Recently, Good discovered the following curious identity (1). Let θ be a real-valued random variable having cumulants Km ≡ Km(θ) (m = 1, 2, …). Take a set of m statistically independent variables θ(1), θ(2), …, θ(m), each with the same distribution as θ, and form the random variable

where ω is any primitive mth root of unity (e.g. ω = e2πi/m). Then

where 〈.〉 denotes expectation.

For high-dimensional simple knots we give two theorems concerning unique factorization into irreducible knots, and provide examples to show that the hypotheses are necessary in each case.

In a 1928 paper [1], Hardy and Littlewood published the following inequality which is valid for all non-negative measurable functions ƒ and g on ( − ∞, ∞):

where 1 < p, q < ∞, 0 < λ < 1, l/p + 1/q + λ = 2 and Cλp, q depends only on λ, p, q; ‖ƒ‖p and ‖g‖q are the Lp and Lq norms of ƒ and g respectively. See also p. 288 of the monograph [2] by Hardy, Littlewood and Pólya for a discussion of this inequality and related matters.

In a recent publication [2], R. Brown and the first author proved a Lifting Theorem for groups (and topological groups) showing that if β: B → H is an epimorphism of groups and H is a certain type of colimit of groups, then this colimit can be lifted (or pulled back) through β; that is B is a colimit of the lifted diagram (see Corollary 2·3 below).

The theory of unfoldings of singularities, or ‘elementary catastrophe theory’ (Thom(10), Poston and Stewart(9), Golubitsky and Guillemin(3), Gibson(2)) has been generalized by Golubitsky and Schaeffer(5,6), providing a powerful method for analysing imperfect bifurcation. One recent application by Golubitsky, Keyfitz, and Schaeffer(4) concerns the ‘explosion peninsula’ in chemical reactions such as that between hydrogen and oxygen. They show that the ‘thermal-chainbranching model’ of Gray and Yang(7,8) is capable of reproducing the necessary qualitative features, a result for which only heuristic arguments had previously been available.

It is shown that each vector in the Hilbert space of certain quasi-free representations of the CCR can be written uniquely in terms of quantum stochastic integrals. As a consequence, we obtain general vector-valued and operator-valued boson quantum martingale representation theorems.

The eigenfunctions for four types of bounded plane domains are constructed by a group-theoretic method. The group theory is then used to prove a completeness theorem and to derive a theta function transformation formula. Rapidly convergent series for the Green's functions of the Laplace operator on these domains are then constructed. Final formulas are summarized without reference to group theory.

If G is a group with a subgroup H and R is a Dedekind domain, then an H-projective RG-lattice is an RG-lattice that is a direct summand of an induced lattice for some RH-lattice N: they have been studied extensively in the context of modular representation theory. If H is the trivial group these are the projective lattices. We define a relative character χG/H on H-projective lattices, which in the case H = 1 is equivalent to the Hattori–Stallings trace for projective lattices (see [5, 8]), and in the case H = G is the ordinary character. These characters can be used to show that the R-ranks of certain H-projective lattices must be divisible by some specified number, generalizing some well-known results: cf. Corollary 3·6. If for example we take R = ℤ, then |G/H| divides the ℤ-rank of any H-projective ℤG-lattice.

For a polynomial $f(x)\in\mathbb{Q}[x]$ and rational numbers c, u, we put $f_c(x)\coloneqq f(x)+c$, and consider the Zsigmondy set $\calZ(f_c,u)$ associated to the sequence $\{f_c^n(u)-u\}_{n\geq 1}$, see Definition 1.1, where $f_c^n$ is the n-st iteration of fc. In this paper, we prove that if u is a rational critical point of f, then there exists an Mf > 0 such that $\mathbf M_f\geq \max_{c\in \mathbb{Q}}\{\#\calZ(f_c,u)\}$.

1. Suppose that f is a homeomorphism of the Euclidean plane E2 onto itself. The set M ⊂ E2 is said to be invariant if f(M) = M and minimal if it is non-void, closed, invariant and irreducible with respect to these properties. In general, invariant and minimal sets in E2 can have a finite or infinite number of components.

