A Frobenius Theorem on Convenient Manifolds

 A Frobenius Theorem for finite dimensional, involutive subbundles of the tangent bundle of a convenient manifold is proved. As first key applications Lie’s second fundamental theorem and Nelson’s theorem are treated in the convenient case.     

Generalized Kneser coloring theorems with combinatorial proofs

The Kneser conjecture (1955) was proved by Lovász (1978) using the Borsuk-Ulam theorem; all subsequent proofs, extensions and generalizations also relied on Algebraic Topology results, namely the Borsuk-Ulam theorem and its extensions. Only in 2000, Matoušek provided the first combinatorial proof of the Kneser conjecture. Here we provide a hypergraph coloring theorem, with a combinatorial proof, which has as special cases the Kneser conjecture as well as its extensions and generalization by (hyper)graph coloring theorems of Dol’nikov, Alon-Frankl-Lovász, Sarkaria, and Kriz. We also give a combinatorial proof of Schrijver’s theorem. Oblatum 17-IV-2001 & 12-IX-2001?Published online: 19 November 2001 An erratum to this article is available at .     

On the Equivalence and Generalized of Weyl Theorem Weyl Theorem

We know that an operator T acting on a Banach space satisfying generalized Weyl＇s theorem also satisfies Weyl＇s theorem. Conversely we show that if all isolated eigenvalues of T are poles of its resolvent and if T satisfies Weyl＇s theorem, then it also satisfies generalized Weyl＇s theorem. We give also a sinlilar result for the equivalence of a-Weyl＇s theorem and generalized a-Weyl＇s theorem. Using these results, we study the case of polaroid operators, and in particular paranormal operators.     

Entropy and the combinatorial dimension

We solve Talagrand’s entropy problem: the L ₂-covering numbers of every uniformly bounded class of functions are exponential in its shattering dimension. This extends Dudley’s theorem on classes of {0,1}-valued functions, for which the shattering dimension is the Vapnik-Chervonenkis dimension. In convex geometry, the solution means that the entropy of a convex body K is controlled by the maximal dimension of a cube of a fixed side contained in the coordinate projections of K. This has a number of consequences, including the optimal Elton’s Theorem and estimates on the uniform central limit theorem in the real valued case. Oblatum 10-XII-2001 & 4-IX-2002?Published online: 8 November 2002     

Retraction of a Compact Subspace onto Its Boundary

 Generalizing the famous Borsuk’s non-retraction theorem, it is proved that there is a retraction from a compact subspace of a euclidean space onto its boundary if and only if it has an empty interior. Received October 31, 2001; in revised form March 14, 2002 Published online September 9, 2002     

Uncertainty principles for the weinstein transform

The Weinstein transform satisfies some uncertainty principles similar to the Euclidean Fourier transform. A generalization and a variant of Cowling-Price theorem, Miyachi’s theorem, Beurling’s theorem, and Donoho-Stark’s uncertainty principle are obtained for the Weinstein transform.     

A simple proof of some ergodic theorems

Some ideas of T. Kamae’s proof using nonstandard analysis are employed to give a simple proof of Birkhoff’s theorem in a classical setting as well as Kingman’s subadditive ergodic theorem.     

On a sheaf-theoretic version of the Witt’s decomposition theorem. A Lagrangian perspective

The approach to a counterpart, in Abstract Geometric Algebra, that is, Geometric Algebra via sheaves of modules, of the classical Witt’s decomposition theoremis based on the axiomatization of the classical context, which however leads to the formulation of a specific subcategory of the category of sheaves of modules: the full subcategory of convenient sheaves of modules. Convenient sheaves of modules turn out, by the very essence of the matter at hand, to be of further importance as far as the setting of results leading to the sheaf-theoretic aspect of several forms of the Witt’s theorem is concerned. Further versions of the Witt’s theorem are still to be treated elsewhere.      

Ordering in mechanical geometry theorem proving

Ordering in mechanical geometry theorem proving is studied from geometric viewpoint and some new ideas are proposed. For Thebault’s theorem which is the most difficult theorem that has ever been proved by Wu’s method, a very simple proof using Wu’s method under a linear order is discovered. Project supported by the National Natural Science Foundation of China.     

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.     

