Search

For the octonion Fourier transform, we establish the real Paley‐Wiener theorem. It relates the mean of derivatives of a function with the support of its octonion Fourier transform via limm→∞‖∂mαf‖L2(ℝ3,핆)1/m=(2π)α‖wα‖L∞(suppF핆{f},핆)for any octonion‐valued Sobolev function f∈H∞(ℝ3,핆). [ABSTRACT FROM AUTHOR]

In this paper, we study the structural properties of Nevanlinna measures, that is, Borel measures that arise in the integral representation of Herglotz–Nevanlinna functions. In particular, we give a characterization of these measures in terms of their Fourier transform, characterize measures supported on hyperplanes including extremal measures, describe the structure of the singular part of the measures when some variable are set to a fixed value, and provide estimates for the measure of expanding and shrinking cubes. Corresponding results are stated also in the setting of the polydisc where applicable, and some of our proofs are actually performed via the polydisc. [ABSTRACT FROM AUTHOR]

Background: Currently, determining procedural endpoints and treatment efficacy of vascular interventions is largely qualitative and relies on subjective visual assessment of digital subtraction angiography (DSA) images leading to large interobserver variabilities and poor reproducibility. Quantitative metrics such as the residual blood velocity in embolized vessel branches could help establish objective and reproducible endpoints. Recently, velocity quantification techniques based on a contrast enhanced X‐ray sequence such as qDSA and 4D DSA have been proposed. These techniques must be robust, and, to avoid radiation dose concerns, they should be compatible with low dose per frame image acquisition. Purpose: To develop and evaluate a technique for robust blood velocity quantification from low dose contrast enhanced X‐ray image sequences that leverages the oscillating signal created by pulsatile blood flow. Methods: The proposed spatiotemporal frequency domain (STF) approach quantifies velocities from time attenuation maps (TAMs) representing the oscillating signal over time for all points along a vessel centerline. Due to the time it takes a contrast bolus to travel along the vessel centerline, the resulting TAM resembles a sheared sine wave. The shear angle is related to the velocity and can be determined in the spatiotemporal frequency domain after applying the 2D Fourier transform to the TAM. The approach was evaluated in a straight tube phantom using three different radiation dose levels and compared to ultrasound transit‐time‐based measurements. The STF velocity results were also compared to previously published approaches for the measurement of blood velocity from contrast enhanced X‐ray sequences including shifted least squared (SLS) and phase shift (PHS). Additionally, an in vivo porcine study (n = 8) was performed where increasing amounts of embolic particles were injected into a hepatic or splenic artery with intermittent velocity measurements after each injection to monitor the resulting reduction in velocity. Results: At the lowest evaluated dose level (average air kerma rate 1.3 mGy/s at the interventional reference point), the Pearson correlation between ultrasound and STF velocity measurements was 99%$99\%$. This was significantly higher (p<0.0001$p < 0.0001$) than corresponding correlation results between ultrasound and the previously published SLS and PHS approaches (91$\hskip.001pt 91$ and 93%$93\%$, respectively). In the in vivo study, a reduction in velocity was observed in 85.7%$85.7\%$ of cases after injection of 1 mL, 96.4%$96.4\%$ after 3 mL, and 100.0%$100.0\%$ after 4 mL of embolic particles. Conclusions: The results show good agreement of the spatiotemporal frequency domain approach with ultrasound even in low dose per frame image sequences. Additionally, the in vivo study demonstrates the ability to monitor the physiological changes due to embolization. This could provide quantitative metrics during vascular procedures to establish objective and reproducible endpoints. [ABSTRACT FROM AUTHOR]

The Lp$L_p$ Busemann–Petty problem for 00$\varepsilon >0$ small enough, if K is an Lp$L_p$ intersection body and L is a star body such that for any u∈Sn−1$u\in S^{n-1}$, the following inequality: ∥u∥IpK−p≤∥u∥IpL−p+ε$$\begin{equation*} \Vert u\Vert _{I_pK}^{-p}\le \Vert u\Vert _{I_pL}^{-p}+\varepsilon \end{equation*}$$implies V(K)n−pn≤V(L)n−pn+Cε,$$\begin{equation*} V(K)^{\frac{n-p}{n}}\le V(L)^{\frac{n-p}{n}}+C\varepsilon , \end{equation*}$$where ∥·∥IpK$\Vert \cdot \Vert _{I_pK}$ is the Minkowski functional of IpK$I_pK$, and C is a positive number depending only on p and n. Moreover, we also prove the linear stability and separation results for the negative answer of the Lp$L_p$ Busemann–Petty problem. [ABSTRACT FROM AUTHOR]

The utilization of handwritten electronic signatures has expanded in various application scenarios, leading to an increased demand for identification. Unlike handwriting signatures, handwritten electronic signatures offer the advantage of extracting dynamic feature data, including writing pressure, velocity, and acceleration. In this study, the Fourier transform was employed to extract 18 characteristics from the time domain and frequency domain of writing pressure, velocity, and acceleration. The experimental findings revealed distinguishable differences between genuine signatures and random forgeries in writing pressure. However, no statistically significant differences were observed in writing velocity and writing acceleration. Moreover, significant differences were detected in most characteristics when comparing genuine signatures with freehand imitation forgeries and tracing imitation forgeries. The canonical discriminant analysis was performed between the genuine and Non‐genuine signatures; the cross‐validation estimated the discriminating power of these characteristics with a satisfactory result. The study proposed a new approach to analyzing handwritten electronic signatures using time‐domain and frequency‐domain characteristics and demonstrated its effectiveness in the examination. [ABSTRACT FROM AUTHOR]

