One construction used to produce a random number table is to take a smooth function F(x) taking values between 0 and 1, to evaluate it at N points spaced 1/M apart, and to ignore the first t decimal digits. With T = 10t this corresponds to taking the fractional part of

where T>M>N. The grounds for assuming this sequence to be random are that it is so difficult to prove anything about it.

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.

§1. Let k be an algebraic number field of finite degree over the field Q of rational numbers and Ki be an extension of k of degree (Ki:k) = ni, i = 1,2. We choose a Hecke Grössencharakter Xi in Ki and consider the L-function

associated with Xi see [1, 2]. It is known to be a meromorphic function on the whole complex plane. We are interested here in the properties of the convolution of functions LK1, LK2 over k defined by

where

and a (and n) run over integral ideals of Ki (and k), whose norm NKi/k a is equal to n. The function (2) is sometimes called the “scalar product of the Hecke L-functions” «3—11». The object of the paper is the following theorem.

Green [8] has shown that a constitutive relation of the form

arises as a special case of an incompressible anisotropic simple fluid, where S is the stress tensor or matrix,

and V is the velocity gradient matrix at time t, all measured in a fixed rectangular cartesian coordinate system. Also, if F is the displacement gradient measured with respect to some curvilinear reference system θi, then

where R is a proper orthogonal matrix, and M and K are positive definite symmetric matrices. In addition

and

It is shown that a separable Hilbert space can be covered by non-overlapping closed convex sets Ci with outer radii uniformly bounded from above and inner radii uniformly bounded from below. This answers a question originating from the work of Klee.

In this paper we shall consider some axisymmetrical problems involving the determination of a harmonic function which satisfies prescribed conditions on two given spherical surfaces. The latter may be exterior to one another or one inside the other. The classical problems which come under this heading are the electrostatic two sphere condenser and the motion of two spheres along the line of centres in an unbounded inviscid fluid. The early investigators of these problems used the well known method of images (Kelvin [1], Hicks [2]). In the electrostatic case (Dirichlet boundary conditions) the images are sets of point monopoles, and in the hydrodynamic case (Neumann boundary conditions) sets of dipole sources. Later Neumann [3] and Jeffery [4] gave solutions using a bipolar coordinate system.

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.

The impaction on symmetrical obstacles placed in uniform streams of aerosols is investigated. The governing equations of motion are nonlinear differential equations involving a parameter called the Stokes number. The study differentiates between the critical value of the Stokes number on the centre-line, kcr, below which no particles reach the stagnation point in finite time, and the critical value of the Stokes number on the obstacle, Kcr, below which no particles may be deposited on the obstacle in finite time. Based on the properties of the centre-line fluid velocity of the potential and viscous flows past a variety of symmetrically shaped obstacles, upper and lower bounds of Kcr and Kcr are established. Furthermore, using a numerical procedure for solving nonlinear differential equations with unknown parameters the critical values of Kcr and Kcr are obtained.

Barannikov and Kontsevich [Frobenius manifolds and formality of Lie algebras of polyvector fields. Int. Math. Res. Not. IMRN1998(4) (1998), 201–215], constructed a DGBV (differential Gerstenhaber–Batalin–Vilkovisky) algebra $\mathbf{t}$ for a compact smooth Calabi–Yau complex manifold $M$ of dimension $m$, which gives rise to the $B$-side formal Frobenius manifold structure in the homological mirror symmetry conjecture. The cohomology of the DGBV algebra $\mathbf{t}$ is isomorphic to the total singular cohomology $H^{\bullet }(M)=\bigoplus _{k=0}^{2m}H^{k}(M,\mathbb{C})$ of $M$. If $M=X_{G}(\mathbb{C})$, where $X_{G}$ is the hypersurface defined by a homogeneous polynomial $G(\text{}\underline{x})$ in the projective space $\mathbb{P}^{n}$, then we give a purely algorithmic construction of a DGBV algebra ${\mathcal{A}}_{U}$, which computes the primitive part $\bigoplus _{k=0}^{m}\mathbf{PH}^{k}$ of the middle-dimensional cohomology $\bigoplus _{k=0}^{m}H^{k}(M,\mathbb{C})$, using the de Rham cohomology of the hypersurface complement $U_{G}:=\mathbb{P}^{n}\setminus X_{G}$ and the residue isomorphism from $H_{\text{dR}}^{k}(U_{G}/\mathbb{C})$ to $\mathbf{PH}^{k}$. We observe that the DGBV algebra ${\mathcal{A}}_{U}$ still makes sense even for a singular projective Calabi–Yau hypersurface, i.e. ${\mathcal{A}}_{U}$ computes $\bigoplus _{k=0}^{m}H_{\text{dR}}^{k}(U_{G}/\mathbb{C})$ even for a singular $X_{G}$. Moreover, we give a precise relationship between ${\mathcal{A}}_{U}$ and $\mathbf{t}$ when $X_{G}$ is smooth in $\mathbf{P}^{n}$.

