A sequence {an} of integers is said to be primitive if ai × aj whenever i ≠ j. Let f be a polynomial with integer coefficients and A a sequence of positive integers. We discuss further a problem considered in [1] in which I. Anderson, W. W. Stothers and the author investigated primitive sequences of the form f(A) = {f(x), x ∈ A}. (Of course, we can assume f(x)→ ∞ as x → ∞.) We shall prove the following theorem in which A(n), as usual, denotes the number of memhers of A that are. less than or equal to n.

In [2], Fabrikant and his colleagues obtain a closed form solution to a generalized potential problem for a surface of revolution. This they specialize to solve three electrostatic problems for a spherical cap, including one for which the boundary conditions are not axisymmetric. In all three the solutions are expressed in terms of elementary functions.

No systematic study seems to have been made of so natural a question as the analogue for matrices of quadratic residues. One generalization of x2 (x an integer) is X2 (X an integral matrix). Another is X′ X, where the prime means “transpose”. We study here the solvability for X of the congruence

where p is a prime, r ≥ 1; I (the identity matrix) and X are n-by-n; and a is an integer not divisible by p2.

The purpose of the paper is to introduce a novel “splitting” procedure which can be helpful in the derivation of explicit formulas for various Bellman functions. As an illustration, we study the action of the dyadic maximal operator on $L^{p}$. The associated Bellman function $\mathfrak{B}_{p}$, introduced by Nazarov and Treil, was found explicitly by Melas with the use of combinatorial properties of the maximal operator, and was later rediscovered by Slavin, Stokolos and Vasyunin with the use of the corresponding Monge–Ampère partial differential equation. Our new argument enables an alternative simple derivation of $\mathfrak{B}_{p}$.

This paper is motivated by Davenport’s problem and the subsequent work regarding badly approximable points in submanifolds of a Euclidean space. We study the problem in the area of twisted Diophantine approximation and present two different approaches. The first approach shows that, under a certain restriction, any countable intersection of the sets of weighted badly approximable points on any non-degenerate ${\mathcal{C}}^{1}$ submanifold of $\mathbb{R}^{n}$ has full dimension. In the second approach, we introduce the property of isotropically winning and show that the sets of weighted badly approximable points are isotropically winning under the same restriction as above.

We introduce a precise framework for transferring strategies from simpler to more complex games, and use it to construct strategies in certain finite and infinite combinations of games. In particular, we give a finitary characterization of finite hypergraphs X such that the first player can win the positional game on infinitely many copies of X. This resolves a conjecture of Leader.

For each $n\geqslant 2$ we construct a measurable subset of the unit ball in $\mathbb{R}^{n}$ that does not contain pairs of points at distance 1 and whose volume is greater than $(1/2)^{n}$ times the volume of the unit ball. This disproves a conjecture of Larman and Rogers from 1972.

We present a first example of a flag vector of a polyhedral sphere that is not the flag vector of any polytope. Namely, there is a unique $3$-sphere with the parameters $(f_{0},f_{1},f_{2},f_{3};f_{02})=(12,40,40,12;120)$, but this sphere is not realizable by a convex $4$-polytope. The $3$-sphere, which is $2$-simple and $2$-simplicial, was found by Werner [Linear constraints on face numbers of polytopes. PhD Thesis, TU Berlin, Germany, 2009]; we present results of a computer enumeration which imply that the sphere with these parameters is unique. We prove that it is non-polytopal in two ways: first, we show that it has no oriented matroid, and thus it is not realizable; this proof was found by computer, but can be verified by hand. The second proof is again a computer-based oriented matroid proof and shows that for exactly one of the facets this sphere does not even have a diagram based on this facet. Using the non-polytopality, we finally prove that the sphere is not even embeddable as a polytopal complex.

Utilizing frameworks developed by Delsarte, Yudin and Levenshtein, we deduce linear programming lower bounds (as $N\rightarrow \infty$) for the Riesz energy of $N$-point configurations on the $d$-dimensional unit sphere in the so-called hypersingular case; i.e., for non-integrable Riesz kernels of the form $|x-y|^{-s}$ with $s>d$. As a consequence, we immediately get (thanks to the poppy-seed bagel theorem) lower estimates for the large $N$ limits of minimal hypersingular Riesz energy on compact $d$-rectifiable sets. Furthermore, for the Gaussian potential $\exp (-\unicode[STIX]{x1D6FC}|x-y|^{2})$ on $\mathbb{R}^{p}$, we obtain lower bounds for the energy of infinite configurations having a prescribed density.

