Then H need not be divisible, i.e., we need not have pH D H . We only know H is contained in pG. Moreover, for x in H , there need not exist an infinite sequence y1;y2;y3:::: such that py2 D y1, py3 D y2; ::. We only know that there exist arbitrarily long finite such sequences. There does exist a unique maximal divisible subgroup D of M , and D is a subgroup of H . Moreover, M DD ̊N where N is a...

The Dirac operators Ly = i 1 0 0 −1 dy dx + v(x)y, y = y1 y2 , x ∈ [0, π], with L 2-potentials v(x) = 0 P (x) Q(x) 0 , P,Q ∈ L 2 ([0, π]), considered on [0, π] with periodic, antiperiodic or Dirichlet boundary conditions (bc), have discrete spectra, and the Riesz projections SN = 1 2πi |z|=N− 1 2 (z − L bc) −1 dz, Pn = 1 2πi |z−n|= 1 2 (z − L bc) −1 dz are well-defined for |n| ≥ N if N is suffi...

This paper is a continuation of [8]. Let A and B be two first order structures of the same vocabulary L. We denote the domains of A and B by A and B respectively. All vocabularies are assumed to be relational. The Ehrenfeucht-Fräıssé-game of length γ of A and B denoted by EFGγ(A,B) is defined as follows: There are two players called ∀ and ∃. First ∀ plays x0 and then ∃ plays y0. After this ∀ pl...

Solution. We suppose that we are given g, g ∈ G, and our task is to find a ∈ Zp. We first check if g is the identity element. If so, we know that a = 0. Otherwise, we set u = g and v = g, where we choose b ∈ Zp randomly (though we make sure that b 6= 0). We then call our hash attacker to find distinct pairs (x1, y1), (x2, y2) ∈ Zp × Zp such that u1 · v1 = u2 · v2 . We can rewrite this as: ux1−x...

A finite image I is a function assigning colors to a finite, rectangular array of discrete pixels. Thus, the information directly encoded by I is purely locational. Such locational information is of little visual use in itself: perception of visual structure requires extraction of relational image information. A very elementary form of relational information about I is provided by its dipole hi...

This is the appendix to Zhu and Wallace [4]. In it, we prove the claims in section 3 of that paper that are made about the model with a non degenerate wealth distribution. We begin by formally setting out the definition of an equilibrium for that model. Let Y = {y = (y1, y2) ∈ Z × Z : y1 + y2 ≤ Z}. An element of Y is an individual portfolio after bond purchases (y1 is the amount of money and y2...

We show that for every 2-coloring of N and every k 2 N, there are infinitely many monochromatic solutions of the system of k2 equations zij = x yj i , 1  i, j  k, where x1, . . . , xk, y1, . . . , yk are distinct positive integers greater than 1. We give similar, but somewhat weaker, results for more than two colors. – Dedicated to the memory of Paul Erdős.

We adopt the following rules: n is a natural number and x1, x2, x3, x4, x5, y1, y2, y3 are sets. Let x1, x2, x3, x4, x5 be sets. We say that x1, x2, x3, x4, x5 are mutually different if and only if: (Def. 1) x1 6= x2 and x1 6= x3 and x1 6= x4 and x1 6= x5 and x2 6= x3 and x2 6= x4 and x2 6= x5 and x3 6= x4 and x3 6= x5 and x4 6= x5. Next we state several propositions: (1) If x1, x2, x3, x4, x5 ...

In game theory and in particular in the theory of stochastic games, we encounter systems of polynomial equalities and inequalities. We start with a few examples. The first example relates to the minmax value and optimal strategies of a two-person zero-sum game with finitely many strategies. Consider a twoperson zero-sum game represented by a k ×m matrix A = (aij), 1 ≤ i ≤ k and 1 ≤ j ≤ m. The n...

In the early eighties, A. Huhn proved that if D, E are finite distributive lattices and ψ : D → E is a {0}-preserving join-embedding, then there are finite lattices K, L and there is a lattice homomorphism φ : K → L such that ConK (the congruence lattice of K) is isomorphic to D, ConL (the congruence lattice of L) is isomorphic to E, and the natural induced mapping extφ : ConK → ConL represents...