Several “classical” results on algebraic complete lattices extend to algebraic posets and, more generally, to so called compactly generated posets; but, of course, there may arise difficulties in the absence of certain joins or meets. For example, the property of weak atomicity turns out to be valid in all Dedekind complete compactly generated posets, but not in arbitrary algebraic posets. The compactly generated posets are, up to isomorphism, the inductive centralized systems, where a system of sets is called centralized if it contains all point closures. A similar representation theorem holds for algebraic posets; it is known that every algebraic poset is isomorphic to the system i(Q) of all directed lower sets in some poset Q; we show that only those posets P which satisfy the ascending chain condition are isomorphic to their own “up-completion” i(P). We also touch upon a few structural aspects such as the formation of direct sums, products and substructures. The note concludes with several applications of a generalized version of the Birkhoff Frink decomposition theorem for algebraic lattices.

In this paper we prove a structure theorem for the class of $\mathcal{AN}$-operators between separable, complex Hilbert spaces which is similar to that of the singular value decomposition of a compact operator. Apart from this, we show that a bounded operator is $\mathcal{AN}$ if and only if it is either compact or a sum of a compact operator and scalar multiple of an isometry satisfying some condition. We obtain characterizations of these operators as a consequence of this structure theorem and deduce several properties which are similar to those of compact operators.

Problem 11 of Hanna Neumann's book [3] asks whether the product variety Β4Β2 has a finite basis for its laws. (For any positive integer k, Βk denotes the variety of all groups of exponent diving k.) I think that Β4Β2 was being suggested as a plausible canditate for a variety without the finite basis property; of course, at a time when no such example was known. It is the primary object of this note to verify the fact that Β4Β2 is not finitely based. Β4Β2 provides, therefore, probably the simplest example known at present of a variety which is not finitely based.

There are infinitely, but at most continuously, many varieties of groups; the precise cardinal is unknown. It is easy to see that if there is no infinite properly descending chain of varieties (equivalently, if the laws of every variety have a finite basis), then the number of varieties is countable infinity; the converse implication does not seem to have been proved. This note presents an argument which implies that if the locally finite or the locally nilpotent varieties fail to satisfy the minimum condition, then there are continuously many such varieties. Alternatively, one can conclude that if a locally finite or locally nilpotent variety has a finite basis for its laws but some subvariety of has none, then there are continuously many varieties between and . This points again to the interesting question: is every locally finite [locally nilpotent] variety contained in a suitable locally finite [locally nilpotent] variety which has a finite basis for its laws? (That is, must be locally finite [locally nilpotent] for some finite n?) For, if the answer were affirmative, it would follow that the number of locally finite [locally nilpotent] varieties is either countable or the cardinal of the continuum, depending exactly on the existence of finite bases for the laws of such varieties.

We give an example of a Banach space X such that K (X, X) is not an ideal in K (X, X**). We prove that if z* is a weak* denting point in the unit ball of Z* and if X is a closed subspace of a Banach space Y, then the set of norm-preserving extensions H B(x* ⊗ z*) ⊆ (Z*, Y)* of a functional x* ⊗ Z* ∈ (Z ⊗ X)* is equal to the set H B(x*) ⊗ {z*}. Using this result, we show that if X is an M-ideal in Y and Z is a reflexive Banach space, then K (Z, X) is an M-ideal in K(Z, Y) whenever K (Z, X) is an ideal in K (Z, Y). We also show that K (Z, X) is an ideal (respectively, an M-ideal) in K (Z, Y) for all Banach spaces Z whenever X is an ideal (respectively, an M-ideal) in Y and X * has the compact approximation property with conjugate operators.

Let W+ denote the Banach algebra of all absolutely convergent Taylor series in the open unit disc. We characterize the finitely generated closed and prime ideals in W+. Finally, we solve a problem of Rubel and McVoy by showing that W+ is not coherent.

This paper defines descriptive, Riemann, and constructive integrals equivalent to the approximately continuous integral of Burkill.

Let S(p) be the family of holomorphic functions f defined on the unit disk D, normalized by f(0) = f1(0) – 1 = 0 and univalent in every hyperbolic disk of radius p. Let C(p) be the subfamily consisting of those functions which are convex univalent in every hyperbolic disk of radius p. For p = ∞ these become the classical families S and C of normalized univalent and convex functions, respectively. These families are linearly invariant in the sense of Pommerenke; a natural problem is to calculate the order of these linearly invariant families. More precisely, we give a geometrie proof that C(p) is the universal linearly invariant family of all normalized locally schlicht functions of order at most coth(2p). This gives a purely geometric interpretation for the order of a linearly invariant family. In a related matter, we characterize those locally schlicht functions which map each hyperbolically k-convex subset of D onto a euclidean convex set. Finally, we give upper and lower bounds on the order of the linearly invariant family S(p) and prove that this class is not equal to the universal linearly invariant family of any order.

