n. The object of this paper is to prove that all semistable elliptic curves over the set of rational numbers are modular. Fermat’s Last Theorem follows as a corollary by virtue of previous work by Frey, Serre and Ribet.

prove three theorems. First, every strategy-proof voting procedure is dictatorial. Second, this paper’s strategy-proofness condition for voting procedures corre-sponds to Arrow’s rationality, independence of irrelevant alternatives, non-negative response, and citizens ’ sovereignty conditions for social

which can be used in proofs of the properties of computer programs. Examples are given of such axioms and rules, and a formal proof of a simple theorem is displayed. Finally, it is argued that important advantages, both theoretical and prac-tical, may follow from a pursuance of these topics.

approach however is closest to Severi's incomplete proof ([Se], Anhang F; the error is on pp. 344-345 and seems to be quite basic) and follows a suggestion of Grothendieck for using the result in char. o to deduce the result in char. p. The basis of both Severi's and Grothendieck's ideas

this article we prove some auxiliary theorems and schemes related to the articles: [1] and [2]. MML Identifier: PARTFUN1. WWW: http://mizar.org/JFM/Vol1/partfun1.html The articles [4], [6], [3], [5], [7], [8], and [1] provide the notation and terminology for this paper. We adopt the following

tiply connected networks: When loops are present, the network is no longer singly connected and local propaga tion schemes will invariably run into trouble . We believe there are general undiscovered theorems about the performance of belief propagation on loopy DAGs. These theo rems, which may have

Abstract. Let G be a word hyperbolic group in the sense of Gromov and P its associated Rips complex. We prove that the fixed point set P H iscontractible for every finite subgroup H of G. Thisisthe main ingredient for proving that P is a finite model for the universal space EG for proper actions. As a corollary we get that a hyperbolic group hasonly finitely many conjugacy classes of finite subgroups.

Following an idea of G. Nguetseng, the author defines a notion of "two-scale" convergence, which is aimed at a better description of sequences of oscillating functions. Bounded sequences in L2(f) are proven to be relatively compact with respect to this new type of convergence. A corrector

at s 0 and then following π: d π (s) = ∞ t=0 γ t P r {s t = s|s 0 , π}. Our first result concerns the gradient of the performance metric with respect to the policy parameter: Theorem 1 (Policy Gradient). For any MDP, in either the average-reward or start-state formulations, Proof: See the appendix

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