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Fourier methods used in two‐ and three‐dimensional image reconstruction can be used also in reconstructability analysis (RA). These methods maximize a variance‐type measure instead of information‐theoretic uncertainty, but the two measures are roughly collinear and the Fourier approach yields results close to that of standard RA. The Fourier method, however, does not require iterative calculations for models with loops. Moreover, the error in Fourier RA models can be assessed without actually generating the full probability distributions of the models; calculations scale with the size of the data rather than the state space. State‐based modeling using the Fourier approach is also readily implemented. Fourier methods may thus enhance the power of RA for data analysis and data mining.

The purpose of this paper is to determine the thermo-gravitational convective state of a non-Fourier fluid layer of the single-phase-lagging type, heated from below. Unlike existing methodologies, the spectral modes are not imposed arbitrarily. They are systematically identified by expanding the spectral coefficients in terms of the relative departure in the post-critical Rayleigh number (perturbation parameter). The number and type of modes is determined to each order in the expansion. Non-Fourier effects become important whenever the relaxation time (delay in the response of the heat flux with respect to the temperature gradient) is of the same order of magnitude as process time.

In the spectral method the flow and temperature fields are expanded periodically along the layer and orthonormal shape functions are used in the transverse direction. A perturbation approach is developed to solve the nonlinear spectral system in the post-critical range.

The Nusselt number increases with non-Fourier effect as suggested in experiments in microscale and nanofluid convection.

Unlike existing nonlinear formulations for RB thermal convection, the present combined spectral-perturbation approach provides a systematic method for mode selection.

The purpose of this paper is to present the method for efficient computation of risk measures using Fourier transform technique. Another objective is to demonstrate that this technique enables an efficient computation of risk measures beyond value-at-risk and expected shortfall. Finally, this paper highlights the importance of validating assumptions behind the risk model and describes its application in the affine model framework.

The method proposed is based on Fourier transform methods for computing risk measures. The authors obtain the loss distribution by fitting a cubic spline through the points where Fourier inversion of the characteristic function is applied. From the loss distribution, the authors calculate value-at-risk and expected shortfall. As for the calculation of the entropic value-at-risk, it involves the moment generating function which is closely related to the characteristic function. The expectile risk measure is calculated based on call and put option prices which are available in a semi-closed form by Fourier inversion of the characteristic function. We also consider mean loss, standard deviation and semivariance which are calculated in a similar manner.

The study offers practical insights into the efficient computation of risk measures as well as validation of the risk models. It also provides a detailed description of algorithms to compute each of the risk measures considered. While the main focus of the paper is on portfolio-level risk metrics, all algorithms are also applicable to single instruments.

The algorithms presented in this paper require little computational effort which makes them very suitable for real-world applications. In addition, the mathematical setup adopted in this paper provides a natural framework for risk model validation which makes the approach presented in this paper particularly appealing in practice.

This is the first study to consider the computation of entropic value-at-risk, semivariance as well as expectile risk measure using Fourier transform method.

The specific objective of the study is to investigate the presence of natural rate of crime rates in selected emerging economies by using panel unit roots. The majority of the literature examines the issue using conventional unit root tests in a country-specific context. Meanwhile, there is no panel unit root investigation has been undertaken considering both cross-sectional dependence (CD) and structural changes.

As a result, this study is to fill the aforementioned gap and validate the natural rate of crime rates for 10 countries by using a Fourier panel unit root test. The advantage of the test is that structural shifts are modelled as gradual or smooth changes with a Fourier approximation, and it also accounts cross-sectional dependency. Thus, the Fourier panel unit root test may have better performance in capturing potential changes in the nature of data.

The result of the conventional unit roots test shows evidence of the hysteresis effect in crime, as it stands does not adequately account for smooth transitions or breaks. On contrary, the Fourier panel unit root test confirms the natural rate hypothesis in crime rates. The present results highlight the detrimental effects of crime cannot be abated by short-run deterrence policies.

Contrary to previous studies, the theoretical implications of the study imply that the empirical models consider the dynamic nature of crime rates should account for natural rate properties instead of the hysteresis assumption.

The purpose of this paper is to study the problem of fashion flat sketches classification and proposed an integrated approach. It aims to propose a fast, reliable method to handle multi-class fashion flat sketches classification problems and lay the foundation for the garment style query in the next step.

The proposed integrated approach adopts wavelet Fourier descriptor (WFD), linear discriminant analysis (LDA) and extreme learning machine (ELM). First, the discrete wavelet and Fourier transform are adopted to extract the shape features of fashion flat sketches. Then, LDA is employed for multi-class classification to reduce dimensionality. Finally, ELM is used as the classifier.

The experimental results show that the classification accuracy of the integrated approach is obtained at about 100 percent. Contrary to the traditional approaches, efficiency and accuracy are the advantages of the present approach.

Fashion concept is conveyed often in the form of the fashion illustration or sketch. This type of sketch is useful to imply the style and overall feel of the design. However, this sketch gives no clue about the parts or sections that make up each garment. For this reason, this paper only studies the classification of flat sketches.

A new shape descriptor named WFD is proposed. The WFD acquires high classification accuracy comparing with Fourier descriptor (FD) and multiscale Fourier descriptor (MFD) without dimensionality reduction and nearly the same classification accuracy comparing with FD while MFD easily causes small sample size problem with dimensionality reduction using LDA. In addition, ELM is first used as the classifier in the textiles field related to the classification problem.

This editorial, the second of a two‐part series, proposes a new measure of risk for analyzing highly non‐normal (i.e. asymmetric and long‐tailed) random variables in the context of both investment and insurance portfolios. The proposed measure replaces the p‐norm‐based definition of “risk” – found wanting in Part 1 – with a cosine‐based alternative.

