Compact Toeplitz operators via the Berezin transform on bounded symmetric domains

Let be an irreducible bounded symmetric domain of genusp, h(x, y) its Jordan triple determinant, andA ² () the standard weighted Bergman space of holomorphic functions on square-integrable with respect to the measureh(z, z) ^–p dz. Extending the recent result of Axler and Zheng for =D, =p=2 (the unweighted Bergman space on the unit disc), we show that ifS is a finite sum of finite products of Toeplitz operators onA ² () and is sufficiently large, thenS is compact if and only if the Berezin transform ofS tends to zero asz approaches . An analogous assertion for the Fock space is also obtained.The author's research was supported by GA AV R grant A1019701 and GA R grant 201/96/0411.     

A note on the position of the centre of flexure of a beam of isotropic material having a section bounded by a parabola and a straight line

Zusammenfassung Die vorliegende Notiz behandelt das Problem der Auffindung des Biegungszentrums eines aus isotropem Material bestehenden Balkens, der einen durch eine parabolische und eine geradlinige Kante begrenzten Querschnitt besitzt. Die Last ist parallel zur geradlinigen Kante angenommen. Der Verfasser benützt zur Lösung der Aufgabe dasselbe Verfahren, dasSokolnikoff [1] für die Auffindung des Biegungszentrums eines halbkreisförmigen Balkens verwendete.

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Nomenclature x, y, z rectangular co-ordinates - l, m, n direction cosines of outward drawn normal - _x , _y , _z normal components of stress parallel tox, y, z axes - _xy , _yz , _zx shearing stress components in rectangular co-ordinates - u, v, w components of displacement - T (x, y) stress function in rectangular coordinates - Poisson's ratio - shear modulus - _yz , _zx , _xy shearing strain components in rectangular coordinates     

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.     

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.     

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.     

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.     

Sources in posets and comparability graphs

A vertex in a poset is a source if its indegree is zero. Further, a vertex in a comparability graph G is a source if there is a transitive orientation of G in which is a source. We characterize sources in comparability graphs in terms of forbidden subgraphs. Certain results follow, including a brief proof of a theorem by Olariu.     

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.     

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.     

Comparative estimation of the degree of crosslinking of styrene-unsaturated polyester copolymers

Copolymers of polydiethylene glycol maleate-sebacates and styrene have been investigated. The degree of crosslinking () has been estimated by a physical method (_p) from the Mooney — Rivlin equation and Wall's formula and also by a chemical method (_c) based on hydrolysis of the copolymer in an alkaline medium. In most cases the difference between the values of calculated from the Mooney — Rivlin equation and Wall's formula does not exceed 20%. It is shown that a linear correlation exists between _p and _c. The fact that the values of _p are lower than those of _c is attributable to the specific characteristics of the molecular network — the presence of short chain segments between neighboring network points and nonload-bearing free ends.Scientific-Research Institute of Plastics, Moscow. Translated from Mekhanika Polimerov, No. 6, pp. 963–968, November–December, 1971.     

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 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.     

Computation of Bessel and Airy Functions and of Related Gaussian Quadrature Formulae

Procedures are described for the high-precision calculation of the modified Bessel function K (x), 0 < < 1, and the Airy function Ai(x), for positive arguments x, as pre-requisites for generating Gaussian quadrature rules having these functions as weight function.     

On the Rate of Local Convergence of High-Order-Infeasible-Path-Following Algorithms for P*-Linear Complementarity Problems

A simple and unified analysis is provided on the rate of local convergence for a class of high-order-infeasible-path-following algorithms for the P_*-linear complementarity problem (P_*-LCP). It is shown that the rate of local convergence of a -order algorithm with a centering step is + 1 if there is a strictly complementary solution and ( + 1)/2 otherwise. For the -order algorithm without the centering step the corresponding rates are and /2, respectively. The algorithm without a centering step does not follow the fixed traditional central path. Instead, at each iteration, it follows a new analytic path connecting the current iterate with an optimal solution to generate the next iterate. An advantage of this algorithm is that it does not restrict iterates in a sequence of contracting neighborhoods of the central path.     

