Finiteness spaces were introduced by Ehrhard as a model of linear logic, which relied on a finitess property of the standard relational interpretation and allowed to reformulate Girard’s quantitative semantics in a simple, linear algebraic setting. We review recent results obtained in a joint work with Christine Tasson, providing a very simple and generic construction of finiteness spaces: basi...

We introduce the notions of open-closed field algebra and open-closed field algebra over a vertex operator algebra V. In the case that V satisfies certain finiteness and reductivity conditions, we show that an open-closed field algebra over V canonically gives an algebra over a C-extension of the Swiss-cheese partial operad. We also give a tensor categorical formulation and categorical construc...

We introduce the notions of open-closed field algebra and open-closed field algebra over a vertex operator algebra V. In the case that V satisfies certain finiteness and reductivity conditions, we show that an open-closed field algebra over V canonically gives an algebra over a C-extension of the Swiss-cheese partial operad. We also give a tensor categorical formulation and categorical construc...

Abstract We consider the standard and k th record values arising in sequences of independent identically distributed continuous positive random variables with finite expectations. determine necessary sufficient conditions on type , its number n moment order r so that value is for every parent distribution. Under we present optimal upper bounds these moments expressed scale units being respectiv...

After results by the author (1980, 1981) and Vinberg (1981), finiteness of the number of maximal arithmetic groups generated by reflections in Lobachevsky spaces was not known in dimensions 2 ≤ n ≤ 9 only. Recently (2005), the finiteness was proved in dimension 2 by Long, Maclachlan and Reid, and in dimension 3 by Agol. Here we use these results in dimensions 2 and 3 to prove finiteness in all ...

We establish conditions for a continuous map of nonzero degree between a smooth closed manifold and a negatively curved manifold of dimension greater than four to be homotopic to a smooth cover, and in particular a diffeomorphism when the degree is one. The conditions hold when the volumes or entropy-volumes of the two manifolds differ by less than a uniform constant after an appropriate normal...

The structure of the commutative Moufang loops (CML) with minimum condition for subloops is examined. In particular it is proved that such a CML Q is a finite extension of a direct product of a finite number of the quasicyclic groups, lying in the centre of the CML Q. It is shown that the minimum conditions for subloops and for normal subloops are equivalent in a CML. Moreover, such CML also ch...

Recently there has been much interest in the Liouville type theorems for harmonic maps. For a detailed survey and progress in this direction, see the works by Hildebrandt [4], Eells and Lemaire [2]. Here we would like to mention that for all known results, the conditions on the harmonic maps can be divided into two kinds. The first of these conditions concerns the finiteness of the energy of th...

We introduce graded $${\mathbb {E}}_{\infty }$$ -rings and modules over them, study their properties. construct projective schemes associated to connective {N}}$$ -graded in spectral algebraic geometry. Under some finiteness conditions, we show that the $$\infty $$ -category of almost perfect quasi-coherent sheaves a scheme $$\text { {Proj}}\,(A)$$ -ring A can be described terms $${{\mathbb {Z}...

In this paper we provide a criteria for geometric finiteness of Kleinian groups in general dimension. We formulate the concept of conformal finiteness for Kleinian groups in space of dimension higher than two, which generalizes the notion of analytic finiteness in dimension two. Then we extend the argument in the paper of Bishop and Jones to show that conformal finiteness implies geometric fini...