We introduce the categories of algebraic σ-varieties and σ-groups over a difference field (K,σ). Under a “linearly σ-closed” assumption on (K,σ) we prove an isotriviality theorem for σ-groups. This theorem yields immediately the key lemma in a proof of the Manin-Mumford conjecture. The present paper uses crucially ideas from [10] but in a model theory free manner. The applications to Manin-Mumf...

Nakayama (Ann. of Math. 42, 1941) showed that over an artinian serial ring every module is a direct sum of uniserial modules. Hence artinian serial rings have the property that each right (left) ideal is a finite direct sum of quasi-injective right (left) ideals. A ring with the property that each right (left) ideal is a finite direct sum of quasi-injective right (left) ideals will be called a ...

Inspired by Fagin’s result that NP = Σ11, we have developed a partial framework to investigate expressibility inside Σ11 so as to have a finer look into NP . The framework uses interesting combinatorics derived from second-order Ehrenfeucht-Fräıssé 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...

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