We prove that any periodic orbit O of a C1 orientation-preserving embedding f of the 2-disc D2 is linked with a fixed point P in the sense that the corresponding periodic orbits {Φt(O)}t ≥ 0 and {Φt(P)}t ≥ 0 of any torus flow Φt suspending f are linked as knots in S3.

A long time ago the concept of H*-algebra was introduced by Ambrose in [1] where the structure of complex associative H*-algebras was given. Since then this theory was extended to such classical types of non-associative algebras as alternative algebras (in [6]), Jordan algebras (in [5, 13, 14]), non-commutative Jordan algebras (in [5]), Lie algebras (in [3, 9, 10]) and Mal'cev algebras (in [2]).

Errors in a previous discussion (4) of the title problem for a perfectly conducting sphere are corrected. An alternative method of determining the ultimate motion is described for both a perfectly conducting and a non-conducting sphere moving along the magnetic field.

For a function from the unit interval to itself with constant slope and one discontinuity, the itineraries of the point of discontinuity are called the critical itineraries. These critical itineraries play a significant role in the study of β-expansions (with positive or negative β) and fractal transformations. A combinatorial characterization of the critical itineraries of such functions is provided.

Let φ (X) denote a function satisfying limx→∞, φ(X) = ∞. Let ℬ be a sequence of positive integers and let T(X) = T(X, ℬ, φ) be the set of real numbers y, 1 ≤ y ≤ X, such that the interval (y, y + φ(y)] contains no member of ℬ.

We prove that if k, F are fields of different characteristics then the permutation module kFn for the affine group AGL(n, F) has simple augmentation submodule. This proves a conjecture of A. R. Camina.

We study singularities of spacelike, constant (non-zero) mean curvature (CMC) surfaces in the Lorentz–Minkowski 3-space L3. We show how to solve the singular Björling problem for such surfaces, which is stated as follows: given a real analytic null-curve f0(x), and a real analytic null vector field v(x) parallel to the tangent field of f0, find a conformally parameterized (generalized) CMC H surface in L3 which contains this curve as a singular set and such that the partial derivatives fx and fy are given by df0/dx and v along the curve. Within the class of generalized surfaces considered, the solution is unique and we give a formula for the generalized Weierstrass data for this surface. This gives a framework for studying the singularities of non-maximal CMC surfaces in L3. We use this to find the Björling data – and holomorphic potentials – which characterize cuspidal edge, swallowtail and cuspidal cross cap singularities.

A classical theory of a spinning particle with charge and dipole moment in an electromagnetic field is obtained by working symmetrically with respect to retarded and advanced fields, and with respect to the ingoing and outgoing fields. The equations are in a simpler form than those of Bhabha and Corben or those of Bhabha, and involve fewer constants. On the assumption that the spin angular momentum tensor θμν satisfies the equation θ2 ≡ θμν θμν = constant, the value of the dipole moment Zμν is chosen to be Cθμν, where C is a constant. The theory is generalized to the case of several particles with charge and dipole moment. By using a suitable Hamiltonian equation, the classical equations of motion, obtained on the assumption that θ is a constant, are put into Hamiltonian form by means of the ‘Wentzel field’ and the λ-limiting process. The passage to the quantum theory is effected by the usual rules of quantization. The theory is extended to the case of particles with charge and dipole moment in the generalized wave field by defining the Wentzel potential in terms of the generalized relativistic δ-function.

We show that the Liouville-Arnol'd theorem concerning knowledge of involutory first integrals for Hamiltonian systems is available for any system of second order ordinary differential equations. In establishing this result we also provide a new proof of the standard theorem in the setting of non-autonomous, regular Lagrangian mechanics on the evolution space ℝ × TM of a manifold M. Both the original theorem and its generalization rely on a certain bijection between symmetries of the system and its first integrals. We give two examples of the use of the theorem for systems on ℝ2 which are not Euler-Lagrange.

Calculations are made to investigate the behaviour of ordering near the critical temperature for alloys with a superlattice of the type AB for various compositions. No fluctuation among the sites of the same kind, which may lead to the possible existence of some other types of superlattice, say, AB3, is considered. Results showing the change of ordering with T near Tc are plotted.

The theory is also applied to the adsorption of gases upon a solid surface, where the sites of accommodation are regularly arranged. It is shown that under certain conditions, a superlattice formation in the adsorbed layer is possible.

