Configuration of subgroups in the special linear group over a skew field with infinite center

Let T be a skew field with infinite center, let be the special linear group over T of degree 3, and let be the subgroup of diagonal matrices with unit Dieudonee determinant. It is proved that for each intermediate subgroup H, H , there exists a net of order n such that ( H N().Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 175, pp. 5–12, 1989.In conclusion, the author would like to thank his instructor Z. I. Borevich, as well as N. A. Vavilov, for their assistance.     

L-fuzzy quantifiers of type determined by fuzzy measures

The aim of this paper is, first, to introduce two new types of fuzzy integrals, namely, -fuzzy integral and →-fuzzy integral. The first integral is based on a fuzzy measure of L-fuzzy sets and the second one on a complementary fuzzy measure of L-fuzzy sets, where L is a complete residuated lattice. Some of their properties and a relation to the fuzzy (Sugeno) integral are investigated. Second, using these integrals, two classes of monadic L-fuzzy quantifiers of type 1 are defined. These L-fuzzy quantifiers can be used for modeling the semantics of natural language quantifiers like “all”, “some”, “many”, “none”, “at most half”, etc. Several semantic properties of these L-fuzzy quantifiers are studied.     

Extensional Σ-Spaces in Type Theory

Synthetic Domain Theory provides a setting for denotational semantics following Dana Scott's slogan domains as sets in which all functions are continuous. Several approaches can be found in the literature, but they are either model-dependent or if they use an axiomatic setting then not uniformly and not explicitly. We present a completely logical approach to Synthetic Domain Theory (SDT), axiomatizing (complete) Extensional PERs. On these grounds some basic domain theory is developed. Special attention is devoted to admissibility. The axiomatic approach is advantageous since it allows for easy formalization and comparison to other axiomatic settings.The consistency of the theory is shown by providing an appropriate realizability model. It is discussed how to get from this special kind of SDT {à la Scott} to a more general form which unifies several approaches.     

Solution of Dirichlet's problem for the equation Δu =−1 in a convex region

Let u be a solution of the following boundary-value problem: u¦ = 0, where is a closed convex curve and u = –1 in the region D bounded by . Then u has only one local maximum, and all its level curves are convex.Translated from Matematicheskie Zametki, Vol. 9, No. 1, pp. 89–92, January, 1971.     

Nearly monotone and nearly convex approximation by smooth splines in ,

Given a monotone or convex function on a finite interval we construct splines of arbitrarily high order having maximum smoothness which are “nearly monotone” or “nearly convex” and provide the rate of -approximation which can be estimated in terms of the third or fourth (classical or Ditzian–Totik) moduli of smoothness (for uniformly spaced or Chebyshev knots). It is known that these estimates are impossible in terms of higher moduli and are no longer true for “purely monotone” and “purely convex” spline approximation.     

Mixed expansion formula for the rectangular Schur functions and the affine Lie algebra

Formulas are obtained that express the Schur S-functions indexed by Young diagrams of rectangular shape as linear combinations of “mixed” products of Schur's S- and Q-functions. The proof is achieved by using representations of the affine Lie algebra of type . A realization of the basic representation that is of “”-type plays the central role.     

Ridge wavelets on the ball

We present a ridge polynomial wavelet-type system on the unit ball in such that any continuous function can be expanded with respect to these wavelets. The order of the growth of the degrees of polynomials is optimal. Coefficient functionals are the inner products of the function and the corresponding elements of a “dual wavelet system”. The “dual wavelets” is also a ridge polynomial system with the same growth of the degrees of polynomials. The system is redundant.     

Sets determined by finitely many X-rays

A measurable set in _n which is uniquely determined among all measurable sets (modulo null sets) by its X-rays in a finite set L of directions, or more generally by its X-rays parallel to a finite set L of subspaces, is called L-unique, or simply unique. Some subclasses of the L-unique sets are known. The L-additive sets are those measurable sets E which can be written E {x _n : _i f _i (x) * 0}. Here, denotes equality modulo null sets, * is either or >, and the terms in the sum are the values of ridge functions f _i orthogonal to subspaces S _i in L. If n=2, the L-inscribable convex sets are those whose interiors are the union of interiors of inscribed convex polygons, all of whose sides are parallel to the lines in L. Relations between these classes are investigated. It is shown that in _n each L-inscribable convex set is L-additive, but L-additive convex sets need not be L-inscribable. It is also shown that every ellipsoid in _n is unique for any set of three directions. Finally, some results are proved concerning the structure of convex sets in _n , unique with respect to certain families of coordinate subspaces.     

Holomorphic functions on bundles over annuli

We consider a family of holomorphic bundles constructed as follows:from any given , we associate a “multiplicative automorphism” of . Now let be a -invariant Stein Reinhardt domain. Then E _m (D, M) is defined as the flat bundle over the annulus of modulus m > 0, with fiber D, and monodromy . We show that the function theory on E _m (D, M) depends nontrivially on the parameters m, M and D. Our main result is that

Wave atoms and sparsity of oscillatory patterns

We introduce “wave atoms” as a variant of 2D wavelet packets obeying the parabolic scaling wavelength(diameter)². We prove that warped oscillatory functions, a toy model for texture, have a significantly sparser expansion in wave atoms than in other fixed standard representations like wavelets, Gabor atoms, or curvelets. We propose a novel algorithm for a tight frame of wave atoms with redundancy two, directly in the frequency plane, by the “wrapping” technique. We also propose variants of the basic transform for applications in image processing, including an orthonormal basis, and a shift-invariant tight frame with redundancy four. Sparsity and denoising experiments on both seismic and fingerprint images demonstrate the potential of the tool introduced.     

