二部图形式的Erd\H{O}s-S\'{o}s猜想

二部图形式的Erd\H{O}s-S\''{o}s猜想     

Pseudo-differential operators involving Watson transform

A class of pseudo-differential operators (p.d.o.'s) associated with the general Fourier kernel studied by Hardy and Titchmarsh is defined. A symbol class T ^m is introduced. It is shown that the p.d.o.'s associated with the symbol are continuous linear mappings of the Braaksma and Schuitman space T(λ,μ) into itself. An integral representation of p.d.o. is obtained. Some special forms of the symbol are considered. It is shown that these p.d.o.'s and their products are bounded in certain Sobolev type space.     

A Note on Scheffès Method of Deriving Confidence Sets for Directions and Ratios of Normals

Scheffé (1970) introduced a method for deriving confidence sets for directions and ratios of normals. The procedure requires use o f an approximation and Scheffé provided evidence that the method performs well for cases in which the variances of the random deviates are known. This paper extends Scheffé's numerical integrations to the case o f unknown variances. Our resultsindi - cate that Scheffg 's method works well when variances are unknown     

Computing the moments of order statistics from nonidentically distributed phase-type random variables

In this paper, we derive a recurrence relation for the single moments of order statistics (o.s.) arising from n independent nonidentically distributed phase-type (PH) random variables (r.v.’s). This recurrence relation will enable one to compute all single moments of all o.s. in a simple recursive manner.     

Credibility theory: a new view from the theory of second order optimal statistics

Second order (s.o.) Bayes estimators, being the main tool in the s.o. optimal statistical theory, provides a natural basis for a new approach to credibility evaluation. For the cases, where the classical credibility formula fails in the sense that it does no longer represent the predicted mean, this approach suggests an s.o. modified credibility formula, which approximately (in some sense) equals the predicted mean even for small size samples. The results are applied to the important class of location dispersion distributions and are illustrated by a number of numerical experiments.     

仿酉对称矩阵的构造及对称正交多小波滤波带的参数化

仿酉矩阵在小波、多小波、框架的构造中发挥了重要的作用.本文给出仿酉对称矩阵(简记为p.s.m.)的显式构造算法,其中仿酉对称矩阵是元素为对称或反对称多项式的仿酉矩阵.基于已构造的p.s.m.和已知的正交对称多小波(简记为o.s.m.),给出o.s.m.的参数化.恰当地选择一些参数,可得到具有一些优良性质的o.s.m.,例如Armlet.最后作这一个算例,构造出一类对称的Chui-Lian Armlet滤波带.     

Moments and sums of squares for polynomial optimization and related problems

We briefly review the duality between moment problems and sums of squares (s.o.s.) representations of positive polynomials, and compare s.o.s. versus nonnegative polynomials. We then describe how to use such results to define convergent semidefinite programming relaxations in polynomial optimization as well as for the two related problems of computing the convex envelope of a rational function and finding all zeros of a system of polynomial equations.     

区域上Besov空间的原子分解和限制定理

本文在区域Ω（Ω 　 Ｒｎ，ｎ≥１）上定义了某类在边界上消失的Ｂｅｓｏｖ空 间Ｂ~（ｓ，ｑ）_（ｐ，ｏ）（Ω）（ｓ∈Ｒ，０＜ｐ，ｑ≤∞），并给出了它的原子分解，然后证明了当区域Ω∈ Ｄε（ｎ）（０＜ε≤１，ｐｓ＜ε）时，得到了限制定理Ｂ~（ｓ，ｑ）_（ｐ，ｏ）（Ω）＝Ｂ~（ｓ，ｑ）_（ｐ）（Ω）．     

A Generalized Coordinate-Momentum Representation in Quantum Mechanics

We obtain a one-parameter family of (q, p)-representations of quantum mechanics; the Wigner distribution function and the distribution function we previously derived are particular cases in this family. We find the solutions o the evolution equations or the microscopic classical and quantum distribution functions in the form of integrals over paths in a phase space. We show that when varying canonical variables in the Green’s function of the quantum Liouville equation, we must use the total increment o the action functional in its path-integral representation, whereas in the Green’s function of the classical Liouville equation, the linear part o the increment is sufficient. A correspondence between the classical and quantum schemes holds only under a certain choice of the value of the distribution family parameter. This value corresponds to the distribution unction previously found.__________Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 143, No. 3, pp. 401–416, June, 2005.     

COMPOSITION OPERATORS FROM B~a TO F(p,q,s)

Necessary and sufficient conditions are given for a composition operator Cof = f o o to be bounded and compact from Ba to F(p, q, s).     

