Let be the self-dual CCR algebra over a triplet (K, r, Γ), where K is assumed to be a Hilbert space, and let {τt}t ∈ R be a Bogoliubov automorphism group of induced by a strongly continuous one-parameter unitary group on K. In this paper, we introduce some continuity for linear functionals on and, under this continuity, we study existence, uniqueness, and non-uniqueness of KMS states for .

A complete and satisfying account of the classification of pencils of binary cubics over an algebraically closed field was given by Newstead (2). Extending these results to the real case is not a matter of mere routine since new questions arise, for example the separation of roots of the cubics in a pencil (as well as their reality).

This paper contains all those formulae which are necessary for the numerical evaluation of the basic confluent hypergeometric functions, including asymptotic formulae, and three skeleton tables which can be used as a basis for the study of the numerical behaviour of these functions over a wide field.

Criteria for singularity and absolute continuity analogous to Zygmund's for Riesz products, are demonstrated for those generalized products introduced by Modica and Mortola.

Let U(1, n; ℂ) be the automorphism group of the Hermitian form

for . We can regard an element of U(1, n; ℂ) as a transformation acting on , where is the closure of the complex unit ball

The non-trivial elements of U(1, n; ℂ) fall into three conjugacy types, depending on the number and the location of their fixed points. Let g be a non-trivial element of U(1, n; ℂ). We call g elliptic if it has a fixed point in Bn and g parabolic if it has exactly one fixed point and this lies on the boundary ∂Bn. An element g will be called loxodromic if it has exactly two fixed points and they lie on ∂Bn.

In the early editions of the Geometry of Three Dimensions Salmon had stated that the equations of any three quadric surfaces could be simultaneously reduced to the sums of five squares. Such a reduction is not possible in general, but can be performed if and only if a certain combinant Λ, of the net of quadrics, vanishes. Algebraically the theory of such a net of quadrics is equivalent, as Hesse(2) showed, to that of a plane quartic curve: and the condition for the equation a quartic to be expressible to the sum of five fourth powers is equivalent to the condition Λ = 0(1). While Clebsch(3) was the first to establish this condition, Lüroth(4) gave it more explicit form by studying the quartic curve

which satisfies the condition. Frahm(5) seems to have been the first to prove the impossibility of the above reduction of three general quadric surfaces, by remarking that the plane quartic curve obtained in Hesse's way from the locus of the vertices of cones of the net of quadrics would be a Lüroth quartic. Frahm further remarked that the three quadrics, so conditioned, could be regarded as the polar quadrics belonging to a cubic surface in ∞2 ways; but that for three general quadrics no such cubic surface exists. An explicit algebraical account of these properties was given by E. Toeplitz(6), who incidentally noticed that certain linear complexes associated with three general quadrics became special linear complexes when Λ = 0. This polar property of three quadrics in [3] was generalized to n dimensions by Anderson (7).

Throughout, the word knot means a subspace of the 3-sphere S3 homeomorphic with the 1-sphere S1. Any knot can be expressed as a connected sum of a finite number of prime knots in a unique way (13), we consider the trivial knot a non-prime knot. (For higher dimensional knots, factorization and uniqueness have been studied in (1).) However given a knot it is difficult to determine if it is prime or not. We prove that totally knotted knots, see definition in §2, are prime in theorem 1, give a class of examples in theorem 2 and investigate how the last result can be applied to the conjecture that the family Y of unknotting number one knots are prime. (See problem 2 in (5).) At the end, prime tangles as defined by W. B. R. Lickerish are used to prove that in a certain family of knots, related somewhat to Y, there is just one non-prime knot: the square knot. The paper should be interpreted as being in the piecewise linear category. Standard definitions of 3-manifolds and knot theory may be found in (6) and (11) respectively.

Second-order perturbation theory, in conjunction with the usual treatment of electron-phonon interaction, allows us to calculate the correction to the first-order result for the scattering of phonons by electrons. It is shown that the second-order term is much smaller, and therefore negligible. This justifies the use of the first-order theory in the treatment of the interaction in metals and semi-conductors.

In this paper we study a generalized Kummer surface associated to the Jacobian of those complex algebraic curves of genus two which admit an automorphism of order three. Such a curve can always be written as y2 = x6 + 2kx3 + 1 and k2 ╪ 1 is the modular parameter. The automorphism extends linearly to an automorphism of the Jacobian and we show that this extension has a 94 invariant configuration, i.e. it has 9 fixed points and 9 invariant theta curves, each of these curves contains 4 fixed points and 4 invariant curves pass through each fixed point. The quotient of the Jacobian by this automorphism has 9 singular points of type A2 and the 94 configuration descends to a 94 configuration of points and lines, reminiscent to the well-known 166 configuration on the Kummer surface. Our ‘generalized Kummer surface’ embeds in ℙ4 and is a complete intersection of a quadric and a cubic hypersurface. Equations for these hypersurfaces are computed and take a very symmetric form in well-chosen coordinates. This computation is done by using an integrable system, the ‘even master system’.

We discuss a minimisation problem of the degree of the Chow–Mumford (CM) line bundle among all possible fillings of a polarised family with fixed general fibers, motivated by the study of the moduli space of K-stable Fano varieties. We show that such minimisation implies the slope semistability of the fiber if the central fiber is smooth.

