Tchebyshev Approximation by S01（△） over Some Special Triangulations

The critical point set plays a central role in the theory of Tchebyshev approximation.Generally,in multivariate Tchebyshev approximation,it is not a trivial task to determine whether a set is critical or not.In this paper,we study the characterization of the critical point set of S 0 1(△) in geometry,where is restricted to some special triangulations(bitriangular,single road and star triangulations).Such geometrical characterization is convenient to use in the determination of a critical point set.     

Tchebyshev Approximation by S~0_1(△) over Some Special Triangulations

The critical point set plays a central role in the theory of Tchebyshev approximation.Generally,in multivariate Tchebyshev approximation,it is not a trivial task to determine whether a set is critical or not.In this paper,we study the characterization of the critical point set of S 0 1(△) in geometry,where is restricted to some special triangulations(bitriangular,single road and star triangulations).Such geometrical characterization is convenient to use in the determination of a critical point set.     

Tchebyshev Approximation by S01(Δ)over Some Special Triangulations

The critical point set plays a central role in the theory of Tchebyshev approximation.Generally,in multivariate Tchebyshev approximation,it is not a trivial task to determine whether a set is critical ...     

CONVERGENCE ANALYSIS FOR SYSTEM OF EQUILIBRIUM PROBLEMS AND LEFT BREGMAN STRONGLY RELATIVELY NONEXPANSIVE MAPPING

In this paper, we prove strong convergence theorems for approximation of a fixed point of a left Bregman strongly relatively nonexpansive mapping which is also a solution to a finite system of equilibrium problems in the framework of reflexive real Banach spaces. We also discuss the approximation of a common fixed point of a family of left Bregman strongly nonexpansive mappings which is also solution to a finite system of equilibrium problems in reflexive real Banach spaces. Our results complement many known recent results in the literature.     

On quasi-weakly almost periodic points

The core problem of dynamical systems is to study the asymptotic behaviors of orbits and their topological structures. It is well known that the orbits with certain recurrence and generating ergodic (or invariant) measures are important, such orbits form a full measure set for all invariant measures of the system, its closure is called the measure center of the system. To investigate this set, Zhou introduced the notions of weakly almost periodic point and quasi-weakly almost periodic point in 1990s, and presented some open problems on complexity of discrete dynamical systems in 2004. One of the open problems is as follows: for a quasi-weakly almost periodic point but not weakly almost periodic, is there an invariant measure generated by its orbit such that the support of this measure is equal to its minimal center of attraction (a closed invariant set which attracts its orbit statistically for every point and has no proper subset with this property)? Up to now, the problem remains open. In this paper, we construct two points in the one-sided shift system of two symbols, each of them generates a sub-shift system. One gives a positive answer to the question above, the other answers in the negative. Thus we solve the open problem completely. More important, the two examples show that a proper quasi-weakly almost periodic orbit behaves very differently with weakly almost periodic orbit.     

Quasisymmetric geometry of the Julia sets of McMullen maps Dedicated to the Memory of Professor Lei Tan

We study the quasisymmetric geometry of the Julia sets of McMullen maps f_λ(z) = z~m+ λ/z~?,where λ∈ C \ {0} and ? and m are positive integers satisfying 1/? + 1/m 1. If the free critical points of f_λ are escaped to the infinity, we prove that the Julia set J_λ of f_λ is quasisymmetrically equivalent to either a standard Cantor set, a standard Cantor set of circles or a round Sierpiński carpet(which is also standard in some sense).If the free critical points are not escaped, we give a sufficient condition on λ such that J_λ is a Sierpiński carpet and prove that most of them are quasisymmetrically equivalent to some round carpets. In particular, there exist infinitely renormalizable rational maps whose Julia sets are quasisymmetrically equivalent to the round carpets.     

APPROXIMATION OF BOCHNER-RIESZ MEANS 0F CONJUGATE FOURIER INTEGRALS BEL0W THE CRITICAL INDEX

In this paper we study the approximation on set of full measure for functions in Sobolev spacesL_m~1 (R~n) (m∈N) by Bochner-Riesz means of conjugale Fourier integrals below the critical index. Atheorem concerning the precise approximation orders with relation to the number m of space L_m~1 (R~n) andthe index of Bochner-Riesz means is obtained.     

渐近φ半压缩映象新的带误差的IshiKawa迭代逼近

Let E be a real Banach space and T be an asymptotically φ-hemicontractive mapping. By properties of a new analytical method, under general cases, the strong convergence of the set sequences {On} of the new Ishikawa iteration approximation with errors to the fixed point of mapping is proved. The paper generalizes and improves the corresponding results in {1},[3-8].     

ON INTERPOLATION AND APPROXIMATION BY RATIONAL FUNCTIONS WITH PREASSIGNED POLES

<正> 1. INTRODUCTION. It is the purpose of this paper to presentsome results,on the problem of interpolation and approximation toa functiou f(z),analytic on a closed limited point set E in thecomplex z-plane whose complement K is connected and regular inthe sense that Green's fumction for K exists,by rational functionsf_n(z) of respective degrees n,n=1,2,…of the form     

Structure of the spectrum of infinite dimensional Hamiltonian operators

This paper deals with the structure of the spectrum of infinite dimensional Hamiltonian operators.It is shown that the spectrum,the union of the point spectrum and residual spectrum,and the continuous spectrum are all symmetric with respect to the imaginary axis of the complex plane. Moreover,it is proved that the residual spectrum does not contain any pair of points symmetric with respect to the imaginary axis;and a complete characterization of the residual spectrum in terms of the point spectrum is then given.As applications of these structure results,we obtain several necessary and sufficient conditions for the residual spectrum of a class of infinite dimensional Hamiltonian operators to be empty.     

