A well known result due to Dixmier and Grothendieck for spaces of continuous scalar-valued functions C(X), X compact Hausdorff, is that C(X) is a Banach dual if, and only if, Xis hyperstonean. Moreover, for hyperstonean X, the predual of C(X) is strongly unique. Here we obtain a formulation of this result for spaces of continuous vector-valued functions. It is shown that if E is a Hilbert space and C(X, (E, σ *) ) denotes the space of continuous functions on X to E when E is provided with its weak * ( = weak) topology, then C(X, (E, σ *) ) is a Banach dual if, and only if, X is hyperstonean. Moreover, for hyperstonean X, the predual of C(X, (E, σ *) ) is strongly unique.

We prove that the sum of two idempotents is in a Banach algebra is itself an idempotent if and only if it is power-bounded.

In this paper we give a description of:

(1) the Hopf algebra structure of k*(G; L) when G is a compact, connected Lie group and L is a ring of type Q(P) so that H*(G; L) is torsion free;

(2) the algebra structure of k*(G2; L) for L = Z2 or Z.

We present a refinement of Iwasawa's approach to Leopoldt's conjecture on the non-vanishing of the p-adic regulator of an algebraic number field K. As an application, the conjecture for K implies the conjecture for a solvable extension L of degree g over K if g is relatively prime to p — 1 and p does not divide g, the discriminant of K, and the quotient of class numbers where is a primitive pth root of unity. This can be viewed as generalizing a theorem of Kummer on cyclotomic units.

Let I = [a, b] ⊆ R and let L be an nth order linear differential operator defined on Cn(I). Let 2 ≦ k ≦ n and let a ≦ x1 < x2 < … < xn = b. A method of forced mono tonicity is used to construct monotone sequences that converge to solutions of the conjugate type boundary value problem (BVP) Ly = f(x, y),y(i-1) = rij where 1 ≦i ≦ mj, 1 ≦ j ≦ k, mj = n, and f : I X R → R is continuous. A comparison theorem is employed and the method requires that the Green's function of an associated BVP satisfies certain sign conditions.

We show that two maps between classifying spaces of compact, connected Lie groups are homotopic after inverting the order of the Weyl group of the source if and only if they induce the same maps on rational cohomology. We shall also give some results on maps from classifying spaces of finite groups to classifying spaces of compact Lie groups. Among other things we construct a map from B(Z/2 + Z/2 4- Z/3) into BSO(3) which is not induced by a homomorphism.

The Pólya-Vinogradov inequality states that for any non-principal character x modulo q and any N ≧ 1,

where c is an absolute constant. We show that (*) holds with c = 2/(3π2) + o(1) in the case x is primitive and x (— 1) =1 with c = 1/(3π) + o(l) in the case x is primitive and x(— 1) = — 1- This improves by a factor 2/3 the previously best-known values for these constants.

We established sufficient conditions for the oscillation of all solutions of the delay differential equation

and of the neutral delay differential equation

where p, q, r and a are nonnegative constants and n is an odd natural number.

A definition of an isometric shift operator on a Banach space is given which extends the usual definition of a shift operator on a separable Hilbert space. It is shown that there is no such shift on many of the common Banach spaces of continuous functions. The associated ideas of a semi-shift and a backward shift are also introduced and studied in the case of continuous function spaces.

The proximal subgradient formula is a refinement due to Rockafellar of Clarke's fundamental proximal normal formula. It expresses Clarke's generalized gradient of a lower semicontinuous function in terms of analytically simpler proximal subgradients. We use the infinite-dimensional proximal normal formula recently given by Borwein and Strojwas to derive a new version of the proximal subgradient formula in a reflexive Banach space X with Frechet differentiable and locally uniformly convex norm. Our result improves on the one given by Borwein and Strojwas by referring only to the given norm on X.

We give three new partition theorems of the classical Rogers-Ramanujan type which are very much in the style of MacMahon. These are a continuation of four theorems of the same kind given recently by the second author. One of these new theorems, very similar to one of the original Rogers-Ramanuj an - MacMahon type theorems is as follows: Let C(n) denote the number of partitions of n into parts congruent to ±2, ± 3, ±4, ± 5, ±6, ±7 (mod 20). Let D(n) denote the number of partitions of n of the form n = b1 + b2 + … + bt, where bt ≧ 2, bt ≧ bi + 1 and if 1 ≦ i ≦ [(t - 2)/2], bi - bi + 1 ≧ 2. Then C(n) = D(n).

A characterization of torsionless rings is given which shows that the integral closure of a torsionless ring need not be Prüfer.

A well-known result of E. Lucas enables one to obtain the residue modulo p of in terms of the base-p digits of M and N. Using a recent result of P. W. Haggard and J. O. Kiltenen, a proof of N. J. Fine has been adapted to yield the corresponding residue modulo pr.

A seven-parameter extension of Askey and Wilson's four parameter q-beta integral is written in a symmetric form as the sum of multiples of two very-well-poised balanced basic hypergeometric 10Φ9 series. Two special cases are considered in which the evaluation of the integral gives single terms by the q-Dixon formula in one case and by a special case of the Verma-Jain formula in the other. An orthogonal polynomial system is obtained in the first case and a system of biorthogonal rational function is obtained in the second. It is also shown that the biorthogonal system represents a generalization of Rogers’ q-ultraspherical polynomials.

Let a ≧ 0, Ia = ﹛a, a + 1, …﹜ and consider the nth order linear difference m ∈ Ia, α0(m) = 1 on Ia. Summability conditions are placed on the coefficients αj(m), 1 ≦ j ≦ n, such that the equation Pu(m) = 0 is eventually disconjugate. Conditions for eventual right disfocality are also given.

Let pij be the number of rooted convex polyhedra with i + 1 vertices and j + 1 faces. We express pij as a singly indexed summation whose terms decrease geometrically. From this we deduce that

uniformly as max(i, j) → ∞.

The Rank Theorem is proved for locally Lipschitz continuous functions f:Rn → Rp with generalized derivative of constant rank.

Let (L, λ) be a locally solid Riesz space. (L, λ) is said to have the Levi property if for every increasing λ-bounded net (xα) ⊂ L+, sup xα exists. The Levi property, appearing in literature also as weak Fatou property (Luxemburg and Zaanen), condition (B) or monotone completeness (Russian terminology), is a classical object of investigation. In this paper we are interested in some variations of the property, their mutual relationships and applications in the theory of topological Riesz spaces. In the first part of the paper we clarify the status of two problems of Aliprantis and Burkinshaw. In the second part we study ideal-injective Riesz spaces.

A prime Goldie ring K, in which each finitely generated left ideal is principal is the endomorphism ring E(F, A) of a free module A, of finite rank, over an Ore domain F. We determine necessary and sufficient conditions to insure that whenever K ≅ E(F, A) ≅ E(G, B) (with A and B free and finitely generated over domains F and G) then (F, A) is semi-linearly isomorphic to (G, B). We also show, by example, that it is possible for K ≅ E(F, A ) ≅ E(G, B), with F and G, not isomorphic.