We characterize some isometric immersions of a compact Riemannian manifold into a tube of Sn(λ) or CPn(λ) (in fact, in some more general spaces in the real case) around a totally geodesic Sq(λ) or CPq(λ) respectively, with the norm of the mean curvature of the immersion bounded from above. This bound depends on the radius of the tube, and is related with the mean curvature of its boundary.

Let R be a ring with unity and let denote the category of unital right R-modules. A preradical γ of is a functor γ: → such that

(i) γ(M) ⊆ M for each R-module M,

(ii) for f:M → N, γ(f) is the restriction of f to γ(M).

The concept of cleft extensions, or equivalently of crossed products, for a Hopf algebra is a generalization of Galois extensions with normal basis and of crossed products for a group. The study of these subjects was founded independently by Blattner-Cohen-Montgomery [1] and by Doi-Takeuchi [4]. In this paper, we determine the isomorphic classes of cleft extensions for a infinite dimensional non-commutative, non-cocommutative Hopf algebra kq[X, X–l, Y], which is generated by a group-like element X and a (1,X)-primitive element Y. We also consider the quotient algebras of the cleft extensions.

By the term “locally convex space”, we mean a locally convex Hausdorff topological vector space (see [17]). We shall denote the algebraic dual of a locally convex space E by E*, and its topological dual by E′. It is convenient to think of the elements of E as being linear functionals on E′, so that E can be identified with a subspace of E′*. The adjoint of a continuous linear map T:E→F will be denoted by T′:F′→E′. If 〈E, F〈 is a dual pair of vector spaces, then we shall denote the corresponding weak, strong and Mackey topologies on E by α(E, F), β(E, F) and μ(E, F) respectively.

It is well known that the lattice Λ(S) of congruences on a regular semigroup S contains certain fundamental congruences. For example there is always a minimum band congruence β, which Spitznagel has used in his study of the lattice of congruences on a band of groups [16]. Of key importance to his investigation is the fact that β separates congruences on a band of groups in the sense that two congruences are the same if they have the same meet and join with β. This result enabled him to characterize θ-modular bands of groups as precisely those bands of groups for which ρ⃗(ρ∨β, ρ∧β)is an embedding of Λ(S) into a product of sublattices.

Let Mn be an n-dimensional manifold immersed in an (n+p)-dimensional unit sphere Sn+p, with mean curvature H and second fundamental form B. We put φ(X, Y) = B(X, Y)–(X, Y)H where X and Y are tangent vector fields on Mn. Assume that the mean curvature is parallel in the normal bundle of Mn in Sn+p. Following Alencar and do Carmo [1] we denote by BH the square of the positive root of

Throughout this note, rings will mean associative rings with identity and all modules are unital. A ring R is called right artinian if R satisfies the descending chain condition for right ideals. It is known that not every ideal of a right artinian ring is right artinian as a ring, in general. However, if every ideal of a right artinian ring R is right artinian then R is called hereditarily artinian. The structure of hereditarily artinian rings was described completely by Kertész and Widiger [5] from which, in the case of rings with identity, we get:

A ring R is hereditarily artinian if and only if R is a direct sum S ⊕ F of a semiprime right artinian ring S and a finite ring F.

The main aim of this article is to discuss the relationship between Pontryagin duality and pro-objects. The basic idea arises from K. H. Hofmann's articles [7] and [8] where it is shown that the elementary abelian (Lie) groups are “dense” in the category of locally compact hausdorff abelian groups.

In this paper we continue the investigation of minimal conditions on semigroups begun by J. A. Green [3] and taken up by Munn [5]. A unified account of the results in [3] and [5], together with some additional material, is presented in the text-book by Clifford and Preston [1, §6.6]. All terminology and notation not introduced explicitly willbe as in [1].

We add Schreier's space S and the Lorentz space d(a, 1) to the list of classical Banach spaces which enjoy the λ-property, investigate the extreme point structure of S, and show that d(a, 1) has a λ-function which is continuous on Sd(a, 1), though not even uniformly so.

For a lifted nontrivial additive character λ' and a multiplicative character λ of the finite field with q2 elements, the “Gauss” sums Σ λ'(trg) over g ∈SU(2n, q2) and Σ λ (detg)λ'(trg) over g ∈ U(2n, q2) are considered. We show that the first sum is a polynomial in q with coefficients involving averages of “bihyperkloosterman sums” and that the second one is a polynomial in q with coefficients involving powers of the usual twisted Kloosterman sums. As a consequence, we can determine certain “generalized Kloosterman sums over nonsingular Hermitian matrices”, which were previously determined by J. H. Hodges only in the case that one of the two arguments is zero.

In [6] Conway and Morrell characterized those operators on Hilbert space that are points of continuity of the spectrum. They also gave necessary and sufficient conditions that a biquasitriangular operator be a point of spectral continuity. Our point of view in this note is slightly different. Given a point T of spectral continuity, we ask what can then be inferred. Several of our results deal with invariant subspaces. We also give some conditions characterizing a biquasitriangular point of spectral continuity (Theorem 3). One of these is that the operator and its adjoint both have the single-valued extension property.

Let f(x) be a complex function of a real variable, defined over the whole real line, which possesses n derivatives (the nth at least almost everywhere) and is such that . Then, if k is any integer for which 0< k < n, Kolmogoroff's inequality may be written as

,

or, by putting ,

The constant K=K (k, n) known explicitly and is the best possible, i.e., there is a (real) function for which equality holds (see Bang [1]).

Let G be a group. We define λ(G) to be the smallest integer n such that every element of the commutator subgroup G′ is a product of n commutators. Ito [4] has shown that λ (An)= 1 for all n. Thompson [7] has shown that λ (SLn(q))= 1 for all n and q. In fact, there is no known simple group G such that λ(G)>1. However, there do exist such perfect groups (cf. [7]).

Let R be a hyperbolic Riemann surface and W an open subset of R with ∂W piecewise analytic. Denote by the space of Dirichlet finite Tonelli functions on R and by π the harmonic projection of . Consider the relative HD–class on W, HD(W;∂W) = {u∈ │ u │ W∈HD(W) and u │ R\W = 0}. The extremization operation μis the linear mapping of HD(W;∂W) into HD(R) defined by μ. Since π preserves values of functions at the Royden harmonic boundary, the maximum principle implies that μis an order preserving injection and that Mμ is an isometry with respect to the supremum norms.

The existence of a function g of hhaving the property that pr divides the order of the automorphism group of a finite group G whenever pg divides the order of G was first established by Ledermann and Neumann [4], who showed that the least such function g(h) satisfies the inequality

Later Green [2] improved this estimate to

In the Present paper this will be revised, for sufficiently large h, to

The previous results on isometrically characterizing ln∞ in terms of operator ideal norms can be summarized as follows.

Let E be an n-dimensional Banach space.

(1) If λ(E) = 1, then (see [3], [5], [6]), where λ(E) is the projection constant of E (see [4]).

(2) If π(E) = n, then (see [1],[2]).

(3) If Δ2(E) = √n, then (see [8]), where Δ2(E) is the 2-dominated constant of E(see [4]).

(4) If for any linear operator T: → E, v1(T) = (T), then (see [7]), where v1 is the 1-nuclear norm (see [4]).