Let $\unicode[STIX]{x1D6FD}>1$ be a real number and define the $\unicode[STIX]{x1D6FD}$ -transformation on $[0,1]$ by $T_{\unicode[STIX]{x1D6FD}}:x\mapsto \unicode[STIX]{x1D6FD}x\,\text{mod}\,1$ . Further, define

$$\begin{eqnarray}W_{y}(T_{\unicode[STIX]{x1D6FD}},\unicode[STIX]{x1D6F9}):=\{x\in [0,1]:|T_{\unicode[STIX]{x1D6FD}}^{n}x-y|<\unicode[STIX]{x1D6F9}(n)\text{ for infinitely many }n\}\end{eqnarray}$$

$$\begin{eqnarray}W(T_{\unicode[STIX]{x1D6FD}},\unicode[STIX]{x1D6F9}):=\{(x,y)\in [0,1]^{2}:|T_{\unicode[STIX]{x1D6FD}}^{n}x-y|<\unicode[STIX]{x1D6F9}(n)\text{ for infinitely many }n\},\end{eqnarray}$$

$\unicode[STIX]{x1D6F9}:\mathbb{N}\rightarrow \mathbb{R}_{{>}0}$

$\unicode[STIX]{x1D6F9}(n)\rightarrow 0$

$n\rightarrow \infty$

A Minkowski class is a closed subset of the space of convex bodies in Euclidean space ℝn which is closed under Minkowski addition and non-negative dilatations. A convex body in ℝn is universal if the expansion of its support function in spherical harmonics contains non-zero harmonics of all orders. If K is universal, then a dense class of convex bodies M has the following property. There exist convex bodies T1, T2 such that M + T1 = T2, and T1, T2 belong to the rotation invariant Minkowski class generated by K. It is shown that every convex body K which is not centrally symmetric has a linear image, arbitrarily close to K, which is universal. A modified version of the result holds for centrally symmetric convex bodies. In this way, a result of S. Alesker is strengthened, and at the same time given a more elementary proof.

The present paper extends the idea of characterizing topological properties of a space X by means of continuous selections for its closed subsets (X) endowed with a “natural” hyperspace topology. In this particular case, it is proved that the property of X to be topologically well-orderable is equivalent to the existence of a selection for (X) which is continuous with respect to the Fell topology.

We study a fast method for computing potentials of advection–diffusion operators $-{\rm\Delta}+2\mathbf{b}\boldsymbol{\cdot }{\rm\nabla}+c$ with $\mathbf{b}\in \mathbb{C}^{n}$ and $c\in \mathbb{C}$ over rectangular boxes in $\mathbb{R}^{n}$. By combining high-order cubature formulas with modern methods of structured tensor product approximations, we derive an approximation of the potentials which is accurate and provides approximation formulas of high order. The cubature formulas have been obtained by using the basis functions introduced in the theory of approximate approximations. The action of volume potentials on the basis functions allows one-dimensional integral representations with separable integrands, i.e. a product of functions depending on only one of the variables. Then a separated representation of the density, combined with a suitable quadrature rule, leads to a tensor product representation of the integral operator. Since only one-dimensional operations are used, the resulting method is effective also in the high-dimensional case.

For a stationary random closed set Ξ in ℝd it is well known that the first-order characteristics volume fraction VV, surface intensity SV and spherical contact distribution function Hs(t) are related by

(1 – VV) Hs′(0) = SV.

When the Helmholz equation

is separated in ellipsoidal coordinates [1; §1.6] and the technique of separation of variables applied, there results the ordinary differential equation

known as the ellipsoidal wave equation or Lamé wave equation. In this equation k is the modulus of the Jacobian elliptic function sn z, and is related to the eccentricity of the fundamental ellipse of the ellipsoidal coordinates; a, b are separation constants, and the parameter q is connected with the wave number χ by

l being a real constant, the dimensional parameter of the coordinate system.

The statement from the title is proved.

We consider positive-definite ternary quadratic forms with integer coefficients. Such a form, f, can be written in matrix notation as

Here x′ is the transpose of the column vector x = {x1, x2, x3) and a,ij = aji is the coefficient of xixj in f. Clearly det A is positive and even and so

is a negative integer.

Let N(ρ; ω) be the number of points of a d-dimensional lattice Γ. where d≥2, inside a ball of radius ρ centred at the point ω. Denote by (ρ) the number N(ρ; ω) averaged over ω in the elementary cell Ω of the lattice Γ. The main result is the following lower bound for for dimensions d ≅ l(mod 4):

We denote by S the unit sphere in ℝ3, and µ is the rotationally invariant measure, generalizing surface area on S; thus µS = 4π. We identify directions (or unit vectors) in ℝ3 with points on S, and prove the following:

Theorem 1. If E is a subset of ℝ3 of Lebesgue measure zero, then for µ almost all directions α, every plane normal to α intersects E in a set of plane measure zero.

If a group G is presented in terms of generators and relations, then the classical Reidemeister-Schreier Theorem [1] gives a presentation for any subgroup of G. If G is a free product of groups Gα each of which is presented in terms of generators and relations, then the main result of this paper is a presentation for any subgroup H of G, which shows the nature of H as a free product of certain subgroups of G. This result is a generalization of the celebrated Kuroš Theorem [2]. It also includes the Reidemeister–Schreier Theorem and the Schreier Theorem [1] which states that any subgroup of a free group is free.

