Halsey-Lucas Function

Ｈａｌｓｅｙ－ＬｕｃａｓＦｕｎｃｔｉｏｎ￥ＺｈａｎｇＺｈｉｚｈｅｎｇ；ＨｕＴｉｎｇｆｅｎｇ（ＬｕｏｙａｎｇＴｅａｃｈｅｒｓ＇Ｃｏｌｌｅｇｅ，，Ｌｕｏｙａｎｇ４７１０２２）／／Ａｔｆｉｒｓｔ，ｗｅｄｅｎｏｔｅｔｈａｔｍ（ｘ）ｉｓｔｈｅｉｎｔｅｇｅｒｄｅ...     

Unification of Filled Function and Tunnelling Function in Global Optimization

In this paper,two auxiliary functions for global optimization are proposed.These two auxiliaryfunctions possess all characters of tunnelling functions and filled functions under certain general assumptions.Thus,they can be considered as the unification of filled function and tunnelling function.Moreover,the processof tunneling or filling for global optimization can be unified as the minimization of such auxiliary functions.Result of numerical experiments shows that such two auxiliary functions are effective.     

On an Arithmetic Function

Ｌｅｔｆ（ｎ）ｄｅｎｏｔｅｔｈｅｎｕｍｂｅｒｏｆｆａｃｔｏｒｉｚａｔｉｏｎｓｏｆｔｈｅｎａｔｕｒａｌｎｕｍｂｅｒｎｉｎｔｏｆａｃｔｏｒｓｌａｒｇｅｒｔｈａｎ１，ｗｈｅｒｅｔｈｅｏｒｄｅｒｏｆｔｈｅｆａｃｔｏｒｓｄｏｅｓｎｏｔｃｏｕｎｔ．Ａｌｓｏｌｅｔ...     

The Whittaker Function M κ ,μ as a Function of κ

The dependence of the Whittaker function M _{κ, μ} (z) on the parameter κ is considered. A convergent expansion in ascending powers and an asymptotic expansion in descending powers of κ are discussed. Some properties of the coefficients of the convergent expansion are shown. February 14, 1997. Date revised: March 5, 1998. Date accepted: March 19, 1998.     

Function spaces on local fields

We study the function spaces on local fields in this paper, such as Triebel B-type and F-type spaces, Holder type spaces, Sobolev type spaces, and so on, moreover, study the relationship between the p-type derivatives and the Holder type spaces. Our obtained results show that there exists quite difference between the functions defined on Euclidean spaces and local fields, respectively. Furthermore, many properties of functions defined on local fields motivate the new idea of solving some important topics on fractal analysis.     

Fuzzy Integrals of Set-vatued Function

ＦｕｚｚｙＩｎｔｅｇｒａｌｓｏｆＳｅｔ－ｖａｌｕｅｄＦｕｎｃｔｉｏｎｓＺｈａｎｇＤｅｌｉａｎｄＷａｎｇＺｉｘｉａｏ（Ｄｅｐｔ．ｏｆＭａｔｈ．ＪｉｌｉｎＰｒｏｖ，Ｉｎｓｔ．ｏｆＥｄｕｃａｔｉｏｎＣｈａｎｇｃｈｕｎ，Ｊｉｌｉｎ，１３００２２，Ｐ．Ｒ．Ｃｈ...     

μ-Statistically Convergent Function Sequences

In the present paper we are concerned with convergence in -density and -statistical convergence of sequences of functions defined on a subset D of real numbers, where is a finitely additive measure. Particularly, we introduce the concepts of -statistical uniform convergence and -statistical pointwise convergence, and observe that -statistical uniform convergence inherits the basic properties of uniform convergence.     

φ-Factors for Function Seminorms

Let (, A, ) be a measure space, a function seminorm on M, the space of measurable functions on , and M the space {f M : (f) < }. Every Borel measurable function : [0, ) [0, ) induces a function : M M by (f)(x) = (|f(x)|). We introduce the concepts of -factor and -invariant space. If is a -subadditive seminorm function, we give, under suitable conditions over , necessary and sufficient conditions in order that M be invariant and prove the existence of -factors for . We also give a characterization of the best -factor for a -subadditive function seminorm when is -finite. All these results generalize those about multiplicativity factors for function seminorms proved earlier.     

On Dual Cosine Operator Function

Some basic properties of daul cosine operator function are given. The concept and characterization of θ-reflexivity with respect to cosine operator function are first studied.     

Note on Hardy-Littlewood Maximal Function

ＮｏｔｅｏｎＨａｒｄｙ－ＬｉｔｔｌｅｗｏｏｄＭａｘｉｍａｌＦｕｎｃｔｉｏｎＣｈｅｎＬｉｙｕａｎ（Ｄｅｐｔ．ｏｆＥｃｏｎｏｍｉｃ．ＨａｎｇｚｏｕＵｎｉｖｅｒｓｉｔｙ，Ｈａｎｇｚｏｕ３１００２８）ＮｏｔｅｏｎＨａｒｄｙ－ＬｉｔｔｌｅｗｏｏｄＭａｘｉｍａｌ...     

