1. Die unendliche Folge der Partialsummen {sn} der Fourierschen Reihe einer integrierbaren und nach 2π periodischen Funktion f(ξ) der reellen Variabeln ξ ist an einer Stelle ξ = x, wo die Funktion stetig ist, (C, 1) limitierbarzu f(x) = s als Grenzwert, d.h. es ist

Some years ago, in connexion with some unpublished work in the theory of queues, the question arose as to whether the characteristic function of a non-negative random variable could vanish identically in an interval. The purpose of this note is to show that such a thing is impossible.

The object of this paper is to establish an integral involving Fox's H-function and employ it to obtain an expansion formula for the H-function involving Bessel functions.

1. Introduction. The class of finite groups in which any two subgroups of the same order are isomorphic will be denoted by (C), and ‘G ∈ (C)’ will mean ‘G belongs to the class (C)’.

Let Mm be a compact connected PL-manifold, we say M is metastably connected if πi(M) = 0 for i ≤ [⅓m] + 1, and m ≥ 6. Our notation will follow (3).

We consider in this paper solutions of the equation

where the primes indicate differentiation with respect to s, and a, b, c are constants.

Let Λ denote the integral lattice in n-dimensional Euclidean space. A classical theorem of Minkowski's states that any bounded closed convex region K symmetrical in the origin 0 and with volume 2n contains a point of Λ other than 0. There will be a lattice point other than 0 in the interior of K except when K has certain forms, of which we will denote an arbitrary one by K*. An example of a K* is the cube |xi| ≤ 1 (i = 1, 2,..., n), and more generally a famous theorem of Hajós (3) states that if K* is a parallelepiped it is defined (except for integral unimodular transformations of the x's) by inequalities of the form |x1| ≤ 1, |a21x1 + x2| ≤ 1, …, |an1x1 + … + an, n-1xn-1+xn| ≤ 1.

The possibility, pointed out in a previous paper, of superlattice formation in an adsorbed layer when the adsorbed atoms tend to repulse each other is developed in detail. Both Bragg and Williams's approximation and Bethe's approximation are used, but restricted to superlattices of the type AB. In a range of θ, the fraction of surface covered, a superlattice is found to be possible. Bragg and Williams's approximation shows further that the state with the lowest free energy is the one with a superlattice when the latter is possible. Rough kinetic expressions are also given. The equations derived from the principle necessary to preserve equilibrium are found to reduce to those of detailed balancing, and they also agree with the formula obtained statistically. As expected, all the corresponding results obtained from the two methods become the same when the number of nearest neighbours of a site approaches infinity, provided that the product of this number and the interaction potential of two adsorbed atoms occupying two neighbouring sites remains finite.

The only significant result is the large value of the slope of isotherms (the rate of change of the pressure with respect to the fraction of adsorption) when there is a nearly complete superlattice.

This is the second part of a two-part paper in which a systematic method is advanced for treating separation of variables problems asymptotically as the frequency becomes large. The method assumes an integral representation in which the contour of integration is the real axis. The contour is then deformed in the neighbourhood of the real axis to derive rigorous asymptotic expansions of the field. In this paper the method is applied to scattering in homogeneous media by the circular and parabolic cylinders.

Blichfeldt (2) gave a method of proving the existence of lattice points in certain regions. His method depends on the following lemma (which he stated in a somewhat different form):

Lemma 1. Let K′ be a closed region of volume V. Let K be another region so related to K′ that the difference of any two points in K′ is a point in K. Let Λ be any lattice whose determinant is positive and does not exceed V.

Motivated by a question of Hirzebruch on the possible topological types of cusp cross-sections of Hilbert modular varieties, we give a necessary and sufficient condition for a manifold M to be diffeomorphic to a cusp cross-section of a Hilbert modular variety. Specialized to Hilbert modular surfaces, this proves that every Sol 3–manifold is diffeo morphic to a cusp cross-section of a (generalized) Hilbert modular surface. We also deduce an obstruction to geometric bounding in this setting. Consequently, there exist Sol 3–manifolds that cannot arise as a cusp cross-section of a 1–cusped nonsingular Hilbert modular surface.