The new forms of Voronovskaya’s theorem in weighted spaces

The Voronovskaya theorem which is one of the most important pointwise convergence results in the theory of approximation by linear positive operators (l.p.o) is considered in quantitative form. Most of the results presented in this paper mainly depend on the Taylor’s formula for the functions belonging to weighted spaces. We first obtain an estimate for the remainder of Taylor’s formula and by this estimate we give the Voronovskaya theorem in quantitative form for a class of sequences of l.p.o. The Grüss type approximation theorem and the Grüss-Voronovskaya-type theorem in quantitative form are obtained as well. We also give the Voronovskaya type results for the difference of l.p.o acting on weighted spaces. All results are also given for well-known operators, Szasz-Mirakyan and Baskakov operators as illustrative examples. Our results being Voronovskaya-type either describe the rate of pointwise convergence or present the error of approximation simultaneously.     

q extension of Wielandt's Theorem

It is well known that the q-Gamma function cannot be characterised by its functional equation o q ( z + 1) =/((1 m q z )(1 m q )) o q ( z ) and the condition o q (1) = 1. In 1980 Richard Askey showed in [1] that the additional assumption of logarithmic convexity yields the uniqueness of o q ( x ) for real x > 0. The goal of this note is to establish a q-extension of Wielandt's theorem which gives a characterization of o q ( z ) for all $ z\in \{z\in{\shadC}:{\rm Re}(z)\gt 0\} $ .     

A fixed point theorem for o‐minimal structures

We prove a definable analogue to Brouwer's Fixed Point Theorem for o‐minimal structures of real closed field expansions: A continuous definable function mapping from the unit simplex into itself admits a fixed point, even though the underlying space is not necessarily topologically complete. Our proof is direct and elementary; it uses a triangulation technique for o‐minimal functions, with an application of Sperner's Lemma. (© 2003 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Approximate equivalence in von Neumann algebras

One formulation of D. Voiculescu's theorem on approximate unitary equivalence is that two unital representations π and ρ of a separable C*-algebra are approximately unitarily equivalent if and only if rank οπ = rank ορ. We study the analog when the ranges of π and ρ are contained in a von Neumann algebra R, the unitaries inducing the approximate equivalence must come from R, and "rank" is replaced with "R -rank" (defined as the Murray-von Neumann equivalence of the range projection).     

Exponents, Symmetry Groups and Classification of?Operator Fractional Brownian Motions

Operator fractional Brownian motions (OFBMs) are zero mean, operator self-similar (o.s.s.) Gaussian processes with stationary increments. They generalize univariate fractional Brownian motions to the multivariate context. It is well-known that the so-called symmetry group of an o.s.s. process is conjugate to subgroups of the orthogonal group. Moreover, by a celebrated result of Hudson and Mason, the set of all exponents of an operator self-similar process can be related to the tangent space of its symmetry group.     

A study of operands in terms of maximal generalized orbits

A left operand X of a monoid S is called saturated if every generalized orbit (g.o.) in X is contained in a union of others. Every operand has a natural decomposition as a union of an operand admitting an irredundant cover by maximal g.o. ′s and of a saturated operand. There is a descending chain of suboperands of an operand, defined in terms of maximal g.o. ′s, which leads to the definition of the saturation length of an operand. S has no saturated operands if and only if S satisfies the a.c.c. on orbits. Full proofs will be given in [2]. Communicated by A. H. Clifford     

An order-reversing duality map for conjugacy classes in Lusztig's canonical quotient

We define a partial order on the set N_o,c of pairs (O,C), where O is a nilpotent orbit and C is a conjugacy class in A(O), Lusztig’s canonical quotient of A(O). We then show that there is a unique order-reversing duality map N_o,c → ^LN_o,c that has certain properties analogous to those of the original Lusztig–Spaltenstein duality map. This generalizes work of E. Sommers.     

On a very weak bernoulli condition †

E. Eberlein has proposed a generalization of D. S. Ornstein's very weak Bernoulli condition for strictly stationary sequences of r.v.'s; Eberlein's generalization is motivated by probability-theoretic (rather than ergodic-theoretic) questions. Dehling, Denker and Philipp showed that the mixing rate in this definition of vwB cannot be o(l/n) except when (X?k$esup;) is i.i.d. Here we construct a class of strictly stationary sequences, and with this class we show that, in essence, any mixing rate that satisfies a mild convexity condition and is not o(1/n) is possible for this vwB condition, and is compatible with a certain condition on maximal correlations which implies nice moment properties useful in proving limit theorems.