Dense Point Sets Have Sparse Delaunay Triangulations or “... But Not Too Nasty”

The spread of a finite set of points is the ratio between the longest and shortest pairwise distances. We prove that the Delaunay triangulation of any set of $n$ points in~$\Real^3$ with spread $\Delta$ has complexity $O(\Delta^3)$. This bound is tight in the worst case for all $\Delta = O(\sqrt{n})$. In particular, the Delaunay triangulation of any dense point set has linear complexity. We also generalize this upper bound to regular triangulations of $k$-ply systems of balls, unions of several dense point sets, and uniform samples of smooth surfaces. On the other hand, for any $n$ and $\Delta = O(n)$, we construct a regular triangulation of complexity $\Omega(n\Delta)$ whose $n$ vertices have spread $\Delta$.     

Unfolding of Surfaces

This paper deals with the approximation of the unfolding of a smooth globally developable surface (i.e. "isometric" to a domain of ) with a triangulation. We prove the following result: let T_n be a sequence of globally developable triangulations which tends to a globally developable smooth surface S in the Hausdorff sense. If the normals of T_n tend to the normals of S, then the shape of the unfolding of T_n tends to the shape of the unfolding of S. We also provide several examples: first, we show globally developable triangulations whose vertices are close to globally developable smooth surfaces; we also build sequences of globally developable triangulations inscribed on a sphere, with a number of vertices and edges tending to infinity. Finally, we also give an example of a triangulation with strictly negative Gauss curvature at any interior point, inscribed in a smooth surface with a strictly positive Gauss curvature. The Gauss curvature of these triangulations becomes positive (at each interior vertex) only by switching some of their edges.     

Compatible triangulations and point partitions by series-triangular graphs

We introduce series-triangular graph embeddings and show how to partition point sets with them. This result is then used to prove an upper bound on the number of Steiner points needed to obtain compatible triangulations of point sets. The problem is generalized to finding compatible triangulations for more than two point sets and we show that such triangulations can be constructed with only a linear number of Steiner points added to each point set. Moreover, the compatible triangulations constructed by these methods are regular triangulations.     

Delaunay Triangulation of Manifolds

We present an algorithm for producing Delaunay triangulations of manifolds. The algorithm can accommodate abstract manifolds that are not presented as submanifolds of Euclidean space. Given a set of sample points and an atlas on a compact manifold, a manifold Delaunay complex is produced for a perturbed point set provided the transition functions are bi-Lipschitz with a constant close to 1, and the original sample points meet a local density requirement; no smoothness assumptions are required. If the transition functions are smooth, the output is a triangulation of the manifold. The output complex is naturally endowed with a piecewise-flat metric which, when the original manifold is Riemannian, is a close approximation of the original Riemannian metric. In this case the output complex is also a Delaunay triangulation of its vertices with respect to this piecewise-flat metric.     

The set of dynamically special points

We call a point ??dynamically special?? if it has a dynamical property, which no other point has. We prove that, for continuous self maps of the real line, all dynamically special points are in the closure of the union of the full orbits of periodic points, critical points and limits at infinity. We completely describe the set of dynamically special points of real polynomial functions. The following characterization for the set of special points is also obtained: A subset of ${\mathbb{R}}$ is the set of dynamically special points for some continuous self map of ${\mathbb{R}}$ if and only if it is closed.     

Triangulations of Surfaces with Boundary and the Homotopy Principle for Functions without Critical Points

We consider triangulations of surfaces with boundary and marked points. These triangulations are classified with respect to flip equivalence. The results obtained are applied to the homotopy classification of functions without critical points on 2-manifolds. It is shown that the set of such functions satisfies the one-parametric h-principle.     

Tchebyshev Posets

We construct for each $n$ an Eulerian partially ordered set $T_n$ of rank $n+1$ whose $ce$-index provides a non-commutative generalization of the $n$th Tchebyshev polynomial. We show that the order complex of each $T_n$ is shellable, homeomorphic to a sphere, and that its face numbers minimize the expression $\max_{|x|\leq 1} |\sum_{j=0}^n (f_{j-1}/f_{n-1})\cdot 2^{-j}\cdot (x-1)^j|$ among the $f$-vectors of all $(n-1)$-dimensional simplicial complexes. The duals of the posets constructed have a recursive structure similar to face lattices of simplices or cubes, offering the study of a new special class of Eulerian partially ordered sets to test the validity of Stanleys conjecture on the non-negativity of the $cd$-index of all Gorenstein$^*$ posets.     

A point set whose space of triangulations is disconnected

By the ``space of triangulations" of a finite point configuration we mean either of the following two objects: the graph of triangulations of , whose vertices are the triangulations of and whose edges are the geometric bistellar operations between them or the partially ordered set (poset) of all polyhedral subdivisions of ordered by coherent refinement. The latter is a modification of the more usual Baues poset of . It is explicitly introduced here for the first time and is of special interest in the theory of toric varieties.

We construct an integer point configuration in dimension 6 and a triangulation of it which admits no geometric bistellar operations. This triangulation is an isolated point in both the graph and the poset, which proves for the first time that these two objects cannot be connected.