The synthesis of 1, 2, 3, 4-tetrahydro-1, 6-naphthyridine and 1, 6-naphthyridine is described and the ultra-violet spectra of these and related substances are discussed.

Certain families of measures on coset-spaces, namely inherited, stable, and pseudo-invariant measures, were defined, and shown to exist, in earlier papers, where Jacobians and factor functions, generalizing the idea of Jacobians in theory of functions of several variables, were also denned. In this paper, the existence is established of exact Jacobians and factor functions, which satisfy certain characteristic identities exactly, without an exceptional set of measure zero. A study is made of how properties of a measure are reflected by properties of the Jacobian or the factor function. Necessary and sufficient conditions are found for a function to be an exact Jacobian for some measure.

In this paper we shall be concerned with the derivation of simple expressions for the sums of some infinite series involving the zeros of Bessel functions of the first kind. For instance, if we denote by γv, n (n = l, 2, 3,…) the positive zeros of Jv(z), then, in certain physical applications, we are interested in finding the values of the sums

and

where m is a positive integer. In § 4 of this paper we shall derive a simple recurrence relation for S2m,v which enables the value of any sum to be calculated as a rational function of the order vof the Bessel function. Similar results are given in § 5 for the sum T2m,v.

The preparation of substituted fluoranthenes by the aromatization of hydrogenated fluoranthene derivatives is described. 8-Nitrofluoranthene, one of the nitration products of fluoranthene, has been synthesized from 2-nitrofluorene and the preparation and properties of some 2-nitrofluorene derivatives of potential synthetic value are reported.

Hurwitz's theorem says that the limit of schlicht functions, in the topology of compact convergence, is again a schlicht function or a constant function. We generalize this to mappings between Riemann surfaces and get more precise information on the relation between the distribution of values of analytic functions and the topology of compact convergence on the space of all analytic maps.

Let G by any given group. A homomorphic mapping μ of a subgroup A of G onto a second subgroup B of G, where A and B need not be distinct, is called a partial endomorphism of G. When μ is defined on the whole of G, that is when A = G, we call μ a total endomorphism of G; or simply an endomorphism of G.

A partial (or total) endomorphism μ* of a supergroup G* of G is said to extend (or continue) μ if μ* is defined on a supergroup A* of A, that is, μ* is defined for at least the elements for which μ. is defined, and moreover μ* coincides with μ on A.

Certain types of rings of infinite matrices are defined and some of their properties are discussed. The main theorems are concerned with the connections between the radical of a ring R and the radicals of rings of infinite matrices over R.

A study of the γ-rays produced during the bombardment of a thick Be9 target by 600 keV deuterons was made to investigate the possible existence of a level at 2·86 MeV in B10, about which contradictory reports have appeared in the literature.

A spectrum of the γ-rays in coincidence with the 0·72 MeV B10 γ-ray (text-fig. 5) was obtained, and is interpreted as providing evidence for a level in B10 at 2·86 MeV. The relative intensities of the γ-rays in an ungated spectrum, and in spectra gated by the 0·72 and 1·02 MeV B10 γ-rays, were found, and a decay scheme consistent with the observations is deduced (text-fig. 6b). The relative intensities of the transitions in this decay scheme are consistent with the intensities of the neutron groups in a spectrum of the neutrons from this reaction. A spin value of 2 or 3 is suggested for the 2.86 MeV level.

In two recent papers [1, 2] the Barnes integral for the E-functions was employed to sum a number of infinite series of E-functions. In §2 of this paper, by making use of the multiplication formula for the gamma function, the method is extended to series of E-functions of a different type.

Suppose throughout that λ > 0, κ > - l. γ is real and that

The series is said to be

(i) summable (C, k) to s if

(ii) strongly summable (C, k + 1) with index λ, or summable |C, k + 1|λ, to s if

(iii) absolutely summable (C, k) with indices γ, λ, or summable |C, k + 1|λ, if

The Laplace transform

has been generalised by Varma [4] by the relation

which reduces to (1.1) when k = -m + ½ by virtue of the identity

We shall define πk, m, λ (p) by the relation

The object of this paper is to obtain some recurrence formulae and series for πk, m, λ (p) and to use them to obtain recurrence formulae and series for MacRobert's E-function.

Let L = L( +, v, ＾) be a lattice-ordered group, or l-group (Birkhoff [1, p. 214]). Two elements a and b of L will be called disjoint if a > 0, b > 0, and a ＾; b = 0. It is easily seen that if L does not contain two disjoint elements, then it is linearly ordered (and, of course, conversely). What can we say about Z-groups containing two but not more than two mutually disjoint elements?

