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The development of financial markets imposes more complex models on the option pricing problems. On the previous papers by the authors, we consider a model under which the underlying asset is driven by two independent Heston-type stochastic volatility processes of multiscale (fast and slow) mean-reverting rates and we compute an approximate solution for the option pricing problem, using asymptotic expansion method. In the present paper, we aim to calibrate the model using the market prices of options on Euro Stoxx 50 index and an equity stock in the European market. Our approach is to use the market implied volatility surface for calibrating directly a set of new parameters required in our second-order asymptotic expansion pricing formula for European options. This secondorder asymptotic expansion formula provides a better approximation formula for European option prices than the first-order formula, as explained in an earlier work of the authors.

There are many models for mortality rates. A well-known problem that complicates modeling of human mortality rates is the “accident hump” occurring in early adulthood. Here, two models of mortality rate based on power-exponential functions are presented and compared to a few other models. The models will be fitted to known data of measured death rates from several different countries using numerical techniques for curve-fitting with the nonlinear least-squares method. The properties of the model with respect to forecasting with the Lee–Carter method will be discussed.

The celebrated Black–Scholes model made the assumption of constant volatility but empirical studies on implied volatility and asset dynamics motivated the use of stochastic volatilities. Christoffersen in 2009 showed that multi-factor stochastic volatilities models capture the asset dynamics more realistically. Fouque in 2012 used it to price European options. In 2013 Chiarella and Ziveyi considered Christoffersen's ideas and introduced an asset dynamics where the two volatilities of the Heston type act separately and independently on the asset price, and using Fourier transform for the asset price process and double Laplace transform for the two volatilities processes, solved a pricing problem for American options. This paper considers the Chiarella and Ziveyi model and parameterizes it so that the volatilities revert to the long-run-mean with reversion rates that mimic fast(for example daily) and slow(for example seasonal) random effects. Applying asymptotic expansion method presented by Fouque in 2012, we make an extensive and detailed derivation of the approximation prices for European options. We also present numerical studies on the behavior and accuracy of our first and the second order asymptotic expansion formulas.

After Black-Scholes proposed a model for pricing European Option in 1973, Cox, Ross and Rubinstein in 1979, and Heston in 1993, showed that the constant volatility assumption in the Black-Scholes model was one of the main reasons for the model to be unable to capture some market details. Instead of constant volatilities, they introduced non-constant volatilities to the asset dynamic modeling. In 2009, Christoffersen empirically showed "why multi-factor stochastic volatility models work so well". Four years later, Chiarella and Ziveyi solved the model proposed by Christoffersen. They considered an underlying asset whose price is governed by two factor stochastic volatilities of mean reversion type. Applying Fourier transforms, Laplace transforms and the method of characteristics they presented an approximate formula for pricing American option.The huge calculation involved in the Chiarella and Ziveyi approach motivated us to investigate another approach to compute European option prices on a Christoffersen type model. Using the first and second order asymptotic expansion method we presented a closed form solution for European option, and provided experimental and numerical studies on investigating the accuracy of the approximation formulae given by the first order asymptotic expansion. In the present chapter we will perform experimental and numerical studies for the second order asymptotic expansion and compare the obtained results with results presented by Chiarella and Ziveyi.

HF analysis of a horizontal dipole antenna buried in lossy ground has been performed in this paper. The soil is treated as a homogenous half-space of known electrical parameters. The authors compare the range of applicability of two forms of transmission line model , a hybrid circuit method, and a point-matching method in such analysis.

Lightning discharge currents and their derivatives have been measured at instrumented tall towers for a few decades already. Results of those measurements are used for lightning research purposes, modelling of lightning discharges and making improvements in lightning protection. Some of the measured lightning currents are represented in this paper by the multi-peaked analytically extended function (MP-AEF). The same function may be used for lightning current derivatives, thus providing their analytical integration. 

Electrostatic discharges (ESD) consequences are not only damages, but also very fast transients in electric circuits and devices, as well as in large power systems. This paper presents frequency spectrum analysis of the IEC 61000-4-2 standard current approximated by multi-peaked analytically extended function (MP-AEF) and of typical lightning currents according to the IEC 62305 standard. This analysis is important for the choice of protective devices and frequency range of measurement equipment used in experiments. A comparison of the measured and theoretical results is also given.

