On Faith's correspondence theorem

A simple proof of an extension of Faith's correspondence theorem for projective modules is given for a Morita context (R, V, W, S) in whichVWV=V andWVW=W.Research supported by N.S.E.R.C. (Canada), Grant No. A 8075.     

On the Mosaic Method for Many-Dimensional Modal Logics: A Case Study Combining Tense and Modal Operators

We present an extension of the mosaic method aimed at capturing many-dimensional modal logics. As a proof-of-concept, we define the method for logics arising from the combination of linear tense operators with an “orthogonal” S5-like modality. We show that the existence of a model for a given set of formulas is equivalent to the existence of a suitable set of partial models, called mosaics, and apply the technique not only in obtaining a proof of decidability and a proof of completeness for the corresponding Hilbert-style axiomatization, but also in the development of a mosaic-based tableau system. We further consider extensions for dealing with the case when interactions between the two dimensions exist, thus covering a wide class of bundled Ockhamist branching-time logics, and present for them some partial results, such as a non-analytic version of the tableau system.     

The schoenflies extension in the analytic case

Summary Let S be an (n−1)-sphere in a euclidean n-space E. Let B be the closed n-ball in E bounded by S. Let z be an arbitrary point of . A real analytic diffeomorphism f of S into E admits a homeomorphic extension F which is defined over some open neighborhood N of B and such that F | (N−z) is an analytic diffeomorphism. We give a new proof of this theorem to serve as a model for a forthcoming theory of analytic families of such extensions. To Enrico Bompiani on his scientific Jubilee.     

Ramanujan’s Master Theorem

S. Ramanujan introduced a technique, known as Ramanujan??s Master Theorem, which provides an explicit expression for the Mellin transform of a function in terms of the analytic continuation of its Taylor coefficients. The history and proof of this result are reviewed, and a variety of applications is presented. Finally, a multi-dimensional extension of Ramanujan??s Master Theorem is discussed.     

Hahn–Banach extension theorems for multifunctions revisited

Several generalizations of the Hahn–Banach extension theorem to K-convex multifunctions were stated recently in the literature. In this note we provide an easy direct proof for the multifunction version of the Hahn–Banach–Kantorovich theorem and show that in a quite general situation it can be obtained from existing results. Then we derive the Yang extension theorem using a similar proof as well as a stronger version of it using a classical separation theorem. Moreover, we give counterexamples to several extension theorems stated in the literature. Dedicated to Jean-Paul Penot with the occasion of his retirement.     

Fenchel duality, Fitzpatrick functions and the Kirszbraun-Valentine extension theorem

We present a new proof of the classical Kirszbraun-Valentine extension theorem. Our proof is based on the Fenchel duality theorem from convex analysis and an analog for nonexpansive mappings of the Fitzpatrick function from monotone operator theory.

     

Proof analysis in intermediate logics

Using labelled formulae, a cut-free sequent calculus for intuitionistic propositional logic is presented, together with an easy cut-admissibility proof; both extend to cover, in a uniform fashion, all intermediate logics characterised by frames satisfying conditions expressible by one or more geometric implications. Each of these logics is embedded by the G?del–McKinsey–Tarski translation into an extension of S4. Faithfulness of the embedding is proved in a simple and general way by constructive proof-theoretic methods, without appeal to semantics other than in the explanation of the rules.     

A remark on a compactness result in electromagnetic theory

Calderon's extension theorem is a crucial tool in the proof of the compactneσs of the resolvent for the Maxwell operator, and whence this result is proved for domains with the strict cone property. However, the proof only requires an extension operator that extends W^2,2-functions compactly as W^1,2-functions. It is shown that this can be achieved under weaker regularity conditions on the domain: the cone may be replaced by some cusp of an appropriate order.     

