A Simple Proof for a Theorem of Kelisky and Rivlin

Theorem (Kelisky and Rivlin) Let f(x) be a function defined in [0,1] and B_n(f(x))=sum from k=o to n (f(k/s)(?)x~k(1-x)~(n-k)) be the nth Bernstein polynomial of f(x). Then lim B~l(f(x))=f(0)+(f(1)-f(0))x. Proof We can assume f(0)=0, Let φ_i(x) and ψ_i(x)(i=1,2,…,n) be Bernstein basis polynomials and Bezier basis polynomials respectively. Let n×n matrices     

A Quantitative Version of the Bishop–Phelps Theorem for Operators in Hilbert Spaces

In this paper, with the help of spectral integral, we show a quantitative version of the Bishop-Phelps theorem for operators in complex Hilbert spaces. Precisely, let H be a complex Hilbert space and 0 ε 1/2. Then for every bounded linear operator T : H → H and x0 ∈ H with ||T|| = 1 = ||x0|| such that ||Tx0|| 1 ε, there exist xε∈ H and a bounded linear operator S : H → H with||S|| = 1 = ||xε|| such that ||Sxε|| = 1, ||xε-x0|| ≤ (2ε)1/2 + 4(2ε)1/2, ||S-T|| ≤(2ε)1/2.     

A New Proof of Calabi-Yau's Theorem

We give a new proof of Calabi-Yau's theorem on the volume growth of Rie- mannian manifolds with non-negative Ricci curvature.     

A Simple Proof of Perelman’s Collapsing Theorem for 3-manifolds

We will simplify earlier proofs of Perelman’s collapsing theorem for 3-manifolds given by Shioya–Yamaguchi (J. Differ. Geom. 56:1–66, 2000; Math. Ann. 333: 131–155, 2005) and Morgan–Tian ( [math.DG], 2008). A version of Perelman’s collapsing theorem states: “Let {M³_i}\{M^{3}_{i}\} be a sequence of compact Riemannian 3-manifolds with curvature bounded from below by (−1) and $\mathrm{diam}(M^{3}_{i})\ge c_{0}>0$\mathrm{diam}(M^{3}_{i})\ge c_{0}>0 . Suppose that all unit metric balls in M³_iM^{3}_{i} have very small volume, at most v _i →0 as i→∞, and suppose that either M³_iM^{3}_{i} is closed or has possibly convex incompressible toral boundary. Then M³_iM^{3}_{i} must be a graph manifold for sufficiently large i”. This result can be viewed as an extension of the implicit function theorem. Among other things, we apply Perelman’s critical point theory (i.e., multiple conic singularity theory and his fibration theory) to Alexandrov spaces to construct the desired local Seifert fibration structure on collapsed 3-manifolds.     

A Bishop–Stone–Weierstrass Theorem for （M2（C））^n

In this paper, we shall give an elementary proof of a Bishop–Stone–Weierstrass theorem for （M2（C））n with respect to its pure states. To be more precise, we shall show that the pure-state Bishop hull of a unital subalgebra （not necessarily self-adjoint） of （M2（C））n is equal to itself.     

A Constructive Proof of the Gelfand—Kolmogorov Theorem

A construction of the Stone—ech compactification of a locale L is presented in this paper as a quotient of the frame of radical ideals of the algebra C ^*(L). As a corollary, a constructive, localic version of the Gelfand—Kolmogorov theorem is obtained.     

A Remark on Chen’s Theorem (Ⅱ)

Let p denote a prime and P2 denote an almost prime with at most two prime factors. The author proves that for suffciently large x,sum from p≤x p 2=P2 1>(1.13Cx)/(log~2x), where the constant 1.13 constitutes an improvement of the previous result 1.104 due to J. Wu.     

A Simpler Proof of Mirzakhani’s Simple Curve Asymptotics

Maryam Mirzakhani (in her doctoral dissertation) has proved the author’s conjecture that the number of simple closed curves of length bounded by L on a hyperbolic surface S is asymptotic to a constant times L^d, where d is the dimension of the Teichmüller space of S. In this note we clarify and simplify Mirzakhani’s argument.     