Lightning discharge currents waveshapes and their derivatives with multiple peaks are measured in artificially triggered lightning experiments and at instrumented tall towers. Such waveshapes are represented in this paper by the N-peaked analytically extended function (NP-AEF) and Marquardt least-squares method is applied for the estimation of its nonlinear parameters. Typical channel-base currents of the first negative, subsequent negative strokes, and positive strokes, based on comprehensive measurements by Berger et al. at Monte San Salvatore in Switzerland, are approximated by NP-AEFs and used in the computation of lightning electromagnetic fields. A new attenuation factor introducing nonlinear current attenuation along the channel is applied within the modified transmission line model (MTLSIN). A lightning return stroke is assumed to have a vertical discharge channel at the perfectly conducting ground. In order to validate this model, calculated lightning electromagnetic fields are compared with the typical, as measured by Lin et al. at various distances from the discharges. Lightning currents and their derivatives measured at the tall towers are also approximated by NP-AEFs. For the measured artificially triggered lightning currents, MTLSIN is applied for the calculation of their electromagnetic fields. These are compared with the measured fields of specific lightning strokes, so as to results of other models.

Genetic algorithm (GA) is applied for the estimation of two-peaked analytically extended function (2P-AEF) parameters in this paper. 2P-AEF is used for approximation of measured and typical lightning discharge currents. Lightning discharge channel is often modeled as thin-wire vertical antenna at perfectly conducting ground. Engineering lightning stroke models assume that the current along that channel is related to the channel-base current which may be measured at the instrumented tall towers and in triggered lightning experiments. Mathematical modeling of lightning currents is important in verification of lightning strokes models based on simultaneously measured electromagnetic fields at various distances, so as in lightning protection studies, computation of lightning induced effects and simulation of overvoltages in power systems. Typical lightning discharge currents of the first positive, first negative, and subsequent negative strokes are defined by IEC 62305 Standard based on comprehensive measurements. Parameters of 2P-AEF’s approximation of the typical negative first stroke current are determined by GA and compared to approximations obtained by other functions. Measured currents at Monte San Salvatore in Switzerland, at Morro de Cachimbo Station in Brazil, and in rocket-triggered lightning experiments at Camp Blanding in Florida are approximated by 2P-AEFs, and good agreement with experimentally measured waveshapes is obtained.

An analytically extended function based on power-exponential functions is used in this paper for approxi- mation of electrostatic discharge (ESD) currents and their derivatives. The Marquardt least-squares method (MLSM) is applied for obtaining nonlinear function parameters. IEC 61000-4-2 Standard ESD current is approximated, as well as some measured ESD currents' wave shapes. Power-exponential terms are extended at the local maxima and minima of the represented wave shape, so that this approximation is done from peak to peak. ESD current derivative is approxi- mated using the same procedure in order to obtain the continuous second order derivative of the current, as all piecewise functions are of differentiability class C¹L . Currents and their derivatives are often measured in ESD experiments so that their analytical representation is needed for simulation of ESD phenomena, better definition of standard requirements, and computation of the transient fields and induced effects.

Analytically extended function (AEF) for modeling measured electrostatic discharge (ESD) currents is presented in this paper. This function is used for the IEC 61000-4-2 Standard current and AEF parameters are calculated using least-squares method. Procedure of obtaining function parameters is simple for experimentally measured multi-peaked functions, such as lightning and other ESD currents. Simulation of ESD currents is done by simple electric circuits and using MATLAB.

Purpose: This paper aims to present the approximation of lightning currents waveshapes by the multi-peaked analytically extended function (MP-AEF) for the experimentally measured channel-base currents in the artificially triggered lightning discharges. Modified transmission line model of lightning return strokes having the channel current both linearly decaying and sinusoidally changing with height (MTLSIN) is used to calculate the lightning electromagnetic field.

