Revision by History

This article proposes a solution to the problem of obtaining plausibility information, which is necessary to perform belief revision: given a sequence of revisions, together with their results, derive a possible initial order that has generated them; this is different from the usual assumption of starting from an all-equal initial order and modifying it by a sequence of revisions. Four semantics for iterated revision are considered: natural, restrained, lexicographic and reinforcement. For each, a necessary and sufficient condition to the existence of an order generating a given history of revisions and results is proved. Complexity is proved coNP complete in all cases but one (reinforcement revision with unbounded sequence length).

SUMSETS AND DIFFERENCE SETS CONTAINING A COMMON TERM OF A SEQUENCE

Let β>1 be a real number, and let {ak} be an unbounded sequence of positive integers such that ak+1/ak≤β for all k≥1. The following result is proved: if n is an integer with n>(1+1/(2β))a1 and A is a subset of {0,1,…,n} with $\vert A\vert \ge (1- 1/({2\beta +1})) n +\frac 12$, then (A+A)∩(A−A) contains a term of {ak }. The lower bound for |A| is optimal. Beyond these, we also prove that if n≥3 is an integer and A is a subset of {0,1,…,n} with $\vert A\vert \gt \frac 45 n$, then (A+A)∩(A−A) contains a power of 2. Furthermore, $\frac 45$ cannot be improved.

2-SYMMETRIC CRITICAL POINT THEOREMS FOR NON-DIFFERENTIABLE FUNCTIONS

In this paper, some min–max theorems for even andC1functionals established by Ghoussoub are extended to the case of functionals that are the sum of a locally Lipschitz continuous, even term and a convex, proper, lower semi-continuous, even function. A class of non-smooth functionals admitting an unbounded sequence of critical values is also pointed out.

On the discreteness of the spectra of the Dirichlet and Neumannp-biharmonic problems

We are interested in a nonlinear boundary value problem for(|u″|p−2u″)′​′=λ|u|p−2uin[0,1],p>1, with Dirichlet and Neumann boundary conditions. We prove that eigenvalues of the Dirichlet problem are positive, simple, and isolated, and form an increasing unbounded sequence. An eigenfunction, corresponding to thenth eigenvalue, has preciselyn−1zero points in(0,1). Eigenvalues of the Neumann problem are nonnegative and isolated,0is an eigenvalue which is not simple, and the positive eigenvalues are simple and they form an increasing unbounded sequence. An eigenfunction, corresponding to thenth positive eigenvalue, has preciselyn+1zero points in(0,1).

INFINITELY MANY SOLUTIONS OF THE NEUMANN PROBLEM FOR ELLIPTIC EQUATIONS INVOLVING THE p-LAPLACIAN

In this paper, we prove the existence of an unbounded sequence of weak solutions under an appropriate oscillating behaviour of φ for a Neumann problem of the type

Tauberian and other theorems concerning Dirichlet's series with non-negative coefficients

The paper is concerned with properties of the Dirichlet series where {λn} is a strictly increasing unbounded sequence of real numbers with λ1 > 0. One of the main Tauberian results proved is that if a1 ≥ 0, an > 0 for n = 2, 3, …, a(x) ≤ ∞ for all x ≥ 0, An:= a1 + a2 + … + an → ∞, an λn = o((λn+1-λn)An), an λn sn > − H(λn+1 − λn) An and then A new summability method Dλ, a based on the Dirichlet series a(x) is defined and its relationship to the weighted mean method Ma investigated.

A simple proof of the Mazur-Orlicz summability theorem

In 1933(1), Mazur and Orlicz stated that, if a conservative (i.e. convergence preserving) matrix sums a bounded nonconvergent sequence, then it must sum an unbounded sequence. Their proof of this result (2) was one of the early applications of functional analysis to summability theory, and it is based on rather deep topological properties of F K-spaces. In (3) Zeller obtained a proof of this important theorem as a consequence of his study of the summability of slowly oscillating sequences. The purpose of this note is to give a simple direct proof of this theorem using only the well-known Silverman-Töplitz conditions for regularity. In order to reduce the details of the argument, we state and prove the result for regular matrices rather than conservative matrices.