Search

We view the (real) Laplace transform through the lens of linear algebra as a continuous analogue of the power series by a negative exponential transformation that switches the basis of power functions to the basis of exponential functions. This approach immediately points to how the complex Laplace transform is a generalisation of the Fourier transform where the pole of the transform realises the linear algebraic intuition. The exponential transformation also motivates the Taylor inversion of the real Laplace transform. [ABSTRACT FROM AUTHOR]

In this paper, we derive some Fourier transforms of confluent hypergeometric functions. We give generalizations of several well-known results involving Fourier transforms of gamma functions. In particular, the generalizations include some Ramanujan's remarkable formulas. [ABSTRACT FROM AUTHOR]

AbstractHigh-throughput mechanical testing based on functionally graded specimens is very promising for accelerating the development of new materials. However, due to the inhomogeneity-induced complexity, most existing analyses on functionally graded materials have recourse to numerical methods to predict their mechanical responses in reaction to external stimuli. This work investigates the surface Green’s functions for an inhomogeneous half-plane with horizontal exponential material gradient subject to both normal and tangential concentrated forces acting on the surface. The governing equations are first simplified by introducing appropriate potential functions, which facilitates the mathematical derivation of displacements via the Fourier transform technique. In the case of normal force, the vertical surface displacement is derived explicitly under the weak gradient assumption while the horizontal surface displacement is derived directly without the same assumption. In particular, the Meijer G-function and Fox H-function are introduced to express and simplify the vertical displacement. In the case of tangential force, the analytical expressions of surface displacements are also derived similarly. It is noted that the surface Green’s functions not only exhibit singularity and asymmetry properties as expected, but also can be reduced to the classical Boussinesq-Flamant solutions for a homogeneous half-plane. In addition, the analytical results are verified through comparison with the finite element analyses. The surface Green’s functions derived here could be a theoretical basis for developing high-throughput mechanical testing methods which use specimens made of functionally graded materials. [ABSTRACT FROM AUTHOR]

Gas turbines play a vital role in gas transportation and power generation, but they are prone to instability phenomena that can lead to vibrations, shorten equipment lifespan, and result in catastrophic failures. To tackle these challenges, a paper introduces an integrated approach that leverages advanced techniques like Fourier transform, Neuro-Fuzzy systems, and wavelet analysis for continuous monitoring of the MS5002C turbine's condition. The proposed method begins by collecting operational data and utilizing the Fourier transform to measure vibratory quantities, accurately representing their evolution through spectral data obtained from the analyzed signals. Adaptive inference-based algorithms of neuro-fuzzy systems are then employed to generate turbine failure indicators. This approach enables the development of a model-based fault detection method that compares the actual turbine operation with the estimated operation derived from a pre-established model, enabling the classification of detected faults. To enhance decision-making quality, evaluation, and validation of the diagnostic strategy's performance, a multi-resolution analysis based on the wavelet transform is applied. The presented results from various implementation and validation tests demonstrate the effectiveness of this intelligent diagnostic approach in detecting and analyzing gas turbine vibrations. The paper exhibits promising outcomes in real-time monitoring, ensuring the operational safety of the turbine. [ABSTRACT FROM AUTHOR]

In this paper, two special-shaped separated heat pipes with different condensation section structures are designed. The working medium is water, and the filling rate is 65%. The evaporation section is placed in the hot water tank, and the condensation section is equipped with a cold water jacket. The effects of different structures of condensing section on the thermal performance of separated heat pipes are experimentally studied, and the temperature changes of the two separated heat pipes during start-up and stable operation under different heating power are analyzed. The results show that the heat transfer characteristics of the two separated heat pipes are different due to the different structures of the condensation section, and the heat transfer capacity of sample 2 is higher. At the same time, the temperature fluctuation in the condensing section of sample 2 is also quite different from that in the condensing section of sample 1, and the temperature fluctuation in the condensing section of sample 2 is greater. The wall temperature fluctuates periodically in a steady-state, and a fast Fourier transform is used to analyze the temperature fluctuation. [ABSTRACT FROM AUTHOR]

The state-space formulation for two-dimensional generalized thermoviscoelasticity has been formulated. In this formulation, the governing equations are transformed into a matrix equation whose solution enables us to write the solution of any two-dimensional problem in terms of the boundary conditions. The resulting formulation is applied to a 2 D problem of a half-space subjected to harmonically varying heating with constant angular frequency. [ABSTRACT FROM AUTHOR]

