Approximating differentiable relationships between delay embedded dynamical systems with radical basis functions

This thesis is about the study of relationships between experimental dynamical systems. The basic approach is to fit radial basis function maps between time delay embeddings of manifolds. We have shown that under certain conditions these maps are generically diffeomorphisms, and can be analysed to determine whether or not the manifolds in question are diffeomorphically related to each other. If not, a study of the distribution of errors may provide information about the lack of equivalence between the two. The method has applications wherever two or more sensors are used to measure a single system, or where a single sensor can respond on more than one time scale: their respective time series can be tested to determine whether or not they are coupled, and to what degree. One application which we have explored is the determination of a minimum embedding dimension for dynamical system reconstruction. In this special case the diffeomorphism in question is closely related to the predictor for the time series itself. Linear transformations of delay embedded manifolds can also be shown to have nonlinear inverses under the right conditions, and we have used radial basis functions to approximate these inverse maps in a variety of contexts. This method is particularly useful when the linear transformation corresponds to the delay embedding of a finite impulse response filtered time series. One application of fitting an inverse to this linear map is the detection of periodic orbits in chaotic attractors, using suitably tuned filters. This method has also been used to separate signals with known bandwidths from deterministic noise, by tuning a filter to stop the signal and then recovering the chaos with the nonlinear inverse. The method may have applications to the cancellation of noise generated by mechanical or electrical systems. In the course of this research a sophisticated piece of software has been developed. The program allows the construction of a hierarchy of delay embeddings from scalar and multi-valued time series. The embedded objects can be analysed graphically, and radial basis function maps can be fitted between them asynchronously, in parallel, on a multi-processor machine. In addition to a graphical user interface, the program can be driven by a batch mode command language, incorporating the concept of parallel and sequential instruction groups and enabling complex sequences of experiments to be performed in parallel in a resource-efficient manner.

Precise orbit determination and analysis from satellite altimetry and laser ranging

For optimum utilization of satellite-borne instrumentation, it is necessary to know precisely the orbital position of the spacecraft. The aim of this thesis is therefore two-fold - firstly to derive precise orbits with particular emphasis placed on the altimetric satellite SEASAT and secondly, to utilize the precise orbits, to improve upon atmospheric density determinations for satellite drag modelling purposes. Part one of the thesis, on precise orbit determinations, is particularly concerned with the tracking data - satellite laser ranging, altimetry and crossover height differences - and how this data can be used to analyse errors in the orbit, the geoid and sea-surface topography. The outcome of this analysis is the determination of a low degree and order model for sea surface topography. Part two, on the other hand, mainly concentrates on using the laser data to analyse and improve upon current atmospheric density models. In particular, the modelling of density changes associated with geomagnetic disturbances comes under scrutiny in this section. By introducing persistence modelling of a geomagnetic event and solving for certain geomagnetic parameters, a new density model is derived which performs significantly better than the state-of-the-art models over periods of severe geomagnetic storms at SEASAT heights. This is independently verified by application of the derived model to STARLETTE orbit determinations.

Computer aided manufacturing of geometric models

As systems for computer-aided-design and production of mechanical parts have developed, there has arisen a need for techniques for the comprehensive description of the desired part, including its 3-D shape. The creation and manipulation of shapes is generally known as geometric modelling. It is desirable that links be established between geometric modellers and machining programs. Currently, unbounded APT and some bounded geometry systems are being widely used in manufacturing industry for machining operations such as: milling, drilling, boring and turning, applied mainly to engineering parts. APT systems, however, are presently only linked to wire-frame drafting systems. The combination of a geometric modeller and APT will provide a powerful manufacturing system for industry from the initial design right through part manufacture using NC machines. This thesis describes a recently developed interface (ROMAPT) between a bounded geometry modeller (ROMULUS) and an unbounded NC processor (APT). A new set of theoretical functions and practical algorithms for the computer aided manufacturing of 3D solid geometric model has been investigated. This work has led to the development of a sophisticated computer program, ROMAPT, which provides a new link between CAD (in form of a goemetric modeller ROMULUS) and CAM (in form of the APT NC system). ROMAPT has been used to machine some engineering prototypes successfully both in soft foam material and aluminium. It has been demonstrated above that the theory and algorithms developed by the author for the development of computer aided manufacturing of 3D solid modelling are both valid and applicable. ROMAPT allows the full potential of a solid geometric modeller (ROMULUS) to be further exploited for NC applications without requiring major investment in new NC processor. ROMAPT supports output in APT-AC, APT4 and the CAM-I SSRI NC languages.

