of the ”eternal return” is examined from the perspective of the topology of spacetime. Constraints on dynamical laws for the periodic evolution of a system or universe are highlighted. In a periodic universe with period T , using a Fourier series expansion for any physical variable A(t) and its time derivatives A(t), an infinite set of simultaneous linear equations is formulated. PACS...

We construct an eternally inflating spacelike brane world model. This model is feasible only if the space dimension of the brane is three (SM2) or six (SM5) for M theory or four (SD3) for superstring theory. A time-dependent n-form field would supply a constant energy density and cause eternal expansion of the spacelike brane. On the other hand, the hyperbolic space perpendicular to the brane w...

A new subdivision method for computing the nearest univariate gcd is described and analyzed. It is based on an exclusion test and an inclusion test. The exclusion test in a cell exploits Taylor expansion of the polynomial at the center of the cell. The inclusion test uses Smale’s α-theorems to certify the existence and unicity of a solution in a cell. Under the condition of simple roots for the...

We introduce the game of Cops and Eternal Robbers played on graphs, where there are infinitely many robbers that appear sequentially over distinct plays game. A positive integer t is fixed, cops required to capture robber in at most time-steps each play. The associated optimization parameter eternal cop number, denoted by ct∞, which equals domination number case t=1, for sufficiently large t. s...