If R is a Noetherian local ring and I = (x1, …, xn)R is an ideal of R then the Rees algebra R[It] can be represented as a homomorphic image of the polynomial ring R[Z1, …, Zn]. The kernel is a homogeneous ideal, and the smallest of the degree bounds among all generating sets, called the relation type of I, is independent of the representation. We derive formulae connecting the relation type of I with the reduction number of I when the analytic spread of I exceeds height(I) by one. In the process we define complete d-sequences with respect to I and use them to help achieve our results. In addition some results on the behaviour of the relation type modulo an element are proved, and examples where the relation type is explicitly computed are presented.

The magnitude of a graph is one of a family of cardinality-like invariants extending across mathematics; it is a cousin to Euler characteristic and geometric measure. Among its cardinality-like properties are multiplicativity with respect to cartesian product and an inclusion-exclusion formula for the magnitude of a union. Formally, the magnitude of a graph is both a rational function over ℚ and a power series over ℤ. It shares features with one of the most important of all graph invariants, the Tutte polynomial; for instance, magnitude is invariant under Whitney twists when the points of identification are adjacent. Nevertheless, the magnitude of a graph is not determined by its Tutte polynomial, nor even by its cycle matroid, and it therefore carries information that they do not.

In this paper we prove several results concerning the complexity of a set relative to compact sets. We prove that for any Polish space X and Borel set B ⊆ X, if B is not , then there exists a compact zero-dimensional P ⊆ X such that p ∩ X is not . We also show that it is consistent with ZFC that, for any A ⊆ ωω, if for all compact K ⊆ ωωA ∩ K is , then A is . This generalizes to in place of assuming the consistency of some hypotheses involving determinacy. We give an alternative proof of the following theorem of Saint-Raymond. Suppose X and Y are compact metric spaces and f is a continuous surjection of X onto Y. Then, for any A ⊆ Y, A is in Y iff f−1(A) is in X. The non-trivial part of this result is to show that taking pre-images cannot reduce the Borel complexity of a set. The techniques we use are the definability of forcing and Wadge games.

In § 7 of [1] we described how a stochastic integral of non-commuting process cesan be constructed. This was achieved via a Doob-Meyer decomposition for the square of a self-adjoint L2-martingale. We introduced a ‘condition D'’ derived from the lsquo;class D’ of stochastic process theory, and showed that if the square of a self-adjoint L2-martingale satisfies this condition then it has a decomposition of the Doob-Meyer type. The purpose of this paper is to extend the results of § 7 of [1] and to introduce another construction of the stochastic integral that does not employ condition D.

A linear connection on a differentiable manifold M defines a horizontal distribution on the tangent bundle T(M). Horizontal distributions on tangent bundles are of some interest even when they are not generated by connections. Much of linear connection theory generalizes to arbitrary horizontal distributions. In particular, there are generalized versions of Bianchi's two identities for the torsion T and curvature R

where C denotes the cyclic sum with respect to X, Y and Z. These identities are derived below by associating with a horizontal distribution a graded derivation of degree 1 in a graded Lie algebra of vertical forms on M. This approach reveals the fundamentally algebraic origin of the Bianchi identities.

Let X be a compact Riemann surface X of genus at–least two. Fix a holomorphic line bundle L over X. Let be the moduli space of Hitchin pairs (E, φ ∈ H0(End0(E) ⊗ L)) over X of rank r and fixed determinant of degree d. The following conditions are imposed:

1. (i) deg(L) ≥ 2g−2, r ≥ 2, and L⊗rKX⊗r;

2. (ii) (r, d) = 1; and

3. (iii) if g = 2 then r ≥ 6, and if g = 3 then r ≥ 4.

We investigate some connectedness properties of the set of points K(f) where the iterates of an entire function f are bounded. We describe a class of transcendental entire functions for which K(f) is totally disconnected if and only if each component of K(f) containing a critical point is aperiodic. Moreover we show that, for such functions, if K(f) is disconnected then it has uncountably many components. We give examples of functions for which K(f) is totally disconnected and we use quasiconformal surgery to construct a function for which K(f) has a component with empty interior that is not a singleton.

Isotropic Gaussian processes, of which we shall give a formal definition presently, arise in various practical problems. The present inquiry arose from the consideration of the variability found in the yields of plots in agricultural field experiments. Samples of such patterns of variability can be obtained from uniformity trials, whereby a piece of land is treated uniformly throughout and the crop is harvested in small units or plots. The results from such trials have been widely used to determine optimum plot sizes for future experiments with the crop concerned, but it has never been clear how valid is the generalization from a uniformity trial to future experiments on other sites. One difficulty arises from the lack of a suitable model to express the variability; another difficulty arises from the formidable analytical problems besetting any attempts to apply deductive reasoning to even simple models. We may mention two approaches that have been made to the uniformity trial problem: Fairfield Smith (3) has supplied an empirical law connecting the variance of contiguous groups of plots with the size of the group, and Quenouille (6) and Whittle (8) have considered the fitting of two-dimensional isotropic Gaussian processes to uniformity trial data. Whittle has pointed out that a satisfactory model may have to incorporate an additional random element at each point, and has outlined the difficulties of estimation which arise when such an element is introduced. The connexion between Fairfield Smith's law and the approach via two-dimensional stochastic processes, if any, seems to be quite unknown.

