Characterizations of Riesz spaces with b-property

A Riesz space E is said to have b-property if each subset which is order bounded in E^~~ is order bounded in E. The relationship between b-property and completeness, being a retract and the absolute weak topology |σ|(E^~, E) is studied. Perfect Riesz spaces are characterized in terms of b-property. It is shown that b-property coincides with the Levi property in Dedekind complete Frechet lattices.      

Symplectic embeddings of ellipsoids

We study the rigidity and flexibility of symplectic embeddings in the model case in which the domain is a symplectic ellipsoid. It is first proved that under the conditionr _n ² ≤2r ₁ ² the symplectic ellipsoidE(r ₁,…,r _n)with radiir ₁≤…≤r _ndoes not symplectically embed into a ball of radius strictly smaller thanr _n.We then use symplectic folding to see that this condition is sharp. We finally sketch a proof of the fact that any connected symplectic 4-manifold of finite volume can be asymptotically filled with skinny ellipoids.     

Generalization of one Poletskii lemma to classes of space mappings

The paper is devoted to investigations in the field of space mappings. We prove that open discrete mappings f ∈ W ^1,n _loc such that their outer dilatation K _O (x, f) belongs to L ⁿ⁻¹ _loc and the measure of the set B _f of branching points of f is equal to zero have finite length distortion. In other words, the images of almost all curves γ in the domain D under the considered mappings f : D → ℝ ⁿ , n ≥ 2, are locally rectifiable, f possesses the (N)-property with respect to length on γ, and, furthermore, the (N)-property also holds in the inverse direction for liftings of curves. The results obtained generalize the well-known Poletskii lemma proved for quasiregular mappings.     

Cuttings for Disks and Axis-Aligned Rectangles in Three-Space

We present new asymptotically tight bounds on cuttings, a fundamental data structure in computational geometry. For n objects in space and a parameter r∈?, a $\frac{1}{r}We present new asymptotically tight bounds on cuttings, a fundamental data structure in computational geometry. For n objects in space and a parameter r∈ℕ, a \frac1r\frac{1}{r} -cutting is a covering of the space with simplices such that the interior of each simplex intersects at most n/r objects. For n pairwise disjoint disks in ℝ³ and a parameter r∈ℕ, we construct a \frac1r\frac{1}{r} -cutting of size O(r ²). For n axis-aligned rectangles in ℝ³, we construct a \frac1r\frac{1}{r} -cutting of size O(r ^3/2). As an application related to multi-point location in three-space, we present tight bounds on the cost of spanning trees across barriers. Given n points and a finite set of disjoint disk barriers in ℝ³, the points can be connected with a straight line spanning tree such that every disk is stabbed by at most O(?n)O(\sqrt{n}) edges of the tree. If the barriers are axis-aligned rectangles, then there is a straight line spanning tree such that every rectangle is stabbed by O(n ^1/3) edges. Both bounds are best possible.     

An integral operator related to the Stokes system in exterior domains

Consider a bounded domain Ω in ?³ with C²-boundary ?Ω. In [1] the Stokes problem in the exterior domain ?³/Ω , with resolvent parameter [λ??\] ? [∞,0], is solved by using the method of integral equations. However, for estimating the corresponding solutions in L^p norms, it turns out that a certain operator defined on the spaces L^r(?Ω)³, for r ?]1, ∞[, has to be evaluated in the norm of L^r(?Ω)³. This estimate is proved in the present paper.     

Free convection above a heated horizontal circular disk

The free convection boundary layer flow above a heated horizontal disk is considered. The equations of motion are solved numerically starting at the circumference of the disk where the flow is basically the same as that above a flat plate. The importance of the curvature effects increases as the centre is approached. It is shown that near the centre, the boundary-layer thickness is very large, and that the flow splits up into two distinct regions. There is a thin viscous region next to the disk of thickness ofO(r ^2/3), wherer measures distance from the centre and a thick outer inviscid region of thickness ofO(r ^?2/3).     

