The object of this paper is to show how a well-known method of demonstrating the independence of one member of a system of axioms can be incorporated into a systematic method of testing for independence, suitable for solution on a logical computer of the type now being built at the University of Nottingham.

A square matrix A, of order n, having complex coefficients can be inverted without the aid of any operations involving complex numbers. This can be done if the coefficients ars + iαrs are replaced by their matrix equivalents and the resulting 2n × 2n real matrix A1 is inverted. The inverse will be a 2n × 2n matrix of similar form in which the subsidiary 2 × 2 matrices can be replaced by the equivalent complex numbers, thus yielding the inverse of A. It is the purpose of this note to show that a similar technique can be employed to evaluate det A.

Let P(x) denote a polynomial with degree n and integer coefficients. By the height h of P we mean the maximum of the absolute values of the coefficients.

A procedure is described to determine phase velocities of a circular helix. Computations are made for three helix configurations using the method of internal constraints. The results so obtained are compared with the results obtained by using the elementary theory proposed by J. H. Michell, and the extended elementary theory proposed by A. B. Basset.

A manifold which admits a reducible genus-2 Heegaard splitting is one of the 3-sphere, S2 × S1, lens spaces or their connected sums. For each of those splittings, the complex of Haken spheres is defined. When the manifold is the 3-sphere, S2 × S1 or a connected sum whose summands are lens spaces or S2 × S1, the combinatorial structure of the complex has been studied by several authors. In particular, it was shown that those complexes are all contractible. In this work, we study the remaining cases, that is, when the manifolds are lens spaces. We give a precise description of each of the complexes for the genus-2 Heegaard splittings of lens spaces. A remarkable fact is that the complexes for most lens spaces are not contractible and even not connected.

We often wish to test the difference of the means of two series of measures of a quantity for a systematic difference. A suitable test, in terms of the theory of probability, can be obtained as follows. We suppose that in each series the probability of error is normally distributed with unknown standard error. Then if we have only one observation in each series, the difference of the two could equally well be interpreted as entirely due to the random error in one series, the other being right, or entirely due to systematic difference, both observations being free from random error. This suggests that we should take as one of our fundamental quantities the expectation of the difference of a single measure in one series from one in the other; the expectation of the square of the difference, however, appears still better.

Introduction. Let Mn+1 denote a closed, orientable, combinatorial (n + 1)-manifold. Let Kn denote an n-dimensional subcomplex of Mn+1. If Mn+1 – Kn is connected, then by ‘cutting’ Mn+1 along Kn we define in a natural way a fundaments domain of a certain regular covering of Mn+1. In case Kn is a spine ((5)) of Mn+1 the any covering space of Mn+1 may be constructed. This construction is quite precis and is combinatorial in spirit.

Let L be a finite-dimensional Lie algebra and U(L) its universal envelope. It is known that U(L) is a Noetherian domain (see (5), theorem v. 3·4) and therefore U(L) has a field of fractions. (Throughout the paper we use the term ‘field’ in the sense of skew field.) We prove in this article the following theorem.

Das Schwarzsche Lemma der klassischen Funktionentheorie einer Veränderlichen kann als vollständige Beschreibung der inneren Abbildungen des Einheitskreises (mit Fixpunkt O) aufgefaßt werden. Ist nämlich f(z) eine in |z| ≤ 1 reguläre Funktion mit |f(z)| ≤ 1 und f(0) = 0, so muß |f′(0)| ≤ 1 sein, und es stellt diese Ungleichung bekanntlich den ersten Schritt zur Lösung des Koeffizientenproblems der im Einheitskreis gleichmäßig beschränkten Funktionen dar. Die Aussage

kann andererseits so gedeutet werden, daß jede innere Abbildung des Einheitskreises mit Fixpunkt O auch innere Abbildung jedes konzentrischen Kreises |z| ≤ |ρ| < 1 ist.

Let k be an algebraically closed field of ch(k) = 0 and G be a simple simply connected algebraic group G over k. By using results about cohomological invariants, we compute the coniveau spectral sequence for classifying spaces BG.

Let X be a real Banach space, X* its dual space and ω a measurable space. Let D be a subset of X, L: Ω × D → X* a random operator and η:Ω →X* a measurable mapping. The random equation corresponding to the double [L, η] asks for a measurable mapping ξ: Ω → D such that

Random equations with operators of monotone type have been studied recentely by Kannan and Salehi [7], Itoh [6] and Kravvarits [8].

