In (3) it is shown that, for a locally compact abelian group G and x∈G, δx has a logarithm in M(G) if and only if x has finite order. Since M(G) can be identified with the multipliers of L1(G), one might expect a similar result for the algebras of multipliers on Lp(G) for 1 < p < ∞. However, in contrast, it is shown in (2) that for a locally compact abelian group G and 1 < p < ∞, every translation operator on Lp(G) has a logarithm in the multiplier algebra. Here we consider whether the same results are true for non-abelian groups.

The notion of an atomic operator between spaces of measurable functions was introduced in 2002 in a paper by Drakhlin, Ponosov and Stepanov in order to provide a reasonable generalization of local operators useful for applications. It has been shown that, roughly speaking, atomic operators amount to compositions of local operators with shifts. A natural problem is then when a continuous-in-measure atomic operator can be represented as a composition of a Nemytskiiˇ (composition) operator generated by a Carathéodory function, and a shift operator. In this paper we will show that the answer to this question is inherently related to the possibility of extending an atomic operator with continuity from a space of functions measurable with respect to some $\sigma$-algebra to a larger space of functions measurable with respect to a larger $\sigma$-algebra, as well as to the possibility of extending any $\sigma$-homomorphism from a smaller-measure algebra to a $\sigma$-homomorphism on a larger-measure algebra. We characterize precisely the condition on the respective $\sigma$-algebras which provides such possibilities and induces the positive answer to the above representation problem.

AMS 2000 Mathematics subject classification: Primary 47B38; 47A67; 34K05

The object of this paper is to discuss conditions for a maximum or minimum of functions of integrals of the type

employing the methods of the Calculus of Variations.

In this note, we determine fields K and groups G that are either nilpotent or FC and such that the set of torsion elements of the group ring KG forms a subgroup.

It is shown that if f is an analytic function of sufficiently small exponential type in the right half-plane, which takes integer values on a subset of the positive integers having positive lower density, then f is a polynomial.

If G is a group, let

m(G)

be the least cardinal such that G may be generated by m(G) elements, and let

m,(G)

The inverse problem we will consider in this paper has its origins in the following problem connected with the scattering of acoustic waves in a nonhomogeneous medium. Let an incoming plane acoustic wave of frequency ω moving in the direction of the z axis be scattered off a “soft” sphere Ω of radius one which is surrounded by a pocket of rarefied or condensed air in which the local speed of sound is given by c(r) where r = |x| for x ∈ R3. Let us(x)eiωt be the velocity potential of the scattered wave and let r, θ, φ be spherical coordinates in R3. Then from a knowledge of the far field pattern f(θ, φ, λ) for λ = ω/c0 contained in some finite interval 0 < λ0 ≤ λ ≤ λ1,, we would like to determine the unknown function c(r).

This paper is a further contribution to the extensive study by a number of authors of the subalgebra lattice of a Lie algebra. We give some necessary and some sufficient conditions for a subalgebra to be upper modular. For algebraically closed fields of any characteristic these enable us to determine the structure of Lie algebras having abelian upper-modular subalgebras which are not ideals. We then study the structure of solvable Lie algebras having an abelian upper-modular subalgebra which is not an ideal and which has trivial intersection with the derived algebra; in particular, the structure is determined for algebras over the real field. Next we classify non-solvable Lie algebras over fields of characteristic zero having an upper-modular atom which is not an ideal. Finally, it is shown that every Lie algebra over a field of characteristic different from two and three in which every atom is upper modular is either quasi-abelian or a $\mu$-algebra.

AMS 2000 Mathematics subject classification: Primary 17B05; 17B50; 17B30; 17B20

In this paper the following expansion will be obtained:

in which n and r are positive integers. The Series in the square brackets are Hypergeometric Series with a finite number of terms.

We prove a technical result required by Ivanov and Shpectorov in their construction of a non-split extension of 34371 by the Baby Monster simple group.

The numerical range in the quaternionic setting is, in general, a non-convex subset of the quaternions. The essential numerical range is a refinement of the numerical range that only keeps the elements that have, in a certain sense, infinite multiplicity. We prove that the essential numerical range of a bounded linear operator on a quaternionic Hilbert space is convex. A quaternionic analogue of Lancaster theorem, relating the closure of the numerical range and its essential numerical range, is also provided.

Given two intersecting domains, we investigate the boundary behaviour of the quotient of Martin kernels of each domain. To this end, we give a characterization of minimal thinness for a difference of two subdomains in terms of Martin kernels of each domain. As a consequence of our main theorem, we obtain the boundary growth of the Martin kernel of a Lipschitz domain, which corresponds to earlier results for the boundary decay of the Green function for a Lipschitz domain investigated by Burdzy, Carroll and Gardiner.

The result of Ballantine [1] to the effect that a singular matrix A is a product of k idempotent matrices if and only if the rank of I – A does not exceed k times the nullity of A is generalized to endomorphisms of a class of independence algebras.

Let G be a non-discrete LCA group with dual group Γ. Denote by M(G) the usual convolution algebra of bounded Borel measures on G and Ma(G) those μ ∈ M(G) which are absolutely continuous with respect to mG—the Haar measure on G.

The paper is concerned with a curve F, the complete intersection of a quadric with a quartic surface, that admits a group of self-projectivities isomorphic to the symmetric group of degree 5. Every generator of the quadric is, as shown at the end of the paper, cut by F equianharmonically. F has 80 stalls, points where its osculating plane is stationary; they are of two kinds, 60 to be labelled ∑, the other 20 Ω. F also has inflections at 24 points which compose a figure encountered on earlier occasions. A search is made for tritangent planes of F of which, when reckoned according to proper multiplicity, there must be 2048. Among them are 60 all of whose three contacts are ∑ while a further 120 each involve a single ∑ among their contacts and 420 each involve a single Ω.

Every discrete definable subset of a closed asymptotic couple with ordered scalar field ${\boldsymbol {k}}$ is shown to be contained in a finite-dimensional ${\boldsymbol {k}}$-linear subspace of that couple. It follows that the differential-valued field $\mathbb {T}$ of transseries induces more structure on its value group than what is definable in its asymptotic couple equipped with its scalar multiplication by real numbers, where this asymptotic couple is construed as a two-sorted structure with $\mathbb {R}$ as the underlying set for the second sort.

Let Un(q) denote the upper triangular group of degree n over the finite field with q elements. It is known that irreducible constituents of supercharacters partition the set of all irreducible characters Irr(Un(q)). In this paper we present a correspondence between supercharacters and pattern subgroups of the form Uk(q) ∩ wUk(q), where w is a monomial matrix in GLk(q) for some k < n.

The object of this paper is firstly to extend the theorem of Pascal concerning six points of a conic to sets of 2 (n + 1) points of the rational normal curve of order n in space of n dimensions; secondly to explain why a wider extension to other sets of 2 (n + 1) points in [n] must be sought; and lastly to give briefly an extension to [3] and [4] which will be further generalised in a later paper. The striking feature of Pascal's theorem—that each of the sixty ways of arranging the points in a cycle, or as vertices of a closed polygon, leads to a different version of the theorem—is retained in the following extension to [n].