Design/methodology/approach: MP-AEF’s parameters for the artificially triggered lightning channel-base currents are calculated by using Marquardt least squares method (MLSM). Lightning electromagnetic fields are calculated based on electromagnetic theory relations, thin-wire antenna model of the vertical lightning channel and the assumption of the perfectly conducting ground. MTLSIN model as an engineering model of lightning strokes is used to obtain the electric field results as these are simultaneously measured in rocket-triggered lightning experiments together with the channel-base currents. Findings: MP-AEF approximates multi-peaked pulse waveshapes. Some important function parameters are chosen prior to the approximation procedure, such as current peaks and the corresponding time moments of those peaks, which presents an advantage in comparison to other functions. The desired accuracy of approximation is obtained by choosing an adequate number of function terms. MLSM is used for the estimation of unknown parameters. Using MTLSIN model, the influence of the channel height and return stroke speed on the lightning electromagnetic field waveshape is analyzed in this paper. Research limitations/implications: MP-AEF may be used for approximation of various multi-peaked waveshapes. It has no errors in the points of maxima which is important for the lightning protection systems design. MTLSIN model may be validated by using simultaneously measured lightning electromagnetic fields at various distances from the channel and for channel heights estimated in the experiments. It is also possible to approximate measured current derivatives by MP-AEF and use them for further computation. Originality/value: MTLSIN model is proposed in this paper for the evaluation of lightning electromagnetic fields induced by artificially triggered lightning discharges. The procedure is based on the approximation of lightning channel-base currents by the multi-peaked analytically extended function previously proposed by the authors. This function may be used not only for representing lightning currents but also for other waveshapes as current derivatives, electric and magnetic fields and their derivatives, which are all important for the lightning protection design. MTLSIN gives lightning electromagnetic fields results which are in better agreement with measured fields than those obtained by other models from literature.

A multi-peaked version of the analytically extended function (AEF) intended for approximation of multi-peaked lightning current wave-forms will be presented along with some of its basic properties. A general framework for estimating the parameters of the AEF using the Marquardt least-squares method (MLSM) for a waveform with an arbitrary (finite) number of peaks as well as a given charge trans-fer and specific energy will also be described. This framework is used to find parameters for some common single-peak wave-forms and some advantages and disadvantages of the approach will be discussed.

Application of the Marquardt least-squares method (MLSM) to the estimation of non-linear parameters of functionsused for representing various lightning current waveshapes is presented in this paper. Parameters are determined for the Pulse,Heidler’s and DEXP function representing the first positive, first and subsequent negative stroke currents as given in IEC62305-1 Standard Ed.2, and also for some other fast- and slow-decaying lightning current waveshapes. The results prove theability of the MLSM to be used for the estimation of parameters of the functions important in lightning discharge modeling.

A multi-peaked form of the analytically extended function (AEF) is used for approximation of lightning current waveforms in this paper. The AEF function's parameters are estimated using the Marquardt least-squares method (MLSM), and the general procedure for fitting the p-peaked AEF function to a waveform with an arbitrary (finite) number of peaks is briefly described. This framework is used for obtaining parameters of 2-peaked waveforms typically present when measuring first negative stroke currents. Advantages, disadvantages and possible improvements of the approach are also discussed.

In order to establish realistic requirements for ESD generators used in equipment and device testing, as well as to ensure and improve the repeatability of tests, an analytical expression of real electrostatic discharge (ESD) currents is sought for. It should be adoptable to various test levels, test set-ups and procedures, and also various ESD conditions (approach speeds, types of electrodes, relative arc length, humidity, etc). A mathematical function is necessary for computer simulation of such phenomena, for verification of test generators and for improving standard waveshape definition.

In this paper the Analytically Extended Function (AEF) with p peaks is used for representation of the electrostatic discharge (ESD) currents and lightning discharge currents. The fitting to data is achieved by interpolation of certain data points. In order to minimize unstable behaviour, the exponents of the AEF are chosen from a certain arithmetic sequence and the interpolated points are chosen according to a D-optimal design. The method is illustrated using several examples of currents taken from standards and measurements.

Multi-peaked analytically extended function (AEF), previously applied by the authors to modelling of lightning discharge currents, is used in this paper for representation of the electrostatic discharge (ESD) currents. The fitting to data is achieved by interpolation of certain data points. In order to minimize unstable behaviour, the exponents of the AEF are chosen from a certain arithmetic sequence and the interpolated points are chosen according to a D-optimal design. ESD currents' modelling is illustrated through two examples: one corresponding to an approximation of the IEC Standard 61000-4-2 waveshape, and the other to representation of some measured ESD current.

