The illumination problem may be phrased as the problem of covering a convex body in Euclidean $n$-space by a minimum number of translates of its interior. By a probabilistic argument, we show that, arbitrarily close to the Euclidean ball, there is a centrally symmetric convex body of illumination number exponentially large in the dimension.

In a complete separable metric space the Souslin sets coincide with the analytic sets and the projection properties of the Souslin sets follow from those of the analytic sets. The projection properties of Souslin sets do not seem to have been discussed in more general circumstances when this equivalence to the analytic sets breaks down. The object of this note is to contribute the following result, that applies in these circumstances, and which will be used by Willmott in forthcoming work on the theory of uniformization.

In this paper, further insight is obtained into the earlier approach of studying residually transcendental extensions of a valuation v of a field K to a simple transcendental extension K(x) of K by means of minimal pairs, thereby introducing new invariants corresponding to any element of an algebraic closure of K. It is also shown that these invariants are of independent interest as well. A characterization of those elements a belonging to is given such that there exists a minimal pair (a, δ) for some δ in the divisible closure of the value group of v.

This paper is devoted to dimensional reductions via the norm-resolvent convergence. We derive explicit bounds on the resolvent difference as well as spectral asymptotics. The efficiency of our abstract tool is demonstrated by its application on seemingly different partial differential equation problems from various areas of mathematical physics; all are analysed in a unified manner, known results are recovered and new ones established.

We consider the fluid motion induced when a circular cylinder performs small-amplitude oscillations about an axis parallel to a generator to which it is rigidly attached as in Fig l(a). In common with other fluid flows dominated by oscillatory motion, a time-independent, or steady streaming develops, and this is the focus of our attention. In particular we relate our results, qualitatively, to the observations that have been made in experiments.

The purpose of this paper is to give some natural examples of Borel-inseparable pairs of coanalytic sets in Polish spaces.

A Polish space is a topological space homeomorphic to a separable complete metric space. In this paper, all spaces are uncountable Polish spaces. A pointset is analytic (or ) if it is the continuous image of a Borel set (in any space), or equivalently, the projection of a Borel set, and is coanalytic (or ) if it is the complement of an analytic set. The class of analytic sets is closed under countable unions and intersections, images and preimages by Borel measurable functions, and projections; it is not closed under complements, hence there is an analytic set which is not Borel.

Let v be a Henselian valuation of any rank of a field K and its unique prolongation to a fixed algebraic closure of K having value group . For any subfield L of , let R(L) denote the residue field of the valuation obtained by restricting to L. Using the canonical homomorphism from the valuation ring of v onto its residue field R(K), one can lift any monic irreducible polynomial with coefficients in R(K) to yield a monic irreducible polynomial with coefficients in K. In an attempt to generalize this concept, Popescu and Zaharescu introduced the notion of lifting with respect to a (K, v)-minimal pair (α, δ) belonging to × . As in the case of usual lifting, a given monic irreducible polynomial Q(y) belonging to R(K(α))[y] gives rise to several monic irreducible polynomials over K which are obtained by lifting with respect to a fixed (K, v)-minimal pair (α, δ). If F, F1 are two such lifted polynomials with coefficients in K having roots θ, θ1, respectively, then it is proved in the present paper that in case (K, v) is a tame field, it is shown that K(θ) and K(θ1) are indeed K-isomorphic.

Every n-dimensional manifold admits an embedding in R2n by the result of H. Whitney [11]. Lie groups are parallelizable and so by the theorem of M. W. Hirsch [5] there is an immersion of any Lie group in codimension one. However no general theorem is known which asserts that a parallelizable manifold embeds in Euclidean space of dimension less than 2n. Here we give a method for constructing smooth embeddings of compact Lie groups in Euclidean space. The construction is a fairly direct one using the geometry of the Lie group, and works very well in some cases. It does not give reasonable results for the group Spin (n) except for low values of n. We also give a method for constructing some embeddings of Spin (n), this uses the embedding of SO(n) that was constructed by the general method and an embedding theorem of A. Haefliger [3]. Although this is a very ad hoc method, it has some interest as it seems to be the first application of Haefliger's theorem which gives embedding results appreciably below twice the dimension of the manifold. The motivation for this work was to throw some light on the problem of the existence of low codimensional embeddings of parallelizable manifolds.

Let $X$ be an infinite-dimensional uniformly smooth Banach space. We prove that $X$ contains an infinite equilateral set. That is, there exist a constant $\lambda \gt 0$ and an infinite sequence $\mathop{({x}_{i} )}\nolimits_{i= 1}^{\infty } \subset X$ such that $\Vert {x}_{i} - {x}_{j} \Vert = \lambda $ for all $i\not = j$.