Satellite laser ranging and the ETALON Geodetic satellites

The technique of Satellite Laser Ranging is today a mature, important tool with applications in many area of geodynamics, geodesy and satellite dynamics. A global network of some 40 stations regularly obtains range observations with sub-cm precision to more than twelve orbiting spacecraft. At such levels of precision it is important to minimise potential sources of range bias in the observations, and part of the thesis is a study of subtle effects caused by the extended nature of the arrays of retro-reflectors on the satellites. We develop models that give a precise correction of the range measurements to the centres of mass of the geodetic satellites Lageos and Etalon, appropriate to a variety of different ranging systems, and use the Etalon values, which were not determined during pre-launch tests, in an extended orbital analysis. We have fitted continuous 2.5 year orbits to range observations of the Etalons from the global network of stations, and analysed the results by mapping the range residuals from these orbits into equivalent corrections to orbital elements over short time intervals. From these residuals we have detected and studied large un-modelled along-track accelerations associated with periods during which the satellites are undergoing eclipse by the Earth's shadow. We also find that the eccentricity residuals are significantly different for the two satellites, with Etalon-2 undergoing a year-long eccentricity anomaly similar in character to that experienced at intervals by Lageos-1. The nodal residuals show that the satellites define a very stable reference frame for Earth rotation determination, with very little drift-off during the 2.5 year period. We show that an analysis of more than about eight years of tracking data would be required to derive a significant value for 2. The reference frame defined by the station coordinates derived from the analyses shows very good agreement with that of ITRF93.

Combination of altimetry data from different satellite missions

Substantial altimetry datasets collected by different satellites have only become available during the past five years, but the future will bring a variety of new altimetry missions, both parallel and consecutive in time. The characteristics of each produced dataset vary with the different orbital heights and inclinations of the spacecraft, as well as with the technical properties of the radar instrument. An integral analysis of datasets with different properties offers advantages both in terms of data quantity and data quality. This thesis is concerned with the development of the means for such integral analysis, in particular for dynamic solutions in which precise orbits for the satellites are computed simultaneously. The first half of the thesis discusses the theory and numerical implementation of dynamic multi-satellite altimetry analysis. The most important aspect of this analysis is the application of dual satellite altimetry crossover points as a bi-directional tracking data type in simultaneous orbit solutions. The central problem is that the spatial and temporal distributions of the crossovers are in conflict with the time-organised nature of traditional solution methods. Their application to the adjustment of the orbits of both satellites involved in a dual crossover therefore requires several fundamental changes of the classical least-squares prediction/correction methods. The second part of the thesis applies the developed numerical techniques to the problems of precise orbit computation and gravity field adjustment, using the altimetry datasets of ERS-1 and TOPEX/Poseidon. Although the two datasets can be considered less compatible that those of planned future satellite missions, the obtained results adequately illustrate the merits of a simultaneous solution technique. In particular, the geographically correlated orbit error is partially observable from a dataset consisting of crossover differences between two sufficiently different altimetry datasets, while being unobservable from the analysis of altimetry data of both satellites individually. This error signal, which has a substantial gravity-induced component, can be employed advantageously in simultaneous solutions for the two satellites in which also the harmonic coefficients of the gravity field model are estimated.

