Application of higher order statistics/spectra in biomedical signals : a review

For many decades correlation and power spectrum have been primary tools for digital signal processing applications in the biomedical area. The information contained in the power spectrum is essentially that of the autocorrelation sequence; which is sufficient for complete statistical descriptions of Gaussian signals of known means. However, there are practical situations where one needs to look beyond autocorrelation of a signal to extract information regarding deviation from Gaussianity and the presence of phase relations. Higher order spectra, also known as polyspectra, are spectral representations of higher order statistics, i.e. moments and cumulants of third order and beyond. HOS (higher order statistics or higher order spectra) can detect deviations from linearity, stationarity or Gaussianity in the signal. Most of the biomedical signals are non-linear, non-stationary and non-Gaussian in nature and therefore it can be more advantageous to analyze them with HOS compared to the use of second order correlations and power spectra. In this paper we have discussed the application of HOS for different bio-signals. HOS methods of analysis are explained using a typical heart rate variability (HRV) signal and applications to other signals are reviewed.

Ejection of energetic photoelectrons in laser-induced autoionisation: effects of high-order ionisation processes

Resonant interaction of an autoionising state with a strong laser field is considered and effects of second-order ionisation processes are investigated. The authors show that these processes play a very important role in laser-induced autoionisation (LIA). They drastically affect the lowest-order peaks in the photoelectron spectrum. In addition to these peaks, high-order peaks due to ejection of energetic photoelectrons appear. For the laser intensities of current interest, second-order peaks are much stronger than the original ones, an important result that, they believe, can be observed experimentally. Moreover, `peak switching', a general feature of above-threshold ionisation, is also manifest in the electron spectrum of LIA.

Automatic extraction of Reduced order models from CFD simulations for building energy modeling

Thermal comfort is defined as “that condition of mind which expresses satisfaction with the thermal environment’ [1] [2]. Field studies have been completed in order to establish the governing conditions for thermal comfort [3]. These studies showed that the internal climate of a room was the strongest factor in establishing thermal comfort. Direct manipulation of the internal climate is necessary to retain an acceptable level of thermal comfort. In order for Building Energy Management Systems (BEMS) strategies to be efficiently utilised it is necessary to have the ability to predict the effect that activating a heating/cooling source (radiators, windows and doors) will have on the room. The numerical modelling of the domain can be challenging due to necessity to capture temperature stratification and/or different heat sources (radiators, computers and human beings). Computational Fluid Dynamic (CFD) models are usually utilised for this function because they provide the level of details required. Although they provide the necessary level of accuracy these models tend to be highly computationally expensive especially when transient behaviour needs to be analysed. Consequently they cannot be integrated in BEMS. This paper presents and describes validation of a CFD-ROM method for real-time simulations of building thermal performance. The CFD-ROM method involves the automatic extraction and solution of reduced order models (ROMs) from validated CFD simulations. The test case used in this work is a room of the Environmental Research Institute (ERI) Building at the University College Cork (UCC). ROMs have shown that they are sufficiently accurate with a total error of less than 1% and successfully retain a satisfactory representation of the phenomena modelled. The number of zones in a ROM defines the size and complexity of that ROM. It has been observed that ROMs with a higher number of zones produce more accurate results. As each ROM has a time to solution of less than 20 seconds they can be integrated into the BEMS of a building which opens the potential to real time physics based building energy modelling.

Automatic extraction of reduced-order models from CFD simulations for building energy modelling

Accurate modelling of the internal climate of buildings is essential if Building Energy Management Systems (BEMS) are to efficiently maintain adequate thermal comfort. Computational fluid dynamics (CFD) models are usually utilised to predict internal climate. Nevertheless CFD models, although providing the necessary level of accuracy, are highly computationally expensive, and cannot practically be integrated in BEMS. This paper presents and describes validation of a CFD-ROM method for real-time simulations of building thermal performance. The CFD-ROM method involves the automatic extraction and solution of reduced order models (ROMs) from validated CFD simulations. ROMs are shown to be adequately accurate with a total error below 5% and to retain satisfactory representation of the phenomena modelled. Each ROM has a time to solution under 20seconds, which opens the potential of their integration with BEMS, giving real-time physics-based building energy modelling. A parameter study was conducted to investigate the applicability of the extracted ROM to initial boundary conditions different from those from which it was extracted. The results show that the ROMs retained satisfactory total errors when the initial conditions in the room were varied by ±5°C. This allows the production of a finite number of ROMs with the ability to rapidly model many possible scenarios.

