Stage-structured models of population growth have been considered in the constant delay and state-dependent delay cases, when modeled by retarded functional differential equations. In the first case we settle a conjecture posed by Aiello and Freedman [1] by showing the existence of oscillatory solutions. In the second case, we show that under suitable criteria, all positive solutions tend to a global attractor.

Let p ≡ 1 mod 8 be a rational prime and let h(—p) be the class number of . In [1], Barrucand and Cohn show that h(-p) = 0 mod 8 iff p = x2 + 32y2 for some x,y € Z. In this article, we generalize their result to a family of relative quadratic extensions K/F, where Fk is the maximum totally real subfield of Q(ζ2k+2 ), and a power of a prime of Fk from a family of positive density.

A residue difference set modulo p is a set A = {a1,a2,...,ak} of integers 1 ≤ ai ≤ p — 1 such that for all i and j with i ≠ j, where is the Legendre symbol. We give a lower and an upper bound for mp—the P maximal cardinality of such set A in the case of p ≡ 1 (mod 4).

A ring R is E-associative if φ(xy) = φ(x)y for all endomorphisms φ of the additive group of R, and all x,y ∊ R. Unital E-associative rings are E-rings. The structure of the torsion ideal of an E-associative ring is described completely. The E-associative rings with completely decomposable torsion free additive groups are also classified. Conditions under which E-associative rings are E-rings, and other miscellaneous results are obtained.

Let X be a Hausdorff space which is the continuous image of an ordered continuum. We prove that every irreducible separator of X is metrizable. This is a far reaching extension of the 1967 theorem of S. Mardešić which asserts that X has a basis of open sets with metrizable boundaries. Our first result is then used to show that, in particular, if Y is an hereditarily locally connected continuum, then for subsets of Y quasi-components coincide with components, and that the boundary of each connected open subset of Y is accessible by ordered continua. These results answer open problems in the literature due to the fourth and third authors, respectively.

TWO comparison theorems, one of pointwise type and one of integral type, will be obtained for linear elliptic equations of order 2m on an exterior domain in Rn

W. Kurc [5] has proved that in the unit sphere of Orlicz space LΦ(μ) generated by an Orlicz function Φ satisfying the suitable Δ2-condition and equipped with the Luxemburg norm every extreme point is strongly extreme. In this paper it is proved in the case of a nonatomic measure μ that the unit sphere of the Orlicz space LΦ(μ) generated by an Orlicz function Φ which does not satisfy the suitable Δ2-condition and equipped with the Luxemburg norm has no strongly extreme point and no H-point.

The subgroup generated by the Bass cyclic and bicyclic units is of infinite index in the group of units of the integral group ring ZG when G is either D or

The local Hopf Bifurcation theorem is extended to implicit differential equations in Rn, of the form ẋ = f(x,ẋ, α), which are not solvable for the variable ẋ. The proof uses the S1 -degree of convex-valued mappings. An example of an implicit differential equation in R3 to which the presented theorem applies is provided.

Nous donnons une expression finie pour la valeur L(1, X) dès lors que X est un caractère de Dirichlet modulo f ≥ 2, impair et non principal. Cette expression, valable même lorsque ce caractère n'est pas primitif, nous permet de généraliser au théorème 2 le résultat de H. Walum sur un comportement en moyenne de ces fonctions L (sa démonstration qui fait usage de sommes de Gauss ne semble pas pouvoir être adaptée au cas de caractères non primitifs.) Nous appliquons ces résultats à l'obtention de bornes pour le nombre de classes relatif des corps cyclotomiques: nous retrouvons celles de T. Metsänkylä et de K. Feng par une méthode nous permettant de les ensuite amender.

A new topology for the closed subsets of a topological space X which are the closure of their interiors is defined and investigated. Some applications to convergence of regular measures are also given.

A canary tree is a tree of cardinality the continuum which has no uncountable branch, but gains a branch whenever a stationary set is destroyed (without adding reals). Canary trees are important in infinitary model theory. The existence of a canary tree is independent of ZFC + GCH.

We prove that for a contact process restricted to the cube [1,n]d and initially fully occupied, the time to die out, when it is suitably normalized, converges to an exponential random variable as n tends to infinity.

A characterization is given of when all kernels (respectively images) of endomorphisms of a module are direct summands, a necessary condition being that the endomorphism ring itself is a left (respectively right) PP-ring. This result generalizes theorems of Small, Lenzing and Colby-Rutter and shows that R is left hereditary if and only if the endomorphism ring of every injective left module is a right PP-ring.

We define pairings and copairings in the category of topological spaces over and under B. We prove a relation between pairings of homotopy sets induced by a pairing and a copairing of topological spaces over and under B. We obtain some properties of the axes of pairings and the homotopy set of the axes of pairings in the category of topological spaces over and under B. We also prove the dual results.

We define an arrangement of pseudohyperplanes as an image of a topological sphere arrangement with appropriate intersections, and prove that the complement components are then homologically trivial. We apply this to extend a formula of Winder and Zaslavsky.

Let λ be a certain measure supported on a curve in R3. We prove that if the curve has nonzero curvature and torsion, then λ * L3/2 ⊂ L2.

We consider the Liénard's equation with f(x) polynomial. By using the generalized polar coordinates we establish the maximum possible number of small amplitude limit cycles of such equation in terms of the degree of f(x).