We use certain derivations of a polynomial ring, that do not leave any proper non-zero ideal invariant, to construct simple non-holonomic modules over the nth Weyl algebra. This approach extends to the rings of differential operators of other smooth affine varieties, like smooth quadric surfaces.

1. Introduction. Throughout the paper θ = (θ1, …, θn), φ = (φ1, …, φn), … denote points of Euclidean space Rn. We write Kn for the set of θ in Rn for which θl, …, θn, 1 are linearly independent over the rational numbers. We denote points of the set of integer n-tuples Zn by x, y, … We write

If α is a real number, ∥α∥ denotes the distance from α to the nearest integer.

Let θ ∈ Rn. By a theorem of Dirichlet ((2), chapter 1, theorem VI).

for all X ≥ 1. We say that θ is singular if

Singular points form a set of n-dimensional Lebesgue measure zero. In fact, H. Davenport and W. M. Schmidt (3) showed that

for almost all θ in Rn. Although there are no singular numbers in Kl ((2), p. 94) there are ‘highly singular’ n-tuples in Kn for n ≥ 2.

Let be a -grading of a simple Lie algebra . The commuting variety associated with such a grading is the variety of pairs of commuting elements from . We study the problem of irreducibility of these varieties. Using invariant-theoretic technique, we present new instances of reducible and irreducible commuting varieties.

Let F(z) be a Hecke–Maass form for SL(m, ℤ) with m ⩽ 3, or be the symmetric power lift of a Hecke–Maass form for SL(2, ℤ) if m = 4, 5 and let AF (n, 1, . . ., 1) be the coefficients of L-function attached to F. We establish

$$\sum_{q\leq Q}\max_{(a,q)=1}\max_{y\leq x}\left|\sum_{n\leq y \atop n\equiv a\bmod q}A_F(n,1, \dots, 1)\Lambda(n)\right| \ll x\log^{-A}x,$$

An integral inequality for a compact Lorentzian manifold which admits a timelike conformal vector field and has no conjugate points along its null geodesics is given. Moreover, equality holds if and only if the manifold has nonpositive constant sectional curvature. The inequality can be improved if the timelike vector field is assumed to be Killing and, in this case, the equality characterizes (up to a finite covering) flat Lorentzian $n(\geq3)$-dimensional tori. As an indirect application of our technique, it is proved that a Lorentzian $2-$torus with no conjugate points along its timelike geodesics and admitting a timelike Killing vector field must be flat.

The aim of this paper is to describe the possible structures for NEC (non-euclidean crystallographic) groups of the hyperbolic plane with non-compact quotient space. The case of compact quotient space was settled by Wilkie (6), and it has been shown by Hoare, Karrass and Solitar (3) that all subgroups of infinite index in a Wilkie group have a certain structure pattern. Their proof depends on showing that a subgroup of infinite index in a Wilkie group has a presentation satisfying certain restrictions on the occurrences of each generator. Our method here is to study the presentation of an arbitrary NEC group which is given by the classical construction of the Dirichlet region, and to prove that it satisfies the same restrictions as were found for the above subgroups of Wilkie groups.

The detailed study of n-dimensional crystallography appears to have been initiated by Hermann but unfortunately only a small fragment of his work was ever published(4). Recently, interest in the subject has been revived by the work of Bülow, Neubüser and Wondratschek(2), who not only obtained definitive results on 4-dimensional lattices and space groups but also proposed general definitions and procedures for the n-dimensional case. In particular they introduced the concept of crystal family and so resolved satisfactorily the controversy on whether there are six or seven 3-dimensional crystal systems (see(1)). Their point of view is arithmetic, that is, they use a coordinate system adapted to the particular lattice under consideration and represent elements of its symmetry group by integer matrices.

In the present paper we suggest as a cobordism invariant of an immersed or embedded submanifold in Euclidean space the singularity set of its projection to a hyperplane. A similar approach has been employed by Banchoff[1] and Koschorke[6]; see also [15]. We consider the range of dimensions n ≤ 3k where n is the dimension and k is the codimension. We prove that in this range (1) our singularity invariant is complete modulo 2-torsion, and (2) modulo-torsion, it can take any value from Thorm's oriented cobordism group of corresponding dimension for k even, while for k odd this invariant is always trivial.

The readings Φ(x) of an instrument with the apparatus function g(x) are related to the true values f(x)of the observed function by the equation

A three-point interpolation operator can be chosen to simulate the spreading action of a class of functions g(x). Under certain conditions the interpolation operator has an exact inverse which, when applied to Φ(x), gives a function which is a better approximation to f(x). The accuracy obtained is discussed, and the effect of the method on the standard errors in the measurement is found.

