Let {\mathrm{OT} }_{d} (n)${\mathrm{OT} }_{d} (n)$ be the smallest integer N$N$ such that every N$N$-element point sequence in { \mathbb{R} }^{d} ${ \mathbb{R} }^{d}$ in general position contains an order-type homogeneous subset of size n$n$, where a set is order-type homogeneous if all (d+ 1)$(d+ 1)$-tuples from this set have the same orientation. It is known that a point sequence in { \mathbb{R} }^{d} ${ \mathbb{R} }^{d}$ that is order-type homogeneous, forms the vertex set of a convex polytope that is combinatorially equivalent to a cyclic polytope in { \mathbb{R} }^{d} ${ \mathbb{R} }^{d}$. Two famous theorems of Erdős and Szekeres from 1935 imply that {\mathrm{OT} }_{1} (n)= \Theta ({n}^{2} )${\mathrm{OT} }_{1} (n)= \Theta ({n}^{2} )$ and {\mathrm{OT} }_{2} (n)= {2}^{\Theta (n)} ${\mathrm{OT} }_{2} (n)= {2}^{\Theta (n)}$. For d\geq 3$d\geq 3$, we give new bounds for {\mathrm{OT} }_{d} (n)${\mathrm{OT} }_{d} (n)$. In particular, we show that {\mathrm{OT} }_{3} (n)= {2}^{{2}^{\Theta (n)} } ${\mathrm{OT} }_{3} (n)= {2}^{{2}^{\Theta (n)} }$, answering a question of Eliáš and Matoušek, and, for d\geq 4$d\geq 4$, we show that {\mathrm{OT} }_{d} (n)${\mathrm{OT} }_{d} (n)$ is bounded above by an exponential tower of height d$d$ with O(n)$O(n)$ in the topmost exponent.

Let

where φ is the notation used by Ramanujan in his notebooks [15], and is the familiar notation of Whittaker and Watson [20, p. 464]. It is well known that [1, p. 102] (with a misprint corrected)

where denotes the ordinary or Gaussian hypergeometric function; k, 0 < k < 1, is the modulus; K is the complete elliptic integral of the first kind; and

where K′=K(k′) and is the complementary modulus. Thus, an evaluation of any one of the functions φ, , or K yields an evaluation of the other two functions. However, such evaluations may not be very explicit. For example, if K(k) is known for a certain value of k, it may be difficult or impossible to explicitly determine K′, and so q cannot be explicitly determined. Conversely, it may be possible to evaluate φ(q) for a certain value of q, but it may be impossible to determine the corresponding value of k. (Recall that [1, p. 102].)

Assume that n points P1,…,Pn are distributed independently and uniformly in the triangle with vertices (0, 1), (0, 0), and (1, 0). Consider the convex hull of (0, 1), P1,…,Pn, and (1, 0). The vertices of the convex hull form a convex chain. Let be the probability that the convex chain consists – apart from the points (0, 1) and (1, 0) – of exactly k of the points P1,…,Pn. Bárány, Rote, Steiger, and Zhang [3] proved that . The values of are determined for k = 1,…,n − 1, and thus the distribution of the number of vertices of a random convex chain is obtained. Knowing this distribution provides the key to the answer of some long-standing questions in geometrical probability.

In 1979 Montgomery and Vaughan [3] showed that for each k > 0

the sum being over the non-principal characters χ modulo q. As a corollary they deduced that, if 0 < θ < 1 then there is a constant c(θ) such that at least θφ(q) of the non-principal characters χ modulo q satisfy

