Division by a holomorphic matrix in strictly pseudoconvex Domains

LetF andG be respectively a vector- and a matrix-function in a bounded strictly pseudoconvex domainD, with entries holomorphic inD and continuous in . We prove that ifF can be divided locally byG with holomorphic factors in a neighborhood of a given pointw inD, and the rank ofG is maximal at all points of , then the division ofF byG holds globally, with some factors which are holomorphic inD and continuous in . This method applies also to other function algebras in pseudoconvex domains.     

Uniform approximation by meromorphic functions on Riemann surfaces

A closed subsetE of a Riemann surfaceS is called a set of uniform meromorphic approximation if every functionf continuous onE and holomorphic onE ⁰ can be approximated uniformly onE by meromorphic functions onS. We show that ifE is a set of uniform meromorphic approximation, then so is for every compact parametric diskD. As a consequence, we obtain a generalization to Riemann surfaces of a well-known theorem of A. G. Vitushkin. Partially supported by a grant from NSERC of Canada.     

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 .     

A relative cohomological invariant for group pairs

We define a cohomological invariantE(G, S, M) whereG is a group,S is a non empty family of (not necessarily distinct) subgroups of infinite index inG andM is a -module ( is the field of two elements). In this paper we are interested in the special case where the family of subgroups consists of just one subgroup, andM is the -module . The invariant will be denoted byE(G, S). We study the relations of this invariant with other endse(G), e(G, S) ande(G,S)), and some results are obtained in the case whereG andS have certain properties of duality.     

Representation of analytic functions by generalized series of exponents in unbounded convex domains

We establish conditions for the representation of functions f(z) analytic in unbounded convex domains D and continuous in via series of the form . Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 6, pp. 812–817, June, 1998.     

On Galois isomorphisms between ideals in extensions of local fields

LetL/K be a totally ramified, finite abelian extension of local fields, let and be the valuation rings, and letG be the Galois group. We consider the powers of the maximal ideal of as modules over the group ring . We show that, ifG has orderp ^m (withp the residue field characteristic), ifG is not cyclic (or ifG has orderp), and if a certain mild hypothesis on the ramification ofL/K holds, then and are isomorphic iffr≡r′ modp ^m . We also give a generalisation of this result to certain extensions not ofp-power degree, and show that, in the casep=2, the hypotheses thatG is abelian and not cyclic can be removed.     

Relative perturbation results for eigenvalues and eigenvectors of diagonalisable matrices

Let and be a perturbed eigenpair of a diagonalisable matrixA. The problem is to bound the error in and . We present one absolute perturbation bound and two relative perturbation bounds. The absolute perturbation bound is an extension of Davis and Kahan's sin θ Theorem from Hermitian to diagonalisable matrices. The two relative perturbation bounds assume that and are an exact eigenpair of a perturbed matrixD ₁ AD ₂ , whereD ₁ andD ₂ are non-singular, butD ₁ AD ₂ is not necessarily diagonalisable. We derive a bound on the relative error in and a sin θ theorem based on a relative eigenvalue separation. The perturbation bounds contain both the deviation ofD ₁ andD ₂ from similarity and the deviation ofD ₂ from identity. This work was partially supported by NSF grant CCR-9400921.     

Eine Verallgemeinerung derCW-Komplexe

In the definition ofCW-complexes, the one-point spaceP, respectively the spaceP∪* with basepoint *, play the roll of the only “building-stone”. Let be a family of compact spaces. Then the definition of a generalizedCW-complex over is obtained from the definition of aCW-complex by replacingP by the spaces of and formation of the mapping cone by a slightly modified construction. LetCW * denote the category of all pointed spaces which have the homotopy type of a generalizedCW-complex over . If , thenCW * is the category of all pointedCW-spaces.CW * is closed under the formation of direct sums and of mapping cones, cylinders and tori, and is formally characterized as the smallest such subcategory of Top * containing the spaces W∪*, . Following the methods of E. H. Brown, it is proved, that any half exact homotopy functor onCW * is representable, and any cohomology theory onCW is naturally equivalent to the cohomology theory of an Ω-spectrum; for example, the singular cohomo logy is representable onCW for any family of compact spaces.      

