A remark on Barth’s connectivity theorem

It has been remarked by Hartshorne, that Barth’s theorem for a smooth projective X follows from the strong Lefschetz theorem for the cohomology of X. Using the strong Lefschetz theorem for intersection cohomology, we give an extension of Barth’s theorem to singular X. This naturally raises several questions concerning possible Barth theorems on the level of intersection cohomology.     

Extension d'un théorème de Harte et applications aux algèbres de Jordan-Banach

In this paper we study some questions related to spectral theory in Jordan-Banach algebras. Firstly, we introduce the notion of exponential spectrum and then we extend to Jordan-Banach algebras a theorem due to Robin Harte in the associative case. Secondly, these results are used to get a theorem on spectral perturbation by inessential elements in Jordan-Banach algebras.

     

Existence of equilibria for generalized games without paracompactness

In this paper, we first prove an existence theorem of maximal elements without any paracompactness. Then, we establish an existence theorem of equilibrium points of generalized games with infinite many players and without any paracompactness, by applying our maximal element existence theorem. Our equilibrium existence theorem answers the two questions raised by Kim [6] in the affirmative.     

Summation of Divergent Series and Ergodic Theorems

In this article questions on the possibility of sharpening classic ergodic theorems is considered. To sharpen these theorems the author uses methods of summation of divergent sequences and series. The main topic is connected with the individual ergodic Birkhoff–Khinchin theorem. The theorem is studied in connection with the Riesz and Voronoi summation methods. These methods are weaker than those of the Cesaro method of arithmetic means. It is shown that already for the Bernoulli transformation of the unit interval, meaningful problems arise. These problems are interesting in connection with the possibility of extension of the strong law of large numbers. The questions of suitable summation factors and of the solution of homological equations by means of divergent series is also discussed.     

On the Hartogs-Phenomenon and Extension of Analytic Hypersurfaces in Non-separated Riemann Domains

In the present paper, we answer two questions raised by Jarnicki and Pflug: First, we show by a counterexample that the Hartogs-Bochner theorem is no longer true for non-separated Riemann domains. Secondly, we generalize a structure theorem of Dloussky, which examines the extension of singularity sets contained in analytic hypersurfaces, to non-separated Riemann domains. Moreover, our method yields a new proof of Dloussky's original result.     

An Easton theorem for level by level equivalence

We establish an Easton theorem for the least supercompact cardinal that is consistent with the level by level equivalence between strong compactness and supercompactness. In both our ground model and the model witnessing the conclusions of our theorem, there are no restrictions on the structure of the class of supercompact cardinals. We also briefly indicate how our methods of proof yield an Easton theorem that is consistent with the level by level equivalence between strong compactness and supercompactness in a universe with a restricted number of large cardinals. We conclude by posing some related open questions. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Some applications of Cartan's theorem to normality and semiduality of gap power series

In this paper we give partial solutions to some questions concerning analytic functions with AP-Gaps raised by Pinto, Ruscheweyh and Salinas (cf. [9], [12]) using a theorem of H. Cartan which extends Montel's theorem on analytic functions omitting the values 0 and 1. Using the same method, we also prove a generalization of a theorem in [9] on the dual hull of sets containing two elements.     

Extensions contained in ideals

We prove a Weyl-von Neumann type absorption theorem for extensions which are not full, and give a condition for constructing infinite repeats contained in an ideal. We also clear up some questions associated with the purely large criterion for full extensions to be absorbing.

     

Anwendungen der De Rhamschen Zerlegung auf Probleme der lokalen Flächentheorie

Various questions of classical differential geometry lead to the problem of determining all those hypersurfaces of euclidean space for which the covariant derivative of the second fundamental form vanishes identically. In the first part of this note, we investigate these hypersurfaces; the result is summarized in the theorem of the opening paragraph, whose proof is based in the decomposition theorem of De Rham (see e.g. [6], pp. 180–186). Applying this result in the second part of the note, we give local characterizations of the Euclidean sphere.     

Sturm’s theorem on zeros of linear combinations of eigenfunctions

Motivated by recent questions about the extension of Courant’s nodal domain theorem, we revisit a theorem published by C. Sturm in 1836, which deals with zeros of linear combination of eigenfunctions of Sturm–Liouville problems. Although well known in the nineteenth century, this theorem seems to have been ignored or forgotten by some of the specialists in spectral theory since the second half of the twentieth-century. Although not specialists in History of Sciences, we have tried to put this theorem into the context of nineteenth century mathematics.     

A metric approach to asymptotic analysis

We introduce a notion of “firm” (or uniform) asymptotic cone to an unbounded subset of a normed space. We relate this notion to a concept of “firm” asymptotic function. We use these notions to study boundedness properties which can be applied to continuity questions for some operations on sets and functions. Such questions arise in stability analysis of Hamilton-Jacobi equations. We present some other applications such as an extension of a theorem of Dieudonné and existence results in optimization and fixed point theory.     

