If we have one birational representation of a threefold locus V on an ordinary space S3, and if we can project V birationally on to another space S3′, then clearly there will be a Cremona correspondence between the two spaces S3 and S3′. This paper deals, in the first place, with threefold loci in higher space which can be represented birationally on ordinary space by means of cubic surfaces; and Cremona transformations for ordinary space are then obtained, as indicated above, by projecting such loci. In particular it will be shown that most of the more important cubic transformations and their reverses can be obtained very simply by this method.

Let G be an infinite connected graph. A ray (from ν) in G is a 1-way infinite path in G (with initial vertex ν). An infinite connected subgraph of a ray R ⊂ G is called a tail of R. If X ⊂ G is finite, the infinite component of R\X will be called the tail of R in G\X.

Some decidability results for presentations in the variety of Clifford monoids are obtained. These results are applied in the solution of the E-unitary problem for presentations in the variety of inverse monoids.

In the first part of the paper a slow two-dimensional motion of viscous fluid is considered which approximates to a motion of uniform shear past an infinite fixed plane, and differs from this motion because there is a gap in the plane (Fig. 1). A simple expression in finite terms is found for the stream function.

1. introduction. A set P of disjoint paths in a graph G is a node-covering of G if every node of G is in a path in P. (For definitions not given here see (5) or (11).) The path (node-covering) number of G is the smallest possible number p(G) of paths in such a node-covering of G. The problem of determining the path number of a graph was considered in (1) and (4) for undirected graphs and in (2) for directed graphs. In particular, an algorithm for determining the path number of a tree was given in (1) and (4). If ℱ denotes some family of trees, let μ(n) = μℱ(n) denote the average value of p(T) over all trees T in ℱ with n nodes. Our object here is to show the existence and to determine the value of the path (node-covering) constant

for certain families ℱ of trees. The arguments are similar to those used in (7), (8), and (9), where some other parameters, associated with families of trees, were investigated.

In this paper, we prove the following:

Theorem. Let C⊂ℙ3be a complex reduced irreducible space curve which is non- degenerate (i.e., C is not contained in any plane). Let p: C→ℙ2be a general projection. Then the topological fundamental group π1(ℙ2/p(C)) is abelian.

For material stratified in parallel planes in local thermodynamical equilibrium we have Milne's well-known result that

where πF is the constant net integrated flux of radiation, τ is the optical depth and B is the “ergiebigkeit” which is related to the temperature T by the relation

σ being the Stefan-Boltzmann constant. If we define the effective temperature by the relation

we have from (1)

where T1 is the temperature ar τ = 1. In this note we establish a similar result for extended photospheres where the curvature of the outer layers is properly taken into account.

In an alternating current motor (of simplest type) the internal E.M.F. e, the terminal tension e, the useful electrical power transformed p and the rotor phase angle φ are connected by the known formula

in which

where r is the resistance and l the self-inductance of the armature and ω the angular velocity of the rotor.

The problem of the equivalence of binary forms is of great importance, both historically and intrinsically, and is also significant for the problems of the so-called canonical and automorphic forms. It consists in deciding whether two given forms are equivalent, i.e. transformable one into the other linearly, and, when they are, in finding all the linear substitutions transforming one form into the other.

The McLaughlin sporadic simple group McL is the flag-transitive automorphism group of a Petersen-type geometry [Gscr ] = [Gscr ](McL) with the diagram

diagram here

where the edge in the middle indicates the geometry of vertices and edges of the Petersen graph. The elements corresponding to the nodes from the left to the right on the diagram P33 are called points, lines, triangles and planes, respectively. The residue in [Gscr ] of a point is the P3-geometry [Gscr ](Mat22) of the Mathieu group of degree 22 and the residue of a plane is the P3-geometry [Gscr ](Alt7) of the alternating group of degree 7. The geometries [Gscr ](Mat22) and [Gscr ](Alt7) possess 3-fold covers [Gscr ](3Mat22) and [Gscr ](3Alt7) which are known to be universal. In this paper we show that [Gscr ] is simply connected and construct a geometry [Gscr ]˜ which possesses a 2-covering onto [Gscr ]. The automorphism group of [Gscr ]˜ is of the form 323McL; the residues of a point and a plane are isomorphic to [Gscr ](3Mat22) and [Gscr ](3Alt7), respectively. Moreover, we reduce the classification problem of all flag-transitive Pmn-geometries with n, m [ges ] 3 to the calculation of the universal cover of [Gscr ]˜.

