α-times Integrated Cosine Operator Functions with Growth ω

For a continuous,increasing functionω:[0,∞)→C of finite exponential type,we establish a Hille-Yosida type theorem for strongly continuous α-times(α>0)integrated cosine operator functions with O(ω).It includes the corresponding results for n-times integrated cosine operator functions that are polynomially bounded and exponentially bounded.

CARATHEODORY INEQUALITY FOR ANALYTIC OPERATOR FUNCTION

Suppose H is a complex Hilbert space, A H(Δ) denotes the set of all analytic operator functions on Δ, and the set N H(Δ)=｛f(z)｜f(z) is an analytic operator function on the open uint disk Δ, f(z)f(w)=f(w)f(z),f *(z)f(z)=f(z)f *(z),z,w∈Δ｝. The note proves that if f(z)∈N H(Δ),(or A H(Δ))‖f(z)‖≤1,z∈Δ then‖f′(T)‖≤(1-‖T‖ 2) -1 ‖I-f *(T)f(T)‖ 12 ‖I-f(T)f *(T)‖ 12 , where T∈L(H)(orT *T=TT *,respectively),‖T‖<1,Tf=fT.

THE PROBABILISTIC REPRESENTATION FORMULAE FOR COSINE OPERATOR FUNCTIONS

In this paper we obtained general representation formulae for strongly continuous cosine operator functions via probabilistic approach,which include Webb's[1]and Shaw's[2]formulae and some new one as special cases.We also give the quantitative estimations for the general formulae.

Band Structure Calculations of Dispersive Photonic Crystals in 3D using Holomorphic Operator Functions

We propose a finite element method to compute the band structures of dispersive photonic crystals in 3D.The nonlinear Maxwell’s eigenvalue problem is formulated as the eigenvalue problem of a holomorphic operator function.The N´ed´elec edge elements are employed to discretize the operators,where the divergence free condition for the electric field is realized by a mixed form using a Lagrange multiplier.The convergence of the eigenvalues is proved using the abstract approximation theory for holomorphic operator functions with the regular approximation of the edge elements.The spectral indicator method is then applied to compute the discrete eigenvalues.Numerical examples are presented demonstrating the effectiveness of the proposed method.

A NOTE ON MEYER-KONIG AND ZELLER OPERATORS FOR FUNCTIONS OF BOUNDED VARIATION

In the present note we give the correct and improved estimate on the rate of convergence of integrated Meyer-Konig and Zetter operators for function of bounded variation.

A SHARP ESTIMATE ON THE DEGREE OF APPROXIMATION TO FUNCTIONS OF BOUNDED VARIATION BY CERTAIN OPERATORS

Recently Guo introduced integrated Meyer -Konig and Zeller operators and studied the rate of convergence for function of bounded variation. In this note we give a sharp estimate for these operators.

On Invertibility of Functional Operators with Shift in Weighted Holder Spaces

In this paper, we consider functional operators with shift in weighted H?lder spaces. We present the main idea and the scheme of proof of the conditions of invertibility for these operators. As an application, we propose to use these results for solution of equations with shift which arise in the study of cyclic models for natural systems with renewable resources.

REFERENCE FUNCTIONAL AND CHARACTERISTIC SPACE FOR LAGRANGE AND BERNSTEIN OPERATORS

This paper deals with the description and the representation of polynomials defined over n-simplices, The polynomials are computed by using two recurrent schemes: the Neville-Aitken one for the Lagrange interpolating operator and the De Casteljau one for the Bernstein-Bezier approximating operator. Both schemes fall intothe framework of transformations of the form where the F iare given numbers (forexample, at the initial step they coincide with the values of the function on a given lattice), and the coefficients (x) are linear polynomials valued in x and x is fixed. A general theory for such sequence of transformations can be found in [2] where it is also proved that these tranformations are completely characterized in term of a linear functional, reference functional. This functional is associated with a linear space., characteristic space.The concepts of reference functionals and characteristic spaces will be used and we shall prove the existence of a characteristic space for the reference functional: associated with these operators.

Operations Improvement Function and Organisational Adaptability of Petroleum Tank Farms in South-South, Nigeria

The study examined the nexus between operations improvement function (dimensioned by contingency planning, benchmarking and continuous improvement processes) and organisational adaptability of Petroleum tank farms in South-South, Nigeria. The contingency theory and the theory of routine dynamics underpinned the study, and positivism was the underlying philosophy. The study adopted the cross-sectional survey through the use of questionnaire. 820 middle and top-level managers constituted the elements of the population, and the Krejcie & Morgan’s formula was used to determine the sample size of 262 respondents. Structural Equation Modeling was deployed to test the hypotheses at a 0.05 significance level. The results showed that contingency planning;benchmarking and continuous improvement processes all have a significant positive relationship with organisational adaptability of Petroleum tank farms in South-South, Nigeria. The study concludes that Petroleum tank farms’ operations should focus on the adoption of contingency planning, benchmarking and continuous improvement processes to enhance organisational adaptability. Therefore, it is recommended that the management of Petroleum tank farms should put in place mechanisms to advance continuous improvement processes by allocating the necessary amount of resources, such as energy, time and money, in order to promote the continuous development of the continuous improvement systems. Furthermore, managers of Petroleum tank farms should make better the adoption of contingency planning, ensuring that there is as much necessary training and information for employees on how to act during a crises situation, in order to evaluate safety and prepare in advance for recovery from disasters.

