SHAPE-PRESERVING BIVARIATE POLYNOMIAL APPROXIMATION IN C([-1,1]×[-1,1])

In thit paper we construct bivariate polynomials attached to a btoariate function, that approximate with Jackson-type rate involving a btoariate Ditzian-Tatik ωξmodulus of smoothness and preserve some natural kinds of bivariate monotonicity and convexity of function.The result-extends that in univariate case-of D. Leviatan in [5-6], improves that in bivariate case of the author in [3] and in some special cases, that in bivariate case of G. Anastassiou in [1].     

二元二次函数逼近的存在性和局部性

This paper analysis the local behavior of the bivariate quadratic function approximation to a bivariate function which has a given power series expansion about the origin. It is shown that the bivariate quadratic Hermite-Pade form always defines a bivariate quadratic function and that this function is analytic in a neighborhood of the origin.     

基于弹性理论的二元样条与黄金分割

Splines are important in both mathematics and mechanics. We investigate the relationships between bivariate splines and mechanics in this paper. The mechanical meanings of some univariate splines were viewed based on the analysis of bending beams. For the 2D case, the relationships between a class of quintic bivariate splines with smoothness 3 and bending of thin plates are presented constructively. Furthermore, the variational property of bivariate splines and golden section in splines are also discussed.     

A Kind of Cubic C~1—Interpolations in the n-dimensional Finite Element Method

As is well known that the multivariate spline function plays an important role in both theory and application. The paper [1]—[11] hove studied the multivariate spline functions and obtained a lot of results concerning this topic. Especially in [3], the existance theorem has been shown for the case of n-dim entional spline functions. A. Zeniek [10] and P. Alfeld [11] have established some of results about the tetrahedron partition. In this paper. we will show a kind of cubic C~1—interpolations for any n-simplicial partition in R~n. Of course, some of the subdivisions will be needed.     

ON THE GENERALIZED INVERSE NEVILLE—TYPE MATRIX—VALUED RATIONAL INTERPOLANTS

A new kind of matrix-valued rational interpolants is recursively established by means of generalized Samelson iverse for matrices,with scalar numerator and matrix-valued denominatror.In this respect,it is essentially different form that of the previous works [7,9],where the matrix-valued rational interpolants is in Thiele-type continued fraction form with matrix-valued numerator and scalar denominator.For both univariate and bivariate cases,sufficient conditions for existence,characterisation and univquenese in some sense are proved respectively,and an error formula for the univariate interpolating function is also given.The results obtained in this paper are illustrated with some numerical examples.     

Dyadic Bivariate Fourier Multipliers for Multi-Wavelets in L2(R2)

The single 2 dilation orthogonal wavelet multipliers in one dimensional case and single A-dilation(where A is any expansive matrix with integer entries and|det A|=2) wavelet multipliers in high dimensional case were completely characterized by the Wutam Consortium(1998) and Z. Y. Li, et al.(2010). But there exist no more results on orthogonal multivariate wavelet matrix multipliers corresponding integer expansive dilation matrix with the absolute value of determinant not 2 in L~2(R~2). In this paper, we choose 2I2=(_0~2 _2~0)as the dilation matrix and consider the 2 I2-dilation orthogonal multivariate waveletΨ = {ψ_1, ψ_2, ψ_3},(which is called a dyadic bivariate wavelet) multipliers. We call the3 × 3 matrix-valued function A(s) = [ f_(i, j)(s)]_(3×3), where fi, jare measurable functions, a dyadic bivariate matrix Fourier wavelet multiplier if the inverse Fourier transform of A(s)( ψ_1(s), ψ_2(s), ψ_3(s)) ~T=( g_1(s), g_2(s), g_3(s))~ T is a dyadic bivariate wavelet whenever(ψ_1, ψ_2, ψ_3) is any dyadic bivariate wavelet. We give some conditions for dyadic matrix bivariate wavelet multipliers. The results extended that of Z. Y. Li and X. L.Shi(2011). As an application, we construct some useful dyadic bivariate wavelets by using dyadic Fourier matrix wavelet multipliers and use them to image denoising.     

Construction and regularity of transition functions on polish spaces under measurability conditions

This paper concerns the construction and regularity of a transition (probability) function of a non-homogeneous continuous-time Markov process with given transition rates and a general state space. Motivating from a lot of restriction in applications of a transition function with continuous (in t≥0) and conservative transition rates q(t, x, Λ), we consider the case that q(t,x,Λ) are only required to satisfy a mild measurability (in t≥0) condition, which is a generalization of the continuity condition. Under the measurability condition we construct a transition function with the given transition rates, provide a necessary and sufficient condition for it to be regular, and further obtain some interesting additional results.     

Semi inherited bivariate interpolation

The bivariate interpolation in two dimensional space R2 is more complicated than that in one dimensional space R, because there is no Haar space of continuous functions in R2. Therefore, the bivariate interpolation has not a unique solution for a set of arbitrary distinct pairwise points. In this work, we suggest a type of basis which depends on the points such that the bivariate interpolation has the unique solution for any set of distinct pairwise points. In this case, the matrix of bivariate interpolation has the semi inherited factorization.     

OPTIMAL BIRKHOFF INTERPOLATION AND BIRKHOFF NUMBERS IN SOME FUNCTION SPACES

This paper investigates the optimal Birkhoff interpolation and Birkhoff numbers of some function spaces in space L_∞[-1,1] and weighted spaces L_p,ω[-1,1],1≤p<∞,with w being a continuous integrable weight function in(-1,1).We proved that the Lagrange interpolation algorithms based on the zeros of some polynomials are optimal.We also show that the Lagrange interpolation algorithms based on the zeros of some polynomials are optimal when the function values of the two endpoin...     

