We consider static spherically symmetric solutions of the Einstein equations with cosmological constant Λ coupled to the SU(2) Yang Mills equations that are smooth at the origin r = 0. We prove that all such solutions have a radius rc at which there is a horizon. However, for any positive integer N , there exists a small positive ΛN , such that whenever 0 < Λ < ΛN , there exist at least N disti...

For a triple (n, t,m) of positive integers, we attach to each t-subset S = {a1, . . . , at} ⊆ {1, . . . , n} the sum f(S) = a1 + · · · + at (modulo m). We ask: for which triples (n, t,m) are the ( n t ) values of f(S) uniformly distributed in the residue classes mod m? The obvious necessary condition, that m divides ( n t ) , is not sufficient, but a q-analogue of that condition is both necessa...

We characterize the structure of maximum-size sum-free subsets of a random subset of an Abelian group G. In particular, we determine the threshold above which, with high probability as |G| → ∞, each such subset is contained in some maximum-size sum-free subset of G, whenever q divides |G| for some (fixed) prime q with q ≡ 2 (mod 3). Moreover, in the special case G = Z2n, we determine the sharp ...

Let K be a perfect field of characteristic 2. In this paper, we classify all hyperplanes of the symplectic dual polar space DW (5,K) that arise from its Grassmann embedding. We show that the number of isomorphism classes of such hyperplanes is equal to 5 + N , where N is the number of equivalence classes of the following equivalence relation R on the set {λ ∈ K |X2 + λX + 1 is irreducible in K[...

Wendt's binomial circulant determinant, W„ , is the determinant of an m by m circulant matrix of integers, with {i, ;')th entry (i,TM.i) whenever 2 divides m but 3 does not. We explain how we found the prime factors of Wm for each even m < 200 by implementing a new method for computations in algebraic number fields that uses only modular arithmetic. As a consequence we prove that if p and q = m...

Rank-width of a graph G, denoted by rw(G), is a width parameter of graphs introduced by Oum and Seymour (2006). We investigate the asymptotic behavior of rank-width of a random graph G(n, p). We show that, asymptotically almost surely, (i) if p ∈ (0, 1) is a constant, then rw(G(n, p)) = dn3 e −O(1), (ii) if 1 n p ≤ 1 2 , then rw(G(n, p)) = d n 3 e − o(n), (iii) if p = c/n and c > 1, then rw(G(n...

In this note, we use a natural desingularization of the conormal variety of the variety of (n × n)-symmetric matrices of rank at most r to find a general formula for the algebraic degree in semidefinite programming. 1. The algebraic degree in semidefinite programming Let P be a general projective space of symmetric (n×n)−matrices up to scalar multiples, and let Yr ⊂ P m be the subvariety of mat...

We give a characterization of the weights u(·) and v(·) for which the fractional maximal operator Ms is bounded from the weighted Lebesgue spaces Lp(lr, vdx) into Lq(lr, udx) whenever 0 ≤ s < n, 1 < p, r < ∞, and 1 ≤ q < ∞.

Let K be a perfect field of characteristic 2. In this paper, we classify all hyperplanes of the symplectic dual polar space DW(5,K) that arise from its Grassmann embedding. We show that the number of isomorphism classes of such hyperplanes is equal to 5+N , where N is the number of equivalence classes of the following equivalence relation R on the set {λ ∈ K |X2 + λX + 1 is irreducible in K[X]}...