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.

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.

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 an arbitrary subset $X$ of a finite-dimensional real Banach space $E$, the ball intersection with parameter $\lambda \gt 0$ is defined as the intersection of all balls of radius $\lambda $ whose centers are in $X$. On the other hand, the intersection of all balls of radius $\lambda $ that contain $X$ is said to be the respective ball hull. We present new results on these two notions and use them to get new insights into complete sets and (pairs of) sets of constant width, e.g., their representation as vector sums of suitable ball intersections and ball hulls. Also in this framework, we give partial answers to the known question, in what finite-dimensional real Banach spaces any complete set is of constant width. For polyhedral norms we obtain characterizations of monotypic balls via constant width properties of pairs formed by the ball intersection and ball hull of the same bounded and non-empty set. Finally, we present some new results on Borsuk numbers of sets of constant width in normed spaces, closely related to (unique) completions of compact sets. For example, the lower estimate on Borsuk numbers of bodies of constant width due to Lenz is extended to arbitrary normed spaces. Furthermore, we also derive the Borsuk number of the normed space with maximum norm.

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.

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.

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.

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.

Before stating the main theorem, we would like to recall the basic properties of the “zeta-functions” attached to cusp forms on SL (2, ℤ). Let k be an even integer ≥ 12 and f a cusp form of weight k on SL (2, ℤ) with q-expansion We shall assume that c1 = 1, and that f is an eigenfunction of the Hecke operators. Define φ(s) as the Dirichlet series The series and the product

over the primes are equal and absolutely convergent for Re (S) > ½(k + 1).

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.

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.

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.

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.

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.

We study the fine-scale $L^{2}$-mass distribution of toral Laplace eigenfunctions with respect to random position in two and three dimensions. In two dimensions, under certain flatness assumptions on the Fourier coefficients and generic restrictions on energy levels, both the asymptotic shape of the variance is determined and the limiting Gaussian law is established in the optimal Planck-scale regime. In three dimensions the asymptotic behaviour of the variance is analysed in a more restrictive scenario (“Bourgain’s eigenfunctions”). Other than the said precise results, lower and upper bounds are proved for the variance under more general flatness assumptions on the Fourier coefficients.

Let $X$ be an infinite-dimensional uniformly smooth Banach space. We prove that $X$ contains an infinite equilateral set. That is, there exist a constant $\lambda \gt 0$ and an infinite sequence $\mathop{({x}_{i} )}\nolimits_{i= 1}^{\infty } \subset X$ such that $\Vert {x}_{i} - {x}_{j} \Vert = \lambda $ for all $i\not = j$.

This paper is devoted to dimensional reductions via the norm-resolvent convergence. We derive explicit bounds on the resolvent difference as well as spectral asymptotics. The efficiency of our abstract tool is demonstrated by its application on seemingly different partial differential equation problems from various areas of mathematical physics; all are analysed in a unified manner, known results are recovered and new ones established.

The main object of this paper is to give a self-contained elementary proof of a result (Theorem 1, below), which could be deduced from a theorem of Siegel ([1], Satz 2). It seems worth while to do so, because Siegel's proof is long and difficult, though his result is deeper and more precise than mine.