In this paper we consider group actions on generalized treelike structures (termed ‘pretrees’) defined simply in terms of betweenness relations. Using a result of Levitt, we show that if a countable group admits an archimedean action on a median pretree, then it admits an action by isometries on an ℝ-tree. Thus the theory of isometric actions on ℝ-trees may be extended to a more general setting where it merges naturally with the theory of right-orderable groups. This approach has application also to the study of convergence group actions on continua.

A random graph on n vertices is a random subgraph of the complete graph on n vertices. By analogy with this, the present paper studies the asymptotic properties of a random submatroid ωr of the projective geometry PG(r−l, q). The main result concerns Kr, the rank of the largest projective geometry occurring as a submatroid of ωr. We show that with probability one, for sufficiently large r, Kr takes one of at most two values depending on r. This theorem is analogous to a result of Bollobás and Erdös on the clique number of a random graph. However, whereas from the matroid theorem one can essentially determine the critical exponent of ωr, the graph theorem gives only a lower bound on the chromatic number of a random graph.

A classical reciprocity formula for Gauss sums due to Cauchy, Dirichlet, and Kronecker states that

formula here

where a, b are nonzero integers and w∈ℚ such that ab+2aw∈2ℤ. For a detailed proof and historical background, see [4, chapter IX]. Various versions of formula (1) have been extensively studied in connection with transformation properties of theta functions. A version of (1) for multivariate Gauss sums was first obtained by A. Krazer [10] in 1912, see also [2, 5, 11, 14]. Krazer's formula generalizes the case w=0 of (1) via replacing one of the numbers a, b by an integer quadratic form of several variables.

Recently, Florian Deloup [6] found a new and most beautiful reciprocity formula for multivariate Gauss sums. His formula is a far reaching generalization of Krazer's result. Roughly speaking, Deloup replaces both numbers a, b with quadratic forms. Deloup involves Wu classes of quadratic forms which allows him to remove the evenness condition appearing in Krazer's formulation. However, Deloup's formula covers only the cases w=0 and w=b/2 of (1).

In this paper we establish a more general reciprocity for Gauss sums including the Krazer and Deloup formulas and formula (1) in its full generality. Our reciprocity law involves two quadratic forms and a so-called rational Wu class, as defined below.

The original proofs of Cauchy, Dirichlet, Kronecker and Krazer are analytical and involve a study of a limit of a transformation formula for theta-functions. Deloup's proof goes by a reduction to the case w=b/2 of (1) based on a careful study of Witt groups of quadratic forms. Our proof is more direct and uses only (a generalization of) the van der Blij computation of Gauss sums via signatures of integer quadratic forms. In particular, our argument provides a new proof of (1).

Families of steady states of the spherically symmetric Einstein–Vlasov system are constructed, which are parametrised by the central redshift. It is shown that as the central redshift tends to zero, the states in such a family are well approximated by a steady state of the Vlasov–Poisson system, i.e., a Newtonian limit is established where the speed of light is kept constant as it should be and the limiting behavior is analysed in terms of a parameter which is tied to the physical properties of the individual solutions. This result is then used to investigate the stability properties of the relativistic steady states with small redshift parameter in the spirit of recent work by the same authors, i.e., the second variation of the ADM mass about such a steady state is shown to be positive definite on a suitable class of states.

In this note, we discuss some questions on the arithmetic properties of recurrence sequences. Our primary purpose is to mention some methods which originated as lemmas in certain transcendence studies, but in order to give context to these ideas, we briefly review other techniques applicable to the problems considered. These problems concern the distribution of zeros in a recurrence, the rate of growth of its terms and the size of the greatest prime factor of the terms.

The object of the present paper is to report upon a case of neutron-produced disintegration of a type which has not previously been observed. The nucleus disintegrated was that of carbon and the peculiarity of the disintegration lies in the fact that three heavy particles resulted from the transformation. Hitherto no example of such disintegration resulting in more than two heavy particles has been obtained, and with carbon, in particular, several experiments agree in showing that this more usual type of disintegration is very rare indeed. In similar circumstances, and with neutrons of less than 12 × 106 electron volts energy, disintegration phenomena occur in oxygen or in nitrogen at least ten times as frequently as in carbon. Thus Harkins, Gans and Newson obtained only two examples of disintegration in 3200 pairs of photographs taken with a source of radiothorium and beryllium and an expansion chamber filled with ethylene, and concluded that in each case an atom of oxygen or some other impurity must have been involved. Likewise, Feather, in 2210 pairs of photographs with the neutrons of polonium-beryllium and an expansion chamber filled with a mixture of acetylene and helium, found only one example of a disintegration which could reasonably be ascribed to the nuclear reaction,

The theory of graduation discusses methods of obtaining a smoothed series of values of a function from a given empirical set of values. This is usually done by replacing each observation by a weighted average of it and neighbouring observations. Thus if {xi} (i = 0, ± 1, …) is a sequence of values, we can replace them by the series

where the A's are constants which are chosen in some suitable manner. If we regard the xi as the sum of a functional part fi and an error term εi we may attempt to choose the A's in such a way that the functional part is reproduced as well as possible, e.g. that

for some desired type of function. If this is so and the error terms εi are independently distributed with zero mean and finite standard deviation σ, we find that

where the ηi are a series of random variables which are no longer independent of each other but which have zero mean and a standard deviation σ1 such that

In this paper we investigate the procedure of blowing up a non-normal variety V in its conductor ideal (denned in Section 1). If V is a hypersurface then corresponds to the subadjunction conditions of Italian algebraic geometry (see (4), section 15), and so we would expect the blow up of V in , denoted , to be closely connected with the normalization Ṽ.

