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Two methods of solving the inverse heat conduction problem with employment of the discrete Fournier transform are presented in this article. The first one operates similarly to the SVD algorithm and consists in reducing the number of components of the discrete Fournier transform which are taken into account to determine the solution to the inverse problem. The second method is related to the regularization of the solution to the inverse problem in the discrete Fournier transform domain. Those methods were illustrated by numerical examples. In the first example, an influence of the boundary conditions disturbance by a random error on the solution to the inverse problem (its stability) was examined. In the second example, the temperature distribution on the inner boundary of the multiply connected domain was determined. Results of calculations made in both ways brought very good outcomes and confirm the usefulness of applying the discrete Fournier transform to solving inverse problems. [ABSTRACT FROM AUTHOR]

New index transforms, involving the real part of the modified Bessel function of the first kind as the kernel are considered. Mapping properties such as the boundedness and invertibility are investigated for these operators in the Lebesgue spaces. Inversion theorems are proved. As an interesting application, a solution of the initial value problem for the second-order partial differential equation, involving the Laplacian, is obtained. It is noted that the corresponding operators with the imaginary part of the modified Bessel function of the first kind lead to the familiar Kontorovich–Lebedev transform and its inverse. [ABSTRACT FROM AUTHOR]

New index transforms are investigated, which contain as the kernel products of the Bessel and modified Bessel functions. Mapping properties and invertibility in Lebesgue spaces are studied for these operators. Relationships with the Kontorovich–Lebedev and Fourier cosine transforms are established. Inversion theorems are proved. As an application, a solution of the initial value problem for the fourth order partial differential equation, involving the Laplacian is presented. [ABSTRACT FROM AUTHOR]

In this paper, for real para-vector-valued signals, we obtain stronger uncertainty principles in terms of covariance and absolute covariance based on Fourier transform in both directional and the spatial cases. We provide certain conditions that give rise to the equal relation between the two uncertainty principles. Examples are presented to verify the results. [ABSTRACT FROM AUTHOR]

For the Fourier transformof a general (non-trivial) self-similar measure μ on the real line, we prove a large deviation estimate [ABSTRACT FROM PUBLISHER]

One-way coupling often occurs in multi-dimensional models in finance. In this paper, we present a dimension reduction technique for Monte Carlo (MC) methods, referred to as drMC, that exploits this structure for pricing plain-vanilla European options under an N-dimensional one-way coupled model, where N is arbitrary. The dimension reduction also often produces a significant variance reduction. The drMC method is a dimension reduction technique built upon (i) the conditional MC technique applied to one of the factors which does not depend on any other factors in the model, and (ii) the derivation of a closed-form solution to the conditional partial differential equation (PDE) that arises via Fourier transforms. In the drMC approach, the option price can be computed simply by taking the expectation of this closed-form solution. Hence, the approach results in a powerful dimension reduction from N to one, which often results in a significant variance reduction as well, since the variance associated with the otherfactors in the original model are completely removed from the drMC simulation. Moreover, under the drMC framework, hedging parameters, or Greeks, can be computed in a much more efficient way than in traditional MC techniques. A variance reduction analysis of the method is presented and numerical results illustrating the method’s efficiency are provided. [ABSTRACT FROM AUTHOR]

In this paper, Cauchy wavelet transform of ultra-distributions in tube domains is defined and its various properties are studied using the theory of Cauchy integrals and Poisson integrals of ultra-distributions. [ABSTRACT FROM PUBLISHER]

Starting from the classical summation formulas and basing on properties of the Mellin transform and Ramanujan's identities, which represent a ratio of products of Riemann's zeta functions of different arguments in terms of the Dirichlet series of arithmetic functions, we obtain a number of new summation formulas of the Poisson, Müntz, Möbius and Voronoi type. The corresponding analogues of the Müntz operators are investigated. Interesting and curious particular cases of summation formulas involving arithmetic functions are exhibited. Necessary and sufficient conditions for the validity of the Riemann hypothesis are derived. [ABSTRACT FROM PUBLISHER]

