Newton polyhedra and the Bergman kernel

The purpose of this paper is to study singularities of the Bergman kernel at the boundary for pseudoconvex domains of finite type from the viewpoint of the theory of singularities. Under some assumptions on a domain in ⁿ⁺¹ , the Bergman kernel B(z) of takes the form near a boundary point p: where (w,) is some polar coordinates on a nontangential cone with apex at p and means the distance from the boundary. Here admits some asymptotic expansion with respect to the variables ^{1/ m} and log(1/) as 0 on . The values of d _F >0, m _F ₊ and m are determined by geometrical properties of the Newton polyhedron of defining functions of domains and the limit of as 0 on is a positive constant depending only on the Newton principal part of the defining function. Analogous results are obtained in the case of the Szegö kernel. Mathematics Subject Classification (2000):32A25, 32A36, 32T25, 14M25.     

Solvability of partial differential equations of infinite order in certain classes of entire functions

In this paper it is shown that under conditions of applicability of the operator to the class [,] =(I,s), ₂ 1, ₂), ₁, ₂< the equation y=f has a particular solution of this class vf[, ]. The general form of a solution of the homogeneous equation y=0 is established. The growth of a solution is investigated by means of a system of conjugate orders and a system of conjugate types. A solvability result is also obtained in the class , where T is a certain set in R ₊ ² depending on the operator .Translated from Matematicheskie Zametki, Vol. 19, No. 2, pp. 225–236, February, 1976.In conclusion, the author would like to express his thanks to his adviser, Yu. F. Korobeinik.     

The conditions of selfadjointness of the operator of singular integration

It is proved that the operator of singular integration along a bounded rectifiable curve is self-adjoint in the weighted spaceL ₂(, ) if and only if is a circle and (t)=const.     

The Hilbert–Poincaré Series for Some Algebras of Invariants of Cyclic Groups

Let be a linear representation of a finite group over a field of characteristic 0. Further, let R be the corresponding algebra of invariants, and let P (t) be its Hilbert–Poincaré series. Then the series P (t) represents a rational function (t)/(t). If R is a complete intersection, then (t) is a product of cyclotomic polynomials. Here we prove the inverse statement for the case where is an almost regular (in particular, regular) representation of a cyclic group. This yields an answer to a question of R. Stanley in this very special case. Bibliography: 3 titles.     

The Whitehead Group of the Novikov Ring

The Bass–Heller–Swan–Farrell–Hsiang–Siebenmann decomposition of the Whitehead group K ₁(A[z,z^-1]) of a twisted Laurent polynomial extension A[z,z^-1] of a ring A is generalized to a decomposition of the Whitehead group K ₁(A((z))) of a twisted Novikov ring of power series A((z))=A[[z]][z^-1]. The decomposition involves a summand W₁(A, ) which is an Abelian quotient of the multiplicative group W(A,) of Witt vectors 1+a₁z+a₂z²+ ··· A[[z]]. An example is constructed to show that in general the natural surjection W(A, )^ab W₁(A, ) is not an isomorphism.     

Taut Distance-Regular Graphs of Odd Diameter

Let denote a bipartite distance-regular graph with diameter D 4, valency k 3, and distinct eigenvalues ₀ > ₁ > ··· > _D. Let M denote the Bose-Mesner algebra of . For 0 i D, let E _i denote the primitive idempotent of M associated with _i . We refer to E ₀ and E _D as the trivial idempotents of M. Let E, F denote primitive idempotents of M. We say the pair E, F is taut whenever (i) E, F are nontrivial, and (ii) the entry-wise product E F is a linear combination of two distinct primitive idempotents of M. We show the pair E, F is taut if and only if there exist real scalars , such that _{i + 1 i + 1} – _{i – 1 i – 1} = _i ( _{i + 1} – _{i – 1}) + _i ( _{i + 1} – _{i – 1}) + (1 i D – 1)where ₀, ₁, ..., _D and ₀, ₁, ..., _D denote the cosine sequences of E, F, respectively. We define to be taut whenever has at least one taut pair of primitive idempotents but is not 2-homogeneous in the sense of Nomura and Curtin. Assume is taut and D is odd, and assume the pair E, F is taut. We show

Some inequalities about the covering radius of Reed-Muller codes

Let R(r, m) be the rth order Reed-Muller code of length 2 ^m , and let (r, m) be its covering radius. We prove that if 2 k m - r - 1, then (r + k, m + k) (r, m + 2(k - 1). We also prove that if m - r 4, 2 k m - r - 1, and R(r, m) has a coset with minimal weight (r, m) which does not contain any vector of weight (r, m) + 2, then (r + k, m + k) (r, m) + 2k(. These inequalities improve repeated use of the known result (r + 1, m + 1) (r, m).This work was supported by a grant from the Research Council of Wright State University.     

