Topology from enrichment: the curious case of partial metrics

For any small quantaloid \Q, there is a new quantaloid \D(\Q) of diagonals in \Q. If \Q is divisible then so is \D(\Q) (and vice versa), and then it is particularly interesting to compare categories enriched in \Q with categories enriched in \D(\Q). Taking Lawvere's quantale of extended positive real numbers as base quantale, \Q-categories are generalised metric spaces, and \D(\Q)-categories are generalised partial metric spaces, i.e.\ metric spaces in which self-distance need not be zero and with a suitably modified triangular inequality. We show how every small quantaloid-enriched category has a canonical closure operator on its set of objects: this makes for a functor from quantaloid-enriched categories to closure spaces. Under mild necessary-and-sufficient conditions on the base quantaloid, this functor lands in the category of topological spaces; and an involutive quantaloid is Cauchy-bilateral (a property discovered earlier in the context of distributive laws) if and only if the closure on any enriched category is identical to the closure on its symmetrisation. As this now applies to metric spaces and partial metric spaces alike, we demonstrate how these general categorical constructions produce the "correct" definitions of convergence and Cauchyness of sequences in generalised partial metric spaces. Finally we describe the Cauchy-completion, the Hausdorff contruction and exponentiability of a partial metric space, again by application of general quantaloid-enriched category theory.Comment: Apart from some minor corrections, this second version contains a revised section on Cauchy sequences in a partial metric spac

Impact of noise on a dynamical system: prediction and uncertainties from a swarm-optimized neural network

In this study, an artificial neural network (ANN) based on particle swarm optimization (PSO) was developed for the time series prediction. The hybrid ANN+PSO algorithm was applied on Mackey--Glass chaotic time series in the short-term $x(t+6)$. The performance prediction was evaluated and compared with another studies available in the literature. Also, we presented properties of the dynamical system via the study of chaotic behaviour obtained from the predicted time series. Next, the hybrid ANN+PSO algorithm was complemented with a Gaussian stochastic procedure (called {\it stochastic} hybrid ANN+PSO) in order to obtain a new estimator of the predictions, which also allowed us to compute uncertainties of predictions for noisy Mackey--Glass chaotic time series. Thus, we studied the impact of noise for several cases with a white noise level ($\sigma_{N}$) from 0.01 to 0.1.Comment: 11 pages, 8 figure

Cluster analysis of multiple planetary flow regimes

A modified cluster analysis method was developed to identify spatial patterns of planetary flow regimes, and to study transitions between them. This method was applied first to a simple deterministic model and second to Northern Hemisphere (NH) 500 mb data. The dynamical model is governed by the fully-nonlinear, equivalent-barotropic vorticity equation on the sphere. Clusters of point in the model's phase space are associated with either a few persistent or with many transient events. Two stationary clusters have patterns similar to unstable stationary model solutions, zonal, or blocked. Transient clusters of wave trains serve as way stations between the stationary ones. For the NH data, cluster analysis was performed in the subspace of the first seven empirical orthogonal functions (EOFs). Stationary clusters are found in the low-frequency band of more than 10 days, and transient clusters in the bandpass frequency window between 2.5 and 6 days. In the low-frequency band three pairs of clusters determine, respectively, EOFs 1, 2, and 3. They exhibit well-known regional features, such as blocking, the Pacific/North American (PNA) pattern and wave trains. Both model and low-pass data show strong bimodality. Clusters in the bandpass window show wave-train patterns in the two jet exit regions. They are related, as in the model, to transitions between stationary clusters

An enhanced linear Kalman filter (EnLKF) algorithm for parameter estimation of nonlinear rational models

In this study, an enhanced Kalman Filter formulation for linear in the parameters models with inherent correlated errors is proposed to build up a new framework for nonlinear rational model parameter estimation. The mechanism of linear Kalman filter (LKF) with point data processing is adopted to develop a new recursive algorithm. The novelty of the enhanced linear Kalman filter (EnLKF in short and distinguished from extended Kalman filter (EKF)) is that it is not formulated from the routes of extended Kalman Filters (to approximate nonlinear models by linear approximation around operating points through Taylor expansion) and also it includes LKF as its subset while linear models have no correlated errors in regressor terms. No matter linear or nonlinear models in representing a system from measured data, it is very common to have correlated errors between measurement noise and regression terms, the EnLKF provides a general solution for unbiased model parameter estimation without extra cost to convert model structure. The associated convergence is analysed to provide a quantitative indicator for applications and reference for further research. Three simulated examples are selected to bench-test the performance of the algorithm. In addition, the style of conducting numerical simulation studies provides a user-friendly step by step procedure for the readers/users with interest in their ad hoc applications. It should be noted that this approach is fundamentally different from those using linearisation to approximate nonlinear models and then conduct state/parameter estimate

Measurements, Models, Systems and Design

531 s.