In the pioneering work, the radiation diagrams of leaky wave antenna arrays can achieve attenuation nulls and gains at specific angles by manually placing the zeros and the poles in the Z domain of the corresponding discrete linear time‐invariant (LTI) system. This handcrafted design procedure does not allow radiation diagrams with wide beams since the interaction between poles involved in the wide beam and their corresponding leaky modes cannot be easily handled. To overcome this limitation, this paper describes a novel method for designing radiation diagrams of leaky‐wave antenna arrays based on the theory of IIR discrete filters. The proposed method relies on the design of discrete filters with the prototypes of analog low‐pass filters defined by Butterworth and Chebyshev type I polynomials, whose roots along with the bilinear transformation provide the location of the poles and the zeros of the discrete LTI system and, therefore, the parameters of the leaky‐wave antenna array. Results with different designs and a comparison with other approaches show the utility and effectiveness of this novel method to design wide‐beam leaky‐wave antenna arrays. [ABSTRACT FROM AUTHOR]

For an expansive real matrix M=ρ−1C0ρ−1$M= \def\eqcellsep{&}\begin{bmatrix} \rho ^{-1} & \mathcal {C}\\ 0& \rho ^{-1} \end{bmatrix}$ and a noncollinear integer digit set D={(0,0)t,(α1,α2)t,(β1,β2)t}$D=\lbrace (0,0)^t,(\alpha _1,\alpha _2)^t,(\beta _1,\beta _2)^t\rbrace$ with α2−2β2∉3Z$\alpha _2-2\beta _2\notin 3\mathbb {Z}$, let μM,D$\mu _{M,D}$ be the self‐affine measure defined by μM,D(·)=13∑d∈DμM,D(M(·)−d)$\mu _{M,D}(\cdot)=\frac{1}{3}\sum _{d\in D}\mu _{M,D}(M(\cdot)-d)$. In this paper, some necessary and sufficient conditions for L2(μM,D)$L^2(\mu _{M,D})$ contains an infinite orthogonal exponential set or μM,D$\mu _{M,D}$ to be a spectral measure are given. [ABSTRACT FROM AUTHOR]

We show how the octonion Fourier transform can be embedded and studied in Clifford geometric algebra of three‐dimensional Euclidean space Cl(3,0)$$ Cl\left(3,0\right) $$. We apply a new form of dimensionally minimal embedding of octonions in geometric algebra that expresses octonion multiplication nonassociativity with a sum of up to four (individually associative) geometric algebra product terms. This approach leads to new polar representations of octonion analytic signals and signal reconstruction formulas. [ABSTRACT FROM AUTHOR]

Summary: In modern integrated circuits (ICs), integrating workload‐agnostic multiphase switched‐capacitor (SC) voltage converters on‐chip is an effective way to reduce the transient response time, improve the security, and boost the power efficiency. However, a stochastic and unbalanced active sequence within the multiphase SC converter may undermine the corresponding output voltage ripple significantly. In this paper, a novel algorithm that is based on Fourier transform is capable of efficiently mitigating the amplitude of output ripple without compromising its security drastically. To achieve this objective, firstly, the output ripple of a single‐phase SC converter needs to be approximated with a Fourier expansion. Then the relationship between the output ripple and active phase sequence of a multiphase SC converter can be modeled precisely under the assistance of Fourier transform. After maximizing the randomness of the active sequences of the multiphase SC converter under different workload conditions and filtering a certain number of phase sequences with unsatisfied output ripples, the optimum control algorithm is generated. As shown in the result, the amplitude of output ripple of a multiphase SC converter can be reduced by 69.5% with negligible security loss and less than 29% power and area overhead under the 55 nm process design kits (PDK). [ABSTRACT FROM AUTHOR]

The octonion Fourier transform (OFT) is a hypercomplex Fourier transform that generalizes the quaternion Fourier transform. However, in octonion algebra, there are two major obstacles that are presented in the loss of associativity and commutativity. Researchers have been trying to extend the results of the Euclidean Fourier transform to quaternion‐valued signals using special techniques to overcome these two problems. In this context, we intend to generalize the Heisenberg uncertainty principles associated with covariance and Hardy's uncertainty principle for octonion multivector valued signals over ℝ3$$ {\mathbb{R}}^3 $$ using the polar form of an octonion, the quaternion decomposition, and the relationship between the OFT and the three‐dimensional (3D)‐Clifford‐Fourier transform. [ABSTRACT FROM AUTHOR]

The surface air temperature seasonal cycle in China has clearly changed in response to global warming, but its future projection has yet to be studied. Here, we investigate the performance and projection of the temperature seasonal cycle in China in 32 global climate models participating in the Coupled Model Intercomparison Project 6 (CMIP6). The characteristics of the temperature seasonal cycle, as represented by the amplitude and phase, are measured by the Fourier transform. The results show that the CMIP6 models can reasonably capture the observed typical temperature seasonal cycle feature with clear north–south gradients both for amplitude and phase in China. In the twenty‐first century, the models robustly project a delay in the phase across the whole country and a reduction in the amplitude in most of China relative to the reference period. Those changes are similar in the two shared socioeconomic pathway (SSP) scenarios, with overall larger magnitudes in the SSP5‐8.5 scenario than in the SSP2‐4.5 scenario. Projected amplitude changes are mainly linked to surface energy flux changes, especially latent heat flux changes. In response to future global warming, larger increases in the latent heat flux and thus stronger surface evaporative cooling lead to weaker surface warming in summer than in winter, contributing to a decrease in the amplitude of the temperature seasonal cycle in most regions of China. [ABSTRACT FROM AUTHOR]

