Generalized Invertibility of Operators through Spectral Sets

If an operator is not invertible, we are interested if there is a subspace such that the reduction of the operator to that subspace is invertible. In this paper we give a spectral approach to generalized inverses considering the subspace determined by the range of the spectral projection associated with an operator and a spectral set containing the point 0. We compare the cases, 0 is a simple pole of the resolvent function, 0 is a pole of order n of the resolvent function, 0 is an isolated point of the spectrum, and 0 is contained in a circularly isolated spectral set.

Near-field multiple super-resolution imaging from Mikaelian lens to generalized Maxwell's fish-eye lens

Super-resolution imaging is vital for optical applications, such as high capacity information transmission, real-time bio-molecular imaging, and nanolithography. In recent years, technologies and methods of super-resolution imaging have attracted much attention. Different kinds of novel lenses, from the superlens to the super-oscillatory lens, have been designed and fabricated to break through the diffraction limit. However, the effect of the super-resolution imaging in these lenses is not satisfactory due to intrinsic loss, aberration, large sidebands, and so on. Moreover, these lenses also cannot realize multiple super-resolution imaging. In this research, we introduce the solid immersion mechanism to Mikaelian lens(ML) for multiple super-resolution imaging. The effect is robust and valid for broadband frequencies. Based on conformal transformation optics as a bridge linking the solid immersion ML and generalized Maxwell's fish-eye lens(GMFEL), we also discovered the effect of multiple super-resolution imaging in the solid immersion GMFEL.

GENERALIZED FINITE SPECTRAL METHOD FOR 1D BURGERS AND KDV EQUATIONS

A generalized finite spectral method is proposed. The method is of highorder accuracy. To attain high accuracy in time discretization, the fourth-order AdamsBashforth-Moulton predictor and corrector scheme was used. To avoid numerical oscillations caused by the dispersion term in the KdV equation, two numerical techniques were introduced to improve the numerical stability. The Legendre, Chebyshev and Hermite polynomials were used as the basis functions. The proposed numerical scheme is validated by applications to the Burgers equation （nonlinear convection- diffusion problem） and KdV equation（single solitary and 2-solitary wave problems）, where analytical solutions are available for comparison. Numerical results agree very well with the corresponding analytical solutions in all cases.

Deep-learning-based methods for super-resolution fluorescence microscopy

The algorithm used for reconstruction or resolution enhancement is one of the factors affectingthe quality of super-resolution images obtained by fluorescence microscopy.Deep-learning-basedalgorithms have achieved stateof-the-art performance in super-resolution fluorescence micros-copy and are becoming increasingly attractive.We firstly introduce commonly-used deep learningmodels,and then review the latest applications in terms of the net work architectures,the trainingdata and the loss functions.Additionally,we discuss the challenges and limits when using deeplearning to analyze the fluorescence microscopic data,and suggest ways to improve the reliability and robustness of deep learning applications.

Image Super-Resolution Based on Generative Adversarial Networks: A Brief Review

Single image super resolution(SISR)is an important research content in the field of computer vision and image processing.With the rapid development of deep neural networks,different image super-resolution models have emerged.Compared to some traditional SISR methods,deep learning-based methods can complete the super-resolution tasks through a single image.In addition,compared with the SISR methods using traditional convolutional neural networks,SISR based on generative adversarial networks(GAN)has achieved the most advanced visual performance.In this review,we first explore the challenges faced by SISR and introduce some common datasets and evaluation metrics.Then,we review the improved network structures and loss functions of GAN-based perceptual SISR.Subsequently,the advantages and disadvantages of different networks are analyzed by multiple comparative experiments.Finally,we summarize the paper and look forward to the future development trends of GAN-based perceptual SISR.

Better Visual Image Super-Resolution with Laplacian Pyramid of Generative Adversarial Networks

Although there has been a great breakthrough in the accuracy and speed of super-resolution(SR)reconstruction of a single image by using a convolutional neural network,an important problem remains unresolved:how to restore finer texture details during image super-resolution reconstruction?This paper proposes an Enhanced Laplacian Pyramid Generative Adversarial Network(ELSRGAN),based on the Laplacian pyramid to capture the high-frequency details of the image.By combining Laplacian pyramids and generative adversarial networks,progressive reconstruction of super-resolution images can be made,making model applications more flexible.In order to solve the problem of gradient disappearance,we introduce the Residual-in-Residual Dense Block(RRDB)as the basic network unit.Network capacity benefits more from dense connections,is able to capture more visual features with better reconstruction effects,and removes BN layers to increase calculation speed and reduce calculation complexity.In addition,a loss of content driven by perceived similarity is used instead of content loss driven by spatial similarity,thereby enhancing the visual effect of the super-resolution image,making it more consistent with human visual perception.Extensive qualitative and quantitative evaluation of the baseline datasets shows that the proposed algorithm has higher mean-sort-score(MSS)than any state-of-the-art method and has better visual perception.