It is well known that the calm which commonly comes on a cold clear night is due to the cooling effect of the ground on the lowest layers of the atmosphere. It frequently happens that the wind at a height of a few hundred feet does not die down at all at night; in fact observations taken at the top of the Eiffel tower show that there is frequently a slight increase in wind at that height. The cooling of the air near the ground makes the lower layers heavier than the higher layers and thus tends to prevent the formation of turbulence at the earth's surface. Since it is only through the medium of turbulence that the higher layers are able to drag the lowest layers over the ground the surface air stops moving.

Noncommutative Radon–Nikodým theorems have attracted a great deal of attention in the theory of operator algebras. There has been considerable work on non-commutative Radon–Nikodým theorems not only for C*-algebras but also for algebras of unbounded operators [3, 5, 8, 9, 12, 13]. In this paper, we will develop a Radon–Nikodým type theorem for completely bounded and completely positive invariant multilinear maps.

The concept of matricial order has turned out to be very important to understand the infinite-dimensional non-commutative structure of operator algebras. As the natural ordering attached to this structure, completely positive maps and completely bounded maps have been studied extensively. Results concerning completely bounded maps have many applications: cohomology of operator algebras, multipliers on group algebras, dilation theory, similarity theory, free product representations, and abstract characterizations of operator algebras.

We denote by b(n) the number of binary partitions of n, that is the number of partitions of n as the sum of powers of 2. As usual, two partitions are not considered to be distinct if they differ only in the order of their summands. It is convenient to define b(0) = 1.

Let $\Lambda\subseteq \{1,\ldots, N\}$ and let $\{a_{n}\}_{n\in\Lambda}$ be a sequence with $|a_n|\leq 1$ for all $n$. It is easy to see that\[ \left\|\sum_{n\in\Lambda}a_{n}e{(n\theta)}\right\|_{p}\leq \left\|\sum_{n\in\Lambda}e{(n\theta)}\right\|_{p}\] for every even integer $p$. We give an example which shows that this statement can fail rather dramatically when $p$ is not an even integer. This answers in the negative a question known as the Hardy–Littlewood majorant conjecture, thereby ruling out a certain approach to the restriction and Kakeya families of conjectures.

It is known † that, if G is any p-group (group whose order is a power of the prime p) the class of which is less than p, then the order of a product of elements of G cannot exceed the orders of all the factors. This result in general ceases to hold when the class of G exceeds p − 1. In the present paper, we prove the following general results.

By the use of a semi-martingale property of the Kolmogorov supremum, the results of Pyke (6) on the weak convergence of the empirical process with random sample size are simplified and extended to the case of p(≥1)-dimensional stochastic vectors.

Let (R, ) be an analytically unramified local ring of positive prime characteristic p. For an ideal I, let I* denote its tight closure. We introduce the tight Hilbert function $$H_I^*\left( n \right) = \Im \left( {R/\left( {{I^n}} \right)*} \right)$$ and the corresponding tight Hilbert polynomial $$P_I^*\left( n \right)$$, where I is an m-primary ideal. It is proved that F-rationality can be detected by the vanishing of the first coefficient of $$P_I^*\left( n \right)$$. We find the tight Hilbert polynomial of certain parameter ideals in hypersurface rings and Stanley-Reisner rings of simplicial complexes.

Let

formula here

be a presentation for the group G where F = 〈a1, …, an〉 is a free group. Put A = {a1, …, an a−11, …, a−1n} and call n the degree of the presentation. For g ∈ G, c ∈ A we let W(k, g, c) be the set of freely reduced words w ∈ F having length k, which end (on the right) in c and which represent the element g ∈ G. Let w(k, g, c) = |W(k, g, c)|. The function

formula here

will be called the cogrowth function for this presentation. Let

formula here.

Throughout, p will be a fixed prime, and will denote the class of all locally finite p-groups. For a fixed Abelian p-group A, we let

where ζ(P) denotes the centre of P. Notice that A is not a class in the usual group-theoretic sense, since it is not closed under isomorphisms.

An alternative derivation of the integral equation of Waller, Heitler and Wilson is given on the basis of the stationary method of the perturbation theory. The previous theoretical treatment of the scattering of charged vector mesons by nuclear particles through the nuclear interaction is extended to pseudoscalar mesons. Numerical results calculated from the theory are given for the total and differential scattering cross-sections.