This article extends the pioneering Johansen cointegration test to allow for structural breaks in a cointegration system. Instead of using usual dummy variables, we utilize a Fourier function to control for an unknown number of multiple breaks in the cointegration system. The underlying presumption of the procedure is that structural breaks often can be captured by using a small number of low‐frequency components from a Fourier approximation. The number of parameters to estimate is reduced significantly, which can lead to a good performance of the tests. Monte Carlo simulations show that the new tests display good size and power properties, except for the cases of sharp breaks. Then, we consider a strategy using a union of rejections. The union test combines our suggested test with a test of the cointegration rank in VAR models in the presence of a possible break in trend at an unknown point. We further consider a procedure that selects a better model using a Schwarz‐type criterion among Johansen, trend break‐point, and Fourier models. The resulting test shows fairly correct sizes in all cases, including sharp breaks, smooth breaks, and no breaks. The power properties are also reasonable in almost all cases. [ABSTRACT FROM AUTHOR]

A new software package, autoPX, for processing X‐ray diffraction data from biomacromolecular crystals is reported. This processing software package is designed on the basis of novel methods such as the location of diffraction spots by an improved Canny operator, indexing by a modified Fourier transform, a novel definition of mosaicity that expresses the dispersion state of reciprocal diffraction spots, and the correction of predicted diffraction spot coordinates by homography transform. New programming of some traditional algorithms necessary for integration and scaling is also included. Several examples of crystal structure determination using data from the SSRF beamlines reduced using autoPX, HKL‐2000, DIALS and XDS are also demonstrated, and indicate that autoPX is capable of processing diffraction data from biomacromolecular crystals and providing adequate solutions to problems encountered at the SSRF beamlines. [ABSTRACT FROM AUTHOR]

Isolation piles are widely used to increase the horizontal resistance of high‐speed railway under adjacent loading in soft soils. This study proposes a theoretical model for horizontal additional stress of pile‐reinforced area subjected to adjacent loading. The isolation piles and the soil between piles in the reinforcement area are equivalent to a uniform medium in the proposed model. The Heaviside function is introduced to avoid the partition solution problem of pile‐reinforced area and the foundation soils. A combined method with the inverse Fourier transform, Gauss integral, Lagrange quadratic interpolation, and Simpson quadrature formula is employed in the calculation process. The theoretical model is validated by the finite element method. According to the calculated results, the distribution of horizontal additional stress is obtained. The horizontal soil displacements at the location of the railway increases in the depth range of 0–2.5 m and then gradually decreases. The proposed model with engineering significance is available for preliminary design of isolation piles. [ABSTRACT FROM AUTHOR]

The small‐angle neutron scattering data of nanostructured magnetic samples contain information regarding their chemical and magnetic properties. Often, the first step to access characteristic magnetic and structural length scales is a model‐free investigation. However, due to measurement uncertainties and a restricted q range, a direct Fourier transform usually fails and results in ambiguous distributions. To circumvent these problems, different methods have been introduced to derive regularized, more stable correlation functions, with the indirect Fourier transform being the most prominent approach. Here, the indirect Fourier transform is compared with the singular value decomposition and an iterative algorithm. These approaches are used to determine the correlation function from magnetic small‐angle neutron scattering data of a powder sample of iron oxide nanoparticles; it is shown that with all three methods, in principle, the same correlation function can be derived. Each method has certain advantages and disadvantages, and thus the recommendation is to combine these three approaches to obtain robust results. [ABSTRACT FROM AUTHOR]

Research in the field of neurobiology and neurochemistry has seen a rapid expansion in the last several years due to advances in technologies and instrumentation, facilitating the detection of biomolecules critical to the complex signaling of neurons. Part of this growth has been due to the development and implementation of high‐resolution Fourier transform (FT) mass spectrometry (MS), as is offered by FT ion cyclotron resonance (FTICR) and Orbitrap mass analyzers, which improves the accuracy of measurements and helps resolve the complex biological mixtures often analyzed in the nervous system. The coupling of matrix‐assisted laser desorption/ionization (MALDI) with high‐resolution MS has drastically expanded the information that can be obtained with these complex samples. This review discusses notable technical developments in MALDI‐FTICR and MALDI‐Orbitrap platforms and their applications toward molecules in the nervous system, including sequence elucidation and profiling with de novo sequencing, analysis of post‐translational modifications, in situ analysis, key advances in sample preparation and handling, quantitation, and imaging. Notable novel applications are also discussed to highlight key developments critical to advancing our understanding of neurobiology and providing insight into the exciting future of this field. © 2020 John Wiley & Sons Ltd. Mass Spec Rev [ABSTRACT FROM AUTHOR]