Foliated control theory, II

The main result is a control theorem for the structure space of E with control near the leaves F in M, where : E M is a fiber bundle over the Riemannian manifold M having a compact closed manifold for fiber and F is a smooth foliation of M, each leaf of which inherits a flat Riemannian geometry from M. A similar result has been proved by the authors under the assumption that each leaf of F is one-dimensional and the fiber of : E M is homotopy stable.Both authors were supported in part by the National Science Foundations.     

Exact Values of Widths of Classes of Analytic Functions on the Disk and Best Linear Approximation Methods

In the Hardy space H _p, (p1, 0< 1, H _p,1 H _p) we develop best linear approximation methods (previously studied by Taikov and Ainulloev) for the classes W(r,,) of analytic functions on the unit disk and calculate the exact values of linear, Gelfand, and informational n-widths of these classes.     

Near boundary behavior of primal—dual potential reduction algorithms for linear programming

This paper is concerned with selection of the-parameter in the primal—dual potential reduction algorithm for linear programming. Chosen from [n + , ), the level of determines the relative importance placed on the centering vs. the Newton directions. Intuitively, it would seem that as the iterate drifts away from the central path towards the boundary of the positive orthant, must be set close ton + . This increases the relative importance of the centering direction and thus helps to ensure polynomial convergence. In this paper, we show that this is unnecessary. We find for any iterate that can be sometimes chosen in a wide range [n + , ) while still guaranteeing the currently best convergence rate of O( L) iterations. This finding is encouraging since in practice large values of have resulted in fast convergence rates. Our finding partially complements the recent result of Zhang, Tapia and Dennis (1990) concerning the local convergence rate of the algorithm.Research supported in part by NSF Grant DDM-8922636.     

On Conservative Confidence Intervals

The subject of the paper – (conservative) confidence intervals – originates in applications to auditing. Auditors are interested in upper confidence bounds for an unknown mean for all sample sizes n. The samples are drawn from populations such that often only a few observations are nonzero. The conditional distribution of an observation given that it is nonzero usually has a very irregular shape. However, it can be assumed that observations are bounded. We propose a way to reduce the problem to inequalities for tail probabilities of certain relevant statistics. Note that a traditional approach involving limit theorems forces to impose additional conditions on regularity of samples and leads to approximate or asymptotic bounds. In the case of , as a statistic we can use sample mean, say , and we have to use Hoeffding [7] inequalities, since currently they are the best available. This leads to upper confidence bounds for which are of (asymptotic) size at most in the case of risk =0.05, where is the unknown standard deviation. We have , where is the bound in a model with normally distributed observations. It seems that the bound is very robust and can be improved replacing Hoeffding's inequalities by more refined ones. The commonly used Stringer bound (it is still not known whether it is an upper confidence bound) is of asymptotic size c with equality only for Bernoulli distributions, and the ratio c / can be arbitrary large already for rather simple distributions. Our bounds can involve a priori information (professional judgment of an auditor) of type ₀ or/and ₀, which leads to improvements. Most of the results also hold for sampling without replacement from finite populations. The i.i.d. condition can be replaced by a martingale-type dependence assumption. Finally, the results can be extended to the noni.i.d. case and for settings with several samples.     

On the best polynomial approximation of entire transcendental functions in Banach spaces. II

We study the behavior of the best approximations of entire transcendental functionsf(z) of the order =0 by polynomials of at mostn th degree in the metric of the spaceE _p(),p1. In particular, we describe the relationship between the best approximations and the logarithmic order _L and type _L of the functionf(z).Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 10, pp. 1318–1322, October, 1994.     

