We establish the existence of positive radially symmetric solutions of Δu + f(r, u, u′) = 0 in the domain R1 <r<R0 with a variety of Dirichlet and Neumann boundary conditions. The function f is allowed to be singular when either u = 0 or u′ = 0. Our analysis is based on Leray-Schauder degree theory.

Let R be a ring. An element p of R is a prime element if pR = Rp is a prime ideal of R. A prime ring R is said to be a Unique Factorisation Ring if every non-zero prime ideal contains a prime element. This paper develops the basic theory of U.F.R.s. We show that every polynomial extension in central indeterminates of a U.F.R. is a U.F.R. We consider in more detail the case when a U.F.R. is either Noetherian or satisfies a polynomial identity. In particular we show that such a ring R is a maximal order, that every height-1 prime ideal of R has a classical localisation in which every two-sided ideal is principal, and that R is the intersection of a left and right Noetherian ring and a simple ring.

Let be a set of disjointly supported, positive functions in the Banach envelope of weak L1. We prove that each fi can be written as ei + gi where ei and gi, are disjointly supported and satisfy these additional properties: the ei's are isometrically the basis in the envelope norm; the envelope norm of a linear combination of the gi's is equal to the envelope norm of the corresponding combination of the fi's.

For a valued field (K, v), let Kv denote the residue field of v and Gv its value group. One way of extending a valuation v defined on a field K to a simple transcendental extension K(x) is to choose any α in K and any μ in a totally ordered Abelian group containing Gv, and define a valuation w on K[x] by w(Σici(x – α)i) = mini (v(ci) + iμ). Clearly either Gv is a subgroup of finite index in Gw = Gv + ℤμ or Gw/Gv is not a torsion group. It can be easily shown that K(x)w is a simple transcendental extension of Kv in the former case. Conversely it is well known that for an algebraically closed field K with a valuation v, if w is an extension of v to K(x) such that either K(x)w is not algebraic over Kv or Gw/Gv is not a torsion group, then w is of the type described above. The present paper deals with the converse problem for any field K. It determines explicitly all such valuations w together with their residue fields and value groups.

In this paper, we prove that if is an increasing sequence of strictly positive and continuous functions on a locally compact Hausdorff space X such that then the Fréchet space C(X) is distinguished if and only if it satisfies Heinrich's density condition, or equivalently, if and only if the sequence satisfies condition (H) (cf. e.g.‵[1] for the introduction of (H)). As a consequence, the bidual λ∞(A) of the distinguished Köthe echelon space λ0(A) is distinguished if and only if the space λ1(A) is distinguished. This gives counterexamples to a problem of Grothendieck in the context of Köthe echelon spaces.

Given a poset (X, ≦), the covering poset (C(X), ≦) consists of the set C(X) of covering pairs, that is, pairs (a, b)∈X2 with a<b such that there is no c∈X with a<c<b, partially ordered by (a, b)≦(a′, b′) if and only if (a, b) = (a′, b′) or b≦a′. There is a natural homomorphism v from the automorphism group of (X, ≦) into the automorphism group of (C(X), ≦). It is shown that given groups G, H and a homomorphism α from G into H there exists a poset (X, ≦) and isomorphisms φψ from G onto Aut(X, ≦), respectively from H onto Aut(C(X), ≦) such that φv = αψ. It is also shown that every group is isomorphic to the automorphism group of a poset all of whose maximal chains are isomorphic to the nationals.

In this paper, we show how to give a geometric interpretation of the modular correspondence T3 on the modular curve X(11) of level 11 using projective geometry. We use Klein's theorem that X(11) is isomorphic to the nodal curve of the Hessian of the cubic threefold Λ defined by V2W + W2X + X2Y + Y2Z + Z2V = 0 in P4(C) and geometry which we learned from a paper of W. L. Edge. We show that the correspondence T3 is essentially the correspondence which associates to a point p of the curve X(11) the four points where the singular locus of the polar quadric of p with respect to Λ meets X(11). Our control of the geometry is good enough to enable us to compute the eigenvalues of T3 acting on the cohomology of X(11). This is the first example of an explicit geometric description of a modular correspondence without valence. The results of this article will be used in subsequent articles to associate two new abelian varieties to a cubic threefold, to desingularize the Hessian of a cubic threefold and to study self-conjugate polygons formed by the quadrisecants of the nodal curve of the Hessian.

Let Cr, be the intersection of n — r quadrics with a common self-polar simplex S in projective n-space [n]. Let Γr be a Cr that can be taken in coordinate form as Every C1 is a Γ1, and its points of hyperosculation have special properties: they are the points of intersection of C1 with the faces of S each counting (n—1)(n — 2)/2 times, and the osculating [s], for s≦n–1, has 2s-point contact. Here we show that if r≧2 and n>2r then every point of Γ, has exceptional higher tangent spaces: the s-tangent space at a point P of an r-dimensional variety V is the intersection of all primes that cut V in a variety having an (s + l)-fold point (at least) at P, and normally has dimension if this is less than n. The s-tangent space to Γ, at a point not in a face of S is an [rs] (provided rs <n). Usually it is the existence of lines on V through P that cause a lower than expected s-tangent dimension. Not so on Γr, since its lines form a subvariety. If n≧5 not every C2 is a Γ2. Take n≧5. We show that C2 is Γ2 if and only if C2 contains a line. Also C2 is a Γ2 if and only if at some one point of C2 off the faces of S the second-tangent space is a [4]. Thus, unexpectedly, we have: if one point of C2 off the faces of S has a [4] for second-tangent space, then so do all such points of C2. We obtain results for points of Γ, in the faces of S.

