The Weyl’s Theorem and Its Perturbations in Semisimple Banach Algebra

Denote a semisimple Banach algebra with an identity e by A.This paper studies the Fredholm,Weyl and Browder spectral theories in a semisimple Banach algebra,and meanwhile considers the properties of the Fredholm element,the Weyl element and the Browder element.Further,for a∈A,we give the Weyl's theorem and the Browder's theorem for a,and characterize necessary and sufficient conditions that both a and f(a) satisfy the Weyl's theorem or the Browder's theorem,where f is a complex-valued function analytic on a neighborhood of σ(a).In addition,the perturbations of the Weyl's theorem and the Browder's theorem are investigated.     

Some Improvement on Hilbert''''s Inequality and Its Application

A new improvement of Hilbert's inequality for double series can be established by means of a strengthened Cauchy's inequality. As application, a quite sharp result on Fejer-Riesz's inequality is obtained.     

Markov and Artin Normal Form Theorem for Braid Groups∗

In this article, we will present the results of Artin–Markov on braid groups by using the Gröbner–Shirshov basis. As a consequence, we can reobtain the normal form of Artin–Markov–Ivanovsky as an easy corollary.     

On Explicit Bounds in Picard's Theorem and Schottky's Theorem

Let λ(τ)be the elliptic function.We define the function t∶(0,∞);→(0,∞) byλ(1 it(α))=-α.The functiorl t(α)is a decreasing functiorl of a with t(1)= 1.In the following we need the function g(a)=ae~(πt).Let S denote the class of functions which are analytic in the unit disc D and omit 0 and 1.Schottky's theorem asserts the existence of a function K(α,γ)sueh that     

The Jentzsch-Szegő Theorem and Balayage Measures

The numerous generalizations of the Jentzsch-Szeg? theorem on the location of zeros of Taylor polynomials have been based so far on the extremal properties satisfied by the corresponding approximants. We do away with those kinds of assumptions and prove the theorem for a general class of interpolating polynomials. This is possible thanks to the discovery presented here that the limit distribution of the zeros of the interpolants is governed by a balayage measure depending on the distribution of the interpolation points and the region of analyticity of the function being approximated.     

A Generalized Comparison Theorem for BSDEs and Its?Applications

This paper establishes a generalized comparison theorem for one-dimensional backward stochastic differential equations (BSDEs) whose generators are uniformly continuous in z and satisfy a kind of weakly monotonic condition in y. As applications, two new existence and uniqueness theorems for solutions of BSDEs are obtained. In the one-dimensional setting, these results generalize some corresponding results in Pardoux and Peng (Syst. Control Lett. 14:55–61, 1990), Mao (Stoch. Process. Their Appl. 58:281–292, 1995), El Karoui et al. (Math. Finance 7:1–72, 1997), Pardoux (Nonlinear Analysis, Differential Equations and Control, Montreal, QC, 1998, Kluwer Academic, Dordrecht, 1999), Cao and Yan (Adv. Math. 28(4):304–308, 1999), Briand and Hu (Probab. Theory Relat. Fields 136(4):604–618, 2006), and Jia (C. R. Acad. Sci. Paris, Ser. I 346:439–444, 2008).     

The “Full Müntz Theorem” Revisited

We give an elementary proof of the convergent sum part of the full Müntz Theorem in L_p(A) and in C(A), together with the Clarkson--Erdös–Schwartz phenomenon for all p (0,), and for all compact A [0,) with positive lower density at 0. This extends earlier results of Müntz, Szász, Clarkson and Erdös, L. Schwartz, P. Borwein and Erdélyi, and Operstein, and offers an arguably shorter and more elementary approach to reprove a large part of the result W. B. Johnson achieved with the author. This approach does not require the usage of Basteros extension of the Krivine–Maurey stable theory. It requires only a standard undergraduate level familiarity with real and complex analysis.     

