CALCULUS ON CANTOR TRIADIC SET （I）——DERIVATIVE

For real-valued functions defined on Cantor triadic ,set. a derivative with corresponding formula of Newton-Leihniz‘s type is given In particular, for the self-simltar functions and alter-nately jumping functions defined in this paper, their derivative and exceptional sets are studied ac-curately by using ergodic theory on Е2 and Duffin-Scbaeffer‘s theorem coneerning metric diophan-tine approximation. In addition, Haar basis of L2(Е2) is constructed and Flaar expansion of stan-drd self-similar function is given.     

FIBONACCI SEQUENCE AND CANTOR＇S TERNARY SET

Let N be the set of positive integers and C be Cantor‘s ternary set.A function ζ：N→[0，1] is established by the help of the Fibonacci sequence such that —↑ζ（N），the closure of the set ζ（N）,is homeomorphic to the set C.     

STRUCTURES OF VALUES FOR SET GAMES RESTRICTED BY PARTITION SYSTEM

§1IntroductionA cooperative game with transferable utility(TU)is a pair(N,v),where N is anonempty,finite set and v∶2N→R is a characteristic function defined on the power set ofN satisfying v()∶=0.LetCGdenote the set of all cooperative TU-games with anarbitrary player set.An element of N(notation:i∈N)and a nonempty subset S of N(notation:S N or S∈2Nwith S≠)are called a player and coalition respectively,andthe associated real number v(S)is called the worth of coalition S to be in…     

ON BIRKHOFF INTERPOLATION (0;2) CASE

P. Turan and his associates considered in detail the problem of (0.2) interpolation based on the zeros of πn(x). Motivated by these results and an earlier result of Szabados and Varma[9] here we consider the problem of existence, uniqueness and explicit representation of the interpolatory polynomial Rn (x) satisfying the function values at one set of nodes and the second derivative on the other set of nodes. It is important to note that this problem has a unique solution provided these two sets of nodes are chosen properly. We also promise to have an interesting convergence theorem in the second paper of this series, which will provide a solution to the related open problem of P. Turan.     

Merit functions for nonsmooth complementarity problems and related descent algorithm

Under some assumptions, the solution set of a nonlinear complementarity problem coincides with the set of local minima of the corresponding minimization problem. This paper uses a family of new merit functions to deal with nonlinear complementarity problem where the underlying function is assumed to be a continuous but not necessarily locally Lipschitzian map and gives a descent algorithm for solving the nonsmooth continuous complementarity problems. In addition, the global convergence of the derivative free descent algorithm is also proved.     

A NEW FRAMEWORK OF PRIMAL-DUAL INFEASIBLE INTERIOR-POINT METHOD FOR LINEAR PROGRAMMING

On the basis of the formulations of the logarithmic barrier function and the idea of following the path of minimizers for the logarithmic barrier family of problems the so called "centralpath" for linear programming, we propose a new framework of primal-dual infeasible interiorpoint method for linear programming problems. Without the strict convexity of the logarithmic barrier function, we get the following results: (a) if the homotopy parameterμcan not reach to zero,then the feasible set of these programming problems is empty; (b) if the strictly feasible set is nonempty and the solution set is bounded, then for any initial point x, we can obtain a solution of the problems by this method; (c) if the strictly feasible set is nonempty and the solution set is unbounded, then for any initial point x, we can obtain a (?)-solution; and(d) if the strictly feasible set is nonempty and the solution set is empty, then we can get the curve x(μ), which towards to the generalized solutions.     

The best quadrature based on given Hermite information for the Sobolev class KW~r[a,b]

As usual, denote by KWr[a,b] the Sobolev class consisting of every function whose (r-1)th derivative is absolutely continuous on the interval [a,b] and rth derivative is bounded by K a.e. in [a, b]. For a function f∈KWr[a, b], its values and derivatives up to r -1 order at a set of nodes x are known. These values are said to be the given Hermite information. This work reports the results on the best quadrature based on the given Hermite information for the class KWr[a. b]. Existence and concrete construction issue of the best quadrature are settled down by a perfect spline interpolation. It turns out that the best quadrature depends on a system of algebraic equations satisfied by a set of free nodes of the interpolation perfect spline. From our another new result, it is shown that the system can be converted in a closed form to two single-variable polynomial equations, each being of degree approximately r/2. As a by-product, the best interpolation formula for the class KWr[a, b] is also obtained.     

STRICTLY WEAK MAJOR EFFICIENT POINT OF SET AND ITS PROPERTIES

Abstract. In this paper, the strictly weak major efficient point of set is introduced. A functional as a separate function is constructed, therefore, a necessary and sufficient condition for the strictly weak major efficient point of set is established.     

EXAMPLES ON EXCEPTIONAL VALUES OF MEROMORPHIC FUNCTIONS

In this paper, by means of the definition of Borel exceptional value method, another exceptional value of meromorphic function which is a T exceptional value is defined by linking the concept of T direction. And we construct a meromorphic function with zero as Borel exceptional value, but not as T exceptional value; and another meromorphic function with zero as T exceptional value, but not as Borel exceptional value.     