Generalized de Bruijn Cycles

For a set of integers , we define a q-ary -cycle to be an assignment of the symbols 1 through q to the integers modulo q ⁿ so that every word appears on some translate of . This definition generalizes that of de Bruijn cycles, and opens up a multitude of questions. We address the existence of such cycles, discuss reduced cycles (ones in which the all-zeroes string need not appear), and provide general bounds on the shortest sequence which contains all words on some translate of . We also prove a variant on recent results concerning decompositions of complete graphs into cycles and employ it to resolve the case of completely.AMS Subject Classification: 94A55, 05C70.     

The Existence of Shilnikov Homoclinic Orbits in the Michelson System: A Computer Assisted Proof

In this paper we present a new topological tool which allows us to prove the existence of Shilnikov homoclinic or heteroclinic solutions. We present an application of this method to the Michelson system

Thin points for Brownian motion

Let Θ(x,r) denote the occupation measure of the ball of radius r centered at x for Brownian motion {W_t}_0≤t≤1 in . We prove that for any analytic set E in [0,1], we have

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, where dim_P(E) is the packing dimension of E. We deduce that for any a≥1, the Hausdorff dimension of the set of “thin points” x for which

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, is almost surely 2−2/a; this is the correct scaling to obtain a nondegenerate “multifractal spectrum” for the “thin” part of Brownian occupation measure. The methods of this paper differ considerably from those of our work on Brownian thick points, due to the high degree of correlation in the present case. To prove our results, we establish general criteria for determining which deterministic sets are hit by random fractals of ‘limsup type' in the presence of long-range correlations. The hitting criteria then yield lower bounds on Hausdorff dimension. This refines previous work of Khoshnevisan, Xiao and the second author, that required decay of correlations.     

On small homotopies of loops

Two natural questions are answered in the negative:

• “If a space has the property that small nulhomotopic loops bound small nulhomotopies, then are loops which are limits of nulhomotopic loops themselves nulhomotopic?”

• “Can adding arcs to a space cause an essential curve to become nulhomotopic?”

The answer to the first question clarifies the relationship between the notions of a space being homotopically Hausdorff and π₁-shape injective.

Quasilinear conflict-controlled processes with additional restrictions

A class of conflict-controlled processes [1–3] with additional (“phase” type) restrictions on the state of the evader is considered. A similar unrestricted problem was considered in [4]. Unlike [5, 6] the boundary of the “phase” restrictions is not a “death line” for the evader. Sufficient conditions for the solvability of the pursuit and evasion problems are obtained, which complement a range of well-known results [5–10].^‡     

Approximation in Lp( ) from Spaces Spanned by the Perturbed Integer Translates of a Radial Function

The problem of approximating smooth L_p-functions from spaces spanned by the integer translates of a radially symmetric function φ is very well understood. In case the points of translation, Ξ, are scattered throughout ^d, the approximation problem is only well understood in the “stationary” setting. In this work, we provide lower bounds on the obtainable approximation orders in the “non-stationary” setting under the assumption that Ξ is a small perturbation of ^d. The functions which we can approximate belong to certain Besov spaces. Our results, which are similar in many respects to the known results for the case Ξ= ^d, apply specifically to the examples of the Gauss kernel and the generalized multiquadric.     

CLASSIFICATION OF SOLUTIONS FOR A CLASS OF SINGULAR INTEGRAL SYSTEM

In this paper, we consider the following integral system: u(x) = R n v q (y) | x y | nα dy, v(x) = R n u p (y) | x y | nμ dy, (0.1) where 0 < α, μ < n; p, q ≥ 1. Using the method of moving planes in an integral form which was recently introduced by Chen, Li, and Ou in [2, 4, 8], we show that all positive solutions of (0.1) are radially symmetric and decreasing with respect to some point under some general conditions of integrability. The results essentially improve and extend previously known results [4, 8].     

On Riemannian 3-manifolds with distinct constant Ricci eigenvalues

The first author was partly supported by the grant GAR 201/93/0469     

Automata and logics over finitely varying functions

We extend some of the classical connections between automata and logic due to Büchi (1960) [5] and McNaughton and Papert (1971) [12] to languages of finitely varying functions or “signals”. In particular, we introduce a natural class of automata for generating finitely varying functions called ’s, and show that it coincides in terms of language definability with a natural monadic second-order logic interpreted over finitely varying functions Rabinovich (2002) [15]. We also identify a “counter-free” subclass of ’s which characterise the first-order definable languages of finitely varying functions. Our proofs mainly factor through the classical results for word languages. These results have applications in automata characterisations for continuously interpreted real-time logics like Metric Temporal Logic (MTL) Chevalier et al. (2006, 2007) [6] and [7].     

Effective Discretization of the Energy Integral and Grunsky Coefficients in Annuli

Let be an analytic Jordan curve in the complex plane. We formulate a discrete minimal energy problem in a suitable class of functions whose solution provides a geometrically fast converging approximation to the equilibrium measure of . For this purpose an extremal point system that was introduced by K. Menke in 1972 is applied. In particular, an explicit error bound for the discretization of the energy integral is computed. The key to this error estimate is a univalence criterion for Laurent series, proved by R. Kuhnau in 1972. Finally, an estimate for the discrepancy between the approximating measures and the equilibrium measure is derived from the discretization error of the energy integral.