In this paper, we consider transmission problems for wave propagation in two inhomogeneous half-spaces with a locally perturbed hyperplane interface. A radiation condition is obtained for the problem in the framework of the spaces B and B*. It is then used, together with the spaces B, B* and the limiting absorption method to prove the existence of the unique solution to the problem.

We first study the minimum total curvature of a knot when it is embedded on the cubic lattice. Let be a knot or link with a lattice embedding of minimum total curvature among all possible lattice embeddings of . We show that there exist positive constants c1 and c2 such that for any knot type . Furthermore we show that the powers of in the above inequalities are sharp hence cannot be improved in general. Our results and observations show that lattice embeddings with minimum total curvature are quite different from those with minimum or near minimum lattice embedding length. In addition, we discuss the relationship between minimal total curvature and minimal ropelength for a given knot type. At the end of the paper, we study the total curvatures of smooth thick knots and show that there are some essential differences between the total curvatures of smooth thick knots and lattice knots.

We consider robust relative homoclinic trajectories (RHTs) for G-equivariant vector fields. We give some conditions on the group and representation that imply existence of equivariant vector fields with such trajectories. Using these results we show very simply that abelian groups cannot exhibit relative homoclinic trajectories. Examining a set of group theoretic conditions that imply existence of RHTs, we construct some new examples of robust relative homoclinic trajectories. We also classify RHTs of the dihedral and low order symmetric groups by means of their symmetries.

In this paper we give a characterization theorem for a subclass of the exponential family whose probability density function is given by

where a(x) ≥ 0, f(ω) = ∫a(x) exp (ωx) dx and ωx is to be interpreted as a scalar product. The random variable X may be an s-vector. In that case ω will also be an s-vector. For obvious reasons we will call (1) as the linear exponential family. It is easy to verify that the moment generating function (m.g.f.) of (1) is given by

The 4-dimensional topological surgery conjecture has been established for a class of groups, including the groups of subexponential growth (see [6], [13] for recent developments), however the general case remains open. The full surgery conjecture is known to be equivalent to the question for a class of canonical problems with free fundamental group [5, chapter 12]. The proof of the conjecture for ‘good’ groups relies on the disk embedding theorem (see [5]), which is not presently known to hold for arbitrary groups. However, in certain cases it may be shown that surgery works even when the disk embedding theorem is not available for a given fundamental group (such results still use the disk-embedding theorem in the simply-connected setting, proved in [3]). For example, this is possible when the surgery kernel is represented by π1-null spheres [4], or more generally by a π1-null submanifold satisfying a certain condition on Dwyer's filtration on second homology [7]. Here we state another instance when the surgery conjecture holds for free groups. The following results are stated in the topological category.

The purpose of this article is to derive a set of ‘easily verifiable’ sufficient conditions for the local asymptotic stability of the trivial solution of

and then examine the ‘size’ of the domain of attraction of the trivial solution of the nonlinear system (1·1) with a countable number of discrete delays.

A recently proved upper bound for the permanents of (0,1) matrices is used to improve the Fowler-Rushbrooke upper bound for the constant λd occurring in the d-dimensional dimer problem, d ≥ 3.

1. The purpose of this note is to prove a result which includes certain classical theorems generally thought of as being unconnected; in explicit terms, a result about the Fourier series of a periodic Lebesgue-integrable function showing that the series is summable at a point by a Nörlund method (N, pn) defined as usual ((2), p. 64) if pn ↓ 0, Σpn = ∞ and the point is in a certain subset of the Lebesgue set. More precisely, the purpose is to prove Theorem I on the Nörlund summability of Fourier series and to derive from it the well-known Theorems A, B which follow and the recent extension of Theorem A in Theorem A' which appears later and is due to Sahney (8).

Several relations between the Cesàro and the Riemann methods of summation are known. For instance, Verblunsky(3) has shown that a series summable (C, k − δ), where k is a positive integer, is also summable (R,k+ 1) and Kuttner (2) has proved that, for k = 1, 2, summability (R, k) implies summability (C, k + δ). In this paper we consider Riesz's typical means and a generalized Riemann summability both of which are intimately connected with almost periodic functions. The result we establish is similar to Verblunsky's, except that we start from a Riesz mean of integral order k*.

The thyratron valve is essentially a triode which contains a small quantity of some inert gas, usually mercury vapour. If a suitable voltage be applied between anode and filament, with sufficient negative grid bias, the valve behaves like an ordinary vacuum tube; but, as the grid is made more positive, at a certain critical voltage an arc strikes between the anode and the filament, with enormous increase in the anode current. The grid then exercises no further control over the anode current, which must be limited by external resistance so as not to exceed the saturation emission current from the filament. If the voltage across the tube exceeds the “disintegration” potential, of some 20–25 volts, the cathode will be disintegrated by positive ion bombardment. Limitation of the anode current to the emission of the filament ensures that the voltage, while higher than the ionization potential of mercury, is considerably below the disintegration potential.

Suppose that w is analytic in the open unit disc Δ in the z-plane. The Schwarzian derivative of w is defined by

(for basic properties of {w, z} we refer to Hille ((3), p. 375)). Let denote the class of functions w which are analytic in Δ and satisfy the conditions

An automorphism σ(G) of a graph G is a permutation of the set of its vertices which preserves adjacency. Under the operation of composition the automorphisms of G form a group Aut(G). The graph G is called asymmetric if Aut(G) is trivial, and symmetric otherwise.