Using a criterion due to J. Tits [4] we shall show that it is easy to give a pair of conjugate matrices in GLn(ℂ) which freely generate a free group of rank two. The difficulty lies in producing two such matrices whose freeness does not follow directly from the known freeness of two 2 × 2 matrices. Finally we show that the well known problem as to whether the Burau representation of Artin's braid group B4 is faithful turns out to be arbitrarily close to being true in a large number of ways. Thanks are due to the referee for his helpful comments.

Solutions for the temperature distribution in a circular pipe have been given by various authors, notably Gretz, Nusselt, Goldstein; all these are cited in (1), section 266. I have already considered in one paper the temperature distribution in a circular pipe when viscous incompressible fluid is flowing through the pipe and rate of heat addition is an oscillatory function of time. In this paper expressions for the temperature distributions are derived when viscous incompressible fluid is flowing through the channel, the dissipation due to friction is neglected and the rate of heat addition is an exponential function of time. From the engineering point of view this situation has some importance.

Let R be a Dedekind domain with field of quotients K. Let A be a finite-dimensional K-algebra. We consider isomorphism classes and genera in a category whose objects are indexed sets of full R-lattices in some ambient A-module and whose morphisms are the A-homomorphisms of the ambient A-modules which map each lattice into its corresponding lattice. We find conditions under which the stable A-isomorphism class of one particular lattice in an indexed set will determine the stable class of the indexed set within its genus. We apply our methods to show that if L/K is a tame Galois extension of algebraic number fields then the stable isomorphism class of the set of ambiguous ideals in L considered as Galois modules over K is determined by the class of the ring of integers in L together with the inertia subgroups and their standard representations over the respective residue fields of R.

Assume that we are given a semifree noncommutative differential graded algebra (DGA for short) whose differential respects an action filtration. We show that the canonical unital algebra map from the homology of the DGA to its characteristic algebra, i.e. the quotient of the underlying algebra by the two-sided ideal generated by the boundaries, is a monomorphism. The main tool that we use is the weak division algorithm in free noncommutative algebras due to P. Cohn.

In this paper we consider the properties of a point-curve correspondence between two (generally distinct) surfaces F and G, paying special attention to the idea of ‘induced’ and ‘extended’ correspondences. We also outline the theory of point-curve corre spondences between a curve and a surface, which is not only of interest in itself, but is of importance because such correspondences arise, as do correspondences between curves, as induced correspondences of point-curve correspondences between two surfaces. The results depend chiefly on Lefschetz's theory of correspondences between algebraic curves(1) and the properties of intersections of cycles on an algebraic surface given by the author (2), as well as on the preceding paper of this series (3). Use is also made of a paper by Severi (4) on the self-correspondences on a variable curve of an algebraic surface. The theory of correspondences on a single surface will be further developed in the next paper on this subject.

We consider a 2-dimensional square lattice which is partitioned into a periodic array of rectangular cells, on which a nearest neighbour random walk with symmetric increments is defined whose transition probabilities only depend on the relative position within a cell. On the basis of a determinantal identity proved in this paper, we obtain a result for finite Markov chains which shows that the diffusion constants for the random walk are monotonic functions of the individual transition probabilities. We point out the similarity of this monotonicity property to Rayleigh's Monotonicity Law for electric networks or, equivalently, reversible random walks.

Let f(z)=u(x, y)+iv(x, y) be regular in a domain containing the unit circle C. If where r < 1, then is regular in |z| ≤ 1 and we have and taking real parts and imaginary parts we obtain Poisson's formulae for the harmonic functions u and v.

In [4] I extended the Plücker formulae to curves in Pn for any n, under fairly strong genericity hypotheses. In this paper we seek to weaken these hypotheses using the theory of excess intersections. The method applies to just one of the formulae (so the results are complete only for n=3), and for this we obtain a number which is a sum of local contributions, given by an explicit rule. We evaluate these contributions in the cases when they are determined by the terms of least order in the respective Taylor expansions, thus obtaining explicit formulae in numerous cases.

In a recent publication the problem of determining the abundance of two interacting species of animals was investigated, and the fundamental equations were established.

