1. During the last sixty years the principal questions presented by the higher singularities of plane algebraic curves have been completely solved, and definite results obtained. The two most successful lines of research have been by expansions and quadratic transformation. By each method it has been shown that a higher singularity may be looked upon as containing concealed or “latent” multiple points or lines in addition to those immediately recognized; and from each, with the help of small quantities, has been constructed a topological explanation of these latent multiple elements, which are accounted for as situated in the immediate vicinity of the point and line base of the singularity. Further, by each method it has been proved that as regards the numerical relations known as Plvicker's equations a singularity produces the same effect as a definite number of nodes, cusps, bitangents, and stationary tangents.

1. The present paper continues the investigation begun in (5). This earlier paper concluded with the description of a pencil P of canonical curves of genus 6 on a del Pezzo quintic surface F, but had first to describe the relevant features of the mapping of prime sections of F on a plane p by cubic curves circumscribing a quadrangle in p. It remains now to consider F and P in their ambient [5] in relation to G, the group, isomorphic to the symmetric group of degree 5, of self-projectivities of F.

1. Young's criterion for the convergence of a Fourier series may be stated in the following form:

The integral

tends to zero as ω → ∞ whenever

(A) ø(t) is absolutely integrable in (0, a),

(B) ø(t) → 0 in mean as t → 0,

(C) tø(t) is of bounded variation to the immediate right of t = 0 and, as t → 0,

Let π be a finite p-group and ${\mathbb{F}_{q}}$ a finite field with q = pn elements. Denote by $\I_{\mathbb{F}_{q}}$ the augmentation ideal of the group ring ${\mathbb{F}_{q}}$[π]. We have found a surprising relation between the abelianization of 1 + $\I_{\mathbb{F}_{q}}$, the Bogomolov multiplier B0(π) of π and the number of conjugacy classes k(π) of π:

$$ \left | (1+\I_{\Fq})_{\ab} \right |=q^{\kk(\pi)-1}|\!\B_0(\pi)|.

$\I_{\mathbb{F}_{q}}$

The algebraic classification problem with which I shall be concerned in this paper is suggested by the algebraic theory of homomorphisms of free crossed modules whose groups of operators are free groups. This theory arises in the study of the homotopy theory of two-dimensional C.W. complexes*. It is shown in (3) that the homotopy theory of such complexes, including the homotopy classification of mappings, is equivalent to this purely algebraic theory.

We consider the ground state of a system of three interacting particles of equal mass. An integro-differential equation is obtained for the optimum pair function f in the product wavefunction Ψ(123) = f(12)f(13)f(23). The solution for harmonic forces reproduces the known exact ground state. Approximate analytic solutions are obtained for inverse-square forces, and for a general force law in the semiclassical limit.

An open continuous function from an open Riemann surface with finite genus and finite number of boundary components into a closed Riemann surface is termed a (p, q)-map, 0 < q < p, if it has a finite number of branch points and assumes every point in the image surface either p or q times, counting multiplicity, with possibly a finite number of exceptions.

The object of this paper is to prove that the geometry of any (p, q)-map resembles that of a (p, q)-map whose q-set (the set of image points of f that are taken on exactly q times, counting multiplicity), constitutes a finite set of Jordan arcs or curves (loops). This leads to interesting geometrie results regarding (p, q)-maps without exceptional points. Further, it yields that every (p, q)-map is homotopic to a simplicial (p, q)-map having the same covering properties.

Few papers presenting a theory of operation of thermostats can be found. This may be explained by mathematical difficulties which are met when considering the rather complicated construction, from the thermal point of view, of most thermostats. There is, however, a relatively great number of papers describing various mechanical constructions of thermostats. In many cases designers obtain very good practical results by making use of some approximate theories only.

The plus and minus norm groups are constructed by Kobayashi as subgroups of the formal group of an elliptic curve with supersingular reduction, and they play an important role in Kobayashi’s definition of the signed Selmer groups. In this paper, we study the cohomology of these plus and minus norm groups. In particular, we show that these plus and minus norm groups are cohomologically trivial. As an application of our analysis, we establish certain (quasi-)projectivity properties of the non-primitive mixed signed Selmer groups of an elliptic curve with good reduction at all primes above p. We then build on these projectivity results to derive a Kida formula for the signed Selmer groups under a slight weakening of the usual µ = 0 assumption, and study the integrality property of the characteristic element attached to the signed Selmer groups.