Orbit and altimetric corrections for the ERS satellites through analysis of single and dual satellite crossovers

Due to the failure of PRARE the orbital accuracy of ERS-1 is typically 10-15 cm radially as compared to 3-4cm for TOPEX/Poseidon. To gain the most from these simultaneous datasets it is necessary to improve the orbital accuracy of ERS-1 so that it is commensurate with that of TOPEX/Poseidon. For the integration of these two datasets it is also necessary to determine the altimeter and sea state biases for each of the satellites. Several models for the sea state bias of ERS-1 are considered by analysis of the ERS-1 single satellite crossovers. The model adopted consists of the sea state bias as a percentage of the significant wave height, namely 5.95%. The removal of ERS-1 orbit error and recovery of an ERS-1 - TOPEX/Poseidon relative bias are both achieved by analysis of dual crossover residuals. The gravitational field based radial orbit error is modelled by a finite Fourier expansion series with the dominant frequencies determined by analysis of the JGM-2 co-variance matrix. Periodic and secular terms to model the errors due to atmospheric density, solar radiation pressure and initial state vector mis-modelling are also solved for. Validation of the dataset unification consists of comparing the mean sea surface topographies and annual variabilities derived from both the corrected and uncorrected ERS-1 orbits with those derived from TOPEX/Poseidon. The global and regional geographically fixed/variable orbit errors are also analysed pre and post correction, and a significant reduction is noted. Finally the use of dual/single satellite crossovers and repeat pass data, for the calibration of ERS-2 with respect to ERS-1 and TOPEX/Poseidon is shown by calculating the ERS-1/2 sea state and relative biases.

An investigation of the pressaccidic spike potential

A large negative spike potential, which is closely related to the onset of saccadic eyemovements, can be recorded from electrodes adjacent to the orbits. This potential, thepresaccadic spike potential, has often been regarded as an artefact related to eyemovement recordings and little work has been performed to establish its normal waveformand parameters. A positive spike potential, exactly coincident with the frontal negativespike, has also been recorded from electrodes positioned over the posterior scalp andthere has been some debate regarding any possible relationship between the twopotentials. The frontal spike potential has been associated with motor unit activity in theextraocular muscles prior to the saccade. This thesis investigates both the large anteriorand smaller posterior spike potentials and relates these recordings to the saccadic eyemovements associated with them. The anterior spike potential has been recorded from normal subjects to ascertain its normallatency and amplitude parameters for both horizontal and vertical saccades. A relationshipbetween saccade size and spike potential amplitude is described, the spike potentialamplitude reducing with smaller saccades. The potential amplitude also reduces withadvancing age. Studying the topographical distribution of the spike potential across thescalp shows the posterior spike activity may arise from potential spread of the larger frontalspike potential. Spike potential recordings from subjects with anomalous eye movements further implicate the extraocular muscles and their innervation in the generation of the spike potential. These recordings indicate that the spike potential may have some use as a clinical recording from patients with disease conditions affecting either their extraocular muscles or the innervational pathways to these muscles. Further recordings of the potential are necessary, however, to determine the exact nature of the changes which may occur with such conditions.

Does money matter in inflation forecasting

This paper provides the most fully comprehensive evidence to date on whether or not monetary aggregates are valuable for forecasting US inflation in the early to mid 2000s. We explore a wide range of different definitions of money, including different methods of aggregation and different collections of included monetary assets. In our forecasting experiment we use two non-linear techniques, namely, recurrent neural networks and kernel recursive least squares regression - techniques that are new to macroeconomics. Recurrent neural networks operate with potentially unbounded input memory, while the kernel regression technique is a finite memory predictor. The two methodologies compete to find the best fitting US inflation forecasting models and are then compared to forecasts from a naive random walk model. The best models were non-linear autoregressive models based on kernel methods. Our findings do not provide much support for the usefulness of monetary aggregates in forecasting inflation.

Enclosing the behavior of a hybrid system up to and beyond a Zeno point

Even simple hybrid systems like the classic bouncing ball can exhibit Zeno behaviors. The existence of this type of behavior has so far forced simulators to either ignore some events or risk looping indefinitely. This in turn forces modelers to either insert ad hoc restrictions to circumvent Zeno behavior or to abandon hybrid modeling. To address this problem, we take a fresh look at event detection and localization. A key insight that emerges from this investigation is that an enclosure for a given time interval can be valid independently of the occurrence of a given event. Such an event can then even occur an unbounded number of times, thus making it possible to handle certain types of Zeno behavior.