二元分形插值的拟合误差估计

A given bivariate continuous function is fitted by using a bivariate fractal interpolation function, and the error of fitting is studied in this paper. The results of error estimates are obtained in two metric cases. This provides a theoretical basis for the algorithms of fractal surface reconstruction.     

GENERALIZED LERAY FORMULA ON POSITIVE COMPLEX LAGRANGE-GRASSMANN MANIFOLDS

A full proof of a matrix lemma stated in[1]is given,and the notions concerningcannonical argument and signature of a triple of the Lagrange planes in a complex phasespace is formulated.Then a formula is established,which generalizes that one of J.Leray'sin real phase space case.Finally,some applications of the formula are given.     

Cluster Partition Function and Invariants of 3-Manifolds

The author reviews some recent developments in Chern-Simons theory on a hyperbolic 3-manifold M with complex gauge group G.The author focuses on the case of G =SL(N,C) and M being a knot complement:M =S3 \ k.The main result presented in this note is the cluster partition function,a computational tool that uses cluster algebra techniques to evaluate the Chern-Simons path integral for G =SL(N,C).He also reviews various applications and open questions regarding the cluster partition function and some of its relation with string theory.     

CRITERIA FOR AN ANALYTIC FUNCTION BELONGING TO THE Q_K SPACES

The author gives a mild integral condition in a nondecreasing function K : [0, ∞) → [0, ∞), which is sufficient and the best possible to ensure that f is a Bloch function if and only if f belongs to QK, a Mbius-invariant space of functions analytic in the unit disk. Their contributions are slight improvements of known results, and the proofs presented here are independently developed. The corresponding results for meromorphic case are also given.     

MATRIX ALGORITHMS AND ERROR FORMULA FOR BIVARIATE THIELE-TYPE RECTANGULAR MATRIX VALUED RATIONAL INTERPOLATION

A new method for the construction of bivariate matrix valued rational interpolants (BGIRI) on a rectangular grid is presented in [6]. The rational interpolants are of Thiele-type continued fraction form with scalar denominator. The generalized inverse introduced by [3]is gen-eralized to rectangular matrix case in this paper. An exact error formula for interpolation is ob-tained, which is an extension in matrix form of bivariate scalar and vector valued rational interpola-tion discussed by Siemaszko[l2] and by Gu Chuangqing [7] respectively. By defining row and col-umn-transformation in the sense of the partial inverted differences for matrices, two type matrix algorithms are established to construct corresponding two different BGIRI, which hold for the vec-tor case and the scalar case.     

α-FUZZY PAIRWISE RETRACT OF L-VALUED PAIRWISE STRATIFICATION SPACES

The notion of a fuzzy retract was introduced by Rodabaugh (1981). The notion of a fuzzy pairwise retract was introduced in 2001. Some weak forms and some strong forms of α-continuous mappings were introduced in 1988 and 1997. The authors extend some of these forms to the L-fuzzy bitopological setting and construct various α-fuzzy pairwise retracts. The concept of weakly induced spaces in the case L = [0,1] was introduced by Martin (1980). Liu and Luo (1987) generalized this notion to the case that L is an arbitrary F-lattice and introduced the notion of induced L-fts. Several results are obtained, especially, for L-valued pairwise stratification spaces.     

ON THE CONVERGENCE OF PADE-TYPE APPROXIMANTS IN TWO VARIABLES

In [1], C. Brezinski raised three unsolved questions in studing multivariable Pade-iype approximation, and the convergence of multivariable Pade-iype approximants under the general conditions is one of them. We have only seen in [2] that the convergence of the mullivariable Fade-type approximants is discussed for a kind of functions defined by Stieltjes integrations under some special conditions. However, the general convergence theorem has not been seen so far. In this paper, some error formulae of bivariate Fade-type approximants in in-legral form are given. By virtue of them, a number of convergence theorems of bivariate Pade-iype approximants are proved under general conditions.     

Subsonic Euler flows in a divergent nozzle

We characterize a class of physical boundary conditions that guarantee the existence and uniqueness of the subsonic Euler flow in a general finitely long nozzle.More precisely,by prescribing the incoming flow angle and the Bernoulli’s function at the inlet and the end pressure at the exit of the nozzle,we establish an existence and uniqueness theorem for subsonic Euler flows in a 2-D nozzle,which is also required to be adjacent to some special background solutions.Such a result can also be extended to the 3-D asymmetric case.     

THE EXTINCTION OF A NON-AUTONOMOUS ALLELOPATHIC PHYTOPLANKTON MODEL WITH FEEDBACK CONTROLS

A non-autonomous allelopathic phytoplankton model with feedback controls is considered in this paper. By constructing some suitable Lyapunov type extinction functions, some sufficient conditions for the extinction of the system are obtained. For the autonomous case, by constructing a suitable Lyapunov function, we show that one species is extinct and the rest species is globally attractive. Our results supplement some known results.     

Driving functions and traces of the Loewner equation

We consider the chordal Loewner differential equation in the upper half-plane,the behavior of the driving functionλ(t)and the generated hull Kt when Kt approachesλ(0)in a fixed direction or in a sector.In the case that the hull Kt is generated by a simple curveγ(t)withγ(0)=0,we prove some sharp relations ofλ(t)/√t andγ(t)/√t as t→0 which improve the previous work.     

ON THE LOCALIZATION AND CONVERGENCE OF MULTIPLE FOURIER INTEGRAL BY BOCHNER-RIESZ MEANS

In this paper we consider lim _(R-) B_R~(f,x_0), in one case that f_x_0 (t) is a ABMV function on [0, ∞], andin another case that f∈L_(m-1)~1(R~) and x~k/~kf∈BV(R) when |k| = m-1 and f(x) = 0 when |x-x_0|<δ for some δ>0. Our theormes improve the results of Pan Wenjie ([1]).