A conjecture of Carlitz on permutation polynomials is as follow: Given an even positive integer n, there is a constant Cn, such that if Fq is a finite field of odd order q with q > Cn, then there are no permutation polynomials of degree n over Fq. The conjecture is a well-known problem in this area. It is easily proved if n is a power of 2. The only other cases in which solutions have been published are n = 6 (Dickson [5]) and n = 10 (Hayes [7]); see Lidl [11], Lausch and Nobauer [9], and Lidl and Niederreiter [10] for remarks on this problem. In this paper, we prove that the Carlitz conjecture is true if n = 12 or n = 14, and give an equivalent version of the conjecture in terms of exceptional polynomials.

In this paper we extend the notion of FP-injective modules to that of complexes and characterize such complexes. We show that some characterizations similar to those for injective complexes exist for FP-injective complexes. We also introduce and study the notion of an FP-injective dimension associated to every complex of left R-modules over an arbitrary ring. We show that there is a close connection between the FP-injective dimension of complexes and flat dimension.

A Chebyshev set is a subset of a normed linear space that admits unique best approximations. In the first part of this paper we present some basic results concerning Chebyshev sets. In particular, we investigate properties of the metric projection map, sufficient conditions for a subset of a normed linear space to be a Chebyshev set, and sufficient conditions for a Chebyshev set to be convex. In the second half of the paper we present a construction of a nonconvex Chebyshev subset of an inner product space.

The groups whose 2-generator subgroups are all nilpotent of class at most 2 are nilpotent of class at most 3 (see Levi [6]). Heineken [3] generalized Levi's result by proving that for n ≧ 3, if the n-generator subgroups of a group are all nilpotent of class at most n, then the group itself is nilpotent of class at most n. Other related problems have been considered by Bruck [1].

We show how to use Jantzen’s sum formula for Weyl modules to prove semisimplicity criteria for endomorphism algebras of $\mathbf{U}_{q}$-tilting modules (for any field $\mathbb{K}$ and any parameter $q\in \mathbb{K}-\{0,-1\}$). As an application, we recover the semisimplicity criteria for the Hecke algebras of types $\mathbf{A}$ and $\mathbf{B}$, the walled Brauer algebras and the Brauer algebras from our more general approach.

Let M, N be Riemannian manifolds, f: M → N a harmonic map with potential H, namely, a smooth critical point of the functional EH(f) = ∫M[e(f)−H(f)], where e(f) is the energy density of f. Some results concerning the stability of these maps between spheres and any Riemannian manifold are given. For a general class of M, this paper also gives a result on the constant boundary-value problem which generalizes the result of Karcher-Wood even in the case of the usual harmonic maps. It can also be applied to the static Landau-Lifshitz equations.

Some geometric properties of Lp spaces are studied which shed light on the prediction of infinite variance processes. In particular, a Pythagorean theorem for Lp is derived. Improved growth rates for the moving average parameters are obtained.

Let $w$ be a multilinear commutator word. We prove that if $e$ is a positive integer and $G$ is a finite group in which any nilpotent subgroup generated by $w$-values has exponent dividing $e$, then the exponent of the corresponding verbal subgroup $w(G)$ is bounded in terms of $e$ and $w$only.

We give algebraic criteria for distinguishing composition operators among all continuous linear operators on spaces of continuous functions with topologies generated by seminorms that are weighted analogues of the supremum norm. In another direction, we also characterize those self maps of the underlying topological space which induce composition operators on such weighted spaces, as well as determine conditions on these self maps which correspond to various basic properties of the induced composition operator. Our results are applied to a question concerning translation invariance which arises in the context of topological dynamics.

Dealing with a problem posed by Kupka we give results concerning the permanence of the almost strong lifting property (respectively of the universal strong lifting property) under finite and countable products of topological probability spaces. As a basis we prove a theorem on the existence of liftings compatible with products for general probability spaces, and in addition we use this theorem for discussing finite products of lifting topologies.

In this paper, we prove that for a transcendental meromorphic function f(z) on the complex plane, the inequality T(r, f) < 6N (r, 1/(f2 f(k)−1)) + S(r, f) holds, where k is a positive integer. Moreover, we prove the following normality criterion: Let ℱ be a family of meromorphic functions on a domain D and let k be a positive integer. If for each ℱ ∈ ℱ, all zeros of ℱ are of multiplicity at least k, and f2 f(k) ≠ 1 for z ∈ D, then ℱ is normal in the domain D. At the same time we also show that the condition on multiple zeros of f in the normality criterion is necessary.

In a previous paper [4] we showed that Γ3,1 = 16/. For the definition of Γr, s for an indefinite quadratic form in n = r + s variables of the type (r, s) see the above paper. Here we shall show that Γ2,2 = 16. More precisely we prove:Theorem. Let Q (x, y, z, t) be an indefinite quaternary quadratic form with determinant D > 0 and signature (2, 2). Then given any real numbers x0, y0, z0, t0 we can find integers x, y, z, t such thatEquality is necessary if and only if either where ρ ≠ 0. For Q1 equality occurs if and only if