Über die Bestimmung der Anzahl dreiachsiger Ellipsoide aus der Anzahl ihrer Schnittellipsen in zufälligen Schnittebenen

Summary A diagram is given for reading off the shape-factor oftri-axial ellipsoids ( = average distance between parallel tangent planes over all orientations).The knowledge of permits the calculation of the number of ellipsoids present per unit volume from the number of elliptical sections observed per unit area.     

The argument principle and holomorphic extendibility

LetD be a bounded domain in the complex plane whose boundary consists of finitely many pairwise disjoint simple closed curves. GivebD the standard orientation, and letA(D) be the algebra of all continuous functions on which are holomorphic onD. We prove that a continuous functionf onbD extends to a function inA(D) if and only if for eachg∈A(D) such thatf+g≠0 onbD, the change of argument off+g alongbD is nonnegative.     

Bergman kernel and metric on non-smooth pseudoconvex domains

A stability theorem of the Bergman kernel and completeness of the Bergman metric have been proved on a type of non-smooth pseudoconvex domains defined in the following way:D = {z∈U|r(z)} <whereU is a neighbourhood of andr is a continuous plurisubharmonic function onU. A continuity principle of the Bergman Kernel for pseudoconvex domains with Lipschitz boundary is also given, which answers a problem of Boas.     

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 .     

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.     

Invariant convex sets and functions in Lie algebras

We say that an invariant convex coneW in a Lie algebras is elliptic if its interior consists of elliptic elements of . If such a cone exists, then has a compactly embedded Cartan subalgebra. The first main result, of this paper is a characterization of those Lie algebras, which contain elliptic invariant cones. If is an invariant domain in such a cone, then we characterize the invariant locally convex functions onD by their restrictions to where is a compactly embedded Cartan subalgebra.     

Natural extensions of holomorphic motions

We consider an arbitrary real analytic family X_z, , over the closed unit disc , of real analytic plane Jordan curves X_z. Ifj _e ^iθ ,e ^iθ ∋ ∂D, is an arbitrary real-analytic family of orientation-reversing homeomorphisms of fixingX _e ^iθ pointwise, we show that there is a unique holomorphic motion of extending the given motion of Jordan curves and consistent with the given family of involutions. If these generalized reflections are defined using the barycentric extension construction of Douady-Earle-Nag, then the resulting extension method for holomorphic motions of X is natural, that is Moebius-invariant and continuous with respect to variation of the given motion of X₀.     

Optimal decompositions for theK-functional for a couple of Banach lattices

Letf=g _t+h _t be the optimal decomposition for calculating the exact value of theK-functionalK(t, f; ) of an elementf with respect to a couple =(X ₀ ,X ₁) of Banach lattices of measurable functions. It is shown that this decomposition has a rather simple form in many cases where one of the spacesX ₀ andX ₁ is eitherL ^∞ orL ¹. Many examples are given of couples of lattices for which |g _t| increases monotonically a.e. with respect tot. It is shown that this property implies a sharpened estimate from above for the Brudnyi-KrugljakK-divisibility constant γ( ) for the couple. But it is also shown that certain couples do not have this property. These also provide examples of couples of lattices for which γ( ). Research supported by the Technion V. P. R. Fund.     

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.     

3-Folds in ℙ5 of degree 12of degree 12

Let be a 3-dimensional submanifold of ℙ⁵ of degree 12. This article gives, up to one case, a complete classification of the deformation classes of those 3-folds. The main tools used are methods already applied in the classification of degrees 9 to 11 and adjunction theoretic results. We show here how the 2^nd reduction of can be applied to analyse the birational structure of or even exclude the existence of .     

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.