Let Mn be a nonpositively curved complete simply connected manifold and D ⊂ Mn be a convex compact subset with non-empty interior and smooth boundary. It is shown that the total mean curvature ∂D can be estimated in terms of volume and curvature bound.

C.C. Chen and G. Grätzer have shown that a Stone lattice is determined by a triple (C, D, ø) where C is a boolean algebra, D is a distributive lattice with 1 and ø is an e-homomorphism from C into D(D), the lattice of dual ideals of D.

It is shown here that any Stone lattice is, up to an isomorphism, a subdirect product of its centre C(L) and a special Stone lattice M(L). Special Stone lattices are characterised, in the terminology of the Chen-Grätzer triple, by the fact that the e-homomorphism Φ is one to one.

In this paper we characterise a special Stone lattice L as a triple (H, C, Do) where H is a distributive lattice with 0 and 1, C is a boolean e-subalgebra of the centre of H and Do is a sublattice of H with o such that d ∈ Do & c ∈ C = d ∧ c ∈ Do, and which separates the elements of C in the sense that for any c1 ≠c2 in C there is a d in Do with d ≤ c1 but d ≰ C2. It then turns out that C is C(L) and Do is the dual of D(L).

Perhaps the simplest example is the lexicographic sum of copies Zn of the integers indexed by the integers. If one performs the shift in the 0-indexed summand m0 → (m+1)0 while leaving all the other summands fixed, and follows this with the shift Zn → Zn+1 in the index set, then the element 00. will be sent on 11; whereas under these automorphisms performed in the other order, it will be sent on 01. This non-commutativity contradicts Theorem 9 in [1].

We investigate the set of limit points of the continued fractions

where x1, x2, … is a given sequence of positive integers. We show that this set is closed, and that it may include any given countable subset of [0, 1] if the integers xk are chosen appropriately. Our main result, which has applications in transcendence theory, is that the sequence of continued fractions has no rational limit point when the sequence {xk} of partial quotients is bounded.

A Fréchet lattice E is an AL-space if its topology can be defined by a family of lattice seminorms that are additive in the positive cone of E. Grothendieck proved that AL-Banach spaces have the Dunford-Pettis property. This result was recently extended by Fernández and Naranjo to AL-Fréchet spaces with a continuous norm and weak order unit. In this note we show how to remove both hypotheses.

Constructive, simple proofs for the existence, regularity, continuous dependence and dynamical properties of a repelling invariant curve for a discrete dynamical system of the plane with an attracting fixed point with real eigenvalues are given. These proofs can be used to generate a numerical algorithm to find these curves and to compute explicitly the dependence of the curve with respect to the system.

Let $n$ be a positive integer. We obtain new Menon’s identities by using the actions of some subgroups of $(\mathbb{Z}/n\mathbb{Z})^{\times }$ on the set $\mathbb{Z}/n\mathbb{Z}$. In particular, let $p$ be an odd prime and let $\unicode[STIX]{x1D6FC}$ be a positive integer. If $H_{k}$ is a subgroup of $(\mathbb{Z}/p^{\unicode[STIX]{x1D6FC}}\mathbb{Z})^{\times }$ with index $k=p^{\unicode[STIX]{x1D6FD}}u$ such that $0\leqslant \unicode[STIX]{x1D6FD}<\unicode[STIX]{x1D6FC}$ and $u\mid p-1$, then

$$\begin{eqnarray}\mathop{\sum }_{x\in H_{k}}(x-1,p^{\unicode[STIX]{x1D6FC}})=\frac{\unicode[STIX]{x1D711}(p^{\unicode[STIX]{x1D6FC}})}{k}\bigg(1+k(\unicode[STIX]{x1D6FC}-\unicode[STIX]{x1D6FD})+u\frac{p^{\unicode[STIX]{x1D6FD}}-1}{p-1}\bigg),\end{eqnarray}$$

$\unicode[STIX]{x1D711}(n)$

An essential limitation for the method of weighting for equality constrained linear least-squares problems is the sub-optimality of the attainable convergence rate. In this paper, we propose a method, related to the method of iterated Tikhonov regularisation, that (under suitable conditions) gives rise to convergence rates which are arbitrarily near the optimal rate. As a by-product, we develop the theory of iterated Tikhonov regularisation for equations with unbounded linear operators.

We classify, modulo the kernel of the action, finite groups G that act 2-transitively on Sylr(G) for some prime r dividing |G|. We furthermore prove that any finite group that acts 2-transitively on Sylr(G) for each prime r is solvable and of nilpotent length at most 3.

The topologies of the density type in Euclidean space of dimension higher than one are introduced. Definitions are based on a notion of density point connected with a set of sequences of real numbers. Our purpose is to study properties of these topologies and connections between them.

Let $a_{1},a_{2},\ldots ,a_{m}$ and $b_{1},b_{2},\ldots ,b_{l}$ be two sequences of pairwise distinct positive integers greater than $1$. Assume also that none of the above numbers is a perfect power. If for each positive integer $n$ and prime number $p$ the number $\prod _{i=1}^{m}(1-a_{i}^{n})$ is divisible by $p$ if and only if the number $\prod _{j=1}^{l}(1-b_{j}^{n})$ is divisible by $p$, then $m=l$ and $\{a_{1},a_{2},\ldots ,a_{m}\}=\{b_{1},b_{2},\ldots ,b_{l}\}$.

Suppose that X is a real or complex Banach space with norm |·|. Then X is a Hilbert space if and only if

for all x in X and all X-valued Bochner integrable functions Y on the Lebesgue unit interval satisfying EY = 0 and |x − Y| ≤ 2 almost everywhere. This leads to the following biconcave-function characterisation: A Banach space X is a Hilbert space if and only if there is a biconcave function η: {(x, y) ∈ X × X: |x − y| ≤ 2} → R such that η(0, 0) = 2 and

If the condition η(0, 0) = 2 is eliminated, then the existence of such a function η characterises the class UMD (Banach spaces with the unconditionally property for martingale differences).

A solution of the equation must be C2 smooth, need not be C3 smooth, but at the same time the ratio zxy: zxx is C1 (provided zxx ≠ 0). An analytical proof of this fact is given; the underlying geometrical interpretation is discussed.

Converses are proved for the Osgood (the Principle of Uniform Boundedness), Dini, and other well known. theorems. The notion of a continuous step function on a topological space is defined and a class of spaces identified for which each lower semicontinuous function is the pointwise limit of a monotonically increasing sequence of step functions.

An operational category is a category of models for an equational theory where the interpretation of some operations is predetermined. Examples include the equational and co-equational categories of Linton, categories of functors preserving some class of limits, and algebras for a prop as defined by MacLane. The chief result is a characterisation of the operational categories and functors in terms of their internal structure.

A variational principle is established to provide a new formulation for convex Hamiltonian systems. Using this formulation, we obtain some existence results for second-order Hamiltonian systems with a variety of boundary conditions, including nonlinear ones.