A flag area measure on an $n$-dimensional euclidean vector space is a continuous translation-invariant valuation with values in the space of signed measures on the flag manifold consisting of a unit vector $v$ and a $(p+1)$-dimensional linear subspace containing $v$ with $0\leqslant p\leqslant n-1$. Using local parallel sets, Hinderer constructed examples of $\text{SO}(n)$-covariant flag area measures. There is an explicit formula for his flag area measures evaluated on polytopes, which involves the squared cosine of the angle between two subspaces. We construct a more general sequence of smooth $\text{SO}(n)$-covariant flag area measures via integration over the normal cycle of appropriate differential forms. We provide an explicit description of our measures on polytopes, which involves an arbitrary elementary symmetric polynomial in the squared cosines of the principal angles between two subspaces. Moreover, we show that these flag area measures span the space of all smooth $\text{SO}(n)$-covariant flag area measures, which gives a classification result in the spirit of Hadwiger’s theorem.

Abstract. Let Φ be in the disc algebra H∞ ∩ C(T) with its restriction to T in the Zygmund space of smooth functions λ*(T). If P(Φ') ⊂ T is the set of Plessner points of Φ' and if F = Φ + Ψ, where Ψ∈C1(T), it is shown that F(P(Φ')) ⊆ C is a set of zero linear Hausdorff measure. Applications are given to the spectral theory of multiplication operators.

Let Ω be a compact metric space of non-σ-finite Λh-measure, where Λh denotes the Hausdorff measure corresponding to some continuous increasing real function h(t), denned for t ≥ 0 and with h(t) > 0 and h(t) > 0 for t > 0. C. A. Rogers has drawn attention to two problems that have remained unsolved for some years, and which seem to present considerable difficulty:

I. Does Ω necessarily have subsets of finite positive Λh-measure?

II. Does Ω. necessarily contain a system ofdisjoint closed subsets each of non-σ-finite Λh-measure ?

The answers are affirmative when Ω is a subset of a Euclidean space (see [1], [2], [3], [4], [5]); recently D. G. Larman ([8], [9]) was able to extend each of these results to the case of the “finite-dimensional” compact metric spaces that he introduced in [7]. Both of his proofs depend on the detailed structure of such spaces; the aim of the present paper is to establish a general theorem, from which it follows in particular that Larman's second result is a consequence of his first, and that an affirmative answer to I would imply an affirmative answer to II. (Since it seems likely that I must be answered negatively in general, and II affirmatively, the present theorem may well be of only ephemeral significance; the nature of the proof reinforces this expectation.)

Much recent attention has been given to geometric representation of elements of the stable homotopy groups of spheres, π*s A particular example concerns non-singular bilinear maps ℝm+1 × ℝn+1 → ℝm+n+1−p; on restriction and normalisation these become biskew maps Sm × Sn ℝ Sm+n-p;. Now the Hopf construction ℋ applied to any map f: Sm × Sn → Sm+n-p yields

In this paper, we prove that if $X$ is an infinite-dimensional real Hilbert space and $J: X\rightarrow \mathbb{R} $ is a sequentially weakly lower semicontinuous ${C}^{1} $ functional whose Gâteaux derivative is non-expansive, then there exists a closed ball $B$ in $X$ such that $(\mathrm{id} + {J}^{\prime } )(B)$ intersects every convex and dense subset of $X$.

Sandwich semigroups were introduced in [4], [5] and [6]. Green's relations (for regular elements) were characterized for these semigroups in [11] and [13]. Sandwich semigroups of continuous functions first made their appearance in [5]. In this paper, we consider only sandwich semigroups of continuous functions and we refer to them simply as sandwich semigroups. We now recall the definition. Let X and Y be topological spaces and fix a continuous function α from Y into X. Let S(X, Y, α) denote the semigroup of all continuous functions from X into Y where the product fg of f, g ε S(X, Y, α) is defined by fg = f ∘ α ∘ g. We refer to S(X, Y, α) as a sandwich semigroup with sandwich function α. If X = Y and α is the identity map then S(X, Y, α) is, of course, just S(X), the semigroup of all continuous selfmaps of X.