The delineation of the linear/nonlinear response of bituminous material in the frequency domain is conducted using the small-amplitude oscillatory shear test, wherein only the peak stress and strain are available. In contrast, the large-amplitude oscillatory shear tests provide complete waveform data. Analysis of such data can help to unravel the linear/nonlinear response with greater accuracy. This investigation reports such an exercise. An experimental investigation is conducted on five different materials at different strain amplitudes, frequencies, and temperatures. The geometry-based parameters obtained from the Lissajous plot gave contradictory linear/nonlinear delineation. Analysis of the higher order harmonics did not show any appreciable presence of nonlinearity. However, the orientation of the Lissajous plots differed from the strain amplitude. The elastic and viscous stress could be predicted using the first-order Chebyshev coefficients indicating the linear response, but the coefficients differed with the strain amplitude. The scaling-superposition of the waveform revealed a clear demarcation of the linear/nonlinear response. This indicated that the response of the bituminous material needs to be analysed for a range of frequency and amplitude. The mere analysis of the waveform data at any test condition can help only to establish the necessary condition for linearity but not a sufficient condition. [ABSTRACT FROM AUTHOR]

Calcium is an essential element in our body and plays a vital role in moderating calcium signalling. Calcium is also called the second messenger. Calcium signalling depends on cytosolic calcium concentration. In this study, we focus on cellular calcium fluctuations with different buffers, including calcium-binding buffers, using the Hilfer fractional advection-diffusion equation for cellular calcium. Limits and start conditions are also set. By combining with intracellular free calcium ions, buffers reduce the cytosolic calcium concentration. The buffer depletes cellular calcium and protects against toxicity. Association, dissociation, diffusion, and buffer concentration are modelled. The solution of the Hilfer fractional calcium model is achieved through utilizing the integral transform technique. To investigate the influence of the buffer on the calcium concentration distribution, simulations are done in MATLAB 21. The results show that the modified calcium model is a function of time, position, and the Hilfer fractional derivative. Thus the modified Hilfer calcium model provides a richer physical explanation than the classical calcium model. [ABSTRACT FROM AUTHOR]

This paper considers D-optimal designs for linear mixed models involving random effects with unknown distributions. From Bayesian point of view, the Dirichlet process as a prior distribution on the space of all distributions is used. Based on the Dirichlet process as a prior, we give the Bayes estimate of the density function of the response variable, which result in a mixture of two normal distributions. An explicit form of the Fisher information matrix for the proposed model is derived by using the Fourier transform and then D-optimal design is obtained by numerical calculations. [ABSTRACT FROM AUTHOR]

A neural network called sinusoidal decomposition neural network (SDNN) is proposed to reconstruct the digital elevation model (DEM) and orbital velocity field (OVF) of sea surface. According to the linear wave theory, DEM can be regarded as the superposition of a series of sine waves, from which OVF can be obtained. The SDNN adopts a fully connected network (FCN) to fit the DEM, which is similar to the inverse discrete Fourier transform (IDFT) model and regression model. The two-dimensional and three-dimensional SDNN are introduced in detail and their validities are demonstrated. A major advantage of the SDNN is that it requires only one scene of the wave height to reconstruct the OVF. By the applications to wind-driven sea surface and ship wake, respectively, the correctness and efficiency of the reconstruction are verified. [ABSTRACT FROM AUTHOR]

Currently, intelligent systems built on a multimodal basis are used to study the functional state of living objects. Its essence lies in the fact that a decision is made through several independent information channels with the subsequent aggregation of these decisions. The method of forming descriptors for classifiers of the functional state of the respiratory system includes the study of the spectral range of the respiratory rhythm and the construction of the wavelet plane of the monitoring electrocardiosignal overlapping this range. Then, variations in the breathing rhythm are determined along the corresponding lines of the wavelet plane. Its analysis makes it possible to select slow waves corresponding to the breathing rhythm and systemic waves of the second order. Analysis of the spectral characteristics of these waves makes it possible to form a space of informative features for classifiers of the functional state of the respiratory system. To construct classifiers of the functional state of the respiratory system, hierarchical classifiers were used. As an example, we took a group of patients with pneumonia with a well-defined diagnosis (radiography, X-ray tomography, laboratory data) and a group of volunteers without pulmonary pathology. The diagnostic sensitivity of the obtained classifier was 76% specificity with a diagnostic specificity of 82%, which is comparable to the results of X-ray studies. It is shown that the corresponding lines of the wavelet planes are correlated with the respiratory system and, using their Fourier analysis, descriptors can be obtained for training neural network classifiers of the functional state of the respiratory system. [ABSTRACT FROM AUTHOR]