Consider a system of polynomials in many variables over the ring of integers of a number field $K$. We prove an asymptotic formula for the number of integral zeros of this system in homogeneously expanding boxes. As a consequence, any smooth and geometrically integral variety $X\subseteq \mathbb{P}_{K}^{m}$ satisfies the Hasse principle, weak approximation, and the Manin–Peyre conjecture if only its dimension is large enough compared to its degree. This generalizes work of Skinner, who considered the case where all polynomials have the same degree, and recent work of Browning and Heath-Brown, who considered the case where $K=\mathbb{Q}$. Our main tool is Skinner’s number field version of the Hardy–Littlewood circle method. As a by-product, we point out and correct an error in Skinner’s treatment of the singular integral.

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

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

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

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

$1\leqslant r\leqslant k-1$

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

$k\geqslant 3$

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

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

We study the distribution of the size of Selmer groups arising from a 2-isogeny and its dual 2-isogeny for quadratic twists of elliptic curves with a non-trivial $2$-torsion point over $\mathbb {Q}$. This complements the work [Xiong and Zaharescu, Distribution of Selmer groups of quadratic twists of a family of elliptic curves. Adv. Math.219 (2008), 523–553] which studied the same subject for elliptic curves with full 2-torsions over $\mathbb {Q}$ and generalizes [Feng and Xiong, On Selmer groups and Tate–Shafarevich groups for elliptic curves $y^2=x^3-n^3$. Mathematika 58 (2012), 236–274.] for the special elliptic curves $y^2=x^3-n^3$. It is shown that the 2-ranks of these groups all follow the same distribution and in particular, the mean value is $\sqrt {\frac {1}{2}\log \log X}$ for square-free positive integers $n \le X$ as $X \to \infty $.

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.

Lyndon's axiomatic methods are used in [1] to show, among other things, that a group G with an integer valued length function satisfying certain conditions is free. At the end of his paper [2] Lyndon gives a method of embedding such a group in a free group whose natural length function extends the function on G. We construct here a simpler embedding with the same property.

A metric space (X, ρ) is called precompact, if, for every ε > 0, there is a finite ε-cover (a covering by sets of diameter ≤ ε). The space (X, ρ) is separable if for every e there is a countable ε-cover. There should be some in-between condition. We say that (X, ρ) has fine covers, if, for every ε > 0, there exists a countable ε-cover (U1, U2, …), such that the diameter ∂(Ui) tends to zero as i → ∞. In fact, Goodey [1] has related this property to Hausdorff dimension. We show that a space with fine covers need not be σ-precompact and that on any complete metrizable non-σ-compact space X there is a metric ρ* such that (X, ρ*) has no fine cover.

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 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 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.

In the first part of the paper we show that the L2-discrepancy with respect to squares is of the same order of magnitude as the usual L2- discrepancy for point distributions in the K-dimensional torus. In the second part we adapt this method to obtain a generalization of Roth's [7] lower bound (log N)(k-1)/2 (for the usual discrepancy) to the discrepancy with respect to homothetic simple convex poly topes.