It is a well-known result in one-dimensional dynamics that if a continuous map of the interval has positive topological entropy, then it has a periodic orbit of period 2i for each integer i ≥ 0 [15] (see also [12]). In fact, one can say rather more: such a map has a sequence of periodic orbits (P)i ≥ 0 with per (Pi) = 2i which form a period-doubling cascade (that is, whose points are ordered and permuted in the way which would occur had the orbits been created in a sequence of period-doubling bifurcations starting from a single fixed point). This result reflects the central role played by period-doubling in transitions to positive entropy in a one-dimensional setting. In this paper we prove an analogous result for positive-entropy orientation-preserving diffeomorphisms of the disc. Using the notion [9] of a two-dimensional cascade, we shall show that such diffeomorphisms always have infinitely many ‘zero-entropy’ cascades of periodic orbits (including a period-doubling cascade, though this need not begin from a fixed point).

The paper is concerned with properties of the Dirichlet series where {λn} is a strictly increasing unbounded sequence of real numbers with λ1 > 0. One of the main Tauberian results proved is that if a1 ≥ 0, an > 0 for n = 2, 3, …, a(x) ≤ ∞ for all x ≥ 0, An:= a1 + a2 + … + an → ∞, an λn = o((λn+1-λn)An), an λn sn > − H(λn+1 − λn) An and then A new summability method Dλ, a based on the Dirichlet series a(x) is defined and its relationship to the weighted mean method Ma investigated.

The wave equation with axial symmetry is reduced, by the assumption of dynamic similarity, to an equation in two variables, r/t and θ A basic solution, having some of the attributes of a source, is introduced, and it is shown that, by use of this solution and a suitable contour integral representation, the solution of any appropriate boundary-value problem may be reduced to the determination of an analytic function, with boundary conditions given by an integral equation.

The method of solution is illustrated by reference to some basic problems in the theory of linearized compressible flow.

It is a classical result of Postnikov [15] that homotopy types X with at most two non-trivial homotopy groups πmX = A and πnX = B, 2 [les ] m < n, are classified by the k-invariant

formula here

Here the cohomology group of the Eilenberg–MacLane space K(A, m) was computed by Eilenberg–MacLane [11] and Cartan [5]. Let p be a prime and let ℤp ⊂ ℚ be the smallest subring of ℚ containing 1/q for all primes q with q ≠ p. We consider finitely generated ℤp-modules A and B and the stable range n < 2m − 1. Hence X is a p-local space with at most two non-trivial homotopy groups in a stable range. Then the homotopy type of X admits a product decomposition

formula here

where all Xi with 1 [les ] i [les ] j are indecomposable and this decomposition is unique up to permutation. We classify in this paper the indecomposable factors in (2) by the following result.

Throughout this paper we work in the smooth category and assume that all knots are oriented and consider two knots K1 and K2 to be equivalent if and only if there is an orientation preserving homeomorphism h:S3→S3 which carries K1 onto K2 so that their orientations match. We write K1 ≅ K2 if K1 and K2 are equivalent and – K denotes the knot obtained from K by inverting its orientation.

Functional analytic properties of the Banach Algebras $A_p^r(G)$ such as Arens regularity, amenability, weak sequential completeness, sets of synthesis and strict containment for fixed $p$ and increasing $r$, are investigated.

It is found that $A_p^r(G)$, $1\leq r<\infty$, behave very differently to the Figa–Talamanca–Herz algebras $A_p(G)$.

Some results are presented which establish connections between shape theory and the theory of multivalued maps. It is shown how to associate an upper-semi-continuous multivalued map F: X → Y to every approximative map f = {fk, X → Y} in the sense of K. Borsuk and it is proved that, in certain circumstances, if F is ‘small’ and admits a selection, then the shape morphism S(f) is generated by a map, and if F admits a coselection then S(f) is a shape domination.

The Dickson algebra Wn+1 of invariants in a polynomial algebra over [ ]2 is an unstable algebra over the mod 2 Steenrod algebra [Ascr ], or equivalently, over the Kudo–Araki–May algebra [Kscr ] of ‘lower’ operations. We prove that Wn+1 is a free unstable algebra on a certain cyclic module, modulo just one additional relation. To achieve this, we analyse the interplay of actions over [Ascr ] and [Kscr ] to characterize unstable cyclic modules with trivial action by the subalgebra [Ascr ]n−2 on a fundamental class in degree 2n – a. This involves a new family of left ideals [Iscr ]a in [Kscr ], which play the role filled by the ideals [Ascr ][Ascr ]n−2 in the Steenrod algebra.

Zygmund (1) proved in 1936 that if f(x) ∈Lp(−∞, ∞) (1 ≤ p ≤ 2), the Fourier transform

exists for almost every λ, in the sense that

exists for almost every λ. The purpose of this note is to provide a short proof of this theorem.

The propagation of electromagnetic waves through isotropic media containing a continuous energy-gain mechanism is considered and the existence of perfect resonance properties is investigated for both homogeneous and inhomogeneous layers.

