Polynomial solutions of a few binomial congruences have been known for a long time. For instance Legendre showed that the congruence has a solution this being the expansion of as far as the term of degree m — 3. [1] It seems that only restricted types, e.g. (1), have been investigated.

Recently K. Mahler [1] introduced a set of φ(2n) matrices of n rows and columns which form under multiplication the abelian group of the residue classes prime to 2n modulo 2n. These remarkable matrices whose elements 0, 1 and −1, have latent roots and determinants which can be given explicitly. Thus we have new examples of matrices with given elements whose powers, roots, inverses and determinants can be written down precisely. Such matrices are often useful in testing the efficacy of methods for finding these functions for a general matrix.

In a recent paper Salem and Zygmund [1] proved the following result: Put and denote the φv(t) Rademacher function. Denote by Ln(t,θ) the unique trigonometric polynominal (in θ) of degree not exceeding n

By using certain fractional integrals and derivatives it is possible to construct a continuum of Hilbert spaces within the space L2 (0, ∞); these are the spaces gλ of functions f(x) for which 1xλf(λ)(x) є L2(0, ∞), and they exhibit invariance properties under generalized Fourier transformations. They are described in (6) and (7).

We consider real-valued functions ƒ(x) which are defined for all sufficiently large real numbers x. In discussing the behaviour of such functions as x → + ∞, it is useful to compare ƒ with the functions of some “comparison scale”. The early work in this field was due to Du Bois-Reymond (see, for example, (2), (3)). This was elaborated by Hardy, (6), who was mainly concerned with what he calls the “logarithmico-exponential” scale of functions. This “scale of Hardy” may be defined as the smallest class ℋ of functions ƒ with the following properties:

(i) if ƒ ∈ ℋ, ƒ, is defined and continuous for all sufficiently large values of x;

(ii) (a) the function ƒ(x) ≡ α, where α is any real constant, is in ℋ;

(b) the function ƒ(x) ≡ x is in ℋ;

(iii) if ƒ and g are in ℋ and g is non-zero for all sufficiently large values of x, then

are also in ℋ.

In this note the theory of the integral equation is reduced to elementary matrix algebra. The Fredholm theorems are proved without the introduction of the usual parameter λ and without using the properties of the resolvent considered as an analytic function of the complex variable λ. There is an exact correspondence between the properties of the equation (1) and the properties of a finite linear system. Here the proofs of these properties are also in extact correspondence with the same for the elementary case.

A positive quadratic form , of determinant and minimum M for integral , is said to be extreme if the ratio is a (local) maximum for small variations in the coefficients .

Minkowski [3] has given a criterion for extreme forms in terms of a fundamental region (polyhedral cone) in the coefficient space. This criterion, however, involves a complete knowledge of the edges of the region and is therefore of only theoretical value.

This paper investigates the existence and equality of the double and repeated integrals of a real function on a plane set. The main result (Theorem 2) is that if a function on a plane Lebesgue measurable set is continuous in one variable and measurable in the other then it is measurable in the plane.

In [2] we studied parametric n-surfaces (f, Mn), where Mn was a compact, oriented, topological n-manifold and f a continuous mapping of Mn into the real euclidean k-space Rn (k≧n). A definition of bounded variation was given and, for each surface with bounded variation and each projection P from Rk to Rn, a signed measure: Was constructed. This measure was used to define a linear type of surface integral: over a “measurable” subset A of Mn, as the Lebesgue-Stieltjes integral: .

We deal with questions about the possible embeddings of two given groups A and B in a group P such that the intersection of A and B is a given subgroup H. The data, consisting of the “constituents” A and B with the “amalgamated” subgroup H, form an amalgam.1 According to a classical theorem of Otto Schreier [5], every amalgam of two groups can be embedded in a group F, the “free product of A and B with amalgamated subgroup H” or the “generalized free product” of the amalgam. This has the property that every group P in which the amalgam is embedded and which is generated by the amalgam, is a homomorphic image of it. Hence theorems on the existence of certain embedding groups P can be interpreted also as theorems on the existence of certain normal subgroups of F.