Purpose: To implement an outcome‐driven frequency filter for improving radiomics‐based modeling of human papillomavirus (HPV) for patients with oropharyngeal squamous cell carcinoma (OPSCC). Methods and materials: One hundred twenty‐eight OPSCC patients with known HPV status (60‐HPV+ and 68‐HPV–, confirmed by immunohistochemistry‐P16 protein testing) were retrospectively studied. A 3D Discrete Fourier Transform was applied on contrast‐enhanced computed tomography (CE‐CT) images of patient gross tumor volumes (GTVs) to transform intensity distributions to the frequency domain and estimate frequency power spectrums of HPV– and HPV+ patient cohorts. Statistical analyses were performed to rank frequency bands contributing toward the prediction of HPV status. An outcome‐driven frequency filter was designed accordingly and applied to GTV frequency information. A 3D inverse discrete Fourier transform was applied to reconstruct HPV‐related frequency‐filtered images. Radiomics features (11 feature‐categories) were extracted from pre‐ and post‐frequency filtered images using our previously published "ROdiomiX" software. Least‐absolute‐shrinkage‐and‐selection‐operation (Lasso) combined with a generalized linear model (Lasso‐GLM) was developed to identify and rank feature subsets with the optimal information for prediction of HPV+/–. Radiomics‐based Lasso‐GLM classifiers (pre‐ and post‐frequency filtered) were constructed and validated using random permutation sampling and nested cross‐validation (CV) techniques. Average area under the receiver operating characteristic (AUC), and positive and negative predictive values (PPV and NPV) were computed to estimate generalization error and prediction performance. Results: Among 192 radiomic features, 15 features were found to be statistically significant discriminators between HPV+/– cohorts on post‐frequency filtered CE‐CT images. Fourteen such radiomic features were observed on pre‐frequency filtered datasets. Discriminant features included tumor morphology and intensity contrast. Performances for prediction of HPV for the pre‐ and post‐frequency filtered Lasso‐GLM classifiers were as follows: AUC/PPV/NPV = 0.789/0.755/0.805 and 0.850/0.808/0.877, respectively. Nested CV performances for prediction of HPV for the pre‐ and post‐frequency filtered Lasso‐GLM classifiers were as follows: AUC/PPV/NPV = 0.814/0.725/0.877 and 0.890/0.820/0.911, respectively. Conclusion: Albeit subject to confirmation in a larger cohort, this pilot study presents encouraging results on the importance of frequency analysis prior to radiomic feature extraction toward enhancement of model performance for characterizing HPV in patients with OPSCC. [ABSTRACT FROM AUTHOR]

In this study, a new series representation of the extended k‐gamma function is investigated. A new representation expresses this function as an infinite sum of delta functions. Several researchers have examined this family of functions, but no study has been conducted dealing with the fractional kinetic equation. A new representation proves useful to solve the fractional kinetic equation involving an extended k‐gamma function. Particular cases involving the original gamma function are discussed as corollaries. Such an application of the gamma function is not possible by using known representations; nevertheless, using this representation, new fractional transform formulae are evaluated. A new representation is also worthwhile to compute new integrals of products of the extended k‐gamma functions, proving to be reliable with known identities. Several distributional properties of the extended k‐gamma function are also discussed using Fourier transform. [ABSTRACT FROM AUTHOR]

A Fourier transform spectrometer (FTS) is recognized as a highly precise analytical instrument for analyzing the constituent elements of matter in the fields of physics, chemistry, aerospace, and so on. With the emergence and development of miniaturized and portable devices in numerous scientific and technological fields, there has been an urgent need for on‐chip FTSs due to their benefits of small size, portability, low energy consumption, and robustness. However, the small size of on‐chip FTSs hinders the acquisition of a large optical path difference, resulting in a low resolution of the spectrometer. In addition, the sampler spacing is not sufficiently small to satisfy Nyquist's sampling theorem, directly influencing the bandwidth of the spectrometer. Therefore, studies have been performed to investigate the trade‐off between the reduced size and performance of spectrometers. This paper aims to systematically review the progress in on‐chip FTSs, especially in on‐chip static FTSs, including spatially modulated, temporally modulated, and space‐time comodulated FTSs, from the aspects of theories, implementations, and performance indicators. Finally, the paper ends with a discussion of the challenges and a view of the prospective development of this exciting field. [ABSTRACT FROM AUTHOR]

An analytical framework was developed to correlate the intrinsic dissipation to the second harmonic of the temperature signal evaluated by means of the Fourier transform. The theoretical model assumes, without loss of generality, total conversion of the input mechanical energy into heat. Elastic‐perfectly plastic or elastic–plastic materials obeying a Ramberg–Osgood law are considered. For the sake of simplicity, analytical solution of the thermoelastic and plastic temperature evolutions are obtained in the case of uniform stress and temperature fields. It was found that the intrinsic dissipation is correlated to the second harmonic of the temperature signal by means of a parameter (the β parameter) that, in the case of an elastic–plastic material, depends on the cyclic hardening exponent n′. [ABSTRACT FROM AUTHOR]

This present survey deals with a novel mathematical model of generalized thermoelasticity which investigates the transient phenomena in a two‐dimensional problem for a transversely isotropic thick plate having a heat source. The upper surface of the plate is stress‐free with prescribed surface temperature while the lower surface of the plate rests on a rigid foundation and is thermally insulated. The study is carried out in the context of the three‐phase‐lag thermoelastic model. The governing equations for displacement and temperature fields are obtained in the Laplace‐Fourier transform domain by applying the Laplace and Fourier transform techniques. The inversion of double transform has been done numerically. The numerical inversion of the Laplace transform is done by using a method based on the Fourier series expansion technique. The numerical estimates of the quantities of physical interest are obtained and depicted graphically. The effect of the memory‐dependent derivative on the solutions has been studied. [ABSTRACT FROM AUTHOR]

In this article, we introduce the notion of biorthgonoal nonuniform multiresolution analysis on the spectrum Λ=0,r/N+2ℤ, where N ≥ 1 is an integer and r is an odd integer with 1 ≤ r ≤ 2N − 1 such that r and N are relatively prime. We first establish the necessary and sufficient conditions for the translates of a single function to form the Riesz bases for their closed linear span. We provide the complete characterization for the biorthogonality of the translates of scaling functions of two nonuniform multiresolution analysis and the associated biorthogonal wavelet families. Furthermore, under the mild assumptions on the scaling functions and the corresponding wavelets associated with nonuniform multiresolution analysis, we show that the wavelets can generate Reisz bases. [ABSTRACT FROM AUTHOR]

