By a well-known theorem of Szemerdi 8 any set of integers that has positive density contains arithmetic progressions of arbitrary length. One might expect that there are conditions of similar generality, under which an integer set contains arbitrarily long strings of consecutive integers, i.e., arithmetic progressions with 1 as common difference. Results of this kind would be of great importance because of potential applications to arithmetically interesting sets such as the set n: (n) = 1, where (n) is the Liouville function, or the sets

where P(n) denotes the greatest prime factor of n and 0< < 1. One naturally expects that such sets contain arbitrarily long strings of consecutive integers, but no results in this direction are known, and the problem seems to be a very difficult one, perhaps comparable in depth to the prime k-tuple conjecture.

Let R be a ring, not necessarily commutative, with an identity element, and let A be a left R-module. We shall describe this situation by writing (RA). If

is an exact sequence of left. R-modules and R-homomorphisms in which each Pi (i ≥ 0) is R-projective, then the sequence

which we denote by P, is called an R-projective resolution of A. Suppose now that A is non-trivial; if Pi = 0 when i > n and if there are no R-projective resolutions of A containing fewer non-zero terms, then A is said to have left projective (or homological) dimension n, and we write 1.dim RA = n. If no finite resolutions of this type exist, we write l.dim RA = ∞. As a convention, we put l.dim R0 = −1. If M denotes a variable left R-module, then is called the left global dimension of the ring R and is denoted by l.gl. dim R. It is well known that l.dim RA < n if and only if for all left R-modules B and that l.gl.dim R < m if and only if regarded as a functor of left R-modules, takes only null values.

Let be a plane bounded convex set whose width in the direction φ is . It has been shown by L. A. Santaló that

is invariant under unimodular affine transformations. Santaló [1] established a number of properties of this invariant and conjectured that if the area of is A() then and that equality characterizes triangles. In this note Santaló's conjecture is shown to be true.

A metric is defined for a plane affine geometry with coordinates in GF(2n) and is applied to the geometry of a triangle.

In a recent article [3] L. Mirsky proved a theorem which gives necessary and sufficient conditions for the existence of a finite integral matrix whose elements, row sums, and column sums all lie within prescribed bounds. Mirsky suggested to me the problem of extending his theorem to infinite matrices, and it is the solution of this problem that is presented in this note. To allow for extra generality, instead of prescribing upper and lower bounds for the row and column sums we shall prescribe upper and lower bounds for the row and column deficiencies (a term to be explained later). The theorem when upper and lower bounds for the row and column sums are prescribed is then a special case of the deficiency theorem. The solution of the problem depends on a construction of Mirsky [3] and a theorem of mine [1] concerning the existence of a partial transversal of a family of sets satisfying certain properties. As will be seen, we shall take a rather broad view of the notion of a matrix.

When K is a convex body in d-dimensional euclidean space E and 0 ≤ s ≤ d, the s-skeleton of K, denoted skel(s)K, consists of those points of K which are not centres of (s + l)-dimensional balls contained in K. The s-skeleton is thus the union of the extreme faces of K having dimension at most s. The s-skeleton is a -set [see 6] and it is therefore measurable with respect to the s-dimensional Hausdorff measure ℋ(s) [see 7]; here we normalize ℋ(s) so that it assigns unit measure to the s-dimensional unit cube.

The aim of this paper is to prove the following

THEOREM. Let M be a C∞ compact and strictly convex surface embedded in the euclidean space E3 or in the hyperbolic space H3. We suppose that all shadow-lines ofM are congruent. Then M is a euclidean 2-sphere or a hyperbolic 2-sphere respectively.

Let $I_{s,k,r}(X)$ denote the number of integral solutions of the modified Vinogradov system of equations

$$\begin{eqnarray}x_{1}^{j}+\cdots +x_{s}^{j}=y_{1}^{j}+\cdots +y_{s}^{j}\quad (1\leqslant j\leqslant k,\;j\neq r),\end{eqnarray}$$

$1\leqslant x_{i},y_{i}\leqslant X\;(1\leqslant i\leqslant s)$

$I_{s,k,r}(X)$

$1\leqslant r\leqslant k-1$

$s,k\in \mathbb{N}$

$k\geqslant 3$

$1\leqslant s\leqslant (k^{2}-1)/2$

$I_{s,k,1}(X)\ll X^{s+\unicode[STIX]{x1D700}}$

Assuming $T_{0}$ to be an m-accretive operator in the complex Hilbert space ${\mathcal{H}}$, we use a resolvent method due to Kato to appropriately define the additive perturbation $T=T_{0}+W$ and prove stability of square root domains, that is,

$$\begin{eqnarray}\text{dom}((T_{0}+W)^{1/2})=\text{dom}(T_{0}^{1/2}).\end{eqnarray}$$

$\text{dom}(T_{0}^{1/2})=\text{dom}((T_{0}^{\ast })^{1/2})$

$$\begin{eqnarray}\text{dom}((T_{0}+W)^{1/2})=\text{dom}(T_{0}^{1/2})=\text{dom}((T_{0}^{\ast })^{1/2})=\text{dom}(((T_{0}+W)^{\ast })^{1/2}),\end{eqnarray}$$

$$\begin{eqnarray}-\text{div}(a{\rm\nabla}\,\cdot )+(\vec{B}_{1}\cdot {\rm\nabla}\,\cdot )+\text{div}(\vec{B}_{2}\,\cdot )+V\end{eqnarray}$$