Multi-peaked analytically extended function (AEF), previously applied by the authors to modelling of lightning discharge currents, is used in this paper for representation of the electrostatic discharge (ESD) currents. The fitting to data is achieved by interpolation of certain data points. In order to minimize unstable behaviour, the exponents of the AEF are chosen from a certain arithmetic sequence and the interpolated points are chosen according to a D-optimal design. ESD currents’ modelling is illustrated through two examples: one corresponding to an approximation of the IEC Standard 61000-4-2 waveshape, and the other to representation of some measured ESD current. 

Examination of how the analytically extended function (AEF) can be used to approximate multi-peaked lightning current waveforms, is performed in the paper. A general framework for estimating the parameters of the AEF using the Marquardt least-squares method (MLSM) for a waveform with an arbitrary (finite) number of peaks is presented. This framework is used to find parameters for some common waveforms with a single peak, such as one-peaked Standard IEC 62305 lightning currents.

An examination of how the analytically extended function (AEF) can be used to approximate multi-peaked lightning current waveforms, is presented in the paper. A general framework for estimating the parameters of the AEF using the Marquardt least-squares method (MLSM) for a waveform with an arbitrary (finite) number of peaks is presented. This framework is used to find parameters for some common waveforms with a single peak, such as Standard IEC 62305 lightning currents. Illustration of fitting a p-peak AEF to recorded lightning current data is also presented.

The Marquardt least-squares method is applied in this paper for estimation of the Pulse function’s non-linear para­meters in order to approximate measured lightning currents. Such procedure is generalized so it could be used for other pulse functions representing lightning current waveshapes. The obtai­ned results show that this method provides good results for the IEC 62305-1 standard lightning current waveshapes of the first and subsequent return stroke currents, and also for some measu­red fast- and slow-decaying current waveshapes.

A multi-peaked analytically extended function (AEF), previously applied by the authors to modeling of lightning dischargecurrents, is used in this paper for representation of the electrostatic discharge (ESD) currents. In order to estimate itsnon-linear parameters, the Marquardt least-squares method (MLSM) is used. ESD currents’ modelling is illustrated through anessential example corresponding to approximation of the IEC Standard 61000-4-2 waveshape.

A new approach to mathematical modelling of lightning current derivative is proposed in this paper. It builds on the methodology, previously developed by the authors, for representing lightning currents and electrostatic discharge (ESD) currents waveshapes. It considers usage of a multi-peaked form of the analytically extended function (AEF) for approximation of current derivative waveshapes. The AEF function parameters are estimated using the Marquardt least-squares method (MLSM) and the framework for fitting the multi-peaked AEF to a waveshape with an arbitrary number of peaks is briefly described. This procedure is validated performing a few numerical experiments, including fitting the AEF to single- and multi-peaked waveshapes corresponding to measured current derivatives.

According to experimental results for lightning discharge currents, they are classified in the IEC 62305 Standard into waveshapes representing the first positive, first and subsequent negative strokes, and long-strokes. These waveshapes, especially shot-term pulses, are approximated with a few mathematical functions in literature, in order to be used in lightning discharge models for calculations of electromagnetic field and lightning induced effects. An analytically extended function (AEF) is presented in this paper and used for lightning currents modeling. The basic properties of this function with a finite number of peaks are examined. A general framework for estimating the parameters of the AEF using the Marquardt least-squares method (MLSM) for a waveform with an arbitrary (finite) number of peaks as well as for the given charge transfer and specific energy is described. This framework is used to find parameters for some common single-peak wave-forms and some advantages and disadvantages of the approach are also discussed.

There are many models for the mortality rates of humans and other organisms. A phenomenon that complicates the modelling of human mortality rates is a rapid increase in mortality rate for young adults (in western Europe this is especially pronounced at the age of 25). We will examine models for mortality rates based on power-exponential functions, compare them to empirical data for mortality rates and other models.

A multi-peaked version of the analytically extendedfunction (AEF) intended for approximation of multi-peaked light-ning current waveforms will be presented along with some of itsbasic properties. A general framework for estimating the parame-ters of the AEF using the Marquardt least-squares method(MLSM) for a waveform with an arbitrary (finite) number ofpeaks as well as a given charge transfer and specific energy willalso be described. This framework is used to find parameters forsome common single-peak wave-forms and some advantages anddisadvantages of the approach will be discussed.