On the radicals of structural matrix rings

The relationship between the radical of a ringR and a structural matrix ring overR has been determined for some radicals. We continue these investigations, amongst others, determining exactly which radicals have the property (M(,R))=M( _s ,(R))+M( _a ,⁺(R))for any structural matrix ringM(,R) and finding (M(,R)) for any hereditary subidempotent radical .     

Properties of solutions of linear functional-differential equations depending on a parameter

On the segment 0 t1 we study the equation A(d/dt, )x(t) + [F()x](t)=f(t), whereA (d/dt, ) x=x( ⁿ )+A ₁ x(^n–1 +...+ ⁿ A _n x, the matrices A₁,...,A_n are of size m × m, x is an unknown and f a given function with values in the m-dimensional space ^m , F() is a linear operator acting from a Hölder space to a Lebesgue space of vectorfunctions with values in ^m and depending on a complex parameter . We find the set of those at which a one-to-one correspondence is established between the solutions of the given equation and the solutions of the equation A(d/dt, )x(t)=0.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 9, pp. 1213–1231, September, 1991.     

Relationship between the Hadamard Theorem on Three Disks and Certain Problems of Polynomial Approximation of Analytic Functions

By using the classical Hadamard theorem, we obtain an exact (in a certain sense) inequality for the best polynomial approximations of an analytic function f(z) from the Hardy space H _p, p 1, in disks of radii , ₁, and ₂, 0 < ₁ < < ₂ < 1.     

Finitely Valued f-Modules, An Addendum

In an -group M with an appropriate operator set it is shown that the -value set (M) can be embedded in the value set (M). This embedding is an isomorphism if and only if each convex -subgroup is an -subgroup. If (M) has a.c.c. and M is either representable or finitely valued, then the two value sets are identical. More generally, these results hold for two related operator sets ₁ and ₂ and the corresponding -value sets and . If R is a unital -ring, then each unital -module over R is an f-module and has exactly when R is an f-ring in which 1 is a strong order unit.     

A Comparison of Two Spaces of Generalized White Noise Functionals

In the literature (see [5, 6, 8]) there are two families of spaces called Kondratiev spaces: (_c)^± and (S _c)^± for 0 1. We investigate the relation between the spaces and show that they are topologically isomorphic when (^d) L2 (^d) (^d) is the underlying Gel'fand triple for (_c)^±. In this case we also give the explicit relation between the S-transform and -transform on (_c)^-1 and (S _c)^-1, respectively.     

Classes of pairs of meromorphic matrix valued functions generated by nonnegative kernels and associated Nevanlinna-Pick problems

Families of pairs of matrix-valued meromorphic functions (,P) depending on two parameters andP are introduced. They are the projective analogues of classes of functions studied in [1] and include as special cases the projective Schur, Nevanlinna and Carathéodory classes. A two sided Nevanlinna-Pick interpolation problem is defined and solved in (,P), using the fundamental matrix inequality method.     

Posterior Sensitivity to the Sampling Distribution and the Prior: More than One Observation

Sensitivity of a posterior quantity (f, P) to the choice of the sampling distribution f and prior P is considered. Sensitivity is measured by the range of (f, P) when f and P vary in nonparametric classes ^f and ^P respectively. Direct and iterative methods are described which obtain the range of (f, P) over f ^f when prior P is fixed, and also the overall range over f ^f and P ^P . When multiple i.i.d. observations X ₁,...,X _k are observed from f, the posterior quantity (f, P) is not a ratio-linear function of f. A method of steepest descent is proposed to obtain the range of (f, P). Several examples illustrate applications of these methods.     

Wave operators in a multistratified strip

One investigates the scattering theory for the positive self-adjoint operatorH=–· acting in with = × and a bounded open set in ^n–1,n2. The real-valued function belongs toL (), is bounded from below byc>0 and there exist real-valued functions ₁ and ₂ inL () such that – _j ,j=1,2 is a short range perturbation of _j when (–1) ^j x _n +. One assumes _j = ^(j) 1_R,j=1,2, with ^(j) L bounded from below byc>0. One proves the existence and completeness of the generalized wave operators _j ^± =s – _j e ^–itHj ,j=1,2, withH _j =–· _j and _j : equal to 1 if (–1) ^j x _n >0 and to 0 if (–1) ^j x _n <0. The ranges ofW _j ^± :=( _j ^± )^* are characterized so that W ₁ ^± =Ran and . The scattering operator can then be defined.