A complementary integrated Transformer network for hyperspectral image classification

In the past,convolutional neural network(CNN)has become one of the most popular deep learning frameworks,and has been widely used in Hyperspectral image classification tasks.Convolution(Conv)in CNN uses filter weights to extract features in local receiving domain,and the weight parameters are shared globally,which more focus on the highfrequency information of the image.Different from Conv,Transformer can obtain the long‐term dependence between long‐distance features through modelling,and adaptively focus on different regions.In addition,Transformer is considered as a low‐pass filter,which more focuses on the low‐frequency information of the image.Considering the complementary characteristics of Conv and Transformer,the two modes can be integrated for full feature extraction.In addition,the most important image features correspond to the discrimination region,while the secondary image features represent important but easily ignored regions,which are also conducive to the classification of HSIs.In this study,a complementary integrated Transformer network(CITNet)for hyperspectral image classification is proposed.Firstly,three‐dimensional convolution(Conv3D)and two‐dimensional convolution(Conv2D)are utilised to extract the shallow semantic information of the image.In order to enhance the secondary features,a channel Gaussian modulation attention module is proposed,which is embedded between Conv3D and Conv2D.This module can not only enhance secondary features,but suppress the most important and least important features.Then,considering the different and complementary characteristics of Conv and Transformer,a complementary integrated Transformer module is designed.Finally,through a large number of experiments,this study evaluates the classification performance of CITNet and several state‐of‐the‐art networks on five common datasets.The experimental results show that compared with these classification networks,CITNet can provide better classification performance.

Results Involving Partial Differential Equations and Their Solution by Certain Integral Transform

In this study,we aimto investigate certain triple integral transformand its application to a class of partial differentialequations.We discuss various properties of the new transformincluding inversion, linearity, existence, scaling andshifting, etc. Then,we derive several results enfolding partial derivatives and establish amulti-convolution theorem.Further, we apply the aforementioned transform to some classical functions and many types of partial differentialequations involving heat equations,wave equations, Laplace equations, and Poisson equations aswell.Moreover,wedraw some figures to illustrate 3-D contour plots for exact solutions of some selected examples involving differentvalues in their variables.

A New Kind of Two-Fold Integration Transformation in Phase Space and Its Uses in Weyl Ordering of Operators

We propose a new two-fold integration transformation in p-q phase space∫∫^∞-∞dpdq/π e^2i（p-x）（q-y）f（p,q）≡G（x,y）,which possesses some well-behaved transformation properties. We apply this transformation to the Weyl ordering of operators, especially those Q-P ordered and P-Q ordered operators.

THE INTEGRAL TYPE GAUGE TRANSFORMATION AND THE ADDITIONAL SYMMETRY FOR THE CONSTRAINED KP HIERARCHY

In this paper, the compatibility between the integral type gauge transformation and the additional symmetry of the constrained KP hierarchy is given. And the string-equation constraint in matrix models is also derived.

New two-fold integration transformation for the Wigner operator in phase space quantum mechanics and its relation to operator ordering

Using the Weyl ordering of operators expansion formula （Hong-Yi Fan, J. Phys.A 25 （1992） 3443） this paper finds a kind of two-fold integration transformation about the Wigner operator △（ q＇,p） q-number transform） in phase space quantum mechanics,∫∫∞-∞dp＇dq＇/π △（q＇,p＇）e-2i（ p-p＇）（ q-q＇）=δ（ p-P）δ（ q-Q）,∫∫∞-∞dqdpδ（p-P）δ（q-Q）e2i（p-p＇）（q-q＇）=△（q＇,p＇）,whereQ,P are the coordinate and momentum operators, respectively. We apply it to study mutual converting formulae among Q-P ordering, P-Q ordering and Weyl ordering of operators. In this way, the contents of phase space quantum mechanics can be enriched. The formula of the Weyl ordering of operators expansion and the technique of integration within the Weyl ordered product of operators are used in this discussion.

MATHEMATICAL PROBLEMS IN THE INTEGRAL-TRANSFORMATION METHOD OF DYNAMIC CRACK

In the investigation on fracture mechanics,the potential function was introduced, and the moving differential equation was constructed. By making Laplace and Fourier transformation as well as sine and cosine transformation to moving differential equations and various responses, the dual equation which is constructed from boundary conditions lastly was solved. This method of investigating dynamic crack has become a more systematic one that is used widely. Some problems are encountered when the dynamic crack is studied. After the large investigation on the problems, it is discovered that during the process of mathematic derivation, the method is short of precision, and the derived results in this method are accidental and have no credibility.A model for example is taken to explain the problems existing in initial deriving process of the integral_transformation method of dynamic crack.

New transformation of Wigner operator in phase space quantum mechanics for the two-mode entangled case

As a natural and important extension of Fan＇s paper （Fan Hong-Yi 2010 Chin. Phys. B 19 040305） by employing the formula of operators＇ Weyl ordering expansion and the bipartite entangled state representation this paper finds a new two-fold complex integration transformation about the Wigner operator A （#, ~） （in its entangled form） in phase space quantum mechanics, and its inverse transformation. In this way, some operator ordering problems regarding to （a1-a2） and （a1＋a2） can be solved and the contents of phase space quantum mechanics can be enriched, where ai,ai are bosonic creation and annihilation operators, respectively.

