We study the position of the computable setting in the “common theory of locality” developed in [4, 5] for local problems on $\Delta $-regular trees, $\Delta \in \omega $. We show that such a problem admits a computable solution on every highly computable $\Delta $-regular forest if and only if it admits a Baire measurable solution on every Borel $\Delta $-regular forest. We also show that if such a problem admits a computable solution on every computable maximum degree $\Delta $ forest then it admits a continuous solution on every maximum degree $\Delta $ Borel graph with appropriate topological hypotheses, though the converse does not hold.

Insects breathe using one or a combination of three gas exchange patterns; continuous, cyclic and discontinuous, which vary in their rates of exchange of oxygen, carbon dioxide and water. In general, there is a trade-off between lowering gas exchange using discontinuous exchange that limits water loss at the cost of lower metabolic rate. These patterns and hypotheses for the evolution of discontinuous exchange have been examined for relatively large insects (>20 mg) over relatively short periods (<4 h), but smaller insects and longer time periods have yet to be examined. We measured gas exchange patterns and metabolic rates for adults of a small insect pest of grain, the red flour beetle, Tribolium castaneum (Coleoptera: Tenebrionidae), using flow-through respirometry in dry air for 48 h. All adults survived the desiccating measurement period; initially they used continuous gas exchange, then after 24 h switched to cyclic gas exchange with a 27% decrease in metabolic rate, and then after 48 h switched to discontinuous gas exchange with increased interburst duration and further decrease in metabolic rate. The successful use of the Qubit, a lower cost and so more common gas analyser, to measure respiration in the very small T. castaneum, may prompt more flow-through respirometry studies of small insects. Running such studies over long durations may help to better understand the evolution of respiration physiology and thus suggest new methods of pest management.

The need to store the trajectory information required by industrial robots, so that they may carry out tasks such as painting, has led to various data compression methods. The method proposed, simple to implement and adaptable to low cost systems, enables a best compromise to be reached for a given application between the detailed description of a complex movement and the use of as little memory as possible. This method was first used on a hydraulic prototype for the French Master-Slave firm PHAREMME.

In December 2006 an outbreak of Campylobacter infection occurred in Forth Valley, Scotland, affecting 48 people over a 3-week period. All cases dined at restaurant A. We conducted a cohort study in a party of 30 who ate lunch at restaurant A on 21 December to identify the vehicle of infection. Of 29 respondents, the attack rate in those who ate chicken liver pâté was 86% (6/7) compared to 0% (0/22) for those who did not. Between 1 December and 1.30 p.m. on 21 December the restaurant had used a different method of cooking the pâté. No cases reported dining at the restaurant after this time. The outbreak's duration suggested a continuous source. This is the first continuous source outbreak of Campylobacter documented in Scotland. Chicken liver pâté was the most likely vehicle of infection. This outbreak illustrates the hazards associated with undercooking Campylobacter-contaminated food.

In the paper we formulate a Covering Property Axiom, $\text{CP}{{\text{A}}_{\text{prism}}}$, which holds in the iterated perfect set model, and show that it implies the following facts, of which (a) and (b) are the generalizations of results of J. Steprāns.

(a) There exists a family $\mathcal{F}$ of less than continuum many ${{C}^{1}}$ functions from $\mathbb{R}$ to $\mathbb{R}$ such that ${{\mathbb{R}}^{2}}$ is covered by functions from $\mathcal{F}$, in the sense that for every $\left\langle x,\,y \right\rangle \,\in \,{{\mathbb{R}}^{2}}$${{\mathbb{R}}^{2}}$ there exists an $f\,\in \,\mathcal{F}$ such that either $f\left( x \right)\,=\,y$ or $f\left( y \right)\,=\,x$.

(b) For every Borel function $f:\,\mathbb{R}\,\to \,\mathbb{R}$ there exists a family $\mathcal{F}$ of less than continuum many “${{C}^{1}}$” functions (i.e., differentiable functions with continuous derivatives, where derivative can be infinite) whose graphs cover the graph of $f$.

(c) For every $n\,>\,0$ and a ${{D}^{n}}$ function $f:\,\mathbb{R}\,\to \,\mathbb{R}$ there exists a family $\mathcal{F}$ of less than continuum many ${{C}^{n}}$ functions whose graphs cover the graph of $f$.

We also provide the examples showing that in the above properties the smoothness conditions are the best possible. Parts (b), (c), and the examples are closely related to work of A. Olevskiî.

We show a result slightly more general than the following. Let $K$ be a compact Hausdorff space, $F$ a closed subset of $K$, and $d$ a lower semi-continuous metric on $K$. Then each continuous function $f$ on $F$ which is Lipschitz in $d$ admits a continuous extension on $K$ which is Lipschitz in $d$. The extension has the same supremum norm and the same Lipschitz constant.

As a corollary we get that a Banach space $X$ is reflexive if and only if each bounded, weakly continuous and norm Lipschitz function defined on a weakly closed subset of $X$ admits a weakly continuous, norm Lipschitz extension defined on the entire space $X$.

We extend various characterizations of scalar-valued lower semicontinuity and determine their relationship to the continuity of vector-valued convex functions. Order completeness of the range space is not assumed.

Let X(t) be a homogeneous and continuous stochastic process with independent increments. The subject of this paper is to characterize the stable process by two identically distributed stochastic integrals formed by means of X(t) (in the sense of convergence in probability). The proof of the main results is based on a modern extension of the Phragmén-Lindelöf theory.

Let X(t) be a continuous, homogeneous stochastic process with independent increments characterized by a, σ, M, N in the Lévy representation formula. In this note we obtain the Lévy canonical representation of the characteristic function of a stochastic integral (in the sense of convergence in probability) of the form (where υ(t) is a non-decreasing, non-negative and left-continuous function) in terms of υ(t), a, σ, M, N.

The following properties of entropies, as measures of expected information, seem natural. The amount of information expected from an experiment does not change if we add outcomes of zero probability (expansibility). The expected information is symmetric in the (probabilities of the) outcomes. The information expected from a combination of two experiments is less than or equal to the sum of the informations expected from the single experiments (subadditivity); equality holds here if the two experiments are independent (additivity).

In this paper it is shown that linear combinations of the Shannon and Hartley entropies and only these have the above properties. The Shannon and the Hartley entropies are also individually characterized.