Spectral Method for Three-Dimensional Nonlinear Klein-Gordon Equation by Using Generalized Laguerre and Spherical Harmonic Functions

In this paper,a generalized Laguerre-spherical harmonic spectral method is proposed for the Cauchy problem of three-dimensional nonlinear Klein-Gordon equation. The goal is to make the numerical solutions to preserve the same conservation as that for the exact solution.The stability and convergence of the proposed scheme are proved.Numerical results demonstrate the efficiency of this approach.We also establish some basic results on the generalized Laguerre-spherical harmonic orthogonal approximation,which play an important role in spectral methods for various problems defined on the whole space and unbounded domains with spherical geometry.

A generalization of Banach's lemma and its applications to perturbations of bounded linear operators

Let X be a Banach space and let P:X→X be a bounded linear operator.Using an algebraic inequality on the spectrum of P,we give a new sufficient condition that guarantees the existence of(I-P)^(-1) as a bounded linear operator on X,and a bound on its spectral radius is also obtained.This generalizes the classic Banach lemma.We apply the result to the perturbation analysis of general bounded linear operators on X with commutative perturbations.

Spectral Gradient Algorithm Based on the Generalized Fiser-Burmeister Function for Sparse Solutions of LCPS

This paper considers the computation of sparse solutions of the linear complementarity problems LCP(q, M). Mathematically, the underlying model is NP-hard in general. Thus an lp(0 p < 1) regularized minimization model is proposed for relaxation. We establish the equivalent unconstrained minimization reformation of the NCP-function. Based on the generalized Fiser-Burmeister function, a sequential smoothing spectral gradient method is proposed to solve the equivalent problem. Numerical results are given to show the efficiency of the proposed method.

Optimal error estimates for Fourier spectral approximation of the generalized KdV equation

A Fourier spectral method for the generalized Korteweg-de Vries equation with periodic boundary conditions is analyzed, and a corresponding optimal error estimate in L^2-norm is obtained. It improves the result presented by Maday and Quarteroni. A modified Fourier pseudospectral method is also presented, with the same convergence properties as the Fourier spectral method.

GENERALIZED JACOBI SPECTRAL GALERKIN METHOD FOR FRACTIONAL-ORDER VOLTERRA INTEGRO-DIFFERENTIAL EQUATIONS WITH WEAKLY SINGULAR KERNELS

For fractional Volterra integro-differential equations(FVIDEs)with weakly singular kernels,this paper proposes a generalized Jacobi spectral Galerkin method.The basis functions for the provided method are selected generalized Jacobi functions(GJFs),which can be utilized as natural basis functions of spectral methods for weakly singular FVIDEs when appropriately constructed.The developed method's spectral rate of convergence is determined using the L^(∞)-norm and the weighted L^(2)-norm.Numerical results indicate the usefulness of the proposed method.

Some new conditions for generalized H-matrices

By using a continuous transition method of a matrix and the estimate for spectral radius of a sub-matrix etc., decision methods for a generalized H-matrix under positive definite matrix conditions are researched. Some new sufficient conditions for generalized H-matrices are obtained. When a block matrix degenerates a point matrix, these conditions namely become sufficient conditions of H-matrix.

Super-resolution processing of passive millimeter-wave images based on conjugate-gradient algorithm

This paper designs a 3 mm radiometer and validate with experiments based on the principle of passive millimeter wave （PMMW） imaging. The poor spatial resolution, which is limited by antenna size, should be improved by post data processing. A conjugate-gradient （CG） algorithm is adopted to circumvent this drawback. Simulation and real data collected in laboratory environment are given, and the results show that the CG algorithm improves the spatial resolution and convergent rate. Further, it can reduce the ringing effects which are caused by regularizing the image restoration. Thus, the CG algorithm is easily implemented for PMMW imaging.

Super-resolution processing of passive millimeter-wave images based on adaptive projected Landweber algorithm

Passive millimeter wave （PMMW） images inherently have the problem of poor resolution owing to limited aperture dimension. Thus, efficient post-processing is necessary to achieve resolution improvement. An adaptive projected Landweber （APL） super-resolution algorithm using a spectral correction procedure, which attempts to combine the strong points of all of the projected Landweber （PL） iteration and the adaptive relaxation parameter adjustment and the spectral correction method, is proposed. In the algorithm, the PL iterations are implemented as the main image restoration scheme and a spectral correction method is included in which the calculated spectrum within the passband is replaced by the known low frequency component. Then, the algorithm updates the relaxation parameter adaptively at each iteration. A qualitative evaluation of this algorithm is performed with simulated data as well as actual radiometer image captured by 91.5 GHz mechanically scanned radiometer. From experiments, it is found that the super-resolution algorithm obtains better results and enhances the resolution and has lower mean square error （MSE）. These constraints and adaptive character and spectral correction procedures speed up the convergence of the Landweber algorithm and reduce the ringing effects that are caused by regularizing the image restoration problem.

New Integrable Couplings of Generalized Kaup-Newell Hierarchy and Its Hamiltonian Structures

A new isospectral problem is firstly presented, then we derive integrable system of soliton hierarchy. Also we obtain new integrable couplings of the generalized Kaup-Newell soliton equations hierarchy and its Hamiltonian structures by using Tu scheme and the quadratic-form identity. The method can be generalized to other soliton hierarchy.