On the realization of symmetries in quantum mechanics

The aim of this paper is to give a simple, geometric proof of Wigner’s theorem on the realization of symmetries in quantum mechanics that clarifies its relation to projective geometry. Although several proofs exist already, it seems that the relevance of Wigner’s theorem is not fully appreciated in general. It is Wigner’s theorem which allows the use of linear realizations of symmetries and therefore guarantees that, in the end, quantum theory stays a linear theory. In the present paper, we take a strictly geometrical point of view in order to prove this theorem. It becomes apparent that Wigner’s theorem is nothing else but a corollary of the fundamental theorem of projective geometry. In this sense, the proof presented here is simple, transparent and therefore accessible even to elementary treatments in quantum mechanics.     

On the realization of symmetries in quantum mechanics

The aim of this paper is to give a simple, geometric proof of Wigner’s theorem on the realization of symmetries in quantum mechanics that clarifies its relation to projective geometry. Although several proofs exist already, it seems that the relevance of Wigner’s theorem is not fully appreciated in general. It is Wigner’s theorem which allows the use of linear realizations of symmetries and therefore guarantees that, in the end, quantum theory stays a linear theory. In the present paper, we take a strictly geometrical point of view in order to prove this theorem. It becomes apparent that Wigner’s theorem is nothing else but a corollary of the fundamental theorem of projective geometry. In this sense, the proof presented here is simple, transparent and therefore accessible even to elementary treatments in quantum mechanics.     

Bordism classes represented by multiple point manifolds of immersed manifolds

We present a geometrical version of Herbert’s theorem determining the homology classes represented by the multiple point manifolds of a self-transverse immersion. Herbert’s theorem and generalizations can readily be read off from this result. The simple geometrical proof is based on ideas in Herbert’s paper. We also describe the relationship between this theorem and the homotopy theory of Thom spaces.     

On some generalizations of Ekeland��s principle and inward contractions in gauge spaces

We give generalizations in complete gauge spaces of the following results: Bishop-Phelps’ theorem, Ekeland’s variational principle, Caristi’s fixed point theorem, the drop theorem and the flower petal theorem. We show that our generalizations are equivalent. We apply those results to obtain fixed point theorems for multivalued contractions defined on a closed subset of a complete gauge space and satisfying a generalized inwardness condition.     

The Stepanov differentiability theorem in metric measure spaces

We extend Cheeger’s theorem on differentiability of Lipschitz functions in metric measure spaces to the class of functions satisfying Stepanov’s condition. As a consequence, we obtain the analogue of Calderon’s differentiability theorem of Sobolev functions in metric measure spaces satisfying a Poincaré inequality. Communicated by Steven Krantz     

Arithmetic height functions over finitely generated fields

In this paper, we propose a new height function for a variety defined over a finitely generated field over ℚ. For this height function, we prove Northcott’s theorem and Bogomolov’s conjecture, so that we can recover the original Raynaud’s theorem (Manin-Mumford’s conjecture). Oblatum 7-VI-1999 & 21-IX-1999 / Published online: 24 January 2000     

Modifications of the Ricci tensor and applications

By using two modified Ricci tensors, we prove some theorems which correspond to Myers’s diameter estimate theorem and Bochner’s vanishing theorem.     

Degree of rational mappings, and the theorems of Sturm and Tarski

We provide an explicit algorithm of computing the mapping degree of a rational mapping from the real projective line to itself. As a corollary we prove Sturm’s theorem and a number of its generalizations. These generalizations are used to prove Tarski’s theorem about real semialgebraic sets. Similarly a version of Tarski’s theorem can be proved for an arbitrary algebraically closed field. To V. I. Arnold on the occasion of his 70th birthday     

Carathéodory, Helly and the Others in the Max-Plus World

Carathéodory’s, Helly’s and Radon’s theorems are three basic results in discrete geometry. Their max-plus or tropical analogues have been proved by various authors. We show that more advanced results in discrete geometry also have max-plus analogues, namely, the colorful Carathéodory theorem and the Tverberg theorem. A conjecture connected to the Tverberg theorem—Sierksma’s conjecture—although still open for the usual convexity, is shown to be true in the max-plus setting.     

Preface

We will interpret a partial theta identity in Ramanujan’s Lost Notebook as a weighted partition theorem involving partitions into distinct parts with smallest part odd. A special case of this yields a new result on the parity of the number of parts in such partitions, comparable to Euler’s pentagonal numbers theorem. We will provide simple and novel proofs of the weighted partition theorem and the special case. Our proof leads to a companion to Ramanujan’s partial theta identity which we will explain combinatorially.