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 collection of examples of first order linear neutral differential delay equations having a nonoscillatory solution with lim sup = ∞ and lim inf = 0 at ∞ is given. This disproves a recent conjecture about the asymptotic behavior of solutions to such equations.

A commutative ring R is said to have the n-generator property if each ideal of R can be generated by n elements. Rings with the n-generator property have Krull dimension at most one. In this paper we consider the problem of determining when a one-dimensional monoid ring R[S] has the n-generator property where R is an artinian ring and S is a commutative cancellative monoid. As an application, we explicitly determine when such monoid rings have the three-generator property.

For any convex body C in ℝd we introduce the notion of the convex shell and we prove that there exists a unique "minimal" convex shell, extending the notion of the minimal spherical shell of C. Then we prove that a "typical" convex body touches the boundary of its minimal convex shell in precisely d + 2 points.

In this paper we study the local bifurcation of critical periods of periodic orbits in the neighborhood of a nondegenerate centre of a vector field with a homogeneous nonlinearity of the third degree. We show that at most three local critical periods bifurcate from a weak linear centre of finite order or from the linear isochrone and at most two local critical periods from the nonlinear isochrone. Moreover, in both cases, there are perturbations with the maximum number of critical periods.

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.

The non-separable Banach space of right continuous functions with left hand limits and the supremum norm is investigated to find the isomorphic types of complemented subspaces. It is shown that every isometric isomorph of c0 is complemented in this space which may be identified as a non-separable C(K) space. Sufficient conditions are given for other isomorphs of C0 to be complemented in the space and the complement of a C0 subspace is characterized isomorphically.

For f analytic in the complex ball having the homogeneous expansion conditions for f to be of Hardy space Hp or of weighted Bergman spaces are expressed in terms of lp properties of the sequence {∥Fk∥p}.

We prove weighted norm inequalities for the Fourier transform of the form

where v is a nonnegative weight function on ℝd and ψ: [— 1,1 ] —> ℝd is a nondegenerate curve. Our results generalize unweighted (i.e. v = 1) restriction theorems of M. Christ, and two-dimensional weighted restriction theorems of C. Carton-Lebrun and H. Heinig.

A known consequence of the theorem of the algebraic subgroup is a lower bound for the rank of matrices whose entries are linear combinations, with algebraic coefficients, of logarithms of algebraic numbers. We extend this kind of result to commutative algebraic groups.

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.

In this paper, we consider the oscillatory behavior of the second order neutral delay differential equation

where t ≥ t0,T and σ are positive constants, a,p, q € C(t0, ∞), R),f ∊ C[R, R]. Some sufficient conditions are established such that the above equation is oscillatory. The obtained oscillation criteria generalize and improve a number of known results about both neutral and delay differential equations.

Unlike for a smooth projective hypersurface, for an isolated hypersurface singularity, the pole order and Hodge filtrations do not in general coincide. This note studies the difference between the two.

Let π: M —> Rn be the blowing-up of Rn at the origin. Then a continuous map-germ f: (Rn — 0,0) —> Rm is called blow analytic if there exists an analytic map-germ such that Then an inverse mapping theorem for blow analytic mappings as a generalization of classical theorem is shown. And the following is shown. Theorem: The analytic family of blow analytic functions with isolated singularities admits an analytic trivialization after blowing-up.

In this note we will study the following problem. For a given partition regular system of equations, which equations can be added to this system without introducing new variables, such that the new augmented system is again partition regular. It turns that the Hindman system on finite sums as well as the Deuber-Hindman system on finite sums of (m, p, c)-sets are maximal in this sense.

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.

Some formulas for multiplicities of pure cubic discriminants are generalized to the case of a pure field of arbitrary odd prime degree.

The Hurwitz zeta function ζ(s, a) is defined by the series

for 0 < a ≤ 1 and σ = Re(s) > 1, and can be continued analytically to the whole complex plane except for a simple pole at s = 1 with residue 1. The integral functions C(s, a) and S(s, a) are defined in terms of the Hurwitz zeta function as follows:

Using integral representations of C(s, a) and S(s, a), we evaluate explicitly a class of improper integrals. For example if 0 < a < 1 we show that

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 will classify those real quadratic fields K having exactly one noninert prime less than where Δ is the discriminant of K. Moreover, we will list all such K and prove that the list is complete with one possible exceptional value remaining (whose existence would be a counterexample to the Riemann hypothesis).