Let p be any prime. It is well known that the modp Dyer-Lashof algebra acts trivially on the mod p homology of an Eilenberg-MacLane space. The main result of this paper is a converse of this fact. Specifically, it is shown that any connected infinite loop space with trivial action of the mod p Dyer-Lashof algebra is (localized at p) homotopy equivalent to a product of Eilenberg-MacLane spaces. It is then shown that this equivalence does not necessarily respect the infinite loop structures involved.

In this paper we will study 2-knots in the 4-sphere S4 which are obtained by roll-spinning 1-knots in the 3-sphere S3. The process of spinning was introduced by Artin[2] and generalized to twist-spinning by Fox [7] and Zeeman [20]. In [7] Fox introduced another variation of the spinning process, called roll-spinning. He only showed that the roll-spun figure-eight knot cannot be obtained by twist-spinning the figure-eight knot. Finally Litherland [14] described the general process of deform-spinning, and he gave a precise definition of roll-spinning. We remark that Fox's roll-spinning is not the same as Litherland's: Fox's roll-spun figure-eight knot is the symmetry-spun figure-eight knot in terms of [14].

In [11], Partington proved that if λ is a Banach sequence space with a monotone basis having the Banach-Saks property, and (Xn) is a sequence of Banach spaces each having the Banach-Saks property, then the vector sequence space ΣλXn has this same property. In addition, Partington gave an example showing that if λ and each Xn, have the weak Banach-Saks property, then ΣλXn need not have the weak Banach-Saks property.

A class of regular semigroups closed under taking direct products, regular subsemigroups and homomorphic images is an e(xistence) -variety of regular semigroups. The class of all combinatorial strict regular semigroups is the e-variety generated by the five element non-orthodox completely 0-simple semigroup and consists of all regular subdirect products of combinatorial completely 0-simple semigroups and/or rectangular bands. The bifree object on the set X in is the natural concept of a ‘free object’ in the class . is generated by the set X and the set of formal inverses X′ under the two binary operations of multiplication · and forming the sandwich element ∧A. Hence is a homomorphic image of the absolutely free algebra of type 〈2, 2〉 generated by X ∪X′. In this paper we shall describe the associated congruence on F〈2, 2〉(X∪X′) and construct a model of in terms of sets and binary relations. As an application, a model of the free strict pseudosemilattice on a set X is obtained.

The main result in this note is as follows. Let E be a Fréchet Montel space which belongs to the large class of decomposable (FG)-spaces introduced by Bonet, Díaz, Taskinen and let F be a Fréchet Montel (resp. distinguished) space. Then the projective tensor product is Montel (resp. distinguished). We also give examples of Montel decomposable (FG)-spaces without an unconditional basis.

In this paper we show that if a group generated by two elements acts freely and without phantom inversions on a Λ-tree, then it is either free or free abelian.

In this paper we show that the cobordism classes of theta curves in S3 form a group under vertex connected sum. We investigate this group by means of knot cobordism and link cobordism.

The Poincaré disc model

of two-dimensional hyperbolic space supports a metric ρ derived from the differential

Geodesics for the metric ρ are arcs of circles orthogonal to the unit circle S, and straight lines through the origin.

Let f be a rational function of degree at least two. We shall prove that the Julia set J(f) of f is uniformly perfect. This means that there is a constant c∈(0, 1) depending on f only such that whenever z∈J(f) and 0 < r < diam J(f) then J(f) intersects the annulus .

A polynomial is defined on signed matroids which contains as specializations the Kauffman bracket polynomial of knot theory, the Tutte polynomial of a matroid, the partition function of the anisotropic Ising model, the Kauffman–Murasugi polynomials of signed graphs. It leads to generalizations of a theorem of Lickorish and Thistlethwaite showing that adequate link diagrams do not represent the unknot. We also investigate semi-adequacy and the span of the bracket polynomial in this wider context.

We characterize those domains Ω in ℝN for which the positive harmonic functions on Ω separate the points of Ω.

