The aim of this paper is to study geometric properties of non-degenerate smooth projective varieties of small degree from a birational point of view. First, using the positivity property of double point divisors and the adjunction mappings, we classify smooth projective varieties in $\mathbb{P}$r of degree d ⩽ r + 2, and consequently, we show that such varieties are simply connected and rationally connected except in a few cases. This is a generalisation of P. Ionescu's work. We also show the finite generation of Cox rings of smooth projective varieties in $\mathbb{P}$r of degree d ⩽ r with counterexamples for d = r + 1, r + 2. On the other hand, we prove that a non-uniruled smooth projective variety in $\mathbb{P}$r of dimension n and degree d ⩽ n(r − n) + 2 is Calabi–Yau, and give an example that shows this bound is also sharp.

The aim of this paper is to answer a question of Coates and Greenberg: let $F$ be a commutative $m$-dimensional formal group over the ring of integers of a local field $k$, and let $K$ be an algebraic extension of $k$ with infinite ramification index. Denote by ${\cal M}_{\Mbar}$ the maximal ideal in the ring of integers of the separable closure of $K$. Suppose that the height of $F$ is greater than $m$. Does $H^1 (K, F ({\cal M}{}^m_{\Mbar}))=0$ imply that $K$ is deeply ramified? The answer is positive.

In some recent papers, the authors considered regular continued fractions of the form \[ \bigg[a_{0};\underbrace{a,\ldots, a}_{m}, \underbrace{a^{2},\ldots, a^{2}}_{m}, \underbrace{a^{3},\ldots, a^{3}}_{m}, \ldots \bigg], \] where $a_{0} \,{\geq}\, 0, a \,{\geq}\, 2$ and $m \,{\geq}\, 1$ are integers. The limits of such continued fractions, for general $a$ and in the cases $m\,{=}\,1$ and $m\,{=}\,2$, were given as ratios of certain infinite series.

However, these formulae can be derived from known facts about two continued fractions of Ramanujan. Motivated by these observations, we give alternative proofs of the results of the previous authors for the cases $m\,{=}\,1$ and $m\,{=}\,2$ and also use known results about other $q$-continued fractions investigated by Ramanujan to derive the limits of other infinite families of regular continued fractions.

We consider the problem of finding the maximum possible size of a family of d-dimensional subcubes of the n-cube {0, 1}n, none of which is contained in the union of the others. (We call such a family ‘irredundant’). Aharoni and Holzman [1] conjectured that for d > n/2, the answer is (which is attained by the family of all d-subcubes containing a fixed point). We prove this in the case where all the subcubes go through either (0, 0, . . . , 0) or (1, 1, . . . , 1). We also give a new proof of an upper bound of Meshulam [6]. Finally, we consider the case d ≤ n/2. We give a probabilistic construction showing that in this case, Meshulam's upper bound is correct to within a constant factor. When n = 2d, we construct an irredundant family of size .

The purpose of this note is to call attention to an error in the paper [1]. The main result was a new proof of the main theorem in [2]. In line 1 of page 446 of [1] one finds the following remark: “This is the result obtained in [2]”. However, a comparison of the two results shows that the generator which is called $d_1 d^{-1}_2$ in the notation of [1] appears in [2] but is missing from the list in [1].

We show that under quite general circumstances a Banach algebra R need not be closed in its tilda algebra . In answer to a query of Katznelson we show that there is a very strongly homogeneous algebra B ≠ C0(R) on R with R as carrier space such that non-analytic functions operate on B.

1. It is trivial that

for all integers n ≥ 0 implies sn = 0 (n = 0, 1, 2, …). This conclusion is no longer true if it is only known that (1·1) is true for an infinity of n. But we shall show that the truth of (1·1) for, roughly speaking, one-half of all positive integers, together with an order condition on the magnitude of sn, ensures sn = 0 for all n. This follows from

Theorem 1. Let a1 < a2 < … be positive integers, n(R) the number of a's not exceeding R, sn a sequence of complex numbers. If

(ii) sn = ο(nK)as n tends to infinity

then sn = P(n), where P(x) is a polynomial of degree less than K.

Clifford's well-known chain of theorems concerning circles in a plane may be stated as follows (the first two links being trivial):

(C2) Given any point A, and through it two circles C1 and C2, these meet again in a point A12, say.

(C3) Given three circles C1, C2, C3 through A, the three points A23, A13, An lie on a circle C123, say.

(C4) Given four circles C1,…,C4 through A, the four circles C234,…,C123 meet in a point A1234 (Wallace(1)).

(C5) Given five circles C1,…,C5 through A, the five points A2345,…,A1234 lie on a circle C12345 (Miquel (2)).

And so on (Clifford(3)).

Grace (4) showed that in three dimensions [3], there is no direct analogue to Clifford's chain, but in [4] he noted the following:

(G4) Given any point A in [4], and through it four hyperspheres S1,…, S4, these meet again in a point A1234, say.

(G5) Given five hyperspheres Sl,…,S5 through a point A, the five points A2345, A1234 lie on a hypersphere S12345, say.

(G6) Given six hyperspheres Sl,…, S7 through A, the six hyperspheres S23456,…, A12345 have in common a point A123456.

This leads naturally to the conjecture

(G7) Given seven hyperspheres Sl,…, S7 through A, the seven points A234567,…, A123456 lie on a hypersphere S1234567.