Simulation and Experimental Study on Influence of Multi⁃port Integrated Hydraulic Transformer Port Number on Transformer Ratio Characteristics

The integrated hydraulic transformer has a compact structure and no throttling loss in the process of pressure regulation.It is widely used in the common pressure rail hydrostatic transmission system.The integrated hydraulic transformer is realized by designing more than three ports on the distribution plate,and the voltage transformation characteristics of the integrated hydraulic transformer with different port numbers are different.In this paper,the influence of port number on the pressure ratio of integrated hydraulic transformer was studied,and the pressure ratio characteristics of 3⁃ports,4⁃ports,and 5⁃ports integrated hydraulic transformer were obtained,and an experimental platform was built for experimental verification,which shows that the simulation results are consistent with the experimental results and provides a theoretical basis for the design of integrated hydraulic transformer.

FOURIER TRANSFORMATION AND SINGULAR INTEGRALS ON SELF-SIMILAR MEASURE

This paper serves two purposes. One is to modify Strichartz's results with respect to the asymptotic averages of the Fourier transform of μ on , self-similar measure defined by Hutchinson. Another purpose is to consider a singular integral operator on μ and show that this op- erator is of type (p,p)(1<p<∞).

材料特性对输送脉动流体的粘弹性轴向功能梯度管非线性动力行为的影响

The nonlinear dynamic behaviors of viscoelastic axially functionally graded material(AFG)pipes conveying pulsating internal flow are very complex.And the dynamic behavior will induce the failure of the pipes,and research of vibration and stability of pipes becomes a major concern.Considering that the elastic modulus,density,and coefficient of viscoelastic damping of the pipe material vary along the axial direction,the transverse vibration equation of the viscoelastic AFG pipe conveying pulsating fluid is established based on the Euler-Bernoulli beam theory.The generalized integral transform technique(GITT)is used to transform the governing fourth-order partial differential equation into a nonlinear system of fourth-order ordinary differential equations in time.The time domain diagram,phase portraits,Poincarémap and power spectra diagram at different dimensionless pulsation frequencies,are discussed in detail,showing the characteristics of chaotic,periodic,and quasi-periodic motion.The results show that the distributions of the elastic modulus,density,and coefficient of viscoelastic damping have significant effects on the nonlinear dynamic behavior of the viscoelastic AFG pipes.With the increase of the material property coefficient k,the transition between chaotic,periodic,and quasi-periodic motion occurs,especially in the high-frequency region of the flow pulsation.

Nonlinear Dynamics of Viscoelastic Pipe Conveying Pulsating Fluid Subjected to Base Excitation

Based on the Euler-Bernoulli beam theory and Kelvin-Voigt model,a nonlinear model for the transverse vibration of a pipe under the combined action of base motion and pulsating internal flow is established.The governing partial differential equation is transformed into a nonlinear system of fourth-order ordinary differential equations by using the generalized integral transform technique(GITT).The effects of the combined excitation of base motion and pulsating internal flow on the nonlinear dynamic behavior of the pipe are investigated using a bifurcation diagram,phase trajectory diagram,power spectrum diagram,time-domain diagram,and Poincare map.The results show that the base excitation amplitude and frequency significantly affect the dynamic behavior of the pipe system.Some new resonance phenomena can be observed,such as the period-1 motion under the base excitation or the pulsating internal flow alone becomes the multi-periodic motion,quasi-periodic motion or even chaotic motion due to the combined excitation action.

On the Fractional Derivatives with an Exponential Kernel

The present article mainly focuses on the fractional derivatives with an exponential kernel(“exponential fractional derivatives”for brevity).First,several extended integral transforms of the exponential fractional derivatives are proposed,including the Fourier transform and the Laplace transform.Then,the L2 discretisation for the exponential Caputo derivative with a∈(1,2)is established.The estimation of the truncation error and the properties of the coefficients are discussed.In addition,a numerical example is given to verify the correctness of the derived L2 discrete formula.

Representation Theory of Three-Dimensional Elliptical Crack Under Dynamic Loading

The relation between the normal displacement on the surface of a dynamical elliptical crack and the normal stress over the crack surface was studied. The three dimensional elastodynamic equations and Fourier Laplace transforms are used. Based on the influence function and the inversion of integral transforms, one can find that if the distribution of normal displacement on the surface of a dynamic elliptical crack is a polynomial of degree n in x 1 and x 2 , then the normal pressure acting over the ellipse is also a polynomial P n(x 1,x 2) of the same degree in x 1 and x 2 .

TRANSIENT RESPONSE OF A PIEZOELECTRIC CERAMIC WITH TWO COPLANAR CRACKS UNDER ELECTROMECHANICAL IMPACT

The transient response of two coplanar cracks in a piezoelectric ceramic under antiplane mechanical and inplane electric impacting loads is investigated in the present paper. Laplace and Fourier transforms are used to reduce the mixed boundary value problems to Cauchy-type singular integral equations in Laplace transform domain, which are solved numerically. The dynamic stress and electric displacement factors are obtained as the functions of time and geometry parameters. The present study shows that the presence of the dynamic electric field will impede or enhance the propagation of the crack in piezoelectric ceramics at different stages of the dynamic electromechanical load. Moreover, the electromechanical response is greatly affected by the ratio of the space of the cracks and the crack length.