An exact solution for the velocity field induced by the slow rolling motion of a sphere in contact with a permeable surface is derived. The porous solid and the viscous fluid each occupies a semi-infinite space.

It is shown that, by the use of conformal mapping, the problem is reduced to a fourth order ordinary linear boundary value problem. As the permeability of the porous body diminishes the important functionals, e.g. the force and torque resisting the rolling motion of the sphere, are divergent. The nature of this divergence is found and the dependence on the permeability of the porous solid is explicitly evaluated.

When a displacement front meets a heterogeneity in a porous medium, its shape will be altered. The amount of distortion depends on the size and shape of the heterogeneity, the amount of variation of the heterogeneous properties, and the mobility ratio between the displaced and displacing fluids. The solution for a circular permeability discontinuity is known when the capillary pressure is uniform and the mobility ratio M is unity [4]. Here, we extend the theory to the case where there is a small change ε in capillary pressure as well as a nearly unit mobility ratio M = 1 + εγ. Corrections can then be found, in closed form, to first order in ε, to the shape of the front and the pressure field. Computations using these expressions are simpler than the full free-boundary problem, and some analytical estimates are possible in further limits. Finally, the theory is extended to the case of a front passing a number of such heterogeneous patches which are widely spaced.

The Local Uniformisation Theorem was proved by O. Zariski [5] in 1940, and, for the general case, it is so far the only existing proof of the theorem.

Let V be an irreducible manifold defined over a ground field of characteristic zero, and let Σ be its function field. Suppose V lies in an affine space An, and D is any subvariety of V not at infinity. Let J be the integral domain of V and ρ be the prime ideal in J defining D. Then we denote the quotient ring of D by Q(D|V), and by this we shall mean the quotient ring Jρ [1; p. 99]. Thus when we deal with subvarieties of two birationally equivalent manifolds V and V', then the quotient rings will always be subrings of the same representation of the function field of V and V'. Let B be any valuation of Σ whose centre on V is C. The Local Uniformisation Theorem states that there exists a birational transform V' of V such that the centre C' of B on V' is simple and Q(C/V) ⊆Q(C'/V').

The results we present were motivated by the product measure problem for Baire measures. For two completely regular Hausdorff spaces X and Y, with totally finite a- additive measures μ and ν defined on the Baire σ- algebras ℬ0(X) and ℬ0(Y) respectively, under what conditions may we define a measure λ on the Baire σ-algebra ℬ0(X × Y), extending the product measure μ ⊗ ν defined on the product σ-algebra ℬ0(X) × ℬ0(Y) and satisfying a Fubini theorem?

In this paper we continue the investigation begun in [6] concerning the number of solutions of the inequality

for almost all α (in the sense of Lebesgue measure on ℝ), where β is a given real number, , and both m and n are confined to sets of numbertheoretic interest. Our aim is to extend existing results ([7], [8], [5] for example), where only n is restricted. Here we shall prove the following result where, as elsewhere in this paper, p denotes a prime, and a square-free integer may be positive or negative.

In this note we give a direct algebraic proof of a theorem of Warfield on algebraically compact modules. It is shorter than the one given by Azumaya in [1], in that it does not use the embedding of a module M into M** (where M* is the character Homz (M, Q/Z)).

We prove that for every strictly convex body C in the Euclidean space of dimension d≥3, some aflfine image of C admits a non-lattice covering of the space, thinner than any lattice covering. We illustrate the general construction with an example of a thin non-lattice covering of with certain congruent ellipsoids.

The Newhouse gap lemma is generalized by finding a geometric condition which ensures that N-fold sums of compact sets, which might even have thickness zero, are intervals. A new proof is also obtained of a lower bound on the thickness of the sum of two Cantor sets.

Consider a real valued Morse function f on a C2 closed connected n-dimensional manifold M. It is proved that a suitable Riemannian metric exists on M, such that f is harmonic outside the set of critical points of f of index 0 and n. The proof is based on a result of Calabi [1], providing a criterion for a closed one-form on a closed connected manifold to be harmonic with respect to some Riemannian metric.

The covariogram of a compact set A⊂ℝn is the function that to each x∈ℝn associates the volume of A∩(A+x). Recently it has been proved that the covariogram determines any planar convex body, in the class of all convex bodies. We extend the class of sets in which a planar convex body is determined by its covariogram. Moreover, we prove that there is no pair of non-congruent planar polyominoes consisting of less than nine points that have equal discrete covariograms.

A new lower bound is established for the distance between two roots of an integer polynomial, and a new upper bound for the distance between a given real number and the set of zeros of an integer polynomial. The latter result is applied to improve a metrical result in Diophantine approximation.