We provide new quantitative versions of Helly’s theorem. For example, we show that for every family \{P_{i}:i\in I\}$\{P_{i}:i\in I\}$ of closed half-spaces in \mathbb{R}^{n}$\mathbb{R}^{n}$ such that P=\bigcap _{i\in I}P_{i}$P=\bigcap _{i\in I}P_{i}$ has positive volume, there exist s\leqslant \unicode[STIX]{x1D6FC}n$s\leqslant \unicode[STIX]{x1D6FC}n$ and i_{1},\ldots ,i_{s}\in I$i_{1},\ldots ,i_{s}\in I$ such that

\begin{eqnarray}\operatorname{vol}_{n}(P_{i_{1}}\cap \cdots \cap P_{i_{s}})\leqslant (Cn)^{n}\operatorname{vol}_{n}(P),\end{eqnarray} $$\begin{eqnarray}\operatorname{vol}_{n}(P_{i_{1}}\cap \cdots \cap P_{i_{s}})\leqslant (Cn)^{n}\operatorname{vol}_{n}(P),\end{eqnarray}$$

\unicode[STIX]{x1D6FC},C>0$\unicode[STIX]{x1D6FC},C>0$

Some examples of the bending of a plane elastic plate by transverse forces applied at isolated points are first considered. The plate is infinite and is bounded internally by a circular edge along which it is clamped; simple expressions are found for the displacement.

The analogous hydrodynamical problem is that of the steady flow of viscous incompressible liquid past a fixed circular cylinder; the equation for the stream function is in the same form as the equation for the displacement of a plate due to a distributed force of amount Z per unit area. The inertia terms in the hydrodynamical problem correspond to Z. In slow motion there is no stream function with the correct form at infinity, because in this case the inertia terms are ignored so that in the plate problem Z = 0. The effect of a transverse force system can be most simply illustrated by supposing that the forces are concentrated at isolated points; in the corresponding stream functions a simple form of allowance for inertia is therefore made, and they display some of the features of steady flow past a cylinder.

The object of this paper is to prove theorems of the following type.

THEOREM 1. If p and q are restricted to odd primes, anddenotes the Legendre symbol, then for x > 2

We show that substantially more than a quarter of the odd integers of the form pq$pq$ up to x$x$, with p,q$p,q$ both prime, satisfy p\equiv q\equiv 3~(\text{mod}\,4)$p\equiv q\equiv 3~(\text{mod}\,4)$.

Among several interesting analogues of Chebyshev's problem about the largest prime factor of

there is the question of the largest prime factor of

where a is a given non-zero integer and the product is taken over positive primes p. The latter subject appears to have been first treated by Goldfeld [2] and Motohashi [6], who showed that, if Px be the greatest prime factor in question, then there exists a constant such that Px > xθ for all sufficiently large x. Their method, which involved the use of both Bombieri's theorem and the Brun-Titchmarsh theorem, had some affinity with the earlier treatments of Chebyshev's original problem.

Euler noted the relation 6^{3}\,=\,3^{3}+4^{3}+5^{3}$6^{3}\,=\,3^{3}+4^{3}+5^{3}$ and asked for other instances of cubes that are sums of consecutive cubes. Similar problems have been studied by Cunningham, Catalan, Gennochi, Lucas, Pagliani, Cassels, Uchiyama, Stroeker and Zhongfeng Zhang. In particular, Stroeker determined all squares that can be written as a sum of at most 50 consecutive cubes. We generalize Stroeker’s work by determining all perfect powers that are sums of at most 50 consecutive cubes. Our methods include descent, linear forms in two logarithms and Frey–Hellegouarch curves.

Let A$A$ be a set of natural numbers. Recent work has suggested a strong link between the additive energy of A$A$ (the number of solutions to a_{1}+a_{2}=a_{3}+a_{4}$a_{1}+a_{2}=a_{3}+a_{4}$ with a_{i}\in A$a_{i}\in A$) and the metric Poissonian property, which is a fine-scale equidistribution property for dilates of A$A$ modulo 1$1$. There appears to be reasonable evidence to speculate a sharp Khinchin-type threshold, that is, to speculate that the metric Poissonian property should be completely determined by whether or not a certain sum of additive energies is convergent or divergent. In this article, we primarily address the convergence theory, in other words the extent to which having a low additive energy forces a set to be metric Poissonian.

