A unified proof of Brooks’ theorem and Catlin’s theorem

Brooks’ theorem is a fundamental result in the theory of graph coloring. Catlin proved the following strengthening of Brooks’ theorem: Let d$d$ be an integer at least 3, and let G$G$ be a graph with maximum degree d$d$. If G$G$ does not contain _Kd+1$K_{d+1}$ as a subgraph, then G$G$ has a d$d$-coloring in which one color class has size α(G)$\alpha \left(G\right)$. Here α(G)$\alpha \left(G\right)$ denotes the independence number of G$G$. We give a unified proof of Brooks’ theorem and Catlin’s theorem.     

A simple proof for the Chernoff-Savage theorem

A proof for the Chernoff-Savage theorem is given based on differentiable statistical functionals. It employs no second order differentiability condition.     

One more simple proof of the Craig–Sakamoto theorem

We give one more elementary proof of the Craig-Sakamoto’s theorem: given such that ; then AB=0.     

A handy proof of Gay's theorem

In 1979, Gay proved that Broyden's methods, when used for n‐square linear systems, terminate in at most 2n iterations (SIAM J. Numer. Anal. 1979; 16 :623–630). Also, the ABS methods were introduced in 1984 (Numer. Math. 1984; 45 :1361–1376). In this paper we show another (handy) proof of Gay's theorem by these algorithms. Copyright © 2008 John Wiley & Sons, Ltd.     

A probabilistic proof of the fundamental theorem of algebra

We use Lévy's theorem on invariance of planar Brownian motion under conformal maps and the support theorem for Brownian motion to show that the range of a non-constant polynomial of a complex variable consists of the whole complex plane. In particular, we obtain a probabilistic proof of the fundamental theorem of algebra.

     

A nonstandard proof of the Eberlein-Smulian theorem

The Eberlein-Smulian theorem on the equivalence of weak compactness and the finite intersection property of bounded closed convex sets is given a short elementary proof by applying Abraham Robinson's nonstandard characterization of compactness.

     

On Kersting's proof of the central limit theorem

By an adaptation of a method originally invented by G. Kersting [1] for the calculation of the limiting distribution of Markovian processes the central limit theorem (CLT) is proven. Only the case of equal variances is considered.     

A nonstandard proof of the spectral theorem for unbounded self-adjoint operators

We generalize Moore’s nonstandard proof of the Spectral theorem for bounded self-adjoint operators to the case of unbounded operators. The key step is to use a definition of the nonstandard hull of an internally bounded self-adjoint operator due to Raab.     

A new proof of a theorem on optimal control of switched systems

Recently Sorin C. Bengea and Raymond A. DeCarlo [Sorin C. Bengea, Raymond A. DeCarlo, Optimal control of switching systems, Automatica J. IFAC 41 (2005) 11-27] have offered a key result that the set of trajectories of the two-switched system is dense in the set of trajectories of the embedded system. This result was proven by means of relaxed controls and the Chattering Lemma. In this paper we use the Lyapunov theorem to give a new simple proof.     

A proof of the normal form theorem for the closed terms of Girard's system F by means of computability

In this paper a proof of the normal form theorem for the closed terms of Girard's system F is given by using a computability method à la Tait. It is worth noting that most of the standard consequences of the normal form theorem can be obtained using this version of the theorem as well. From the proof-theoretical point of view the interest of the proof is that the definition of computable derivation here used does not seem to be well founded. MSC: 03F05, 03B15.     

A constructive proof of the Peter‐Weyl theorem

We present a new and constructive proof of the Peter‐Weyl theorem on the representations of compact groups. We use the Gelfand representation theorem for commutative C*‐algebras to give a proof which may be seen as a direct generalization of Burnside's algorithm [3]. This algorithm computes the characters of a finite group. We use this proof as a basis for a constructive proof in the style of Bishop. In fact, the present theory of compact groups may be seen as a natural continuation in the line of Bishop's work on locally compact, but Abelian, groups [2]. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A new proof of Poltoratskii's theorem

We provide a new simple proof to the celebrated theorem of Poltoratskii concerning ratios of Borel transforms of measures. That is, we show that for any complex Borel measure μ on and any a.e. w.r.t. μ_sing, where μ_sing is the part of μ which is singular with respect to Lebesgue measure and F denotes a Borel transform, namely, and F_μ(z)=∫(x−z)⁻¹dμ(x).     

A combinatorial proof of Marstrand’s theorem for products of regular Cantor sets

In a paper from 1954 Marstrand proved that if K⊂R² has a Hausdorff dimension greater than 1, then its one-dimensional projection has a positive Lebesgue measure for almost all directions. In this article, we give a combinatorial proof of this theorem when K is the product of regular Cantor sets of class C^1+α, α>0, for which the sum of their Hausdorff dimension is greater than 1.     

关于Atiyah-Singer指标定理的随机方法证明

运用随机分析中的广义Wiener泛函给出Atiyah-Singer指标定理的新的随机方法的证明, 这一证明在形式上是比较自然简单的. 广义Wiener泛函由Watanabe (1989, 1990) 建立和运用.     

A new proof and generalizations of Gearhart's theorem

Let be a Hilbert space, let be the space of almost periodic functions from to , and let be a closed densely defined linear operator on . For a closed subset , let be the subspace of consisting of functions with spectrum contained in . We prove that the following properties are equivalent: (i) for every function there exists a unique mild solution of equation ; (ii) and . The case yields a new proof of the well-known Gearhart's spectral mapping theorem.

     

A proof of the trace theorem of Sobolev spaces on Lipschitz domains

A complete proof of the trace theorem of Sobolev spaces on Lipschitz domains has not appeared in the literature yet. The purpose of this paper is to give a complete proof of the trace theorem of Sobolev spaces on Lipschitz domains by taking advantage of the intrinsic norm on . It is proved that the trace operator is a linear bounded operator from to for .

     

Berry–Esseen theorem and local limit theorem for non uniformly expanding maps

In Young towers with sufficiently small tails, the Birkhoff sums of Hölder continuous functions satisfy a central limit theorem with speed , and a local limit theorem. This implies the same results for many non uniformly expanding dynamical systems, namely those for which a tower with sufficiently fast returns can be constructed.