Let p be a prime and let Pi denote the Steenrod p-power if p ≠ 2 and the Steenrod square Sqi if p = 2. If p ≠ 2 let β denote the Bockstein co-boundary (defined so as to anticommute with suspension).

We study the prime ideal structure of the Iwasawa algebra $\Lambda_G$ of an almost simple compact $p$-adic Lie group $G$. When the Lie algebra of $G$ contains a copy of the two-dimensional non-abelian Lie algebra, we show that the prime ideal structure of $\Lambda_G$ is somewhat restricted. We also provide a potential example of a prime c-ideal of $\Lambda_G$ in the case when the Lie algebra of $G$ is ${\frak s}{\frak l}_2(\mathbb{Q}p)$.

1. Ganea has shown (Theorem 2·2 of (2)) that h-cogroup structures on a CW space coincide, in the homotopy category, with coalgebra structures defined with respect to the cotriple determined by SΩ. The construction used by Ganea to prove this does not dualize. It seems possible moreover that the dual statement is false. In this note I show, using the results of (4), that, for suitable spaces, there is a surjection from the set of algebra structures defined with respect to the triple determined by ΩS to the set of h-group structures, and I construct a right inverse for this surjection.

If H(ξ, η) is a harmonic function which is defined and positive in η > 0, then there is a non-negative number D and a bounded non-decreasing function G(x) such that

(For a proof, see Loomis and Widder, Duke Math. J. 9 (1942), 643–5.) If we write

where λ > 1, then the equation

defines a harmonic function h which is positive in υ > 0. Hence there is a non-negative number d and a bounded non-decreasing function g(x) such that

The problem of finding the connexion between the functions G(x) and g(x) has been mentioned by Loomis (Trans. American Math. Soc. 53 (1943), 239–50, 244).

Let k be a positive integer and F(x, k) denote the number of integers n < x which have a divisor in every residue class prime to k. Erdös (1) proved that for every fixed ε > 0, we have F(x, k) ˜ x when

and conjectured the following result, which we prove in this paper.

It was first proved by R. Lashof in [4], using the work of S. Cappell and J. Shaneson on four-dimensional surgeryu (see [1]), that there exist locally flat topological knots S3 ∪ S5 which are not smoothable. In [2] (compare also [6]) S. Cappell and J. Shaneson have constructed infinitely many non-smoothable locally fat topological knots as the fixed points of locally nice (= locally smoothable) Zp actions on S5, therefore giving non-trivial examples of locally smoothable but equivariantly non-smoothable actions of Zp on S5.

Some theorems in the operational calculus, taking definite integration as the fundamental operator, are proved for discrete systems. It is suggested that it is physically more satisfactory to regard the solution for a continuous system as the limit of the solutions for a set of discrete systems, rather than as the solution of a partial differential equation. The necessary and sufficient condition for the validity in this sense of the usual operational method for continuous systems appears to be that the discrete systems considered shall not tend to infinite instability; that is, that all poles of the Bromwich integrand shall always have real parts less than some fixed positive quantity. The correction for finiteness of the number of degrees of freedom is examined for a case analogous to heat conduction and found to be unimportant.

We study the asymptotics of the higher dimensional Reidemeister torsion for torus knot exteriors, which is related to the results by W. Müller and P. Menal–Ferrer and J. Porti on the asymptotics of the Reidemeister torsion and the hyperbolic volumes for hyperbolic 3-manifolds. We show that the sequence of 1/(2N)2) log | Tor(EK; ρ2N)| converges to zero when N goes to infinity where TorEK; ρ2N is the higher dimensional Reidemeister torsion of a torus knot exterior and an acyclic SL2N(ℂ)-representation of the torus knot group. We also give a classification for SL2(ℂ)-representations of torus knot groups, which induce acyclic SL2N(ℂ)-representations.

We study the pullback of the apolarity invariant of complex polynomials in one variable under a polynomial map on the complex plane. As a consequence, we obtain variations of the classical results of Grace and Walsh in which the unit disk, or a circular domain, is replaced by its image under the given polynomial map.