The possibility of constructing a quantum field theory by means of fields on Euclidean space is based on works by Schwinger and Symanzik (1). Probabilistic methods were used and Nelson has shown (2) that from so-called ‘Markoff fields’ one can construct Wightman fields. This idea turned out to be unusually fruitful as it made available the statistical mechanic's techniques for consideration of quantumfield theory problems. See for example (7). However as in the Minkowski space approach, the only two-dimensional space-time nontrivial models have been proved to fulfil all Wightman axioms. Since the problem for higher dimension theories is still open and extremely difficult, it is useful to have at least some criterion which allows us to eliminate certain procedures as leading to trivial theories. One such criterion existing in the Minkowski space approach is described by a notion of Borchers class (5).

Several methods have been developed in recent years in the Cavendish Laboratory for the absolute measurement of fast neutron flux. All depend on observing the effects of elastic collisions between neutrons and light nuclei. The methods fall into two categories according as individual recoil nuclei are counted (1, 2), or the total ionization current due to the recoils is measured (3). In order completely to interpret the experimental results, it is necessary to know the cross-section for scattering and the angular distribution of the recoil nuclei. The total number of recoil nuclei and their energy distribution are then determined for a known incident neutron spectrum. For precise neutron flux measurements, recoil protons are invariably studied, as the neutron-proton scattering cross-section has been measured over a wide range of energies (4, 5), and the angular distribution of the recoils is isotropic in the centre of gravity space for neutrons of energy less than about 10 MeV. (6, 7). This makes the reduction of the experimental results particularly simple and certain.

Let denote a sequence of integers exceeding 1, and let τ(n, ) be the number of those divisors of n which belong to . We say that is a Behrend sequence if

where, here and in the sequel, we use the notation p.p. to indicate that a relation holds on a set of asymptotic density one.

1. Introduction. In 1948, H. Heibronn (6) improved Vinogradov's result (11) and obtained that for any ε > 0 there exists some positive constant C(ε) which depends on ε only such that, for any real number θ and any integer N ≥ 1, there is an integer x satisfying

Here ∥t∥ means the distance from t to the nearest integer.

1. Saks calls an integral complete if it satisfies two conditions, of which one, the condition (C), is as follows.

If f(x) is defined in (a, b) and is integrable in each interval (a + ε, b − η), where a < a + ε < b − η < b, and

exists, then f(x) is integrable in (a, b) and the integral over (a, b) is equal to the above limit.

In automatic temperature-controlling apparatus a temperature-sensitive element, commonly called a thermostat, serves to increase or decrease a supply of heat to the body or oven whose temperature is to be prevented from changing. The thermostat and heating elements are commonly both electrical, the thermostat comprising a “master coil” whose resistance changes with temperature, the heating element consisting of a “slave coil” carrying the heating current. The temperature of the master coil is made to control the current in the slave coil, usually by the interposition of some form of relay with the appropriate translating and amplifying apparatus. Since every form of relay exhibits “backlash”—the critical values of the operative signal at make and at break are not exactly equal—such a system necessarily oscillates or “hunts” through a range at least as great as the backlash of the relay; and since further there is thermal separation between the master coil and the slave coil, the hunting has a range exceeding the backlash of the relay. It is a common experience with such thermostatic apparatus that, owing to this action, continued improvement towards constancy of temperature is not attainable by increasing the delicacy with which the temperature of the master coil controls the current in the slave coil.

Let G be a locally compact topological group, and let μ be the left Haar measure on G, with μ the corresponding outer measure. If R' denotes the non-negative extended real numbers, B (G) the Borel subsets of G, and V = {μ(C):C ∈ B(G)}, then we can define ΦG: V × V → R' by

where AB denotes the product set of A and B in G. Then clearly

so that a knowledge of ΦG will give us some idea of how the outer measure of the product set AB compares with the measures of the sets A and B.

This note exhibits an infinite independent set of laws for monoids (viz. (xpyp)2 = (ypxp)2, p prime) and a variety of groups defined by one law (which is x2y2 = y2x2) such that the smallest containing variety of monoids is not finitely definable.

We prove the existence of unique limits and establish inequalities for matrix generalizations of the arithmetic–geometric mean of Lagrange and Gauss. For example, for a matrix A = (aij) with positive elements aij, define (contrary to custom) A½ elementwise by [A½]ij = (aij)½. Let A(0) and B(0) be d × d matrices (1 < d < ∞) with all elements positive real numbers. Let A(n + 1) = (A(n) + B(n))/2 and B(n + 1 ) = (d−1A(n)B(n))½. Then all elements of A(n) and B(n) approach a common positive limit L. When A(0) and B(0) are both row-stochastic or both column-stochastic, dL is less than or equal to the arithmetic average of the spectral radii of A(0) and B(0).

The modern theory of integrable systems rests on two fundamental pillars: the classification of Lax [13] and zero-curvature equations [14, 1, 2]; and algebraic models of the classical calculus of variations [9,5] specialized to the residue calculus in modules of differential forms over rings of matrix pseudo-differential operators [9, 6]. Both these aspects of the theory are by now very well understood for integrable systems in one space dimension.

Let f(x, y) be a binary cubic form with real coefficients and determinant D ≠ 0. In a recent paper, Mordell has proved that there exist integral values of x, y, not both zero, for which

These inequalities are best possible, since they cannot be satisfied with the sign of strict inequality when f(x, y) is equivalent to

for the case D < 0, or to

for the case D > 0.