The molecular dynamics method is used to study the process of development of dynamic instability of a thin film, leading to its destruction. The calculations are performed for a thin aluminum film using the interatomic interaction potential tested by comparing the numerical results with the analytical ones from the elasticity theory. An original approach which allows one to calculate the dispersion law of long‐wave phonons in ultrathin films is developed. The temperatures (<600 K) at which the system remains stable for 0.6 ns are found. This makes it possible to analyze the low‐frequency part of the spectrum down to the minimum frequency νmin = 0.0166 THz, and to determine the vibration frequency of the longest bending wave ν0 = 0.033 THz which decreases with increasing temperature, and therefore, its period grows. Once the vibration period becomes comparable with the time of simulation, there occurs a continuous increase in the amplitude of this mode which will be referred to as "retarded mode." It is shown that the film destruction begins with the attainment of a certain critical value of the bending wave amplitude. [ABSTRACT FROM AUTHOR]

The classic eye‐color gene white (w) in Drosophila melanogaster (fruitfly) has unexpected behavioral consequences. How w affects locomotion of adult flies is largely unknown. Here, we show that a mutant allele (w1118) selectively increases locomotor components at relatively high frequencies (> 0.1 Hz). The w1118 flies had reduced transcripts of w+ from the 5′ end of the gene. Male flies of w1118 walked continuously in circular arenas while the wildtype Canton‐S walked intermittently. Through careful control of genetic and cytoplasmic backgrounds, we found that the w1118 locus was associated with continuous walking. w1118‐carrying male flies showed increased median values of path length per second (PPS) and 5‐min path length compared with w+‐carrying males. Additionally, flies carrying 2–4 genomic copies of mini‐white+ (mw+) in the w1118 background showed suppressed median PPSs and decreased 5‐min path length compared with controls, and the suppression was dependent on the copy number of mw+. Analysis of the time‐series (i.e., PPSs over time) by Fourier transform indicated that w1118 was associated with increased locomotor components at relatively high frequencies (> 0.1 Hz). The addition of multiple genomic copies of mw+ (2–4 copies) suppressed the high‐frequency components. Lastly, the downregulation of w+ in neurons but not glial cells resulted in increased high‐frequency components. We concluded that mutation of w modified the locomotion in adult flies by selectively increasing high‐frequency locomotor components. [ABSTRACT FROM AUTHOR]

In the current research the effect of the radius of a transducer, vibration frequency and anisotropic properties of the host structure on the occuring wave fields is analyzed. Semi‐analytical approach based on the Green's matrix representations and the Fourier transform is applied for calculation of dispersion curves and wave fields in two types of composite layups in a wide frequency‐thickness product range. The contribution of each wave mode in the excited wave field is analyzed. A method for decomposition of the Lamb waves is proposed and examined for the example of a combination of two ring‐shaped actuators. [ABSTRACT FROM AUTHOR]

The annual temperature cycle affects atmospheric circulation, biomass cycling, and human life. Its variation is influenced by surface solar radiation, which plays a key role in surface energy budgets. In this study, the Fourier transform method is used to analyse the spatial distribution and long‐term variations in temperature phase, amplitude, and baseline terms between 1960 and 2016 in China. Mean surface air temperature has a north–south gradient for phase, amplitude, and baseline. The regional mean phase values of the maximum, minimum, and mean temperatures in South China are larger than in North China, indicating that the timing of the annual cycle in Southern China is several days later than in Northern China. At the national scale, the annual maximum, minimum, and mean surface air temperature cycles are advancing, decreasing in range, and have a rising baseline. Linear regression is used to quantify the effect of surface solar radiation on temperature by removing its contribution on the raw maximum and mean temperatures. The surface solar radiation increases the phase of maximum and mean daily air temperatures, but reduces the amplitude and baseline. The surface solar radiation effect on maximum temperature is especially significant on the Yunnan‐Guizhou Plateau located in the south‐southwest. The phase of maximum temperature adjusted for solar radiation for the Plateau is similar to South China at the same latitude and later than the raw maximum temperature. [ABSTRACT FROM AUTHOR]

To reduce the computation required in determining the proper scale of salient object, a fast visual saliency based on multi‐scale difference of Gaussians fusion in frequency domain (MDF) is proposed. First, based on the phenomenon that the foreground energy is highlighted and densely distributes on certain band of spectrum, the scale coefficients of foreground in an image can be literately approximated on the amplitude spectrum. Next, relying on the linear integration property of Fourier transform, the feature spectrum is obtained through the weighted infinite integral of difference of Gaussian feature maps with respect to the scale of object. Then, the saliency of each channel is obtained from feature spectrum by the inverse Fourier transform and scale filtering. Finally, through the channel integration, the MDF saliency map is obtained. Experiments on Li‐Jian data set demonstrate that combined with most appropriate colour space and scale filter, MDF achieves obvious acceleration (5.4 times faster than frequency domain analysis and spatial information) while getting desired accuracy (area under the curve, 0.8814 at Li‐Jian data set), which achieves the best accuracy efficiency trade‐off. [ABSTRACT FROM AUTHOR]

To reduce the computation required in determining the proper scale of salient object, a fast visual saliency based on multi-scale difference of Gaussians fusion in frequency domain (MDF) is proposed. First, based on the phenomenon that the foreground energy is highlighted and densely distributes on certain band of spectrum, the scale coefficients of foreground in an image can be literately approximated on the amplitude spectrum. Next, relying on the linear integration property of Fourier transform, the feature spectrum is obtained through the weighted infinite integral of difference of Gaussian feature maps with respect to the scale of object. Then, the saliency of each channel is obtained from feature spectrum by the inverse Fourier transform and scale filtering. Finally, through the channel integration, the MDF saliency map is obtained. Experiments on Li-Jian data set demonstrate that combined with most appropriate colour space and scale filter, MDF achieves obvious acceleration (5.4 times faster than frequency domain analysis and spatial information) while getting desired accuracy (area under the curve, 0.8814 at Li-Jian data set), which achieves the best accuracy efficiency trade-off. [ABSTRACT FROM AUTHOR]

