James' Theorem Fails for Starlike Bodies

Starlike bodies are interesting in nonlinear functional analysis because they are strongly related to bump functions and to n-homogeneous polynomials on Banach spaces, and their geometrical properties are thus worth studying. In this paper we deal with the question whether James' theorem on the characterization of reflexivity holds for (smooth) starlike bodies, and we establish that a feeble form of this result is trivially true for starlike bodies in nonreflexive Banach spaces, but a reasonable strong version of James' theorem for starlike bodies is never true, even in the smooth case. We also study the related question as to how large the set of gradients of a bump function can be, and among other results we obtain the following new characterization of smoothness in Banach spaces: a Banach space X has a C¹ Lipschitz bump function if and only if there exists another C¹ smooth Lipschitz bump function whose set of gradients contains the unit ball of the dual space X*. This result might also be relevant to the problem of finding an Asplund space with no smooth bump functions.     

A Metric Version of Milyutin Theorem

Open covering and metric regularity are properties playing a crucial role in several topics of modern variational analysis. Here their stability behaviour in the presence of perturbations is investigated in a purely metric setting. Some results in this sense are obtained, which lead to extend the known Milyutin theorem, and then to expand the concept of radius of regularity, a quantitative measure of the open covering stability. Estimations for the latter in terms of covering moduli are provided.     

A Version of the Poincare-Birkhoff-Witt Theorem

In this note I shall prove that if L is a finite-dimensionalLie algebra over a field F of characteristic zero which is generatedas an algebra by a set of elements {e₁, e₂,...,e_k}, then theuniversal enveloping algebra U(L) of L is linearly generatedby monomials spanned by the elements {e_i} of an a priori boundedwidth. As an application, a criterion of Kostant for a leftideal of U(L) to be of finite codimension is proved by purelyalgebraic means.     

一个一般的拓扑型极大极小定理

本文得出一个一般形式的拓扑型的极大极小定理，它包含的主要结果为特例，而且回答了［３］中提出的一个未解决问题。     

A Nonlinear Version of Noether's Type Theorem

Abstract

Let L be a line bundle on a smooth curve C, which defines a birational morphism onto Φ(C) ? P ^r . We prove that, under suitable assumptions on L, which are satisfied by Castelnuovo's curves, a generic section in H ⁰(C, L ²) can be written as α² + β² + γ², with α, β, γ ∈ H ⁰(C, L). If there are no quadrics of rank 3 containing Φ(C), this is true for any section. For canonical curves, this gives a non linear version of Noether's Theorem.     

A Quantitative Version of a de Bruijn-Post Theorem

A theorem due to de Bruijn and Post states that if a real valued function f defined on [0, 1] is not Riemann-integrable, then there exists a uniformly distributed sequence { x_i} such that the averages do not admit a limit. In this paper we will prove a quantitative version of this result and we will extend it to functions with values in ℝ^d.     

A Double Bounded Version of Schur's Partition Theorem

Dedicated to the memory of Paul Erdős Schur's partition theorem states that the number of partitions of n into distinct parts (mod 3) equals the number of partitions of n into parts which differ by 3, where the inequality is strict if a part is a multiple of 3. We establish a double bounded refined version of this theorem by imposing one bound on the parts (mod 3) and another on the parts (mod 3), and by keeping track of the number of parts in each of the residue classes (mod 3). Despite the long history of Schur's theorem, our result is new, and extends earlier work of Andrews, Alladi-Gordon and Bressoud. We give combinatorial and q-theoretic proofs of our result. The special case L=M leads to a representation of the generating function of the underlying partitions in terms of the q-trinomial coefficients extending a similar previous representation of Andrews. Received November 18, 1999 Research of the first author supported in part by NSF Grant DMS-0088975.     

A Homological Version of the Implicit Function Theorem

Our objective in this paper is to prove an Implicit Function Theorem for general topological spaces. As a consequence, we show that, under certain conditions, the set of the invertible elements of a topological monoid X is an open topological group in X and we use the classical topological group theory to conclude that this set is a Lie group.     