It is well known that, in the study of quotient spaces it suffices to consider a topological space (X, ), an equivalence relation R on X and the projection mapping p: X → X/R (where X/R is the family of R-classes of X) defined by p(x) = Rx (where Rx is the R-class to which x belongs) for each x ∈ X. A topology may be defined for the set X/R by agreeing that U ⊆ X/R is -open if and only if p-1 (U) is -open in X. The topological space is known as the quotient space relative to the space ) and projection p. If (or simply ) since the symbol ≤ denotes all partial orders and no confusion arises) is a topological ordered space (that is, X is a set for which both a topology and a partial order ≤ is defined) then, providing the projection p satisfies the property

a partial order may be defined in X/R by agreeing that p(x) < p(y) if and only if x < y in x. The topological ordered space is known as the quotient ordered space relative to the ordered space and projection p.

The generalization of the Dirac and quaternion algebras to Riemannian spaces is outlined. The components of various elements of the algebras are interpreted as physical quantities (tensors) and their symmetries and algebraic properties are linked with the properties of the algebra. The generalization of the quaternion algebra is of particular interest in that it resolves the anomalies that arise in the usual identification of quaternions with rank two spinors. Algebraic expressions for the electromagnetic energy momentum tensor, the Ricci tensor and Einstein tensor are obtained in both E-number and quaternion form. Extension of the principles to EF-numbers yields a proof of the symmetry properties of Bel's tensor and a simple expression for its divergence.

It was remarked in (1) that many of the results valid for compact (this term is taken to imply Hausdorff) semigroups with jointly continuous multiplication also hold if the multiplication is only separately continuous. It is the purpose of this note to show that this is true of the structure theorem for compact simple semigroups. Any unsubstantiated facts about semigroups which we use may be found in the second section of (1) (the results given there do not depend on the assumption that a semigroup should contain an identity).

Every nonsingular projective real algebraic curve C has a unique, up to isomorphism over ℝ, nonsingular projective complexification V. If V\C is disconnected, then C is said to be dividing (we identify C with the set of real points of V). Classical results supply numerous examples of dividing real curves. This class of curves was first studied by Felix Klein [11]. A characterization of dividing real curves is given in [6].

In higher dimensions the situation is more complicated. First of all, every nonsingular projective real algebraic variety X of dimension d always has several nonisomorphic nonsingular projective complexifications, provided that d[ges ]2. Furthermore, if d[ges ]2 and W is a nonsingular projective complexification of X, then W\X is connected (we identify X with the set of real points of W). What does it then mean for X to be dividing? An answer can be given in terms of homology theory. Let K be a principal ideal domain. Assume that X, regarded as a topological manifold, is orientable over K. We say that X is dividing over K if there exists a fundamental homology class of X over K whose image in Hd(W, K) under the homomorphism induced by the inclusion map X[rarrhk ]W is zero. In the present paper we prove that this definition does not depend on the choice of W, and give a characterization of real varieties dividing over ℚ. For more information on real varieties dividing over ℤ/2 the reader may consult [18] (please note that terminology in [18] is different than here). We also discuss the relationship between real varieties dividing over ℚ (or ℤ) and dividing over ℤ/2. It follows from well-known facts that a real curve is dividing over K if and only if it is dividing.

For a real number λ, ‖λ‖ is the absolute value of the difference between λ and the nearest integer. Let X represent the m-tuple (x1, x2, … xm) and let

be any n linear forms in m variables, where the Θij are real numbers. The following is a classical result of Khintchine (1):

For all pairs of positive integers m, n there is a positive constant Г(m, n) with the property that for any forms Lj(X) there exist real numbers α1, α2, …, αn such that

for all integers x1, x2, …, xm not all zero.

Let ${\frak M}_{g}$ be the moduli space of smooth, integral curves of genus $g$ over the complex field ${\mathbb C}$ and let $\overline{{\frak M}}_{g}$ be its Mumford–Deligne compactification. If $C$ is a smooth, integral curve of genus $g$ we denote by [$C$] its isomorphism class in ${\frak M}_{g}$.

(1) The cataphoresis of ions and colloid particles is discussed in so far as it is affected by the ionic atmosphere.

(2) The ions in the atmosphere which carry a charge opposite in sign to that on the central particle are attracted to the central particle. However, when an electric field is applied to the liquid, they are able to drift away in virtue of their molecular energy, and the migration of the central particle is dependent on this fact. The relaxation force is the resultant of the forces between central particle and ions during this separation.

(3) Such a force draws the ions in the atmosphere after the central particle to some extent. From a consideration of the energy involved in the separation of particle and atmosphere, and of the molecular energies of the ions, we are able to calculate the number of ions which this relaxation force could draw through the liquid as though they were bound to the particle, and hence deduce the magnitude of the force in terms of the friction constant of the ions. The expression is the same as that given by Debye and Hückel.

(4) The cataphoresis equation usually employed for colloid particles takes no account of this relaxation force during migration. The corrected equation is given.

Let X denote the general point with coordinates (x1, x2, x3) in 3-dimensional space; and let P(X) be the function defined by