In this study, the authors propose a fractional derivative‐based image decomposition and denoising model which decomposes the image into the cartoon component (the component formed by homogeneous regions and with sharp boundaries) and the texture (or noise) component. The cartoon component is modelled by a function of the fractional‐order total bounded variation, while the texture component is modelled by an oscillatory function, bounded in the negative Sobolev space norm. The authors give the corresponding minimisation functional, after some transformations, and then the resulting fractional‐order partial differential equation can be solved using the Fourier transform. By symmetry and asymmetry of the fractional‐order derivative, some generalisations and variants of the proposed model are also introduced. Finally, the authors implement the algorithm by the fractional‐order finite difference in the frequency‐domain. The experimental results demonstrate that the proposed models make objective and visual improvements compared with other standard approaches in the task of decomposition and denoising. [ABSTRACT FROM AUTHOR]

The present article deals with the similarity method to tackle the fractional Schrӧdinger equation where the derivative is defined in the Riesz sense. Moreover, the procedure of reducing a fractional partial differential equation (FPDE) into an ordinary differential equation (ODE) has been efficiently displayed by means of suitable scaled transform to the proposed fractional equation. Furthermore, the ODEs are treated effectively via the Fourier transform. The graphical solutions are also depicted for different fractional derivatives α. [ABSTRACT FROM AUTHOR]

After 2000, an interest in the Hausdorff operators grew, first of all in the sense of a diversity of spaces on which these operators were considered. We try to introduce a 'correct' definition of the Hausdorff operator on Euclidean spaces. This is supplemented by a variety of known and possible applications. [ABSTRACT FROM AUTHOR]

We study the free boundary problem for a plasma–vacuum interface in ideal incompressible magnetohydrodynamics. Unlike the classical statement when the vacuum magnetic field obeys the div‐curl system of pre‐Maxwell dynamics, to better understand the influence of the electric field in vacuum, we do not neglect the displacement current in the vacuum region and consider the Maxwell equations for electric and magnetic fields. Under the necessary and sufficient stability condition for a planar interface found earlier by Trakhinin, we prove an energy a priori estimate for the linearized constant coefficient problem. The process of derivation of this estimate is based on various methods, including a secondary symmetrization of the vacuum Maxwell equations, the derivation of a hyperbolic evolutionary equation for the interface function, and the construction of a degenerate Kreiss‐type symmetrizer for an elliptic‐hyperbolic problem for the total pressure. [ABSTRACT FROM AUTHOR]

Summary: This article is devoted to the efficient numerical solution of the Helmholtz equation in a two‐ or three‐dimensional (2D or 3D) rectangular domain with an absorbing boundary condition (ABC). The Helmholtz problem is discretized by standard bilinear and trilinear finite elements on an orthogonal mesh yielding a separable system of linear equations. The main key to high performance is to employ the fast Fourier transform (FFT) within a fast direct solver to solve the large separable systems. The computational complexity of the proposed FFT‐based direct solver is 𝒪(NlogN) operations. Numerical results for both 2D and 3D problems are presented confirming the efficiency of the method discussed. [ABSTRACT FROM AUTHOR]

In this note, we prove that the uni‐modular mono‐components from Blaschke products in the unit disc are eigenvectors of the Sturm‐Liouville operators and investigate a kind of wave equations with some initial conditions, which give rise to mono‐components. [ABSTRACT FROM AUTHOR]

In many practical applications, spatial data are often collected at areal levels (i.e., block data), and the inferences and predictions about the variable at points or blocks different from those at which it has been observed typically depend on integrals of the underlying continuous spatial process. In this paper, we describe a method based on Fourier transforms by which multiple integrals of covariance functions over irregular data regions may be numerically approximated with the same level of accuracy as traditional methods, but at a greatly reduced computational expense. [ABSTRACT FROM AUTHOR]

A differential equation‐based framework is suitable for the modeling of nonequilibrium complex systems if its solution is able to reach, as time goes to infinity, the related nonequilibrium steady states. The thermostatted kinetic theory framework has been recently proposed for the modeling of complex systems subjected to an external force field. The present paper is devoted to the mathematical proof of the convergence of the solutions of the thermostatted kinetic framework towards the related nonequilibrium stationary states. The proof of the main result is gained by employing the Fourier transform and distribution theory arguments. [ABSTRACT FROM AUTHOR]

Spatial heterodyne Raman spectroscopy has become a useful spectroscopic detection technique that is particularly suitable for Raman measurements. This method uses the Fourier transform of the interferogram imaged on the detector by stationary diffraction gratings. Spatial heterodyne Raman spectroscopy has the same characteristic of conventional Fourier‐transform spectroscopy, which is that the field of view is limited. We propose a two‐dimensional spatial heterodyne Raman spectrometer (SHRS) that uses a field widening prism to effectively increase the throughput or sensitivity without sacrificing the spectral resolution for Raman measurements while broadening the bandpass. The signal‐to‐noise ratio is measured under different integrations or laser powers and shows that the system exhibits good stability and fair repeatability. Raman spectra obtained from the field‐widened SHRS and a commercialized Raman instrument are compared, and the field‐widened SHRS and the SHRS using the same instrumental parameters without field widening are also compared. A higher signal‐to‐noise ratio can be achieved using the field‐widened SHRS. In our measurements, the field‐widened SHRS can be operated under the slightly more complex conditions required for wide‐field measurements with short integration times and low laser power. Raman spectra of a pure inorganic target or a target contained in glass and a plastic bottle are observed. A sulfate is investigated in two states: solid salt and aqueous solution. Several mixture liquids, mixture solids, minerals, and anti‐Stokes Raman shifts are also investigated. The results show that the field‐widened SHRS exhibits a good performance to successfully perform wide‐field Raman measurements. [ABSTRACT FROM AUTHOR]