The topology of a topological group $G$ is called an ${\cal L}_{\infty}$-topology if it can be represented as the intersection of a decreasing sequence of locally compact Hausdorff group topolgies on $G$. If ${\cal L}_1 < {\cal L}_2$ are two distinct ${\cal L}_{\infty}$-topologies on an Abelian group $G$, it is shown that the quotient of the corresponding character groups has cardinality ${\geqslant} 2^{\rm c}$. A conjecture in this sense announced by J. B. Reade in his paper [6] is thereby proved.

We use the theory of sectorial sesquilinear forms to characterize the closure of the Creation Operator of Quantum Mechanics in the classical set-up. Further, we bring that theory to bear upon the class of unbounded cyclic subnormal operators that admit analytic models; in particular, we provide a sufficient condition for the existence of complete sets of eigenvectors for certain sectorial operators related to unbounded subnormals. The relevant theory is illustrated in the context of a class of analytic models of which the classical Segal–Bargmann space is a prototype. The framework of sectorial sesquilinear forms is also shown to be specially useful for treating questions related to the existence, uniqueness and stability of certain parabolic evolution equations naturally associated with such analytic models.

Before the introduction of interpolation formulae, beyond linear interpolation by proportional parts, the presentation of the numerical values of mathematical functions was much restricted, for the labour of computation and the cost of printing, to say nothing of the inconvenience of handling a bulky volume, had to be increased quite disproportionately with every increase in accuracy. A four-figure logarithm table occupies two small pages, Chambers's seven-figure table takes 150 pages, while Vega's ten figure table requires 300 pages twelve inches long.

We show that any smooth projective cubic hypersurface of dimension at least 29 over the rationals contains a rational line. A variation of our methods provides a similar result over p-adic fields. In both cases, we improve on previous results due to the second author and Wooley.

We include an appendix in which we highlight some slight modifications to a recent result of Papanikolopoulos and Siksek. It follows that the set of rational points on smooth projective cubic hypersurfaces of dimension at least 29 is generated via secant and tangent constructions from just a single point.

The algebraic tensor product A1⊗A2 of two Banach algebras is an algebra in a natural way. There are certain norms α on this tensor product for which the multiplication is continuous so that the completion, A1αA2, is a Banach algebra. The representation theory of such tensor products is the subject of this paper. It will be shown that, under certain simple conditions, the tensor product of two semi-simple Banach algebras is semi-simple although, without these conditions, the result fails.

For infinitely many d, Hassett showed that special cubic fourfolds of discriminant d are related to polarised K3 surfaces of degree d via their Hodge structures. For half of the d, each associated K3 surface (S, L) canonically yields another one, (Sτ, Lτ). We prove that Sτ is isomorphic to the moduli space of stable coherent sheaves on S with Mukai vector (3, L, d/6). We also explain for which d the Hilbert schemes Hilbn (S) and Hilbn (Sτ) are birational.

We consider generalisations of the so-called Euler adic and investigate dynamical properties like ergodicity and total ergodicity. We prove the existence of a unique fully-supported ergodic measure for these generalisations. We also investigate the structure of non-fully-supported ergodic measures and in addition show that each of these measures (fully- and non-fully-supported) is also totally ergodic. In order to determine these dynamical properties we find closed-form expressions for the generalised Eulerian numbers. Additionally we extend a result given by Frick and Petersen to a wider class of adic transformations.

In a paper from 2010, Budarina, Dickinson and Levesley studied the rational approximation properties of curves parametrised by polynomials with integral coefficients in Euclidean space of arbitrary dimension. Assuming the dimension is at least three and excluding the case of linear dependence of the polynomials together with P(X) ≡ 1 over the rational number field, we establish proper generalisations of their main result.

The ring A = $\mathbb{F}$r[T] and its fraction field k, where r is a power of a prime p, are considered as analogues of the integers and rational numbers respectively. Let K/k be a finite extension and let φ be a Drinfeld A-module over K of rank d and Γ ⊂ K be a finitely generated free A-submodule of K, the A-module structure coming from the action of φ. We consider the problem of determining the number of primes ℘ of K for which the reduction of Γ modulo ℘ is equal to $\mathbb{F}$℘ (the residue field of the prime ℘). We can show that there is a natural density of primes ℘ for which Γ mod ℘ is equal to $\mathbb{F}$℘. In certain cases, this density can be seen to be positive.