What is a least integer upper bound on van der Waerden number W (r, k) among the powers of the integer r? We show how this can be found by expanding the integer W (r, k) into powers of r. Doing this enables us to find both a least upper bound and a greatest lower bound on W (r, k) that are some powers of r and where the greatest lower bound is equal to or smaller than W (r, k). A finite series ...

the idea of separation of powers has been recognized as a model which was first developed and favored by thomas hobbes, john locke, montesquieu and rousseau and was progressively adopted by political systems and constitutional law of various countries. few other countries can be found today that their structure – although seemingly – are not organized on the basis of separation of powers. this ...

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 590 R I. Multidistrict Consolidation and the Powers of the Transferee Court . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 593 R A. The MDL Statute . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 594 R B. Multidistrict Consolidation and the Declin...

Let T be a T-set, i.e., finite set of nonnegative integers satisfying 0 ? T, and G graph. In the paper we study relations between T-edge spans espT (G) and espd?T (G), where d is positive integer and d ? = {0 ? t (max + 1): |t ? t/d T} . We show that espd?T (G) espT (G) ? r, r 1, an integer depends on G. Next focus case {0} and show that espd?{0} (G) ?d(?c(G) 1)?, where ?c(G) circular chromatic...

This paper is to discuss powers of class wF (p, r, q) operators for 1 ≥ p > 0, 1 ≥ r > 0 and q ≥ 1; and an example is given on powers of class wF (p, r, q) operators.

Synchronization of coupled oscillators on a d-dimensional lattice with the power-law coupling G(r) = g0/rα and randomly distributed intrinsic frequency is analyzed. A systematic perturbation theory is developed to calculate the order parameter profile and correlation functions in powers of ϵ = α/d-1. For α ≤ d, the system exhibits a sharp synchronization transition as described by the conventio...

The vacuum entanglement entropy in quantum field theory provides nonperturbative information about renormalization group flows. Most studies so far have focused on the universal terms, related to Weyl anomaly even space-time dimensions, and sphere free energy $F$ odd dimensions. In this work we study a of radius $R$ large limit, for theories with gravity duals. At admits geometric expansion pow...

1 I n t r o d u c t i o n Powers of graphs play an impor t an t role for solving cer tain problems re la ted to dis tances in graphs : p -cen te r and q-dispers ion (el. [7, 3]), k d o m i n a t i o n and k -s tab i l i ty (of. [8, 3]), d iamete r (of. [13]), k -colour ing (cf. [26, 20]) and approx ima t ion of bandwid th (el. [27]). For instance, consider the k-co lour ing problem. The vertice...

We consider the Friedmann-Robertson-Walker cosmologies of theories of gravity that generalise the Einstein-Hilbert action by replacing the Ricci scalar, R, with some function, f(R). The general asymptotic behaviour of these cosmologies is found, at both early and late times, and the effects of adding higher and lower powers of R to the Einstein-Hilbert action is investigated. The assumption tha...