A theorem for quantum operator correspondence to the solution of the Helmholtz equation

We propose a theorem for the quantum operator that corresponds to the solution of the Helmholtz equation, i.e., ∫∫∫V （x1 ,x2,x3）〈X1 ,x2,x3〉〈x1 ,x2,x3〉d3x = V （X1 ,X2,X3） = e-λ2/4 ：V （X1 ,X2,X3）：,where V （X1 ,X2,X3） is the solution to the Helmholtz equation △2V ＋λ2V = 0, the symbol ： ： denotes normal ordering, and X1, X2, X3 are three-dimensional coordinate operators. This helps to derive the normally ordered expansion of Dirac＇s radius operator functions. We also discuss the normally ordered expansion of Bessel operator functions.

DIFFERENTIAL OPERATORS OF INFINITE ORDER IN THE SPACE OF RAPIDLY DECREASING SEQUENCES

We consider the space of rapidly decreasing sequences s and the derivative operator D defined on it. The object of this article is to study the equivalence of a differential operator of infinite order; that is φ（D） =^∞∑k=0φkD^k.φk constant numbers an a power of D.Dn, meaning, is there a isomorphism X （from s onto s） such that Xφ（D） = D^nX？. We prove that if φ（D） is equivalent to Dn, then φ（D） is of finite order, in fact a polynomial of degree n. The question of the equivalence of two differential operators of finite order in the space s is addressed too and solved completely when n = 1.

New useful special function in quantum optics theory

By virtue of the operator Hermite polynomial method [Fan H Y and Zhan D H 2014 Chin. Phys. B 23 060301] we find a new special function which is useful in quantum optics theory, whose expansion involves both power-series and Hermite polynomials, i.e.,min（m,n）∑l=0^min（m,n）n!m!（-1）^l/l!（n-1）!（m-l）!/Hn-l（x）y^m-l≡ n,m（x,y）.By virtue of the operator Hermite polynomial method and the technique of integration within ordered product of operators（IWOP） we derive its generating function. The circumstance in which this new special function appears and is applicable is considered.

Weak Type Weighted Inequalities for the Commutators of the Multilinear Calderón-Zygmund Operators

For the commutators of multilinear Calder ′on-Zygmund singular integral operators with B MO functions, the weak type weighted norm inequalities with respect to A^P weights are obtained.

WEAK BOUNDS FOR ROUGH SQUARE AND MAXIMAL OPERATORS

With Ω ∈ L (log + L)(S n-1 ) and suitable h ∈ L γ (R 1)(1<γ≤2),the weak type (1,1) of the square function g(f)(x) =k｜ψ k*f｜ 2 12(x) and the maximal operator M ψ(f)(x) = sup k｜ψ k｜*｜f｜(x) where ψ(x)=｜x｜ -n Ω(x)h(｜x｜),ψ k(x)=ψ 2 k (x), are studied in this paper.As a corollary,the weak bounds of M Ω(f) proved by Christ in 1988 are given and the previous weak type results for M ψ(f)(x) are improved.In addition,the weighted weak type (1,1) estimates of the Littlewood Paley function g ψ(f) with power weights is also proved.

AN APPLICATION OF HARDY-BOEDEWADT'S THEOREM TO ITERATED FUNCTIONAL EQUATIONS

In this paper an iterated functional equation of polynomial type which does not possess the firt order iterative term g(x) is to be discussed. The difficulties resulted from loss of the first order term are overcome by utilization of Hardy-Boedewadt's theorem.

Lucas Symbolic Formulae and Generating Functions for Chebyshev Polynomials

This work shows that each kind of Chebyshev polynomials may be calculated from a symbolic formula similar to the Lucas formula for Bernoulli polynomials. It exposes also a new approach for obtaining generating functions of them by operator calculus built from the derivative and the positional operators.

Esophageal nasopharyngeal catheter in the wake-up anesthesia of the brain function area operations

Objective To evaluate the feasibility and safety of the self developed sound outside the ventilation device-esophageal nasopharynx catheter in brain functional areas surgery applications. Methods 13 patients involved functional areas of brain surgery were chosed. After induction of general anesthesia,the catheters were placed in the esophagus,then connected to anesthesia machines to an external

MAXIMAL SUBSPACES FOR SOLUTIONS OF THE SECOND ORDER ABSTRACT CAUCHY PROBLEM

For a continuous, increasing function ω ： R^＋ →R^＋／{0} of finite exponential type, this paper introduces the set Z（A, ω） of all x in a Banach space X for which the second order abstract differential equation （2） has a mild solution such that [ω（t）]^-1u（t,x） is uniformly continues on R^＋, and show that Z（A, ω） is a maximal Banach subspace continuously embedded in X, where A ∈ B（X） is closed. Moreover, A[z（A,ω） generates an O（ω（t）） strongly continuous cosine operator function family.

Multilinear singular integrals and commutators in variable exponent Lebesgue spaces

Boundedness of multilinear singular integrals and their commutators from products of variable exponent Lebesgue spaces to variable exponent Lebesgue spaces are obtained. The vector-valued case is also considered.

A fast and practical method to pack spheres for mesh generation

Sphere packing is an attractive way to generate high quality mesh. Several algorithms have been proposed in this topic, however these algorithms are not sufficiently fast for large scale problems. The paper presents an efficient sphere packing algorithm which is much faster and appears to be the most practical among all sphere packing methods presented so far for mesh generation. The algorithm packs spheres inside a domain using advancing front method. High efficiency has resulted from a concept of 4R measure, which localizes all the computations involved in the whole sphere packing process.