Let Aand B be linearly ordered groups (o-groups), and let A ⋏ B be the cardinal sum of A and B. That is, A ⋏ B is the direct sum of A and B, and (a, b) is positive in A + B if and only if a is positive in A and b is positive in B. An l-group L containing A ⋏ B as a convex normal subgroup (or Z-ideal) is called a lexico-extension of A ⋏ B if every positive element of L not in A ⋏ B exceeds every element of A ⋏ B. It then follows (subsection 2.9 below) that L/(A ⋏ B) is an o-group. Such an l-group L is easily seen to satisfy the following condition: (D)

There exists a pair of disjoint elements in L, but no triple of pairwise disjoint elements exists in L.

When the theory of Hankel transforms is applied to the solution of certain mixed boundary value problems in mathematical physics, the problems are reduced to the solution of dual integral equations of the type

where α and ν are prescribed constants and f(ρ) is a prescribed function of ρ [1]. The formal solution of these equations was first derived by Titchmarsh [2]. The method employed by Titchmarsh in deriving the solution in the general case is difficult, involving the theory of Mellin transforms and what is essentially a Wiener-Hopf procedure. In lecturing to students on this subject one often feels the need for an elementary solution of these equations, especially in the cases α = ± 1, ν = 0. That such an elementary solution exists is suggested by Copson's solution [3] of the problem of the electrified disc which corresponds to the case α = –l, ν = 0. A systematic use of a procedure similar to Copson's has in fact been made by Noble [4] to find the solution of a pair of general dual integral equations, but again the analysis is involved and long. The object of the present note is to give a simple solution of the pairs of equations which arise most frequently in physical applications. The method of solution was suggested by a procedure used by Lebedev and Uflyand [5] in the solution of a much more general problem.

Investigations based on gas masses, bright star counts, and luminosity-mass ratios of galaxies lead to one of two conclusions. If the galaxies are all of the same age, the faint ends of the initial luminosity functions of stars at formation differ greatly from one galaxy to another. On the other hand consistent results in the analysis are obtained with luminosity functions that are more nearly constant and ages which range from one to thirty thousand-million years. The various possibilities can be tested by observations on the Magellanic Clouds.

Equations are set up which describe, as functions of time, the integrated properties of a galaxy as a system of stars and gas.

1. All operators considered in this paper are bounded operators on a Hilbert space. In case A and B are self-adjoint, certain conditions on A, B and their difference

assuring the unitary equivalence of Aand B,

have recently been obtained by Rosenblum [6] and Kato [2]. The present paper will consider the problem of investigating consequences of an assumed relation of type (2) for some unitary U together with an additional hypothesis that the difference H of (1) be non-negative, so that

First, it is easy to see that if only (2) and (3) are assumed, thereby allowing H = 0, relation (2) can hold for A arbitrary with U = I (identity) and B = A. If H = 0 in (3) is not allowed, however (an impossible assumption in the finite dimensional case, incidentally, since then the trace of H is zero and hence H = 0), it will be shown, among other things, that any unitary operator U for which (2) and (3) hold must have a spectrum with a positive measure (as a consequence of (i) of Theorem 2 below). Moreover A (hence B) cannot differ from a completely continuous operator by a constant multiple of the identity (Theorem 1). In case 0 is not in the point spectrum of H, then U is even absolutely continuous (see (iv) of Theorem 2). In § 4, applications to semi-normal operators will be given.

The probability that each of n equally correlated normal random variables shall not fall short of a given value h is obtained as the product of the joint density function of the variables at the cut-off point (h, h,…, h) and an infinite power series in h. The coefficients in the latter series may be interpreted geometrically as the moments of a regular (n – l)-dimensional spherical simplex with common dihedral angles arc cos –p relative to a certain plane of symmetry. These moments may in their turn be expressed as linear functions of the measures of regular hyperspherical simplices of various dimensionalities, tables of which are available elsewhere.

In the preceding paper of the same title (cf. [1]) I defined the notion of the principal genus GK of a finite number field K as the least ideal group, which contains the group IK of totally positive principal ideals and is characterized rationally. The quotient group of the group AK of ideals in K modulo GK is the genus group, its order (Ak: GK) = gK is the genus number, which is thus a factor of the class number hK (in the narrow sense). Associated with the genus group is the genus-field, of K, which is defined as the maximal non-ramified extension of K composed of K and of some absolutely Abelian field.