In this paper, a concentration identification problem of the time fractional inhomogeneous diffusion equation (TFIDE) is investigated. The TFIDE is obtained from the classical diffusion equation by replacing the first-order time derivative by the Caputo fractional derivative of order. Our purpose is to recover the solute concentration from source measurement and boundary data. We show that this problem is severely ill-posed and further apply a convolution-type regularization method to solve it. Convergence estimates are given under a priori bound assumptions for the exact solution. Finally, we provide two numerical examples to confirm our theoretical analysis and show that the corresponding numerical method works effectively. [ABSTRACT FROM PUBLISHER]

In their paper Barndorff-Nielsen et al. [4] employ so called ambit fields to model electricity spot-forward dynamics. We briefly introduce and discuss ambit fields, and introduce a novel method for approximating general ambit fields by a linear combination of ambit fields driven by exponential kernel functions (as has already been done in the null-spatial case of Lévy semistationary processes by Benth et al. [11]) by approximating the deterministic kernel function by a carefully selected finite sum. Moreover, we shall study examples, in the setting of modelling electricity forward markets, of ambit fields with singular kernel functions to illustrate the usefulness of our method for pricing purposes. [ABSTRACT FROM AUTHOR]

Letbe inand let,be a locally Lipschitz function with respect to the variableand. In this article, we consider the following backward heat problem with a nonlinear heat sourceThe problem is ill-posed in the sense of Hadamard. Hence a regularization is in order. We use the-dimensional Fourier transform to construct an approximate solution for this problem. We also prove error estimates for our problem and give comments on similar problems. [ABSTRACT FROM PUBLISHER]

This article treats the problem of introducing transform theory (Fourier, Laplace, z) to undergraduate students and we suggest avectorapproach which means that signals (functions of time) should be treated as vectors from the beginning and that transforms are introduced as a scalar product; the transform should be presented as a tool to analyse the signal exactly in the same way as the dot product is used to analyse an ‘arrow’ vector in a Cartesian space. Hence, the transform becomes a tool to find the signal's magnitude in the directions of the basis vectors. [ABSTRACT FROM PUBLISHER]

In this paper, we consider the Fourier-type functionals introduced in [Chung HS, Tuan VK. Fourier-type functionals on Wiener space. Bull Korean Math Soc. 2012;49:609–619] and used in [Lee IY, Chung HS, Chang SJ. Series expansions of the analytic Feynman integral for the Fourier-type functional. J Korean Soc Math Educ Ser B. 2012;19:87–102]. We first establish the existence of the transform with respect to the Gaussian process for the Fourier-type functionals. We then obtain the various relationships for the transform with respect to the Gaussian process of the Fourier-type functionals which involved the ⋄-product and in addition, establish various integration formulas. [ABSTRACT FROM AUTHOR]

In this work, we establish a Babenko–Beckner-type inequality for the Dunkl transform of a radial function and Dunkl transform associated to a finite reflection group generated by the sign changes. As applications, we establish a new version of Young’s-type inequality and uncertainty relation for Renyi entropy. [ABSTRACT FROM PUBLISHER]

This paper proposes a pricing method for path-dependent derivatives with discrete monitoring when an underlying asset price is driven by a time-changed Lévy process. The key to our method is to derive a backward recurrence relation for computing the multivariate characteristic function of the intertemporal joint distribution of the time-changed Lévy process. Using the derived representation of the characteristic function, we obtain semi-analytical pricing formulas for geometric Asian, forward start, barrier, fader and lookback options, all of which are discretely monitored. [ABSTRACT FROM AUTHOR]

We use real Paley–Wiener theorems to give a characterization of the functions whose Fourier transform has support in antipodal points, and apply the results to study Roe's theorem. [ABSTRACT FROM AUTHOR]