We study approximation properties of multi-resolution analyses in the context of matrix-valued function spaces. Here, we generalize the notions of approximation order and density order given by the reference [de Boor C, DeVore RA, Ron A. Approximation from shift-invariant subspaces of L 2 (R d). Trans Am Math Soc. 1994;341(2):787–806]. Indeed, we prove a characterization of the closed subspaces generated by the shifts of a single matrix-valued function that provide approximation order and/or density order α ≥ 0. To give our conditions, we need the classical notion of approximate continuity. As a consequence, we prove necessary and sufficient conditions on a matrix-valued function to be a scaling function in a multi-resolution analysis. [ABSTRACT FROM AUTHOR]

Abstract The mathematical theory of elasticity helps in the study of physical quantities in the problem of structures. The structures face the problem of crack’s presence, which makes the problem difficult but not impossible to deal. Integral equations are useful in a variety of problems. Integral equations are used to solve problems like fracture mechanics or fracture design. The physical interest in the fracture design criterion is due to stress and crack opening displacement components. We have an accurate form of stress and displacement components for a pair of longitudinal crack propagations in the bone fracture of the human body at the interface of an isotropic and orthotropic half-space that are bounded together in the proposed study. The expression was calculated using the Fourier transform approach near the crack tips, but these components were evaluated using Fredholm integral equations and subsequently reduced to coupled Fredholm integral equations. We employ the Lowengrub and Sneddon problem in this research and reduce it to triple integral equations. The Srivastava and Lowengrub method reduces the solution of these equations to a coupled Fredholm integral equation. The problem is further reduced to a decoupled Fredholm integral equation of the second kind. Triple integral equations are solved, and the problem is reduced to a coupled Fredholm integral equation. The Fredholm integral equation is solved and reduced to a decoupled Fredholm integral equation of the second kind. Stress and crack opening displacement components drive physical interest in fracture design criteria. Finally, the stress and displacement components may be simply calculated in their exact form. [ABSTRACT FROM AUTHOR]

This paper presents a comprehensive study on developing a novel mathematical model of generalized thermoelasticity to investigate the transient phenomena for an infinite thermoelastic half-space under the action of ramp-type thermal loading and due to the influence of a gravitational field. The bounding plane of the half-space is subjected to rough and rigid foundation so that the rough surface prevents the vertical displacement. Due to the shortcomings of power-law distributions, some other forms of derivatives with few other kernel functions are proposed. The present analysis deals with the heat transport which involves the memory-dependent derivative (MDD) on a slipping interval in the context of Lord–Shulman model to describe the physical phenomena which is defined in the form of convolution with the kernels in the form of power functions. Employing the Laplace and the Fourier transform techniques as tools, the analytical expressions for different physical fields have been obtained on the transformed domain. The numerical inversion of the Fourier transforms have been performed analytically, whereas numerical inversion of the Laplace transform is carried out using the Riemann-sum approximation method. Excellent predictive capability is demonstrated due to the presence of MDD, delay time and gravitational field also. [ABSTRACT FROM AUTHOR]

In this paper, sampling Theorems for perfect signal reconstruction using samples of a signal taken below the Nyquist rate involving nonlinear signal reconstruction technique are presented. The presented results are in contrast with the linear time-invariant filtering based signal reconstruction in the celebrated Shannon sampling theory. It is shown that using kth order nonlinearity in the signal reconstruction system, the required sampling rate for perfect signal reconstruction can be reduced by the same factor k for positive odd values of it. The extensions of the proposed sampling expansion to the fractional Fourier transform and linear canonical transform domains are also derived. Simulation results of the proposed technique are given to validate the proposed approach. [ABSTRACT FROM AUTHOR]

The main concern of this article is to deal with the thermoelastic interaction in a functionally graded thermoelastic rod being enlightened by the memory-dependent derivative. This article investigates the transient phenomena due to the influence of an induced magnetic field of constant intensity and due to the presence of a moving heat source of constant velocity in the context of three-phase lag model of generalized thermoelasticity. Employing the Laplace and the Fourier transforms as tool, the problem has been constructed in the transformed domain. The inversions of the Fourier transform have been carried out using residual calculus whereas the numerical inversions of the Laplace transform have been performed employing the Riemann sum approximation method. Numerical computations for stress, displacement, and temperature within the rod are carried out and have been demonstrated graphically. The results also demonstrate how the nonhomogeneity parameter and the speed of the moving heat source influence the thermophysical quantities. It is observed that the temperature, thermally induced displacement, and stress of the rod are found to decrease at large source speed. Also, significant differences on the thermophysical quantities are revealed due to the influence of magnetic field, nonhomogeneity, and memory effect also. [ABSTRACT FROM AUTHOR]