Let n,b≥2 be two positive integers. For D={0,1,⋯,b−1}, let the self‐similar measure μbn,D be defined by μbn,D=1b∑d∈Dμbn,D(bnx−d). It is known [18] that μbn,D is a spectral measure with a spectrum Λ(bn,C)=∑j=0finiteajbnj:aj∈C,where C=bn−1{0,1,⋯,b−1}. In this paper, we give some conditions on τ∈Z under which the scaling set τΛ(bn,C) is also a spectrum of μbn,D. [ABSTRACT FROM AUTHOR]

This article proposes a relatively new methodology named empirical wavelet transform (EWT) to analyse various power quality (PQ) indices of current and voltage at the point of common coupling (PCC) in a power system. EWT provides information in frequency as well as time domain; however, Fourier transform gives information in frequency domain only. This makes EWT more powerful technique compared with conventional techniques. PQ indices such as RMS value of voltage and current, active power and harmonic pollution are analysed with different loading conditions. The proposed approach is simulated on MATLAB environment which is analysed using real‐time signal. The results obtained are compared with discrete wavelet transform (DWT), wavelet packet transform (WPT), and the actual value. It is noted that the EWT produces a better result than DWT and WPT in terms of percentage error. [ABSTRACT FROM AUTHOR]

In the present paper, a fractional wavelet transform of real order α is introduced, and various useful properties and results are derived for it. These include (for example) Perseval's formula and inversion formula for the fractional wavelet transform. Multiresolution analysis and orthonormal fractional wavelets associated with the fractional wavelet transform are studied systematically. Fractional Fourier transforms of the Mexican hat wavelet for different values of the order α are compared with the classical Fourier transform graphically, and various remarkable observations are presented. A comparative study of the various results, which we have presented in this paper, is also represented graphically. [ABSTRACT FROM AUTHOR]

In this paper, the three‐dimensional initial value problem for elastic system in inhomogeneous orthotropic media is considered and an analytical method is studied to solve this problem. The system is written in terms of Fourier images of displacements with respect to lateral variables. The resulting problem is reduced to integral equations of the Volterra type, whose solution is obtained by the method of successive approximations. Finally, using the real Paley‐Wiener theorem, it is shown that the solution of the initial value problem can be found by the inverse Fourier transform. [ABSTRACT FROM AUTHOR]

The Fourier transform method for analytical determination of the two‐center Coulomb integrals needed for evaluation of the electrostatic interaction energies between pseudoatom‐based charge distributions is presented, and its Fortran‐based implementation using the 128‐bit floating‐point arithmetic in the XDPROP module of the XD software is described. In combination with mathematical libraries included in the Lahey/Fujitsu LF64 Linux compiler, the new implementation outperforms the previously reported Löwdin α‐function technique [Nguyen et al. (2018). Acta Cryst. A74, 524–536] in terms of precision of the determined individual Coulomb integrals regardless of whether the latter uses the 64‐, 80‐ or 128‐bit precision floating‐point format, all the while being only marginally slower. When the Löwdin α‐function or Fourier transform method is combined with a multipole moment approximation for large interatomic separations (such a hybrid scheme is called the analytical exact potential and multipole moment method, aEP/MM) the resulting electrostatic interaction energies are evaluated with a precision of ≤5 × 10−5 kJ mol−1 for the current set of benchmark systems composed of H, C, N and O atoms and ranging in size from water–water to dodecapeptide–dodecapeptide dimers. Using a 2012 4.0 GHz AMD FX‐8350 computer processor, the two recommended aEP/MM implementations, the 80‐bit precision Löwdin α‐function and 128‐bit precision Fourier transform methods, evaluate the total electrostatic interaction energy between two 225‐atom monomers of the benchmark dodecapeptide molecule in 6.0 and 7.9 s, respectively, versus 3.1 s for the previously reported 64‐bit Löwdin α‐function approach. [ABSTRACT FROM AUTHOR]

Summary: In actual geotechnical and civil engineering, dynamic stress concentrations around cavities generated by wave sources widely exist. In this study, based on the complex variable theory and Fourier transform method, the expression of the dynamic stress concentration factor (DSCF) around a circular cavity in infinite homogeneous media subjected to transient waves with arbitrary waveform is obtained. The relationships between both steady‐state and transient DSCF and their waveform parameters are investigated quantitatively. The results indicate that a relatively large tensile stress is generated with low Poisson's ratio under steady‐state incidence. Under the condition of transient incidence, the position of the wave peak has a minor effect on the DSCF in the case of small wavenumber, but it has a significant effect in the opposite case. It is found that when the wavenumber is high, such as 0.5, the stress response lags behind the stress wave. In addition, the closer the wave peak to the center of the waveform, the greater the potential damage of the transient incidence. [ABSTRACT FROM AUTHOR]

In this paper, we derive an inversion of the weighted Radon transform by Fourier transform, Riesz potential, and integral transform. We extend results of Rigaud and Lakhal to the n‐dimensional Euclidean space. Furthermore, we obtain some properties of the weighted Radon transform. Finally, we develop some estimate results of the weighted Radon transform under Sobolev space. [ABSTRACT FROM AUTHOR]