The theory of generalized magnetothermoelasticity is applied to a two-dimensional problem of a homogeneous isotropic perfectly conducting thick plate with heat source subjected to a time-dependent compression under constant magnetic and electric intensities. Double integral transforms (Laplace transform for time variable and Fourier transform for space variable) are used, and the resulting equations are written in the form of a vector-matrix differential equation, which was solved by the eigenvalue approach. The inversion of the Laplace transform is carried out numerically by the Zakian method. Finally, numerical computations of the displacement and stress components, temperature distribution, induced magnetic and electric fields have been done and are represented graphically. [ABSTRACT FROM AUTHOR]

The Wiener-Hopf factorization plays a crucial role in studying various mathematical problems. Unfortunately, in many situations, the Wiener-Hopf factorization cannot provide closed form solutions and one has to employ some approximation techniques to find its solutions. This article provides several Lp(R), 1 < p ≤ 2, approximation for a given Wiener- Hopf factorization problem. Application of our finding in spectral factorization and Lévy processes have been given. [ABSTRACT FROM AUTHOR]

A two-dimensional problem for a thermoelastic thick plate is considered within the context of the theory of generalized thermoelastic diffusion with one relaxation time. The upper surface of the thick plate is taken to be traction free and heated by a laser beam. The chemical potential is also assumed to be a known function of time on the bounding plane. The lower surface is taken to be in contact with a rigid surface and is thermally insulated and impermeable. Laplace and Fourier transform techniques are used. The solution in the transformed domain is obtained by a direct approach. Numerical inversion techniques are used to obtain the inverse double transforms; numerical results are discussed and represented graphically. [ABSTRACT FROM AUTHOR]

The problem involving scattering of oblique waves by small undulation on the porous ocean bed in a two-layer fluid is investigated within the framework of linearised theory of water waves where the upper layer is free to the atmosphere. In such a two-layer fluid, there exist waves with two different wave numbers (modes): wave with lower wave number propagates along the free surface whilst that with higher wave number propagates along the interface. When an oblique incident wave of a particular mode encounters the undulating bottom, it gets reflected and transmitted into waves of both modes so that some of the wave energy transferred from one mode to another mode. Perturbation analysis in conjunction with Fourier transform technique is used to derive the first-order corrections of velocity potentials, reflection and transmission coefficients at both modes due to oblique incident waves of both modes. One special type of undulating bottom topography is considered as an example to evaluate the related coefficients in detail. These coefficients are shown in graphical forms to demonstrate the transformation of water wave energy between the two modes. Comparisons between the present results with those in the literature are made for particular cases and the agreements are found to be satisfactory. In addition,energy identity, an important relation in the study of water wave theory, is derived with the help of the Green’s integral theorem. [ABSTRACT FROM PUBLISHER]

The solvability conditions of the over-determined boundary value problems for PDE with constant coefficients in the form of connection between Fourier transforms of boundary functions are obtained. The jump problem on the hyperplane for PDE is formulated and investigated. Two-dimensional Helmholtz equation is considered as an example. It is proved that the mixed type conjugation problem for Helmholtz equation, to which the diffraction problem of the electromagnetic wave on thin conducting ribbons is reduced, is equivalent to one-sided mixed type boundary value problem. The uniqueness theorem is proved for this problem. [ABSTRACT FROM AUTHOR]

We are interested in estimating a density function f of i.i.d. random variablesX1, …,Xnfrom the modelYj=Xj+Zj, whereZjare unobserved error random variables, distributed with the density function g and independent ofXj. This problem is known as the deconvolution problem in nonparametric statistics. The most popular method of solving the problem is the kernel one in which, we assumegft(t) ≠ 0, for all, wheregft(t) is the Fourier transform of g. The more general case in whichgft(t) may have real zeros has not been considered much. In this article, we will consider this case. By estimating the Lebesgue measure of the low level sets ofgftand combining with the Tikhonov regularization method, we give an approximationfnto the density function f and evaluate the rate of convergence of. A lower bound for this quantity is also provided. [ABSTRACT FROM PUBLISHER]