In this paper, we deduct some properties of the Fourier transform on arbitrary time scales. We define the generalized shifting problem and we prove the existence of solutions. We define a generalized convolution on anarbitrary time scale and we deduct and prove the generalized convolution theorem. [ABSTRACT FROM AUTHOR]

Abstract A wide variety of methods have been proposed for clustering of stochastic processes. However, for clustering of periodically correlated processes (PC) it is demanding to introduce some similarity measures that take into account the inherent periodicity of these processes. The frequency-domain based methods seem more desirable to determine groups of PC processes with similar frequency characterizations. In this article, we present new similarity measures based on Hilbert-Schmidt inner product of finite Fourier transforms of PC processes. Based on simulated stochastic processes and a real gene expression dataset we illustrate the accuracy of the methods. [ABSTRACT FROM AUTHOR]

This paper presents the three-dimensional analysis of the stress distribution arising in an isotropic infinite slab with a triaxial ellipsoidal cavity, the surface of which is subjected to the three principal stresses σ 1 , σ 2 , and σ 3 . To satisfy both boundary conditions on the surface of slab and the cavity, harmonic functions in rectangular coordinates are used and double Fourier transform is applied. The problem is reduced to the solution of three integro-differential equations. The existence and uniqueness of the solution is investigated. [ABSTRACT FROM AUTHOR]

Differential privacy mechanisms vary in modalities, and there have been many methods implementing differential privacy on unimodal data. Few studies focus on unifying them to protect multimodal data, though privacy protection of multimodal data is of great significance. In our work, we propose a multimodal differential privacy protection framework. Firstly, we use multimodal representation learning to fuse different modalities and map them to the same subspace. Then based on this representation, we use the Local Differential Privacy (LDP) mechanism to protect data. We propose two protection methods for low-dimensional and high-dimensional fusion tensors respectively. The former is based on Binary Encoding, and the latter is based on multi-dimensional Fourier Transform. To the best of our knowledge, we are the first to propose LDP-based methods for the representation learning of multimodal fusion. Experimental results demonstrate the flexibility of our framework where both approaches show efficient performance as well as high data utility. [ABSTRACT FROM AUTHOR]

The dynamic responses of an unsaturated ground with or without an embankment are analysed using 2.5D finite element method (FEM) to investigate the effects of the soil saturation and the embankment. The Galerkin method is utilised to establish the weak form governing equations for an unsaturated poroviscoelastic medium in the framework of 2.5D FEM. To reduce the reflected waves from the truncated boundaries of the 2.5D FE model, the formulations of the perfectly match layer (PML) technique are coupled with the governing equations of the unsaturated medium. The unsaturated ground vibrations with an embankment under moving loads are analysed by the proposed 2.5D FE approach, to investigate the influences of an embankment on unsaturated ground vibrations. The results of this research found that the PML technique can effectively reduce the reflected waves in the 2.5D FE model of unsaturated grounds. The increased soil saturation would increase the ground vibration amplitude, as well as the pore water pressure. Constructing embankment on the unsaturated ground could reduce the vibrations amplitude under the moving load, however increase the ground vibrations at the foot of the embankment. [ABSTRACT FROM AUTHOR]

The high-power hydraulic rock drill is the key to the development of modern hydraulic drill rig. With the increase of power and frequency of new type rock drill, the rebound of drill tool will be obvious during drilling, which will seriously affect the structural safety and operation efficiency. For this phenomenon, it is necessary to research the characteristics of double damping system to absorb the rebound energy of drill tool. A new model of incident stress wave is revised on the basis of rectangular wave. By combining the reflected stress wave with the model of damping system, the pressure fluctuation of damping system is analysed. Based on the theory of stress wave transmission, the rebound model of drill tool is established, and the transmission law of incident wave and reflection wave in drilling process is analysed. The stress wave experiment was designed and the incident wave shape was obtained by testing. Based on the experimental results, the incident wave in the rebound model is corrected by Fourier transform principle. The accumulator model and the double damping internal structure model are established. The two parameters that can independently change the energy absorption effect of the double damping system are summarised. [ABSTRACT FROM AUTHOR]

