Dirichlet forms and their associated function spaces have been studied by a number of authors [4, 6, 7, 12, 15–18, 22, 25, 26]. Important motivation for the study has been the connection of Dirichlet forms with Markov processes [16–18, 25, 26]: for example, to every regular symmetric Dirichlet form, there is an associated Hunt process [13, 20]. This makes the theory of Dirichlet forms a convenient source of examples of Hunt processes. In the non-commutative setting, Markov fields have been studied by several authors [1–3, 14, 19, 24, 28]. It is therefore interesting to develop a non-commutative extension of the theory of Dirichlet forms and to study their connection with non-commutative Markov processes.

We obtain a formula for the essential supremum of the packing dimensions of the sections of sets parallel to a given subspace. This depends on a variant of packing dimension defined in terms of local projections of sets.

Introduction. If ξ is a real number we denote by ∥ ξ ∥ the difference between ξ and the nearest integer, i.e.

It is well known (e.g. Koksma (3), I, Satz 4) that if θ1, θ2, …, θn are any real numbers, the inequality

has infinitely many integer solutions q > 0. In particular, if α is any real number, the inequality

has infinitely many solutions.

Let Q be a loop; then the left and right translations La(x) = ax and Ra(x) = xa are permutations of Q. The permutation group M(Q) = 〈La, Ra | a ε Q〉 is called the multiplication group of Q; it is well known that the structure of M(Q) reflects strongly the structure of Q (cf. [1] and [8], for example). It is thus an interesting question, which groups can be represented as multiplication groups of loops. In particular, it seems important to classify the finite simple groups that are multiplication groups of loops. In [3] it was proved that the alternating groups An are multiplication groups of loops, whenever n ≥ 6; in this paper we consider the finite classical groups and prove the following theorems

1. Let f(x1, x2, …, xn) be a homogeneous form with real coefficients in n variables x1, x2, …, xn. Let a1, a2, …, an be n real numbers. Define mf(a1, …, an) to be the lower bound of | f(x1 + a1, …, xn + an) | for integers x1, …, xn. Let mf be the upper bound of mf(a1, …, an) for all choices of a1, …, an. For many forms f it is known that there exist estimates for mf in terms of the invariants alone of f. On the other hand, it follows from a theorem of Macbeath* that no such estimates exist if the region

has a finite volume. However, for such forms there may be simple estimates for mf dependent on the coefficients of f; for example, Chalk has conjectured that:

If f(x,y) is reduced binary cubic form with negative discriminant, then for any real a, b there exist integers x, y such that

It is shown that anomalously large and anomalously small values of the constant A in the thermionic current formula for metals can be explained by the usual statistical equilibrium theory provided proper account is taken of the fact that free electrons may be shared by two competing overlapping energy bands. When this theory is developed there arises a comparatively large temperature variation of the statistical parameter ζ which occurs in the Fermi function, and when the exact formula is forced into the empirical form with a constant work function, the temperature factor appears as a multiplying factor of the current constant, this factor being greater or less than unity depending upon the nature of the overlapping bands. In particular the observed values of the constant A for nickel and for hafnium are well explained.

This paper is concerned with expressions of the form

where fa(x) = ax. Thus we have

Such expressions have been described variously as continued exponentials, iterated exponentials, towers of exponents or hyperpowers. We shall frequently make use of their elementary properties, as for example

and

This paper discusses a problem in traffic flow by the method of kinematic waves developed by Lighthill and Whitham(1,2). The theory of kinematic waves introduced by Lighthill and Whitham has been extended here to include the case when the flow q varies with the time, and it is seen that the expression for the wave velocity is of the same form as before, namely, ∂q/∂k, where q is the flow (quantity passing a given point in unit time) and k is the concentration (quantity per unit distance). The theory is applied to the problem of estimating how a uniform oncoming flow behaves on entering a bottleneck, the capacity of which varies with time. This capacity has initially a higher value than the oncoming flow but falls at a uniform rate to a lower value, where it remains constant for a time, and again rises at a uniform rate to the original value. A shock wave is found to move back from the bottleneck, and later forward again and through it, much as in the case of a bottleneck of constant capacity with varying oncoming flow studied by Lighthill and Whitham.

Let L be a ring additively isomorphic to ℤd. The zeta function of L is defined to be

where the sum is taken over all subalgebras H of finite index in L. This zeta function has a natural Euler product decomposition:

These functions were introduced in a paper of Grunewald, Segal and Smith [5] where the local factors ζL[otimes ]ℤp(s) were shown to always be rational functions in p−s. The proof depends on representing the local zeta function as a definable p-adic integral and then appealing to a general result of Denef’s [1] about the rationality of such integrals. The proof of Denef relies on Macintyre’s Quantifier Elimination for ℚp [8] followed by techniques developed by Igusa [6] which employ resolution of singularities.

We obtain a result compatible with Bloch's conjecture for Mumford's fake plane. In the process some new surfaces associated with Mumford's surface are found.

The polynomial invariants of classical links that relate to the Jones polynomial have various inter-connections. Here a new one is given: the square of the Jones polynomial is obtained from the F-polynomial by means of a certain substitution of variables. This explains and corrects some similarities between known evaluations of these polynomials. The proof uses the ‘Dubrovnik’ version of the F-polynomial; this is first proved to be equivalent to the F-polynomial. Finally a correlation between the F and P polynomials is included as a provocative curiosity.