Jacobi Collocation Methods for Solving Generalized Space-Fractional Burgers Equations

The aim of this paper is to obtain the numerical solutions of generalized space-fractional Burgers' equations with initial-boundary conditions by the Jacobi spectral collocation method using the shifted Jacobi-Gauss-Lobatto collocation points. By means of the simplifed Jacobi operational matrix, we produce the diferentiation matrix and transfer the space-fractional Burgers' equation into a system of ordinary diferential equations that can be solved by the fourth-order Runge-Kutta method. The numerical simulations indicate that the Jacobi spectral collocation method is highly accurate and fast convergent for the generalized space-fractional Burgers' equation.

Nonlinear spectral cleaning effect in cross-polarized wave generation

The underlying mechanism of the spectral cleaning effect of the cross-polarized wave（XPW） generation process was theoretically investigated. This study shows that the spectral noise of an input spectrum can be removed in the XPW generation process and that the spectral cleaning effect depends on the characteristics of the input pulses, such as the chirp and Fourier-transform-limited duration of the initial pulse, and the modulation amplitude and frequency of the spectral noise. Though these factors codetermine the output spectrum of the XPW generation process, the spectral cleaning effect is mainly affected by the initial pulse chirp. The smoothing of the spectrum in the XPW generation process leads to a significant enhancement of the coherent contrast.

Time Series-, Time-Frequency- and Spectral Analyses of Sensor Measurements in an Offshore Wave Energy Converter Based on Linear Generator Technology

Inside the second experimental wave energy converter (WEC) launched at the Lysekil research site on the Swedish west coast in March 2009 a number of sensor systems were installed for measuring the mechanical performance of the WEC and its mechanical subsystems. One of the measurement systems was a set-up of 7 laser triangulation sensors for measuring relative displacement of the piston rod mechanical lead-through transmission in the direct drive. Two measurement periods, separated by 2.5 month, are presented in this paper. One measurement is made two weeks after launch and another 3 months after launch. Comparisons and correlations are made between different sensors measuring simultaneously. Noise levels are investigated. Filtering is discussed for further refinement of the laser triangulation sensor signals in order to separate noise from actual physical displacement and vibration. Measurements are presented from the relative displacement of the piston rod mechanical lead-through, from magnetic flux in the air gap, mechanical strain in the WEC structure, translator position and piston rod axial displacement and active AC power. Investigation into the measurements in the time domain with close-ups, in the frequency domain with Fast Fourier transform (FFT) and with time-frequency analysis with short time Fourier transform (STFT) is carried out to map the spectral content in the measurements. End stop impact is clearly visible in the time-frequency analysis. The FFT magnitude spectra are investigated for identifying the cogging bandwidth among other vibrations. Generator cogging, fluctuations in the damping force and in the Lorenz forces in the stator are distinguished and varies depending on translator speed. Vibrations from cogging seem to be present in the early measurement period while not so prominent in the late measurement period. Vibration frequencies due to wear are recognized by comparing with the noise at generator standstill and the vibration sources in the generator. It is concluded that a moving average is a sufficient filter in the time domain for further analysis of the relative displacement of the piston rod mechanical lead-through transmission.

Experimental Measurement of the Generalized Stokes Parameters of a Radially Polarized Random Electromagnetic Beam

Utilizing the Young’s double slits and Mach-Zehnder interferometer, we proposed an experimental method to measure the generalized Stokes parameters of a radially polarized random electromagnetic beam. After the partially coherent beam propagating through the Young’s double slits, the interference fringe is obtained by the help of a Mach-Zehnder interferometer consisting of apertures, quarter-wave plates and polarizers. The electric cross-spectral density matrix is detected by the coherence degree of interference fringe and the density of each single slit. The generalized Stokes parameters can be obtained from the electric cross-spectral density matrix. This experiment measures the generalized Stokes parameters of the random electromagnetic beam successfully. The results show that the spectral degree of coherence for copolarized cases (xx and yy) is similar with that for cross-polaried cases (xy and yx) for the radially polarized random electromagnetic beam. This method will help us determine the change of the polarization and coherence of the light in propagation by detecting the change of the generalized Stokes parameters.

面向天文多普勒差分测速的太阳/行星光谱对生成方法

为了提供天文多普勒差分测速所需的同步太阳/行星光谱对,提出了一种变分自编码器(VAE)和对偶生成对抗网络(Dual GAN)相融合的VAE-Dual GAN。首先,实测太阳光谱经过VAE编码到隐空间,实现了光谱到光谱域的扩充;然后,由Dual GAN将隐空间映射到伪行星光谱;最后,利用伪行星光谱生成重构太阳光谱。此外,利用编码和生成重建损失加强对网络的约束。VAE-Dual GAN利用Dual GAN的转换学习能力完成了两个光谱域的转换,生成同步太阳/行星光谱对。实验结果表明,VAE-Dual GAN可生成高质量的太阳/行星光谱对,将天文多普勒差分测速精度提高60%以上。