A Quantitative Version of the Non-Abelian Idempotent Theorem

Suppose that G is a finite group and f is a complex-valued function on G. f induces a (left) convolution operator from L ²(G) to L ²(G) by g ? f *g{g \mapsto f \ast g} where

A Version of the Bartle–Graves Theorem for Temperate Functions

We prove a version of the Bartle–Graves theorem for temperate functions in the category of the b-spaces of L. Waelbroeck. As a consequence, we give a characterization of some spaces of functions with values in quotients which appear in L. Waelbroeck's holomorphic functional calculus.     

S-fuzzy Version of Stone's Theorem for Distributive Lattices

In this paper,we initiate a study of S-fuzzy ideal(filter) of a lattice where S stands for a meet semilattice.A S-fuzzy prime ideal(filter) of a lattice is defined and it is proved that a S-fuzzy ideal(filter) of a lattice is S-fuzzy prime ideal(filter) if and only if any non-empty α-cut of it is a prime ideal(filter).Stone's theorem for a distributive lattice is extended by considering S-fuzzy ideals(filters).     

A Central Limit Theorem for a Localized Version of the SK Model

In this note, we consider a SK (Sherrington–Kirkpatrick)-type model on ℤ ^d for d≥1, weighted by a function allowing to any single spin to interact with a small proportion of the other ones. In the thermodynamical limit, we investigate the equivalence of this model with the usual SK spin system, through the study of the fluctuations of the free energy. This author’s research partially supported by CAPES.     

An Effective Version of Belyi's Theorem

We compute bounds on covering maps that arise in Belyi's Theorem. In particular, we construct a library of height properties and then apply it to algorithms that produce Belyi maps. Such maps are used to give coverings from algebraic curves to the projective line ramified over at most three points. The computations here give upper bounds on the degree and coefficients of polynomials and rational functions over the rationals that send a given set of algebraic numbers to the set {0,1,∞} with the additional property that the only critical values are also contained in {0,1,∞}.     

One Boundary Version of Morera's Theorem

Let D be a bounded domain in C ⁿ (n>1) with a connected smooth boundary D and let f be a continuous function on D. We consider conditions (generalizing those of the Hartogs–Bochner theorem) for holomorphic extendability of f to D. As a corollary we derive some boundary analog of Morera's theorem claiming that if the integrals of f vanish over the intersection of the boundary of the domain with complex curves in some class then f extends holomorphically to the domain.     

New Version of the Daniell-Stone-Riesz Representation Theorem

The traditional representation theorems after Daniell-Stone and Riesz were in a kind of separate existence until Pollard-Topsøe 1975 and Topsøe 1976 were the first to put them under common roofs. In the same spirit the present article wants to obtain a unified representation theorem in the context of the author’s work in measure and integration. It is an inner theorem like the previous ones. The basis is the recent comprehensive inner Daniell-Stone theorem, so that in particular there are no a priori assumptions on the additive behaviour of the data.     

Infinite Matroidal Version of Hall's Matching Theorem

Hall's theorem for bipartite graphs gives a necessary and sufficientcondition for the existence of a matching in a given bipartitegraph. Aharoni and Ziv have generalized the notion of matchabilityto a pair of possibly infinite matroids on the same set andgiven a condition that is sufficient for the matchability ofa given pair (M, W) of finitary matroids, where the matroidM is SCF (a sum of countably many matroids of finite rank).The condition of Aharoni and Ziv is not necessary for matchability.The paper gives a condition that is necessary for the existenceof a matching for any pair of matroids (not necessarily finitary)and is sufficient for any pair (M, W) of finitary matroids,where the matroid M is SCF.     

A Filtered Version of the Bipolar Theorem of Brannath and Schachermayer

We extend the Bipolar Theorem of Kramkov and Schachermayer⁽¹²⁾ to the space of nonnegative càdlàg supermartingales on a filtered probability space. We formulate the notion of fork-convexity as an analogue to convexity in this setting. As an intermediate step in the proof of our main result we establish a conditional version of the Bipolar theorem. In an application to mathematical finance we describe the structure of the set of dual processes of the utility maximization problem of Kramkov and Schachermayer⁽¹²⁾ and give a budget-constraint characterization of admissible consumption processes in an incomplete semimartingale market.