Limits of balayage measures

LetA be a subset of a balayage space (X,W) and a measure onX. It is shown that for every sequence _n of measures such that lim_nn and lim_{n n} ^A = the limit measure is of the formf+[(1-f)]^A for some (unique) Borel function 0f1_Cb(A). Furthermore, conditions are given such that any such functionf occurs.     

On the E- and MV-optimality of block designs having k≥v

In this paper we consider the problem of determining and constructing E- and MV-optimal block designs to use in experimental settings where treatments are applied to experimental units occurring in b blocks of size k, k. It is shown that some of the well-known methods for constructing E- and MV-optimal unequally replicated designs having k fail to yield optimal designs in the case where . Some sufficient conditions are derived for the E- and MV-optimality of block designs having and methods for constructing designs satisfying these sufficient conditions are given.     

Decktransformationen transnormaler Mannigfaltigkeiten

For every transnormal m-manifold V (see [3] or [7]) in ⁿ :VW, mapping pV into its normal plane (p) is a covering map onto a submanifold W of the open Grassmannian H_n,n–m of all (n–m)-dimensional planes in ⁿ. The transnormal frame T:=^–1((p)) admits a transitive operation by a group J of isometries. The group action of the covering transformations of (V,,W) on T commutes with the action of J. The elements of J, which are restrictions of covering transformations to T, are exactly the elements of the centre of J. This property is applied to show the existence of nontrivial covering transformations of (V,,W) for n–m3.

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Diese Arbeit faßt die Kapitel 5, 6 und 7 der von der Fakultät für Allgemeine Ingenieurwissenschaften der TU Berlin genehmigten Dissertation [6] zusammen.     

Weak Sequential Completeness of β-duals of Double Sequence Spaces

We consider dual pairs E,E ⁽⁾ of double sequence spaces E and E ⁽⁾, where E ⁽⁾ is the -dual space of E with respect to the -convergence of double sequences for = p (Pringsheim convergence), bp (bounded p-convergence) and r (regular convergence). Motivated by Boos, Fleming and Leiger [3], we introduce two oscillating properties (signed P_OSCP(k), k {1,2}) for a double sequence space E such that the signed P_OSCP(1) guarantees the (E ^(p), E)-sequential completeness of E ^(p), whereas the signed P_OSCP(2) implies the equalities E ^(r) = E ^(bp) = E ^(p) and the (E ⁽⁾, E)-sequentialcompleteness of E ⁽⁾ for = bp and r.     

On the strongly spherical Sasakian metric of a spherical tangent bundle

Let Mⁿ denote an n-dimensional Riemannian manifold. Its metric is called -strongly spherical if at every point Q Mⁿ there exists a -dimensional subspace _Q T_QMⁿ such that the curvature operator of the metric of Mⁿ satisfies R(X, Y) Z = k(< Y, Z > X < X, Z > Y), where k = const > 0, Y _Q , X, Z #x2208; T_QMⁿ. The number is called the index of sphericity and k the exponent of sphericity. The following theorems are proved in the paper.THEOREM 1. Let the Sasakian metric of T₁Mⁿ be -strongly spherical with exponent of sphericity k. The following assertions hold: a) = 1 if and only if M² has constant Gaussian curvature K 1 and k = K²/4; b) = 3 if and only if M² has constant curvature K = 1 and k = 1/4; c) = 0, otherwise.THEOREM 2. Let the Sasakian metric of T₁Mⁿ (n Mⁿ) be -strongly spherical with exponent of sphericity k. If k > 1/3 and k 1, then = 0. Let us denote by (Mⁿ, K) a space of constant curvatureK. THEOREM 3. Let the Sasakian metric of T₁(Mⁿ, K) (n 3) be -strongly spherical with exponent of sphericity k. The following assertions hold: a) = 1 if and only if K = 1/4; b) = 0, otherwise. In dimension n = 3 Theorem 2 is true for k {1/4, 1}.Translated from Ukrainskii Geometricheskii Sbornik, No. 35, pp. 150–159, 1992.     

Aperture-Angle Optimization Problems in Three Dimensions

Let [a,b] be a line segment with end points a, b and a point at which a viewer is located, all in R ³. The aperture angle of [a,b] from point , denoted by (), is the interior angle at of the triangle (a,b,). Given a convex polyhedron P not intersecting a given segment [a,b] we consider the problem of computing _max() and _min(), the maximum and minimum values of () as varies over all points in P. We obtain two characterizations of _max(). Along the way we solve several interesting special cases of the above problems and establish linear upper and lower bounds on their complexity under several models of computation.     

Brownian Motion in a Hilbert Space with a Semipermeable Membrane on a Hyperplane

In a separable Hilbert space, we construct a continuous Markov process whose behavior coincides everywhere, except for a hyperplane S orthogonal to a given unit vector , with the behavior of a homogeneous Gaussian process with a given correlation operator tB, where B is a nonsingular nuclear operator. As the process hits the hyperplane, it receives an impulse infinite in modulus in the direction A such that |(A, )| (B, ).We obtain a stochastic differential equation whose solutions are trajectories of the process constructed.     