We introduce the translation operators and convolution product connected with some singular partial differential operators defined on the half plane [0,+∞[×ℝ. We investigate Calderon's reproducing formula associated with the convolution * involving finite Borel measures, leading to results onLp-norm for functions on half plane. [ABSTRACT FROM PUBLISHER]

An opening crack model is proposed by considering the effects of various physical properties inside a crack of thermomagnetoelectroelastic solids in the present article. The thermal flux, electric displacement, and magnetic induction at the crack surfaces are assumed to be dependent on crack opening displacement. Applying the Fourier transform technique, the thermal stress intensity factor for an opening Griffith crack in thermomagnetoelectroelastic solids is given in the explicit form. The effects of applied magnetoelectromechanical loadings and the physical properties of crack interior on the normalized thermal stress intensity factor are investigated. The obtained results reveal that the thermal stress intensity factor depends not only on applied thermal loadings but also on applied magnetoelectromechanical loadings and various physical properties of crack interior for a crack filled with a medium. [ABSTRACT FROM AUTHOR]

This paper considers the boundary-value inverse heat conduction problem with steady boundary. To solve this problem, different approaches based on the Laplace and Fourier transforms are proposed. Application of the Laplace transform makes it possible to obtain an operator equation describing the explicit dependence of the desired boundary-value function on the initial data at the other boundary. This approach to solving the problem is used for the first time. The method based on the direct and inverse Fourier transforms with respect to the time variable provides stable solutions. The estimation of error of these solutions is the best with respect to the order. The range of application of each of the proposed methods and the stability of the solutions obtained by these methods was evaluated by a computational experiment. [ABSTRACT FROM AUTHOR]

In this paper, we present a nonparametric estimator for ruin probability in the classical risk model with unknown claim size distribution. We construct the estimator by Fourier inversion and kernel density estimation method. Under some conditions imposed on the kernel, bandwidth and claim size density, we present some large sample properties of the estimator. Some simulation studies are also given to show the finite sample performance of the estimator. [ABSTRACT FROM AUTHOR]

This paper investigates the implementation of Field Dislocation Mechanics (FDM) theory for media with a periodic microstructure (i.e. the Nye dislocation tensor and the elastic moduli tensor are considered as spatially periodic continuous fields). In this context, the uniqueness of the stress and elastic distortion fields is established. This allows to propose an efficient numerical scheme based on Fourier transform to compute the internal stress field, for a given spatial distribution of dislocations and applied macroscopic stress. This numerical implementation is assessed by comparison with analytical solutions for homogeneous as well as heterogeneous elastic media. A particular insight is given to the critical case of stress-free dislocation microstructures which represent equilibrium solutions of the FDM theory. [ABSTRACT FROM AUTHOR]

We derive a new, efficient closed-form formula approximating the price of discrete lookback options, whose underlying asset price is driven by an exponential semimartingale process, which includes ( jump) diffusions, Lévy models, affine processes and other models. The derivation of our pricing formula is based on inverting the Fourier transform using B-spline approximation theory. We give an error bound for our formula and establish its fast rate of convergence to the true price. Our method provides lookback option prices across the quantum of strike prices with greater efficiency than for a single strike price under existing methods. We provide an alternative proof to the Spitzer formula for the characteristic function of the maximum of a discretely observed stochastic process, which yields a numerically efficient algorithm based on convolutions. This is an important result which could have a wide range of applications in which the Spitzer formula is utilized. We illustrate the numerical efficiency of our algorithm by applying it in pricing fixed and floating discrete lookback options under Brownian motion, jump diffusion models, and the variance gamma process. [ABSTRACT FROM AUTHOR]