Let be the Jacobi polynomial as defined by Szegö in (7) (see equation (4) below.) Carlitz in (2) presented among others the following formula

Although, as Carlitz claims, this formula may be derived directly from the definition of Jacobi polynomials, a probabilistic proof such as presented below may shed some new light on formula (1), as well as suggest probabilistic proofs for other similar formulas of Jacobi polynomials, e.g. those given by Manocha and Sharma in (4) and (5) and by Manocha in (3). In addition, it is quite possible that this method of proof will result in the derivation of some new formulas for Jacobi polynomials.

The solution for an infinite cylinder filled with viscous fluid, rotating about its axis under the action of a constant couple, has already been obtained by the present writer by the operational method of Heaviside and Bromwich, and previously by other means by T. H. Havelock.

This paper is concerned with the electric and magnetic field close to conductors which carry an alternating current. Hertz's famous solution of Maxwell's equations gave the field at a great distance from a magnetic or electric dipole whose moment was alternating harmonically: but from its very character it could not describe the field at points which were insufficiently remote from the origin for the source to appear as a dipole. The great development of radio communication has made important certain problems about the field in the vicinity of wires which are carrying an alternating current. Thus imagine that an aerial of some thousand feet in height and very many acres in extent has been erected. Before the designer of this vast and costly structure can calculate the rate of energy output, he must suppose himself so far removed that the aerial has dwindled to a mere speck in the aether. Such a process must seem both unsatisfactory and unsatisfying to those who take the responsibility of obtaining radio communication. It is already known that the field at any point can be expressed formally when the distribution of current and charge in the source is postulated: it is shown in this paper that the field at all points very near to the source can be expressed by simple formulae. Of course it is not an exact solution for the field due to a conductor of specified shape, because the postulated distribution of current and charge is incorrect.

The planes of a given S5 can be represented by the points of a locus V9 in space of nineteen dimensions. This locus is a double locus on a certain other manifold V14, and the tangent spaces of the V9 generate a W18. Segre has a memoir on the subject of these loci, in which he arrives at his results by a series of short steps, the argument being mainly geometrical.

In the Transactions of the American Math. Society, vol. I, 1900, and vol. IV, 1904, Mr E. Kasner treats exhaustively the (2, 2) double binary form and discusses the theory connected with double binary forms and multiple binary forms in general terms. He shows the relations between the systems of multiple binary forms with digredient variables and the forms with cogredient variables. Hitherto nothing seems to have been written on systems of triple binary forms (with regard to higher forms see a paper on the (1, 1, 1, 1) form by C. Segre); so here I propose to discuss the complete system of a (1, 1, 1) binary form which consists of six forms connected by one syzygy. When two of the variables are the same we naturally get the (2, 1) form and when the three are the same We get the (3) or cubic binary form.

1. Introduction. The aim of this paper is to show that, in every complex Banach algebra with a one-sided or two-sided bounded approximate identity, there exists another bounded approximate identity of the same sort whose spectra lie close to the unit interval [0, 1].

The solution of linear differential equations by the method of Frobenius is a straightforward matter consisting merely in the substitution of a power series and obtaining the coefficients from a recurrence relation. The corresponding method when applied to difference equations is laborious and leads in general to divergent series from which convergent solutions can be obtained by a method due to Birkhoff(1). Actually a more appropriate method is to find a factorial series, but even here the direct substitution of such a series requires repeated transformations and throws little light on the structure of the equation. A method of symbolic operators was originated by Boole(2), but owing to the restricted definition of the operators it has a very limited scope. In the present paper a more general interpretation is given to the operators, and their application to a certain class of linear difference equations with rational coefficients is discussed.

Let T be a linear operator on a complex normed space X. Its spatial numerical range V(T) is denned as

The object of this paper is to show that some important classes of stochastic processes can be proved to be stochastic integrators, in the sense of Bichteler[1], by entirely elementary methods. Let us recall what this means.