Conservation Laws and Soliton Solutions for Generalized Seventh Order KdV Equation

With the assistance of the symbolic computation system Maple,rich higher order polynomial-type conservation laws and a sixth order t/x-dependent conservation law are constructed for a generalized seventh order nonlinear evolution equation by using a direct algebraic method.From the compatibility conditions that guaranteeing the existence of conserved densities,an integrable unnamed seventh order KdV-type equation is found.By introducing some nonlinear transformations,the one-,two-,and three-solition solutions as well as the solitary wave solutions are obtained.

Fractional description of mechanical property evolution of soft soils during creep

The motion of pore water directly influences mechanical properties of soils, which are variable during creep. Accurate description of the evolution of mechanical properties of soils can help to reveal the internal behavior of pore water. Based on the idea of using the fractional order to reflect mechanical properties of soils, a fractional creep model is proposed by introducing a variable-order fractional operator, and realized on a series of creep responses in soft soils. A comparative analysis illustrates that the evolution of mechanical properties, shown through the simulated results, exactly corresponds to the motion of pore water and the solid skeleton. This demonstrates that the proposed variable-order fractional model can be employed to characterize the evolution of mechanical properties of and the pore water motion in soft soils during creep. It is observed that the fractional order from the proposed model is related to the dissipation rate of pore water pressure.

Second Order Nonlinear Evolution Inclusions Ⅱ： Structure of the Solution Set

We contimle the work initiated in [1] （Second order nonlinear evolution inclusions I： Existence and relaxation results. Acta Mathematics Science, English Series, 21（5）, 977-996 （2005）） and study the structural properties of the solution set of second order evolution inclusions which are defined in the analytic framework of the evolution triple. For the convex problem we show that the solution set is compact Rs, while for the nonconvex problem we show that it is path connected, Also we show that the solution set is closed only if the multivalued nonlinearity is convex valued. Finally we illustrate the results by considering a nonlinear hyperbolic problem with discontinuities.

哈耶克《自由宪章》简评

从哈耶克自由主义的三大基石"无知论"、"进化论"、"自生自发秩序"三个方面简要分析哈耶克论证中存在的基本矛盾和错误,同时指出,在任何一种社会制度下,都不可能取消理性的整体建构;我们所能做的仅仅是运用理性的批判,一方面确保每种理性的建构都保持在其特定的领域内,一方面在特定的领域内不断推动理性建构的修正与进步。

The evolution of coal, examining the transitions from anthracite to natural graphite: a spectroscopy and optical microscopy evaluation

Coal-derived natural graphite(CDNG)has multiple industrial applications.Here,ten metamorphic coals from anthracite to CDNG were obtained from Lutang and Xinhua in the Hunan Province and Panshi in the Jilin Province.Bulk characterization(proximate and ultimate analyses,X-Ray powder diffraction(XRD),and powder Raman spectroscopy),along with optical microscopy,scanning electron microscope(SEM)and micro-Raman spectroscopy were utilized to examine the transitions from anthracite to semi-graphite to CDNG.The XRD and Raman spectroscopy data indicate that from anthracite to highly ordered graphite the average crystal diameter(La)and height(Lc)increased from 6.1 and 4.6 nm to 34.8 and 27.5 nm,respectively.The crystalline parameters of the CDNG samples from Panshi and Lutang varied slightly when closer to the intrusive body.Optical microscopy and SEM indicated that in the anthracite samples there were thermoplastic vitrinite,devolatilized vitrinite,and some“normal”macerals.In the meta-anthracite,pyrolytic carbon,mosaic structure,and crystalline tar were present.In the CDNG there were flake graphite,crystalline aggregates,and matrix graphite.The crystalline aggregates show the highest structural ordering degree as determined from Raman spectral parameters(full-width at half maxima(G-FWHM)~20 cm^(−1),D1/(D1+D2+G)area ratio(R2)value<0.5).The flake graphite is less ordered with G-FWHM~28 cm^(−1) and 0.5<R2<1,but a larger grain size(up to 50μm).The mosaic structures were likely the precursors of the matrix graphite through in situ solid-state transformation.The pyrolytic carbon and crystalline tars are the transient phase of gas-state and liquid-state transformations.This study is beneficial to realize the rational utilization of CDNG.

POLAR COORDINATES FOR THE GENERALIZED BAOUENDI-GRUSHIN OPERATOR AND APPLICATIONS

In this parer, by using the polar coordinates for the generalized Baouendi- Grushin operatorLα=∑i=1^n 偏d^2/偏dxi^2＋∑j=1^m｜x｜^2α偏d^2/偏dy^2j,where x = （x1,x2,……,Xn）∈R^n,y = （y1,y2,… ,ym） ∈,α 〉 0, we obtain the volume of the ball associated to Lα and prove the nonexistence for a second order evolution inequality which is relative to Lα.

ON THE NONLINEARTIMOSHENKO-KIRCHHOFF BEAM EQUATION

When an elastic string with fixed ends is subjected to transverse vibrations, its length varies with the time: this introduces changes of the tension in the string. This induced Kirchhoff to propose a nonlinear correction of the classical D’Alembert equation. Later on, WoinowskyKrieger (Nash & Modeer) incorporated this correction in the classical Euler-Bernoulli equation for the beam (plate) with hinged ends.Here a new equation for the small transverse vibrations of a simply supported beam is proposed. Such equation takes into account Kirchhoff’s correction, as well as the correction for rotary inertia of the cross section Of the beam and the influence of shearing strains, already present in the Timoshenko beam equation (of the Mindlin-Timoshenko equation for the plate).The model is inspired by a remark of Rayleigh, and by a joint paper with Panizzi & Paoli. It looks more complicated than the one proposed by Sapir & Reiss, but as a matter of fact it is easier to study if a suitable change of variables is performed.The author proves the local well-posedness of the initial-boundary value problem in Sobolev spaces of order ≥2.5. The technique is abstract, i.e. the equation is rewritten as a fourth order evolution equation in Hilbert space (thus the results could be applied also to the formally analogous equation for the plate).

FREE LONG WAVES INDUCED BY WEAKLY NONLINEAR STOKES WAVES PROPAGATING OVER STRONG AMBIENT CURRENT AND AN UNEVEN BOTTOM

This paper concerns the generation of forced and free long waves. The free long waves are due to uneven bottoms and ambient currents. The pure wave evolution equation of Liu & Dingemans is extended to include the effects of strong ambient currents, leading to more general third order governing equations for the evolution of the envelope of the short waves and for the generation and scattering of the long waves.