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. Using a Fourier series expansion, an infinite set of simultaneous linear equations is formulated for the periodic evolution of a system/universe. PACS: 05.70.-a, 05.90.+m, 04.20.Jb, 04.20.Gz

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...

in descartes theological writing, he promotes two jointly puzzling theses: t1) god freely creates the eternal truths (i.e. the creation doctrine) and t2) the eternal truths are necessarily true. according to t1 god freely chooses which propositions to make necessary, contingent and possible. however the creation doctrine makes the acceptance of t2 tenuous for the creation doctrine implies that ...

A vertex u of a graph “t-dominates” a vertex v if there are at most t vertices different from u, v that are adjacent to v and not to u; and a graph is “t-dominating” if for every pair of distinct vertices, one of them t-dominates the other. Our main result says that if a graph is t-dominating, then it is close (in an appropriate sense) to being 0-dominating. It was already known that every t-do...

We define and study the following two-player game on a graph G. Let k ∈ N∗. A set of k guards is occupying some vertices of G while one spy is standing at some node. At each turn, first the spy may move along at most s edges, where s ∈ N∗ is his speed. Then, each guard may move along one edge. The spy and the guards may occupy same vertices. The spy has to escape the surveillance of the guards,...

We present the gravitational action and Hamiltonian for a spatially bounded region of an eternal black hole. The Hamiltonian is of the general form H = H+ − H−, where H+ and H− are respectively the Hamiltonian for the regions M+ and M− located in the left and right wedges of the spacetime. We construct explicitly the quasilocal energy for the system and discuss its dependence on the time direct...

Eternal domination is a problem that asks the following question: Can we eternally defend graph ? The principle to against an attacked vertex, changes every turn, by moving guards along edges of graph. In classical version game (Burger et al., 2004), only one guard can move at time, but in m-eternal (Goddard 2005), any number single turn. This led introduction two parameters: eternal and number...

In the Eternal Domination game, a team of guard tokens initially occupies a dominating set on a graph G. A rioter then picks a node without a guard on it and attacks it. The guards defend against the attack: one of them has to move to the attacked node, while each remaining one can choose to move to one of his neighboring nodes. The new guards’ placement must again be dominating. This attack-de...