A generalized Padé approximation method of solving homoclinic and heteroclinic orbits of strongly nonlinear autonomous oscillators

An intrinsic extension of Pad′e approximation method, called the generalized Pad′e approximation method, is proposed based on the classic Pad′e approximation theorem. According to the proposed method, the numerator and denominator of Pad′e approximant are extended from polynomial functions to a series composed of any kind of function, which means that the generalized Pad′e approximant is not limited to some forms, but can be constructed in different forms in solving different problems. Thus, many existing modifications of Pad′e approximation method can be considered to be the special cases of the proposed method. For solving homoclinic and heteroclinic orbits of strongly nonlinear autonomous oscillators, two novel kinds of generalized Pad′e approximants are constructed. Then, some examples are given to show the validity of the present method. To show the accuracy of the method, all solutions obtained in this paper are compared with those of the Runge–Kutta method.

Degeneracy Cases of Generalized Inverse Function-Valued Padé Approximation

A new generalized inverse function-valued Padé approximation (GIFPA) was defined. Existence condition of GIFPA was given and its uniqueness theorem was proved. All possible degeneracy cases of GIFPA were discussed and constructed. An example was given to illustrate its application.

Assignment of terahertz vibrational modes of L-glutamine using density functional theory within generalized-gradient approximation

The crystal structure of L-glutamine is stabilized by a three-dimensional network of intermolecular hydrogen bonds.We utilize plane-wave density functional theory lattice-dynamics calculations within the generalized-gradient approximation（GGA）, Perdew–Burke–Ernzerhof（PBE）, PBE for solids（PBEsol）, PBE with Wu–Cohen exchange（WC）, and dispersion-corrected PBE, to investigate the effect of these intermolecular contacts on the absorption spectra of glutamine in the terahertz frequency range. Among these calculations, the solid-state simulated results obtained using the WC method exhibit a good agreement with the measured absorption spectra, and the absorption features are assigned with the help of WC. This indicates that the vibrational modes of glutamine were related to the combination of intramolecular and intermolecular motions, the intramolecular modes were dominated by rocking or torsion involving functional groups; the intermolecular modes mainly result from the translational motions of individual molecules, and the rocking of the hydrogenbonded functional groups.

GENERALIZED SIMULTANEOUS CHEBYSHEV APPROXIMATION

In this note,we develop,without assuming the Haar condition,a generalized simultaneous Chebyshev approximation theory which is similar to the classical Chebyshev theory and con- rains it as a special case.Our results also contain those in[1]and[3]as a special case,and the two conjectures proposed by C.B.Dunham in[2]are proved to be true in the case of simulta- neous approximation.

A New Class of Generalized Landau Linear Positive Operator Sequence and Its Properties of Approximation

In this paper two sequences of generalized Landau linear positive operators are introduced. They can be applied in approximating continuous functions with arbitrary growth order, defined on a finite interval or the whole real axis. The properties of approximation are studied and their asymptotic formulae are presented. These results show that their degrees of approximation are the best among existing operator sequences of Landau type, for example, their degrees of approximation for C 2[0, 1] are O(1/n 2) but corresponding degree of ordinary Landau operators are only O(1/n).

Electronic and Optical Properties of TiS2 Determined from Generalized Gradient Approximation Study

The electronic and optical properties of TiS2 are studied of density functional theory. A linearized and augmented by using an ab-initio calculation within the frame plane wave basis set with the generalized gradient approximation as proposed by Perdew et al. is used for the energy exchange-correlation determination. The results show a metallic character of TiS2, and the plots of total and partial densities of states of TiS2 show the metallic character of the bonds and a strong hybridization between the states d of Ti and p of S below the Fermi energy. The optical properties of the material such as real and imaginary parts of dielectric constant （ε（w） = ε1（w） ＋ iε2（w））, refractive index n（w）, optical reflectivity R（w）, for E / /x and E / /z are performed for the energy range of 0-.14 eV.

APPROXIMATION OF CONTINUOUS FUNCTIONS BY GENERALIZED KARAMATA MEANS

A sufficient condition for the order of approximation of a continuous 2π periodic function with a given majorant for the modulus of continuity by the [F, d_n] means of its Fourier serier to be of Jackson order is obtained. This sufficient condition is shown to be not enough for the order of approximation by partial sums of their Fourier series to be of Jackson order. The error estimate is shown to be the best possible.

Computation of waves scattered by inhomogeneous velocity gradient using generalized Born approximation

The generalized Born approximation is an approximation method that represents the scattering term by the error between the exact Green's function and the approximate Green's function,mainly for the gradient scattering problem.However,so far,the research on the generalized Born approximation has only stayed in theory,and its implementation techniques are rarely reported.In order to fill this gap,the basic theory of generalized Born approximation is reviewed,and the implementation method of generalized Born approximation is discussed in this paper.In particular,the problem of calculating boundary effect elimination is discussed in detail.Finally,through model trial calculation,the calculation of gradient scattering,by comparing Born approximation and finite difference method,shows that using the generalized Born approximation to calculate gradient scattering achieves higher computational accuracy.

Approximate Generalized Conditional Symmetries for Perturbed Evolution Equations

The concept of approximate generalized conditional symmetry （AGCS） for the perturbed evolution equations is introduced, and how to derive approximate conditional invariant solutions to the perturbed equations via their A GCSs is illustrated with examples.

EXPONENTIALLY BOUNDED C-SEMIGROUP AND INTEGRATED SEMIGROUP WITH NONDENSELY DEFINED GENERATORS I: APPROXIMATION:

In this paper, by Laplace transform version of the Trotter-Kato approximation theorem and the integrated C-semigroup introduced by Myadera, the authors obtained some Trotter-Kato approximation theorems on exponentially bounded C-semigroups, where the range of C (and so the domain of the generator) may not be dense. The authors deduced the corresponding results on exponentially bounded integrated semigroups with nondensely generators. The results of this paper extended and perfected the results given by Lizama, Park and Zheng.