By “sensitive” time of a cloud chamber is meant the length of time for which the supersaturation in the chamber remains sufficient to cause condensation along the tracks of ionizing particles. If the cooling expansion is made rapidly this sensitive time depends on the rate at which the temperature of the gas in the chamber rises after the expansion. This problem of the warming of a gas in a vessel, after an initial cooling, has been previously considered by Rayleigh (1). To follow the whole history of the warming until the gas reaches the temperature of the vessel requires an elaborate mathematical treatment which can only be carried out for specially shaped vessels. Rayleigh considers two special cases, viz. a gas contained between two parallel infinite planes, and a gas contained in a spherical vessel. The results obtained by Rayleigh for these cases are, however, not very helpful in dealing with the cloud-chamber problem under consideration. This is so, not so much because specially shaped vessels are assumed, but because the treatment does not lend itself to an evaluation of the rate of warming in the early stages of the process. It is this initial warming that we are concerned with here, since a cloud chamber ceases to be sensitive when the average temperature of the gas has risen by only 5% or less of the cooling produced by the expansion. Rayleigh's result for the average rise in temperature of the gas at any instant is represented by an infinite series which converges very slowly when the rise in temperature is small.

Many problems in physics, chemistry and other subjects give rise to differential equations which are difficult or impossible to treat mathematically. It has long been possible to obtain numerical solutions by arithmetic methods, and the development of adding and multiplying machines to their present state of perfection has made these much simpler to carry out. The amount of work involved is, however, still very great.

We give an example of a univalent orientation-preserving harmonic mapping f = h + g¯ defined on the unit disc U which is real on the real axis, satisfies and is not typically real. Furthermore, we give a geometric characterization for univalent, harmonic and typically real mappings.

If F is a free profinite group, it is well known that the closed subgroups of F need not be free profinite; however, if p is a prime number, every closed subgroup of a free pro-p-group is free pio-p (cf. [2, 8, 7]). In this paper we show that there is an analogous contrast regarding the closed subgroups of free products in the category of profinite groups, and the closed subgroups of free products in the category of pro-p-groups, at least for (topologically) finitely generated subgroups.

In this paper, we are going to introduce an elliptic analogue of the classical Mahler measure of an integral polynomial. The measure is shown to vanish if and only if the roots of the polynomial are attached to division points of the curve. This is the exact analogue of the statement that the Mahler measure of an integral polynomial vanishes if and only if all of its roots are division points of the circle (in other words, roots of unity). The proof of this result exploits an integral representation for the local canonical heights on the elliptic curve. The integral is that of a simple polynomial function over the complete curve in the appropriate valuation. Attention then shifts to the calculation of local integrals of arbitrary rational functions on elliptic curves. Results are proved which show these integrals may be computed as effective limits of Riemann sums. Finally, consideration is given to analogous behaviour for abelian varieties. We give a method, in principle, for computing the global canonical height of a rational point on an abelian variety denned over an algebraic number field, once again exploiting the integral representation of the local heights. The methods used include recent inequalities for linear forms in elliptic and abelian logarithms.

Let A1,A2,…,An be events on a given probability space, and let mn be the number of those Aj which occur. Put S0 = S0,n = 1, and

where the summation is over all subscripts satisfying 1 ≤ i1 < i2 < … < ik ≤ n. For convenience in some formulae we adopt the convention Sk, n = 0 if k > n. By turning to indicator variables one immediately finds that

Inequalities of the form

where ck = ck(r, n) and dk = dk(r, n) are constants (not dependent on the events Aj, 1 ≤ j ≤ n), possibly zero, are called Bonferroni-type inequalities. This same name applies if P(mn = r) is replaced by P(mn ≥ r) in the middle. The best known such inequalities are the method of inclusion and exclusion

where j ≥ 0 is an arbitrary integer. An extension of (4) to arbitrary r, called Jordan's inequalities (see Takács [16]), is as follows: for 0 ≤ r ≤ n and for any integer j ≥ 0,

It is observed by Galambos and Mucci[9] that (5) follows from (4), and indeed, one can always generate inequalities of the form (3) for arbitrary r from the special case r = 0 if one utilizes only Sr,Sr+1… in the case of P(mn = r). As a matter of fact, from the instructions of Galambos and Mucci one has that the inequalities

hold for an arbitrary sequence A1,A2,…,An of events if, and only if, for an arbitrary sequence A1,A2,…,An−r of events,

where

Let G be a finite soluble group with Fitting subgroup F(G). The Fitting series of G is defined, as usual, by F0(G) = 1 and Fi(G)/Fi−1(G) = F(G/Fi−1(G)) for i ≥ 1, and the Fitting height h = h(G) of G is the least integer such that Fn(G) = G. Suppose now that a finite soluble group A acts on G. Let k be the composition length of A, that is, the number of prime divisors (counting multiplicities) of |A|. There is a certain amount of evidence in favour of the

CONJECTURE. |G:Fk(G)| is bounded by a number depending only on |A| and |CG(A)|.

In 1956 Toda [5] introduced two fibrations localized at p > 2:

where is a subcomplex of the James construction . The construction of H′ was somewhat difficult and was discussed by Moore (see [3]). Moore's definition however is not natural (in the sense of Theorem 1 (b) below). It is our purpose to give another definition, closer to Toda's original definition, which is natural, is an H map* and behaves well with respect to the Dyer-Lashof map λ: BΣp → Q(So). We use this to settle an unanswered question in [2].

Spatially homogeneous cosmological models of the Bianchi type-V are considered in the Brans-Dicke scalar-tensor theory of gravitation. Exact solutions are given in the vacuum case as well as for models filled with dust, radiation or stiff matter.