Let {Xn: n ≥ 1} be a sequence of independent random variables and write Suppose that the random vairables Xn are uniformly bounded by a random variable X in the sense that

Set qn(x) = Pr(|Xn| > x) and q(x) = Pr(|Xn| > x). If qn ≤ q and E|X|r < ∞ with 0 < r < 2 then we have (see Loève(4), 242)

where ak = 0, if 0 < r < 1, and = EXk if 1 ≤ r < 2 and ‘a.s.’ stands for almost sure convergence. the purpose of this paper is to study the rates of convergence of

to zero for arbitrary ε > 0. We shall extend to the present context, results of (3) where the case of identically distributed random variables was treated. The techniques used here are strongly related to those of (3).

Let A1 denote the algebra generated by one pair of canonical operators. It is shown that the realizations of sl(2) in A1 divide into two non-empty classes determined by the spectrum of the semisimple element of sl(2) in A1.

The annihilator $J^{\perp}$ of a weak*-closed inner ideal $J$ in the JBW*-triple $A$ consists of elements $b$ of $A$ for which $\{J\,b\,A\}$ is equal to zero, and the kernel $\mathrm{Ker}(J)$ of $J$ consists of those elements $b$ in $A$ for which $\{J\,b\,J\}$ is equal to zero. The annihilator $J^{\perp}$ is also a weak*-closed inner ideal in $A$, and $A$ enjoys the Peirce decomposition \begin{eqnarray*} A &=& J \oplus_M J^{\perp} \oplus J_1\\ &=& J_2 \oplus_M J_0 \oplus J_1, \end{eqnarray*} where $J_1$ is the intersection of the kernels of $J$ and $J^{\perp}$. When all of the usual Peirce arithmetical relations hold, $J$ is said to be a Peirce inner ideal. A pair $(J,K)$ of weak*-closed inner ideals is said to be compatible when \[A = \bigoplus_{j,k = 0}^2 J_j \cap K_k.\] By analysing the biannihilator $J^{\perp\perp}$ of a Peirce inner ideal $J$ it is shown that, if $J$ is a Peirce inner ideal, then $J^{\perp}$ is a Peirce inner ideal. Furthermore, if $K$ is a weak*-closed inner ideal in $A$ such that $(J,K)$ is a compatible pair, then $(J^{\perp},K)$ is a compatible pair.

In applying the general theory of intermolecular forces to the complicated problem of ionic solvation, the minimum requirements are an exact knowledge of (1) the force laws governing the interaction of two isolated solvent molecules as a function of their separation and the various angles of mutual inclination, (2) the force laws governing the interaction of an ion with one solvent molecule, also as a function of their distance apart and of their mutual inclinations, (3) the average disposition of solvent molecules in the pure liquid and (4) the disposition of the solvent molecules in the force field of the ion, particularly in its immediate vicinity. None of these theoretical requirements is available for systems of practical interest. The established laws of intermolecular force are those relating to spherically symmetrical fields, applied to gases (J. E. Lennard-Jones; see (1)), in which the molecular arrangements are random, and to crystals (M. Born (2)), in which the arrangements are orderly. In seeking to extend these laws to liquids and solutions, we are hampered at the start by our ignorance of directional forces in general and of the degree of disorder prevailing in condensed fluid systems. What is attempted here is a simplification of the problem of applying intermolecular force theory to ionic hydration by the introduction of assumptions which can claim some experimental warrant. How far the introduction of these assumptions may swamp other factors must remain an open question. At any rate, the attempt removes many anomalies, and allows of the absolute calculation of ionic dimensions in aqueous solution from intermolecular force constants determined without reference to the condensed states of matter.

We consider 3-webs, hyper-para-complex structures and integrable Segre structures on manifolds of even dimension and generalise the second heavenly Plebański equation in the context of higher-dimensional hyper-para-complex structures. We also characterise the Segre structures admitting a compatible hyper-para-complex structure in terms of systems of ordinary differential equations.

Consider A and B to be two independent and non-degenerate random variables. It is conjectured by D. G. Kendall (private communication) that the distribution of the vector (A2, 2AB, B2) is indecomposable. The purpose of the present paper is to show that this is true.

Let M be a C∞ manifold smoothly embedded as a convex hypersurface of Euclidean n-space n. M has constant width if the distance between each pair of parallel supporting hyperplanes in n is equal. This is equivalent to the condition that any chord of M normal to M at one end-point be normal to M at both end-points.

1. Many experiments have shown that the natural heterogeneous β-rays from any radioactive source are absorbed according to a law which is exponential for small thicknesses of absorber but which falls below the exponential for larger thicknesses, giving, finally, a quite definite ‘kink’. The positions of these kinks have been measured by the author(1), by Feather (2) and by others. Experiments on the continuous spectra of β-rays (e.g. Gurney(3)) seem to indicate a definite high-velocity limit to the continuous spectrum; and experiments on the absorption of homogeneous β-rays (e.g. Madgwick(4)) show a definite extrapolated range for the β-rays. So one might expect the ‘kinks’ found to represent the extrapolated ranges of the β-rays of highest velocity. The author (1), using the extrapolated ranges of Madgwick(4) and Varder(5), finds this interpretation of the kinks to give values of the end-points agreeing with the direct experimental values of Gurney(6). Feather(2), on the other hand, uses the results of the kinks in certain cases to give an empirical relationship between the kinks and the end-points. Feather's relationship is purely empirical and is independent of any interpretation given to the kinks; for the purpose of using the kinks to find the end-points, Feather's relationship is sufficient, but the actual interpretation of the kinks is of interest. Feather(7) has considered the matter from the theoretical standpoint, and he finds no reason to expect any abrupt discontinuity; he interprets the apparent kink as the limit of the measurable β-ray effect.

We investigate the first-order Thom–Boardman singularity sets of the dual mapping for an arbitrary (and then for a generic) smooth hypersurface in the complex projective space ℙn. Our results focus on nonemptiness, connectedness, regular stratifications and numerical invariants for these sets.