Among other limitations, the celebrated Black--Scholes option pricingmodel assumes constant volatility and constant interest rates, which is not supportedby empirical studies on for example implied volatility surfaces. Studiesby many researchers such as Heston in 1993, Christoffersen in 2009, Fouque in2012, Chiarella--Ziveyi in 2013, and the authors' previous work removed the constantvolatility assumption from the Black--Scholes model by introducing one ortwo stochastic volatility factors with constant interest rate. In the present paperwe follow this line but generalize the model by considering also stochasticinterest rate. More specifically, the underlying asset process is governed by amean-reverting interest rate process in addition to two mean-reverting stochasticvolatility processes of fast and slow mean-reverting rates respectively. The focusis to derive an approximating formula for pricing the European option using adouble asymptotic expansion method.

To solve the electromagnetic scattering problem in two dimensions, the Finite Difference Time Domain (FDTD)method is used. The order of convergence of the FDTD algorithm, solving the two-dimensional Maxwell’s curl equations,is estimated in two different computer implementations: with and without an obstacle in the numerical domain of the FDTDscheme. This constitutes an electromagnetic scattering problem where a lumped sinusoidal current source, as a source ofelectromagnetic radiation, is included inside the boundary. Confined within the boundary, a specific kind of AbsorbingBoundary Condition (ABC) is chosen and the outside of the boundary is in form of a Perfect Electric Conducting (PEC)surface. Inserted in the computer implementation, a semi-norm has been applied to compare different step sizes in the FDTDscheme. First, the domain of the problem is chosen to be the free-space without any obstacles. In the second part of thecomputer implementations, a PEC surface is included as the obstacle. The numerical instability of the algorithms can berather easily avoided with respect to the Courant stability condition, which is frequently used in applying the general FDTDalgorithm.

Approximate evaluation of Sommerfeld type integrals has been of great interest for researchers in the field of electromagnetics, in particular in the areas of antenna theory and grounding systems analysis. These integrals arise in the expressions describing the electromagnetic field in the surroundings of such structures when they are located above/inside a semi-conducting media. The fact that these integrals don’t have a closed form solution, enticed researchers to approximately evaluate them either by employing a numerical integration technique, or using some kind of procedure that will approximate them and allow their analytical evaluation.

A simple procedure for approximate calculation of one type of Sommerfeld integrals occurring in cases of wire conductors buried in semi-conducting ground is proposed. It considers approximation of a part of the integrand using a weighted exponential function with an additional unknown constant complex term. This kind of modification allows the obtained integral to be calculated analytically.

In this paper, the authors perform HF analysis of a thin horizontal conductor fed in its center, and arbitrarily positioned above lossy homogeneous ground of known electrical parameters. The approach is based on the electric-field integral equation method, and formulation of the Hallén’s integral equation. This equation is then solved for the current using the point-matching method. The Sommerfeld’s integrals that express the influence of the lossy ground, and that appear in these calculations, are solved approximately. Thorough analysis is performed in order to observe the influence of different parameters of the geometry and the ground on current distribution in the specified frequency range. Furthermore, the verification of the method is done by comparison with the exact model based on the full-wave theory.

Increase of the radiation power in different frequency bands during the last decades, has called for a study of harmful effects on the living organisms and electronic equipment of the radio frequency energy. An accurate determination of the near field strength, electric as well as magnetic, in the vicinity of higher-power transmitting antennas is necessary for assessing any possible radiation hazard. In that sense, it is of great importance to account for the influence of the finite ground conductivity on the electromagnetic field structure in the surroundings of these emitters. The estimation of this influence has been intensively studied, and a number of approaches has been applied in that sense, ranging from the exact full-wave based ones to different forms of approximate, less time-consuming, ones. Although the approximate methods introduce a certain level of calculation error, their simplicity is of interest in the electomagnetic compatibility (EMC) studies. For that reason, finding an approximate, but satisfyingly accurate method, applicable to wide range of parameters is often a goal of researches done in this field.