Let where a0≠0, m≥2, n = e1 + … + em and the ξi(1≤i≤m) are the distinct zeros of f in some algebraic closure of the p-adic field . Then K = (ξ1, ξ2, …, ξm) is a finite separable extension of and we denote by “ord” the unique extension of the (additive) p-adic valuation on to K, normalized with ord p = 1.

The notion of generalized X-ray for star sets in a Riemannian manifold is introduced to prove uniqueness theorems for convex bodies contained in a simply convex neighbourhood of a two-manifold. These results extend to the whole space and to arbitrary dimension when spaces of constant curvature are considered. As a consequence, a characterization of centrally symmetric convex bodies is obtained.

Let A⊆ℕ, let p be a prime and w a word over ℤ pℤ ending with a non-zero digit. The relationship is investigated between the density of A. the length of w and the density of the set of numbers n for which the base p expansion of ends with w0n for some a ∈ A. Also considered is the analogous problem on Pascal's triangle. This leads in particular to answering a question of Granville and Zhu [7] regarding the asymptotic frequency of sums of 3 squares in Pascal's triangle.

There exists a family of pairwise disjoint congruent infinite circular cylinders such that no two cylinders of are parallel and the density of is greater than zero.

One of the substantial differences between real and complex analysis is the behaviour of pointwise sequential limits of functions. It is well known that, if f(z) is a bounded analytic function in D = {z∈ C: |z| < 1}, then there exists a sequence {pn(z): n = 1,2,…} of polynomials such that

(i) ‖Pn‖ ≤ ‖f‖ for all n = 1,2,3,…, and

(ii) for each z ∈ D, Pn(Z) → f(z) as n → ∞,

where we have used the notation

The object of this paper is to give a new characterization of the set of Pisot-Vijayaraghavan numbers (P.V.-numbers for short). As usual, [x] and [x] represent respectively the integer part and the fractional part of the real number x, and

Let $C\in \mathbb{Z}[x_{1},\ldots ,x_{n}]$ be a cubic form. Assume that $C$ splits into four forms. Then $C(x_{1},\ldots ,x_{n})=0$ has a non-trivial integer solution provided that $n\geqslant 10$.

Let G1,… Gn be groups, let *Gi be their free product, and let ´ Gi be their direct product. A homomorphism may be defined by requiring it to be trivial on Gj and the identity on Gi for i ¹ j. Let [G1 … Gn] = Çker pj. P. J. Hilton [2] proves that [G1, …, Gn] is a free group, and, if HiÌGi, i = 1, …, n, that [H1,… Hn] is a free factor of [G1 … Gn]. He asks whether, if Hλi Ì Gi, λ = 1, …, k, i = 1, …, n, and Hλ = [Hλ1, …, Hλn] the group generated by the Hλ is a free factor of [G1, …, Gn].

One may perhaps doubt whether in the geometry of numbers any particular family of lattices deserves such an attention as, for example, BCH codes receive in coding theory. However, only recently a quite interesting family has emerged. The general case of these lattices considered by Rosenbloom and Tsfasman [5, Section 2] parallels Goppa's construction of codes from algebraic curves. Here we shall take a closer look at the case of genus zero where some special features of Goppa's early codes will show up again: There is a lattice Λ (L, g) in n-dimensional euclidean space associated with a subset L of the field, and a polynomial g satisfying g(є) ≠ for all λ є L. For g = zd previously known sphere packings are recovered and generalized. A nonconstructive argument shows that for n → ∞ and some irreducible polynomials g Minkowski's lower packing bound is met (this being not achieved in [5] where q is fixed, but the genus grows; cf. also [4]).

The usual definition for vertex-criticality with respect to the chromatic index is that a multigraph G is vertex-critical if G is Class 2, connected, and χ'(G\υ) <χ'(G) for all υ ε V(G). We consider here an allied notion, that of vertex-criticality with respect to the chromatic class–in this case G is vertex critical if G is Class 2 and connected, but G\υ is Class 1 for all υ ε V(G). We also investigate the analogues of these two notions for edge-criticality.