Let X be a ball quasi-Banach function space on R n and H X (R n) the associated Hardy space. In this article, under the assumptions that the Hardy–Littlewood maximal operator satisfies some Fefferman–Stein vector-valued inequality on X and is bounded on the associated space of X as well as under a lower bound assumption on the X-quasi-norm of the characteristic function of balls, the authors show that the Fourier transform of f ∈ H X (R n) coincides with a continuous function g on R n in the sense of tempered distributions and obtain a pointwise inequality about g and the Hardy space norm of f. Applying this, the authors further conclude a higher order convergence of the continuous function g at the origin and then give a variant of the Hardy–Littlewood inequality in the setting of Hardy spaces associated with X. All these results have a wide range of applications. Particularly, the authors apply these results, respectively, to mixed-norm Lebesgue spaces, variable Lebesgue spaces, and Orlicz spaces. Even in these special cases, the obtained results for variable Hardy spaces and Orlicz–Hardy spaces are totally new. [ABSTRACT FROM AUTHOR]

The measurement of spatial dependence within a set of observations or the residuals from a regression is one of the most common operations within spatial analysis. However, there appears to be a lack of appreciation for the fact that these measurements are generally based on an a priori definition of a spatial weights matrix and hence are limited to detecting spatial dependence at a single spatial scale. This paper highlights the scale-dependence problem with current measures of spatial dependence and defines a new, multi-scale approach to defining a spatial weights matrix based on a discrete Fourier transform. This approach is shown to be able to detect statistically significant spatial dependence which other multi-scale approaches to measuring spatial dependence cannot. The paper thus serves as a warning not to rely on traditional measures of spatial dependence and offers a more comprehensive, and scale-free, approach to measuring such dependence. [ABSTRACT FROM AUTHOR]

Ground surface settlement (GSS) is one of the major concerns in design and construction of a deep excavation. This paper proposes an analytical approach for the prediction of GSS induced by a braced deep excavation. Considering wall deformation and stress release, the problem investigated is formulated as a system of two second-order partial differential equations (Lame equations) with mixed boundary conditions based on the elastic theory. Taking advantage of the superposition principle, the mixed boundary conditions are decomposed into displacement and stress boundary. The separation of variables method is applied to solve the governing equations with displacement boundary, while the Fourier Transform Method is employed to derive the solution for the governing equations with stress boundary. A novel least-squares based method is proposed to transform the scatter data of wall deflection into a continuous function, which is used to determine the unknown coefficients in the solution. The validity of the proposed solution is checked by predicting the GSS of two well-documented cases and by comparing with some empirical approaches. Parametric studies are conducted to demonstrate the impact of the modulus ratio on the excavation responses. [ABSTRACT FROM AUTHOR]

Differential privacy mechanisms vary in modalities, and there have been many methods implementing differential privacy on unimodal data. Few studies focus on unifying them to protect multimodal data, though privacy protection of multimodal data is of great significance. In our work, we propose a multimodal differential privacy protection framework. Firstly, we use multimodal representation learning to fuse different modalities and map them to the same subspace. Then based on this representation, we use the Local Differential Privacy (LDP) mechanism to protect data. We propose two protection methods for low-dimensional and high-dimensional fusion tensors respectively. The former is based on Binary Encoding, and the latter is based on multi-dimensional Fourier Transform. To the best of our knowledge, we are the first to propose LDP-based methods for the representation learning of multimodal fusion. Experimental results demonstrate the flexibility of our framework where both approaches show efficient performance as well as high data utility. [ABSTRACT FROM AUTHOR]

In this article, we non-trivially prove the higher dimensional version of uniqueness theorem that established by M. M. Dzhrbashyan in the complex plane C . We further prove the growth property involving of the Fourier transform of functions in L 2 in the upper half-space of R n , which partly generalizes the result in Levi [Lectures on entire functions. Providence (RI): American Mathematical Society; 1996]. [ABSTRACT FROM AUTHOR]

In this paper, we price a compound option with log asset price following an extended variance gamma process. The extended variance gamma process can control the skewness and kurtosis. The parameters of the model are estimated via the maximum likelihood method from historical data. We start with finding the risk neutral Esscher measure under which the discounted asset price process is a martingale. Then we derive an analytical pricing formula for compound option in terms of the Fourier integral of the characteristic function of extended variance gamma process, and we use this formula, in combination with the FFT algorithm, to calculate the compound option price across the whole spectrum of the exercise price. Finally, we present some numerical results for illustration. [ABSTRACT FROM AUTHOR]

In this paper, a novel formulation for scattering from one-dimensional (1D) rough surface problem has been presented. The formulation is based on the representation of the scattering field in spectral domain with an unknown spectral coefficient which is solved by applying Taylor series and Fourier Transforms techniques. It is also shown that forward–backward (FB) acceleration technique can be adapted to the spectral formulation to improve the effectiveness in a limited region of convergence. The obtained results are compared with the conventional method of moments (MoM) and the limitations of the formulation are analyzed in detail. [ABSTRACT FROM AUTHOR]