An alternating boundary integral based method for a Cauchy problem for the Laplace equation in semi-infinite regions

We consider a Cauchy problem for the Laplace equation in a two-dimensional semi-infinite region with a bounded inclusion, i.e. the region is the intersection between a half-plane and the exterior of a bounded closed curve contained in the half-plane. The Cauchy data are given on the unbounded part of the boundary of the region and the aim is to construct the solution on the boundary of the inclusion. In 1989, Kozlov and Maz'ya [10] proposed an alternating iterative method for solving Cauchy problems for general strongly elliptic and formally self-adjoint systems in bounded domains. We extend their approach to our setting and in each iteration step mixed boundary value problems for the Laplace equation in the semi-infinite region are solved. Well-posedness of these mixed problems are investigated and convergence of the alternating procedure is examined. For the numerical implementation an efficient boundary integral equation method is proposed, based on the indirect variant of the boundary integral equation approach. The mixed problems are reduced to integral equations over the (bounded) boundary of the inclusion. Numerical examples are included showing the feasibility of the proposed method.

Exploring trust in E-B2B relationships

The Internet’s unbounded opportunity for inter-firm exchanges is limited by concerns about trust, however, little is known about the development and formation of trust in the area of electronic business-to-business relationships. Drawing on existing research the aim of paper is to explore differences arising between trust in the traditional and in the electronic business to business relationship. In addition, an overall research framework is developed which links trust formation to three dimensions: the type of interaction, the stage of the relationship lifecycle and the type of transacting organizations. These dimensions are further examined in different relationship scenarios. Finally, implications for the agri-food industry are examined.

On Graph-Based Cryptography and Symbolic Computations

We have been investigating the cryptographical properties of in nite families of simple graphs of large girth with the special colouring of vertices during the last 10 years. Such families can be used for the development of cryptographical algorithms (on symmetric or public key modes) and turbocodes in error correction theory. Only few families of simple graphs of large unbounded girth and arbitrarily large degree are known. The paper is devoted to the more general theory of directed graphs of large girth and their cryptographical applications. It contains new explicit algebraic constructions of in finite families of such graphs. We show that they can be used for the implementation of secure and very fast symmetric encryption algorithms. The symbolic computations technique allow us to create a public key mode for the encryption scheme based on algebraic graphs.

Algorithms for Finding Unitals and Maximal Arcs in Projective Planes of Order 16

The paper has been presented at the International Conference Pioneers of Bulgarian Mathematics, Dedicated to Nikola Obreshkoff and Lubomir Tschakalo ff , Sofia, July, 2006.

On a New Approach to Williamson's Generalization of Pólya's Enumeration Theorem

Pólya’s fundamental enumeration theorem and some results from Williamson’s generalized setup of it are proved in terms of Schur- Macdonald’s theory (S-MT) of “invariant matrices”. Given a permutation group W ≤ Sd and a one-dimensional character χ of W , the polynomial functor Fχ corresponding via S-MT to the induced monomial representation Uχ = ind|Sdv/W (χ) of Sd , is studied. It turns out that the characteristic ch(Fχ ) is the weighted inventory of some set J(χ) of W -orbits in the integer-valued hypercube [0, ∞)d . The elements of J(χ) can be distinguished among all W -orbits by a maximum property. The identity ch(Fχ ) = ch(Uχ ) of both characteristics is a consequence of S-MT, and is equivalent to a result of Williamson. Pólya’s theorem can be obtained from the above identity by the specialization χ = 1W , where 1W is the unit character of W.

Neural Control of Chaos and Aplications

Signal processing is an important topic in technological research today. In the areas of nonlinear dynamics search, the endeavor to control or order chaos is an issue that has received increasing attention over the last few years. Increasing interest in neural networks composed of simple processing elements (neurons) has led to widespread use of such networks to control dynamic systems learning. This paper presents backpropagation-based neural network architecture that can be used as a controller to stabilize unsteady periodic orbits. It also presents a neural network-based method for transferring the dynamics among attractors, leading to more efficient system control. The procedure can be applied to every point of the basin, no matter how far away from the attractor they are. Finally, this paper shows how two mixed chaotic signals can be controlled using a backpropagation neural network as a filter to separate and control both signals at the same time. The neural network provides more effective control, overcoming the problems that arise with control feedback methods. Control is more effective because it can be applied to the system at any point, even if it is moving away from the target state, which prevents waiting times. Also control can be applied even if there is little information about the system and remains stable longer even in the presence of random dynamic noise.