We denote by F the field R of real numbers, the field C of complex numbers, or the skew field H of real quaternions, and by Fn an n dimensional left vector space over F. If A is a matrix with elements in F, we denote by A* its conjugate transpose. In all three cases of F, an n × n matrix A is said to be hermitian if A = A*, and we say that two n × n hermitian matrices A and B with elements in F can be diagonalized simultaneously if there exists a non singular matrix U with elements in F such that UAU* and UBU* are diagonal matrices. We shall regard a vector u ∈ Fn as a l × n matrix and identify a 1 × 1 matrix with its single element, and we shall denote by diag {A1, …, Am} a diagonal block matrix with the square matrices A1, …, Am lying on its diagonal.

We use a simple approximation method to prove the Holder continuity of the generalized de Rham functions.

1. Consider the following dilatation equation

where |α|<l/2. Suppose that f is an integrable solution of (1); then f must satisfy

where is the Fourier transform of f, and

which immediately leads to

Over the last few years, various extensions of the topological degree of a mapping have been made so as to include non-compact perturbations of the identity. One such extension, which employs compactness conditions, has been to the class of limit compact maps which were extensively studied by Sadovsky [7]. The class is a large one as it contains all compact mappings, contraction mappings and, more generally, condensing mappings. Sadovsky [7] gives a theory of degree for maps of the form I-f, where f is limit compact, and this was extended independently and with different methods by Petryshyn and Fitzpatrick [4] and the author [9] to allow f to be a multi-valued mapping. A refinement of the methods of [9] was given by Vanderbauwhede [8].

A group G is said to be an FC-group if each element of G has only a finite number of conjugates in G. We are concerned with the class of periodic locally soluble. FC-groups. Clearly subgroups and factor groups of -groups are also -groups.

Every finite soluble group is a -group, and we consider here the generalization of a concept from the theory of finite soluble groups.

Relationships between injectivity or generalized injectivity and chain conditions on a module category have been studied by several authors. A well-known theorem of Osofsky [14, 15] asserts that a ring all of whose cyclic right modules are injective is semisimple Artinian. Osofsky's proofs in [14, 15] essentially used homological properties of injective modules, and, later, her arguments were applied by other authors in their studies of rings for which cyclic right modules are quasi-injective, continuous or quasi-continuous (see e. g. [1, 10, 12]). Following [5] (cf. [4]), a module M is called a CS-module if every submodule of M is essential in a direct summand of M. In the recent paper [17], B. L. Osofsky and P. F. Smith have proved a very general theorem on cyclic completely CS-modules from which many known results in this area follow rather easily. In another direction, it was proved in [8] that a finitely generated quasi-injective module with ACC (respectively DCC) on essential submodules is Noetherian (respectively Artinian). This result was also extended to CS-modules in [3, 16], and weak CS-modules in [19].

In spectral theory on Banach spaces, certain more incisive results hold when the underlying space is weakly complete (that is, weakly sequentially complete). The standard proofs rely on the following deep theorem: any bounded linear map from the algebra of all complex continuous functions on a compact Hausdorff space to a weakly complete Banach space is weakly compact. The proof of this result depends in turn on a considerable amount of measure-theoretic machinery (see [4, Section VI.7]). We present here some alternative methods which avoid these technicalities. The results are then used to give an example of a set of projections, each having unit norm, which generate an unbounded Boolean algebra.

Let f(x) be an irreducible polynomial of degree n with coefficients in a field L and r be an integer prime to the characteristic of L. The object of this paper is to describe the galois group g of f(xr) over L when the galois group G of f(x) itself over L is either the full symmetric group Snor the alternating group An. We shall call f standard if G = Sn or An with |G|>2.

Let k be an algebraic number field and Ck its ideal class group (in the wider sense). Suppose K is a finite extension of k. Then we say that an ideal class of k capitulates in K if this class is in the kernel of the homomorphism

induced by extension of ideals from k to K (See Section 2 below). In [4], Iwasawa gives examples of real quadratic number fields, with distinct primes Pi ≡ 1 (mod 4), for which all the ideal classes of the 2-class group, Ck,2 (the 2-Sylow subgroup of Ck), capitulate in an unramified quadratic extension of k. In these examples, Ck,2 is abelian of type (2,2), i.e. isomorphic to ℤ/2ℤ×ℤ/2ℤ and so all four ideal classes capitulate.

To find a “ description of the structure of bands which is complete modulo semilattices ” (from page 25 of [1]) seems to be a very difficult problem. As far as the author is aware, the only class of bands (except for rectangular bands) for which this problem has been solved (see [4] and [3]) is the class of all bands satisfying a generalization of commutativity, namely the condition that efgh = egfh for all elements e, f, g and h.