Modelling and Analysis of Some Time Series

The thesis deals with some of the non-linear Gaussian and non-Gaussian time models and mainly concentrated in studying the properties and application of a first order autoregressive process with Cauchy marginal distribution. In this thesis some of the non-linear Gaussian and non-Gaussian time series models and mainly concentrated in studying the properties and application of a order autoregressive process with Cauchy marginal distribution. Time series relating to prices, consumptions, money in circulation, bank deposits and bank clearing, sales and profit in a departmental store, national income and foreign exchange reserves, prices and dividend of shares in a stock exchange etc. are examples of economic and business time series. The thesis discuses the application of a threshold autoregressive(TAR) model, try to fit this model to a time series data. Another important non-linear model is the ARCH model, and the third model is the TARCH model. The main objective here is to identify an appropriate model to a given set of data. The data considered are the daily coconut oil prices for a period of three years. Since it is a price data the consecutive prices may not be independent and hence a time series based model is more appropriate. In this study the properties like ergodicity, mixing property and time reversibility and also various estimation procedures used to estimate the unknown parameters of the process.

Analysis of plate bending by means of high order finite hyperelements

The possibilities and limitations of high order hyperelements in plate bending analysis are discussed. Explicit shape functions for some types of triangular elements are given. These elements are applied to simple cases to assess their computational efficiency.

Synthesis and evaluation of novel functionalised indoles as anticancer agents

This thesis outlines the design and application of new routes towards a range of novel bisindolylmaleimide and indolo[2,3-a]carbazole derivatives, and evaluation of their biological effects and their chemotherapeutic potential. A key part of this work focussed on utilising a hydroxymaleimide as a replacement for the prevalent lactam/maleimide functionality and forming a series of novel derivatives through substitution on the indole nitrogens. To achieve this, a robust synthetic strategy was developed which allowed access to key maleic anhydride intermediates using Perkin-type methodology. These hydroxymaleimides were further modified via a Lossen rearrangement to furnish a series of analogues containing a 6-membered F-ring. The theme of F-ring modulation was further expanded through the utilisation of a second route involving the design and synthesis of β-keto ester intermediates, which afforded novel derivatives containing pyrazolone and isocytosine headgroups, and various N-substituents. Work on a further route involving a dione intermediate resulted in the isolation of a bisindolyl derivative with a novel imidazole F-ring. Following the synthesis of 42 novel compounds, extensive screening was undertaken using the NCI-60 cell line screen, with twelve candidates progressing to evaluation via the five dose assay. This led to the identification of several lead compounds with high cytotoxicity and excellent selectivity profiles, which included derivatives with low nanomolar GI50 values against specific cancer cell lines, and also derivatives with selective cytotoxicity. Preliminary results from a kinase screen indicated noteworthy selectivity towards GSK3α/β and PIM1 kinases, with low micromolar IC50 values being observed for these enzymes.

