q-Pseudoconvexity and Regularity at the Boundary for Solutions of the \bar \partial -problem

For a domain of we introduce a fairly general and intrinsic condition of weak q-pseudoconvexity, and prove, in Theorem 4, solvability of the -complex for forms with -coefficients in degree . All domains whose boundary have a constant number of negative Levi eigenvalues are easily recognized to fulfill our condition of q-pseudoconvexity; thus we regain the result of Michel (with a simplified proof). Our method deeply relies on the L ²-estimates by Hörmander (with some variants). The main point of our proof is that our estimates (both in weightened-L ² and in Sobolev norms) are sufficiently accurate to permit us to exploit the technique by Dufresnoy for regularity up to the boundary.     

Prime ideal decomposition inF({}^m\sqrt \mu )

LetF be an algebraic number field and F such thatx ^m– is irreducible, wherem is an integer. Let be a prime ideal inF with . The prime decomposition of in is explicitly obtained in the following cases. Case 1: , (a,m) = 1 (where means , 0 ). Case 2:m lt, wherel is a prime andl 0 . Case 3:m 0 and every prime that dividesm also dividespf–1. It is not assumed that thev ^th roots of unity are inF for anyv 2.     

Euler Characteristics of Local Systems on \mathcal{M}> 2

We calculate the Euler characteristics of the local systems S ^k S ² on the moduli space ₂ of curves of genus 2, where is the rank 4 local system R ¹ _* .     

Twists and Singular Vectors in \widehat{sl}\left( {2\left| 1 \right.} \right) Representations

We propose new formulas for singular vectors in Verma modules over the affine Lie superalgebra . We analyze the coexistence of singular vectors of different types and identify the twisted modules arising as submodules and quotient modules of Verma modules. We show that with the twists (spectral flow transformations) properly taken into account, a resolution of irreducible representations can be constructed consisting of only the modules.     

Structure of Injective Real Factors of Type III \textIIIl ,0 < l < 1{\text{III}}_\lambda ,0 < \lambda < 1

We consider ^*-automorphisms and ^*-antiautomorphisms of real and complex factors. We establish both the uniqueness of the class of ^*-automorphisms (with ) of a real injective factor and the uniqueness of the class of ^*-antiautomorphisms (with ) of a complex injective factor. It is well known that, for complex factors, the notions of hyperfiniteness and injectivity are equivalent. Here we prove that, for real factors, the two notions are no longer equivalent. Bibliography: 15 titles.     

Approximation of $$\bar \omega $$ -integrals of continuous functions defined on the real axis by Fourier operators

We obtain asymptotic formulas for the deviations of Fourier operators on the classes of continuous functions and in the uniform metric. We also establish asymptotic laws of decrease of functionals characterizing the problem of the simultaneous approximation of -integrals of continuous functions by Fourier operators in the uniform metric.Translated from Ukrainskyi Matematychnyi Zhurnal, Vol. 56, No. 5, pp. 663–676, May, 2004.     

L_{\infty _\omega }free algebras

In this paper the study of which varieties, in a countable similarity type, have non-free (or equivalently ₁-free) algebras is completed. It was previously known that if a variety satisfies a property known as the construction principle then there are such algebras. If a variety does not satisfy the construction principle then either every -free algebra is free or for every infinite cardinalk, there is a k⁺-free algebra of cardinality k⁺ which is not free. Under the set theoretic assumption V=L, for any varietyV in a countable similarity type, either the class of free algebras is definable in or it is not definable in any .In Memory of Evelyn NelsonPresented by Ralph McKenzie.Research partially supported by NSERC of Canada Grant #A8948.Research partially supported by NSERC of Canada. The research for this paper was begun while the second author was visiting Simon Fraser University.     

Fast-decaying potentials on the finite-gap background and the\bar \partial - problem on the Riemann surfaces

The direct and the inverse scattering problems for the heat-conductivity operator are studied for the following class of potentials:u(x,y)=u _o (x,y)+u ₁(x,y), whereu _o (x,y) is a nonsingular real finite-gap potential andu ₁(x,y) decays sufficiently fast asx ²+y². We show that the scattering data for such potentials is the data on the Riemann surface corresponding to the potentialu _o (x,y). The scattering data corresponding to real potentials is characterized and it is proved that the inverse problem corresponding to such data has a unique nonsingular solution without the small norm assumption. Analogs of these results for the fixed negative energy scattering problem for the two-dimensional time-independent Schrödinger operator are obtained.L. D. Landau Institute for Theoretical Physics, Kosygina 2, GSP-1, Moscow 177940, Russia. E-mail: pgg@cpd.landau.free.net. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 99, No. 2, pp. 300–308, May, 1994.     

Discussing Knarr's 2-Surface of \mathbb{R} 4 which Generates the First Single Shift Plane

The generating line of the first single shift plane (cf. [11, p. 435]) is a 2-surface of ⁴ which we call the the affine part of Knarr's surface. We compute all affinities leaving invariant. After embedding ⁴ into PG(4, ) we calculate the uniquely determined projective closure _Kn of . Using a suitable projection we transform questions on Knarr's surface to questions on Cayley's surface in PG(3, ). In this way we determine all planes carrying 1-dimensional algebraic varieties of _Kn . We exhibit all automorphic collineations of _Kn .     

