Universal elements for non-linear operators and their applications

We prove that under certain topological conditions on the set of universal elements of a continuous map T acting on a topological space X, that the direct sum T and M_g is universal, where M_g is multiplication by a generating element of a compact topological group. We use this result to characterize R_+-supercyclic operators and to show that whenever T is a supercyclic operator and z_1,...,z_n are pairwise different non-zero complex numbers, then the operator z_1T\oplus ... \oplus z_n T is cyclic. The latter answers affirmatively a question of Bayart and Matheron.

Sequentially continuous non-linear fundamental systems of solutions of affine equations in locally convex spaces

According to the Mickael's selection theorem any surjective continuous linear operator from one Fr\'echet space onto another has a continuous (not necessarily linear) right inverse. Using this theorem Herzog and Lemmert proved that if $E$ is a Fr\'echet space and $T:E\to E$ is a continuous linear operator such that the Cauchy problem $\dot x=Tx$, $x(0)=x_0$ is solvable in $[0,1]$ for any $x_0\in E$, then for any $f\in C([0,1],E)$, there exists a continuos map $S:[0,1]\times E\to E$, $(t,x)\mapsto S_tx$ such that for any $x_0\in E$, the function $x(t)=S_tx_0$ is a solution of the Cauchy problem $\dot x(t)=Tx(t)+f(t)$, $x(0)=x_0$ (they call $S$ a fundamental system of solutions of the equation $\dot x=Tx+f$). We prove the same theorem, replacing "continuous" by "sequentially continuous" for locally convex spaces from a class which contains strict inductive limits of Fr\'echet spaces and strong duals of Fr\'echet--Schwarz spaces and is closed with respect to finite products and sequentially closed subspaces. The key-point of the proof is an extension of the theorem on existence of a sequentially continuous right inverse of any surjective sequentially continuous linear operator to some class of non-metrizable locally convex spaces.

Model selection approaches for non-linear system identification: a review

The identification of non-linear systems using only observed finite datasets has become a mature research area over the last two decades. A class of linear-in-the-parameter models with universal approximation capabilities have been intensively studied and widely used due to the availability of many linear-learning algorithms and their inherent convergence conditions. This article presents a systematic overview of basic research on model selection approaches for linear-in-the-parameter models. One of the fundamental problems in non-linear system identification is to find the minimal model with the best model generalisation performance from observational data only. The important concepts in achieving good model generalisation used in various non-linear system-identification algorithms are first reviewed, including Bayesian parameter regularisation and models selective criteria based on the cross validation and experimental design. A significant advance in machine learning has been the development of the support vector machine as a means for identifying kernel models based on the structural risk minimisation principle. The developments on the convex optimisation-based model construction algorithms including the support vector regression algorithms are outlined. Input selection algorithms and on-line system identification algorithms are also included in this review. Finally, some industrial applications of non-linear models are discussed.

K-Theory of non-linear projective toric varieties

We define a category of quasi-coherent sheaves of topological spaces on projective toric varieties and prove a splitting result for its algebraic K-theory, generalising earlier results for projective spaces. The splitting is expressed in terms of the number of interior lattice points of dilations of a polytope associated to the variety. The proof uses combinatorial and geometrical results on polytopal complexes. The same methods also give an elementary explicit calculation of the cohomology groups of a projective toric variety over any commutative ring.

Modelling the impact of antibiotic use and infection control practices on the incidence of hospital-acquired methicillin-resistant Staphylococcus aureus: a time-series analysis

Objectives: Methicillin-resistant Staphylococcus aureus (MRSA) is a major nosocomial pathogen worldwide. A wide range of factors have been suggested to influence the spread of MRSA. The objective of this study was to evaluate the effect of antimicrobial drug use and infection control practices on nosocomial MRSA incidence in a 426-bed general teaching hospital in Northern Ireland.

Methods: The present research involved the retrospective collection of monthly data on the usage of antibiotics and on infection control practices within the hospital over a 5 year period (January 2000–December 2004). A multivariate ARIMA (time-series analysis) model was built to relate MRSA incidence with antibiotic use and infection control practices.

Results: Analysis of the 5 year data set showed that temporal variations in MRSA incidence followed temporal variations in the use of fluoroquinolones, third-generation cephalosporins, macrolides and amoxicillin/clavulanic acid (coefficients = 0.005, 0.03, 0.002 and 0.003, respectively, with various time lags). Temporal relationships were also observed between MRSA incidence and infection control practices, i.e. the number of patients actively screened for MRSA (coefficient = -0.007), the use of alcohol-impregnated wipes (coefficient = -0.0003) and the bulk orders of alcohol-based handrub (coefficients = -0.04 and -0.08), with increased infection control activity being associated with decreased MRSA incidence, and between MRSA incidence and the number of new patients admitted with MRSA (coefficient = 0.22). The model explained 78.4% of the variance in the monthly incidence of MRSA.

Conclusions: The results of this study confirm the value of infection control policies as well as suggest the usefulness of restricting the use of certain antimicrobial classes to control MRSA.

A meta-analysis of latitudinal variations in life-history traits of roach, Rutilus rutilus, over its geographical range: linear or non-linear relationships?

1. We collated information from the literature on life history traits of the roach (a generalist freshwater fish), and analysed variation in absolute fecundity, von Bertalanffy parameters, and reproductive lifespan in relation to latitude, using both linear and non-linear regression models. We hypothesized that because most life history traits are dependent on growth rate, and growth rate is non-linearly related with temperature, it was likely that when analysed over the whole distribution range of roach, variation in key life history traits would show non-linear patterns with latitude.

Genetic Algorithm Search for Predictive Patterns in Multidimensional Time Series

Based on an algorithm for pattern matching in character strings, we implement a pattern matching machine that searches for occurrences of patterns in multidimensional time series. Before the search process takes place, time series are encoded in user-designed alphabets. The patterns, on the other hand, are formulated as regular expressions that are composed of letters from these alphabets and operators. Furthermore, we develop a genetic algorithm to breed patterns that maximize a user-defined fitness function. In an application to financial data, we show that patterns bred to predict high exchange rates volatility in training samples retain statistically significant predictive power in validation samples.