We derive simple conditions, which are both necessary and sufficient, for the validity of sharp local embeddings $H^{\sigma,\alpha} X({\bf R}^{n}) \hookrightarrow Y({\bf {\Omega})$ and sharp global embeddings $H^{\sigma,\alpha}, X({\bf R}^{n}) \hookrightarrow Z({\bf R}^{n})$. Here $H^{\sigma,\alpha}$, stands for a Bessel potential operator involving the classical smoothness $\alpha$ and logarithmic smoothness $\alpha$, X, Y and Z are (generalized) Lorentz–Zygmund spaces, and $\Omega\subset {\bf R^{n}}$ is an open subset whose n-dimensional Lebesgue measure is finite. Our results extend those of [18] and improve them especially in the extreme cases when X is close to L1, or Y and Z are close to $L^{\infty}$, or when $\sigma=0$ and $\alpha > 0$.

Let N be a positive integer. We enquire whether there is a quadratic f(t)=at2 + bt + c, not identically square, which takes square values for N consecutive integers t. In [1] it was shown that this was possible, in what we shall call the symmetric case, if N = 8. A better treatment of this case appears here, together with a discussion of the nonsymmetric case, where we may take N = 7. There is no evidence that these are the best possible results.

Given a presentation of a finitely presented group, there is a natural way to represent the group as the fundamental group of a 2-complex. The first part of this paper demonstrates one possible way to represent a finitely presented algebra $S$ in a similarly compact form. From a presentation of the algebra, we construct a quiver with relations whose path algebra is finite dimensional. When we adjoin inverses to some of the arrows in the quiver, we show that the path algebra of the new quiver with relations is $M_n(S)$ where $n$ is the number of vertices in our quiver. The slogan would be that every finitely presented algebra is Morita equivalent to a universal localization of a finite dimensional algebra.

The general Tchebysheffian approximation problem is the following: Given a real continuous function g(x) in the interval a ≤ x ≤ b, and a real function f(x; Pj) of prescribed form, continuous in the variable x and the parameters Pj, determine the values of the parameters so that the measure of absolute error or ‘deviation’ of f, shall be as small as possible.

The notion of statistical convergence of a sequence (xk) in a locally convex Hausdorff topological linear space X was introduced recently by Maddox[5], where it was shown that the slow oscillation of (sk) was a Tauberian condition for the statistical convergence of (sk).

We consider links in an orientable 3-manifold M which have an alternating diagram on a closed orientable surface F of positive genus. We see that if the diagram is ‘complex’ enough and if F gives a Heegaard splitting of M, then such a link L is prime and M—L does not contain an essential torus.

In this paper a precise distribution-theoretic formalism for the description of point charges interacting through a classical electromagnetic field is given. The distribution solutions of the Maxwell equations are shown to reduce to the Lienard–Wiechert fields, which are summable over the whole of space-time. It is not possible to obtain directly a unique field intensity, on the world lines, from these distribution solutions. A brief analysis shows that analytic continuation methods also do not give a unique field intensity on the world lines. The energy tensor for the field is a distribution which can only be rigorously defined in terms of the motions of the charges. Thus only the value of the field energy or momentum residing in any finite region of space-time can be given any meaning. Equations of motion for the charges may be derived from the field energy tensor, using the principle of conservation of total energy and momentum for the system. These equations agree with those obtained previously by Dirac (2). Also any canonical formalism in which the field potentials and the particle variables enter in an equivalent manner is not possible. The difficulty of the definition of products of distributions in the stress tensor does not occur.

1. Definition 1. A linear set E is said to possess the property C if, to any sequence of positive numbers {ln} (n = 1, 2, …), there corresponds a set of intervals, of length not greater than l1, l2, …, which includes all the points of E.

The investigations of von Karman dealing with the unsymmetrical double row of vortices in an infinite sea of liquid are well known. He found that the unsymmetrical double row is stable when, and only when, cosh2πa/b = 2, where 2a is the distance between the two rows and 2b is the distance between consecutive vortices on the same row. A detailed account of the stability of the Karman street and of the symmetrical double row has been given by Lamb, and it has been shown that the symmetrical double row is unstable for all values of the ratio a/b. The object of this paper is to investigate the stability of a double row of vortices of arbitrary stagger. We define a double row of stagger 2l to be the system formed by positive vortices at the points (2nb + l, a) and negative vortices at (2mb − l, − a), where m and n assume all integral values from − ∞ to + ∞. The vortices are thus neither exactly “in step” nor exactly “out of step.” When l = 0 the system reduces to the symmetrical double row and when the system is the unsymmetrical double row.

In [13] T. H. Pate has recently proved an important inequality between immanants of Hermitian positive semi-definite matrices. In this paper we introduce some more inequalities with the eventual aim of better understanding the permanental dominance conjecture. Indeed the results in this paper together with known results show that the permanental dominance conjecture for immanants is true for all n × n Hermitian positive semi-definite matrices with n ≤ 8, except perhaps for the case of the partition (24) of 8.

A Markov branching process in either discrete time (the Galton–Watson process) or continuous time is modified by the introduction of a process of catastrophes which remove some individuals (and, by implication, their descendants) from the population. The catastrophe process is independent of the reproduction mechanism and takes the form of a sequence of independent identically distributed non-negative integer-valued random variables. In the continuous time case, these catastrophes occur at the points of an independent Poisson process with constant rate. If at any time the size of a catastrophe is at least the current population size, then the population becomes extinct. Thus in both discrete and continuous time we still have a Markov chain with stationary transition probabilities and an absorbing state at zero. Some authors use the term ‘emigration’ as an alternative to ‘catastrophe’.

Four-index tensors with all possible terms either (a) quadratic in the Riemann curvature tensor, or (b) linear in its second derivatives, with coefficients constructed from products of metric tensors, are enumerated for the case of Riemannian space-time. All linear combinations of these which have zero divergence, and which might therefore be associated with the super-energy in gravitational theory, are found by elementary methods. The only independent such combination reduces in empty space-time to the tensor previously found by Bel and Robinson.