The expectation, variance and covariance for different states of a k-state Markov chain have been given by Patankar (6), Whittle (7), Good (3) and Bhat (l). Patankar's results involve the k latent roots of the determinantal equation. As it is not easy to determine the latent roots when k > 2, the actual asymptotic values of variances and covariances cannot be readily evaluated. Whittle gives exact probability distributions for the transitions, but the moments have been obtained after some gross approximations. Good (3) and Bhat(l) have given the first two moments and product moment for the frequency of different states. By using certain methods developed by Iyer and Kapur(4), the first four cumulants and product cumulants for the transition numbers of a two-state Markov chain were calculated and presented in an earlier publication(5).

In the paper [12] Shcherbak studied some duality properties of projective curves and applied them to obtain information concerning central projections of surfaces in projective three space. He also states some interesting results relating the contact of a generic surface with lines and the contact of its dual with lines in the dual space. In this paper we extend this duality to cover non-generic surfaces. Our proof is geometric, and uses deformation theory. The basic idea is the following. Given a surface X in projective 3-space we can consider the lines tangent to X, and measure their contact. The points on the surface with a line yielding at least 4-point contact are classically known as the flecnodal. (The reason is that the tangent plane meets the surface in a nodal curve, one branch of which has an inflexion at the point in question; see Proposition 7 below. The line in question is the inflexional tangent, which is clearly asymptotic.)

A function f:X → ℝ is countably decomposable (into continuous functions) if the topological space X can be partitioned into countably many sets An with each restriction f│ An continuous. According to L. V. Keldysh(2), the question whether every Baire function is countably decomposable was first raised by N. N. Luzin, and answered by P. S. Novikov. The answer is negative even for Baire-1 functions, as is shown in (2) (see also (1). In this paper we develop a characterization of the countably decomposable functions on a separable metric space X (see Corollary 1). We deduce that when X is complete they include all functions possessing the property P defined by D. E. Peek in (3): each non-empty σ-perfect set H contains a point at which f│ H is continuous. The example given by Peek shows that not every countably decomposable Baire-1 function has property P.

Let p be an odd prime, and h*( − ) a multiplicative mod p cohomology theory with a complex orientation satisfying the 2(p − 1)-sparseness condition. A formula concerning the formal group law for such h*( − ) is used to investigate the structure of h*(BG), G being a finite group whose p-Sylow subgroups are cyclic.

1. The postulation of a multiple curve for primals of sufficiently large order in space of any number of dimensions has been obtained recently by J. A. Todd, by a simple and elegant degeneration argument which, however, is not deemed to be a conclusive proof by the author himself. And, indeed, in order to make sure of the unconditional validity of such an argument, one should ascertain whether

(i) the postulation θk of an irreducible non-singular curve ϲ, of order c and genus p, for the primals of sufficiently large order n of [r + 2] (r ≥ 1), required to go through it with multiplicity k (≥ 1), is a function of k, c, p, n, r only;

(ii) it is possible, by means of a continuous variation of ϲ, to reduce this curve to connected polygon ϲ′ having the same virtual characters as ϲ, in such a way that each intermediate position of ϲ is still irreducible and non-singular;

(iii) the postulation θk of ϲ equals the similarly defined postulation of ϲ′.

Given a surface homeomorphism isotopic to the identity which is pseudo-Anosov relative to a finite set, we show that the sum of the Lefschetz numbers of periodic points of any period greater than one is non-negative. If this period is odd and greater than a number which depends only on the surface, the sum is zero. If we consider sequences of periods such that each element is twice that of its predecessor, then this sum is increasing beyond a certain point also depending on the surface. As a corollary, for each periodic orbit contained within the boundary of the surface there exists one of the same period contained in the interior.

In the development of homological algebra, one has to prove at some point that, in defining the derived functors of ⊗ and of Hom, it makes no difference whether we resolve both variables or only one of them. Taking ⊗ ( = ⊗R) as a typical example, what has to be proved is

(A) If the complex F is projective, or even flat, as a right R-module, and if f: P → A is a projective resolution of the left R-module A, then

is an isomorphism.

The principal purpose of this note is to give an elementary proof of an inequality, due to Pisier (proposition 2·4 of (4)), for martingales taking values in a Banach space X. This proof yields an explicit estimate, sharp when X is a Hilbert space, for one of the constants involved. The notation follows (1, 4).

The following is a simple example of what Poincaré calls probabilité du continu.

Two points are marked at random on a unit line and it is assumed that all positions of either point are equally probable. What is the average or probable positive distance between the two points?

Much recent attention has been devoted to those nonlinear partial differential equations admitting higher-order conservation laws (e.g. [2] and references therein) or equivalently admitting higher-order symmetries. In particular the sine-Gordon equation possesses such symmetries [5, 7] where is the two-dimensional d'Alembertian operator. The question posed and solved here is whether such behaviour is possible in higher dimensions. We therefore consider the ‘Generalized Klein–Gordon’ (GKG) equation

in N dimensions where and N ≥ 3.