This review is devoted to trapping mass spectrometry wherein ions are confined by electromagnetic fields for prolonged periods of time within limited volume, with mass measurement taking place within the same volume. Three major types of trapping mass spectrometers are discussed, specifically radiofrequency ion trap, Fourier transform ion cyclotron resonance and Orbitrap. While these three branches are intricately interwoven with each other over their recent history, they also differ greatly in their fundamentals, roots and historical origin. This diversity is reflected also in the difference of viewpoints from which each of these directions is addressed in this review. Following the theme of the issue, we focus on developments mainly associated with the country of Germany but, at the same time, we use this review as an illustration of the rapidly increasing globalization of science and expanding multi‐national collaborations. [ABSTRACT FROM AUTHOR]

In this article, various convolution theorems involving certain weight functions and convolution products are derived. The convolution theorems we obtain are more general, convenient, and efficient than the complicated convolution theorem of the Hartley transform. Further results involving new variants of generalizations of Fourier and Hartley transforms are also discussed. [ABSTRACT FROM AUTHOR]

Studying nonlinear and potentially chaotic phenomena in geophysics from measured signals is problematic when system noise interferes with the dynamic processes that one is trying to infer. In such circumstances, a framework for statistical hypothesis testing is necessary but the nonlinear nature of the phenomena studied makes the formulation of standard hypothesis tests, such as analysis of variance, problematic as they are based on underlying linear, Gaussian assumptions. One approach to this problem is the method of surrogate data, which is the technique explained in this paper. In particular, we focus on (i) hypothesis testing for nonlinearity by generating linearized surrogates as a null hypothesis, (ii) a variant of this that is perhaps more appropriate for image data where structural nonlinearities are common and should be retained in the surrogates, and (iii) gradual reconstruction where we systematically constrain the surrogates until there is no significant difference between data and surrogates and use this to understand geophysical processes. In addition to time series of sunspot activity, solutions to the Lorenz equations, and spatial maps of enstrophy in a turbulent channel flow, two examples are considered in detail. The first concerns gradual wavelet reconstruction testing of the significance of a specific vortical flow structure from turbulence time series acquired at a point. In the second, the degree of nonlinearity in the spatial profiles of river curvature is shown to be affected by the occurrence of meander cutoff processes but in a more complex fashion than previously envisaged. Plain Language Summary: Complexity and nonlinearity in physics and the geosciences are at the heart of understanding a great number of processes. It can therefore be difficult to infer exactly what is happening in a system due to this complexity. This paper examines various methods for generating surrogate data for testing hypotheses about nonlinearity in data. That is, the surrogates provide the null model and one looks to see if the real data has a value significantly different to the surrogates. Examples are given for how a variable changes in time, how several variables measured at the same point change in time, and how the spatial values for a variable change. In particular, we consider a method called gradual wavelet reconstruction where we can systematically change the properties of the surrogates until there is no longer a difference between data and surrogates and use this as a way to understand geophysical processes. Examples of the use of this technique are given for understanding scour caused by turbulent flows transporting sediment and how river meandering dynamics effect the curvature of the river channel. Key Points: Hypothesis testing with surrogate data is describedTemporal and spatial data are consideredExamples make use of gradual reconstruction tools [ABSTRACT FROM AUTHOR]

Grid cells in rat medial entorhinal cortex are widely thought to play a major role in spatial behavior. However, the exact computational role of the population of grid cells is not known. Here we provide a descriptive model, which nonetheless considers biologically feasible mechanisms, whereby the grid cells are viewed as a two‐dimensional Fourier basis set, in hexagonal coordinates, with restricted availability of basis functions. With known properties imposed in the model parameters, we demonstrate how various empirical benchmark findings are straight‐forward to understand in this model. We also explain how complex computations, inherent in a Fourier model, are feasible in the medial entorhinal cortex with simple mechanisms. We further suggest, based on model experiments, that grid cells may support a form of lossy compression of contextual information, enabling its representation in an efficient manner. In sum, this hexagonal Fourier model suggests how the entire population of grid cells may be modeled in a principled way, incorporates biologically feasible mechanisms and provides a potentially powerful interpretation of the relationship between grid‐cell activity and contextual information beyond spatial knowledge. This enables various phenomena to be modeled with relatively simple mechanisms, and leads to novel and testable predictions. [ABSTRACT FROM AUTHOR]

Coherence is one common metric for cross‐dependence in multichannel signals. However, standard coherence does not sufficiently model many biological signals with complex dependence structures such as cross‐oscillatory interactions between a low‐frequency component in one signal and a high‐frequency component in another. The notion of cross‐dependence between low‐ and high‐frequency components, as defined in classical harmonizable processes, is still inadequate because it assumes time invariance and thus cannot capture cross‐frequency interactions that evolve over time. We construct a novel framework for modeling and estimating these dependencies under the replicated time series setting. Under this framework, we establish the novel concept of evolutionary dual‐frequency coherence and develop time‐localized estimators based on dual‐frequency local periodograms. The proposed nonparametric estimation procedure does not suffer from model misspecification. It uses the localized fast Fourier transform and hence is able to handle massive data. When applied to electroencephalogram data recorded in a motor intention experiment, the proposed method uncovers new and interesting cross‐oscillatory interactions that have been overlooked by the standard approaches. [ABSTRACT FROM AUTHOR]

This study introduces an algorithm for video denoising based on improved dual‐domain filtering and 3D block matching. The wavelet thresholding based on 3D block matching is introduced to make full use of the correlation of video sequence in order to apply dual‐domain filtering to the video. A layered approach is used that attempts denoising in both a base layer and a detail layer. The result of wavelet thresholding based on 3D block matching is used as a guide image to make the base layer smoother. Shrinkage of short‐time Fourier transform coefficients further decreases the noise in the detail layer. Experimental results show that the authors' algorithm generates a better base layer and detail layer than the traditional dual‐domain filtering algorithms. The subjective and objective comparisons of different algorithms also prove that the proposed algorithm performs better for video denoising. [ABSTRACT FROM AUTHOR]