In this paper, an asymmetric cryptosystem with unequal modulus decomposition in the Fourier domain is presented. The input-colour image is decomposed into its red, green, and blue components. Each component is bounded with random phase mask and undergoes Fourier Transform followed by unequal modulus decomposition. One of resulting masks acts as first private key and other one is again Fourier Transformed and undergoes unequal modulus decomposition. Further two masks are obtained, where one acts as second private key and other is phase truncated to obtain encrypted image. Encrypted image is attenuated by a factor and appended with host image to obtain watermarked image. Numerical simulations on MATLAB are performed for authenticating and validating proposed scheme. Statistical, correlation distribution, information entropy and histogram analyses are performed to demonstrate scheme efficacy. The results illustrate that the scheme resists classical cryptographic, special and occlusion attacks. The proposed scheme is also highly sensitive to its private keys and attenuation factor. [ABSTRACT FROM AUTHOR]

Fourier transform profilometry (FTP) is widely used for real-time three-dimensional (3-D) surface shape measurement with a single frame of projection fringe pattern. The band-pass filter is very important to the shape retrieval accuracy of this technique. On the basis of the Hanning band-pass filter, a cross-shaped Hanning filter is developed in the paper to extract the first-order spectrum of deformed fringe pattern, and the effect of fringe angle on the proposed filter is analysed. The phase retrieval results using the proposed filter and Hanning filter are compared in the simulation and measurement experiment. The results show that the cross-shaped Hanning filter coordinating with the fringe pattern of about 45 degrees holds the higher accuracy of 3-D shape retrieval. [ABSTRACT FROM AUTHOR]

In this paper, we present a testing procedure for fractional orders of integration in the context of non-linear terms approximated by Fourier functions. The test statistic has an asymptotic standard normal distribution and several Monte Carlo experiments conducted in the paper show that it performs well in finite samples. Various applications using real life time series, such as US unemployment rates, US GNP and Purchasing Power Parity (PPP) of G7 countries are presented at the end of the paper. [ABSTRACT FROM AUTHOR]

The dynamic in-plane behavior of two-dimensional finite lattices with periodic microskeletal truss structures, made either by bar or beam elements, is studied adopting a homogenization asymptotic technique. The resulting equations are then analyzed for deriving relationships between reference cell properties and homogenized continuum properties. Specific selection criteria are found for geometric and material parameters of the reference cell in view of getting orthotropic as well as isotropic homogenized plates. Finally, the effects of microstructural properties on the vibrating behavior of the homogenized plate are investigated in terms of modal analysis, studying their influence on natural frequencies. [ABSTRACT FROM AUTHOR]

Although W. L. Bragg's law can be easily derived for beginners in the field of crystallography, its interpretation however seems to cause some difficulties which lies essentially in the relation between the concept of lattice planes and the unit cell constants characterizing the lattice periodicity of the crystal structure. Our approach is certainly not new and is based on a more physical approach where every single point in the crystal participates in the diffraction process. From the early stages of developing a model of diffraction, we make abundant use the dual reference frames namely the direct and reciprocal reference frames. With this approach, W. L. Bragg's law can be reformulated directly in terms of the reciprocal unit cell constants avoiding thus the necessity to introduce a priori the notion of lattice planes. Following the derivation of the diffraction law, different steps and methods leading to the complete determination of a crystal structure are derived. We present also some simulation tools to explain in particular the crystal diffraction phenomenon based on the Ewald sphere and the solution of crystalline structures based on the dual space iteration techniques which are currently used. [ABSTRACT FROM AUTHOR]

We consider a prototype class of Lévy-driven stochastic differential equations (SDEs) with McKean–Vlasov (MK–V) interaction in the drift coefficient. It is assumed that the drift coefficient is affine in the state variable, and only measurable in the law of the solution. We study the equivalent functional fixed-point equation for the unknown time-dependent coefficients of the associated linear Markovian SDE. By proving a contraction property for the functional map in a suitable normed space, we infer existence and uniqueness results for the MK–V SDE, and derive a discretized Picard iteration scheme that approximates the law of the solution through its characteristic function. Numerical illustrations show the effectiveness of our method, which appears to be appropriate to handle the multi-dimensional setting. [ABSTRACT FROM AUTHOR]

In this paper, we recall the main results of the Fourier transform on the quaternionic Heisenberg group, then we introduce the notion of the Gabor transform on this group and study some of its properties. Finally, we prove some uncertainty principles associated with this transform. [ABSTRACT FROM AUTHOR]