On the structure of level sets of uniform and Lipschitz quotient mappings from $$ \mathbb{R}^n $$ to $$ \mathbb{R} $$

We study two questions posed by Johnson, Lindenstrauss, Preiss, and Schechtman, concerning the structure of level sets of uniform and Lipschitz quotient mappings from . We show that if , is a uniform quotient mapping then for every has a bounded number of components, each component of separates and the upper bound of the number of components depends only on and the moduli of co-uniform and uniform continuity of .Next we prove that all level sets of any co-Lipschitz uniformly continuous mapping from to are locally connected, and we show that for every pair of a constant and a function with , there exists a natural number , so that for every co-Lipschitz uniformly continuous map with a co-Lipschitz constant and a modulus of uniform continuity , there exists a natural number and a finite set with card so that for all has exactly components, has exactly components and each component of is homeomorphic with the real line and separates the plane into exactly 2 components. The number and form of components of for are also described - they have a finite tree structure.     

A Difference Set in \left( {\mathbb{Z}/4\mathbb{Z}} \right)^3 \times \mathbb{Z}/5\mathbb{Z}

We present a (320, 88, 24)-difference set in , the existence of which was previously open. This new difference set improves a theorem of Davis-Jedwab with the removal of the exceptional case. It also enables us to state a theorem of Schmidt on Davis-Jedwab difference sets more neatly.     

The Laplacian on $$ C(\overline \Omega) $$ with generalized Wentzell boundary conditions

In this note we prove that the Laplacian with generalized Wentzell boundary conditions on an open bounded regular domain in defined by generates an analytic semigroup of angle on for every > 0 and (for the definition of cf. (1.3)).Received: 13 July 2002     

Beckman-Quarles type theorems for mappings from $$ \mathbb{R}^n $$ to $$ \mathbb{C}^n $$

Summary. Let We say that preserves the distance d 0 if for each implies Let A _n denote the set of all positive numbers d such that any map that preserves unit distance preserves also distance d. Let D _n denote the set of all positive numbers d with the property: if and then there exists a finite set S _xy with such that any map that preserves unit distance preserves also the distance between x and y. Obviously, We prove: (1) (2) for n 2 D _n is a dense subset of (2) implies that each mapping f from to (n 2) preserving unit distance preserves all distances, if f is continuous with respect to the product topologies on and      

On the \Gamma-limit of the Mumford-Shah functional

We study by means of -convergence the asymptotics of the rescaled Mumford-Shah functional

A note on the isotopy problem for $$ \mathbb{R}^2 $$ -planes

For a class of stable planes we define a notion of isotopy equivalence with respect to that class and prove that any two planes of a certain class of -planes comprising all affine -planes are isotopy equivalent. Furthermore we obtain that all affine -planes are isotopy equivalent in the class of affine -planes. Finally we give an example which shows that this approach cannot be easily generalized to 2-dimensional projective planes, and we outline a different way for a possible generalization.Received: 27 April 2001     

Microlocalisation et estimations pour\bar \partial _b dans quelques hypersurfaces pseudoconvexes

Résume Nous étudions des estimations précises pour sur des hypersurfaces dont la matrice de Levi dans une base a une certaine forme. En particulier, nous donnons, par la méthode de microlocalisation, une démonstration très simple des estimations dites maximales lorsque les valeurs propres sont comparables. Nous obtenons en fait une précision supplémentaire (inégalité 3.17) qui permet de considérer des cas plus généraux. Finalement, de telles estimations, on déduit des estimations sous elliptiques optimales pour .

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Summary We study precise estimates for on hypersurfaces the Levi matrix of which has certain form. In particular, we give and simple proof of the so-called maximal estimates, using the microlocal analysis. In fact, we obtain a refined version (inequality 3.17), which permits the study of more general cases. Finally, from such estimates, we deduce optimal subelliptic estimates for .

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Oblatum 3-IX-1990     

Cohomology of Algebras of Dihedral Type. IV: the Family D\left( {{2\mathcal{B}}} \right)

For algebras of dihedral Type that form the family (from K. Erdmann's list), the Yoneda algebras are described in terms of quivers with relations. Bibliography: 9 titles.     

Multiple Fourier Sums on Sets of \bar \psi-Differentiable Functions (Low Smoothness)

We investigate the behavior of deviations of rectangular partial Fourier sums on sets of -differentiable functions of many variables.     

The type set for some measures on \mathbb{R}^{2n} with n-dimensional support

Let be realhomogeneous functions in ofdegree and let bethe Borel measure on given by

A Completion of \mathbb{Z} is a Field

We define various ring sequential convergences on and . We describe their properties and properties of their convergence completions. In particular, we define a convergence on by means of a nonprincipal ultrafilter on the positive prime numbers such that the underlying set of the completion is the ultraproduct of the prime finite fields Further, we show that is sequentially precompact but fails to be strongly sequentially precompact; this solves a problem posed by D. Dikranjan.