Dr. A. D. Sands has pointed out that the following sentence, occurring in (2, p. 60, lines 2 to 4), should be deleted: “Hence Sinkov's group of order 1344 … is the holomorph of the Abelian group {pi}, of order 8 (1, pp. 111–117).” In fact, Sinkov's group and the holomorph of C2×C2×C2 are not isomorphic. For, the holomorph is representable on 8 letters by definition (1, p. 87), whereas Sinkov's group is not representable on 8 letters. To see this, we recall that Sinkov's group (3, p. 584) is generated by two elements of periods 2 and 3 (namely, QP3 and QP2) whose commutator is of period 8. If these two generators could be represented as permutations of 8 letters, their commutator would be an even permutation and thus could not be of period 8.

Statement of the formula and numerical examples of it.

The object of the present note is to communicate the following formula for the solution of algebraic or transcendental equations:

The root of the equation

We define

where ap ≠ aq when p ≠ q. If N = Σλi, then the partition (λ1, λ2, …, λn) of N with λ1 ≥ λ2 ≥ … ≥ λn is denoted by (λ) and we set

All partitions will be in descending order and the usual notation for repeated parts will be used.

Hadamard defines the “elementary solution” of the general linear partial differential equation of the second order, namely

(Aik, BiC being functions of the n variables x1, x2, .., xn, which may be regarded as coordinates in a space of n dimensions), to be one of those solutions which are infinite to as low an order as possible at a given point and on every bicharacteristic through that point.

A semigroup is said to be congruence-free if and only if its only congruences are the universal relation and the identical relation. Congruence-free inverse semigroups were studied by Baird [2], Trotter [19], Munn [15,16] and Reilly [18]. In addition, results on congruence-free regular semigroups have been obtained by Trotter [20], Hall [4] and Howie [7].

Let X be a complex normed space, with dual space X′. Let T be a bounded linear operator on X. The numerical range V(T) of T is defined as {f(Tx): x ∈ X, f ∈ X′, ‖ x ‖ = ‖ f ‖ = f(x) = 1}, and the numerical radius v(T) of T is defined as sup {|z|: z ∈ V(T)}. For a unital Banach algebra A, the numerical range V(a) of a ∈ A is defined as V(Ta), where Ta is the operator on A defined by Tab = ab. It is shown in (2, Chapter 1.2, Lemma 2) that V(a) = {f(a): f ∈ D(1)}, where D(1) = {f ∈ A′: ‖f‖ = f(1) = 1}.

We prove the following theorem, which was established by Abel (1) for the case u = 1.

Theorem. If u, k are positive integers and x, 1, , u,are real numbers, then

where the sum is taken over all distinct ordered solutions (s1, , su) in non-negative integers of the equation .

An integer n may be partitioned into the set of integers (α1, α2, …, αl), where Whenever αi−1>αi the substitution of αi+1 for αi, where αi may be αi+1≡0, would give a partition of n+1 whilst the substitution of αi−1—1 for αi−1 would give a partition of n—1; partitions of n+1 and of n—1 so arising will be said to be associated with the given partition of n.

Let L(H) be the algebra of all bounded linear operators on a separable complex Hubert space H. In a recent paper [7], explicit expressions for solutions of a boundary value problem in the Hubert space H, of the type

are given in terms of solutions of an algebraic operator equation

Let 2* denote the dual of the mod two Steenrod algebra. In [5] an algebraic filtration B*(n) of H*(BO; ℤ2) was constructed such that each B*(n) is a bipolynomial sub Hopf algebra and sub 2*-comodule of H*(BO; ℤ2). In Lemma 3.1 we prove that the Thom isomorphism determines a corresponding filtration of H*(MO;ℤ2) by polynomial subalgebras and sub 2*-comodules M*(n). Let (n) denote the subalgebra of 2 generated by Sq2k, 0 ≦ k < n, and let *(n) be its dual, a quotient Hopf algebra of 2*. In Section 3 we construct a polynomial algebra and *(n)-comodule R(n) such that M*(n)≃2*□*(n)R(n) as algebras and 2*-comodules. Here □ denotes the cotensor product defined in [9, §2]. Dually it will follow that M*(n) has a sub (n)-module and subcoalgebra T(n) such that M*(n)≃2⊗n)T(n) as coalgebras and 2-modules. We also show that M*(n) can not be realised as the homology of a spectrum for n≧4. Of course M*(0)=H*(MO;ℤ2), M*(1)=H*(MSO;ℤ2), M*(2)=H* (MSpin;ℤ2) and M*(3)=H*(MO<8>;ℤ2). Moreover, it follows from [4; Thm. 2.10, Cor. 2.11] that M*(n)=Images[H*(MO<ϕ(n)>;ℤ2)→H*(MO;ℤ2)] and M*(n) ≃ Image [H*(MO;ℤ2)→ H*(MO<ϕ(n)>;ℤ2)]. Here MO<k> id the Thom spectrum of BO<k>, the (k−1)-connected covering of BO, and ϕ(n)=8s + 2t where n = 4s + t, 0≦t≦3. In Section 4 we sketch the odd primary analogue—a filtration pM*(n) of H*(MUp, 0;ℤp) for p an odd prime. MUp, 0 is the Thom spectrum of the (2p-3)-connected factor of the Adams splitting [2] of BU(p).

At the 29th Meeting of the British Mathematical Colloquium held at Edinburgh in March 1977, D. H. Fremlin announced the following result.

Let Φ denote the proposition that there exists a K-analytic Hausdorff space, having metrizable compacta, which is not Souslin. (We refer the reader to Fremlin's paper (1) for the relevant definitions)

Theorem 1.1. (Fremlin) (i) Assume . (ii) Assume Martin's Axiom together with . Then Φ is false.