In this paper, the authors perform an analysis of a thin horizontal dipole antenna (HDA) above real ground of known electrical parameters. The approach is based on the electric-field integral equation method, and formulation of the Hallén’s integral equation (HIE). This equation is then solved for the current, which is assumed in a polynomial form, using the point-matching method (PMM). This way obtained system of linear equations involves improper Sommerfeld’s integrals, which express the influence of the real ground and are here solved approximately using simple, so-called OIA and TIA, approximations (one- and two-image approximations). Both types of approximations are in an exponential form, and therefore are similar to those obtained applying the method of images. It should be kept in mind that the goal of this approach is to develop approximations that have a simple form, whose application yields satisfyingly accurate calculations of the Sommerfeld`s type of integrals, and are widely applicable, i.e. their employment is not restricted by the values of electrical parameters of the ground, or the geometry.

Thorough analysis is performed in order to observe the influence of different parameters of the geometry, and the ground, on current distribution and the input impedance/admittance of the HDA in a wide frequency range. Furthermore, the verification of the method is done by comparison to the exact model based on the full-wave theory, and experimental data. Obtained results indicate a possibility of applying the described methodology to inverse problem involving evaluation of electrical parameters of the ground (or detection of ground type change) based on measured input impedance/admittance of the antenna.

In risk management, foreign investors or multinational corporations are highly interested in knowing how volatile a currency is in order to hedge risk. In this chapter, using daily exchange rates and the exponential weighted moving average (EWMA) model, we perform volatility forecasting. We will investigate how the use of the available time series affects the forecasting, i.e. how reliable our forecasting is depending on the period of available data used. We will also test the effects of the decay factor appearing in the model used on the forecasts. The results show that, forthe data used, it is optimal to use a larger value of the decay factor and also, for longer out-of-sample periods, the forecasts get closer to reality.

In risk management, foreign investors or multinational corporations are highly interested in knowing how volatile acurrency is in order to hedge risk. In this paper, using daily exchange rates and the Exponential Weighted Moving Average (EWMA) model, we perform volatility forecasting. We will investigate how used available time series affect the forecastings, i.e. how reliable our forecasting is depending on the period of used available data. We will also experiment the effects of the decay factor appearing in the model used on the forecasts.The results show that for the data used, it is optimal to use a smaller value of the decay factor and also for longer out-of-sample periods the forecasts get closer to the reality.

A shielded coupled microstrip line with partial dielectric support is analysed using the hybrid boundary element method (HBEM) and the finite difference method (FDM). The HBEM is a combination of the equivalent electrodes method (EEM) and the boundary element method (BEM). The microstrip line characteristic parameters: the effective relative permittivity and the characteristic impedance are deter­mined. “Odd” and “even” modes are taken into account. The results are compared with corresponding ones found in the literature.

The paper reviews the frequency domain modeling of buried dipole antenna using both the Pocklington integrodifferential equation and Hallén integral equation formulation. The details pertaining to the numerical solution procedures are outlined as, well. Some strengths and weaknesses of both approaches are presented. A typical computational example for the current distribution induced along the dipole embedded in a lossy ground illustrating both methods is given, as well.

The paper reviews the frequency domain modeling of dipole antenna using both the Pocklington integro-differential equation and the Hallén integral equation formulation. The details pertaining to the numerical solution procedures are outlined as well. Some strengths and weaknesses of both approaches are presented. A typical computational example for the current distribution induced along the dipole radiating over a lossy ground illustrating both methods is given, as well.

The paper reviews the frequency domain modeling of buried dipole antenna using both the Pocklington integro-differential equation and Hallen integral equation formulation. The details pertaining to the numerical solution procedures are outlined as, well. Some strengths and weaknesses of both approaches are presented. A typical computational example for the current distribution induced along the dipole embedded in a lossy ground illustrating both methods is given, as well.

Tutorial describes some approaches to mathematical modelling of physical problems. Applications will be illustarted on examples from the areas of antenna theory, grounding systems analysis, modelling of discharge currents and actuarial mathematics. 