Novel analytical and numerical methods for solving fractional dynamical systems

During the past three decades, the subject of fractional calculus (that is, calculus of integrals and derivatives of arbitrary order) has gained considerable popularity and importance, mainly due to its demonstrated applications in numerous diverse and widespread fields in science and engineering. For example, fractional calculus has been successfully applied to problems in system biology, physics, chemistry and biochemistry, hydrology, medicine, and finance. In many cases these new fractional-order models are more adequate than the previously used integer-order models, because fractional derivatives and integrals enable the description of the memory and hereditary properties inherent in various materials and processes that are governed by anomalous diffusion. Hence, there is a growing need to find the solution behaviour of these fractional differential equations. However, the analytic solutions of most fractional differential equations generally cannot be obtained. As a consequence, approximate and numerical techniques are playing an important role in identifying the solution behaviour of such fractional equations and exploring their applications. The main objective of this thesis is to develop new effective numerical methods and supporting analysis, based on the finite difference and finite element methods, for solving time, space and time-space fractional dynamical systems involving fractional derivatives in one and two spatial dimensions. A series of five published papers and one manuscript in preparation will be presented on the solution of the space fractional diffusion equation, space fractional advectiondispersion equation, time and space fractional diffusion equation, time and space fractional Fokker-Planck equation with a linear or non-linear source term, and fractional cable equation involving two time fractional derivatives, respectively. One important contribution of this thesis is the demonstration of how to choose different approximation techniques for different fractional derivatives. Special attention has been paid to the Riesz space fractional derivative, due to its important application in the field of groundwater flow, system biology and finance. We present three numerical methods to approximate the Riesz space fractional derivative, namely the L1/ L2-approximation method, the standard/shifted Gr¨unwald method, and the matrix transform method (MTM). The first two methods are based on the finite difference method, while the MTM allows discretisation in space using either the finite difference or finite element methods. Furthermore, we prove the equivalence of the Riesz fractional derivative and the fractional Laplacian operator under homogeneous Dirichlet boundary conditions – a result that had not previously been established. This result justifies the aforementioned use of the MTM to approximate the Riesz fractional derivative. After spatial discretisation, the time-space fractional partial differential equation is transformed into a system of fractional-in-time differential equations. We then investigate numerical methods to handle time fractional derivatives, be they Caputo type or Riemann-Liouville type. This leads to new methods utilising either finite difference strategies or the Laplace transform method for advancing the solution in time. The stability and convergence of our proposed numerical methods are also investigated. Numerical experiments are carried out in support of our theoretical analysis. We also emphasise that the numerical methods we develop are applicable for many other types of fractional partial differential equations.

Fractional generalization of memristor and higher order elements

Fractional calculus generalizes integer order derivatives and integrals. Memristor systems generalize the notion of electrical elements. Both concepts were shown to model important classes of phenomena. This paper goes a step further by embedding both tools in a generalization considering complex-order objects. Two complex operators leading to real-valued results are proposed. The proposed class of models generate a broad universe of elements. Several combinations of values are tested and the corresponding dynamical behavior is analyzed.

On a Differential Equation with Left and Right Fractional Derivatives

Mathematics Subject Classification: 26A33; 70H03, 70H25, 70S05; 49S05

Computational methods in the fractional calculus of variations and optimal control

The fractional calculus of variations and fractional optimal control are generalizations of the corresponding classical theories, that allow problem modeling and formulations with arbitrary order derivatives and integrals. Because of the lack of analytic methods to solve such fractional problems, numerical techniques are developed. Here, we mainly investigate the approximation of fractional operators by means of series of integer-order derivatives and generalized finite differences. We give upper bounds for the error of proposed approximations and study their efficiency. Direct and indirect methods in solving fractional variational problems are studied in detail. Furthermore, optimality conditions are discussed for different types of unconstrained and constrained variational problems and for fractional optimal control problems. The introduced numerical methods are employed to solve some illustrative examples.

Operational Rules for a Mixed Operator of the Erdélyi-Kober Type

2000 Mathematics Subject Classification: 26A33 (main), 44A40, 44A35, 33E30, 45J05, 45D05

On the Riemann-Liouville Fractional q-Integral Operator Involving a Basic Analogue of Fox H-Function

2000 Mathematics Subject Classification: 33D60, 26A33, 33C60

Numerical Solution of Fractional Diffusion-Wave Equation with two Space Variables by Matrix Method

Mathematics Subject Classi¯cation 2010: 26A33, 65D25, 65M06, 65Z05.

Fractional Derivatives in Spaces of Generalized Functions

MSC 2010: 26A33, 46Fxx, 58C05 Dedicated to 80-th birthday of Prof. Rudolf Gorenflo