Many positive results are known to hold for the class of Banach spaces known as Asplund spaces and it was for a time conjectured that Asplund spaces should admit equivalent norms with good smoothness and strict convexity properties. A counterexample to these conjectures, in the form of a space of continuous real-valued functions on a suitably chosen tree, was presented in [5]. In this paper we show that the bad behaviour of that example is shared by a wider class of Banach spaces, associated with a wider class of trees. The immediate aim of this extension of the original result is to answer a question posed by Deville and Godefroy [3]. They introduced and studied a subclass of Asplund spaces, those with Corson compact bidual balls, and asked whether this additional assumption is enough to guarantee the existence of nice renormings. We show that it is not.

Let P be a (convex) d-polytope in the Euclidean space Ed and p a point of Ed not contained in P or in a supporting hyperplane of a facet of P (we use the terminology of Grunbaum [2]). The part of the boundary of P which is “visible” from p, i.e. the union of those facets whose supporting hyperplanes separate p and P, form a (d – l)-ball, whose boundary is a (d – 2)-sphere S (all balls, spheres and manifolds to be considered are piecewise-linear). The boundary-complex ℬ(P) induces a subdivision of S, which we call a (sharp) shadow-boundary ofP. is combinatorially isomorphic to the boundary complex of a poly tope which is a central projection of P from the point p on a hyperplane.

We consider the minimization of Dirichlet eigenvalues \unicode[STIX]{x1D706}_{k}$\unicode[STIX]{x1D706}_{k}$, k\in \mathbb{N}$k\in \mathbb{N}$, of the Laplacian on cuboids of unit measure in \mathbb{R}^{3}$\mathbb{R}^{3}$. We prove that any sequence of optimal cuboids in \mathbb{R}^{3}$\mathbb{R}^{3}$ converges to a cube of unit measure in the sense of Hausdorff as k\rightarrow \infty$k\rightarrow \infty$. We also obtain an upper bound for that rate of convergence.

In Hausdorff topological spaces there are currently three definitions of analytic sets due respectively to Choquet [1], Sion [8], and Frolīk [3, 4]. Here it is shown that these definitions are equivalent.

Let V(x) denote the number of distinct values not exceeding x taken by Euler's ø-function, so that we have π(x) ≤ V(x) ≤ x. It was shown by Erdős and Hall [1] that for each fixed B > 2√(2/log 2), the estimate

holds; moreover we stated that the ratio V(x)/π(x) tends to infinity with x, faster than any fixed power of log log x. Our aim in the present paper is to prove the following result.

If(X1, …, Xn), n ≥ 3, is a non-singular quadratic form with rational integral coefficients whose greatest common divisor is 1, then G. L. Watson [1] showed that f(x1, …, xn) = 1, for suitable algebraic integers x1, …, xn. In the present paper we extend this result to forms of arbitrary degree, with algebraic integers as coefficients (see Theorem 3). In fact we prove the stronger result (Theorem 2) that, if f(X1, …, Xn) is any polynomial with relatively prime algebraic integers as coefficients, then f(x1, …, xn) is a unit, for suitable algebraic integers x1, …, xn. Unfortunately, our result is just an existence theorem. We cannot limit the size of the field which x1, …, xn generate, as Watson could.

We consider the second order linear differential equation

where p and q are real-valued and p(t) > 0 for all t ≥ T. Our interest here is the oscillatory nature of solutions of (1.1). More particularly we consider the following questions, (I), (II) and (III).

Denote by N(t) the number of zeros β + iγ of ζ(s), s = σ + it, with 0 < β < 1 and 0 < γ < t. It is well known that

where

For each odd prime p there is a finite regular abstract 4-dimensional polytope of type {3, 3, p}. Its cells are simplices, and its vertex figures belong to an infinite family of regular polyhedra. We also give a geometric realization for these polytopes.