We show in this note the following statement which is an improvement over a result of R. M. Dudley and which is also of independent interest. Let X be a set of a Hilbert space with the property that there are constants ρ, σ>0, and for each n∈ℕ, the set X can be covered by at most n balls of radius ρn−σ. Then, for each n∈ℕ, the convex hull of X can be covered by 2n balls of radius cn−½−σ. The estimate is best possible for all n∈ℕ, apart from the value c=c(ρ, σ, X). In other words, let N(ε, X), ε>0, be the minimal number of balls of radius ε covering the set X. Then the above result is equivalent to saying that if N(ε, X)=O(ε−1/σ) as ε↓0, then for the convex hull conv (X) of X, N(ε, conv (X)) =O(exp(ε−2/(1+2σ))). Moreover, we give an interplay between several covering parameters based on coverings by balls (entropy numbers) and coverings by cylindrical sets (Kolmogorov numbers).

The precise indeterminacy of Milnor's concordance μ invariants is determined. It is shown that the (nilpotent quotient) Artin representation of string links is surjective. Orr's computation of the number of linearly independent μ invariants of a fixed length is recovered. A structure theorem for the set of links up to concordance is proven. We define an action of 2l-component string links on l-component string links, which passes to concordance classes. It is shown that the set of links up to concordance is in bijection with the orbit space of the restriction of this action to the stabilizer of the identity. Via the Artin representation, the action passes to a unipotent action, defined purely algebraically and consequently algorithmically computable, on the corresponding automorphism groups.

This work is an investigation of the extent to which convex bodies are determined by the sizes of their projections. It is shown, on the one hand, that there are non-congruent bodies for which all corresponding projections have the same size. On the other hand, it is proved that in the usual Baire category sense, most convex bodies are determined, up to translation or reflection, by the combination of their widths and brightnesses in all directions.

it is shown that, for all large $x$, there are more than $x^{0.33}$ carmichael numbers up to $x$, improving on the ground-breaking work of alford, granville and pomerance, who were the first to demonstrate that there are infinitely many such numbers. the same basic construction as that employed by these authors is used, but a slight modification enables a stronger result on primes in arithmetic progressions based on a sieve method to be employed.

Based on computing spherical lengths of polygonal curves and spherical areas of domains bounded by polygonal curves on the complex plane, this paper proves a property of holomorphic functions and, as an application, gives a very brief proof of a famous inequality obtained by L. V. Ahlfors which easily implies Picard's theorem. Taken together with the arguments given by Alhfors, therefore, this paper in fact gives an elementary, brief and geometrical proof of Picard's theorem.

A bounded operator $T$ acting on a Banach space $\cal B$ is said to be supercyclic if there is a vector $x\,{\in}\,\cal B$ such that the projective orbit$\{\lambda T^n x\,{:}\,n\,{\geq}\,0\text{ and } \lambda\,{\in}\,\Bbb C \}$ is dense in $\cal B$. Examples of supercyclic operators are hypercyclic operators, in which the orbit itself is dense without the help of scalar multiples. Supercyclic operators are, in turn, a special case of cyclic operators. An operator is called cyclic if the linear span of the orbit of some vector is dense in the underlying space. This survey describes some recent results on linear subspaces in which all elements, except the zero vector, are supercyclic for a given supercyclic operator.

A base of an action of a group G on a set $\Omega$ is a subset $B \subseteq \Omega$ such that the pointwise stabiliser of $B$ in $G$ is the identity. We prove that if $\Omega$ is the set of partitions of $[ 1, kl ]$ into $l$ subsets of size $k$, then the action of $S_{kl}$ on $\Omega$ has a base of size two if and only if $k \geq 3$ and $l \geq \max \{ k + 3, 8 \}$. This result completes a classification of the primitive base 2 actions of the symmetric groups. During the proof we show that there exists a $k$-regular bipartite graph $\mathcal{G}$ on $2l$ vertices with no non-trivial automorphisms fixing the bipartite blocks if and only if $k \geq 3$ and $l \geq \max \{ k + 3, 8 \}$.

A relationship is established between the factorization of $24 n + 1$ and the 5-divisibility of $Q(n)$ , where $Q(n)$ is the number of partitions of $n$ into distinct parts. As an application, an abundance of infinite families of congruences for $Q(n)$ modulo powers of 5 are explicitly exhibited.