Note On The Utility Of A Theorem Of Leindler–Meir–Totik

Our purpose is to generalize and to extend a theorem of S. Sharma and S. K. Varma [15] concerning the order of approximation by Abel means in the Lipschitz norm. The proof is basically based on a simple extension of a general theorem of L. Leindler, A. Meir and V. Totik [6] related to approximation by finite summability methods.     

New proof of the cobordism invariance of the index

We give a simple proof of the cobordism invariance of the index of an elliptic operator. The proof is based on a study of a Witten-type deformation of an extension of the operator to a complete Riemannian manifold. One of the advantages of our approach is that it allows us to treat directly general elliptic operators which are not of Dirac type.

     

相对整体维数有限的扩张

代数的扩张是指两个代数之间保持单位元的同态映射,设f:B→A是代数的扩张,扩张f的相对整体维数是指所有A-模的相对投射维数的上确界.我们给出了扩张的相对整体维数有限的一个充分必要条件,作为应用,还获得了Hochschild的文[Relative homological algebra, Trans. Am. Math. Soc.,1956,82:246-269]中一个结果的简洁证明.     

KKM in an arbitrary product of simplices

Summary In this note we present an extension of Peleg's version of the well known KKM lemma to an arbitrary product of simplices. In the case of intending an extension to a finite or even a countably infinite product of simplices, a product topology on that set is enough to get a proof. In the general case we need to use a weak^*-topology.     

Note On The Utility Of A Theorem Of Leindler–Meir–Totik

Our purpose is to generalize and to extend a theorem of S. Sharma and S. K. Varma [15] concerning the order of approximation by Abel means in the Lipschitz norm. The proof is basically based on a simple extension of a general theorem of L. Leindler, A. Meir and V. Totik [6] related to approximation by finite summability methods. This revised version was published online in June 2006 with corrections to the Cover Date.     

Simple proofs of occupancy tail bounds

We give a simple proof of an occupancy tail bound in the balls and bins experiment using the method of bounded differences in expected form. We also indicate how a short, calculation-free proof of another occupancy tail bound follows from an extension of the standard Chernoff–Hoeffding bounds to variables that satisfy some general notions of negative dependence. © 1997 John Wiley & Sons, Inc. Random Struct. Alg., 11 , 119–123 (1997)     

A simple proof of some generalized principal ideal theorems

Using symmetric algebras we simplify and slightly strengthen the Bruns-Eisenbud-Evans ``generalized principal ideal theorem' on the height of order ideals of nonminimal generators in a module. We also obtain a simple proof and an extension of a result by Kwiecinski, which estimates the height of certain Fitting ideals of modules having an equidimensional symmetric algebra.

     

There is no low maximal d. c. e. degree – Corrigendum

We give a corrected proof of an extension of the Robinson Splitting Theorem for the d. c. e. degrees. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Hausdorff's moment problem and expansions in Legendre polynomials

A new proof is given for Hausdorff's condition on a set of moments which determines when the function generating these moments is in L². The proof uses Legendre polynomials and their discrete extensions found by Tchebychef. Then an extension is given to a weighted L² space using Jacobi polynomials and their discrete extensions.     

A priori estimates for higher order multipliers on a circle

We present an elementary proof of an a priori estimate of Bourgain for a general class of multipliers on a circle using an extension of methods developed in our previous work. The main tool is a suitable version of a counting argument of Zygmund for unbounded regions.

     

Nakai's conjecture for varieties smoothed by normalization

The notion of D-simplicity is used to give a short proof that varieties whose normalization is smooth satisfy Ishibashi's extension of Nakai's conjecture to arbitrary characteristic. This gives a new proof of Nakai's conjecture for curves and Stanley-Reisner rings.

     

Isomorphisms of general linear groups over associative rings graded by an Abelian group

In this paper, we give a simpler proof of the Golubchik–Mikhalev–Zelmanov theorem on the structure of isomorphisms between general linear groups over associative rings, and also prove an extension of this theorem for linear groups over rings graded by an Abelian group.