The extension E of degree n over the Galois field F={\text GF}(q)} is called regular over F, if {\text ord}_r(q) and n have greatest common divisor 1 for all prime divisors r of n which are different from the characteristic p of F (here, \order{r}{q} denotes the multiplicative order of q modulo r). Under the assumption that E is regular over F and that q-1 is divisible by 4 if q is odd and n is even, we prove the existence of a primitive element w \in E which is also completely normal over F (the latter means that w simultaneously generates a normal basis for E over every intermediate field K of E/F). Our result achieves, for the class of extensions under consideration, a common generalization of the theorem of Lenstra and Schoof on the existence of primitive normal bases [12] and the theorem of Blessenohl and Johnsen on the existence of complete normal bases [1].

In 1997, M. Cho, M. Ito and S. Oshiro showed that Weyl's theorem holds for p-hyponormal operators, for any p>0. In this note we give another proof of this result. Also, it is shown that Weyl's theorem holds for M-hyponormal operators. Further, in 1962, Stampfli showed that if T is hyponormal and its Weyl spectrum is {0} then T is compact and normal. We show that this result remains true if the hypothesis of hyponormality is replaced by either (a) p-hyponormality or (b) M-hyponormality.

This article is to discuss the automatic continuity properties and the representation of disjointness preserving linear mappings on certain normal Fréchet algebras of complex-valued functions. This class of operators is defined by the condition that any pair of functions with disjoint cozero sets is mapped to functions with disjoint cozero sets, and subsumes the class of local operators. It turns out that such operators are always continuous outside some finite singularity set of the underlying topological space. Our main emphasis is on disjointness preserving operators from Fréchet algebras of differentiable functions. Such operators are shown to admit a canonical representation that involves weighted composition for the derivatives. This result extends the classical characterization of local operators as linear partial differential operators.

Let X be a proper geodesic metric space which is \delta-hyperbolic in the sense of Gromov. We study a class of functions on X, called horofunctions, which generalize Busemann functions. To each horofunction is associated a point in the boundary at infinity of X. Horofunctions are used to give a description of the boundary. In the case where X is the Cayley graph of a hyperbolic group \Gamma, we show, following ideas of Gromov sketched in his paper Hyperbolic groups, that the space of cocycles associated to horofunctions which take integral values on the vertices is a one-sided subshift of finite type.

In this paper we prove that the ideal property and the projection property do not conincide in general even in the separable case (despite the fact that, as we proved before, they are the same for GAH algebras-and, in particular, for AH algebras-and for separable LB algebras). We also study the behaviour of the projection property with respect to several natural operations.

We compute the precise bounds, for every positive integer n, for the nilpotency class of nilpotent subgroups of {\text GL}(n,{\tf="P7DC6"Q}) and {\text GL}(n,{\tf="P7DC6"Z}).

We prove a natural factorization of supersolvable groups and then we give another characterization of them in connection with the Fitting subgroup. Applying these theorems we describe the structure of some subclasses of supersolvable groups.

Suppose that G is a finite p-solvable group and let \chi \in {\rm Irr}(G) be of p^\prime -degree. In this note, we investigate when \chi remains irreducible when restricted to {\bf {N)}_{G}(P).

The investigation of general F-abundant semigroups is initiated. After obtaining some properties of such semigroups, the structure of a class of F-abundant semigroups is established. In addition, a problem raised in [2] is positively answered.

We introduce and study the type I-, II-, and III-Λ-complete continuity property of Banach spaces, where Λ is a subset of the dual group of a compact metrizable abelian group G.

Let T be a dominant operator that is a quasi-affine transform of an M-hyponormal operator. In this paper we show that if f is a function analytic on a neighborhood of the spectrum of T, then Weyl's theorem holds for f(T{\hskip1}).

The Kadets path distance between Banach spaces X and Y is defined to be the infimum of the lengths with respect to the Kadets distance of all curves joining X and Y. If there is no curve joining X and Y, the Kadets path distance between X and Y is defined to be infty .

Some approaches to estimates of the Kadets path distance from above and from below are developed. In particular, the Kadets path distances between the spaces l_p^n,\ p\in [1,+\rm (inf)ty ], n\in {\b (N)} are estimated.

We prove that the module categories of Noether algebras (i.e., algebras module finite over a noetherian center) and affine noetherian PI algebras over a field enjoy the following product property: whenever a direct product \prod _(n \in ℕ) M_n of finitely generated indecomposable modules M_n is a direct sum of finitely generated objects, there are repeats among the isomorphism types of the M_n. The rings with this property satisfy the pure semisimplicity conjecture which stipulates that vanishing one-sided pure global dimension entails finite representation type.

It is shown that an immersion of n dimensional compact oriented manifold without boundary into the n+1 dimensional Euclidean space, hyperbolic space or open half sphere is a totally umbilic immersion if one of the mean curvature function H_l does not vanish and the ratio H_k/H_l is constant, 1\leq k, l \leq n, k\ne l.

Every compact well-bounded operator has a representation as a linear combination of disjoint projections reminiscent of the representation of compact self-adjoint operators. In this note we show that the converse of this result holds, thus characterizing compact well-bounded operators. We also apply this result to study compact well-bounded operators on some special classes of Banach spaces such as hereditarily indecomposable spaces and certain spaces constructed by G. Pisier.

For a nontrivial additive character \lambda and a multiplicative character \chi of the finite field with q elements (q a power of an odd prime), and for each positive integer r, the exponential sums \sum \lambda ((\tr w)^r) over w\in {SO}(2n+1,q) and \sum \chi (\det w)\lambda ((\tr w)^r) over {O}(2n+1,q) are considered. We show that both of them can be expressed as polynomials in q involving certain exponential sums. Also, from these expressions we derive the formulas for the number of elements w in {SO}(2n+1,q) and {O}(2n+1,q) with (\tr w)^r=\beta , for each \beta in the finite field with q elements.

It is shown that the truncated multidimensional moment problem is more general than the full multidimensional moment problem.

In this paper, we show that if T=S+N, where S is similar to a hyponormal operator, S and N commute and N is a nilpotent operator of order m (i.e., N^m=0), then T is a subscalar operator of order 2m. As a corollary, we get that such a T has a nontrivial invariant subspace if its spectrum \sigma(T\hskip1) has the property that there exists some non-empty open set U such that \sigma(T\hskip1)\capU is dominating for U.

Methods of calculating discriminant and image Milnor numbers in terms of the Milnor numbers of multiple point spaces are described for cases f:{ℂ}^3 \to {ℂ}^4 and F:{ℂ}^3 \to {ℂ}^3.

In this paper we prove that if M_C=\pmatrix {A&#TAB;C\cr0&#TAB;B} is a 2\times 2 upper triangular operator matrix on the Hilbert space H\bigoplus K and if \sigma (A)\cap \sigma (B)=\emptyset , then \sigma is continuous at A and B if and only if \sigma is continuous at M_C, for every C\in B(K,H{\hskip1}).