EPSILON-ALGORITHM AND ETA-ALGORITHM OF GENERALIZED INVERSE FUNCTION-VALUED PAD APPROXIMANTS USING FOR SOLUTION OF INTEGRAL EQUATIONS

Two efficient recursive algorithms epsilon_algorithm and eta_algorithm are introduced to compute the generalized inverse function_valued Padé approximants. The approximants were used to accelerate the convergence of the power series with function_valued coefficients and to estimate characteristic value of the integral equations. Famous Wynn identities of the Pad approximants is also established by means of the connection of two algorithms.

Approximate homotopy similarity reduction for the generalized Kawahara equation via Lie symmetry method and direct method

This paper studies the generalized Kawahara equation in terms of the approximate homotopy symmetry method and the approximate homotopy direct method. Using both methods it obtains the similarity reduction solutions and similarity reduction equations of different orders, showing that the approximate homotopy direct method yields more general approximate similarity reductions than the approximate homotopy symmetry method. The homotopy series solutions to the generalized Kawahara equation are consequently derived.

The scattering states of the generalized Hulthén potential with an improved new approximate scheme for the centrifugal term

This paper finds the approximate analytical scattering state solutions of the arbitrary 1-wave Schrodinger equation for the generalized Hulthen potential by taking an improved new approximate scheme for the centrifugal term. The normalized analytical radial wave functions of the 1-wave SchrSdinger equation for the generalized Hulthen potential are presented and the corresponding calculation formula of phase shifts is derived. Some useful figures are plotted to show the improved accuracy of the obtained results and two special cases for the standard Hulthen potential and Woods-Saxon potential are also studied briefly.

GENERALIZED RAY APPROXIMATE METHOD TO INVERSE MEDIUM PARAMETERS

In this paper, the inverse problem of the medium parameters in an inhomogeneous medium is studied and a generalized ray approximate form of the total wave field is described. First, the acoustic wave equation derived from the elastic wave equation is studied, the referential variables and perturbational variables are introduced, and the integral equation of the medium perturbational parameters is obtained. Then from the point of view of the local principles of the wave function in an inhomogeneous medium, a generalized ray approximate form of the total wave field in an inhomogeneous medium is described, and attention is focused on the Fredholm integral equation of the first kind. Finally, the medium parameters in half-plane are inversed. Numerical examples show when the perturbations of the medium parameters are about 0.5, this method can effectively inverse its variation. Apparently, this method is better than the conventional Born weak scattering approximation.

Approximate analytic solutions for a generalized Hirota—Satsuma coupled KdV equation and a coupled mKdV equation

This paper applies the variational iteration method to obtain approximate analytic solutions of a generalized Hirota-Satsuma coupled Korteweg-de Vries （KdV） equation and a coupled modified Korteweg-de Vries （mKdV） equation. This method provides a sequence Of functions which converges to the exact solution of the problem and is based on the use of Lagrange multiplier for identification of optimal values of parameters in a functional. Some examples are given to demonstrate the reliability and convenience of the method and comparisons are made with the exact solutions.

L^p APPROXIMATION BY GENERAL BERNSTEIN-DURRMEYER OPERATOR DEFINED ON SIMPLEX

As a generalization of the Bernstein-Durrmeyer operatora defined on the simplex, a class of general Bernstein-Durrmeyer operators is introduced. With the weighted moduli of smoothness as a metric, we prove a strong direct theorem and an inverse theorem of weak type for these operators by using a decom-position way. From the theorems the characterization of Lp approximation behavior is derived.

Generalized Nonconvex Low-Rank Algorithm for Magnetic Resonance Imaging (MRI) Reconstruction

In recent years,utilizing the low-rank prior information to construct a signal from a small amount of measures has attracted much attention.In this paper,a generalized nonconvex low-rank(GNLR) algorithm for magnetic resonance imaging(MRI)reconstruction is proposed,which reconstructs the image from highly under-sampled k-space data.In the algorithm,the nonconvex surrogate function replacing the conventional nuclear norm is utilized to enhance the low-rank property inherent in the reconstructed image.An alternative direction multiplier method(ADMM) is applied to solving the resulting non-convex model.Extensive experimental results have demonstrated that the proposed method can consistently recover MRIs efficiently,and outperforms the current state-of-the-art approaches in terms of higher peak signal-to-noise ratio(PSNR) and lower high-frequency error norm(HFEN) values.

Carlson iterating rational approximation and performance analysis of fractional operator with arbitrary order

The performance analysis of the generalized Carlson iterating process,which can realize the rational approximation of fractional operator with arbitrary order,is presented in this paper.The reasons why the generalized Carlson iterating function possesses more excellent properties such as self-similarity and exponential symmetry are also explained.K-index,P-index,O-index,and complexity index are introduced to contribute to performance analysis.Considering nine different operational orders and choosing an appropriate rational initial impedance for a certain operational order,these rational approximation impedance functions calculated by the iterating function meet computational rationality,positive reality,and operational validity.Then they are capable of having the operational performance of fractional operators and being physical realization.The approximation performance of the impedance function to the ideal fractional operator and the circuit network complexity are also exhibited.

THE GENERALIZED REFLEXIVE SOLUTION FOR A CLASS OF MATRIX EQUATIONS (AX-B,XC=D)

In this article, the generalized reflexive solution of matrix equations （AX = B, XC = D） is considered. With special properties of generalized reflexive matrices, the necessary and sufficient conditions for the solvability and the general expression of the solution are obtained. Moreover, the related optimal approximation problem to a given matrix over the solution set is solved.