A Simple Proof of the Riemann Mapping Theorem for Domains with Spherical Boundary

We give a simple proof of a Riemann mapping theorem for domains in a Stein manifold with a spherical boundary. Those boundaries are real hypersurfaces which are locally CR equivalent to the sphere inside complex space. The main observation is that for higher dimensional Stein manifolds, the fundamental group of its boundary coincides with that of its interior.     

A Simple Proof of the Borel Extension Theorem and Weak Compactness of Operators

Let T be a locally compact Hausdorff space and let C ₀(T) be the Banach space of all complex valued continuous functions vanishing at infinity in T, provided with the supremum norm. Let X be a quasicomplete locally convex Hausdorff space. A simple proof of the theorem on regular Borel extension of X-valued -additive Baire measures on T is given, which is more natural and direct than the existing ones. Using this result the integral representation and weak compactness of a continuous linear map u: C ₀(T) X when c ₀ X are obtained. The proof of the latter result is independent of the use of powerful results such as Theorem 6 of [6] or Theorem 3 (vii) of [13].     

A Simple Proof for a Spectral Factorization Theorem

It is shown using simple methods that the transform Z(z), z C, of the coefficient sequence of the Wold decomposition ofany full-rank wide-sense stationary purely non-deterministicstochastic process satisfies (i) Z(z) H² (D) and (ii) Z^–1(z) H(D). Further it is shown that all spectral factors satisfying(i) and (ii) are equal up to right multiplication by orthogonalmatrices, and that among these the normalized (Z(0) =I) spectralfactors are equal to the transform of the Wold decomposition.An elementary proof of Youla's Theorem is then given togetherwith a simple proof that the rows of a Cholesky factor of abanded block Toeplitz matrix converge to the coefficients ofa stable matrix polynomial. Computer and Automation Institute of the Hungarian Academyof Sciences, Budapes, Hungary.     

A Simple Proof of a Theorem of Whyte

Kevin Whyte showed that all Baumslag–Solitar groups BS(p,q) with 1 < p < q are quasi-isometric [Whyte, K., Geom. Funct. Anal. 11 (2001), 1327–1343]. We provide an elementary geometric proof.     

A Simple Proof of Tyurin's Babylonian Tower Theorem

Using the method of Coand? and Trautmann [4 Coand? , I. , Trautmann , G. ( 2006 ). The splitting criterion of Kempf and the Babylonian tower theorem . Comm. Algebra 34 : 2485 – 2488 .[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]], we give a simple proof of a theorem due, in the smooth case, to Tyurin [9 Tyurin , A. N. ( 1976 ). Finite dimensional vector bundles over infinite varieties . Math. USSR Izv. 10 : 1187 – 1204 .[Crossref], [Web of Science ®] , [Google Scholar]]: if a vector bundle E on a c-codimensional locally Cohen–Macaulay closed subscheme X of ? ⁿ extends to a vector bundle F on a similar closed subscheme Y of ? ^N , for every N > n, then E is the restriction to X of a direct sum of line bundles on ? ⁿ . Using the same method, we also provide a proof of the Babylonian tower theorem for locally complete intersection subschemes of projective spaces.     

关于稳定矩阵分解定理的一个简单证明

一个矩阵称为稳定的,如果这个矩阵的特征值全包含在单位开圆盘内.利用Parker关于复方阵的分解定理给出了稳定矩阵分解定理的一个简单证明,并对奇异值全部严格小于1的矩阵给出了类似的结论.     

泛圈图方面的著名Bondy定理的简单证明

本文用极好的新方法给出泛图图方面的Bondy定理的简捷证明。     

A Theorem for Generalized Extension ofOperators and Some Applications

In this note it is shown that every bounded linear operator T Э B(Y,Z)defined on a closed subspace Y of a Banach space X admits a generalized extension T Э B(X, V). Some examples of the applications are given; especially, a characterization of H.I. spaces is obtained.     

Cauchy微分中值定理的推广的一个简单证明

给出Cauchy微分中值定理的推广的一个简单证明.     

A New Simple Proof of the Aztec Diamond Theorem

The Aztec diamond of order n is the union of lattice squares in the plane intersecting the square \(|x|+|y|<n\). The Aztec diamond theorem states that the number of domino tilings of this shape is \(2^{n(n+1)/2}\). It was first proved by Elkies et al. (J. Algebraic Comb. 1(2):111–132, 1992). We give a new simple proof of this theorem.     

关于Shimoji and Takahashi一个定理的简单证明

目的是在一般Banach空间中建立一个无限可数多个非扩张映像族的公共不动点的强收敛定理．使用一些新的分析技巧,尤其是Dotson引理和W映像技巧,给出了Shimoji and Takahashi的一个定理的简单证明．     

矩阵秩分解定理的一个证明及其应用

矩阵的秩分解定理是矩阵论中的一个基础性定理.本文给出了秩分解定理的一个证明,描述了其中可逆矩阵P,Q的构造,讨论了秩分解定理及其P,Q的构造在解线性方程组中的应用,以及在判别Sylvester不等式等号成立中的应用.     

外平面图全色数定理的简单证明

1988年,张忠输等证明了对于△(G)≥3的外平面图G,全色数X_T(G)=△(G)+1.本文给出此结论一个简单证明,方法是全新的.     

El Karoui的半鞅上穿定理的简单证明

El Karoui在[1]及[2]中给出了半鞅局部时的上穿刻划,这部分地推广了有关Brown运动的Lévy下穿定理。本文用随机积分的控制收效定理直接推出了这一结果,从而大大简化了原证明。     

A Simple Proof of the Matrix-Valued Fejér-Riesz Theorem

A very short proof of the Fejér-Riesz lemma is presented in the matrix case.      

Browder不动点定理的推广及应用

研究 Ｂｒｏｗｄｅｒ 不动点定理, 得到 Ｂｒｏｗｄｅｒ 不动点定理之一新的推广定理及其几种等价形式·　作为应用,研究了最近点和不动点的存在性问题·