Distance–regular graphs having the M-property

We analyse when the Moore–Penrose inverse of the combinatorial Laplacian of a distance–regular graph is an M-matrix; that is, it has non-positive off-diagonal elements or, equivalently when the Moore–Penrose inverse of the combinatorial Laplacian of a distance–regular graph is also the combinatorial Laplacian of another network. When this occurs we say that the distance–regular graph has the M-property. We prove that only distance–regular graphs with diameter up to three can have the M-property and we give a characterization of the graphs that satisfy the M-property in terms of their intersection array. Moreover, we exhaustively analyse strongly regular graphs having the M-property and we give some families of distance–regular graphs with diameter three that satisfy the M-property. Roughly speaking, we prove that all distance–regular graphs with diameter one; about half of the strongly regular graphs; only some imprimitive distance–regular graphs with diameter three, and no distance–regular graphs with diameter greater than three, have the M-property. In addition, we conjecture that no primitive distance–regular graph with diameter three has the M-property.     

Reiter’s condition for amenable hypergroups

We study modifications of Reiter’s condition (P _r ) which are generated by certain power and root procedures. In that way we can illustrate the difference between the (P ₁)- and the (P ₂)-property. Furthermore we present equivalent conditions to (P ₂). In order to have examples we discuss the results for polynomial hypergroups.     

Global boundary regularity for the $ \bar \partial $‐equation on q‐pseudoconvex domains

We introduce a notion of q ‐pseudoconvex domain of new type for a bounded domain of ?ⁿ and prove that for given a ‐closed (p, r)‐form, r ≥ q, that is smooth up to the boundary, there exists a (p, r – 1)‐form smooth up to the boundary which is a solution of ‐equation on a bounded q ‐pseudoconvex domain. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Picard values of <Emphasis Type="Italic">p</Emphasis>-adic meromorphic functions

We investigate Picard-Hayman behavior of derivatives of meromorphic functions on an algebraically closed field K, complete with respect to a non-trivial ultrametric absolute value. We present an analogue of the well-known Hayman’s alternative theorem both in K and in any open disk. Here the main hypothesis is based on the behaviour of |f|(r) when r tends to +∞ on properties of special values and quasi-exceptional values.We apply this study to give some sufficient conditions on meromorphic functions so that they satisfy Hayman’s conjectures for n = 1and for n = 2. Given a meromorphic transcendental function f, at least one of the two functions f′f and f′f ² assumes all non-zero values infinitely often. Further, we establish that if the sequence of residues at simple poles of a meromorphic transcendental function on K admits no infinite stationary subsequence, then either f′ + af ² has infinitely many zeros that are not zeros of f for every a ∈ K* or both f′ + bf ³ and f′ + bf ⁴ have infinitely many zeros that are not zeros of f for all b ∈ K*. Most of results have a similar version for unbounded meromorphic functions inside an open disk.     

Invariant Subspaces for Banach Space Operators with a Multiply Connected Spectrum

We consider a multiply connected domain where denotes the unit disk and denotes the closed disk centered at with radius r _j for j = 1, . . . , n. We show that if T is a bounded linear operator on a Banach space X whose spectrum contains ∂Ω and does not contain the points λ₁, λ₂, . . . , λ _n , and the operators T and r _j (T − λ _j I)⁻¹ are polynomially bounded, then there exists a nontrivial common invariant subspace for T ^* and (T − λ _j I)^*-1.     

Approximation of Cauchy-Type Integrals

We investigate approximations of analytic functions determined by Cauchy-type integrals in Jordan domains of the complex plane. We develop, modify, and complete (in a certain sense) our earlier results. Special attention is given to the investigation of approximation of functions analytic in a disk by Taylor sums. In particular, we obtain asymptotic equalities for upper bounds of the deviations of Taylor sums on the classes of -integrals of functions analytic in the unit disk and continuous in its closure. These equalities are a generalization of the known Stechkin's results on the approximation of functions analytic in the unit disk and having bounded rth derivatives (here, r is a natural number).On the basis of the results obtained for a disk, we establish pointwise estimates for the deviations of partial Faber sums on the classes of -integrals of functions analytic in domains with rectifiable Jordan boundaries. We show that, for a closed domain, these estimates are exact in order and exact in the sense of constants with leading terms if and only if this domain is a Faber domain.     

Radiation boundary conditions for wave-like equations

In the numerical computation of hyperbolic equations it is not practical to use infinite domains. Instead, one truncates the domain with an artificial boundary. In this study we construct a sequence of radiating boundary conditions for wave-like equations. We prove that as the artificial boundary is moved to infinity the solution approaches the solution of the infinite domain as O(r^?m?1/2) for the m-th boundary condition. Numerical experiments with problems in jet acoustics verify the practical nature and utility of the boundary conditions.     

