SOME APPROXIMATION THEOREMS FOR MODIFIED BERNSTEIN-DURRMEYER OPERATORS

The modified Bernstein-Durrmeyer operators discussed in this paper are given by M_nf≡M_n(f,x)=(n+2)P_(n,k)∫_0~1p_n+1.k(t)f(t)dt, where We will show,for 0<α<1 and 1≤p≤∞ M,f-f_p=O(n^-a)ω_Φ~2(f,t)_p=O(t^(2a)), |M_n f-f(x)|≤M(x(1-x)/n+1/_n2)~a/2ω(f,t)=O(t^a), where otherwise.

ALGEBRAIC EXTENSION OF *-A OPERATOR

In this paper, we study various properties of algebraic extension of ＊-A operator.Specifically, we show that every algebraic extension of ＊-A operator has SVEP and is isoloid.And if T is an algebraic extension of ＊-A operator, then Weyl＇s theorem holds for f（T）, where f is an analytic functions on some neighborhood of σ（T） and not constant on each of the components of its domain.

EQUIVALENCE THEOREM ON WEIGHTED SIMULTANEOUS L-APPROXIMATION BY MEANS OF SZASZ-KANTOROVICH OPERATORS AND BASKAKOV-KANTOROVICH OPERATORS

The intention of this paper is to give direct and converse results on weighted simultaneous approximation by means of Sz(?)sz-Kantorovich operators and Baskakov-Kantorovich operators in L_p-norms using the weighted Ditzian-Totik modulus of smoothness.

On the Spectrum of a Class of Differential Operators and Embedding Theorems

The author considers the embedding problem of weighted Sobolev spaces Hnp in weighted Ls spaces Ls,r,and some sufficient conditions and necessary conditions are given, when weight functions satisfy certain conditions.The author uses the results obtained to the qualitative analysis of the spectrum of 2n-order weighted differential operator,and gives some sufficient conditions and necessary conditions to ensure that the spectrum is discrete.

Thermal vacuum state for the two-coupled-oscillator model at finite temperature: Derivation and application

Following the spirit of thermo field dynamics initiated by Takahashi and Umezawa, we employ the technique of integration within an ordered product of operators to derive the thermal vacuum state （TVS） for the Hamiltonian H of the two-coupled-oscillator model. The ensemble averages of the system are derived conveniently by using the TVS. In addition, the entropy for this system is discussed based on the relation between the generalized Hellmann-Feynman theorem and the entroy variation in the context of the TVS.

A Note of Duality Theorem for Decomposable Operators

St. Frunza"3 laid the foundation of-duality theorem for decomposable operators. He proved that if y￡J?(X) is a 2-decomposable operator, then T* is also a 2-decomposable operator. His proof is fundamental, but it is rather complicated. Here we give a very simple proof, using Dieudonn^’s duality theorem.Wang Shengwang proved that if T* is a decomposable operator, and F a closed set, then -STJ.^) is weak* closed. In this paper we prove if T* has the single-valued extension property, then xt*(f~) is weak* closed whenever it is closed in the strong topology.

QCD factorization Drell–Yan process

An alternative proof of factorization theorem for Drell–Yan process that works at operator level is presented in this paper. Contributions of interactions after the hard collision for such inclusive processes are proved to be canceled at operator level according to the unitarity of time evolution operator. After this cancellation, there are no longer leading pinch singular surface in Glauber region in the time evolution of electromagnetic currents. Effects of soft gluons are absorbed into Wilson lines of scalar-polarized gluons. Cancelation of soft gluons is attribute to unitarity of time evolution operator and such Wilson lines.