STOCHASTIC APPROXIMATION UNDER A CLASS OF MEASUREMENT ERRORS

In this paper we shall consider the cases where the zeroes or extreme points of regression function h(x) are not singal, the measurement errors are quite general. When it is known a priori that the zero set or extreme set of h(x) lie in some bounded close set, a truncated Bobbins-Monro (RM) or Kiefer-Wolfowitz (KW) algorithms is suggested, and the strong convergence of the algorithms is proved under simple conditions. For the same mea surement errors to search zero set of linear fuction or extreme set of quadratic function usual RM or KW procedures can guarantee their strong convergence under rather simple conditions.     

随机级数的例外函数

On random series, people usually study the case of equally distributed random variable sequences, such as Rademacher, Steinhaus and Gauss sequences, and discuss the exceptional constant values. In this paper, we extend the Lemma of PaleyZygmund to more general case, in order to study the random Taylor series of non equal distribution. Then we prove that for the random Taylor series almost surely (a. s.), there is no almost surely exceptional function, and that every point on the circumferen…     

ON THE 2ND POWER OF PRIME MEROMOPRHIC FUNCTIONS

In this paper, the author respectively constructs the prime entire function and the prime non-entire meromorphic function of finite order whose 2nd power and derivative are not pseudo-prime so that the answer to the Problem (A) Song-Huang posed is in the affirmative. And it is exhibited that none but periodic meromorphic functions has the property mentioned in Problem (A).     

Invariant sets and solutions to the generalized thin film equation

The invariant sets and the solutions of the 1 2-dimensional generalized thin film equation are discussed. It is shown that there exists a class of solutions to the equations, which are invariant with respect to the set E0 = {u : ux = vxF(u), uy = vyF(u)}, where v is a smooth function of variables x, y and F is a smooth function of u. This extends the results of Galaktionov (2001) and for the l l-dimensional nonlinear evolution equations.     

涉及分担集的整函数的唯一性

In this paper,uniqueness of entire function related to shared set is studied.Let f be a non-constant entire function and k be a positive integer,d be a finite complex number.There exists a set S with 3 elements such that if f and its derivative f（k）satisfy E（S,f）= E（S,f（k））,and the zeros of f（z）-d are of multiplicity ≥ k ＋ 1,then f = f（k）.     

The pointwise convergence of p-adic Mbius maps

The convergence of linear fractional transformations is an important topic in mathematics.We study the pointwise convergence of p-adic Mbius maps,and classify the possibilities of limits of pointwise convergent sequences of Mbius maps acting on the projective line P1(C p),where C p is the completion of the algebraic closure of Q p.We show that if the set of pointwise convergence of a sequence of p-adic Mbius maps contains at least three points,the sequence of p-adic Mbius maps either converges to a p-adic Mbius map on the projective line P1(C p),or converges to a constant on the set of pointwise convergence with one unique exceptional point.This result generalizes the result of Piranian and Thron(1957)to the non-archimedean settings.     

THE STRUCTURE AND APPROXIMATION OF A.S. SELF-SIMILAR SET

The structure of any a.s. self-similar set K(w) generated by a class of random elements {gn,wσ} taking values in the space of contractive operators is given and the approximation of K(w) by the fixed points {Pn,wσ} of {gn,ow} is obtained. It is useful to generate the fractal in computer.     

SOME EQUAL DIVISIONS ON ORDERED TREE SET

We say a divisien (S=SA∪SB, g) is equal if it divides set S into two subsets SA and SB (?)atf(?)fying f(SA)=f(SB), where f is som attribute function on these sets and SA∩SB need not be empty. In the present paper, we have obtained some equal divisions on set of ordered tree with n nodes: two equal divisions on tree set; two equal divisions on node set; two divisions on leaf set, one is equal, another one is "asymptotically" equal We conclude that, ordered tree set is on its good behavior in equal dividing.     

Manakov方程和导数Manakov方程的有限参数解

Finite-dimensional integrable Hamiltonian systems, obtained through the non- linearization of the 3×3 spectral problems associated with the Manakov and the derivative Manakov equations, are investigated. A generating function method is used to give a simple and effective way to prove the involutivity of integrals. Finite-parameter solutions of the Manakov and the derivative Manakov equations are calculated based on the commutative systems of ordinary differential equations with these integrals as Hamiltonians.     

符号混合控制问题的某些NP-完全性结果

Let G =(V, E) be a simple graph with vertex set V and edge set E. A signed mixed dominating function of G is a function f:V∪E→ {-1, 1} such that ∑_(y∈N_m(x)∪{x})f(y)≥ 1for every element x∈V∪E, where N_m(x) is the set of elements of V∪E adjacent or incident to x. The weight of f is w(f) =∑_(x∈V∪E)f(x). The signed mixed domination problem is to find a minimum-weight signed mixed dominating function of a graph. In this paper we study the computational complexity of signed mixed domination problem. We prove that the signed mixed domination problem is NP-complete for bipartite graphs, chordal graphs, even for planar bipartite graphs.     

THE LIMIT SET OF A CONTINUOUS SELF-MAP OF THE TREE

Let f denote a continuous map of a tree T to itself. A point x ∈ T is called a 7-limit point of f if it is both an ω-limit point and an α-limit point. In the present paper, we show that (1) Ω-Γ is countable, (2) A -Γ and P - Γ are either empty or countably infinite, where P denotes the closure of the set of periodic points P.