In many radio investigations of the ionosphere it is desirable to use senders working on different wave-lengths and situated in different positions with respect to the receiver. For this purpose it is very convenient to make use of commercial senders which work more or less continuously at high power. It so happens, however, that these senders usually emit Morse signals consisting of dots and dashes of variable lengths and with variable intervals between them, so that an instrument which records the mean signal will give an indication which depends on the speed of sending and on the spacing and length of the signs. In order to obtain a reading which depends only on the signal amplitude, it is possible to employ a quick moving recorder such as a string galvanometer or a cathode ray oscillograph, which will follow the individual Morse signs. With both these types of recorder the record must be made photographically so that changes in the signal strength are not noticed until the record is developed. This is a serious disadvantage in investigations of short-lived disturbances of the “catastrophic” type.

P. M. Cohn has given a characterization of pseudovaluation on a field in (1). There he defines the notion of an integral element over a multiplicatively closed set and states that the association of the set A to the set A* of all elements that are integral over it is a closure operator. There is an error in the statement that A* = (A*)*. In this note we show, by an example, that A* and (A*)* are not equal in general. We also give an example to show that even for the special type of sets that Cohn needs, namely the gauge sets, this property fails to hold. However, since the other conditions for a closure operator (A ⊆ A* and A ⊆ B implies A* ⊆ B*) are satisfied, one obtains a Kuratowski closure operator by iterating *. If we denote by Ā the closure of A by iterating * then theorem 10·3, its corollary and theorem 10·4 in (1) that have a bearing on this error, are valid as stated with the same proof replacing only K* by . Further theorem 13·3 is valid without modification since Cohn considers only a special class of pseudovaluations namely regular pseudo valuations for which K* = .

The proof of Corollary 4⋅3 in our paper [1] is not correct because there is a mistake in the expression given for ∥X* ∧ Y*∥2 on page 374. In fact, the correct expression for this term is

\begin{eqnarray*} \norm{X^*\wedge Y^*}^2 & = & \norm{X^*}^2\norm{Y^*}^2-\pair{X^*,Y^*}^2\\ {} & = & 1+\pair{X,T}^2+\pair{Y,T}^2\geq 1, \end{eqnarray*}

1. A general threefold in [5] has an apparent double surface whose projection on to a [4] is a surface which contains a triple curve and which is characterised by the fact that it has no improper nodes; such surfaces have been considered in a previous paper. In the present note we consider a class of surfaces in [5] which possess triple curves and project into surfaces in [4] having improper nodes: these are the surfaces which represent the chords of a general curve of ordinary space upon a quadric primal Ω of [5]. For the representation of a line congruence of [3] by the points of a surface on Ω, reference may be made to a previous note, the results of which are used in the present work.

Modifications to the Feynman propagation functions Sp and Dp are suggested in order to deal with field theoretic problems involving the presence in the initial and the final states, of free fermions or bosons, having all values of momentum (and energy) within a certain specified range.

Let p(x) be periodic with mean value c and let α > 0. Then the investigation of the asymptotic solution of the second-order equation

can be split into four cases: (i) α > 2, (ii) α > 1 and c = 0, (iii) 0 < α ≤ 1 and c = 0, (iv) 0 < α ≤ 2 and c ≠ 0. The first two cases are straightforward because then there are solutions y1 and y2 of (1·1) such that y1(x) ~ x and y2(x) → 1 as x → ∞ [5, theorems 2·7·2 and 2·7·5; 7, p. 375; 8]. Thus the coefficient of y in (1·1) represents a perturbation which has no effect on the dominant asymptotic behaviour of solutions. Case (iii) is less simple because here the nature of p(x) does influence the asymptotic form of solutions and, indeed, when α < 1 the solutions of (1·1) are oscillatory. This case has been fully covered by a number of authors: Atkinson, Eastham and McLeod[2, §3], Cassell[3, 4], Eastham[5, §4·13], Mahoney[10], Willett[13] and Wong[14, 15]. In addition, corresponding results for a higher-order self-adjoint analogue of (1·1) have been obtained recently by Al-Hammadi and Eastham [1, 6] as an application of general asymptotic methods for differential systems.

In (2), indefinite Kählerian manifolds have been examined from the point of view of holomorphic sectional curvature. Examples in (2) show that the analogue of Kulkarni's theorem (see (4), p. 173) for the holomorphic sectional curvature is false and the best possible result in the direction is:

Theorem 1 (known, (2)). Let M be a connected indefinite Kählerian manifold with complex dimension n ≥ 2. If the holomorphic sectional curvature of M is bounded above and bounded below, then M is an indefinite complex space form.

We study the spectral gaps of the Schrödinger operatorwhere κ∈(0,2π) and are parameters. Let τ=2π−κ. Suppose that the ratio κ0:=τ/κ is irrational. We denote the jth gap of the spectrum of H by Gj, its length by |Gj|. We obtain a relationship between the asymptotic behaviour of |Gj| as j→∞ and the Diophantine properties of κ0. In particular, we show that if β1+β2=0, thenwhere M(κ0) stands for the Markov constant of κ0.

Let unity and the irrational components ri of a d-dimensional vector r be linearly independent, and consider the integers n for which

where {x} denotes the fractional part of x. It is a trivial restriction to assume that ri and φi lie between 0 and 1. This note is concerned with the question: what are the gaps between the successive n?