Polygon similarity can play an important role in geographic information retrieval, map matching and updating, and spatial data mining applications. Geographic information science (GIS) represents various spatial objects as polygons, including simple polygons and polygons with holes, as well as multipolygons. Spatial objects of multipolygons possess complex structure which makes it difficult to assess their similarity. This study develops a method based on convex hulls and position graphs to measure the similarity between multipolygons. The proposed method first finds correspondences between subpolygons in the two multipolygons based on a control polygon. Thereafter, the method constructs a position graph to denote the distribution of these subpolygons and applies a turning function to compute the similarity between various graphs. Fourier transformation and moment invariants were combined to characterize the different matching relationships among subpolygons. The experiments involve three different kinds multipolygons to verify the effectiveness and robustness of proposed method. The experiments show that this approach effectively measures similarity between multipolygons. Moreover, the proposed method accounts for the relationships across the entire complex geometrical shape and components of multipolygon during measuring similarity. [ABSTRACT FROM AUTHOR]

For an efficient and directional representation of signals in higher dimensions, we propose the non-isotropic angular Stockwell transform in the context of time-frequency analysis. The proposed transform is aimed at rectifying the conventional Stockwell transform by employing an angular and scalable localized window which offers directional flexibility and thereby results in the multi-scale and directional analysis of signals in higher dimensions. The basic properties of the proposed transform such as orthogonality relation, reconstruction formula, derivation of the admissibility condition and characterization of the range are discussed using the machinery of operator theory and Fourier transforms. In addition, we introduce the discrete version of the non-isotropic angular Stockwell transform and establish a sufficient condition for the corresponding discrete family to be a frame in L 2 (R 2). Furthermore, some generalizations of the well-known Heisenberg-type inequalities are derived for the non-isotropic angular Stockwell transform in the Fourier domain. [ABSTRACT FROM AUTHOR]

The main concern of this article is to deal with the thermoelastic interaction in a thick plate subjected to a moving heat source and being enlightened by memory-dependent derivative (MDD). Due to the shortcomings of power law distributions in Taylor's series, some other forms of derivatives with few other kernel functions have been proposed. The present literature deals with a novel mathematical model of generalized thermoelasticity to investigate the transient phenomena for a thermoelastic thick plate in which, the memory-dependent heat transport equation involves the Dual-phase (DP) lag model of generalized thermoelasticity. Employing the Laplace transform and Fourier transforms, the analytical results for the distributions of the thermophysical quantities have been derived. The numerical inversions of the respective transforms have been carried out using a suitable numerical scheme based on the Fourier series expansion technique. Numerical computations for stress, displacement and temperature within the plate have been carried out and have been demonstrated graphically. The results also demonstrate how the heat source moves with time and influences the thermophysical quantities according to its respective position by that time. Also, significant differences on the thermophysical quantities are revealed due to the influence of magnetic field, memory effect and time-delay also. [ABSTRACT FROM AUTHOR]

In order to calculate the Wiener integrals for functionals on Wiener space, one can usually apply the change of variable theorem. But, there are many functionals that are difficult or impossible to calculate even when using the change of variable formula. In order to solve this problem, we establish an evaluation formula via the Fourier-type functionals on Wiener space. We then present various examples to which our evaluation formula can be applied and with the corresponding numerical approximations. [ABSTRACT FROM AUTHOR]

Quantitative Shapiro's dispersion uncertainty principle and umbrella theorem are proved for the multivariate continuous shearlet transform S H ψ introduced earlier in Dahlke et al. [The continuous shearlet transform in arbitrary space dimensions. Preprint Nr. 2008-7, Philipps-Universität Marburg; 2008; The continuous shearlet transform in arbitrary space dimensions. J Fourier Anal Appl. 2010;16:340–364]. Also, we extend local uncertainty principles for a set of finite measure to S H ψ . [ABSTRACT FROM AUTHOR]

This paper designs a general type time–frequency transform whose kernel function involves a nonlinear modulation and a frequency-varying dilation. The corresponding inversion formula is established. [ABSTRACT FROM AUTHOR]

We prove the existence of solutions for the stochastic differential equation d X t = b (t , X t −) d Z t + a (t , X t) d t , X 0 ∈ I R , t ≥ 0 , with the measurable coefficients a and b satisfying the condition 0 < μ ≤ | b (t , x) | ≤ ν and | a (t , x) | ≤ K for all t ≥ 0 , x ∈ I R , where μ , ν , and K are constants. The driving process Z is a symmetric stable process of index 1 < α < 2. This generalizes the result of Krylov [Controlled Diffusion Processes, Springer, New York, 1980] for the case of α = 2 , that is, when Z is a Brownian motion. The proof is based on integral estimates of the Krylov type for the given equation, which are also derived in this note and are of independent interest. Moreover, unlike in Krylov [Controlled Diffusion Processes, Springer, New York, 1980], we use a different approach to derive the corresponding integral estimates. [ABSTRACT FROM AUTHOR]