Geometric methods in the study of irregularities of distribution

Let be a signed measure on E ^d with E ^d =0 and ¦¦E^d<. DefineD _s() as sup ¦H¦ whereH is an open halfspace. Using integral and metric geometric techniques results are proved which imply theorems such as the following.Theorem A. Let be supported by a finite pointsetp _i. ThenD _s()>c _d(₁/ ₂)^1/2{ _{i(p i})²}^1/2 where ₁ is the minimum distance between two distinctp _i, and ₂ is the maximum distance. The numberc _d is an absolute dimensional constant. (The number .05 can be chosen forc ₂ in Theorem A.)Theorem B. LetD be a disk of unit area in the planeE ², andp ₁,p ₂,...,p _n be a set of points lying inD. If m if the usual area measure restricted toD, while nP _i=1/n defines an atomic measure _n, then independently of _n,nD _s(m – _n) .0335n ^1/4. Theorem B gives an improved solution to the Roth disk segment problem as described by Beck and Chen. Recent work by Beck shows thatnD _s(m – _n)cn ^1/4(logn)^–7/2.     

Integral points on strictly convex closed curves

A negative answer is given to Swinnerton-Dyer's question: Is it true that for any > 0 there exists a positive integer n such that for any planar closed strictly convex n-times differentiable curve , when it is blown up a sufficiently large number of times, the number of integral points on the resultant curve will be less than . An example has been constructed when this number for an infinite number is not less than ^1/2, while is infinitely differentiable.Translated from Matematicheskie Zametki, Vol. 21, No. 6, pp. 799–805, June, 1977.The author thanks S. B. Stechkin for attention to the work.     

Criteria for the Boundedness and Compactness of Operators with Power-Logarithmic Kernels

In the present work, necessary and sufficient conditions are given in terms of a nonnegative Borel measure which ensure the boundedness and compactness of operators with power-logarithmic kernels from L ^p (0, a) to L ^p (0, a) (or to L ^q (0, a)), where 0 < a < , 1 < p, q < , > 1/p and 0.     

Multidimensional Hardy inequalities and the absence of positive eigenvalues for the Schrödinger operator with a complex potential

The operator –(d/dx)²+ where is a complex function, |J(x)(1+|x|¹⁺|>, has no positive eigenvalues. This follows from the existence of a triangular Green function for the operator –(d/dx)^z–,gl>0, and from Hardy type inequalities. We give the multidimensional analogues of these inequalities and we prove the absence of positive eigenvalues for the operator –+.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 138, pp. 33–34, 1984.     

An automatic integration procedure for infinite range integrals involving oscillatory kernels

Let the real functionsK(x) andL(x) be such thatM(x)=K(x)+iL(x)=e^ix g(x), whereg(x) is infinitely differentiable for all largex and is non-oscillatory at infinity. We develop an efficient automatic quadrature procedure for numerically computing the integrals _a K(t)f(t) and _a L(t)f(t)dt, where the functionf(t) is smooth and nonoscillatory at infinity. One such example for which we also provide numerical results is that for whichK(x)=J (x) andL(x)=Y (x), whereJ (x) andY (x) are the Bessel functions of order . The procedure involves the use of an automatic scheme for Fourier integrals and the modified W-transformation which is used for computing oscillatory infinite integrals.     

Analytic structures for subnormal operators

For a compactly supported measure on , we construct a mutually absolutely continuous measure so thatP ²() has analytic bounded point evaluations, and the operator of multiplication byz onP ²() has every invariant subspace hyperinvariant. We also construct an equivalent measure so thatR ²(K, ) has as analytic bounded point evaluations precisely the interior of the set of weak-star continuous point evaluations ofR (K, ). In the course of the proof, we classify weak-star closed super-algebras ofR (K, ) whenR(K) is hypo-Dirichlet.     

On the immersion of metrics close to immersible metrics

For a C^r,-immersion z:X E, r 2, 0 < < 1, of an n-dimensional (n 1) simply-connected C^r+2,-manifold X into Euclidean space E, the metric I(z) induced by z has a neighborhood in C^r,-topology in which every metric from a given subbundle of metrics is C^r,-immersible into E. In particular, it is proved that metric ds ₀ ² of the Riemannian product of p spheres of dimensions ₁, , _p 2 has a neighborhood in C^2,-topology from which any conformally equivalent metric to ds ₀ ² , is immersible into E with dimE = ₁ + + _p + p. The proofs are based on the investigation of a varied system of Gauss—Codazzi—Ricci equations for an infinitely small deformation of surface z(X) in E with a prescribed variation of the metric.Translated from Ukrainskii Geometricheskii Sbornik, No. 35, pp. 49–67, 1992.     