Ein Teil von Philip Halls bekannter Theorie der auflösbaren Gruppen läßt sich, wie im folgenden gezeigt wird, auf beliebige endiliche Gruppen G ausdehnen. Wir fragen: Was kann man über die Normalstruktur (Hauptreihen, Kompositionsfaktoren) von G aussagen, wenn man mehrere Zer legungen von G in Produkte von Untergruppen mit gegebener Normalstruktur kennt? Eine Antwort gibt der folgende Satz, das Hauptergebnis dieser Note:

Let ρ and d be two positive numbers. We shall be concerned in this paper with two classes of entire functions, namely: (i) the class C (ρ, d) of all entire functions of order ρ and type not exceeding d — we note that this class includes all entire functions of order less than ρ;

The ultraspherical polynomial (x) of degree n and order λ is defined by for n = 0, 1, 2, …. Of these polynomials, the most commonly used are the Chebyshev polynomials Tn(x) of the first kind, corresponding to λ = 0; the Legendre polynomials Pn(x) for which λ = ½; and the Chebyshev polynomials Un(x) of the second kind (λ = 1). In the first case the standardisation is different from that given in equation (1), since.

The most elementary problem of the calculus of variations consists in finding a single-valued function y(x), defined over an interval [a, b] and taking given values at the end points, such that the integral is stationary relative to all small weak variations of the function y(x) consistent with the boundary conditions. Since y′ occurs in the integrand, it is clear that I is only defined when y(x) is differentiable and accordingly when y(x) is continuous. Usually y′(x) is also continuous. Occasionally, however, the boundary conditions can only be satisfied and a stationary value of I found, by permitting y′ (x) to be discontinuous at a finite number of points. The arc y = y(x) will then possess ‘corners’ and the well-known Weierstrass-Erdmann corner conditions [1]must be satisfied at all such points by any function y (x) for which I is stationary. Arcs y = y (x) for which y′(x) is continuous except at a finite number of points, are referred to as admissible arcs. In this paper, we shall extend the range of admissible arcs to include those for which y(x) is discontinuous at a finite number of points.

Two sets of r + 2 points, Pi, P'i, each spanning a projective space of r + 1 dimensions, [r + 1], which has no solid ([3]) common with that spanned by the other, are said to be projective from an [r — 1], if here is an [r — 1] which meets the r + 2 joins Pi ′i. It is to be proved that the two sets are projective, if and only if the r + 2 intersections Ai of their corresponding [r]s lie in a line a. Ai are said to be the arguesian points and a the arguesian line of the sets. When r= 1, the proposition becomes the well- known Desargues' two-triangle theorem (3) in a plane. Therefore in analogy with the same we name it as the Desargues' theorem in [2r]. Following Baker (1, pp. 8—39), we may prove this theorem in the same synthetic style by making use of the axioms and the corresponding proposition of incidence in [2r + 1] or with the aid of the Desargues' theorem in a plane and the axioms of [2r] only. But the use of symbols makes its proof more concise; the algebraic approach adopted here is due to the referee (Arts. 2, 3, 5, 6, 7). Pairs of sets of r + p points each projective from an [r— 1] are also introduced to serve as a basis for a much more thorough investigation.

We will consider a reflexive Banach space 𝔅, with real or complex scalars, and a bounded operator in 𝔅 with a real spectrum.

As in [5] a parametric n-surface in Rk (where k ≧ n) will be a pair (f, Mn), consisting of a continuous mapping f of an oriented topological manifold Mn into the euclidean k-space Rk. (f, Mn) is said to be closed if Mn is compact. The main purpose of this paper is to use the method of [4] to prove a general form of Cauchy's Integral Theorem (Theorem 5.3) for those closed parametric n-surfaces (f, Mn) in Rn+1, which have bounded variation in the sense of [5] and for which f(Mn) has a finite Hausdorff n-measure. As in [4], the proof is carried out by approximating the surface with a simpler type of surface. However, when n > 1, a difficulty arises in that there are entities, which occur in a natural way, but are not parametric surfaces. We therefore introduce a concept which we call an S-system and which forms a generalisation (see 2.2) of the type of closed parametric n-surface that was studied in [5] II, 3 in connection with a proof of a Gauss-Green Theorem. The surfaces of [5] II, 3 include those that are studied in this paper.

The main result obtained in this paper is Theorem 1. The symplectic group on the skew matrix T of 2m rows and columns over GF (2)** can be generated by the two matrices Q, R, whereTi,j being the substitution matrix which interchanges the elements numbered i and j,.

Uniform asymptotic formulae are obtained for solutions of the differential equation for large positive values of the parameter u. Here p is a positive integer, θ an arbitrary parameter and z a complex variable whose domain of variation may be unbounded. The function ƒ (u, θ, z) is a regular function of ζ having an asymptotic expansion of the form for large u.

The results obtained include and extend those of earlier writers which are applicable to this equation.