一类伪polar环（英文）

受拟polar环和伪polar环概念的启发,引入根polar环的定义.称环R中的元素α是根polar元,如果存在p~2=p∈R使得p∈comm~2(α),α+p∈U(R)且ap∈J(R);称环R是根polar的,如果R中每个元素都是根polar元.本文研究了根polar环的基本性质并构造了许多例子,同时借助根polar环研究了相关环类.证明了阶数大于1的任意矩阵环都不是根polar的,因此给出局部环上2阶矩阵是根polar元的判定准则.     

拟polar环的注记（英文）

称一个环R中的元素a是拟polar元,若存在p2=P∈R满足p∈comm_R~2（a）,a＋P∈U（R）并且ap∈R~（qnil）;且称环R是拟polar的如果R中每一个元素都是拟polar元.本文证明了,任一环R中强π-正则元是拟polar的,而拟polar元是强clean的.拟polar环的一些扩张性质也作了探讨.     

Theorems of Erd?s-Ko-Rado type in polar spaces

We consider Erd?s-Ko-Rado sets of generators in classical finite polar spaces. These are sets of generators that all intersect non-trivially. We characterize the Erd?s-Ko-Rado sets of generators of maximum size in all polar spaces, except for H(4n+1,q²) with n?2.     

Apéry base and polar invariants for real valuations

The main objective of this paper is to study real valuations?μ defined on the polynomial ring K[T] with coefficients in a field K such that the restriction of?μ to K is a proper valuation on K. For this, we consider the Apéry base and the iterated sequence of valuations associated with?μ and we show the main properties and the relation between both invariants. We also prove a factorization theorem for elements ${f\in K[T]}$ of the suitable degree such that?μ(f) is in the Apéry base of?μ, obtaining strong properties of their irreducible factors. In particular, some results of M. Merle, R. Ephraim and A. Granja for irreducible algebroid plane curve singularity are a special case of this theorem.     

An Erdős-Ko-Rado theorem for finite classical polar spaces

Consider a finite classical polar space of rank \(d\ge 2\) and an integer n with \(0<n<d\). In this paper, it is proved that the set consisting of all subspaces of rank n that contain a given point is a largest Erd?s-Ko-Rado set of subspaces of rank n of the polar space. We also show that there are no other Erd?s-Ko-Rado sets of subspaces of rank n of the same size.     

Symmetric Monge–Kantorovich problems and polar decompositions of vector fields

We address the problem of whether a bounded measurable vector field from a bounded domain Ω into \({\mathbb{R}^d}\) is N-cyclically monotone up to a measure preserving N-involution, where N is any integer larger than 2. Our approach involves the solution of a multidimensional symmetric Monge–Kantorovich problem, which we first study in the case of a general cost function on a product domain Ω ^N . The polar decomposition described above corresponds to a special cost function derived from the vector field in question (actually N ? 1 of them). The problem amounts to showing that the supremum in the corresponding Monge–Kantorovich problem when restricted to those probability measures on Ω ^N which are invariant under cyclic permutations and with a given first marginal μ, is attained on a probability measure that is supported on a graph of the form x → (x, Sx, S ² x,..., S ^N-1 x), where S is a μ-measure preserving transformation on Ω such that S ^N  = I a.e. The proof exploits a remarkable duality between such involutions and those Hamiltonians that are N-cyclically antisymmetric.     

A topological characterization for closed sets under polar duality in ℚ n

A topological characterization is given for closed sets in ⁿ under the restriction of (cone) polar duality to ⁿ .     

s-Representations for involutions¶on affine Kac–Moody algebras are polar

Let denote the eigenspace decomposition of a twisted affine Kac–Moody algebra with respect to an involution , where is a twisted loop algebra, is the center and d is the scaling element of . We endow with the standard bilinear symmetrical form.Then with and carries a Lorentzian signature. Let denote the group that corresponds to , then the adjoint representation of on can be restricted to and this submanifold is isometrical to the Hilbert space E _ε, where is the decomposition of the twisted loop algebra with respect to the induced involutionρ₀.We thus obtain an affine representation on E _ε and we show that this representation is polar, i. e., there exists a submanifold that intersects all orbits, and intersects them orthogonally. Received: 16 February 2000 RID=" ID="Supported by a DFG grant.     

Measures not charging polar sets and Schrödinger equations in L p

We study the SchrSdinger equation （q -￡）u ＋μu = f, where ￡ is the generator of a Borel right process and μ is a signed measure on the state space. We prove the existence and uniqueness results in Lp, 1 ≤p 〈∞ . Since we consider measures μcharging no polar set, we have to use new tools： the Revuz formula with fine versions and the appropriate Revuz correspondence, the perturbation （subordination） operators （in the sense of G Mokobodzki） induced by the regular strongly supermedian kernels. We extend the results on the SchrSdinger equation to the case of a strongly continuous sub-Markovian resolvent of contractions on Lp. If the measure μ is positive then the perturbed process solves the martingale problem for ￡- μ and its transition semigroup is given by the Feynman-Kac formula associated with the left continuous additive functional having μ as Revuz measure.     

The largest Erdős-Ko-Rado sets of planes in finite projective and finite classical polar spaces

Erd?s-Ko-Rado sets of planes in a projective or polar space are non-extendable sets of planes such that every two have a non-empty intersection. In this article we classify all Erd?s-Ko-Rado sets of planes that generate at least a 6-dimensional space. For general dimension (projective space) or rank (polar space) we give a classification of the ten largest types of Erd?s-Ko-Rado sets of planes. For some small cases we find a better, sometimes complete, classification.     

Commutator inequalities associated with polar decompositions of <Emphasis Type="Italic">τ</Emphasis>-measurable operators

We prove commutator inequalities associated with polar decompositions of τ-measurable operators.     

A new upper bound for constant distance codes of generators on hermitian polar spaces of type H(2d − 1, q 2)

We provide new bounds for the maximum size of a set of generators of H(2d ? 1, q ²) which pairwise intersect in codimension i by applying a multiplicity bound by C. D. Godsil. This implies a new bound on the maximum size of partial spreads of H(2d ? 1, q ²), d even.