By considering a tree-like decomposition of an arbitrary set we prove the existence of an associated series with the property that its partial sums approximate uniformly all elements in a relevant space of bounded functions. In a topological setting we show that these sums are dense in the space of continuous functions, hence in turn any Borel measurable function is the almost everywhere limit of an appropriate sequence of partial sums of the same series. The coefficients of the series may be chosen in c0, or in a weighted ℓp with 1 < p < ∞, but not in the corresponding weighted ℓ1.

We prove that the set of all endpoints of the Julia set of $f(z)=\exp\!(z)-1$ which escape to infinity under iteration of f is not homeomorphic to the rational Hilbert space $\mathfrak E$ . As a corollary, we show that the set of all points $z\in \mathbb C$ whose orbits either escape to $\infty$ or attract to 0 is path-connected. We extend these results to many other functions in the exponential family.

Clifford's well-known chain of theorems concerning circles in a plane may be stated as follows (the first two links being trivial):

(C2) Given any point A, and through it two circles C1 and C2, these meet again in a point A12, say.

(C3) Given three circles C1, C2, C3 through A, the three points A23, A13, An lie on a circle C123, say.

(C4) Given four circles C1,…,C4 through A, the four circles C234,…,C123 meet in a point A1234 (Wallace(1)).

(C5) Given five circles C1,…,C5 through A, the five points A2345,…,A1234 lie on a circle C12345 (Miquel (2)).

And so on (Clifford(3)).

Grace (4) showed that in three dimensions [3], there is no direct analogue to Clifford's chain, but in [4] he noted the following:

(G4) Given any point A in [4], and through it four hyperspheres S1,…, S4, these meet again in a point A1234, say.

(G5) Given five hyperspheres Sl,…,S5 through a point A, the five points A2345, A1234 lie on a hypersphere S12345, say.

(G6) Given six hyperspheres Sl,…, S7 through A, the six hyperspheres S23456,…, A12345 have in common a point A123456.

This leads naturally to the conjecture

(G7) Given seven hyperspheres Sl,…, S7 through A, the seven points A234567,…, A123456 lie on a hypersphere S1234567.

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

Let f be an entire function of the form

where ø is a function in L2(ℝ) with compact support. If f|ℝ is real-valued then, for obvious reasons, (a) the supporting interval for ø is symmetric with respect to the origin, and

Assuming that f has no zeros in {Im z > 0}, we prove that the converse is also true: (a) and (b) together imply that f|ℝ takes values in αℝ, where α is a fixed complex number.

The proof relies on a certain formula involving the Dirichlet integral, which may be interesting on its own.

Introduction. Hall ((3), (4)) introduced the concept of a Sylow system and its normalizer into the theory of finite soluble groups. In (4) he showed that system normalizers may be characterized as those subgroups D of G minimal subject to the existence of a chain of subgroups from D up to G in which each subgroup is maximal and non-normal in the next; he also showed that a system normalizer covers all the central chief factors and avoids all the eccentric chief factors of G (for definitions of covering and avoidance, and an account of their elementary properties), the reader is referred to Taunt ((5)). This note arises out of an investigation into the question to what extent this covering/ avoidance property characterizes system normalizers; it provides a partial answer by means of two elementary counter-examples given in section 3 which seem to indicate that the property ceases to characterize system normalizers as soon as the ‘non-commutativity’ of the group is increased beyond a certain threshold. For the sake of completeness we include proofs in Theorems 1 and 2 of generalizations of two known results communicated to me by Dr Taunt and which as far as we know have not been published elsewhere. Theorem 1 shows the covering/avoidance property to be characteristic for the class of soluble groups with self-normalizing system normalizers introduced by Carter in (1), while Theorem 2 shows the same is true for A -groups (soluble groups with Abelian Sylow subgroups investigated by Taunt in (5)).

It is pointed out that there is some confusion concerning the meaning of the sign of a mutual inductance. The question is examined from first principles and it is shown that since the sign of a mutual inductance depends upon conventions as to sign of current directions, these, as well as the sign of the mutual inductance, must be given in order to specify what is really required—the relation between the directions of winding of the primary and secondary coils. It is also shown that in many cases it is possible to replace mutual inductance by a quantity which is numerically equal to the mutual inductance but which is independent of current directions. The two methods applied to a Wheatstone network yield consistent results.

The spontaneous disappearance of an electron and proton in the form of radiation has often been suggested as a possible source of energy to maintain the emission of radiation from hot stars. It can be calculated that the energy of a single quantum formed in this way should be about 1000 million electron-volts. In order to account for the absorption of the ultra-penetrating radiation observed by Regener in deep lakes, and interpreted on the basis of the Klein-Nishina formula, a quantum of still higher energy is required. Sir James Jeans has recently calculated that the transformation of a helium atom into a single quantum would provide a radiation of about the required penetrating power.