A New Proof of the Assmus–Mattson Theorem for Non-Binary Codes

C. Bachoc gave a new proof of theAssmus–Mattson theorem for linear binary codes using harmonicweight enumerators which she defined B. We give a new proof ofthe Assmus–Mattson theorem for linear codes over any finitefield using similar methods.     

An Analogue of Beurling’s Theorem for NA Groups

In this paper, we prove Beurling’s theorem for NA groups, from which we derive some other versions of uncertainty principles.     

Graphs With Least Eigenvalue —2: A New Proof of the 31 Forbidden Subgraphs Theorem

Generalized line graphs were introduced by Hoffman Proc. Calgary Internat. Conf. on Combinatorial Structures and their applications, Gordon and Breach, New York (1970); they were characterized in 1980 by a collection of 31 forbidden induced subgraphs, obtained independently by Cvetkovi et al., Comptes Rendus Math. Rep. Acad. Sci. Canada (1980) and S. B. Rao et al., Proc. Second Symp., Indian Statistical Institute, Calcutta, Lecture Notes in Math., (1981). Here a short new proof of this characterization theorem is given, based on an edge-colouring technique.     

The Approximation Theorem of Convolution Operator in △~p Set-valued Function Space

The paper is a contribution to the problem of approximating random set with values in a separable Banach space. This class of set-valued function is widely used in many areas.We investigate the properties of p-bounded integrable random set. Based on this we endow it with △p metric which can be viewed as a integral type hausdorff metric and present some approximation theorem of a class of convolution operators with respect to △p metric. Moreover we also can establish analogous theorem for other integral type operator in △p space.     

A New Proof of Diophantine Equation ■

By using algebraic number theory and p-adic analysis method,we give a new and simple proof of Diophantine equation ■.     

Successful couplings for a class of stochastic differential equations driven by Lvy processes

By constructing proper coupling operators for the integro-differential type Markov generator,we establish the existence of a successful coupling for a class of stochastic differential equations driven by L’evy processes.Our result implies a new Liouville theorem for space-time bounded harmonic functions with respect to the underlying Markov semigroups,and it is sharp for Ornstein-Uhlenbeck processes driven by α-stable L’evy processes.     

Strong Convergence Theorems for a Family of Quasi-φ-Asymptotically Nonexpansive Mappings

The purpose of this article is to propose a modified hybrid projection algorithm and prove a strong convergence theorem for a family of quasi-φ-asymptotically nonexpansive mappings.Its results hold in reflexive,strictly convex,smooth Banach spaces with the property(K).The results of this paper improve and extend recent some relative results.     

A Generalized Form of Whitney＇s Theorem

Whitney's theorem is a famous theorem in the local singularity theory.In this paper,as an application of Malgrange preparation theorem,a generalized form of Whitney's theorem will be derived.     

On Ha＇s Version of Set-valued Ekeland＇s Variational Principle

By using the concept of cone extensions and Dancs-Hegedus-Medvegyev theorem, Ha [Some variants of the Ekeland variational principle for a set-valued map. J. Optim. Theory Appl., 124, 187–206 (2005)] established a new version of Ekeland’s variational principle for set-valued maps, which is expressed by the existence of strict approximate minimizer for a set-valued optimization problem. In this paper, we give an improvement of Ha’s version of set-valued Ekeland’s variational principle. Our proof is direct and it need not use Dancs-Hegedus-Medvegyev theorem. From the improved Ha’s version, we deduce a Caristi-Kirk’s fixed point theorem and a Takahashi’s nonconvex minimization theorem for set-valued maps. Moreover, we prove that the above three theorems are equivalent to each other.     

A $\overline{\partial}$ -Theoretical Proof of Hartogs’ Extension Theorem on Stein Spaces with Isolated Singularities

Let X be a connected normal Stein space of pure dimension d≥2 with finitely many isolated singularities. By solving a weighted -equation with compact support on a desingularization of X, we derive Hartogs’ Extension Theorem on X by the -idea due to Ehrenpreis.