We define a class of equations that are not amenable but are type K and are therefore solvable over torsion-free groups. Moreover, we show that these new equations are solvable over all groups.

Lutwak (Adv. Math., vol. 118(2), 1996, pp. 244–294) defined the notion of Lp-geominimal surface area based on Lp-mixed volumes. Recently, Wang and Qi (J. Inequal. Appl., vol. 2011, 2011, pp. 1–10) introduced the concept of Lp-dual geominimal surface area based on Lp-dual mixed volumes. In this paper, based on Lp-dual mixed quermassintegrals, we define the concept of Lp-dual mixed geominimal surface area and establish several inequalities for this new notion.

On présente des exemples de représentations de de dimension 2, de déterminant pair, qui sont de type diédral (I) ou de conducteur premier et de type quelconque (II), en imitant la construction de Tate (Serre [11]) de représentations de déterminant impair.

Let q be a power of a prime p, and let Sqbe the set of permutations of {0, 1,…, q – 1). As Sq is isomorphic to the group of permutations of Fq, the field of q elements, each element of 5, can be regarded as a polynomial over Fq. Various authors (e.g. [1], [2], [3]) have considered functions f(x) such that

f(x) ∈ Sq, and (f(x) + λ×) ∈ Sq

for some λ ∈ Fq when λ = 1, f(x) is a complete mapping polynomial ([3]).

On a semigroup S let the relation ℛ*, sometimes denoted by ℛ, be defined by xℛ*y[(sx = txsy = ty]. A semigroup S is called left type-A, iff the set Es of idempotents of S forms a semilattice under multiplication, each element x of Sis ℛ* related to a (necessarily unique) idempotent x+, and xe = (xe)+x for all x ∈ S, е ∈ Es.

The first discussion of the propagation of elastic waves in a thick plate was given by Lamb [1] for the two-dimensional problem of a harmonic wave travelling in a direction parallel to the medial plane of the plate. Lamb derived equations relating the thickness of the plate to the phase velocities of two types of wave, one symmetric with respect to the medial plane and the other antisymmetric. The symmetric modes of propagation introduced by Lamb have been studied by Holden [2] and the antisymmetric modes have been studied by Osborne and Hart [3]. More recently Pursey [4] has shown how the amplitude of the disturbance is related to a given distribution of stress, varying harmonically with time, applied to the free surfaces of the plate; two types of source are considered by Pursey, one producing a two-dimensional field of the Lamb type, and the other having circular symmetry about an axis normal to the surface of the plate.

Let L = L( +, v, ＾) be a lattice-ordered group, or l-group (Birkhoff [1, p. 214]). Two elements a and b of L will be called disjoint if a > 0, b > 0, and a ＾; b = 0. It is easily seen that if L does not contain two disjoint elements, then it is linearly ordered (and, of course, conversely). What can we say about Z-groups containing two but not more than two mutually disjoint elements?

Let Aand B be linearly ordered groups (o-groups), and let A ⋏ B be the cardinal sum of A and B. That is, A ⋏ B is the direct sum of A and B, and (a, b) is positive in A + B if and only if a is positive in A and b is positive in B. An l-group L containing A ⋏ B as a convex normal subgroup (or Z-ideal) is called a lexico-extension of A ⋏ B if every positive element of L not in A ⋏ B exceeds every element of A ⋏ B. It then follows (subsection 2.9 below) that L/(A ⋏ B) is an o-group. Such an l-group L is easily seen to satisfy the following condition: (D)

There exists a pair of disjoint elements in L, but no triple of pairwise disjoint elements exists in L.

In this note we consider transcendental entire functions

whose power series contain gaps, i.e.

where Λ = {λk} is a suitable set of positive integers. We denote the set of all such functions f(z) by E(Λ). As usual M(r) = M(r, f) denotes the maximummodulus of f(z) on the circle |z| = r. The order p and the lower order λ of f(z) are defined by

respectively.

Let Rbe an associative ring, Jan infinite index set, and Rj the ring of all J × J row-finite matrices over R. The Jacobson radical of Rwill be denoted by Γ(R).

The fundamental work on approximation in weighted spaces of continuous functions on a completely regular space has been done mainly by Nachbin ([5], [6]). Further investigations have been made by Summers [10], Prolla ([7], [8]), and other authors (see the monograph [8] for more references). These authors considered functions with range contained in the scalar field or a locally convex topological vector space. In the present paper we prove some approximation results without local convexity of the range space.

