For a completely regular Hausdorff topological space X, let Z(X) denote the lattice of zero-sets of X. If T is a continuous map from X to Y, then there is a lattice homomorphism T” from Z(Y) to Z(X) induced by T which is defined by τ‘(A) = τ←(A). A characterization is given of those lattice homomorphisms from Z(Y) to Z(X) which are induced in the above way by a continuous function from X to Y.

Suppose throughout that

and that {μn}(n≥ 0) is a sequence of real numbers. The (generalized) Hausdorff moment problem is to determine necessary and sufficient conditions for there to be a function x in some specified class satisfying

.

In 1857, Kronecker [10] showed that if θ1,…, θn are the roots of the polynomial P(z)= zn|cn-1+ … + cn, where c1, …, cn are integers with cn≠0, and if |θ1| ≤ 1, …, |θ1| ≤1, then θ1, …, θn are roots of unity. The proof is short and ingenious: Consider the polynomials Pm(z) whose roots are The condition on the size of the roots and the fact that the ci are integers implies that there can only be a finite number of different Pm. Thus two distinct powers of each root must coincide and this means that each root is a root of unity.

About a hundred years ago, the author of Through the Looking-Glass wrote another book called Euclid and his Modern Rivals. Were these rivals Lobachevsky and Bolyai, Riemann and Schlafli? No, they were merely the authors of dull school textbooks that would soon be forgotten. The sad truth is that, in 1873, hardly anyone in England knew of the breakthrough that had occurred on the Continent some fifty years before: only Cayley in Cambridge, Clifford in London, and a few students. Even if Cayley or Clifford had visited Oxford and given a lecture there, it is doubtful that he would have succeeded in convincing the conservative Dodgson that Euclid's postulates could be modified to yield two new worlds, surpassing in strangeness the worlds of the two Alice books and yet just as logically consistent as Euclid.

Desarguesian affine Hjelmslev planes (D.A.H. planes) were introduced by Klingenberg in [1] and generalized by Lane and Lorimer in [2]. D.A.H. planes are coordinatized by affine Hjelmslev rings (A.H. rings) which are local rings whose radicals are equal to their sets of two-sided zero divisors and whose principal right ideals are totally ordered. In [5], ordered D.A.H. planes were defined and the induced orderings of their A.H. rings were discussed. In this note an ordered non-Desarguesian A.H. plane is constructed from an arbitrary ordered D.A.H. plane. The existence of such planes ensures that the discussion of ordered non-Desarguesian A.H. planes by J. Laxton in [3] is meaningful. The basic idea employed is essentially the same as the one used in the construction of the classical Moulton plane from the real affine plane (cf. [4])

We extend the Picard's theorem to ordinary differential equation of generalized order α, 0 ≤ α ≤ l, and prove a global existence and uniqueness theorem by using the Banach contraction principle.

Let R denote an associative ring with 1, let n be a positive integer, and let k = 1, 2, or 3. The ring R will be called an (n, k)-ring if it satisfies the identities

for all integers m with n ≤ m ≤ n + k - 1. It was shown years ago by Herstein (See [2], [9], and [10]) that for n >1, any (n, l)-ring must have nil commutator ideal C(R). Later Luh [12] proved that primary (rc, 3)-rings must in fact be commutative, and Ligh and Richoux [11] recently showed that all (n, 3)-rings are commutative.

We show that, in a complete metric space, every selfmap that is a “weak directional contraction” admits a fixed point.

We give a necessary and sufficient condition on weight functions u and v such that for l≤p≤q≤∞ there exists a constant C for which

A corresponding dual result is also given. This extends a result of B. Muckenhoupt which appeared in Studia Math., 34 (1972).

Let be a separable complex infinite dimensional Hilbert space, the algebra of bounded operators in the ideal of compact operators, and the quotient map. Throughout this paper A denotes a separable nuclear C*-algebra with unit. An extension of A is a unital *-monomorphism of A into . Two extensions τ1 and τ2 are strongly (weakly) equivalent if there exists a unitary (Fredholm partial isometry) U in such that

for all a in A.

