For a 3-connected graph G imbedded in the plane let νk(G) denote the number of k-valent vertices, and let pk(G) denote the number of k-gonal faces (countries) of the map defined by G. (We consider only graphs without loops or multiple edges; hence, νk(G) and pk(G) are defined only for k ≥ 3.) It is a wellknown consequence of Euler's formula that

We consider the random functions $S_{N}(z):=\sum _{n=1}^{N}z(n)$, where $z(n)$ is the completely multiplicative random function generated by independent Steinhaus variables $z(p)$. It is shown that $\mathbb{E}|S_{N}|\gg \sqrt{N}(\log N)^{-0.05616}$ and that $(\mathbb{E}|S_{N}|^{q})^{1/q}\gg _{q}\sqrt{N}(\log N)^{-0.07672}$ for all $q>0$.

A. Bezdek and W. Kuperberg constructed a nonlattice packing of congruent ellipsoids in Euclidean 3-space E3 with density 0·7459 …, which exceeds the density σL2 = 0·74048… of the densest lattice packing of spheres and hence of ellipsoids in E3. G. Kuperberg improved this to 0·7533… We improve this slightly to 0·7549…. In our case the quotient of the largest and the smallest halfaxis of the ellipsoids is <42, so the ellipsoids are not too degenerate. If one combines G. Kuperberg's refinement and ours, one obtains a packing density of 0·7585…

We investigate the distribution of the hitting time T defined by the first visit of the Brownian motion on the Sierpiński gasket at geodesic distance r from the origin. For this purpose we perform a precise analysis of the moment generating function of the random variable T. The key result is an explicit description of the analytic behaviour of the Laplace- Stieltjes transform of the distribution function of T. This yields a series expansion for the distribution function and the asymptotics for t →0.

A tournament is a relational structure on the non-empty set T such that for x, y ∈ T exactly one of the three relations

holds. Here x → y expresses the fact that {x, y} ∈ → and we sometimes write this in the alternative form y ← x. Extending the notation to subsets of T we write A → B or B ← A if a → b holds for all pairs a, b with a ∈ A and b ∈ B. is a subtournament of , and is an extension of , if T′ ⊂ T and →′ is the restriction of → to T′; we will usually write 〈′, → 〉 instead of 〈 ′, → ′〉. In particular, if |T − T′| = k, we call a k-poinf extension of .

In [2], Sawyer considers a closed, central, convex region K which is such that, however it is displaced in the plane, a point of the integral lattice is covered. He shows that the area A(K) of K satisfies . We prove here a result in the opposite direction.

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].)

We prove that there are permutation classes (hereditary properties of permutations) of every growth rate (Stanley–Wilf limit) at least λ≈2.48187, the unique real root of x5−2x4−2x2−2x−1, thereby establishing a conjecture of Albert and Linton.

A method of “inner and outer expansions” employed by Cox and Brenner (1967) is used to calculate the hydrodynamic force experienced by either of two identical small solid spheres, approaching each other, at low Reynolds number, with the same velocity along their line of centres, for the limiting case when the gap width between them tends to zero. Numerical results are compared with those furnished by the exact solution and an improved expression is then obtained for a similar problem arising from the motion of a small sphere towards an identical stationary sphere, in the limit of small gap widths.

A new type of convergence (called uniformly pointwise convergence) for a sequence of scalar valued functions is introduced. If (fn) is a uniformly bounded sequence of functions in l∞(Γ), it is proved that:

(i) (fn) converges uniformly pointwise on Γ to some function f if, and only if, every subsequence of (fn) is Cesaro summable in l∞(Γ); and

(ii) there exists a subsequence (f′n) of (fn) such that either (f′n) converges uniformly pointwise on Γ to some f or no subsequence of (f′n) is Cesaro-summable in l∞(Γ).

Applications of the above results in Banach space theory are given.

This note is a continuation of the articles [6] and [2]. In [1], trees with a given partition α = (a1; a2, …), where ai is the number of vertices (points) of valency (degree) i were enumerated. After the determination of the number of plane trees in [2], the number of planted plane trees with a given partition α was found explicitly in [6]. In the present note, the number of plane trees with a given partition is expressed as a function of the number of planted trees with a given partition. The method, which is not new, consists of an application of the enumeration techniques of Otter [3] and Pólya [4]; it was used in [1] and also by Riordan [5].

Multiplications on spheres are studied in [11], [12] from the standpoint of homotopy theory. These multiplications are products with a unit element. The present paper deals with products in general. The investigation involves proving some results on the toric construction and the Whitehead product. These results also lead to theorems about the Stiefel manifold of unit tangent vectors to a sphere, originally proved by M. G. Barratt, which clear up some points in the homotopy theory of sphere bundles over spheres (see [15], [16]). They also enable us to prove that certain of the classical Lie groups are not homotopy-commutative.