The Function Concept(二)

The domain of linear function f(x)=ax b(a≠0) is R, its range is R. The domain of inverse proportion function f(x)=(k/x)(k≠0) is {x|x≠0}, its range is B ={y|y≠0}. The domain of quadratic function, f(x)=ax~2 bx c(a≠0)is R, its range is B. B={y|y≥(4ac-b~2)/4a},when a>0;B={y|y≤(4ac-b~2/4a)},     

Subdifferentials with Autoconjugate Fitzpatrick Function

In the framework of real Banach spaces, the present paper provides a necessary and sufficient condition for the Fitzpatrick function of the subdifferential of a proper lower semicontinuous convex function to be autoconjugate. This enables us to: obtain a new proof of the fact that subdifferentials of indicator and sublinear functions have autoconjugate Fitzpatrick functions; characterize those classes of functions whose subdifferentials fulfill the condition under study in the same special way as indicator and sublinear functions do; prove that, in the one-dimensional case, the functions of these classes are the only ones whose subdifferentials have autoconjugate Fitzpatrick functions, while this is not true in higher dimensions.     

Function Spaces and Dieudonné Completeness

Abstract

For a completely regular space X and a normed space E let C_k (x, E) (resp., C_p (x, E)) be the set of all E-valued continuous maps on X endowed with the compact-open (resp., pointwise convergence) topology. It is shown that the set of all F-valued linear continuous maps on C_k (x, E) when equipped with the topology of uniform convergence on the members of some families of bounded subsets of C_k (x, E) is a complete uniform space if F is a Band space and X is Dieudonné complete. This result is applied to prove that Dieudonné completeness is preserved by linear quotient surjections from C_k (x, E) onto C_k (Y, E) (resp., from C_p (x, E) onto C_p (x, E)) provided E, F are Band spaces and Y is a k-space.     

Rate of Convergence for the Empirical Distribution Function and the Empirical Characteristic Function of Mixing Processes

Ｒａｔｅ　ｏｆ　Ｃｏｎｖｅｒｇｅｎｃｅ　ｆｏｒ　ｔｈｅ　Ｅｍｐｉｒｉｃａｌ　Ｄｉｓｔｒｉｂｕｔｉｏｎ　Ｆｕｎｃｔｉｏｎ　ａｎｄ　ｔｈｅ　Ｅｍｐｉｒｉｃａｌ　Ｃｈａｒａｃｔｅｒｉｓｔｉｃ　Ｆｕｎｃｔｉｏｎ　ｏｆ　Ｍｉｘｉｎｇ　ＰｒｏｃｅｓｓｅｓＨｕＳｈ...     

On the Relations Between the Numbers of Deficient Function and the Growth of an Algebroidal Function

Let f(z) be an algebroidal function defined by the following irreducible eq-uation A_ny~n A_(n-1)y~(n-1) … A_1y A_0=0 (1)where A_j(j=0,1,…,n) are entire functions in the finite complex plane.     

Teichmüller Spaces and Function Spaces

ＴｅｉｃｈｍüｌｅｒＳｐａｃｅｓａｎｄＦｕｎｃｔｉｏｎＳｐａｃｅｓＧｕｏＨｕｉ（郭辉）（ＳｃｈｏｌｏｆＭａｔｈｅｍａｔｉｃａｌＳｃｉｅｎｃｅ，ＰｅｋｉｎｇＵｎｉｖｅｒｓｉｔｙ，Ｂｅｉｊｉｎｇ，１００８７１）ＣｏｍｍｕｎｉｃａｔｅｄｂｙＬｉＺｈｏｎｇＲ...     

The Dehn Function of Stallings’ Group

We prove that the Dehn function of a group of Stallings that is finitely presented but not of type F₃{\mathcal{F}_3} is quadratic.     

S-harmonic Function and RieszDecomposition Theorem of Superprocesses

RecentlyItheremanyprogressinthetheoryofsuperprocesses(seeLZ'4--gi).Inthispaper,wewillestablishtheRicszdecompositiontheoremofexcessivefunctionsofaclassofsuperprocesses,andgeneralizetheclassicalRicszdecompositiontheoremofHuntprocess.FortheconstructionofsupcrprocesscswecanseeL2,4,sj.ConsiderthefollowingrealfunctionPX,9(x,z)=a(x)z b(x)zZ J,(e":--I "z)n.(du),(1)Pcowheren.(du)isakernelfromR'lto(0,~)and'a(x),b(x)andA(x)~I(uZAu)n.(du)areJ0positiveboundedBorclfunctions.AssumingthatE~{Et,11.}is…     

Function S-rough Sets and Their Law Characteristic

Function S-rough sets(Function Singular rough sets) are defined by R-function equivalence class which has dynamic characteristic, and a function is s law, function S-rough sets have law characteristic. Function S-rough sets has these forms: function one direction S-rough sets, function two direction S-rough sets and dual of function one direction S-rough sets. This paper presents the law characteristic of function one direction S-rough sets and puts forward the theorems of law-chain-attribute and law-belt. Function S-rough sets is s new research direction of the rough sets theory.     

The Fractional Derivatives of a Fractal Function

The present paper investigates the fractional derivatives of Weierstrass function, proves that there exists some linear connection between the order of the fractional derivatives and the dimension of the graphs of Weierstrass function.