Analytical solution for stress and deformation of the mining floor based on integral transform

Following exploitation of a coal seam, the final stress field is the sum of in situ stress field and an excavation stress field. Based on this feature, we firstly established a mechanics analytical model of the mining floor strata. Then the study applied Fourier integral transform to solve a biharmonic equation,obtaining the analytical solution of the stress and displacement of the mining floor. Additionally, this investigation used the Mohr–Coulomb yield criterion to determine the plastic failure depth of the floor strata. The calculation process showed that the plastic failure depth of the floor and floor heave are related to the mining width, burial depth and physical–mechanical properties. The results from an example show that the curve of the plastic failure depth of the mining floor is characterized by a funnel shape and the maximum failure depth generates in the middle of mining floor; and that the maximum and minimum principal stresses change distinctly in the shallow layer and tend to a fixed value with an increase in depth. Based on the displacement results, the maximum floor heave appears in the middle of the stope and its value is 0.107 m. This will provide a basis for floor control. Lastly, we have verified the analytical results using FLAC3 Dto simulate floor excavation and find that there is some deviation between the two results, but their overall tendency is consistent which illustrates that the analysis method can well solve the stress and displacement of the floor.

Dynamic response of axially moving Timoshenko beams: integral transform solution

The generalized integral transform technique （GITT） is used to find a semianalytical numerical solution for dynamic response of an axially moving Timoshenko beam with clamped-clamped and simply-supported boundary conditions, respectively. The implementation of GITT approach for analyzing the forced vibration equation eliminates the space variable and leads to systems of second-order ordinary differential equations （ODEs） in time. The MATHEMATICA built-in function, NDSolve, is used to numerically solve the resulting transformed ODE system. The good convergence behavior of the suggested eigenfunction expansions is demonstrated for calculating the transverse deflection and the angle of rotation of the beam cross-section. Moreover, parametric studies are performed to analyze the effects of the axially moving speed, the axial tension, and the amplitude of external distributed force on the vibration amplitude of axially moving Timoshenko beams.

Analytical Solution for Wave-Induced Response of Seabed with Variable Shear Modulus

A plane strain analysis based on the generalized Biot＇s equation is utilized to investigate the wave-induced response of a poro-elastic seabed with variable shear modulus. By employing integral transform and Frobenins methods, the transient and steady solutions for the wave-inducod pore water pressure, effective stresses and displacements are analytically derived in detail. Verification is available through the reduction to the simple case of homogeneous seabed. The numerical results indicate that the inclusion of variable shear modulus significantly affects the wave-induced seabed response.

SMALL HANKEL OPERATORS ON WEIGHTEDBERGMAN SPACES OF BOUNDED SYMMETRICDOMAINS

The small Hankel operators on weighted Bergman space of bounded symmetric domains Omega in C-n with symbols in L-2(Omega,dV(lambda)) are studied. Characterizations for the boundedness, compactness of the small Hankel operators h(Phi) are presented in terms of a certain integral transform of the symbol Phi.

Axisymmetric fundamental solutions for a finite layer with impeded boundaries

Axisymmetric fundamental solutions that are applied in the consolidation calculations of a finite clay layer with impeded boundaries were derived. Laplace and Hankel integral transforms were utilized with respect to time and radial coordinates, respectively in the analysis. The derivation of fundamental solutions considers two boundary value problems involving unit point loading and ring loading in the vertical. The solutions are extended to circular distributed and strip distributed normal load. The computation and analysis of settlements, vertical total stress and excess pore pressure in the consolidation layer subject to circular loading are presented.

THE SCATTERING OF GENERAL SH PLANE WAVE BY INTERFACE CRACK BETWEEN TWO DISSIMILAR VISCOELASTIC BODIES

The scattering of general SH plane wave by an interface crack between two dissimilar viscoelastic bodies is studied and the dynamic stress intensity factor at the crack-tip is computed. The scattering problem can be decomposed into two problems: one is the reflection and refraction problem of general SH plane waves at perfect interface (with no crack); another is the scattering problem due to the existence of crack. For the first problem, the viscoelastic wave equation, displacement and stress continuity conditions across the interface are used to obtain the shear stress distribution at the interface. For the second problem, the integral transformation method is used to reduce the scattering problem into dual integral equations. Then, the dual integral equations are transformed into the Cauchy singular integral equation of first kind by introduction of the crack dislocation density function. Finally, the singular integral equation is solved by Kurtz's piecewise continuous function method. As a consequence, the crack opening displacement and dynamic stress intensity factor are obtained. At the end of the paper, a numerical example is given. The effects of incident angle, incident frequency and viscoelastic material parameters are analyzed. It is found that there is a frequency region for viscoelastic material within which the viscoelastic effects cannot be ignored.