We show that the abc conjecture implies that the number of terms of any arithmetic progression consisting of almost perfect ”inhomogeneous” powers is bounded, moreover, if the exponents of the powers are all ≥ 4, then the number of such progressions is finite. We derive a similar statement unconditionally, provided that the exponents of the terms in the progression are bounded from above.

I argue that the leading quantum corrections, in powers of the energy or inverse powers of the distance, may be computed in quantum gravity through knowledge of only the low energy structure of the theory. As an example, I calculate the leading quantum corrections to the Newtonian gravitational potential.

We study a result of Postnikov-Shapiro concerning the Hilbert series of a family of ideals Jφ generated by powers of linear forms in k[x1, . . . , xn]. Using the results of Emsalem-Iarrobino, we formulate this as a problem about fatpoints in P. In the three variable case this is equivalent to studying the dimension of a linear series on a blow up of P. The ideals that arise have the points in v...

The explicit formulas for the sums of positive powers of the integers s i unrepresentable by the triple of integers d 1 , d 2 , d 3 ∈ N, gcd(d 1 , d 2 , d 3) = 1, are derived. For short we denote the tuple (d 1 ,. .. , d m) by d m and consider the generating function Φ (d m ; z)

We obtain estimates for Vinogradov’s integral which for the first time approach those conjectured to be the best possible. Several applications of these new bounds are provided. In particular, the conjectured asymptotic formula in Waring’s problem holds for sums of s kth powers of natural numbers whenever s > 2k + 2k − 3.

We introduce the notion of the P -sequences and apply their properties in studying representability of real numbers. Another application of P -sequences we find in generating the Prouhet–Tarry–Escott pairs.

A "quantum" field-theoretic formulation of the dynamics of the contact process on a regular graph of degree z is introduced. A perturbative calculation in powers of 1/z of the effective potential for the density of particles phi(t) and an instantonic field psi(t) emerging from the formalism is performed. Corrections to the mean-field distribution of densities of particles in the out-of-equilibr...

Parliamentary and Presidential Democracies: Which One is Best? With the exception of the United States, where a system of separation of executive and legislative powers exists, all countries that are considered to be stable democracies adopt a constitution that is, at least partially, parliamentary. A parliamentary constitution is characterized by the fusion of executive and legislative powers,...