Introduction. Agarwalin 1953 ((1)) gave the most general transformations involving bilateral cognate trigonometrical series of the hypergeometric type. Later, Shukla in 1957 ((2)) gave certain relations involving the products of two bilateral hypergeometric series. In this paper an attempt has been made to establish certain new transformations involving the products of two and more bilateral hypergeometric type of series. (For notations and definitions see Agarwal ((1)).)

The problem of completing the mathematical structures abstracted from the time ordering of the events in the space-times of general relativity is motivated and solved. The new construction is compared with that of Geroch, Kronheimer and Penrose (5).

Suppose that (ni) = (n1, n2,…) and (mi) are infinite sequences of positive integers with ni < mi < ni+1. It is well-known and easily proved that, if a series σak is (C, 1) summable to s and has lacunae (ni, mi) such that ak = 0 (ni < k < mi) with

then where

The paper contains some ionization-height curves for an isothermal ionized atmosphere in which the mechanism of electron disappearance is attachment to oxygen atoms at a rate proportional to the product of the number of oxygen atoms and of electrons present. Chapman's formula for the rate of production of ions is used. The curves were obtained by means of a differential analyser.

In [6], p. 80, Serre showed that the group SL2(Z[1/p]) is a free product of two copies of SL2(Z) amalgamated along a sub-group of index p + 1 in both components. This result was obtained from the structure theory of groups acting on trees and the construction of a tree on which SL2 over a field with discrete valuation acts in a natural way. This result can be extended to other SL2 examples. Using the above description, Serre [6] also gave presentations for the groups SL2(Z[1/p]) where p = 2, 3.

In this paper we formalize the two-valued propositional calculus with implication and the logical constant 0 (denoting a false statement), as primitives. We can define the logical constant 1 (denoting a true statement), by

and we can define negation by NP = df. CP0.

1. In his book, A Theory of Time and Space, the writer showed that, contrary to the generally accepted view, the ideas of Congruence can be built up from purely ordinal considerations, provided that we do not confine ourselves to linear order, but admit what, for convenience, he has called Conical Order. This Conical Order is built up from the purely abstract asymmetrical relations of before and after and does not pre-suppose the existence of the cones which are used to illustrate it.

Let R be a commutative ring which may not contain a multiplicative identity. A set of elements a1,…,ak in R will be called an H1-set (this notation is explained in section 1) if for each relation r1a1 + … +rkak = 0 (ri ∈ R) there exist elements sij ∈ R such that

where Xl,…,Xk are indeterminates. Any R-sequence is an H1-set, but there do exist H1-sets which are not R-sequences (see section 1). Throughout this note we consider an H1-set a1,…,ak which we suppose to be partitioned into two non-empty sets bl…, br and cl,…, cs. Our main purpose is to show that the ideals B = Rb1 + … + Rbr and C = Rc1 + … + Rcs satisfy Bm ∩ Cn = BmCn for all positive integers m and n (Corollary 1). This generalizes Lemma 2 of Caruth(2) where the result is proved when a1,…, ak is a permutable R-sequence. Our proof involves more detail than is necessary just for this, and we obtain various other properties of H1-sets. In particular we extend the main results of Corsini(3) concerning the symmetric and Rees algebras of a power of the ideal Ra1 +… + Rak (Corollary 3).

We construct stably nontrivial vector bundles of dimension 5 and also stably nontrivial vector bundles of dimension 6 on real projective spaces of arbitrarily high dimensions. Consequently, we answer a problem posed by J. F. Adams.

In this paper, we study “the ring of component functions” of SL(2, C)-representations of free abelian groups. This is a subsequent research of our previous work [11] for free groups. We introduce some descending filtration of the ring, and determine the structure of its graded quotients.

Then we give two applications. In [30], we constructed the generalized Johnson homomorphisms. We give an upper bound on their images with the graded quotients. The other application is to construct a certain crossed homomorphisms of the automorphism groups of free groups. We show that our crossed homomorphism induces Morita's 1-cocycle defined in [22]. In other words, we give another construction of Morita's 1-cocyle with the SL(2, C)-representations of the free abelian group.

Let G be a locally compact Abelian group; Γ the dual group of G; C0(Γ) the algebra of continuous functions on Γ which vanish at infinity; CB(Γ) the continuous, bounded functions on Γ; M (G) the algebra of bounded Borel measures on G; L1(G) the algebra of absolutely continuous measures; and M(G)∩ the algebra of Fourier–Stieltjes transforms.