The structure of modules over polycyclic group rings RG is investigated using the idea of a link P ⇝ Q between prime ideals of RG. The representation theory of RG splits into two parts – the part we discuss is determined by the representation theory of certain Noetherian polynomial identity factor rings of subrings of RG.

A system of equations is derived for determining the elastic fields in an inclusion and its surrounding finite matrix when the inclusion suffers a physical change and, if not constrained by the matrix, would undergo a deformation . A method for obtaining the exact solution of these equations, when the matrix and inclusion have the same elastic constants, is described and the particular problem of the square inclusion in an infinite matrix solved.

1. The method introduced by Dirac (1) in his classical theory of the interaction of an electron with an electromagnetic field has since been extended to the interaction of point particles with more general wave fields, such as occur in the neutral meson theory. Bhabha (2, 3) considered the scattering of neutral mesons by nucleons and gave arguments to show that typically quantum-mechanical effects should be unimportant if the energy of the incident meson is less than the rest energy of a nucleon, so that a classical theory with damping is appropriate to such problems. Now only the charged mesons are observable, and therefore a discussion of the classical theory of the interaction of a nucleon with a charge-bearing field is desirable. Apart from a paper by Fierz (4) which omits the dipole-like coupling terms, there is little published work on this subject.

One usually assumes that the Schrödinger operator on a thin quantum network (“fattened graph”) can be simulated by the ordinary Schrödinger operator on the corresponding one-dimensional quantum graph. On the other hand, each quantum graph can be constructed of standard elements – star graphs. We prove that a thin star-shaped quantum network which consists of a compact domain and a few straight semi-infinite quantum wires, with a two-dimensional Schrödinger operator on it, can be simulated by the corresponding solvable model in form of a one-dimensional star graph: an outer space, with an ordinary Schrödinger operator on the leads, a resonance vertex supplied with an inner space and a finite matrix in it and an appropriate boundary condition connecting the inner and outer components of elements from the domain of the model. The model is (locally) quantitatively consistent: the scattering matrix of the model on a certain spectral interval serves an approximation of the scattering matrix of the network. The role of the constructed star-graph model as a “jump-start” in analytic perturbation procedure on continuous spectrum is discussed.

In a pure loss GI/G/1 queueing system, necessary and sufficient conditions are given for the output to be a renewal process. These conditions involve dependence between the service distribution and the renewal function of the arrival process: for example, if pr {service time < ξ} = 0 for some ξ > 0, then it is sufficient for the renewal function to be that of a quasi-Poisson process with index ξ.

We give a condition on a full coaction (A, G, δ) of a (possibly) non-amenable group G and a closed normal subgroup N of G which ensures that Mansfield imprimitivity works; i.e. that A×δ[mid ]G/N is Morita equivalent to A×δG×δˆrN. This condition obtains if N is amenable or δ is normal. It is preserved under Morita equivalence, inflation of coactions, the stabilization trick of Echterhoff and Raeburn, and on passing to twisted coactions.

Each number a in the segment (0, ½) produces a Cantor set, Ca, by putting b = 1 − 2a and recursively removing segments of relative length b from the centres of the interval [0, 1] and the intervals which are subsequently generated. The distribution function of the uniform probability measure on Ca is a Cantor function, fa. When a = 1/3 = b, Ca is the standard Cantor set, C, and fa is the standard Cantor function, f. The upper derivative of f is infinite at each point of C and the lower derivative of f is infinite at most points of C in the following sense: the Hausdorff dimension of C is ln(2)/ln(3) and the Hausdorff dimension of S = {x ∈ C: the lower derivative of f is finite at x} is [ln(2)/ln(3)]2. The derivative of f is zero off C, the derivative of f is infinite on C — S, and S is the set of non-differentiability points of f. Similar results are established in this paper for all Ca: the Hausdorff dimension of Ca is ln (2)/ln (1/a) and the Hausdorff dimension of Sa is [ln (2)/ln (1/a)]2. Removing k segments of relative length b and leaving k + 1 intervals of relative length a produces a Cantor set of dimension ln(k + l)/ln(1/a); the dimension of the set of non-differentiability points of the corresponding Cantor function is [ln (k + l)/ln (1/a)]2.

We give a generalisation of the duality of a zero-dimensional complete intersection for the case of one-dimensional almost complete intersections, which results in a Gorenstein module M = I/J. In the real case the resulting pairing has a signature, which we show to be constant under flat deformations. In the special case of a non-isolated real hypersurface singularity f, with a one-dimensional critical locus, we relate the signature on the Jacobian module I/Jf to the Euler characteristic of the positive and negative Milnor fibre, generalising the result for isolated critical points. An application to real curves in ℙ2(ℝ) of even degree is given.