Soit $f$ une fonction arithmétique. Posons $$m(n;f) \,{\coleq}\, \min_{d|n}\!{\Vert}f(d){\Vert}\qquad(n\,{\geq}\,1),$$ oú ${\Vert} u {\Vert}$ désigne la distance du nombre réel $u$ à l'ensemble des entiers. Le comportement normal de $m$($n$;$f$) est une mesure de la nature probabiliste de l'ensemble des diviseurs d'un entier [Lt ]statistique[Gt ]. Une hypothèse standard d'équirépartition conduit à l'évaluation $$m(n;f) = 1/\tau(n)^{1+o(1)}\quad {\rm pp}$$, oú $\tau$($n$) désigne le nombre total des diviseurs d'un entier naturel $n$, et oú, ici et dans la suite, nous utilisons la mention pp (presque partout) pour indiquer qu'une propriété relative à un entier générique est valable sur une suite de densité naturelle unité.

Theorem 1. Let M3be a compact 3-manifold and let M3 = E3 ∪ R be a standard decomposition where R is a subcomplex in a triangulation of M3((2)). If M3is a homology 3-sphere while R contains no arc that is wild in M3then M3 = S3

This paper presents the asymptotic expansion, with respect to the large real parameter λ, of the singular Fourier integral

formula here

where 0 < b [les ] + ∞, the symbol fp designates an integration in the finite part sense of Hadamard and complex function K(x, u) belongs to a specific set of pseudo-functions which may present a singular behaviour of logarithmic nature at the endpoints.

Little is known about definite integrals for the Whittaker functions. The object of this paper is to establish such results and the following formulae are established

where the symbol means that in the expression following it i is to be replaced by –i and the two expressions are to be added.

We study the pullback of the apolarity invariant of complex polynomials in one variable under a polynomial map on the complex plane. As a consequence, we obtain variations of the classical results of Grace and Walsh in which the unit disk, or a circular domain, is replaced by its image under the given polynomial map.

1. The determination of an explicit formula for the postulation of a multiple variety of given characters for primals of sufficiently large order is one which seems to have received attention only in special cases. The postulation of a multiple curve for surfaces in ordinary space is known*, and Roth† has obtained the postulation of a multiple surface in [4], free from singularities, for primals in that space. Apart from these results, and the trivial case of isolated multiple points, the only general results seem to be those of Torelli‡ and Giambelli§ for the case in which the multiple variety is a Vh which is the complete intersection of r − h primals in [r].

1. Many investigations have been made to determine the wave resistance acting on a body moving horizontally and uniformly in a heavy, perfect fluid. Lamb obtained a first approximation for the wave resistance on a long circular cylinder, and this was later confirmed to be quite sufficient over a large range. In 1926 and 1928, Havelock (4, 5) obtained a second approximation for the wave resistance and a first approximation for the vertical force or lift. Later, in 1936(6), he gave a complete analytical solution to this problem, in which the forces were expressed in the form of infinite series in powers of the ratio of the radius of the cylinder to the depth of the centre below the free surface of the fluid. General expressions for the wave resistance and lift of a cylinder of arbitrary cross-section were found by Kotchin (7) using integral equations, and the special case of a flat plate was evaluated. He continued with a discussion of the motion of a three-dimensional body. More recently, Haskind (3) has examined the same problem when the stream has a finite depth.

Many measurements have been made of the complete absorption curves for β-rays in various substances, and it has been found that, for homogeneous β-rays, the curve is not exponential, and, in particular, when aluminium is the absorber, the curve is linear over a considerable portion of its range (e.g. Schonland, etc.).

In a previous paper, the authors introduced a filtration on the ring ${\cal O}_{V,0}$ of germs of functions on a germ $(V,0)$ of a complex analytic variety defined by arcs on the singularity and called the arc filtration. The Poincaré series of this filtration were computed for simple surface singularities in the 3-space. Here they are computed for surface singularities from Arnold's lists including uni- and bimodular ones. The classification of the unimodular singularities by these Poincaré series turns out to be in accordance with their hierarchy defined by E. Brieskorn using the adjacency relations. We also give a general formula for the Poincaré series of the arc filtration for isolated surface singularities which are stabilizations of plane curve ones.

A square matrix A, of order n, having complex coefficients can be inverted without the aid of any operations involving complex numbers. This can be done if the coefficients ars + iαrs are replaced by their matrix equivalents and the resulting 2n × 2n real matrix A1 is inverted. The inverse will be a 2n × 2n matrix of similar form in which the subsidiary 2 × 2 matrices can be replaced by the equivalent complex numbers, thus yielding the inverse of A. It is the purpose of this note to show that a similar technique can be employed to evaluate det A.

The paper studies the motion of an electro-conductive non-viscous incompressible fluid in a generator of rectangular section having finite electrodes and the rest of its walls being insulating. It is assumed that in the electrodes region a homogeneous magnetic field is applied, whose orientation is normal to the plane of the motion. If the induced magnetic field is neglected, the current function ψ(x, y) and the electrical potential ϕ(x, y) satisfy a system of equations with partial derivatives of the second order, with distributions. Since the system depends upon the parameter λ = RM RH a method of successive approximations is developed, which permits one to obtain the exact solution of the problem. In the last paragraph the results are extended to the case of a generator with N pairs of symmetrical electrodes.

This article deals with the currents set up in thin-walled cylinders by two-dimensional field variation in the plane of the section, and is applied particularly to the problem of the distribution of alternating currents in strip conductors, a matter of some technical importance.