That the monoid of all transformations of any set and the monoid of all endomorphisms of any vector space over a division ring are regular (in the sense of von Neumann) has been known for many years (see [6] and [16], respectively). A common generalization of these results to the endomorphism monoid of an independence algebra can be found in [13]. It also follows from [13] that the endomorphism monoid of a free G-act is regular, where G is any group. In the present paper we use a version of the wreath product construction of [8], [9] to determine the projective right S-acts (S any monoid) whose endomorphism monoid is regular.

The main object of this paper is to give a self-contained elementary proof of a result (Theorem 1, below), which could be deduced from a theorem of Siegel ([1], Satz 2). It seems worth while to do so, because Siegel's proof is long and difficult, though his result is deeper and more precise than mine.

We study the fine-scale $L^{2}$-mass distribution of toral Laplace eigenfunctions with respect to random position in two and three dimensions. In two dimensions, under certain flatness assumptions on the Fourier coefficients and generic restrictions on energy levels, both the asymptotic shape of the variance is determined and the limiting Gaussian law is established in the optimal Planck-scale regime. In three dimensions the asymptotic behaviour of the variance is analysed in a more restrictive scenario (“Bourgain’s eigenfunctions”). Other than the said precise results, lower and upper bounds are proved for the variance under more general flatness assumptions on the Fourier coefficients.

The method of the large sieve has played a very important role in number theory. It turns out that estimates of exponential sums are of basic importance for large sieve inequalities. Let

be an exponential sum with complex coefficients c(n). It follows from Theorem 1 of Bombieri and Davenport [1] that

In this paper we shall be concerned with the following problem. Let k1 ≤ k2 ≤…≤ ks be natural numbers, λ1,…, λs be nonzero real numbers, not all of the same sign. Is it then true that the values taken by

at integer points (x1,…, xs) ∈ ℤk are dense on the real line, provided at least one of the ratios λi/λj, is irrational? We shall refer to this, for brevity, as the inequality problem for k1,…, ks. Optimistically one may conjecture that the inequality problem is true whenever

The velocity potential near the boundary of the disturbance is determined for steady supersonic flow past a simple body-and-wing model. It is found that the disturbance decays exponentially as it spreads round the body; the alteration caused by changing the radius of curvature is discussed. A universal formula for the potential away from the fuselage is also derived.

Let λ1, …, λs be nonzero real numbers and suppose that λ1/λs is irrational. In 1955, Davenport and Roth showed [6] that the values taken by

at integer points (x1, …, xs) are dense on the real line, provided that s≥8. In the present paper we obtain the same result with seven variables.

The prototype of isoperimetric problems is to minimize the surface area of a convex body with given volume. The minimal body is naturally the suitable ball. The solution to this problem in the planar case was already known to the ancient Greeks. In the higher dimensional cases, the first proofs were provided with the help of Steiner's symmetrization method towards the end of the last century. Important later contributors are, among others, Minkowski, Blaschke, Hadwiger. By their work, the optimality of the ball has been also verified for a much wider class of sets (see [14]).

In his Tata Lecture Notes, Igusa conjectured the validity of a strong uniformity in the decay of complete exponential sums modulo powers of a prime number and determined by a homogeneous polynomial. This was proved for non-degenerate forms by Denef–Sperber and then by Cluckers for weighted homogeneous non-degenerate forms. In a recent preprint, Wright has proved this for degenerate binary forms. We give a different proof of Wright’s result that seems to be simpler and relies upon basic estimates for exponential sums mod $p$as well as a type of resolution of singularities with good reduction in the sense of Denef.

We prove a “special point” result for products of elliptic modular surfaces, elliptic curves, multiplicative groups and complex lines, and deduce a result about vanishing linear combinations of singular moduli and roots of unity.

An important result of Weyl states that for every sequence $(a_{n})_{n\geqslant 1}$ of distinct positive integers the sequence of fractional parts of $(a_{n}{\it\alpha})_{n\geqslant 1}$ is uniformly distributed modulo one for almost all ${\it\alpha}$. However, in general it is a very hard problem to calculate the precise order of convergence of the discrepancy of $(\{a_{n}{\it\alpha}\})_{n\geqslant 1}$ for almost all ${\it\alpha}$. In particular, it is very difficult to give sharp lower bounds for the speed of convergence. Until now this was only carried out for lacunary sequences $(a_{n})_{n\geqslant 1}$ and for some special cases such as the Kronecker sequence $(\{n{\it\alpha}\})_{n\geqslant 1}$ or the sequence $(\{n^{2}{\it\alpha}\})_{n\geqslant 1}$. In the present paper we answer the question for a large class of sequences $(a_{n})_{n\geqslant 1}$ including as a special case all polynomials $a_{n}=P(n)$ with $P\in \mathbb{Z}[x]$ of degree at least 2.