This paper extends the integral transform approach of McKean [Ind. Manage. Rev., 1965,6, 32–39] and Chiarella and Ziogas [J. Econ. Dyn. Control, 2005,29, 229–263] to the pricing of American options written on more than one underlying asset under the Black and Scholes [J. Polit. Econ., 1973,81, 637–659] framework. A bivariate transition density function of the two underlying stochastic processes is derived by solving the associated backward Kolmogorov partial differential equation. Fourier transform techniques are used to transform the partial differential equation to a corresponding ordinary differential equation whose solution can be readily found by using the integrating factor method. An integral expression of the American option written on any two assets is then obtained by applying Duhamel’s principle. A numerical algorithm for calculating American spread call option prices is given as an example, with the corresponding early exercise boundaries approximated by linear functions. Numerical results are presented and comparisons made with other alternative approaches. [ABSTRACT FROM PUBLISHER]

A new global optimisation strategy in frequency domain (GOFD) is presented and applied in medical image elastic registration. The method is consists of a global optimisation phase for rough searching and a local optimisation phase for fine searching. Rough searching is based on the random sampling technique in the frequency domain. According to the sampling theory, when the sampling frequency is higher than twice the maximum frequency of a function, the function can be completely reconstructed from these finite sampling points. The maximum (or minimum) value of the function at these finite sampling points is approximately in the global extreme. To obtain the exact global extreme, fine searching is performed in the small neighbourhood of the point corresponding to the approximate global maximum value. The new method presented can theoretically ensure that the global optimisation solution is found. The experiments show that our new method is more robust and accurate than other elastic registration algorithms. [ABSTRACT FROM AUTHOR]

The adoption of a angular discretization scheme in the solution of the linear transport equation is introducing a wide range of distortions associated with the propagation of particles along discrete directions, usually called ray effects. These numerical artifacts and the consequent distortion of the physical description are observed in multidimensional configurations in steady-state and also in the one-dimensional case when the time-dependence is considered. Through the analysis of simplified configurations, this work aims at a characterization of the ray effect phenomenon. Steady-state and time-dependent problems are solved analytically and then compared to the results obtained by applying discrete ordinates, clearly isolating ray effects from other numerical effects and allowing the quantification of the physical distortion. The analysis of the transport kernel in the Fourier-transformed space permits the identification of a proper integral parameter, called ray effect indicator, to measure the distortion introduced in the transport model by the angular discretization. The efficiency of alternative schemes to the standard SN approach for the reduction of ray effects is also discussed. [ABSTRACT FROM AUTHOR]

In this paper we obtain solutions in closed form for some special cases of the Toeplitz plus Hankel integral equation on the whole real line with the help of a new Hartley–Fourier cosine generalized convolution. [ABSTRACT FROM AUTHOR]

In this paper, we generalize the theory of variation diminishing kernels introduced by Hirschman, Jr and van de Wetering for a class of singular second-order differential operators introduced by Chébli and Trimèche. We give a necessary and sufficient condition such that a function Λ is a variation diminishing convolution kernel. [ABSTRACT FROM PUBLISHER]

Pseudospectral simulations of homogeneous turbulence provide an important class of benchmark flow problems used for fundamental studies of turbulence and for numerical validation purposes. Depending on the numerical resolution, fully resolved computations of homogeneous turbulence can consume large amounts of central processing unit (CPU) time. Here, we present an approach analogous to adaptive mesh refinement for computations performed in physical space to adaptively refine the spectral resolution for pseudospectral computations of isotropic homogeneous turbulent flows. The method is applied to simulations of two-dimensional and three-dimensional isotropic homogeneous turbulence, and the results are compared with direct numerical simulations (DNS) performed using a fixed fine mesh. Significant savings in computational time are found in each case, with little to no compromise in the accuracy of the solutions. [ABSTRACT FROM AUTHOR]

By using Fourier transforms of two symmetric sequences of finite orthogonal polynomials, we introduce two new classes of finite orthogonal functions and obtain their orthogonality relations via Parseval's identity. [ABSTRACT FROM PUBLISHER]

