Let p be a prime, q a prime divisor in p−1, and g an element of order q in Z∗ p . Suppose a prover P has chosen w in Zq at random and has published h = gw mod p. A verifier V who gets p, q, g, h can check that p, q are prime, and that g, h have order q. Since there is only one subgroup of order q in Z∗ p , this automatically means that h ∈< g >, i.e. there exists w such that h = gw. But this do...

Inspired by Fagin’s result that NP = Σ 1 , we have developed a partial framework to investigate expressibility inside Σ 1 so as to have a finer look into NP . The framework uses interesting combinatorics derived from second-order EhrenfeuchtFräıssé games and the notion of game types. Some of the results that have been proven within this framework are: (1) for any k, divisibility by k is not exp...

We study axially symmetric static solitons of O(3) nonlinear σ model coupled to (2+1)-dimensional anti-de Sitter gravity. The obtained solutions are not self-dual under static metric. The usual regular topological lump solution cannot form a black hole even though the scale of symmetry breaking is increased. There exist nontopological solitons of half integral winding in a given model, and the ...

In the noncommutative geometry of Artin, Van den Bergh, and others, the twisted homogeneous coordinate ring is one of the basic constructions. Such a ring is defined by a σ-ample divisor, where σ is an automorphism of a projective scheme X. Many open questions regarding σ-ample divisors have remained. We derive a relatively simple necessary and sufficient condition for a divisor on X to be σ-am...

All compact AC(σ) operators have a representation analogous to that for compact normal operators. As a partial converse we obtain conditions which allow one to construct a large number of such operators. Using the results in the paper, we answer a number of questions about the decomposition of a compact AC(σ) into real and imaginary parts.

Some aspects and applications of σ-models in particle and condensed matter physics are briefly reviewed. ∗Contribution to Salamfest, 8 -12 March 1993, Trieste 1

To understand nonabelian Fourier analysis, we first need to introduce some notions from group representation theory. For further information on this subject, a good basic reference is the book Linear Representations of Finite Groups by Serre. A linear representation (or simply representation) of a group G over the vector space Cn is a homomorphism σ : G → GL(Cn), i.e., a map from group elements...

1 CES Utility In many economic textbooks the constant-elasticity-of-substitution (CES) utility function is defined as: U (x, y) = (αx ρ + (1 − α)y ρ) It is a tedious but straightforward application of Lagrangian calculus to demonstrate that the associated demand functions are: x(p x , p y , M) = α p x σ M α σ p 1−σ x + (1 − α) σ p 1−σ y and y(p x , p y , M) = 1 − α p y σ M α σ p 1−σ x + (1 − α)...

1. 1.1. Loops and Fundamental group. Let X be an arcwise connected topological space. Consider the homotopy classes of loops starting at a fixed point b 0 ∈ X : Ω(X, b 0). Recall that a loop is a continuous mapping σ : (I, ∂I) → (X, b 0). Composition σ * τ of two loops σ, τ is defined as (σ * τ)(t) =    σ(2t), 0 ≤ t ≤ 1/2 τ (2t − 1), 1/2 ≤ t ≤ 1 Let c be the constant loop at b 0 and σ −1 be ...

Given a compact orientable surface Σ, let S(Σ) be the set of isotopy classes of essential simple loops on Σ. We determine a complete set of relations for a function from S(Σ) to Z to be a geometric intersection number function. As a consequence, we obtain explicit equations in R S(Σ) and P (R S(Σ)) defining Thurston's space of measured laminations and Thurston's compactification of the Teichmül...