The closure diagram for nilpotent orbits of the split real form of E8

Let and be adjoint nilpotent orbits in a real semisimple Lie algebra. Write ≥ if is contained in the closure of . This defines a partial order on the set of such orbits, known as the closure ordering. We determine this order for the split real form of the simple complex Lie algebra, E ₈. The proof is based on the fact that the Kostant-Sekiguchi correspondence preserves the closure ordering. We also present a comprehensive list of simple representatives of these orbits, and list the irreeducible components of the boundaries and of the intersections .     

A class of multipliers for $$\mathcal{W}^ \bot $$

Let denote the class of ergodic probability preserving transformations which are disjoint from every weakly mixing system. Let be the class of multipliers for , i.e. ergodic transformations whose all ergodic joinings with any element of are also in . Fix an ergodic rotationT, a mildly mixing actionS of a locally compact second countable groupG and an ergodic cocycle ϕ forT with values inG. The main result of the paper is a sufficient (and also necessary by [LeP] whenG is countable Abelian andS is Bernoullian) condition for the skew product build fromT, ϕ andS to be an element of . Moreover, the self-joinings of such extensions ofT are described with an application to study semisimple extensions of rotations. Dedicated to Hillel Furstenberg on the occasion of his retirement The first-named author was supported in part by CRDF, grant UM1-2546-KH-03. The second-named author was supported in part by KBN grant 1P03A 03826.     

Equivariant D-modules attached to nilpotent orbits in a semisimple Lie algebra

Let _u be a compact Lie algebra and let _u be its complexification. Let ζ^−1/2 be the inverse on the set of regular elements of _u of a square root of the discriminant of . Generalizing a result of W. Lichtenstein in the case _u = (n, ℂ) or (nℝ), we prove that ∂(q).ζ^1/2 is non zero for all harmonic polynomialsq ∈S( ) \ {0}. This fact is deduced from results about equivariantD-modules supported on the nilpotent cone of .     

Blaschke-decomposable convex bodies

For a given centred convex bodyK of ℝ,n≥3, let be the class of all convex bodies with the same projection body asK. The question whetherK can be expressed as a Blaschke average of two non-homothetic bodies from is considered. Necessary and sufficient conditions onK to be Blaschke decomposable in are given. The paper provides also a characterization of the bodiesK such that the Blaschke indecomposable bodies in are dense in itself.     

Minimal number of idempotent generators for certain algebras

It is known [KRS] that for each finitely generated Banach algebra there exists a numberN such that for eachn>N the matrix algebras can be generated by three idempotents. In this paper we show that the same statement is true for direct sums and , where is a finitely generated free algebra, i.e. polynomials in several non-commuting variables. These results are new even for algebras because the numberN we obtain here improves known estimates (see for example [R]). We show that the algebra can be generated by two idempotents if and only ifn _j =2 for eachj and is singly generated. Also we give an example of a free singly generated algebra for which can not be generated by two idempotents. But% MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWefv3ySLgznf% gDOfdaryqr1ngBPrginfgDObYtUvgaiuaacuWFSeIqgaacaaaa!409A!\[{\tilde {\cal B}}\] can be generated by three idempotents for each singly generated free algebra .     

An optimal bound for high-quality conforming triangulations

This paper shows that, for any plane geometric graph withn vertices, there is a triangulation that conforms to , i.e., each edge of is the union of some edges of , where hasO(n²) vertices with each angle of its triangles measuring no more than 11/15π. Additionally, can be computed inO(n ² logn) time. This research was partially supported by the National University of Singapore under Grant RP940641.     

On residue complexes,dualizing sheaves and local cohomology modules

According to Grothendieck Duality Theory [RD], on each varietyV over a fieldk, there is a canonical complex of -modules, theresidue complex . These complexes satisfy (and are characterized by) functorial properties in the categoryV ofk-varieties. In [Ye] a complex is constructed explicitly (when the fieldk is perfect). The main result of this paper is that the two families of complexes, and , which carry certain additional data (such as trace maps…), are uniquely isomorphic. As a corollary we recover Lipman’s canonical dualizing sheaf of [Li], and we obtain formulas for residues of local cohomology classes of differential forms.     

Martin boundary of unlimited covering surfaces

LetW be an open Riemann surface and ap-sheeted (1<p<∞) unlimited covering surface ofW. Denote by Δ¹ (resp., ) the minimal Martin boundary ofW (resp., ). For ζ ∈ Δ, let ζ be the (cardinal) number of the set of pionts which lie over ζ and the class of open connected subsetsM ofW such thatM∪{ζ} is a minimal fine neighborhood of ζ. Our main result is the following: , where is the number of components of π^-1 M and π is the projection of ontoW. Moreover, some applications of the above results are discussed whenW is the unit disc.     

Property ∏σ and tensor products of operator algebras

In this paper, we introduce Property ∏σ of operator algebras and prove that nest subalgebras and the finite-width CSL subalgebras of arbitrary von Neumann algebras have Property ∏σ.Finally, we show that the tensor product formula alg ML1-(×)algNL2 = algM-(×)N(L1 (×) L2) holds for any two finite-width CSLs L1 and L2 in arbitrary von Neumann algebras M and N, respectively.     

Differential inequalities associated with weighted symmetrization processes on the real line

We consider the differential operator in a symmetric intervalJ ofR, where ϕ is the density function of a finite measure supported inJ satisfying a certain regularizing property. Using as main tools left symmetrization with respect to the weight ϕ, and certain measure preserving transformations, we prove differential inequalities involving the differential operators . As applications, we give comparison results for solutions to non-linear differential equations related to the operator, and with Neumann boundary data, generalizing in different ways results by C. Borell in [Bor 85b]. Research supported in part by a Grant from El Ministerio de Educación y Cultura (Spain) PB97-1081, and from La Junta de Andalucía.     