In this study, a subwavelength aperture antenna with multiple corrugations and a superstrate layer was investigated based on Fourier transform and mode matching, demonstrating a rigorous and fast convergent series solution. The aperture antenna with a corrugated conducting flange and a dielectric superstrate was fabricated in the X-band frequency range to demonstrate the validity of the derived series solution. The measured reflection coefficient and radiation pattern of the proposed antenna are also presented and compared to the computational results to assess the effect of the dielectric superstrate. A maximum gain enhancement of 5.19 dB was achieved by employing the superstrate. [ABSTRACT FROM AUTHOR]

In this paper, we construct and study a new polyconvolution for the Hartley, Fourier cosine and Fourier sine transforms. Here, it is the first time that four different integral transforms appear in the factorization equalities. Next, we get the norm inequalities for this polyconvolution operator in the function spacesL1(ℝ) andLp(ℝ). The applications in solving integral equations and systems of two integral equations are presented. [ABSTRACT FROM AUTHOR]

We define convolution of two distributions in the spaces, in two ways, which turn out to be equivalent, and show that the space is closed under convolution. We also define the Fourier transform of elements of, and prove the Parseval equality. [ABSTRACT FROM AUTHOR]

We consider the numerical solution, in a three-dimensional bounded domain, of the inverse problem for identifying the location of small electromagnetic imperfections in a medium with homogeneous background. Our numerical algorithm is based on the coupling of a discontinuous Galerkin method for the time-dependent Maxwell's equations, on the exact controllability method and on a Fourier inversion. Several numerical results are given with one and two imperfections and the robustness and accuracy of the numerical method used for the dynamic detection problem are shown. [ABSTRACT FROM PUBLISHER]

In this paper, we show that slight modifications of the Kontorovich–Lebedev (KL) transform lead to an automorphism of the vector space of polynomials. This circumstance along with the Mellin transformation property of the modified Bessel functions perform the passage of monomials to central factorial polynomials. A special attention is driven to the polynomial sequences whose KL transform is the canonical sequence, which will be fully characterized. Finally, new identities between the central factorials and the Euler polynomials are found. [ABSTRACT FROM PUBLISHER]

We present an extension of our recent examination of the method of moments when moments are computed on a spatial grid instead of through integration over the entire domain to include time dependence. For the problem we consider, we show that (i) the flux-weighted average ofxremains equal to its initial value but (ii) the flux-weighted average of (x−xa)2differs from its initial value by an additional error term when moments are determined in this manner, wherexis the spatial variable andxais an arbitrary point. We also present numerical examples that confirm the accuracy of our results. [ABSTRACT FROM PUBLISHER]

The method of moments is a well-known technique for determining exact expressions for spatial and angular moments of radiation distributions in infinite, homogeneous media. These moments can further be used to calculate other quantities of interest. We examine how the method of moments is altered when the underlying transport problem is spatially continuous but involves a grid and moments are computed on this grid instead of through integration over the entire domain. For the problem we consider, we employ both singular-eigenfunction and Fourier-transform approaches to show that when moments are evaluated in this manner (i) the flux-weighted average of x remains equal to the source-weighted average of x, but (ii) the flux-weighted average of (x−xa )2 is greater than the source-weighted average of (x−xa )2 by an additional error term, where x is the spatial variable and xa is an arbitrary point. We also demonstrate that the two resulting expressions for this error term are equivalent. [ABSTRACT FROM PUBLISHER]

In the present study, we consider the problem of classifying spatial data distorted by a linear transformation or convolution and contaminated by additive random noise. In this setting, we show that classifier performance can be improved if we carefully invert the data before the classifier is applied. However, the inverse transformation is not constructed so as to recover the original signal, and in fact, we show that taking the latter approach is generally inadvisable. We introduce a fully data-driven procedure based on cross-validation, and use several classifiers to illustrate numerical properties of our approach. Theoretical arguments are given in support of our claims. Our procedure is applied to data generated by light detection and ranging (Lidar) technology, where we improve on earlier approaches to classifying aerosols. This article has supplementary materials online. [ABSTRACT FROM PUBLISHER]