The polynomial invariants of a certain classical linear group of order 336 arise naturally in studying error-correcting codes over GF(7). An incomplete description of these invariants was given by Maschke in 1893. With the aid of the Poincaré series for this group, found by Edge in 1947, we complete Maschke's work by giving a unique representation for the invariants in terms of 12 basic invariants. A conjecture is made concerning the relationship between the Poincaré series and the degrees of the basic invariants for any linear group. A partial answer to this conjecture, due to E. C. Dade, is given.

Two kinds of duality arising in studies of interaction models are discussed. The first kind, which has not previously been investigated, is related to algebraic properties of the coefficient ring. The second kind is the well-known geometric duality for planar graphs. The two dualities together lead to a perfectly symmetrical relationship for a general form of partition function.

In this paper we prove the generalized version of the same n-type conjecture posed by McGibbon and Møller in [15, page 287]: it is shown that the set of all the same homotopy n-types of the suspension of the smash products of the Eilenberg-MacLane spaces is the one element set consisting of a single homotopy type.

In finite elasticity a homogeneous strain is not always determined uniquely by the conjugate stress. The phenomenon is investigated for any constitutive work-function, and in both its global and local aspects. Uniqueness is lost in specific critical states. Basic properties of their aggregates in the stress and strain spaces are derived, and shown to be fundamental for the material response near individual critical states. The geometry of local branches is analyzed in detail. Further properties resulting from material isotropy are obtained. The general theory is illustrated by examples typifying the responses of metal crystals and rubberlike polymers.

A Riesz space E has the (sequential) order-continuity property if every positive linear map from E to an Archimedean Riesz space is (sequentially) order-continuous. This is the case if and only if the canonical maps from E to its Archimedean quotient spaces are all (sequentially) order-continuous. I relate these properties to others that have been described elsewhere.

Let A ⊂ ℝ be finite. We quantitatively improve the Balog–Wooley decomposition, that is A can be partitioned into sets B and C such that

$ \begin{equation*} \max\{E^+(B) , E^{\times}(C)\} \lesssim |A|^{3 - 7/26}, \ \ \max \{E^+(B,A) , E^{\times}(C, A) \}\lesssim |A|^{3 - 1/4}. \end{equation*} $

$ \begin{equation*} |A+A| + |A A| \gtrsim |A|^{4/3 + 5/5277}. \end{equation*} $

In the theory of locales (or ‘pointless topology'), several authors have tried to find a suitable form of a T1-separation axiom. Their proposals are unsatisfactory in the sense that they do not coincide for topological spaces with the T1-axiom. Our main result is that sublocales of sober T1-spaces are exactly locales in which primes are dual atoms. Hence, these locales should be called T1, despite the fact that they include any locale without points. Our T1-locales are also closed with respect to products; in fact they form the smallest epireflective subcategory of locales containing all sober T1-spaces. Besides T1-locales we will also consider T2-locales and epireflective sub-categories of locales in general.

Two different calculations are made of lower bounds for the electronic energy of . In the first the method of truncated Hamiltonians due to Bazley and Fox is adapted in such a way that the nuclear charge rather than the energy becomes the eigenvalue. Lower bounds are calculated for the energies of the six lowest σg and six lowest σu states, as well as of the three lowest of both πg and πu symmetries. This approach gives better convergence than when the energy is used as eigenvalue. In the second calculation the method of Temple and Kato is shown to give a satisfactory value for the energy of the ground state, provided that some necessary knowledge of the energy of the first-excited state is available.

Let X be a non-negative discrete random variable with distribution {Px} and Y be a random variable denoting the undestroyed part of the random variable X when it is subjected to a destructive process such that

It is known that, for a transcendental entire function f, the Hausdorff dimension of the Julia set of f satisfies 1[les ]dim J(f)[les ]2. In this paper we introduce a family of transcendental entire functions fp, K for which the set {dim J(fp, K)[ratio ]0<p, K<∞} has infemum 1 and supremum 2.

1. The reflexion of X-rays from an ideal crystal would be independent of temperature if the particles forming the crystal were at rest. Because of the motion of the particles a temperature effect does exist. It can easily be seen that only the component of the displacement at right angles to the reflecting plane is of importance, since it is the path difference between such planes which matters. The complete theory has been worked out by Debye and by Waller. The result is that, owing to the influence of the temperature, the intensity is multiplied by a factor e−2m, where

Here and μs is the mean square of the component of the displacement (in a direction at right angles to the reflecting planes) due to a normal mode π is the glancing angle of incidence of the X-rays on the reflecting planes, and λ is the wave-length.