凸幂凝聚算子的不动点定理及其对抽象半线性发展方程的应用

从应用问题的需要出发,给出了一类新的算子-凸幂凝聚算子的定义,推广了凝聚算子的概念,并证明了这类新算子的不动点定理,从而推广了著名的Schauder不动点定理和Sadovskii不动点定理.作为应用,获得了Banach空间中一类具有非紧半群的半线性发展方程初值问题整体mild解和正mild解的存在性.     

Random Partitions by Semigroup Methods

P of all partitions of {1,2,3,...}, or rather its distribution. There is a natural compact metrizable topology on P taking care of measurability questions. With respect to the maximum operation P becomes an abelian semigroup, and our first theorem characterizes random partitions as normalized positive definite functions on the subsemigroup of partitions "with finite support". We then present a new proof of Kingman's theorem stating that the exchangeable random partitions form a simplex whose extreme points are the so-called "paint-box distributions". An interesting moment problem which is still open arises in this connection.     

Product formulas,nonlinear semigroups,and accretive operators

A special case of our main theorem, when combined with a known result of Brezis and Pazy, shows that in reflexive Banach spaces with a uniformly Gâteaux differentiable norm, resolvent consistency is equivalent to convergence for nonlinear contractive algorithms. (The linear case is due to Chernoff.) The proof uses ideas of Crandall, Liggett, and Baillon. Other applications of our theorem include results concerning the generation of nonlinear semigroups (e.g., a nonlinear Hille-Yosida theorem for “nice” Banach spaces that includes the familiar Hilbert space result), the geometry of Banach spaces, extensions of accretive operators, invariance criteria, and the asymptotic behavior of nonlinear semigroups and resolvents. The equivalence between resolvent consistency and convergence for nonlinear contractive algorithms seems to be new even in Hilbert space. Our nonlinear Hille-Yosida theorem is the first of its kind outside Hilbert space. It establishes a biunique correspondence between m-accretive operators and semigroups on nonexpansive retracts of “nice” Banach spaces and provides affirmative answers to two questions of Kato.     

On a theorem of Hermite and Hurwitz

The Hermite-Hurwitz theorem computes the degree, over $\text{R}$, of a real rational function ? in terms of the signature of an associated quadratic form—known today as the Hankel matrix of ?. This formula, which Hermite was led to by his work on the problem of representing integers as sums of squares, gave rise to striking applications in the theory of equations and in the stability theory of ordinary differential equations. In this paper, this theorem and various generalizations to the matrix-valued case are discussed and described in terms of signature formulae. These include its relation to stability theory and the matrix Hermite-Hurwitz theorem of Bitmead-Anderson as applied to questions of circuit synthesis. This also includes a global form of Hörmander's signature formula for the Maslov index of a rational loop in a Lagrangian Grassmannian, due to Byrnes and Duncan, and applications to the topology of spaces of rational matrix-valued functions, following the work of Brockett, Byrnes, and Duncan. This includes, in particular, a topological proof of the matrix Hermite-Hurwitz theorem.     

An example of a finitely defined solvable group without the maximality condition for normal subgroups

There is a finitely defined solvable group which does not satisfy the maximality condition for normal subgroups. This theorem gives a negative answer to one of the questions raised by P. Hall.Translated from Matematicheskie Zametki, Vol.l2,No. 3, pp. 287–293, September, 1972.     

An elliptic analogue of a theorem of Hecke

We revisit the classical theorem of Euler regarding special values of the Riemann zeta function as well as Hecke’s generalization of this to Dirichlet’s \(L\)-functions and derive an elliptic analogue. We also discuss transcendence questions that arise from this analogue.     

On Koosis' approach to the proof of the Carleson interpolation theorem

The paper is devoted to Koosis' approach to the Carleson theorem. We show that this approach also works for other related questions. The main emphasis in this paper is on the method. Bibliography: 10 titles. Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 355, 2008, pp. 5–36.     

On the number of conjugacy classes of zeros of characters and the length of solvable groups

Abstract

We apply an orbit theorem to a few questions about character degrees. We investigate the relation of the number of conjugacy classes where characters vanish and the length of the solvable groups. As another application, we give a bound for the size of defect groups of blocks of solvable groups.

Communicated by J. Zhang     

Minimax and interlacing thoerems for matrices

An exposition is given of some recent and important work by E. Marques de Sá and R. C. Thompson, which characterized the relationship between the similarity invariants of a square matrix over a field and those of a principal submatrix of that matrix. This work is then related to results on eigenvalues of complex hermitian matrices (the Courant-Fischer minimax theorem and the Cauchy interlacing theorem), on singular values of rectangular complex matrices (due to Thompson), and on invariant factors of rectangular matrices over a principal ideal domain (due to Sá and Thompson). A partial unification of these theories is presented, and several open questions stated.