The paper proposes an analysis method to predict the optical performance of an LCD in random vibration environment. The dynamic response analysis is firstly conducted and then inverse discrete Fourier transform converts the response signal from frequency domain to time domain. Based on the visual staying phenomenon, the displacements of two points on CF/TFT substrates are taken out every 1/47 s to get LC thickness changes for further analysis on optical transmittance. Some practical vibration tests were performed and the coincidence between the practical and the simulation results indicates the effectiveness and practicability of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the definitions of the Fourier transform and spherical Fourier transform on quaternionic Heisenberg group, then we present some of their properties, in particular, we will determine the spherical functions and the heat kernel of quaternionic Heisenberg group. Finally, we prove a qualitative uncertainty principle 'Donoho–Stark's uncertainty principle'. [ABSTRACT FROM AUTHOR]

We prove a Heisenberg type uncertainty principle for the Gabor spherical mean transform, and we study its generalization. Next, we extend local uncertainty principle for sets of finite measure to the latter transform. [ABSTRACT FROM AUTHOR]

The music-theoretic idea of a Tonnetz can be generalized at different levels: as a network of chords relating by maximal intersection, a simplicial complex in which vertices represent notes and simplices represent chords, and as a triangulation of a manifold or other geometrical space. The geometrical construct is of particular interest, in that allows us to represent inherently topological aspects to important musical concepts. Two kinds of music-theoretical geometry have been proposed that can house Tonnetze: geometrical duals of voice-leading spaces and Fourier phase spaces. Fourier phase spaces are particularly appropriate for Tonnetze in that their objects are pitch-class distributions (real-valued weightings of the 12 pitch classes) and proximity in these space relates to shared pitch-class content. They admit of a particularly general method of constructing a geometrical Tonnetz that allows for interval and chord duplications in a toroidal geometry. This article examines how these duplications can relate to important musical concepts such as key or pitch height, and details a method of removing such redundancies and the resulting changes to the homology of the space. The method also transfers to the rhythmic domain, defining Zeitnetze for cyclic rhythms. A number of possible Tonnetze are illustrated: on triads, seventh chords, ninth chords, scalar tetrachords, scales, etc., as well as Zeitnetze on common cyclic rhythms or timelines. Their different topologies – whether orientable, bounded, manifold, etc. – reveal some of the topological character of musical concepts. [ABSTRACT FROM AUTHOR]

In this paper some new classes of two-variable orthogonal functions by using Fourier transforms of two-variable orthogonal polynomials are introduced. Orthogonality relations are obtained by using the Parseval identity. Recurrence relations for new families of orthogonal functions are also presented. [ABSTRACT FROM AUTHOR]

Surface Fourier transform profilometry (FTP) is a technique used for reconstructing three-dimensional (3-D) surface topography using light. It has been widely used in machine vision for biomedical and biometric automations, providing solutions beyond conventional 2-D imaging. This paper proposes an implementation of the handheld pocket-size FTP for 3-D surface profile imaging using a projector-enabled Samsung Galaxy Beam smartphone. In the implementation, a crossed-optical-axes geometry of the FTP is formed by using a mirror positioned over the phone's projector via an adjustable tilt mounting bracket. Experimental proof-of-concept of the proposed profilometry is done by implementing conventional and non-phase shifting FTPs with different diffuse test objects. The experimental results obtained by using the non-phase shifting technique are in good agreement with those of the direct contact measurement. Besides having superiority of compactness, the proposed profilometry paves the way for the development of real-time 3-D profiling and printing through internet or Bluetooth interconnection. [ABSTRACT FROM AUTHOR]

In this paper, we study Boas transforms of wavelets and obtain a sufficient condition under which the Boas transform of a wavelet is the derivative of another wavelet. Also, a characterization of the Boas transform of a wavelet ψ ∈ B R − [ − 1 , 1 ] is given. A sufficient condition is given to obtain higher order vanishing moments of Boas transforms of wavelets. Further, we study the Boas transform of wavelets in L 2 (R 2). Finally, higher order vanishing moments of Boas transforms of wavelets have been used to approximate finite energy functions. [ABSTRACT FROM AUTHOR]