We generalise to Jordan–Banach algebras some results of Aupetit on perturbation by inessential elements. This enables us to show that the kh-socle is the largest inessential ideal, and to give some spectral characterisations of it.

Let f be a 1-periodic C1-function whose Fourier coefficients satisfy the condition [sum ]n[mid ]n[mid ]3[mid ] fˆ(n[mid ]2<∞. For every α∈R\Q and m∈Z\{0}, we consider the Anzai skew product T(x, y) = (x+α, y+mx+f(x)) acting on the 2-torus. It is shown that T has infinite Lebesgue spectrum on the orthocomplement L2(dx)⊥ of the space of functions depending only on the first variable. This extends some earlier results of Kushnirenko, Choe, Lemańczyk, Rudolph, and the author.

In this note we discuss the radial positive solutions of

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where D is the annulus {s∈Rn [ratio ]b<|s|<1}. Here 0<b<1, n[ges ]2, and g[ratio ]R→R is a suitable C1 function with g(0)[ges ]0 but with g changing sign. We answer a question on page 223 of the original Gidas–Ni–Nirenberg paper [8], by showing that the maximum of these solutions occurs at a point sε, where |sε|→b as ε→0. This also shows that the non-negativity condition in the result on page 223 of [8] is necessary.

Requirements for boundary conditions are found which imply sharp by order estimates in L∞-metrics of eigenfunctions of the ordinary differential operator

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It is shown that the inequalities

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for the eigenfunctions {yλ} yield certain restrictions on the spectrum.

In [2], H. Furstenberg studied a distal action of a locally compact group G on a compact metric space X, and established a structure theorem. As a consequence, he showed that if G is abelian, then a simply connected space X does not admit a minimal distal G-action.

In this paper we concern ourselves with a nonsingular flow ϕ={ϕt} on a closed 3-manifold M. Recall that ϕ is called distal if for any distinct two points x, y ∈ M, the distance d(ϕtx, ϕty) is bounded away from 0. The distality depends strongly upon the time parametrization. For example, there exists a time parametrization of a linear irrational flow on T2 which yields a nondistal flow [4, 6].

A Banach algebra [afr ] is AMNM if whenever a linear functional ϕ on [afr ] and a positive number δ satisfy [mid ]ϕ(ab)−ϕ(a)ϕ(b)[mid ] [les ]δ|a|·|b| for all a, b∈[afr ], there is a multiplicative linear functional ψ on [afr ] such that |ϕ−ψ|=o(1) as δ→0. K. Jarosz [1] asked whether every Banach algebra, or every uniform algebra, is AMNM. B. E. Johnson [3] studied the AMNM property and constructed a commutative semisimple Banach algebra that is not AMNM. In this note we construct uniform algebras that are not AMNM.

A famous problem of Zariski, stated in 1971 [21], is to decide whether the classical algebro-geometric multiplicity mo(f) of a complex hypersurface f−1(0) at an isolated singularity O is an invariant of the topological type of the embedded germ. That is, if f[ratio ]Cn, O→C, 0 and g[ratio ]Cn, O→C, 0 are two germs of polynomial functions, and there is a homeomorphism ϕ defined between two open neighbourhoods U and V of O in Cn such that ϕ(U) = V, ϕ(U∩f−1(0)) = V∩g−1(0) and ϕ(O) = O, then is it always the case that mo(f) = mo(g)? Here the multiplicity mo(f) is defined to be the intersection number of f−1(0) with a generic complex line L passing through O in Cn, that is, the number of points of intersection of f−1(0) with a line which is an arbitrarily small perturbation of L. If f is reduced, then mo(f) is equal to the order of the polynomial f(z) at O.

For each d[ges ]2 we construct a connected open set Ω⊂ℝd such that Ω=int (clos(Ω)), and for each k[ges ]1 and each p∈[1, +∞), the subset Wk, ∞(Ω) fails to be dense in the Sobolev space Wk, p(Ω), in the norm of Wk, p(Ω).

This work is an investigation of the extent to which convex bodies are determined by the sizes of their projections. It is shown, on the one hand, that there are non-congruent bodies for which all corresponding projections have the same size. On the other hand, it is proved that in the usual Baire category sense, most convex bodies are determined, up to translation or reflection, by the combination of their widths and brightnesses in all directions.

John Leslie Britton died in a climbing accident on the Isle of Skye on 13 June 1994 at the age of 66. He became a member of the London Mathematical Society in 1957, served the society as Meetings and Membership Secretary from 1973 to 1976, and at the same time was Editor of the Society's Newsletter.

In this paper the domain in which any two points are joined by finite numbers of arbitrarily small homeomorphisms is studied, and two examples are given.

In this paper we are going to study the solutions of a complex T-periodic Riccati equation

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and we are going to determine all the possible dynamics of this type of equation. Also, we can say that generically there are two different T-periodic solutions.

In the 1970s, a question of Kaplansky about discontinuous homomorphisms from certain commutative Banach algebras was resolved. Let A be the commutative C*-algebra C(Ω), where Ω is an infinite compact space. Then, if the continuum hypothesis (CH) be assumed, there is a discontinuous homomorphism from C(Ω) into a Banach algebra [2, 7]. In fact, let A be a commutative Banach algebra. Then (with (CH)) there is a discontinuous homomorphism from A into a Banach algebra whenever the character space ΦA of A is infinite [3, Theorem 3] and also whenever there is a non-maximal, prime ideal P in A such that [mid ]A/P[mid ]=2ℵ0 [4, 8]. (It is an open question whether or not every infinite-dimensional, commutative Banach algebra A satisfies this latter condition.)

The well-known inequality of Hardy for Fourier coefficients of functions f(t)∼[sum ]∞n=−∞bneint in the real Hardy space is [sum ]∞n= −∞[mid ]bn[mid ]/([mid ]n[mid ]+1)<∞. We shall establish analogues of this inequality for the Hermite function expansions and also for the Laguerre function expansions.

It is known [10, 11] that if T is an integral operator with an extended totally positive kernel, then T has a countably infinite family of simple, positive eigenvalues. We prove a similar result for a rather larger class of kernels and, writing the eigenvalues of T in decreasing order as (λn)n∈IN, we use results obtained in [4] and [5] to give a formula for the ratio λn+1/ lambda;n analogous to that given in [3] for the case of a strictly totally positive matrix, and to the spectral radius formula

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This may be regarded as a generalisation of inequalities due to Hopf [8, 9].

A C*-algebra A is said to be monotone (respectively monotone σ-) complete if every increasing net (respectively increasing sequence) of elements in the ordered space Ah of all hermitian elements of A has a supremum in Ah. It is straightforward to verify that every monotone complete C*-algebra is an AW*-algebra. For type I AW*-algebras, the converse is known to be true. However, for general AW*-algebras, this question is still open, although an impressive attack on the problem was made by E. Christensen and G. K. Pedersen, who showed that properly infinite AW*-algebras are monotone σ-complete [4].