Fourier series of additive vector measures and their term-by-term differentiation

On a measurable space (T, , ) we choose an additive measure: Z (Z is a Banach space) with the following property: for alle , we have ; this measure defines an indefinite integral over the measure onL ² (T, ,). We prove that if { _n (t)} _n ^=1/ is an orthonormal basis inL ² and _n (e)=e _n (t) d, then any additive measure: Z whose Radon-Nikodým derivatived/d belongs toL ² is uniquely expandable in a series(e)= _n ^=1/ _n n(e) that converges to(e) uniformly with respect toe can be differentiated term-by-term, and satisfies _n ^=1/ _n ^/2 <. In the caseL ²[0,2],Z=, the Fourier series of a 2-periodic absolutely continuous functionF(t) such thatF'(t) L ²[0, 2] is superuniformly convergent toF(t).Translated fromMatematicheskie Zametki, Vol. 64, No. 2, pp. 180–184, August, 1998.     

Two Tree-Width-Like Graph Invariants

In this paper we introduce two tree-width-like graph invariants. The first graph invariant, which we denote by ⁼(G), is defined in terms of positive semi-definite matrices and is similar to the graph invariant (G), introduced by Colin de Verdière in [J. Comb. Theory, Ser. B., 74:121–146, 1998]. The second graph invariant, which we denote by (G), is defined in terms of a certain connected subgraph property and is similar to (G), introduced by van der Holst, Laurent, and Schrijver in [J. Comb. Theory, Ser. B., 65:291–304, 1995]. We give some theorems on the behaviour of these invariants under certain transformations. We show that ⁼(G)=(G) for any graph G with ⁼(G)4, and we give minimal forbidden minor characterizations for the graphs satisfying ⁼(G)k for k=1,2,3,4.This paper is extracted from two chapters of [7]. This work was done while the author was at the Centrum voor Wiskunde en Informatica, Kruislaan 413, 1098 SJ Amsterdam, The Netherlands.     

Nonnegative Hall Polynomials

The number of subgroups of type and cotype in a finite abelian p-group of type is a polynomialg with integral coefficients. We prove g has nonnegative coefficients for all partitions and if and only if no two parts of differ by more than one. Necessity follows from a few simple facts about Hall-Littlewood symmetric functions; sufficiency relies on properties of certain order-preserving surjections that associate to each subgroup a vector dominated componentwise by . The nonzero components of (H) are the parts of , the type of H; if no two parts of differ by more than one, the nonzero components of – (H) are the parts of , the cotype of H. In fact, we provide an order-theoretic characterization of those isomorphism types of finite abelian p-groups all of whose Hall polynomials have nonnegative coefficients.     

Maximum Norm Resolvent Estimates for Elliptic Finite Element Operators

We study a finite element approximation A _h, based on simplicial Lagrange elements, of a second order elliptic operator A under homogeneous Dirichlet boundary conditions in two and three dimensions, where h is thought of as a meshsize. The main result of the paper is a new resolvent estimate for the operator A _h in the L -norm. This estimate is uniform with respect to h for the case with at least quadratic elements. In the case with linear elements, the estimate contains on the right a factor proportional to (log log ), where = 1 or = in two or three dimensions, respectively.This revised version was published online in October 2005 with corrections to the Cover Date.     

The Kronecker Product of Schur Functions Indexed by Two-Row Shapes or Hook Shapes

The Kronecker product of two Schur functions s and s , denoted by s * s , is the Frobenius characteristic of the tensor product of the irreducible representations of the symmetric group corresponding to the partitions and . The coefficient of s in this product is denoted by , and corresponds to the multiplicity of the irreducible character in .We use Sergeev's Formula for a Schur function of a difference of two alphabets and the comultiplication expansion for s [XY] to find closed formulas for the Kronecker coefficients when is an arbitrary shape and and are hook shapes or two-row shapes.Remmel (J.B. Remmel, J. Algebra 120 (1989), 100–118; Discrete Math. 99 (1992), 265–287) and Remmel and Whitehead (J.B. Remmel and T. Whitehead, Bull. Belg. Math. Soc. Simon Stiven 1 (1994), 649–683) derived some closed formulas for the Kronecker product of Schur functions indexed by two-row shapes or hook shapes using a different approach. We believe that the approach of this paper is more natural. The formulas obtained are simpler and reflect the symmetry of the Kronecker product.     

Generalization of some classical inequalities in the theory of orthogonal series

Let {X_i} _– be a sequence of random variables, E(X_i) 0. If 1, estimates for the -th moments can be derived from known estimates of the -th moment. Here we generalized the Men'shov-Rademacher inequality for =2 for orthonormal X_i, to the case 1 and dependent random variables. The Men'shov-Payley inequality >2 for orthonormal X_i) is generalized for >2 to general random variables. A theorem is also proved that contains both the Erdös -Stechkin theorem and Serfling's theorem withv > 2 for dependent random variables.Translated from Matematicheskie Zametki, Vol. 17, No. 2, pp. 219–230, February, 1975.This article was written while the author was working in the V. A. Steklov Mathematics Institute, Academy of Sciences of the USSR.