$L^{2}({\rm\Omega})$

${\rm\Omega}\subseteq \mathbb{R}^{n}$

$n\in \mathbb{N}$

$\partial {\rm\Omega}$

$\partial {\rm\Omega}$

Let $\Vert \cdots \Vert$ denote distance from the integers. Let $\unicode[STIX]{x1D6FC}$, $\unicode[STIX]{x1D6FD}$, $\unicode[STIX]{x1D6FE}$ be real numbers with $\unicode[STIX]{x1D6FC}$ irrational. We show that the inequality

$$\begin{eqnarray}\Vert \unicode[STIX]{x1D6FC}p^{2}+\unicode[STIX]{x1D6FD}p+\unicode[STIX]{x1D6FE}\Vert <p^{-3/17+\unicode[STIX]{x1D700}}\end{eqnarray}$$

$p$

$\unicode[STIX]{x1D6FC}p$

$\unicode[STIX]{x1D6FD}=0$

We establish asymptotic formulae for the number of $k$-free values of square-free polynomials $F(x_{1},\ldots ,x_{n})\in \mathbb{Z}[x_{1},\ldots ,x_{n}]$ of degree $d\geqslant 2$ for any $n\geqslant 1$, including when the variables are prime, as long as $k\geqslant (3d+1)/4$. This generalizes a work of Browning.

We provide a unified approach that encompasses some integral formulas for functions of the visual angle of a compact convex set due to Crofton, Hurwitz and Masotti. The basic tool is an integral formula that also allows us to integrate new functions of the visual angle. Also, we establish some upper and lower bounds for the considered integrals, generalizing, in particular, those obtained by Santaló for Masotti’s integral.

We show that integral monodromy groups of Kloosterman $\ell$-adic sheaves of rank $n\geqslant 2$ on $\mathbb{G}_{m}/\mathbb{F}_{q}$ are as large as possible when the characteristic $\ell$ is large enough, depending only on the rank. This variant of Katz’s results over $\mathbb{C}$ was known by works of Gabber, Larsen, Nori and Hall under restrictions such as $\ell$ large enough depending on $\operatorname{char}(\mathbb{F}_{q})$ with an ineffective constant, which is unsuitable for applications. We use the theory of finite groups of Lie type to extend Katz’s ideas, in particular the classification of maximal subgroups of Aschbacher and Kleidman–Liebeck. These results will apply to study hyper-Kloosterman sums and their reductions in forthcoming work.

I am grateful to Professor K. Prachar for pointing out to me that there is a mistake in the proof of Theorem 2 in my paper “On the distribution of primes in short intervals” [Mathematika, 23 (1976), 4–9]. The mistake is in the assertion on p. 6 that, if 1 ≤ μ/λ < 4, the result is trivial. The corrected version reads as follows.

Throughout this paper A is a commutative noetherian ring (with identity) and M is an A-module. We use to denote, for i ≥ 0, the i-th right derived functor of the local cohomology functor L with respect to an ideal a of A [8; 2.1].

A natural and welcomed decomposition theorem for elements in the positive cone of the tensor product of Archimedean vector lattices leads to substantial simplifications in the theory of tensor products of Archimedean vector lattices.

This paper is concerned with the determination of the small deflexions and stresses in a thin isotropic homogeneous semi-circular plate subject to hydrostatic pressure distributed over the entire plate. The semi-circular plate is clamped along the bounding diameter and elastically restrained, against rotation along the circular edge, according to a general boundary condition including the simply supported and rigidly clamped boundaries as particular cases. The boundary conditions around the circular edge and the condition of zero deflexion on the straight edge are exactly satisfied while the remaining condition of zero slope on the bounding diameter is approximately satisfied. Calculations have been carried out corresponding to three cases of boundary constraint; namely, those for which the edge is fully restrained, the edge is simply supported and an intermediate case. Positions and magnitudes of maximum deflexions are determined. Numerical results are presented in tabular and graphical forms.

A finite family (Ci)i ∊ I of at least two convex subsets of ℝn is said to have the intersection property provided that the set is non-empty for all families (ai)i ∊ I of points in ℝn. Previously, D. G. Larman [2, Theorem 3] has given a sufficient condition (which is not necessary) and an “almost” necessary condition (which is not sufficient) for to have the intersection property.

A hemisphere, resting on a horizontal plane, is initially at rest relative to an incompressible, inviscid, non-diffusive fluid whose density is vertically stratified. The hemisphere is then given, impulsively, a small constant horizontal velocity which is maintained thereafter. Assuming that the Froude number is small, and using the Boussinesq approximation, the equations of motion are linearised and solved using a Laplace transform. The disturbance in the fluid is analysed for large times and is found to contain a steady component of purely horizontal flow, an internal wave field and internal oscillations at the Brunt-Väisälä frequency, together with their various interactions. The effects of viscosity and diffusivity are discussed qualitatively by considering their effects on an internal wave.

The purpose of this paper is to call attention to a very simple example of intersections on a singular variety, and to its effect on intersection theories relating to ambient varieties with singularities; in particular it will be shown how the example—to which we refer throughout as Example A—invalidates an aspect of the theory put forward by P. Samuel ([1], Ch. VI).