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.

The aim of this article is to present a general “large deviations approach” to the geometry of polytopes spanned by random points with independent coordinates. The origin of our work is in the study of the structure of ±1-polytopes, the convex hulls of subsets of the combinatorial cube . Understanding the complexity of this class of polytopes is important for the “polyhedral combinatorics” approach to combinatorial optimization, and was put forward by Ziegler in [20]. Many natural questions regarding the behaviour of ±1-polytopes in high dimensions are open, since, for many important geometric parameters, low-dimensional intuition does not help to identify the extremal ±1-polytopes. The study of random ±1-polytopes sheds light to some of these questions, the main reason being that random behaviour is often the extremal one.

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.

Let x1, …, xn be linear forms in u1, …, un with real coefficients, and (for simplicity) determinant 1. Given a form (that is, a homogeneous polynomial) F(x1, …, xn), we can ask the following question: do there exist, for arbitrary real α1, …, αn, integers ul, …, un such that

where C is a suitable number independent of α1, …, αn and of the particular linear forms x1 …, xn? In two well-known cases this is true: namely when

and when

where 1 ≤ r ≤ n − 1.

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?

Let p be a prime, t a positive integer, and P = P(x1, …, xn) a polynomial over the rational field K, in any number n of variables, of degree k = 2, 3, or 4. We shall consider the congruence

Let T be a nonlinear Volterra integral operator of the form

(with I compact, a < b), whose kernel K = K(x, t, u) satisfies the following conditions:

with

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.

Suppose that θ is a positive irrational number and α is an arbitrary real number. Then Kronecker's Theorem in diophantine approximation (Theorem 440 of [7[) can be stated as follows.

A strong quantitative form of Manin’s conjecture is established for a certain variety in biprojective space. The singular integral in an application of the circle method involves the third power of the integral sine function and is evaluated in closed form.

A method of solving Oseen's equations for the flow of viscous fluid past a cylinder was devised by Bairstow, Cave and Lang [1], who found a doublet solution of the equations and reproduced the flow by a distribution of these doublets over the surface of the cylinder. The same method is used here to deal with axisymmetric flow and an integral equation is given to determine the density of the distribution. The particular case of the flow past a sphere at low Reynolds numbers is solved by this method.

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.

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').

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 begin by introducing some notations which will be used throughout the paper. Let denote a sequence of distinct positive integers. Let B denote a measurable subset of [0, 1) and {x} the fractional part of x. We write

so that f has period 1 on the real line; and we write

for the number of times {mkx} falls in B, minus the expected value of this number (m(B) ≤ Lebesgue measure of B).

Let K be an algebraic number field, [K: Q] = κ є N; only the case κ > 1 is of interest in this paper. Let f be any non-zero ideal in ZK, the ring of integers of K, and let b be any ray-class (modx f) of K. In this paper we answer a question of P. Erdös (private communication) about the “maximum-growth-rate” of the functions

and

the sum here taken over all ray-classes (modx f), while N(a) is the absolute norm of a. Let

and

where, as usual, for x є R, log+ x - log max {1, x}. We prove

THEOREM 1. For every f, b we have

The principal objective of this work is to investigate various classes of centrally symmetric convex sets. These classes range from the zonoids at one extreme to the class of all centrally symmetric bodies at the other. The defining properties of these classes involve inequalities between mixed volumes. Various other characterizations will be found in response to a number of questions in a recent survey article by Rolf Schneider and Wolfgang Weil. Some of these are concerned with measures on a Grassmannian manifold while others relate to the intermediate surface area measures of convex bodies. We shall also show these classes are characterized by certain extremal geometric inequalities. The work concludes with a brief discussion of related results concerned with generalized zonoids.

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)).

An ordered triangle in the plane S2 is defined as a sextuple (PlP2, P3; l1, l2, l3) consisting of three points Pi and three lines lj restricted by the relations of incidence Pi Ì lj (i ¹ j) If we map unrestricted sextuples by the points of a V12—Segre product of six planes—we obtain an image-manifold Ω6 for ordered triangles as the appropriate subvariety of V12. The variety Ω6 possesses an ordinary double threefold ɸ3 whose points map the totally degenerate triangles (i.e. those for which P1 ═ P2 ═ P3 and l1 ═ l2 ═ l3); Ω6 is therefore unsuitable as a basis for the construction of an enumerative calculus for triangles, for equivalence theory is as yet developed satisfactorily only on non-singular varieties.