Let w(z) be regular in the unit disc U:|z|<l, with w(0) = 0 and let h(r, s, t) be a complex function defined in a domain D of C3. The author determines conditions on h such that if

z∈U, then |w(z)|< 1 for z ∈ U and n= 0, 1, 2, …. Here Dnw(z) = (z/(l-z)n+1*w(z), where * stands for the Hadamard product (convolution). Some applications of the results to certain differential equations are given.

Let R be a prime ring possessing a nontrivial automorphism T such that xTx = ±xxT. If R is not of characteristic 3 or R has nonzero center, then R is a commutative integral domain.

Sums of the form where f(n) is a multiplicative arithmetical function and denotes summation over those values of n for which f(n)>0 and f(n) ≠1, were studied by De Koninck [2], De Koninck and Galambos [3], Brinitzer [1] and Ivič [5]. The aim of this note is to give an asymptotic formula for a certain class of multiplicative, positive, primeindependent functions (an arithmetical function is prime-independent if f(pv) = g(v) for all primes p and v = 1, 2, …). This class of functions includes, among others, the functions a(n) and τ(e)(n), which represent the number of nonisomorphic abelian groups of order n and the number of exponential divisors of n respectively, and none of the estimates of the above-mentioned papers may be applied to this class of functions. We prove the following.

Let G be a group and ρ and σ two irreducible unitary representations of G in complex Hilbert spaces and assume that dimp ρ= n < ∞. D. Poguntke [2] proved that is a sum of at most n2 irreducible subrepresentations. The case when dim a is also finite he attributed to R. Howe.

We shall prove analogous results for arbitrary finite-dimensional representations, not necessarily unitary. Thus let F be an algebraically closed field of characteristic 0. We shall use the language of modules and we postulate that allour modules are finite-dimensional as F-vector spaces. The field F itself will be considered as a trivial G-module.

Let E be a locally convex tvs, F a normed space and the space of continuous linear mappings from E into F In this paper, we investigate the continuous convergence structure (c-structure) on. denotes the resulting convergence vector space (cvs).

The c-structure is by definition the coarsest cvs structure on making evaluation a continuous mapping.

Several non-absolutely convergent integrals have been defined which generalize the Perron integral. The most significant of these integrals from the point of view of application to trigonometric series are the Pn- and pn-integrals of R. D. James [10] and [11]. The theorems relating the Pn -integral to trigonometric series state essentially that if the series

1.1

Let 0<a1<…<an be integers and (a, b) denotes the greatest common divisor of a, b. R. L. Graham [1] has conjectured that

for some i and j. In a recent paper Weinstein [2] has improved Winterle's result [3] and has proven the following interesting theorem:

If A is the sequence a1< … <an where ak = P, a prime for some k and , then

.

In [7], Hedstrom and Houston introduce a type of quasilocal integral domain, therein dubbed a pseudo-valuation domain (for short, a PVD), which possesses many of the ideal-theoretic properties of valuation domains. For the reader′s convenience and reference purposes, Proposition 2.1 lists some of the ideal-theoretic characterizations of PVD′s given in [7]. As the terminology suggests, any valuation domain is a PVD. Since valuation domains may be characterized as the quasilocal domains of weak global dimension at most 1, a homological study of PVD's seems appropriate. This note initiates such a study by establishing (see Theorem 2.3) that the only possible weak global dimensions of a PVD are 0, 1, 2 and ∞. One upshot (Corollary 3.4) is that a coherent PVD cannot have weak global dimension 2: hence, none of the domains of weak global dimension 2 which appear in [10, Section 5.5] can be a PVD.

The following problem arose in connection with the study of Poincaré polynomials for homogeneous spaces. Let

gi hi positive integers and set d = (h1, h1,…, hk), the greatest common divisors of the exponents in the denominator. Let hi = dri and assume that the ri′s are comprime in pairs, i.e.

An intrinsic characterization is given of those von Neumann algebras which are injective objects in the category of C*-algebras with completely positive maps. For countably generated von Neumann algebras several such characterizations have been given, so it is in fact enough to observe that an injective von Neumann algebra is generated by an upward directed collection of injective countably generated sub von Neumann algebras. The present work also shows that three of the intrinsic characterizations known in the countably generated case hold in general.