We show that if a collection of lines in a vector space over a finite field has “dimension” at least $2(d-1)+\unicode[STIX]{x1D6FD}$ , then its union has “dimension” at least $d+\unicode[STIX]{x1D6FD}$ . This is the sharp estimate of its type when no structural assumptions are placed on the collection of lines. We also consider some refinements and extensions of the main result, including estimates for unions of $k$ -planes.

Let $Q$ be an infinite subset of $\mathbb{N}$. For any ${\it\tau}>2$, denote $W_{{\it\tau}}(Q)$ (respectively $W_{{\it\tau}}$) to be the set of ${\it\tau}$ well-approximable points by rationals with denominators in $Q$ (respectively in $\mathbb{N}$). We consider the Hausdorff dimension of the liminf set $W_{{\it\tau}}\setminus W_{{\it\tau}}(Q)$ after Adiceam. By using the tools of continued fractions, it is shown that if $Q$ is a so-called $\mathbb{N}\setminus Q$-free set, the Hausdorff dimension of $W_{{\it\tau}}\setminus W_{{\it\tau}}(Q)$ is the same as that of $W_{{\it\tau}}$, i.e. $2/{\it\tau}$.

We prove that for any positive integers $k,n$ with $n>\frac{3}{2}(k^{2}+k+2)$, prime $p$, and integers $c,a_{i}$, with $p\nmid a_{i}$, $1\leqslant i\leqslant n$, there exists a solution $\text{}\underline{x}$ to the congruence

$$\begin{eqnarray}\mathop{\sum }_{i=1}^{n}a_{i}x_{i}^{k}\equiv c\hspace{0.6em}({\rm mod}\hspace{0.2em}p)\end{eqnarray}$$

$1\leqslant {x_{i}\ll }_{k}p^{1/k}$

$1\leqslant i\leqslant n$

$n$

$\unicode[STIX]{x1D700}>0$

$c_{\unicode[STIX]{x1D700}}$

$n>c_{\unicode[STIX]{x1D700}}k$

$1\leqslant {x_{i}\ll }_{\unicode[STIX]{x1D700},k}p^{1/k+\unicode[STIX]{x1D700}}$

Suppose that X is a set, and E a Riesz subspace of ℝx. Let f : E → ℝ be an increasing linear functional which is smooth in the sense of [1, 72C]; i.e.

We recall that if S is a d - simplex then each facet and each vertex figure of S is a (d − 1)-simplex and S is a self-dual. We introduce a d-polytope P, called a d-multiplex, with the property that each facet and each vertex figure of P is a (d − 1)-multiplex and P is self-dual.

We consider the Falkner–Skan equation

in the special case β = – 1. It is known that this equation may be integrated twice to get the Riccati equation

Let A be a subset of a compact metric space Ω, and suppose that A has non-σ-finite h-measure, where h is some Hausdorff function. The following problem was suggested to me by Professor C. A. Rogers:

If A is analytic, is it possible to construct 2ℵodisjoint closed subsets of A which also have non-σ-finite h-measure?

At this level of generality the problem, like others which involve selection of subsets, appears to offer some difficulty. Here we prove two results which were motivated by it.

The results developed by Watson [1] are interpreted to indicate how the slow viscous flow due to the rotation of a small circular cylinder in the presence of a stationary cylinder can be calculated. It is shown how the stream function is given as a combination of the force-free representations corresponding to a line rotlet and a line stokeslet outside the stationary body, plus the streaming flow past the body. The coefficients which multiply these representations are calculated by techniques already described by Watson.

§1. With respect to coordinates {z, x1, …, xq+1, …, x2q} a contact transformation is a local diffeomorphism of ℝ2q+1, which preserves the 1-form

up to multiplication by a non-zero real valued function. The family of such maps contains identities, inverses and admits a partial composition law; denote it by the contact pseudogroup. Passing to germs of local diffeomorphisms we obtain a topological groupoid Γ2q+1, ω, to which by any of several constructions, see [1] for example, there corresponds a classifying space BΓ2q+1, ω, By analogy with BΓq this space classifies codimension (2q + l)-foliations, which locally admit a contact structure normal to the leaves. In particular, at least when q is odd, the structural group of the normal bundle reduces to Uq. We shall be most interested in the case, when the foliation is by points, and the underlying manifold M2q+1 admits a global 1-form co such that

From time to time results about partitioning a given set into subsets have been established. (See for example [2], [3].) We consider here the reverse problem of forming the union of three sets in a certain best possible way. For simplicity we work in Euclidean n-space, En. Let mX denote the measure of the set X.

Let λ be a random variable with the distribution function F(λ). A transform of F which has, in effect, been used in several recent papers ([1], [2], [3], [4]; see also [6]) is

defined formally by the equation

It is the main purpose of this paper to prove the inversion formulae given in the two theorems below.

Let denote the set of convex bodies of Ed, i.e. the set of all compact convex subsets of Ed. Let Bi be the unit i-ball, ωi its volume and Si-1 the unit sphere bd Bi. For an arbitrary denote the j-th quermassintegral (for definition and properties compare [7]). A different normalization of the functionals Wo, …, Wd leads to the intrinsic volumes Vj (which were introduced in [9]) defined by The intrinsic volumes are independent of the dimension of the space in which K is embedded. In particular, V0(K) = 1, Vd-1 (K) is half the surface area of K and Vd{K) = V(K) is its volume.