Let (Q,.) be a finite quasigroup, i.e. a finite set Q with a binary operation. called multiplication such that in the equation x.y = z any two elements determine the third uniquely. Then the mappings R(x): Q → Q; q ↦ q.x and L(x): Q → Q; q ↦ x.q are permutations of Q. The multiplication group G of Q is the subgroup of the symmetric group on Q generated by {R(x), L(x) | x ∈ Q}. If S is a field, G has a faithful representation Ḡ by permutation matrices acting on the S-vector space with Q as basis. The set of matrices commuting with Ḡ forms an S-algebra (under the usual operations) called the centraliser ring V(G, Q) of G on Q. The purpose of this note is to show how the permutation-theoretic object ‘centraliser ring’ may be expressed in terms of the quasigroup structure of Q, both to prepare one tool for the long-term programme of classifying finite quasigroups by means of their multiplication groups, and for comparison with the Schur ring method of group theory.

The concept of an m-full ideal was introduced and studied first by D. Rees (unpublished). In 1983, after having considered and discussed the concept with Professor Rees, the author showed some properties of these ideals in [11], and other authors also have obtained results related to them (cf. [5, 7]). The purpose of this paper is to seek syzygies of m-full ideals and try to analyze their structure. Let a be an m-full ideal, and ᾱ the reduction by a general element. Then it is possible to determine the number of basic syzygies of a in terms of ᾱ. We show that this leads to a method for obtaining a set of basic syzygies of a provided that one for ᾱ is known (Theorem 6). Moreover the entire structure of the syzygy module is known when it is reduced by a general element. It turns out that a/za is the direct sum of ᾱ and copies of the residue field (Corollary 7).

Let Fn be a free group of finite rank n with basis {x1,…, xn}. Let be a variety of groups and write for the verbal subgroup of Fn corresponding to . (See [11] for information on varieties and related concepts.) Every automorphism of Fn induces an automorphism of the relatively free group Fn/V, and those automorphisms of Fn/V arising in this way are called tame. If is the variety of all metabelian groups and n ╪ 3 then every automorphism of Fn/V is tame [2, 4, 12]. But this is an exceptional situation. For many (and probably most) other varieties , Fn/V has non-tame automorphisms for all sufficiently large n. This holds for the variety of all nilpotent groups of class at most c where c ≥ 3 [1, 3] and for nearly all product varieties including, in particular, the variety of all groups whose derived groups are nilpotent of class at most c, where c > 2 [10, 13].

In this paper explicit expressions in closed forms are first obtained for the complex potentials and deflexion at any point of a circular annular plate under various edge conditions when the plate is acted upon by general line loadings distributed along the circumference of a concentric circle. These solutions are then used to discuss the bending of a circular plate with a central hole under a concentrated load or a concentrated couple acting at any point of the plate. Solutions for singularly loaded sectorial plates bounded by two arcs of concentric circles and two radii are also derived when the plate is simply supported along the straight edges. The boundary conditions along the circular edges include the cases of a free boundary as well as the elastically restrained boundary which covers the usual rigidly clamped and simply supported boundaries as special cases. The usual restrictions relating to the small deflexion theory of thin plates of constant thickness are assumed. Limiting forms of the resulting solutions are investigated.

In [1] we used the alternating group A5 to produce a set of five permutations of order three acting on 12 letters; this set was normalized by A5 and generated the Mathieu group M12. We analogously used the linear group PSL2(7) to produce a set of seven involutions acting on 24 letters which was normalized by our PSL2(7) and generated M24. In this paper we describe the combinatorial and geometric interpretations of our generators, and aim to justify our use of the word ‘natural’. The generators for M12 are defined both as permutations of twelve 5-cycles in a conjugacy class of A5, and as permutations of the faces of a dodecahedron. The outer automorphism of M12 emerges as a bonus. The generators of M24 are defined analogously both as permutations of twenty-four 7-cycles in the action of PSL2(7) on seven letters, and as permutations of the 24 faces of a certain cubic graph drawn on a surface of genus three. In addition we mention the connection between the latter construction and the binary Golay code.

In the paper (4) of Green and Rees it was established that the finiteness of finitely generated semigroups satisfying xr = x is equivalent to the finiteness of finitely generated groups satisfying xr−1 = 1 (Burnside's Problem). A group satisfying x2 = 1 is abelian and if it is generated by n elements, it has at most 2n elements. The free finitely generated semigroups satisfying x3 = x are thus established to be finite, and in fact the connexion with the corresponding problem for groups can be used to give an upper bound on the size of these semigroups. This is a long way from an algorithm for a solution of the word problem however, and providing such an algorithm is the purpose of the present paper. The case x = x3 is of interest since the corresponding result for x = x2 was done by Green and Rees (4) and independently by McLean(6).

When an atomic system emits or absorbs radiation, the frequency of the radiation is connected with the change of energy of the system by Bohr's frequency condition δE = hv. This equation is usually applied to the transition of an atom from one stationary state to another, the frequency of the radiation being measured in a frame of reference in which the atom as a whole is at rest, both before and after the transition. It has been pointed out by Bohr that difficulties arise when no natural frame of reference presents itself in which to measure the frequency, which is the case for non-isolated systems, or even for isolated systems when the process of radiation is connected with a change of momentum.

The object of this note is to obtain necessary and sufficient conditions for the existence of

on the assumption that ø is integrable in any finite positive interval not including zero (0 < α ≤ x ≤ β < ∞). On reference to the literature on the topic, I find no exhaustive analysis of this problem anywhere.

1. Introduction. The Nullstellensatz in commutative algebraic geometry may be described as a means of studying certain commutative rings (viz. affine algebras) by their homomorphisms into algebraically closed fields, and a number of attempts have been made to extend the result to the non-commutative case. In particular, Amitsur and Procesi have studied the case of general rings, with homomorphisms into matrix rings over commutative fields ((1), (2)) and Procesi has obtained more precise results for homomorphisms of PI-rings (11). Since a finite-dimensional division algebra can always be embedded in a matrix ring over a field, this includes the case of skew fields that are finite-dimensional over their centre, but it tells us nothing about general skew fields.