It was shown by Vinogradov (see Theorem 7, Chapter 4 of [18]) that, for ε > 0, there are infinitely many solutions in primes p of the inequality

where {x} denotes the fractional part of x and y = 0.1. The value of γ was improved to

by Kaufman [9]. On the Riemann Hypothesis he showed that one can take γ =¼ The method used actually shows that, for any real β and any δ with 0 < 8 > 1, the number of primes p ≤ x satisfying

is

where π(x) denotes the number of primes not exceeding x. The sequence √p is, of course, a subsequence of the sequence n½ whose distribution modulo one has also been investigated (see Chapter 2, Section 3 of [10]: the argument for sequences nσ (0 < σ < 1) is entirely elementary). It is useful in this context to define the discrepancy (modulo one) of a sequence an by

The classical theory of analytic sets works well in metric spaces, but the analytic sets themselves are automatically separable. The theory of K-analytic sets, developed by Choquet, Sion, Frolik and others, works well in Hausdorff spaces, but the K-analytic sets themselves remain Lindelof. The theory of k-analytic sets developed by A. H. Stone and R. W. Hansell works well in non-separable metric spaces, especially in the special case, when k is ℵ0, with which we shall be concerned, see [9, 10 and 16–20]. Of course the k-analytic sets are metrizable. For accounts of these theories, see, for example, [15].

The 2-ball property is shown to be transitive. Combining this with some results on the decomposability of convex bodies, we produce new examples of Banach spaces which contain proper semi-M-ideals. These semi-M-ideals are not hyperplanes, nor are they the direct sums of examples which are hyperplanes.

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

We consider point distributions in compact connected two-point homogeneous spaces (Riemannian symmetric spaces of rank one). All such spaces are known: they are the spheres in the Euclidean spaces, the real, complex and quaternionic projective spaces and the octonionic projective plane. For all such spaces the best possible bounds for the quadratic discrepancies and sums of pairwise distances are obtained in the paper (Theorems 2.1 and 2.2). Distributions of points of $t$-designs on such spaces are also considered (Theorem 2.3). In particular, it is shown that the optimal $t$-designs meet the best possible bounds for quadratic discrepancies and sums of pairwise distances (Corollary 2.1). Our approach is based on the Fourier analysis on two-point homogeneous spaces and explicit spherical function expansions for discrepancies and sums of distances (Theorems 4.1 and 4.2).

For each algebraic integer α, let ℤα denote the ring of integers of the algebraic number field ℚ(α). There has been continuing interest in finding ring-theoretic conditions characterizing when ℤα coincides with its subring ℤ[α] (cf.[15,18,1,13,12]). One way to extend such work is to consider the intermediate ring ℤ[α]+, the seminormalization (in the sense of [17]) of ℤ[α] in ℤα. Indeed, if we let Iα denote the conductor (ℤ[α]: ℤα), then it is easy to see (cf. Proposition 3.1) that ⅂[α] = ℤα, if, and only if, ℤ[α]+ = ℤα and Iα is a radical ideal of Zα. The condition ℤ[α]+ = ℤα seems worthy of separate attention in view of recent results (cf. [3]) that seminormal rings generated by algebraic integers are “often” automatically of the form ℤα. We show in Proposition 3.3 that the condition ℤ[α]+ = ℤα is equivalent to several universal properties, including notably that the canonical closed surjection Spec (ℤα) → Spec (ℤ[α]) be universally open, be universally going-down, or be a universal homeomorphism.

Fermat gave the first example of a set of four positive integers {a1, a2, a3, a4} with the property that aiaj+1 is a square for 1≤i<j≤4. His example was {1, 3, 8, 120}. Baker and Davenport [1] proved that the example could not be extended to a set of 5 positive integers such that the product of any two of them plus one is a square. Kangasabapathy and Ponnudurai [6], Sansone [9] and Grinstead [4] gave alternative proofs. The construction of such sets originated with Diophantus who studied the problem when the ai are rational numbers. It is conjectured that there do not exist five positive integers whose pairwise products are all one less than the square of an integer. Recently Dujella [3] proved that there do not exist nine such integers. In this note we address the following related problem. Let V denote the set of pure powers, that is, the set of positive integers of the form xk with x and k positive integers and k>1. How large can a set of positive integers A be if aa′ + 1 is in V whenever a and a′ are distinct integers from A? We expect that there is an absolute bound for |A|, the cardinality of A. While we have not been able to establish this result, we have been able to prove that such sets cannot be very dense.

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