A distributive p-algebra is an algebra 〈L; ∨, ∧, *, 0, 1〉 for which 〈L, ∨, ∧, 0, 1〉 is a bounded distributive lattice and * is a unary operation on L such that a ∧ x = 0 if and only if x ≤ a* (i.e. a pseudocomplementation). A distributive double p-algebra is an algebra 〈L; ∨, ∧, *, +, 0, 1〉 in which the deletion of + gives a distributive p-algebra and the deletion of * gives a dual distributive p-algebra, that is a ∨ (x = 1 if and only if x ≥ a+.

In a recent paper [5] one of us has given a sufficient condition to be satisfied by a given property of radical classes within a universal class w in order that, for any subclass ℳ of w, there should be a smallest radical class having the given property and containing ℳ. The sufficient condition is that the classof all radical classes with the given property can be characterised as the class of all radical classes fixed by an admissible function F (see Section 1 below). In this paper a necessary and sufficient condition is derived and the corresponding result for semi-simpleclasses is also presented. These results are given in Section 2.

We show that the sum and the product of two commuting operators with the single-valued extension property need not inherit this property.

The genus spectrum of a finite group G is the set of all g such that G acts faithfully on a compact Riemann surface of genus g. It is an open problem to find a general description of the genus spectrum of the groups in interesting classes, such as the Abelian p-groups. Motivated by earlier work of Talu for odd primes, we develop a general combinatorial method, for arbitrary primes, to obtain a structured description of the so-called reduced genus spectrum of Abelian p-groups, including the reduced minimum genus. In particular, we determine the complete genus spectrum for a large subclass, namely, those having ‘large’ defining invariants. With our method we construct infinitely many counterexamples to a conjecture of Talu, which states that an Abelian p-group is recoverable from its genus spectrum. Finally, we give a series of examples of our method, in the course of which we prove, for example, that almost all elementary Abelian p-groups are uniquely determined by their minimum genus, and that almost all Abelian p-groups of exponent p2 are uniquely determined by their minimum genus and Kulkarni invariant.

Using variational arguments we study the non-existence and multiplicity of non-negative solutions for a class equations of the form where Ω is a bounded domain in N, N ≧ 3, f is a sign-changing Carathéodory function on Ω × [0, +∞) and λ is a positive parameter.

This paper continues our study of non-linear multiparameter eigenvalue problems. In recent work [3], [4], [5] we have discussed bifurcation from eigenvalues in both abstract and concrete multiparameter problems. In this note we present further simple conditions which will ensure bifurcation from eigenvalues of a multiparameter Sturm-Liouville problem. We consider the regular case and the asymptotic case of bifurcation from infinity.

We show that if A is a compact C*-algebra without identity that has a faithful *-representation in the C*-algebra of all compact operators on a separable Hilbert space and its multiplier algebra admits a minimal central projection p such that pA is infinite-dimensional, then there exists a Hilbert A1-module admitting no frames, where A1 is the unitization of A. In particular, there exists a frame-less Hilbert C*-module over the C*-algebra $K(\ell^2) \dotplus \mathbb{C}I_{\ell^2}$.

Throughout, R denotes a commutative domain with 1, and Q (≠R) its field of quotients, which is viewed here as an R-module. The symbol K will stand for the R-module Q/R, while R denotes the multiplicative monoid R/0.

The integral operator which we will consider in this paper is the operator T denned for suitably restricted functions f on (0, ∞) by

where x >0 and the integral is taken in the Cauchy principal value sense at t = x. This operator plays a considerable role in Wiener–Hopf theory; see [2; Chapter 5].

Since T is clearly the restriction to (0, ∞) of minus the Hilbert transformation applied to functions which vanish on (−∞, 0), it follows easily from the theory of the Hilbert transformation, as given in say [6; Theorem 101], that T is a bounded operator from Lp(0, ∞) to itself for 1 < p < ∞.

Numerous formulae have been given which exhibit the asymptotic behaviour as t → ∞solutions of

where F(t) is essentially positive and Several of these results have been unified by a theorem of F. V. Atkinson [1]. It is the purpose of this paper to establish results, analogous to the theorem of Atkinson, for the third order equation

and for the fourth order equation