Monge-Kantorovitch Measure Transportation and Monge-Ampère Equation on Wiener Space

Let (W,,H) be an abstract Wiener space assume two _i ,i=1,2 probabilities on (W,(W)). We give some conditions for the Wasserstein distance between ₁ and ₂ with respect to the Cameron-Martin space to be finite, where the infimum is taken on the set of probability measures on W×W whose first and second marginals are ₁ and ₂. In this case we prove the existence of a unique (cyclically monotone) map T=I _W +, with :WH, such that T maps ₁ to ₂. Moreover, if ₂, then T is stochastically invertible, i.e., there exists S:WW such that ST=I _{W 1} a.s. and TS=I _{W 2} a.s. If, in addition, ₁=, then there exists a 1-convex function in the Gaussian Sobolev space such that =. These results imply that the quasi-invariant transformations of the Wiener space with finite Wasserstein distance from can be written as the composition of a transport map T and a rotation, i.e., a measure preserving map. We give also 1-convex sub-solutions and Ito-type solutions of the Monge-Ampère equation on W. Mathematics Subject Classification (2000)60H07, 60H05,60H25, 60G15, 60G30, 60G35, 46G12, 47H05, 47H1, 35J60, 35B65,35A30, 46N10, 49Q20, 58E12, 26A16, 28C20cf. Theorem 6.1 for the precise hypothesis about ₁ and ₂.In fact this hypothesis is too strong, cf. Theorem 6.1. AcknowledgementThe authors are grateful to Françoise Combelles for all the bibliographical help that she has supplied for the realization of this research. We thank also the anonymous referee for his particular attention and valuable remarks.     

Packing circuits in eulerian digraphs

LetG be an eulerian digraph; let (G) be the maximum number of pairwise edge-disjoint directed circuits ofG, and (G) the smallest size of a set of edges that meets all directed circuits ofG. Borobia, Nutov and Penn showed that (G) need not be equal to (G). We show that (G)=(G) provided thatG has a linkless embedding in 3-space, or equivalently, if no minor ofG can be converted toK ₆ by –Y andY– operations.     

The Near Resolvable 2-(13, 4, 3) Designs and Thirteen-Player Whist Tournaments

A -player whist tournament is a schedule of games, where in each round the players are partitioned into games of four players each with at most one player left over. In each game two of the players play as partners against the other two. All pairs of players must play in the same game exactly three times during the tournament; of those three times, they are to play as partners exactly once. Whist tournaments for players are known to exist for all 0,1 (mod 4). The special cases of directed whist tournaments and triplewhist tournaments are known to exist for all sufficiently large , but for small several open cases remain. In this paper we introduce a correspondence between near resolvable 2-(, k, designs and a particular class of codes. The near resolvable 2-(13, 4, 3) designs are classified by classifying the corresponding codes with an orderly algorithm. Finally, the thirteen-player whist tournaments are enumerated starting from the near resolvable 2-(13, 4, 3) designs.Both authors were supported by Helsinki Graduate School in Computer Science and Engineering (HeCSE). In addition, the first author was supported by a grant from the Nokia Foundation and the second author by a grant from the Foundation of Technology, Helsinki, Finland (Tekniikan Edistämissäätiö).AMS classification: primary 05B05, secondary 05B30, 05-04, 94B25     

Directed packings with block size 5 and even ν

Let andk be positive integers. A transitively orderedk-tuple (a ₁,a ₂,...,a _k) is defined to be the set {(a _i, a_j) 1i<jk} consisting ofk(k–1)/2 ordered pairs. A directed packing with parameters ,k and index =1, denoted byDP(k, 1; ), is a pair (X, A) whereX is a -set (of points) andA is a collection of transitively orderedk-tuples ofX (called blocks) such that every ordered pair of distinct points ofX occurs in at most one block ofA. The greatest number of blocks required in aDP(k, 1; ) is called packing number and denoted byDD(k, 1; ). It is shown in this paper that for all even integers , where [x] is the floor ofx.     

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.     

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.