The Hausdorff and box dimensions for measures associated with recurrent self-similar sets generated by similitudes is explicitly given. The box dimension of the attractor associated with a class of two-dimensional affine maps is also computed.

An important example of a model category is the category of unbounded chain complexes of R-modules, which has as its homotopy category the derived category of the ring R. This example shows that traditional homological algebra is encompassed by Quillen's homotopical algebra. The goal of this paper is to show that more general forms of homological algebra also fit into Quillen's framework. Specifically, a projective class on a complete and cocomplete abelian category [Ascr ] is exactly the information needed to do homological algebra in [Ascr ]. The main result is that, under weak hypotheses, the category of chain complexes of objects of [Ascr ] has a model category structure that reflects the homological algebra of the projective class in the sense that it encodes the Ext groups and more general derived functors. Examples include the ‘pure derived category’ of a ring R, and derived categories capturing relative situations, including the projective class for Hochschild homology and co-homology. We characterize the model structures that are cofibrantly generated, and show that this fails for many interesting examples. Finally, we explain how the category of simplicial objects in a possibly non-abelian category can be equipped with a model category structure reflecting a given projective class, and give examples that include equivariant homotopy theory and bounded below derived categories.

In this paper the mixed problem for an isotropic, elastic half-space is considered. Boundary conditions are prescribed interior and exterior to a circular region of unit radius, and the state of stress is assumed to be axially symmetric. Several authors have treated this problem. Mossakovskii(1) considered a punch adhering to and indenting an elastic half-space. Has solution was obtained by introducing certain operators that transformed the half-space problem into a problem in plane potential theory. The method of linear relationship was used to solve this auxiliary problem and inverse operators returned the plane to the half-space. The general case of a circular, rigid punch adhering to a half-space was treated by Ufliand (2,3) and a solution was obtained through the use of toroidal coordinates and the Mehler-Fok integral transforms.

In this note the problem of random flights is considered for the case of a small number of vectors, each vector possessing the same constant magnitude. A formula is developed which permits the calculation of the probability function of the sum of N + 1 such vectors if the function of the sum of N vectors is known. The formulae for the probability distributions up to N = 8 are quoted and some of their practical applications mentioned.

A general method is given of calculating the effect of damping on the collision cross-sections for problems involving free electrons and mesons. The result is expressed in the form of an integral equation which can only be solved if certain simplifying assumptions are made. It is shown that radiation damping has a negligible effect on the scattering of light by free electrons, so that the Klein-Nishina formula is unchanged by the inclusion of damping effects. The effect of damping for mesons, on the other hand, is extremely large and reduces the cross-sections considerably. The main problems considered are the nuclear scattering of mesons and the energy radiated by mesons during collisions. The revised cross-sections are much more reasonable than those calculated previously, but on account of the inadequacy of the data no detailed comparison with the experimental results is possible.

1. The power possessed by water and air in motion of lifting and carrying solids considerably denser than themselves is familiar to all; but it does not seem to have been pointed out that classical hydrodynamics provides a simple explanation. If a solid rests on the bottom of a stream, the points of contact are points of zero velocity; and the velocity just above the solid, by the equation of continuity, must be greater than the general velocity. Hence the velocities produce high pressures under the solid and low ones above it, and the difference tends to lift the solid up. If the resulting thrust exceeds the weight of the solid in the liquid, the solid will be raised, and will be unable to rest in equilibrium on the floor of the stream.

Let M be a smooth manifold in . One may ask whether , the restriction of the Fourier transform of f to M makes sense for every f in . Since, for does not make sense pointwise, it is natural to introduce a measure σ on M and ask for an inequality

for every f in (say) the Schwartz class. Results of this kind are called restriction theorems. An excellent survey article on this subject is to be found in Tomas[13].

With the neglect of collisions, a relativistic form of Boltzmann's transport equation is derived and its invariance under a Lorentz transformation is established.

The assumption is often made that when competition is extremely intense at any stage in a life cycle, natural selection is bound to be intense also. This assumption will be examined quantitatively and it will be shown that the intensity of selection may diminish and become negative at high rates of elimination, while at its best its increase is extremely slow.

A variational principle is obtained for the Boltzmann transport equation in the presence of a magnetic field and modified by boundaries, when the standard formulation of Kohler breaks down because of the lack of microscopic reversibility. The connexion with entropy production is discussed.

We present, starting from a set of canonical axioms, a complete classification of the notions of non-commutative stochastic independence. Our result originates from a first contribution and a conjecture by M. Schürmann and is based on a fundamental paper by R. Speicher.

The unitary irreducible representations on a Hilbert space of the universal covering group G of the 3 + 2 deSitter group are determined, using Harish-Chandra's theory of semi-simple Lie groups. These irreducible representations of G are reduced out with respect to representations of the universal covering group K of the direct product R2 × R3 of a two-dimensional and a three-dimensional rotation group. It is convenient to distinguish between two types of irreducible representations of G, called ‘irreducible case’ and ‘reducible case’ representations. The former are more numerous in a certain sense, while the latter may be regarded as special cases.

Let (j, µ) be an irreducible unitary representation of K, where j denotes the (ray) representation of R3 involved (2j is a non-negative integer) and µ that of R2 (µ is a real number). Then in an ‘irreducible case’ representation of G, each (j, µ) that occurs at all occurs with multiplicity j + ½ if 2j is odd, or j + ½ ± ½ if 2j is even. In a ‘reducible case’ representation of G, the multiplicity of the (j, µ)'s is smaller, and may even be 1 for all (j, µ)'s which occur. (The analogous problem for the 4 + 1 deSitter group has been solved by L. H. Thomas by reducing out the irreducible representations of the whole group with respect to those of the four-dimensional rotation group R4. However, each irreducible representation of the 4 + 1 deSitter group contains any irreducible representation of R4 at most once, so that the more general Harish-Chandra theory is not then needed to carry out the classification.)