Extending earlier work, we establish the finiteness of the number of two-generator arithmetic Kleinian groups with one generator parabolic and the other either parabolic or elliptic. We also identify all the arithmetic Kleinian groups generated by two parabolic elements. Surprisingly, there are exactly 4 of these, up to conjugacy, and they are all torsion free.

This paper describes – in terms of potential-theoretic properties – the necessary and sufficient conditions required for a bounded domain satisfying the capacity density condition to be, respectively: a John domain, a uniform domain and an inner uniform domain.

In this paper, the existence problem is studied for extremals of the Sobolev trace inequality $W^{1,p}(\Omega)\to L^{p_*}(\partial\Omega)$, where $\Omega$ is a bounded smooth domain in $\RR^N$, $p_*=p(N-1)/(N-p)$ is the critical Sobolev exponent, and $1<p<N$.

Let $G_0$ and $G$ be the heat kernels of $\Delta$ and $\Delta - V$ respectively. Under some sharp conditions on $V$, a proof is given that $G(x, t; y, 0)/G_0(x, t; y, 0)$ can be bounded from above and below by two positive constants. This largely improves the well-known bounds $C_1 e^{-c_1 |x-y|^2/t} \le G(x, t; y, 0)/G_0(x, t; y, 0) \le C_2 e^{c_2 |x-y|^2/t}$, $t \in (0, T]$. The result answers an open question of V. Liskevich and Yu. Semenov, in the case where $V \ge 0$. A sharp global bound in time is also obtained when $V \ge 0$.

Effective formulae are found for the complex geodesics in the symmetrized bidisc.

The quantum stochastic differential equation dkt = kt ∘ θαβdΛβα(t) is considered on a unital C*-algebra, with separable noise dimension space. Necessary conditions on the matrix of bounded linear maps θ for the existence of a completely positive contractive solution are shown to be sufficient. It is known that for completely positive contraction processes, k satisfies such an equation if and only if k is a regular Markovian cocycle. ‘Feller’ refers to an invariance condition analogous to probabilistic terminology if the algebra is thought of as a non-commutative topological space.

A famous problem of Zariski, stated in 1971 [21], is to decide whether the classical algebro-geometric multiplicity mo(f) of a complex hypersurface f−1(0) at an isolated singularity O is an invariant of the topological type of the embedded germ. That is, if f[ratio ]Cn, O→C, 0 and g[ratio ]Cn, O→C, 0 are two germs of polynomial functions, and there is a homeomorphism ϕ defined between two open neighbourhoods U and V of O in Cn such that ϕ(U) = V, ϕ(U∩f−1(0)) = V∩g−1(0) and ϕ(O) = O, then is it always the case that mo(f) = mo(g)? Here the multiplicity mo(f) is defined to be the intersection number of f−1(0) with a generic complex line L passing through O in Cn, that is, the number of points of intersection of f−1(0) with a line which is an arbitrarily small perturbation of L. If f is reduced, then mo(f) is equal to the order of the polynomial f(z) at O.

The simplest example of the sort of representation formula that we shall study is the following familiar inequality for a smooth, real-valued function f(x) defined on a ball B in N-dimensional Euclidean space IRN:

formula here

where ∇f denotes the gradient of f, fB is the average [mid ]B[mid ]−1∫ Bf(y)dy, [mid ]B[mid ] is the Lebesgue measure of B, and C is a constant which is independent of f, x and B. This formula can be found, for example, in [4] and [12]; see also the closely related estimates in [20, pp. 228{231]. Indeed, such a formula holds in any bounded convex domain.

This paper provides a unified and elementary proof of four Thompson–Wielandt-like theorems.

Explicit relationships are given connecting ‘almost’ isospectral Sturm–Liouville problems with eigen-value dependent, and independent, boundary conditions, respectively. Application is made to various direct and inverse spectral questions.

The authors solve the following problem: for which connected Lie groups is the group of homotopy classes of self maps commutative?

In this paper, a growth result is obtained for normalized $k$ -fold symmetric convex biholomorphic mappings on the unit ball of a complex Banach space.