On the Rational Spline Functions

Since Schoenberg's fundamental paper was published in 1946, the theory of polynomial spline functions developed rapldly and was rich in contents. Spline functions have wide range of applications to numerical approximation.But to certain functions, for example, to the function with singular points, the interpolation by polynomial spline is not always efficient and reasonable. Since 1973 some authors have investigated the rational spline functions, and abtained a lot of results. But     

A Class of Rational Arithmetical Functions with Combinatorial Meanings

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On the Complexity of the Julia Sets of Rational Functions

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Erdős-Turʼan-Type Discrepancy for the Zeros of Rational Functions

In 1950 P. Erdős and P. Turán published a discrepancy theorem for the zeros of a polynomial. Therein, the maximum deviation of the normalized zero counting measure from the equilibrium measure of the unit circle is estimated. Many other discrepancy theorems and related propositions about weak-star-convergence of the zero distribution of a sequence of polynomials were proved during the last decades. For several years the weak-star-convergence of the zero distribution of a sequence of rational functions is also studied. The main result of this paper is a discrepancy theorem for the zero distribution of a rational function which generalizes and sharpens previous propositions about weak-star-convergence of the zero counting measure of sequences of rational functions and known discrepancy theorems for polynomials.     

The Uniqueness of Meromorphic Functions Concerning Small Functions

Inthe1920s,Nevanlinna[1]establishedthefamoussecondfUndamentaltheorem,whosepreciseformisasfollows:Letf(z)beanonconstantmeromorphicfunctiononthecomplexplaneCanddistinctcomplexvaluesintheextendedcomplexplaneC,thenhere,T(r,f),N(r,.),(r,.)andS(r,f)arethestandardnotationsoftheNevanlinnatheor}(See[2]forexample).Oneoftheimportantapplicationsof(l)(butnot(2))isthefollowingwelLknownNevanlinna5-piont--theorem.ontheuniquenessofmeromorphicfunctions:TheoremA[1]Letf(z)andg(z)benonconstantmeromorphicfunct…     

Note on Rational Interpolants

<正> In this note we present a constructive proof of symmetrical determinantal formulas forthe numerator and denominator of an ordinary rational interpolant,consider the confluencecase and give new determinantal formulas of the rational interpolant by means of Lagrange'sbasis functions.     

On θ -Preirresolute Functions

We introduce a new class of functions called -preirresolute functions which is contained in the class of quasi preirresolute functions and contains the class of preirresolute functions. It is shown that p-closedness and -connectedness are preserved by -preirresolute surjections.     

Functions Defined by Sequences

We construct real valued functions from infinite sequences.We also considersome properties of such functions and null sequences.     

Combinatorics of Rational Functions and Poincaré- Birchoff-Witt Expansions of the Canonical U ({\frak n}\_)-Valued Differential Form

We study the canonical U -valued differential form, whose projections to different Kac-Moody algebras are key ingredients of the hypergeometric integral solutions of KZ-type differential equations and Bethe ansatz constructions. We explicitly determine the coefficients of the projections in the simple Lie algebras A_r, B_r, C_r, D_r in a conveniently chosen Poincaré- Birchoff-Witt basis.Received June 28, 2004     

Approximation of Multivariate Functions with a Certain Mixed Smoothness by Entire Functions

ＡｐｐｒｏｘｉｍａｔｉｏｎｏｆＭｕｌｔｉｖａｒｉａｔｅＦｕｎｃｔｉｏｎｓｗｉｔｈ　ａＣｅｒｔａｉｎＭｉｘｅｄＳｍｏｏｔｈｎｅｓｓｂｙＥｎｔｉｒｅＦｕｎｃｔｉｏｎｓＷａｎｇＨｅｐｉｎｇ（汪和平）；ＳｕｎＹｏｎｇｓｈｅｎｇ（孙永生）（Ｄｅｐａｒｔｍｅｎｔ...     

Bivariate Blending Thiele-Werner's Osculatory Rational Interpolation

Both the expansive Newton's interpolating polynomial and the Thiele-Werner's in- terpolation are used to construct a kind of bivariate blending Thiele-Werner's oscula- tory rational interpolation.A recursive algorithm and its characteristic properties are given.An error estimation is obtained and a numerical example is illustrated.     

Rational Homotopy Theory and Nonnegative Curvature

in this note, we answer positively a question by Belegradek and Kapovitch about the relation between rational homotopy theory and a problem in Riemannian geometry which asks that total spaces of which vector bundles over compact non-negative curved manifolds admit （complete） metrics with non-negative curvature.     

Mean-square Almost Periodic Random Functions

In this paper, we present a basic theory of mean-square almost periodicity, apply the theory in random differential equation, and obtain mean-square almost periodic solution of some types stochastic differential equation.     

Isochronous Centers and Isochronous Functions

Abstract We study isochronous centers of two classes of planar systems of ordinary differential equations.Forthe first class which is the Linard systems of the form =y-F(x),=-g(x) with a center at the origin, we provethat if g is isochronous(see Definiton 1.1),then the center is isochronous if and only if F≡0.For the secondclass which is the Hamiltonian systems of the form =-g(y),=f(x) with a center at the origin,we prove thatif f or g is isochronous,then the center is isochronous if and only if the other is also isochronous.     

Building New Functions from Old

The functions we have considered so far, such as x ,x10, 10x, or logx, can be thought of as building blocks out of which we construct other,more complicated,functions. In a simple case,we can take the power functions x and x2 and the constants 3,4, and -5 to create a new function f(x) = 3x2 4x-5. Thus we have produced a quadratic function as a linear combination of power functions. In fact,we have already seen that any polynomial can be thought of as a linear combination of power functions. In this section,we investigate in more de-     

Rational double points on Enriques surfaces

We classify,up to some lattice-theoretic equivalence,all possible configurations of rational double points that can appear on a surface whose minimal resolution is a complex Enriques surface.     

A Recursive Algorithm on Rational Interpolation

In this paper we introduce a so called C-Matrix w.r.t a rational interpolation problem and study the relationship between the unattainable points and C-Matrix. Finally, we present a recursive algorithm on rational interpolation. 1 Introduction Rational interpolation, as an important supplement of polynomial interpolation, has some     

Quasi-convex Functions in Carnot Groups

In this paper, the authors introduce the concept of h-quasiconvex functions on Carnot groups G. It is shown that the notions of h-quasiconvex functions and h-convex sets are equivalent and the L∞estimates of first derivatives of h-quasiconvex functions are given. For a Carnot group G of step two, it is proved that h-quasiconvex functions are locally bounded from above. Furthermore, the authors obtain that h-convex functions are locally Lipschitz continuous and that an h-convex function is twice differentiable almost everywhere.     

Some Properties of Bent Functions

First, this paper discusses and sums up some properties of a pair of functions p（x）, q（x） that makes （y ＋ 1）p（x） ＋ yq（x） into a bent function. Then it discusses the properties of bent functions. Also, the upper and lower bounds of the number of bent functions on GF（2）2k are discussed.     

Vector Spaces of Non-measurable Functions

We show that there exists an infinite dimensional vector space every non-zero element of which is a non-measurable function. Moreover, this vector space can be chosen to be closed and to have dimensionβ for any cardinalityβ. Some techniques involving measure theory and density characters of Banach spaces are used.