Quaternion-valued functions have been used as a model for colour images and have recently been studied using various Fourier-type transforms. We develop some fundamental wavelet theory for quaternionic signals using the Fourier kernel introduced by Brackx, De Schepper, and Sommen in [1]. We present several analogs to classical wavelet theory, such as the quadrature mirror filter condition. We also include necessary design conditions for a wavelet basis to have desired regularity and sufficient design conditions which will guarantee compact support. Due to the difficulty in constructing orthogonal wavelet bases, we present some theory for biorthogonal wavelet bases and construct a few examples. [ABSTRACT FROM AUTHOR]

Phase characterization with a good spatial resolution is crucial for focused beams in nonlinear media. The phase-shifting interferometry technique, using the least-squares error criterion for several interferograms, is implemented using a reflective spatial light modulator (SLM). The method provides a convenient calibration for any phase-shift steps. The reliability of the proposed method is checked by direct comparison with results obtained by the Fourier transform method as well as using a previously characterized circular phase object. [ABSTRACT FROM AUTHOR]

Two periodogram-like functions, called quantile periodograms, are introduced for spectral analysis of time series. The quantile periodograms are constructed from trigonometric quantile regression and motivated by different interpretations of the ordinary periodogram. Analytical and numerical results demonstrate the capability of the quantile periodograms for detecting hidden periodicity in the quantiles and for providing an additional view of time-series data. A connection between the quantile periodograms and the so-called level-crossing spectrum is established through an asymptotic analysis. [ABSTRACT FROM AUTHOR]

The goal of this article is to model cognitive control related activation among predefined regions of interest (ROIs) of the human brain while properly adjusting for the underlying spatio-temporal correlations. Standard approaches to fMRI analysis do not simultaneously take into account both the spatial and temporal correlations that are prevalent in fMRI data. This is primarily due to the computational complexity of estimating the spatio-temporal covariance matrix. More specifically, they do not take into account multiscale spatial correlation (between-ROIs and within-ROI). To address these limitations, we propose a spatio-spectral mixed-effects model. Working in the spectral domain simplifies the temporal covariance structure because the Fourier coefficients are approximately uncorrelated across frequencies. Additionally, by incorporating voxel-specific and ROI-specific random effects, the model is able to capture the multiscale spatial covariance structure: distance-dependent local correlation (within an ROI), and distance-independent global correlation (between-ROIs). Building on existing theory on linear mixed-effects models to conduct estimation and inference, we applied our model to fMRI data to study activation in prespecified ROIs in the prefontal cortex and estimate the correlation structure in the network. Simulation studies demonstrate that ignoring the multiscale correlation leads to higher false positive error rates. [ABSTRACT FROM AUTHOR]

Although research involving economic time series forecasting based on virtual market models is frequently conducted, long-term forecasting is difficult due to many factors that affect actual markets. However, as exemplified by the business cycle and Elliot Wave theories in economics, it is assumed that fluctuations in economic time series forecasting have various periodicities, ranging from short-term to long-term. Accordingly, we used a new high-resolution frequency analysis (Non-Harmonic Analysis (NHA)) method, which we have recently developed, to conduct analysis of the periodicity of economic time series forecasting. We also attempted a long-term economic time series forecast by combining multiple periodic signals. In the verification experiment, we analysed the National Association of Securities Dealers Automated Quotations (NASDAQ) closing price data for a time period of approximately 20 years using nonharmonic analysis with an analysis window of the previous 2 years, and forecasted price fluctuations for the following 2 years. [ABSTRACT FROM AUTHOR]

The Fourier exchange theorem is generalized to 𝒮′-convolvable kernels, i.e. temperate distributions K(x, y) ∈ 𝒮′(ℝ2n ) for which K(x − y, y) is semitemperate with respect to x and partially summable with respect to y. [ABSTRACT FROM AUTHOR]

In this paper, we obtain solutions in closed form for some special cases of the Toeplitz plus Hankel integral equation with the help of generalized convolutions. [ABSTRACT FROM AUTHOR]