Group actions on ℝ-trees may be split into different types, and in Section 1 of this paper five distinct types are defined, with one type splitting into two sub-types. For a group G acting as a group of isometries on an ℝ-tree, conditions are considered under which a subgroup or a factor group may inherit the same type of action as G. In Section 2 subgroups of finite index are considered, and in Section 3 normal subgroups and also factor groups are considered. The results obtained here, Theorems 2.1 and 3.4, allow restrictions on possible types of actions for hypercentral, hypercyclic and hyperabelian groups to be given in Theorem 3.6. In Section 4 finitely generated subgroups are considered, and this gives rise to restrictions on possible actions for groups with certain local properties. The results throughout are stated in terms of group actions on trees. Using Chiswell's construction in [3], they could equally be stated in terms of restrictions on possible types of Lyndon length functions.

Let $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\mathbb{F}_q$ be the finite field of $q$ elements. An analogue of the regular continued fraction expansion for an element $\alpha $ in the field of formal Laurent series over $\mathbb{F}_q$ is given uniquely by

$$\begin{equation*} \alpha = A_0(\alpha )+\cfrac {1}{A_1(\alpha )+\cfrac {1}{A_2(\alpha )+\ddots }}, \end{equation*}$$

$(A_n(\alpha ))_{n=0}^\infty $

$\mathbb{F}_q$

$\deg (A_n(\alpha ))\ge 1$

$n\ge 1.$

$(p_n)_{n=1}^\infty $

$$\begin{equation*} \lim _{n\to \infty }\frac {1}{n}\sum _{j=1}^n \deg (A_{p_j}(\alpha )) = \frac {q}{q-1} \end{equation*}$$

$\alpha $

$(p_n)_{n=1}^\infty $

$(n)_{n=1}^\infty ,$

We consider exponential sums with x-coordinates of points qG and q−1G where G is a point of order T on an elliptic curve modulo a prime p and q runs through all primes up to N (with gcd (q,T)=1 in the case of the points q−1G). We obtain a new bound on exponential sums with q−1G and correct an imprecision in the work of W. D. Banks, J. B. Friedlander, M. Z. Garaev and I. E. Shparlinski on exponential sums with qG. We also note that similar sums with g1/q for an integer g with gcd (g,p)=1 have been estimated by J. Bourgain and I. E. Shparlinski.

Let S be a point set of the Euclidean plane, such that

(i) S is bounded,

(ii) the closure of S has unit Lebesgue measure.

Let P be an arbitrary set of n points contained in S, and let l(P) denote the total length of the shortest system of lines connecting the points of P together. Define ln to be the supremum of l(P), taken over all sets P of n points in S. Beardwood, Halton, and Hammersley [1[ proved that there exists an absolute constant α, independent of S, such that

Introduction. All the sets X considered in this paper are assumed to be compact convex sets in the Euclidean plane. We shall let K denote this class of sets. Problems concerning the division of such sets by three non-concurrent lines have been considered by Eggleston [1] on page 118 and by Grünbaum [2].

A tiling of a convex m-gon by a finite number r of convex n-gons is said to be of type <m, n, r>. The Main Theorem of this paper gives necessary and sufficient conditions on m, n and r for a tiling of type <m, n, r> to exist.

This paper is devoted to the establishment of explicit bounds on the rational function solutions of a general class of equations in several variables. The first general result on Diophantine equations over function fields was discovered in 1930 [1], as an analogy on the work of Thue on number fields: it was shown that the degrees of polynomial solutions X, Y in k[z] of f(X, Y) = c are bound for each c: here f denotes an irreducible binary form over k[z] of degree at least three, and k is an algebraically closed field of characteristic zero. The Manin-Grauert theorem [7] extended this conclusion to the rational function solutions X, Y in k(z) of any equation in two variables over k(z), provided that the curve corresponding to the equation has genus two or more: this is the analogue for function fields of the Mordell conjecture for number fields, proved by Faltings in 1983. Parsin [6] made the Manin-Grauert theorem effective by furnishing explicit bounds on the degrees of X and Y: this approach followed Grauert and Shafarevitch in relying heavily on algebraic geometry. In 1976 a different attack was made by Schmidt [8] using the theory of algebraic differential equations, first developed by Kolchin and Osgood. This method produced very good bounds for equations such as the Thue equation discussed above: for example, Schmidt provides the bound

for any rational function solution X, Y in k(z) of f(X, Y) = 1, where f denotes a binary form over k(z) of degree in X and Y at least five, and which factorises into distinct linear factors. Here deg zf is the degree of f in z. In 1983 [2] the power of Diophantine approximation provided another approach to attack Dlophantine equations.over function fields. By first establishing an inequality on solutions of the unit equation in two variables, it was shown in [3, p. 122] that the bound (1) could be improved to 36 degz f, and the condition on the degree in X and Y could be relaxed to include quartic forms. In fact the principal purpose of this new approach using Diophantine approximation had been to provide algorithms for the effective construction of all the polynomial solutions of various equations: it was a surprise to find that not only were these algorithms extremely efficient, but also that the technique extended to include rational function solutions. A full account of this approach and its consequences is given in a recent tome [3].

Let Q denote the field of rational numbers, and let p be an odd prime number. Let K be a cyclic extension of Q of degree p, and let a be a generator of Gal (KQ). Let CK denote the p-class group of K (i.e., the Sylow p-subgroup of the ideal class group of K), and let for i = 1, 2, 3, . It is well known that is an elementary abelian p-group of rank tt1, where t is the number of ramified primes in KQ. So we focus our attention on . We let