A constructive proof of the Chang-Marshall theorem

In the paper one gives constructive proofs (without the use of the space of maximal ideals of the algebra H) of the Marshall-Chang theorem and of a theorem of T. H. Wolff. These theorems are proved in the paper in a unique manner with the use of a lemma which connects the level lines of a positive harmonic function in a circle with the level lines of the inner function.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematieheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 141, pp. 149–153, 1985.     

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 constructive proof of Ky Fan’s coincidence theorem

We present a constructive proof for the well-known Ky Fan’s coincidence theorem through a simplicial algorithm. In a finite number of steps the algorithm generates a simplex containing an approximate coincidence point. In the limit, when the mesh size converges to zero, the sequence of approximations converges to a coincidence point. This research was carried out while the second author was visiting the CentER for Economic Research, Tilburg University. He would like to thank both CentER and the Netherlands Organization for Scientific Research (NWO) for their financial support.     

A division theorem for real analytic functions

We characterize those homogeneous polynomials P [z₁, ... ,z_d] for which the principal ideal (P) = P · A(^d) is complementedin A(^d) or, equivalently, those which admit a continuous lineardivision operator. The condition is the same as that which characterizes,among the homogeneous polynomials, those which are nonellipticand for which P(D) is surjective in A(^d), and those for whichP(D) admits a continuous linear right inverse in C(^d). It dependsonly on the type of real singularities.     

A constructive proof of a Whitney extension theorem in one variable

We prove a new stability property of the classical Hermite interpolation scheme in one variable. This stability property refines the usual convergence property of this interpolation scheme. Consequently, we obtain a natural proof of a Whitney extension theorem in one variable.     

A short and constructive proof of Tarski’s fixed-point theorem

I give short and constructive proofs of Tarski’s fixed-point theorem, and of Zhou’s extension of Tarski’s fixed-point theorem to set-valued maps.I thank Charles Blair, William Thomson, an associated editor and a referee for their helpful suggestions.I have taught Tarski’s Theorem with F continuous to Caltech undergraduates.     

A subanalytic triangulation theorem for real analytic orbifolds

Let X be a real analytic orbifold. Then each stratum of X is a subanalytic subset of X. We show that X has a unique subanalytic triangulation compatible with the strata of X. We also show that every Cr-orbifold, 1?r?∞, has a real analytic structure. This allows us to triangulate differentiable orbifolds. The results generalize the subanalytic triangulation theorems previously known for quotient orbifolds.     

A simple proof of the approximation by real analytic Lipschitz functions

A theorem in Azagra et al. (preprint) [1] asserts that on a real separable Banach space with separating polynomial every Lipschitz function can be uniformly approximated by real analytic Lipschitz function with a control over the Lipschitz constant. We give a simple proof of this theorem.     

A constructive proof for a theorem on contractive mappings in power sets

Recently I proved the following theorem: To every positive integer m there exists a positive integer h such that the following holds: If S is a set of h elements and f a mapping of the power set B of S into B such that f(T)?T for all T?B, then there exists a strictly increasing sequence T₁?…?T_m of subsets of S such that one of the following three possibilities holds: (a) All sets f(T_i), i=1,…,m, are equal. (b) For all i=1,…,m we have f(T_i)=T_i (c) T_i=f(T_i+1) for all i=1,…,m?1.The proof given in [2] was non-constructive. In this paper now we give a constructive proof. By the way, this also yields a solution of a problem of Rado [3, p. 106].     

A constructive proof of the Cartan-Dieudonné-Scherk Theorem in the real or complex case

The Cartan-Dieudonné-Scherk Theorem states that for fields of characteristic other than 2, every orthogonality can be written as the product of a certain minimal number of reflections across hyperplanes. The earliest proofs are not constructive, and constructive proofs either do not achieve minimal results or have been restricted to special cases. This paper presents a constructive proof in the real or complex field of the decomposition of a generalized orthogonal matrix into the product of the minimal number of generalized Householder matrices.     

Quantization of supermanifolds and an analytic proof of the Atiyah-Singer index theorem

An analytic proof of the Atiyah-Singer index, theorem is given with the help of the tools of supermathematics. The index formula for the Dirac operator on a spinor manifold is obtained here by direct calculation. A large portion of the paper is devoted to questions of quantization on supermanifolds, using spinors as example.Translated from Itogi Nauki i Tekhniki, Seriya Sovremennye Problemy Matematiki, Noveishie Dostizheniya, Vol. 38, pp. 3–118, 1991.     

A constructive version of Birkhoff's theorem

A version of Birkhoff's theorem is proved by constructive, predicative, methods. The version we prove has two conditions more than the classical one. First, the class considered is assumed to contain a generic family, which is defined to be a set‐indexed family of algebras such that if an identity is valid in every algebra of this family, it is valid in every algebra of the class. Secondly, the class is assumed to be closed under inductive limits. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A proof of the Dvoretzky-Rogers theorem

We give a new proof of the famous Dvoretzky-Rogers theorem ([2], Theorem 1), according to which a Banach spaceE is finite-dimensional if every unconditionally convergent series inE is absolutely convergent.     

A constructive proof of Tucker's combinatorial lemma

Tucker's combinatorial lemma is concerned with certain labellings of the vertices of a triangulation of the n-ball. It can be used as a basis for the proof of antipodal-point theorems in the same way that Sperner's lemma yields Brouwer's theorem. Here we give a constructive proof, which thereby yields algorithms for antipodal-point problems. Our method is based on an algorithm of Reiser.