We start with problems related to numerical analysis of sources in presence of a lossy medium. A well-known problem of dealing with so-called Sommerfeld type integrals occurs in these analysis. Their approximate evaluation has been of great interest for researchers in the areas of antenna theory and grounding systems analysis. These integrals arise in the expressions describing the electromagnetic field in the surroundings of such structures when they are located above/inside a semi-conducting media. The fact that these integrals don’t have a closed form solution, enticed researchers to approximately evaluate them either by employing a numerical integration technique, or using some kind of procedure that will approximate them and allow their analytical evaluation. 

Second part of the tutorial deals with modelling of lightning and electrostatic discharge currents. A general function that would be able to reproduce desired waveshapes of theses currents is needed, such that analytical solutions for their derivatives, integrals, and integral transformations, exist. We present a review of existing models, their advatages and disadvartages and possible extensions. 

Finally, we discuss modelling of mortality rates of living organisms or equipment. Variation of mortality over a life span has different characteristics that put constraints and requirements on a model developed to represent it. A well-know problem that complicates modelling of human mortality rates is the "accident hump" occurring in early adulthood. We review existing models and discuss their properties and application to mortality forcasting and pricing life insurances. 

A new simple approximation that can be used for modeling of one type of Sommerfeld integrals typically occurring in the expressions that describe sources buried in the lossy ground, is proposed in the paper. The ground is treated as a linear, isotropic and homogenous medium of known electrical parameters. Proposed approximation has a form of a weighted exponential function with an additional complex constant term. The derivation procedure of this approximation is explained in detail, and the validation is done applying it in the analysis of a bare conductor fed in the center and immersed in the lossy ground at arbitrary depth. Wide range of ground and geometry parameters of interest has been taken into consideration.

This book highlights the latest advances in engineering mathematics with a main focus on the mathematical models, structures, concepts, problems and computational methods and algorithms most relevant for applications in modern technologies and engineering. In particular, it features mathematical methods and models of applied analysis, probability theory, differential equations, tensor analysis and computational modelling used in applications to important problems concerning electromagnetics, antenna technologies, fluid dynamics, material and continuum physics and financial engineering.  The individual chapters cover both theory and applications, and include a wealth of figures, schemes, algorithms, tables and results of data analysis and simulation. Presenting new methods and results, reviews of cutting-edge research, and open problems for future research, they equip readers to develop new mathematical methods and concepts of their own, and to further compare and analyse the methods and results discussed.The book consists of contributed chapters covering research developed as a result of a focused international seminar series on mathematics and applied mathematics and a series of three focused international research workshops on engineering mathematics organised by the Research Environment in Mathematics and Applied Mathematics at Mälardalen University from autumn 2014 to autumn 2015: the International Workshop on Engineering Mathematics for Electromagnetics and Health Technology; the International Workshop on Engineering Mathematics, Algebra, Analysis and Electromagnetics; and the 1st Swedish-Estonian International Workshop on Engineering Mathematics, Algebra, Analysis and Applications. It serves as a source of inspiration for a broad spectrum of researchers and research students in applied mathematics, as well as in the areas of applications of mathematics considered in the book. 

This book highlights the latest advances in engineering mathematics with a main focus on the mathematical models, structures, concepts, problems and computational methods and algorithms most relevant for applications in modern technologies and engineering. It addresses mathematical methods of algebra, applied matrix analysis, operator analysis, probability theory and stochastic processes, geometry and computational methods in network analysis, data classification, ranking and optimisation.

The individual chapters cover both theory and applications, and include a wealth of figures, schemes, algorithms, tables and results of data analysis and simulation. Presenting new methods and results, reviews of cutting-edge research, and open problems for future research, they equip readers to develop new mathematical methods and concepts of their own, and to further compare and analyse the methods and results discussed.

The book consists of contributed chapters covering research developed as a result of a focused international seminar series on mathematics and applied mathematics and a series of three focused international research workshops on engineering mathematics organised by the Research Environment in Mathematics and Applied Mathematics at Mälardalen University from autumn 2014 to autumn 2015: the International Workshop on Engineering Mathematics for Electromagnetics and Health Technology; the International Workshop on Engineering Mathematics, Algebra, Analysis and Electromagnetics; and the 1st Swedish-Estonian International Workshop on Engineering Mathematics, Algebra, Analysis and Applications.

It serves as a source of inspiration for a broad spectrum of researchers and research students in applied mathematics, as well as in the areas of applications of mathematics considered in the book.