Barlow-wu continuum systems: A discussion on the model

Summary Barlow-Wu continuum structure functions have been introduced in the past as one particularly interesting family of continuum structure functions. In this paper we provide an alternative characterization for such continuum structure functions, showing other interesting properties. Research supported by Dirección General de Investigación Científica y Técnica (DGICYT), national grants number PB91-0389 and number BE91-225.     

Barlow-wu continuum systems: A discussion on the model

Barlow-Wu continuum structure functions have been introduced in the past as one particularly interesting family of continuum structure functions. In this paper we provide an alternative characterization for such continuum structure functions, showing other interesting properties. Research supported by Dirección General de Investigación Científica y Técnica (DGICYT), national grants number PB91-0389 and number BE91-225.     

Axiomatic characterizations of continuum structure functions

A continuum structure function is a non-decreasing mapping from the unit hypercube to the unit interval. Axiomatic characterizations of the continuum structure functions based on the Barlow-Wu and Natvig multistate structure functions are derived.     

On the geometric character of stress in continuum mechanics

This paper shows that the stress field in the classical theory of continuum mechanics may be taken to be a covector-valued differential two-form. The balance laws and other fundamental laws of continuum mechanics may be neatly rewritten in terms of this geometric stress. A geometrically attractive and covariant derivation of the balance laws from the principle of energy balance in terms of this stress is presented. Jerrold G. Marsden: Research partially supported by the California Institute of Technology and NSF-ITR Grant ACI-0204932.     

A continuum whose hyperspace of subcontinua is not -contractible

A topological space is said to be -contractible provided that there exists a continuous onto function such that is homotopic to a constant function. Answering a question by Sam B. Nadler, Jr., in this paper we construct a metric continuum such that its hyperspace of subcontinua is not -contractible.

     

Critical path analysis for continuum percolation

We prove the validity of the Critical path analysis for a continuum percolation model close to Golden–Kozlov one. This is obtained in the limit of strong disorder.

Résumé

On montre la validité de “Critical path analysis” pour un modèle de percolation continue proche de celui de Golden–Kozlov. Le résultat est obtenu à la limite de grand désordre.     

Subdifferential characterization of s-lower regular functions

As for Moreau envelopes of primal lower nice as well as prox-regular functions, Moreau s-envelopes of s-lower regular functions have been proved recently to have several remarkable differential properties and to have many important applications. Here, we provide a subdifferential characterization of extended real-valued s-lower regular functions on Banach spaces in terms of a hypomonotonicity-like property of the subdifferential.     

On Chebyshev functions and Klee functions

Associated to a lower semicontinuous function, one can define its proximal mapping and farthest mapping. The function is called Chebyshev (Klee) if its proximal mapping (farthest mapping) is single-valued everywhere. We show that the function f is 1/λ-hypoconvex if its proximal mapping Pλf is single-valued. When the function f is bounded below, and Pλf is single-valued for every λ>0, the function must be convex. Similarly, we show that the function f is 1/μ-strongly convex if the farthest mapping Qμf is single-valued. When the function is the indicator function of a set, this recovers the well-known Chebyshev problem and Klee problem in Rn. We also give an example illustrating that a continuous proximal mapping (farthest mapping) needs not be locally Lipschitz, which answers one open question by Hare and Poliquin.     

On the structure of homothetic functions

Summary Homothetic function is a term which refers to some extension of the concept of a homogeneous function. We study different hierarchies of generalized homogeneous functions. The main result is a general classification of those functions.     

On the coefficients of binary bent functions

We prove a 2-adic inequality for the coefficients of binary bent functions in their polynomial representations. The 2-adic inequality implies a family of identities satisfied by the coefficients. The identities also lead to the discovery of some new affine invariants of Boolean functions on .

     

On the continuity of biconjugate convex functions

We show that a Banach space is a Grothendieck space if and only if every continuous convex function on has a continuous biconjugate function on , thus also answering a question raised by S. Simons. Related characterizations and examples are given.

     

On a characterization of the support functions

Letf:S ^n–1 be a support function. Then, for everya , if the functionu f(u)–a, u has a negative minimum, then a unique argument exists for which this minimum is attained. It is shown that the converse holds true under some obvious restrictions onf. A perturbation theorem for the space (S ^n–1) is given as an application.     

On the integral representation of superbiharmonic functions

We consider a nonnegative superbiharmonic function w satisfying some growth condition near the boundary of the unit disk in the complex plane. We shall find an integral representation formula for w in terms of the biharmonic Green function and a multiple of the Poisson kernel. This generalizes a Riesz-type formula already found by the author for superbihamonic functions w satisfying the condition 0 ⩽ w(z) ⩽ C(1-|z|) in the unit disk. As an application we shall see that the polynomials are dense in weighted Bergman spaces whose weights are superbiharmonic and satisfy the stated growth condition near the boundary. Research supported in part by IPM under the grant number 83310011.     

On extensions of partial functions

The problem of extending partial functions is considered from the general viewpoint. Some aspects of this problem are illustrated by examples, which are concerned with typical real-valued partial functions (e.g. semicontinuous, monotone, additive, measurable, possessing the Baire property).     

On the constructive characterization of threshold functions

In this paper, the structure of the set of threshold functions and complexity problems are considered. The notion of the signature of a threshold function is defined. It is shown that if a threshold function essentially depends on all of its variables, then the signature of this function is unique. The set of threshold functions is partitioned into classes with equal signatures. A theorem characterizing this partition is proved. The importance of the class of monotone threshold functions is emphasized. The complexity of transferring one threshold function specified by a linear form into another is examined. It is shown that in the worst case this transfer would take exponential time. The structure of the set of linear forms specifying the same threshold function is also examined. It is proved that for any threshold function this set of linear forms has a unique basis in terms of the operation of addition of linear forms. It is also shown that this basis is countable.     

On measurability of fuzzy-number-valued functions

This paper gives a new definition of the measurability of fuzzy-number-valued functions, and shows the relationship among this measurability and those derived from the corresponding set-valued functions.     

On the unique range set of meromorphic functions

This paper studies the unique range set of meromorphic functions and shows that there exists a finite set such that for any two nonconstant meromorphic functions and the condition implies . As a special case this also answers an open question posed by Gross (1977) about entire functions and improves some results obtained recently by Yi.