The treatment of the Fourier Series, that is, of the series which proceeds according to sines and cosines of multiples of the variable, is in most English text-books very unsatisfactory; in many cases it shows almost no advance upon that of Poisson and, even where a more or less accurate reproduction of Dirichlet's investigations is given, there is no attempt at indicating the advantages it possesses over the so-called proof of Poisson. Nor is the uniformity of the convergence of the series so much as mentioned, not to say discussed. I have therefore thought it might be useful to give a fairly complete outline of the historical development of the series so far as the materials at my disposal allow. I do not think that any important contribution to the theory is omitted, but, as I indicate at one or two places, there are some memoirs to which I have not had access and which I only know at second hand.

If we have given two equations φ(x) = 0 and ψ(y) = 0, it is possible to express in the form of a determinant the equation whose roots are f(x, y), where f is any given rational integral function.

A ring R is called a right QI-ring if every quasi-injective right R-module is injective. The well-known Boyle's Conjecture states that any right QI-ring is right hereditary. In this paper we show that if every continuous right module over a ring R is injective, then R is semisimple artinian. In fact, if every singular continuous right R-module satisfying the restricted semisimple condition is injective, then R is right hereditary. Moreover, in this case, every singular right R-module is injective.

In the last six lines of Turnbull's 1948 paper, he left an enigmatic statement on a Capelli-type identity for skew-symmetric matrix spaces. In the present paper, on Turnbull's suggestion, we show that certain Capelli-type identities hold for this case. Our formulae connect explicitly the central elements in U(gln) to the invariant differential operators, both of which are expressed with permanent. This also clarifies the meaning of Turnbull's statement from the Lie-theoretic point of view.

An affine connection in an n-dimensional manifold Xn defines a system of paths, but conversely a connection is not defined uniquely by a system of paths. It was shown by H. Weyl that any two affine connections whose components are related by an equation of the form

where is the unit affinor, give the same system of paths. In the geometry of a system of paths, a particular parameter on the paths, called the projective normal parameter, plays an important part. This parameter, which is invariant under a transformation of connection (1), was introduced by J. H. .C. Whitehead. It can be defined by means of a Schwarzian differential equation and it is determined up to linear fractional transformations. In § 1 this method is briefly discussed.

Foguel (8) and Fixman (7) independently proved that an invertible spectral operator, which is power-bounded, is of scalar type. Their proofs rely heavily on a result of Dunford on spectral operators whose resolvents satisfy a growth condition. (See Lemma 3.16 of (6, p. 609).) Observe that the resolvent of an invertible power-bounded operator T satisfies an inequality of the form

For distributive double p-algebras, a close connection is established between being congruence coherent and congruence regular. Every congruence regular distributive double p-algebra is congruence coherent. Even though every congruence coherent distributive double p-algebra that has either a non-empty core or finite range is congruence regular, an example is given that is congruence coherent but not congruence regular.

In a previous paper it is supposed that if A is a Bernstein algebra, every maximal subalgebra, M, verifies that dim M = dim A − 1. This is not true in general. Therefore Proposition 2 in this paper is not correct. However other results there, where this assertion was used, are correct but their proofs need some modifications now.

This is a note on the theory of continued fractions, in which the chief feature is the use made of the successive remainders or divisors which occur in the reduction of any given ratio to a continued fraction.

The treatment of the Pellian equation also differs from that which is generally given.

It is well known that the canonical system of curves on an algebraic surface is only relatively invariant under birational transformations of the surface. That is, if we have a birational transformation T between two surfaces F and F′, and if K and K′ denote curves of the unreduced canonical systems on F and F′, then

where E and E′ denote the sets of curves, on F and F′ respectively which are transformed into the neighbourhoods of simple points on the other surface.

Let S be a nontrivial monoid with zero and let F be a field. A sufficient condition, on the 0-simple principal factors of S and the characteristic of F, is given for the contracted monoid algebra of S over F to be directly finite.

1. V. Bernstein, N. Levinson, R. P. Boas, Jr., and others have investigated under what conditions on the sequence

for all functions f(z) regular and of suitably restricted growth in the halfplane x≥0. (For references see [1].)

In this note the restrictions on f(z) are that f(z) is regular, and not identically zero in continuous in and that for some positive B

If G is a hyperfinite locally soluble group and A an artinian ZG-module then Zaĭcev proved that A has an f-decomposition. For G being a hyper-(cyclic or finite) locally soluble group, Z. Y. Duan has shown that any periodic artinian ZG-module A has an f-decomposition. Here we prove that: if G is a hyper-(cyclic or finite) group, then any artinian ZG-module A has an f-decomposition.

We obtain a formula relating the Conway potential functions of links in S3 which are connected by a framed surgery operation. Using this formula we extend the theory of Conway potential functions to links in all oriented ℚ-homology 3-spheres.