Let G be a locally compact connected topological group. Let Aut0G be the identity component of the group of all bi-continuous automorphisms of G topologized by Birkhoff topology. We give a necessary and sufficient condition for Aut0G to be locally compact.

Consider the nonautonomous delay logistic difference equation

where (pn)n≧0 is a sequence of nonnegative numbers, (ln)n≧0 is a sequence of positive integers with limn→∞(n−ln) = ∞ and K is a positive constant. Only solutions which are positive for n≧0 are considered. We established a sharp condition under which all solutions of (E0) are oscillatory about the equilibrium point K. Also we obtained sufficient conditions for the existence of a solution of (E0) which is nonoscillatory about K.

(i) Let R be a semiprime right Goldie ring with dim R = n. Then a maximal chain of right annihilators in R has exactly n terms, (ii) A semiprime locally right Goldie ring with ACC and DCC on right annihilators is a right Goldie ring.

For a fixed integer q≧2, every positive integer where each ar(q, k) ∈ {0, 1, 2, …, q–1}. The sum of digits function α(q, k) = behaves rather erratically but on averaging has a uniform behaviour. In particular if A(q, n) = , where n > 1, then it is well known that A(q, n)∼½ ((q – 1)/log q) n log n as n→∞. For even values of q, a lower bound is now given for the difference ½S(q, n) = A(q, n)–½(q–1)[logn/logq] n, where [log n/log q] denotes the greatest integer ≦ log n/log q, complementing an earlier result for odd values of q.

It is shown that if G is a group and Aut G is a countable periodic CC-group then Aut G is FC.

We consider interpolation at 2n equidistant nodes in [0,π) from the space ℱN spanned by sines and cosines of odd multiples of x. This interpolation problem is shown to be correct for an arbitrary sequence of derivatives specified at all the nodes. Explicit expressions for the fundamental polynomials are obtained and it is shown that under mild smoothness assumptions on the function f interpolant from ℱN converges uniformly to f as the node spacing goes to zero.

Recent research on aspects of distributive lattices, p-algebras, double p-algebras and de-Morgan algebras (see [2] and the references therein) has led to the consideration of the classes (n≧1) of distributive lattices having no n + 1-element chain in their poset of prime ideals. In [1] we were obliged to characterize the members of by a sentence in the first-order theory of distributive lattices. Subsequently (see [2]), it was realised that coincides with the class of distributive lattices having n+1-permutable congruences. This result is hereby employed to describe those distributive p-algebras and double p-algebras having n-permutable congruences. As an application, new characterizations of those distributive p-algebras and double p-algebras having the property that their compact congruences are principal are obtained. In addition, those varieties of distributive p-algebras and double p-algebras having n-permutable congruences are announced.

In this paper we examine the dependence of the solutions of an evolution inclusion on a parameter λ We prove two dependence theorems. In the first the parameter appears only in the orientor field and we show that the solution set depends continuously on it for both the Vietoris and Hausdorff topologies. In the second the parameter appears also in the monotone operator. Using the notion of G-convergence of operators we prove that the solution set is upper semicontinuous with respect to the parameter. Both results make use of a general existence theorem which we also prove in this paper. Finally, we present two examples. One from control theory and the other from partial differential inclusions.

A well known lemma of Burnside is generalised, to give necessary and sufficient conditions for a finite p-group K to be normally embedded in a nilpotent group V, with K⊆ω(V). (Here, ω denotes a single word and ω(V) is the corresponding verbal subgroup.) Our main result is related to earlier work of Blackburn, Gaschütz and Hobby.

The class of non-associative algebras over a field of characteristic zero named in the title is studied using a result of Ouattara [9]. As an application, the differential equation for overlapping generations in the time-continuous model is solved.

An obvious question occurs at the very start of equivariant homotopy theory. What is the relationship between maps equivariant up to homotopy and strictly equivariant maps? This question has been studied by various people, usually away from the group order ([8, 11, 22, 25, 26]). We consider the problem stably and answer it by giving a spectral sequence proceeding from homotopy equivariant to strictly equivariant information. The form of the spectral sequence is not surprising, but there are three distinctive features of our approach: (1) we show that the spectral sequence may be viewed as an Adams spectral sequence based on nonequivariant homotopy, (2) we show how to exploit the product structure, and (3) we give a treatment showing how Dress's algebra of induction theory [13] applies to give non-normal subgroups equal status. As a spinoff from (3) we also obtain spectral sequences for calculating homology and cohomology of universal spaces (3.5).