Compactness of minimal closed invariant sets of actions of unipotent groups

Let G be a connected Lie group, let Γ be a lattice in G, and let be a unipotent subgroup of G. It is proved that, for the natural action of on G/Γ, every minimal closed -invariant subset is compact. Dedicated to Professor Jacques Tits on the occasion of his sixtieth birthday     

Operator inequalities and construction of Krein spaces

Let be a Hilbert space. A continuous positive operatorT on uniquely determines a Hilbert space which is continuously imbedded in and for which with the canonical imbedding . A Kreîn space version of this result, however, is not valid in general. This paper provides a necessary and sufficient condition for that a continuous selfadjoint operatorT uniquely determines a Kreîn space ( ) which is continuously imbedded in and for which with the canonical imbedding .     

A Helly-type theorem for simple polygons

Let be a family of simple polygons in the plane. If every three (not necessarily distinct) members of have a simply connected union and every two members of have a nonempty intersection, then {P:P in } . Applying the result to a finite family of orthogonally convex polygons, the set {C:C in } will be another orthogonally convex polygon, and, in certain circumstances, the dimension of this intersection can be determined.Supported in part by NSF grant DMS-9207019.     

Asymptotic properties of the maximum likelihood estimate in the first order autoregressive process

Summary In this paper we obtain an asymptotic expansion of the distribution of the maximum likelihood estimate (MLE) based onT observations from the first order Gaussian process up to the term of orderT ⁻¹. The expansion is used to compare with a generalized estimate including the least square estimate (LSE) , based on the asymptotic probabilities around the true value of the estimates up to the terms of orderT ⁻¹. It is shown that (or the modified MLE ) is better than (or the modified estimate ). Further, we note that does not attain the bound for third order asymptotic median unbiased estimates.     

Sui fibrati con struttura quaternionale generalizzata

Summary Quaternion generalized fiber bundles are studied, both isomorphic to global tensorial product ordinary quaternion fiber bundles right and left respectively) and quite general ones. A cohomology class is considered which represents the obstruction in order the fiber bundle be a tensorial product. Several properties and a splitting principle are proved for bundles . On this ground and founding on a convenient bundle B_E → X associated to jaz (that we call Bonan's bundle and for which ɛ( =ɛ(B_E)) relations are stated among Stiefel-Whitney classes of , B_E and the class ɛ.

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Entrata in Redazione il 14 agosto 1974.

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Lavoro eseguito con contributo del C.N.R., nell'ambito del Gruppo Nazionale per le Strutture Algebriche e Geometriche e loro Applicazioni.     

Construction of conjugate quadrature filters with specified zeros

Let ℂ denote the complex numbers and denote the ring of complex-valued Laurent polynomial functions on ℂ\{0}. Furthermore, we denote by the subsets of Laurent polynomials whose restriction to the unit circle is real, nonnegative, respectively. We prove that for any two Laurent polynomials , which have no common zeros in ℂ\{0} there exists a pair of Laurent polynomials satisfying the equation Q ₁ P ₁ + Q ₂ P ₂ = 1. We provide some information about the minimal length Laurent polynomials Q ₁ and Q ₂ with these properties and describe an algorithm to compute them. We apply this result to design a conjugate quadrature filter whose zeros contain an arbitrary finite subset Λ⊂ℂ\{0} with the property that for every implies and . This revised version was published online in June 2006 with corrections to the Cover Date.     

Notes on Brown-McCoy-radicals for Γ-near-rings

The Brown-McCoy radical is known to be an ideal-hereditary Kurosh-Amitsur radical in the variety of zerosymmetric near-rings. We define the Brown-McCoy and simplical radicals, and , respectively, for zerosymmetric -near-rings. Both and are ideal-hereditary Kurosh-Amitsur radicals in that variety. IfM is a zerosymmetric -near-ring with left operator near-ringL, it is shown that , with equality ifM has a strong left unity. is extended to the variety of arbitrary near-rings, and and are extended to the variety of arbitrary -near-rings, in a way that they remain Kurosh-Amitsur radicals. IfN is a near-ring andA N, then , with equality ifA if left invariant.     

Isometries and direct decompositions of pseudo MV-algebras

In the paper isometries in pseudo MV-algebras are investigated. It is shown that for every isometry f in a pseudo MV-algebra = (A, ⊕, ⁻, ^∼, 0, 1) there exists an internal direct decomposition of with commutative such that and for each x ∈ A. On the other hand, if is an internal direct decomposition of a pseudo MV-algebra = (A, ⊕, ⁻, ^∼, 0, 1) with commutative, then the mapping g: A → A defined by is an isometry in and .      

Projective surfaces withk-very ample line bundles of genus ≤3k+1

Let be the set of surfaces,S, polarized by a k-very ample line bundle,L, with genus≤3k+1. All the elements (S, L) of are listed. The classification of surfaces polarized by ak-very ample line bundle of degree ≤4k+4 is completed by proving that this class of surfaces is a subset of .     

Baer cotorsion Pairs

LetR be a unital associative ring and two classes of leftR-modules. In [St3] the notion of a ( ) pair was introduced. In analogy to classical cotorsion pairs, a pair (V,W) of subclasses is called a ( ) pair if it is maximal with respect to the classes and the condition Ext _R ¹ (V, W)=0 for all . In this paper we study pairs whereR = ℤ and is the class of all torsion-free abelian groups andT is the class of all torsion abelian groups. A complete characterization is obtained assumingV=L. For example, it is shown that every pair is singly cognerated underV=L. The author was supported by a DFG grant.