The Berry-Esseen Theorem and Non-Uniform Estimate for U-Statistics

Let X_1,X_2,…,X_n be independent random variables. Define a U-statistic by U_n(?)~(-1)sum from 1≤i≤j≤n (h(X_i,X_j), where h(x,y) is a symmetric function of two variables x,y and that Eh(X_i,X_j)=0(i≠j, i,j=1,2,…,n). Write g_j(X_i)=E(h(x_i,x_j)|x_i),g(X_1)=1/n-1 sum from j=1 j≠i to n g_j(X_i) We give the following two theorem: Theorem 1 Suppore that     

The Hillam–Thron Theorem in Higher Dimensions

Let D be a disc in the complex plane, and let s _n be a sequence of Möbius maps each of which maps D into itself. In 1965 Hillam and Thron proved (essentially) that if the points s _n (), n=12,..., lie in a compact subset of D then the functions s ₁^...^s _n converge locally uniformly to a constant on D. They then made applications to continued fractions. In this paper, the corresponding results are proved in all dimensions.     

The Proof of a Theorem in DC Problem

ＴｈｅＰｒｏｏｆｏｆａＴｈｅｏｒｅｍｉｎＤＣＰｒｏｂｌｅｍＸｉａＺｈｏｎｇｈａｎｇＸｉａＺｕｎｑｕａｎ（ＤｅｐｔｏｆＡｐｐｌｉｅｄＭａｔｈｅｍａｔｉｃｓ，ＤａｌｉａｎＵｎｉｖｅｒｓｉｔｙｏｆＴｅｃｈｎｏｌｏｇｙ，１１６０２４）ＴｈｅＰｒｏｏｆｏｆａＴ...     

Discrete Arzelà-Ascoli Theorem on the Half Line and Its Application

The compactness criterion in l_◇={x(·)∈l~∞(0,∞))|lim t→∞ x(t)=x(∞)} is given and as an application,we study the existence of positive solutions of second order boundary value problems of difference equation on the half line by the fixedpoint theorem in cones.     

On Hardy－Hilbert‘s Integral Inequality and Its Equivalent Form

In this paper, by introducing three parameters a, b and λ, we give some new generalizations of Hardy-Hilbert‘s integral inequality. As applications, we con-sider its equivalent form and some particular results.     

Deterministic “Snakes and Ladders” Heuristic for the Hamiltonian cycle problem

We present a polynomial complexity, deterministic, heuristic for solving the Hamiltonian cycle problem (HCP) in an undirected graph of order $n$ . Although finding a Hamiltonian cycle is not theoretically guaranteed, we have observed that the heuristic is successful even in cases where such cycles are extremely rare, and it also performs very well on all HCP instances of large graphs listed on the TSPLIB web page. The heuristic owes its name to a visualisation of its iterations. All vertices of the graph are placed on a given circle in some order. The graph’s edges are classified as either snakes or ladders, with snakes forming arcs of the circle and ladders forming its chords. The heuristic strives to place exactly $n$ snakes on the circle, thereby forming a Hamiltonian cycle. The Snakes and Ladders Heuristic uses transformations inspired by $k$ -opt algorithms such as the, now classical, Lin–Kernighan heuristic to reorder the vertices on the circle in order to transform some ladders into snakes and vice versa. The use of a suitable stopping criterion ensures the heuristic terminates in polynomial time if no improvement is made in $n^3$ major iterations.     

The Variable Coefficient ABE-I Method and Its Stability

The ABE-I (Alternating Block Explicit-Implicit) method for diffusion problem is extended to solve the variable coefficient problem and the unconditional stability of the ABE-I method is proved by the energy method.     

Takács’ Asymptotic Theorem and Its Applications: A Survey

The book of Lajos Takács Combinatorial Methods in the Theory of Stochastic Processes has been published in 1967. It discusses various problems associated with

The Brezis–Browder Theorem in a general Banach space

During the 1970s Brezis and Browder presented a now classical characterization of maximal monotonicity of monotone linear relations in reflexive spaces. In this paper, we extend (and refine) their result to a general Banach space. We also provide an affirmative answer to a problem posed by Phelps and Simons.