Minima of decomposable forms of degree n in n variables for n ≥ 3

The following theorem is proved: if for all (X0)one has ¦ F(x) ¦ >0, where F(x) is a decomposable form of degree n of n variables, then, for n 3, F(x) is proportional to an integral form.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 183, pp. 142–154, 1990.     

Flat deformations of ℙ n

In this paper we study projective flat deformations of ? ⁿ . We prove that the singular fibers of a projective flat deformation of ? ⁿ appear either in codimension 1 or over singular points of the base. We also describe projective flat deformations of ? ⁿ with smooth total space, and discuss flatness criteria.     

Balanced path decomposition of λ K n,n and λ K n,n *

Let P _k denote a path with k edges and λ K _n,n denote the λ-fold complete bipartite graph with both parts of size n. In this paper, we obtain the necessary and sufficient conditions for λ K _n,n to have a balanced P _k -decomposition. We also obtain the directed version of this result.     

Rigid Sets of Dimension n – 1 in ℝ n

We give conditions allowing an intrinsic isometry on a dense subset to be extended to an isometry of the whole set. This enables us to find examples of (n-1)-dimensional sets rigid in ⁿ .     

Global Behavior of Solutions of x n + 1 = max {x n , A}/x n x n – 1

We investigate the asymptotic behavior, the oscillatory character, and theperiodic nature of solutions of the difference equation

A family of à n -groups

For each integern≥2 and for each prime powerq, we exhibit a group Γ which acts simply transitively on the set of vertices of the building of type à _n associated with the local field $\mathbb{F}^q ((Y))$ . This generalizes work in [2] and [8] (wheren=2) and in [1] (wheren≤4).     

About the Cohomology of the Lie Superalgebra of Vector Fields on ℝ n|n

In this article, we compute the first space of cohomology of Vect (? ^n|n ), the Lie superalgebra of vector fields on the supermanifold ? ^n|n with coefficients in 𝒻 (? ^n|n ), the space of smooth functions on ? ^n|n . We give a super analog of the cohomologies of vector fields that where studied for instance by Fuchs [2 Fuchs , D. B. ( 1986 ). Cohomology of Infinite Dimensional Lie Algebras . New York : Consultants Bureau . [Google Scholar]]. This work allows us to classify the deformations of the action of Vect(? ^n|n ) on 𝒻 (? ^n|n ).     

On the (n − 2)-transversals of n convex subsets of the plane

In this paper we consider families of distinct ovals in the plane, with the property that certain subfamilies have stabbing lines (transversals). Our main result says that if any k member of the family can be stabbed by a line avoiding all the other ovals and k is large enough, then the family consists of at most k+1 ovals. For any n4 we show a family of n ovals, whose n–2 element subfamilies have, but the n–1 element subfamilies do not have, transversals.     

The local rank of an ergodic symmetric power T ⊙n does not exceed n!n −n

The infinity of the rank of ergodic symmetric powers of automorphisms of the Lebesgue space is proved, and sharp upper bounds for their local rank are found.     

The actions ofS n+1 andS n on the cohomology ring of a coxeter arrangement of typeA n−1

Let be the complexified Coxeter arrangement of hyperplanes of typeA _n−1. In this paper we construct anS _n+1 extension of the naturalS _n action on the complex cohomology ring of the complement ofA _n−1. Recurrence formulas connecting characters with respect to theS _n and theS _n+1 action are given.     

2-Arc-transitive cyclic covers of K n,n −nK 2

Du et al. (in J. Comb. Theory B 74:276–290, 1998 and J. Comb. Theory B 93:73–93, 2005), classified regular covers of complete graph whose fiber-preserving automorphism group acts 2-arc-transitively, and whose covering transformation group is either cyclic or isomorphic to $\mathbb{Z}_{p}^{2}$ or $\mathbb{Z}_{p}^{3}$ with p a prime. In this paper, a complete classification is achieved of all the regular covers of bipartite complete graphs minus a matching K _n,n ?nK ₂ with cyclic covering transformation groups, whose fiber-preserving automorphism groups act 2-arc-transitively.     

Boundary regularity of correspondences in ℂn

LetM, M′ be smooth, real analytic hypersurfaces of finite type in ℂⁿ and a holomorphic correspondence (not necessarily proper) that is defined on one side ofM, extends continuously up toM and mapsM to M′. It is shown that must extend acrossM as a locally proper holomorphic correspondence. This is a version for correspondences of the Diederich-Pinchuk extension result for CR maps.     

Hochschild cohomology of ? n -Galois coverings of an algebra

We consider the ? _n -Galois covering ?? _n of the algebra A introduced by F. Xu [Adv. Math., 2008, 219: 1872?C1893]. We calculate the dimensions of all Hochschild cohomology groups of ?? _n and give the ring structure of the Hochschild cohomology ring modulo nilpotence. As a conclusion, we provide a class of counterexamples to Snashall-Solberg??s conjecture.