Lowengrub [l] has considered simultaneous dual integral equations of the form

where i = 1,2 …n, I1= {x:0 ≦ x x <1}, I2 {= x:0 ≦ x >1}, the cIJ are constants, the f1(x) are known functions and the functions φ(x) are to be determined.

denotes the modified operator of the Hankel transform with the inversion formula

Let be the complex linear space of all infinitely differentiable functions φ on the interval J = (a,b)(− ∞ ≦ a < 0 < b ≦ + ∞) such that φ(k)(0) = 0 for all non-negative integers k. Krabbe ([2], [3]) has defined a class of generalized functions on Jas an algebra ℳ of linear operators on and has developed an operational calculus for these operators. Shultz ([6], Theorem 2.18) has recently shown that ℳ is isomorphic to , where (resp. ) is the set of all distributions on J whose supports are contained in J+ = [0, b) (J- = (a,0]). In this paper we combine some of the ideas developed in [4] with results established in an earlier paper by Shultz to give an easier proof of the above result. Our methods also give a more direct proof of the main result (Theorem 1.22) of [2].

The Lucas numbers υn and the Fibonacci numbers υn are defined by υ1 = 1, υ2= 3, υn+2 = υn+1 + υn and u1 = u2 = 1, un+2 = un+1 + un for all integers n. The elementary properties of these numbers are easily established; see for example [2].However, despite the ease with which many such properties are proved, there are a number of more difficult questions connected with these numbers, of which some are as yet unanswered. Among these there is the well-known conjecture that un is a perfect square only if n = 0, ± 1, 2 or 12. This conjecture was proved correct in [1]. The object of this paper is to prove similar results for υn, ½un and ½υn, and incidentally to simplify considerably the proof for un. Secondly, we shall use these results to solve certain Diophantine equations.

A recent paper of Rhemtulla and Wilson [4] is concerned with elliptically embedded subgroups of groups. A subgroup H of a group G is elliptically embedded in G if, for each subgroup K of G, there is some integer m such that (H, K) = (HK)m. Some sufficient conditions for elliptic embedding are given in Section 2 of [4], and some consequences of the presence of this property are to be found in Theorems 1 and 2 of the same paper and in the main theorem of [5]. It is evident from all of these results that the property of being elliptically embedded is closely related to the nilpotency and subnormality of certain subgroups. One of the questions considered here i s the following.

Let Fg be a closed orientable surface of genus g > 1 and let be the Teichmuller space of Fg, i.e., the space of marked hyperbolic structures on Fg We shall also denote by the space of marked hyperbolic structures on Fgwith one distinguished point; by this, we mean a distinguished point on the universal cover gof Fg. This space is isomorphic to the space of marked complete hyperbolic structures on a genus g surface with 1 cusp which is the usual interpretation of . Choose a decomposition of Fginto pairs of pants by a collection of non–intersecting, totally geodesic simple closed curves. The Fenchel–Nielsen coordinates for relative to this decomposition are given by the lengths of the curves as well as twist parameters defined on each curve. Varying the length and twist parameters gives deformations of the marked hyperbolic structures.

The following result is found quite widely. Suppose f(z) is a non-constant entire function such that |f(z)| = 1 along |z| = 1. Then, f (z) has form czm, |c| = 1, m ≧ 1. See Ahlfors [1, p. 172, exercise 3], Dienes [4, p. 172, exercise 23], Hille [6, p. 317, exercise 2]. It is natural to inquire about a generalization of this result.

The purpose of this short note is to make an observation about Dunford–Pettis operators from L1[0, 1] to C0. Recall that an operator T:E→F (where E and F are Banach spaces) is called Dunford–Pettis if T takes weakly convergent sequences of E into norm convergent sequences of F. A Banach space F has the Compact Range Property (CRP) if every operator T:L1]0, 1]→F is Dunford–Pettis. Talagrand shows in his book [2] that C0 does not have the CRP. It is of interest (especially in mathematical economics [3]) to note that every positive operator from L1[0, 1] to C0 is Dunford–Pettis.

Let R be a commutative ring, with an identity element. It is the purpose of this note to establish conditions for an arbitrary but fixed ideal a of R to satisfy the distributive law

for all ideals b and c of R. In particular, in the Noetherian case, this will be related to the decomposition of a into prime ideals. We start with

Proposition 1. For a fixed ideal a in a commutative ring R with an identity element, the following conditions are equivalent.

An ideal extension of one semigroup by another is determined by a partial homomorphism into the translational hull of the first semigroup [3, §2, Theorem 5]. In most ins tances, the development of the theory of ideal extensions has been hindered by inadequate knowledge of the translational hull; it is our purpose here to characterize certain basic structures in the translational hull of an arbitrary inverse semigroup.