Rank one transformations with singular spectral type

We show that a certain class of measures arising from generalized Riesz products is singular. In particular, cutting and stacking (i.e. rank one) transformations whose cuts do not grow too rapidly, have singular maximal spectral type. The precise condition is , wherew _h is the number of cuts at stagen. Supported by a Canada NSERC grant. Supported by an NSF grant.     

Nonclassical multidimensional singular integral operators on R + n+1

Banach algebras of certain bounded operators acting on the half-spaceL _p (R ₊ ⁿ⁺¹ ,x ₀ ) (1<p<, –1<<p–1) are defined which contain for example Wiener-Hopf operators, defined by multidimensional singular convolution integral operators, as well as certain singular integral operators with fixed singularities. Moreover the symbol may be a positive homogeneous function only piecewise continuous on the unit sphere. Actually these multidimensional singular integral operators may be not Calderón-Zygmund operators but are built up by those in lower dimensions. This paper is a continuation of a joint paper of the author together with R.V. Duduchava [10]. The purpose is to investigate invertibility or Fredholm properties of these operators, while the continuity is given by definition. This is done in [10] forp=2 and –1<<1, and in the present paper forL _p (R ₊ ⁿ⁺¹ ,x ₀ ) with 1<p< and –1<<p–1.     

Variational calculus with sums of elementary tensors of fixed rank

In this article we introduce a calculus of variations for sums of elementary tensors and apply it to functionals of practical interest. The survey provides all necessary ingredients for applying minimization methods in a general setting. The important cases of target functionals which are linear and quadratic with respect to the tensor product are discussed, and combinations of these functionals are presented in detail. As an example, we consider the solution of a linear system in structured tensor format. Moreover, we discuss the solution of an eigenvalue problem with sums of elementary tensors. This example can be viewed as a prototype of a constrained minimization problem. For the numerical treatment, we suggest a method which has the same order of complexity as the popular alternating least square algorithm and demonstrate the rate of convergence in numerical tests.     

Finite Intervals in the Lattice of Topologies

We discuss the question whether every finite interval in the lattice of all topologies on some set is isomorphic to an interval in the lattice of all topologies on a finite set – or, equivalently, whether the finite intervals in lattices of topologies are, up to isomorphism, exactly the duals of finite intervals in lattices of quasiorders. The answer to this question is in the affirmative at least for finite atomistic lattices. Applying recent results about intervals in lattices of quasiorders, we see that, for example, the five-element modular but non-distributive lattice cannot be an interval in the lattice of topologies. We show that a finite lattice whose greatest element is the join of two atoms is an interval of T ₀-topologies iff it is the four-element Boolean lattice or the five-element non-modular lattice. But only the first of these two selfdual lattices is an interval of orders because order intervals are known to be dually locally distributive.     

Approximationseigenschaft,Lifting und Kohomologie bei lokalkonvexen Produktgarben

Motivated by examples like spaces of solutions of hypoelliptic operators, we treat product sheaves, prove density theorems in connection with the approximation property and use them for results on liftings and the vanishing of cohomology groups. Theorems of this type (2.4,2.9,3.3) are derived on regular subsets (2.3) of a product for product sheaves, where one factor has essentially a partition of unity. In the case of the compact open topology, we obtain the approximation property on arbitrary open subsets by a localization principle (4.5,4.9). The nuclearity of a sheaf in the co-topology turns out to imply strong nuclearity (1.11); the same is shown for the sheaf of holomorphic functions on the dual of a strongly nuclear (F)-space (1.12).

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Herrn Professor Gottfried Köthe zum 70. Geburtstag gewidmet     

Self–regularization by projection for noisy pseudodifferential equations of negative order

It is well known, that pseudodifferential equations of negative order considered in Sobolev spaces with small smoothness indices are ill–posed. On the other hand, it is known that efficient discretization schemes with properly chosen discretization parameters allow to obtain a regularization effect for such equations. The main accomplishment of the present paper is the principle for the adaptive choice of the discretization parameters directly from noisy discrete data. We argue that the combination of this principle with wavelet–based matrix compression techniques leads to algorithms which are order–optimal in the sense of complexity.     

Synthesis of the selective antimuscarinic agent 4-{[cyclohexylhydroxy(2-methoxyphenyl)silyl]-methyl}-1,1-dimethylpiperazinium methyl sulfate (o-methoxy-sila-hexocyclium methyl sulfate)

The synthesis of the potent and highly selective silicon-containing antimuscarinic agent o-methoxysila-hexocyclium methyl sulfate and its corresponding tertiary amine (isolated as the dihydrochloride) is described. The quarternary compound is an o-methoxy derivative of sila-hexocyclium methyl sulfate, which represents one of the tools currently used in experimental pharmacology for the subclassification of muscarinic receptors. The o-methoxy derivative, the pharmacological profile of which differs substantially from that of the nonmethoxy compound, is also recommended as a tool for the investigation of muscarinic receptor heterogeneity.     

The synthesis and the crystal and molecular structure of the fungicide bis(4-fluorophenyl)-methyl(1H-1,2,4-triazol-1-yl-methyl)silane (flusilazole,DPX H 6573)

Detailed procedures (based on the respective patent literature) for the preparation of the broadspectrum fungicide flusilazole (DPX H 6573) on the laboratory scale are described using as the starting material dichloro(chloromethyl)methylsilane [CH₃(ClCH₂)SiCl₂]. In addition, the crystal and molecular structure of flusilazole (determined by means of X-ray diffraction analysis) is described.