In an irreducible representation of G, only those irreducible representations (j, µ) of K may occur for which the fractional part of µ is fixed throughout the representation of G. This fractional part of µ helps to characterize the representations of G, as do the two polynomials in the infinitesimal elements, one of degree 2 and the other of degree 4, which generate all those polynomials which have the property of commuting with all members of the Lie algebra of G.

Representations of the inhomogeneous Lorentz group may be obtained from representations of G by the process of contraction, as defined by Inonu and Wigner, just as representations of the former group may in turn be contracted to obtain representations of the Galilei group.

The work of H. M. Srivastava (9) on generalizations of an interesting identity of Carlson is expanded upon in order to obtain series identities analogous to all four of the Appell functions and their generalizations. Reduction formulas for some Kampé de Fériet functions are derived as consequences of the various double-series identities considered in this paper.

Boron and carbon activated by diplon and proton bombardment have been used as positron emitting sources in experiments on the annihilation radiation.

By coincidence counting with two Geiger-Müller tubes it has been established that the radiation is emitted in pairs.

By the coincidence method of Becker and Bothe it has been shown that the annihilation radiation consists only of “soft” quanta.

By ordinary absorption measurements it has been shown that the annihilation radiation is homogeneous with a hardness corresponding to 0·5 million e-volt.

In order to study 3-manifolds with some particular group as fundamental group, one likes to have examples. There is an absence in the literature of means of constructing such examples. In this paper, theorems are proved which give necessary and sufficient conditions for the canonical 2-complex which corresponds to a group presentation to be a spine (2-dimensional skeleton in a cell decomposition with one 3-cell) of a connected closed† orientable 3-manifold. By enumerating all presentations of a group given by a particular presentation, two of the theorems provide an effective algorithm which permits one to attempt in a systematic way to construct 3-manifolds with a given group as fundamental group. Since every closed 3-manifold has a spine which corresponds to a group presentation, if the group is the fundamental group of an orientable 3-manifold, then the algorithm will eventually yield a 3-manifold, if not, then one is out of luck. There is no way out of this since Stallings has shown (1) that no algorithm exists which can, for every finitely presented group G, answer the question: Is G the fundamental group of a 3-manifold?

The idea of a net will be familiar to anyone who has made a model of a three-dimensional convex polytope (3-polytope) out of a flat sheet of card or similar material. To begin with, one cuts out a polygon, and then the model is formed by folding this and joining its edges in an appropriate manner. For example, Fig. 1 illustrates two nets for a regular tetrahedron. The folds are indicated by broken lines and lettering (either of the edges or of the vertices) indicates how the edges are to be joined.

Let n be a nonzero integer. A set of m positive integers {a1, a2, …, am} is said to have the property D(n) if aiaj+n is a perfect square for all 1 [les ] i [les ] j [les ] m. Such a set is called a Diophantine m-tuple (with the property D(n)), or Pn-set of size m.

Diophantus found the quadruple {1, 33, 68, 105} with the property D(256). The first Diophantine quadruple with the property D(1), the set {1, 3, 8, 120}, was found by Fermat (see [8, 9]). Baker and Davenport [3] proved that this Fermat’s set cannot be extended to the Diophantine quintuple, and a famous conjecture is that there does not exist a Diophantine quintuple with the property D(1). The theorem of Baker and Davenport has been recently generalized to several parametric families of quadruples [12, 14, 16], but the conjecture is still unproved.

On the other hand, there are examples of Diophantine quintuples and sextuples like {1, 33, 105, 320, 18240} with the property D(256) [11] and {99, 315, 9920, 32768, 44460, 19534284} with the property D(2985984) [19]].

The concept of random set, though vaguely known for a long time (possibly since Buffon's needle problem), did not develop until Robbins [25, 26] provided for the first time a solid mathematical formulation of this concept and investigated relationships between random sets and geometric probabilities. Later on (in a different context) this concept gave rise to a more general concept of set-valued function in topology, and applications were also found in several areas such as economics (see, for example, Aumann[3] and Debreu[10]) and control theory (see, for example, Hermes [13]), among others. Recently in two independent formulations, D. G. Kendall [17] and Matheron[21] provided a comprehensive mathematical theory of this concept influenced by the geometric probability point of view. Actually, much of the research in this area falls under the heading of stochastic geometry (see, for example, M. G. Kendall and P. A. P. Moran [18]). The D. G. Kendall and Matheron theories have been compared and ‘reconciled’ by Ripley [24]. In the past few years random sets have been investigated as extensions of random variables and random vectors, and in this framework the problems of deriving limit theorems have received a great deal of attention. This approach is benefiting greatly from probability results in Banach spaces.

Let the probability density of observations be denoted by φ(x | θ), where x stands for the variables and θ for the parameters. A function t of the observations is called an unbiased estimate of the function ψ(θ) of the parameters if

where dx stands for the product of differentials.

Let ν(M) denote the number of integers N in the range (0 ≤ N ≤ M) for which

where {y} denotes the fractional part of y. It is a convenient but trivial restriction to assume always that θ and φ lie between 0 and 1. There is a well-known result, which will be called ‘Weyl's Theorem’, that if θ is irrational,