Stable Local Bases for Multivariate Spline Spaces

We present an algorithm for constructing stable local bases for the spaces ^r_d(Δ) of multivariate polynomial splines of smoothness r1 and degree dr2ⁿ+1 on an arbitrary triangulation Δ of a bounded polyhedral domain Ω ⁿ, n2.     

The Katětov property for finite degree seminormal functors

A seminormal functor kF enjoys the Katěetov property (K-property) if for every compact set X the hereditary normality of kF(X) implies the metrizability of X. We prove that every seminormal functor of finite degree n>3 enjoys the K-property. On assuming the continuum hypothesis (CH) we characterize the weight preserving seminormal functors with the K-property. We also prove that the nonmetrizable compact set constructed in [1] on assuming CH is a universal counterexample for the K-property in the class of weight preserving seminormal functors.     

Reducing Subspaces on the Annulus

We study reducing subspaces for an analytic multiplication operator M_zⁿ{M_{z^{n}}} on the Bergman space L_a²(A_r){L_{a}^{2}(A_{r})} of the annulus A _r , and we prove that M_zⁿ{M_{z^{n}}} has exactly 2 ⁿ reducing subspaces. Furthermore, in contrast to what happens for the disk, the same is true for the Hardy space on the annulus. Finally, we extend the results to certain bilateral weighted shifts, and interpret the results in the context of complex geometry.     

Crack Singularities for General Elliptic Systems

We consider general homogeneous Agmon‐Douglis‐Nirenberg elliptic systems with constant coefficients complemented by the same set of boundary conditions on both sides of a crack in a two‐dimensional domain. We prove that the singular functions expressed in polar coordinates (r, θ) near the crack tip all have the form r^{k + 1/2}φ(θ) with k ≥ 0 integer, with the possible exception of a finite number of singularities of the form r^k log r φ(θ). We also prove results about singularities in the case when the boundary conditions on the two sides of the crack are not the same, and in particular in mixed Dirichlet‐Neumann boundary value problems for strongly coercive systems: in the latter case, we prove that the exponents of singularity have the form with real η and integer k. This is valid for general anisotropic elasticity too.     

A Reflexivity Result Concerning Banach Space Operators with a Multiply Connected Spectrum

We consider a multiply connected domain Ω which is obtained by removing n closed disks which are centered at λ _j with radius r _j for j = 1, . . . , n from the unit disk. We assume that T is a bounded linear operator on a separable reflexive Banach space whose spectrum contains ∂Ω and does not contain the points λ₁, λ₂, . . . , λ _n , and the operators T and r _j (T − λ _j I)⁻¹ are polynomially bounded. Then either T has a nontrivial hyperinvariant subspace or the WOT-closure of the algebra {f(T) : f is a rational function with poles off [`(W)]{\overline\Omega}} is reflexive.     

A remark on level curves for domains convex in one direction † ‡

Let D be a plane domain which is convex in the v-direction, i.e.t the intersection of D with each vertical line is connected (or empty). It has been an open question whether level curves of a domain convex in the v-direction bound a domain with the same property. In this note we construct an example which settles the question in the negative.Closely related is a family ? of analytic functions g in the unit disk with the property that g(0) = 0 and Re{(1 - z²)g(z)/z} ? 0. For univalent functions we show that membership in ? is essentially characterized by the geometric condition that Im g(eⁱ⁰)? 0 for a.e. ? ? (0, ?) and Im g(eⁱ⁰)? 0 for a.e. We conclude with a coefficient theorem     

Matroids with No (q+2)-Point-Line Minors

It is known that a geometry with rankrand no minor isomorphic to the (q+2)-point line has at most (q^r−1)/(q−1) points, with strictly fewer points ifr>3 andqis not a prime power. Forqnot a prime power andr>3, we show thatq^r−1−1 is an upper bound. Forqa prime power andr>3, we show that any rank-rgeometry with at leastq^r−1points and no (q+2)-point-line minor is representable overGF(q). We strengthen these bounds toq^r−1−(q^r−2−1)/(q−1)−1 andq^r−1−(q^r−2−1)/(q−1) respectively whenqis odd. We give an application to unique representability and a new proof of Tutte's theorem: A matroid is binary if and only if the 4-point line is not a minor.