Just as p‐norm‐based risk measures were derived as generalizations of the standard deviation in Part 1, the paper now extend this approach to a cosine‐based risk measure. This involves computing the Fourier transform of the underlying random variable for a given frequency value. Methods for selecting an appropriate frequency are then discussed.

The cosine‐based risk measure provides an effective alternative to p‐norm‐based measures because the Fourier transform is always well defined, even for long‐tailed random variables. The frequency parameter necessary for the Fourier transform may be computed according to several interesting criteria, including the maximization of marginal Shannon information, as well as consideration of “Planck boundaries” in human cognition.

The editorial explores the use of cosine‐based measures in constructing a general definition of “risk” that is equally applicable to asymmetric and long‐tailed random variables as to normal random variables.

This study aims to attempt to construct a new mathematical model of the generalized thermoelasiticity theory based on the memory-dependent derivative (MDD) considering three-phase-lag effects. The governing equations of the problem associated with kernel function and time delay are illustrated in the form of vector matrix differential equations. Implementing Laplace and Fourier transform tools, the problem is sorted out analytically by an eigenvalue approach method. The inversion of Laplace and Fourier transforms are executed, incorporating series expansion procedures. Displacement component, temperature and stress distributions are obtained numerically and illustrated graphically and compared with the existing literature.

This study is to analyze the influence of MDD of three-phase-lag heat conduction interaction in an isotropic semi-infinite medium. The current model has been connected to generalize two-dimensional (2D) thermoelasticity problem. The governing equations are shown in vector matrix form of differential equation concerning Laplace-transformed domain and solved by using the eigenvalue technique. The combined Laplace Fourier transform is applied to find the analytical interpretations of temperature, stresses, displacement for silicon material in a non-dimensional form. Inverse Laplace transform has been found by applying Fourier series expansion techniques introduced by Honig and Hirdes (1984) after performing the inverse Fourier transform.

The main conclusion of this current study is to demonstrate an innovative generalized concept for heat conducting Fourier’s law associated with moderation of time parameter, time delay variable and kernel function by applying the MDDs. However, an important role is played by the time delay parameter to characterize the behavioral patterns of the physical field variables. Further, a new categorization for materials may be created rendering to this new idea along MDD for the time delay variables to develop a new measure of its potential to regulate heat in the medium.

Generalized thermoelasticity is hastily undergoing modification day-by-day from basic thermoelasticity. It has been progressed to get over from the limitations of fundamental thermoelasticity, for instance, infinite velocity components of thermoelasticity interference, in the adequate thermoelastic response of a solid to short laser pulses and deprived illustrations of thermoelastic performance at low temperature. In the past few decades, the fractional calculus is used to change numerous existing models of physical procedure, and its applications are used in various fields of physics, continuum mechanics, fluid mechanics, biology, viscoelasticity, biophysics, signal and image processing, control theory, engineering fields, etc.

In this paper, the authors take the first step in the study of constructive methods by using Sobolev polynomials.

To do that, the authors use the connection formulas between Sobolev polynomials and classical Laguerre polynomials, as well as the well-known Fourier coefficients for these latter.

Then, the authors compute explicit formulas for the Fourier coefficients of some families of Laguerre–Sobolev type orthogonal polynomials over a finite interval. The authors also describe an oscillatory region in each case as a reasonable choice for approximation purposes.

In order to take the first step in the study of constructive methods by using Sobolev polynomials, this paper deals with Fourier coefficients for certain families of polynomials orthogonal with respect to the Sobolev type inner product. As far as the authors know, this particular problem has not been addressed in the existing literature.

The paper presents the derivation of the Fourier coefficients of the magnetisation wave form in closed form for periodic triangular field excitation. The mathematical model used here to approximate the hysteresis loop applies exponential functions and it is presented briefly in the first part of the paper. This method of calculation is applicable to a group of excitation wave forms, constructed of straight lines, such as square, triangular and trapezoid wave forms. The criteria for the Rayleigh region is given and an appropriate formulation of the Fourier coefficients in closed form for the Rayleigh region is also described. It is shown that the calculation is also applicable to anhysteretic processes both in the saturation and in the “small” signal (Rayleigh) region. The Fourier components of the periodic magnetisation wave form resulting from an anhysteretic magnetisation process are also calculated in closed form for triangular field excitation.

To provide an explicit representation for wide‐sense stationary stochastic fields which can be used in stochastic finite element modelling to describe random material properties.

This method represents wide‐sense stationary stochastic fields in terms of multiple Fourier series and a vector of mutually uncorrelated random variables, which are obtained by minimizing the mean‐squared error of a characteristic equation and solving a standard algebraic eigenvalue problem. The result can be treated as a semi‐analytic solution of the Karhunen‐Loève expansion.

According to the Karhunen‐Loève theorem, a second‐order stochastic field can be decomposed into a random part and a deterministic part. Owing to the harmonic essence of wide‐sense stationary stochastic fields, the decomposition can be effectively obtained with the assistance of multiple Fourier series.

The proposed explicit representation of wide‐sense stationary stochastic fields is accurate, efficient and independent of the real shape of the random structure in consideration. Therefore, it can be readily applied in a variety of stochastic finite element formulations to describe random material properties.

This paper discloses the connection between the spectral representation theory of wide‐sense stationary stochastic fields and the Karhunen‐Loève theorem of general second‐order stochastic fields, and obtains a Fourier‐Karhunen‐Loève representation for the former stochastic fields.