Let $K$ be a convex body in $\mathbb{R}^{d}$ which slides freely in a ball. Let $K^{(n)}$ denote the intersection of $n$ closed half-spaces containing $K$ whose bounding hyperplanes are independent and identically distributed according to a certain prescribed probability distribution. We prove an asymptotic formula for the expectation of the difference of the volumes of $K^{(n)}$ and $K$, and an asymptotic upper bound on the variance of the volume of $K^{(n)}$. We obtain these asymptotic formulas by proving results for weighted mean width approximations of convex bodies that admit a rolling ball by inscribed random polytopes and then using polar duality to convert them into statements about circumscribed random polytopes.

The shape of large densest sphere packings in a lattice L ⊂ Ed (d ≥ 2), measured by parametric density, tends asymptotically not to a sphere but to a polytope, the Wulff-shape, which depends only on L and the parameter. This is proved via the density deviation, derived from parametric density and diophantine approximation. In crystallography the Wulff-shape describes the shape of ideal crystals. So the result further indicates that the shape of ideal crystals can be described by dense lattice packings of spheres in E3.

The problem of a concentrated normal force at any point of a thin clamped circular plate was treated in terms of infinite series by Clebsch [1], who gave the general solution of the biharmonic equation D∇4w = p. Using the method of inversion Michell [2] found a solution for the same problem in finite terms. The method of complex potentials was used by Dawoud [3] to solve the problem of an isolated load on a circular plate under certain boundary conditions. Applying Muskhelishvili's method Washizu [4] obtained the same results for clamped and hinged boundaries. The complex variable method was applied by the authors [5] to obtain solutions for a thin circular plate having an eccentric circular patch symmetrically loaded with respect to its centre under a particular form of boundary condition defining certain types of boundary constraints which include the usual clamped and hinged boundaries as well as other special cases. Flügge [6] gave the solution for a linearly varying load over the complete simply supported circular plate. Using complex variable methods Bassali [7] found the solution for the same load distributed over the area of an eccentric circle under the boundary conditions mentioned before [5], and the authors [8] obtained the solutions for general loads of the type cos nϑ(or sinnϑ), spread over the area of a circle concentric with the plate. In this paper the solutions for a circular plate subjected to the same boundary conditions are obtained when the plate is acted upon by the following types of loading: (a) a concentrated load at an arbitrary point; (b) a line load spread on any part of a diameter; (c) a load distributed over the area of a sector of the plate; (d) a concentrated couple at an arbitrary point of the plate. As a limiting case we find the deflexion at any point of a thin elastic plate having the form of a half plane clamped along the straight edge and subject to an isolated couple at any point.

In this paper, it is proved that, for any m unit vectors. x1…, xm in any n-dimensional real Hilbert space, there exists a unit vector x0 such that

for any y∈Sn−1. The exact value of the above integral is calculated, and these results used to improve some lower bounds for multilinear forms on real Hilbert spaces. An integral expression is also given for the complex case.

We study translation invariant, real-valued valuations on the class of convex polytopes in Euclidean space and discuss which continuity properties are sufficient for an extension of such valuations to all convex bodies. For this purpose, we introduce flag support measures of convex bodies via a local Steiner formula and derive some of the properties of these measures.

At high frequency, the leading terms in the virtual mass and wavemaking coefficients for a heaving, axisymmetric body depend only on the limit potential (Rhodes-Robinson, 1971). Here this result is applied to closed and open tori and solutions found in closed form. Some numerical values of the coefficients are tabulated.

The notion of nonatomicity of a measure on a Boolean σ-algebra is an important concept in measure theory. What could be an appropriate analogue of this notion for charges defined on Boolean algebras is one of the topics dealt with in this paper. Analogous to the decomposition of a measure on a Boolean σ-algebra into atomic and nonatomic parts, no decomposition of charges is available in the literature. We provide a simple proof of such a decomposition. Next, we study the conditions under which a Boolean algebra admits certain types of charges. These conditions lead us to give a characterisation of superatomic Boolean algebras. Babiker' [1] almost discrete spaces are connected with superatomic Boolean algebras and a generalisation of one of his theorems is obtained. A counterexample is also provided to disprove one of his theorems. Finally, denseness problems of certain types of charges are studied.

A study complementary to Part 1 (Smith 1979) is made of the linear stability characteristics, at high Reynolds number (R), of Poiseuille How through tubes with closed cross-sections. The first significant deviation of the upper branch of the neutral stability curve (Part 1 having described the lower bilanch) from that of plane Poiseuille flow arises when the aspect ratio is decreased from infinity to O(R1/11). The axial wavenumber α on the upper branch is then O(R-1/11). A further decrease of the aspect ratio, to a finite value, forces this α to fall sharply to O(R-1). A similar phenomenon occurs for the lower branch (Part 1). Thus the two branches are likely to meet only when the aspect ratio becomes finite, with the neutrally stable disturbances then having very large axial length scales.

In this paper, further insight is obtained into the earlier approach of studying residually transcendental extensions of a valuation v of a field K to a simple transcendental extension K(x) of K by means of minimal pairs, thereby introducing new invariants corresponding to any element of an algebraic closure of K. It is also shown that these invariants are of independent interest as well. A characterization of those elements a belonging to is given such that there exists a minimal pair (a, δ) for some δ in the divisible closure of the value group of v.