Given any domain Ω in ℝN we consider spaces X = X(Ω),Y = Y(Ω) which are Banach function spaces in the sense of Luxemburg [12]. From these we form what we call the abstract Sobolev space W(X, Y), which is denned to be the linear space of all ƒ ∈ X such that for i = 1, …, N, the distributional derivative ∂ƒ/∂xi belongs to Y; equipped with the norm

W(X, Y) is a Banach space. The closure of (Ω) in W(X, Y) is denoted by W0(X, Y). Let w to bea weight function on Ω, that is, a measurable function which is positive and finite almost everywhere on Ω. We say that [w, X, Y] supports the (weighted) Poincaré inequality if there is a positive constant K such that for all u ∈ W(X, Y),

analogously, [X, Y] is said to support the Friedrichs inequality if there is a positive constant K such that for all u ∈ W0(X, Y),

The purpose of this paper is to settle in the negative an open problem in operator theory, which asks whether in a finite distributive subspace lattice ℒ on a Hilbert space, every finite rank operator of Alg ℒ can be written as a finite sum of rank one operators from Alg ℒ. The counter-example constructed is on a specific Hilbert space realization of the free distributive lattice on three generators and the operator which fails the above property has rank two.

In Baxter and Williams [1] we began a study of Abel averages,

as opposed to the oft-studied Cesàro averages In Baxter and Williams [2], hereinafter referred to as [BW2], we studied the large-deviation behaviour of these averages. In the case where X is an irreducible Markov chain on a finite state-space S = {1,…, n}, we observed that

and

where π is the invariant distribution of X. We noted that

where v is an n-vector, δ(v):= sup{Re(z): z ∈ spect(Q+ V)}, and where spect(·) denotes spectrum (here the set of eigenvalues), Q is the Q-matrix of X, and V denotes the diagonal matrix diag (vi). It is also true that the large-deviation property holds for Ct with rate function I denned on M = {(xt)i ∈ S: xi ≥ 0, Σixi = 1}.

Let γn denote n-dimensional Lebesgue measure. It follows easily from the well-known volume mean value property of harmonic functions that if h is an integrable harmonic function on an open ball B of centre ξ0 in ℝn, where n ≥ 2, then

A converse of this result is due to Kuran [8]: if D is an open subset of ℝn such that γn(D) < + ∞ and if there exists a point ξo∈D such that

for every integrable harmonic function h on D, then D is a ball of centre ξ0. Armitage and Goldstein [2], theorem 1, showed that the same conclusion holds under the weaker hypothesis that (1·2) holds for all positive integrable harmonic functions h on D.

Let U(1, n; ℂ) be the automorphism group of the Hermitian form

for . We can regard an element of U(1, n; ℂ) as a transformation acting on , where is the closure of the complex unit ball

The non-trivial elements of U(1, n; ℂ) fall into three conjugacy types, depending on the number and the location of their fixed points. Let g be a non-trivial element of U(1, n; ℂ). We call g elliptic if it has a fixed point in Bn and g parabolic if it has exactly one fixed point and this lies on the boundary ∂Bn. An element g will be called loxodromic if it has exactly two fixed points and they lie on ∂Bn.

For a class of kernels, we prove the Lp estimate for the exotic Riesz potential, with which a restriction theorem of the Fourier transform to surfaces of half the ambient dimension is proved. A simpler proof of Barcelo's result is given. We also find that it is possible to combine the Hausdorff–Young theorem with the Fefferman–Zygmund method to prove some optimal results on the restriction theorem.

In this paper we obtain criteria for a reflecting Brownian motion in the first orthant of the plane to reach an arbitrary open neighbourhood of the origin in finite time, or in finite mean time. The reflecting Brownian motion (RBM) is assumed to have a constant non-zero drift, and a constant non-singular covariance, and the directions of reflection on the two sides of are constant along each side, but not necessarily normal. We explain why the criteria we find are to be expected.

Recent efforts have been focused on using numerical methods to estimate the Titchmarsh–Weyl m-coefficient. In this paper we look at interval analytic methods to provide provable bounds for these values.

Recently Kauffman[2, 3] has classified the (strongly) spin-preserving solutions of the Yang—Baxter equation, and in particular discussed two solutions. One of these can be used to obtain the Jones polynomial, while the other (with care) leads to the Alexander polynomial. In this paper we shall complete this analysis, and shall describe all strongly spin-preserving invertible solutions of the Yang-Baxter equation which lead to link invariants. Having done this, we investigate what sort of link invariants can be obtained from these solutions. It turns out that these invariants are precisely those which have been described in Hennings [l].