The Superalgebra spl(p,q) and Differential Operators

Series of finite-dimensional representations of the superalgebrasspl(p,q) can be formulated in terms of linear differentialoperators acting on a suitable space of polynomials. We sketch the generalingredients necessary to construct these representations and presentexamples related to spl(2,1) and spl(2,2). By revisiting the products ofprojectivised representations of sl(2), we are able to construct new sets ofdifferential operators preserving some space of polynomials in two or morevariables. In particular, this allows us to express the representation ofspl(2,1) in terms of matrix differential operators in two variables. Thecorresponding operators provide the building blocks for the construction ofquasi-exactly solvable systems of two and four equations in two variables.We also present a quommutator deformation of spl(2,1) which, by constructionprovides an appropriate basis for analyzing the quasi exactly solvablesystems of finite difference equations.     

Unitary Quasi-finite Representations of W∞

We classify the unitary quasi-finite highest-weight modules over the Lie algebra W and realize them in terms of unitary highest-weight representations of the Lie algebra of infinite matrices with finitely many nonzero diagonals.     

Graded Hochschild cohomology of a path algebra with oriented cycles

The aim of this paper is to characterize the first graded Hochschild cohomology of a hereditary algebra whose Gabriel quiver is admitted to have oriented cycles. The interesting conclusion we have obtained shows that the standard basis of the first graded Hochschild cohomology depends on the genus of a quiver as a topological object. In this paper, we overcome the limitation of the classical Hochschild cohomology for hereditary algebra where the Gabriel quiver is assumed to be acyclic. As preparation, we first investigate the graded differential operators on a path algebra and the associated graded Lie algebra.     

RMVT- and PVD-based finite cylindrical layer methods for the three-dimensional buckling analysis of multilayered FGM cylinders under axial compression

The unified formulations of finite cylindrical layer methods (FCLMs) based on the Reissner mixed variational theorem (RMVT) and the principle of virtual displacements (PVD) are developed for the three-dimensional (3D) linear buckling analysis of simply-supported, multilayered functionally graded material (FGM) circular hollow cylinders and laminated composite ones under axial compression. The material properties of the FGM layer are assumed to obey the power-law distributions of the volume fraction of the constituents through the thickness coordinate. In these formulations, the cylinder is divided into a number of finite cylindrical layers, in which the trigonometric functions and Lagrange polynomials are used to interpolate the in- and out-of-surface variations of the primary variables of each individual layer, respectively, as well as the related order of each primary variable can be freely chosen, such as the layerwise linear, quadratic or cubic function distribution through the thickness coordinate. The accuracy and convergence of the RMVT- and PVD-based FCLMs developed in this article are assessed by comparing their solutions with the exact 3D solutions available in the literature.     

Elastic solutions of a functionally graded cantilever beam with different modulus in tension and compression under bending loads

This paper considers a functionally graded cantilever beam with different modulus in tension and compression. The beam is subjected to bending loads, including pure bending, shear force at the free end and uniform pressure on the upper lateral, respectively. Its modulus values in tension and compression both change with the thickness coordinate as arbitrary functions, which could bring the beam a broader range of applications in engineering. The problem is treated as a plane stress case and described by Airy stress function. By using semi-inverse method, the elastic solutions for the beam are obtained, which can be easily degenerated into the ones for homogeneous beams. An example is finally presented to show the effect of nonhomogeneous materials with different modulus on the elastic field in a cantilever beam.     

程度与精度的逻辑差近似算子的性质

目的是结合精度与程度,探索新的粗糙集拓展模型.从程度与精度的逻辑差运算出发,定义了程度与精度的逻辑差粗糙集模型.模型中,通过变精度近似与程度近似的转化公式,研究了程度与精度的逻辑差近似算子,并得到了近似算子的幂作用等性质.用程度与精度的逻辑差粗糙集模型拓展了程度粗糙集模型、经典粗糙集模型,并在这些模型中得到了近似算子幂作用的相应性质.     

Thermal stresses around a circular inclusion with functionally graded interphase in a finite matrix

Based on the theory of the complex variable functions, the analysis of non-axisymmetric thermal stresses in a finite matrix containing a circular inclusion with functionally graded interphase is presented by means of the least square boundary collocation technique. The distribution of thermal stress for the functionally graded interphase layer with arbitrary radial material parameters is derived by using the method of piece-wise homogeneous layers when the finite matrix is subjected to uniform heat flow. The effects of matrix size, interphase thickness and compositional gradient on the interfacial thermal stress are discussed in detail. Numerical results show that the magnitude and distribution of interfacial thermal stress in the inclusion and matrix can be designed properly by controlling these parameters.     

Interpenetrating Polymer Networks with Spatially Graded Morphology Controllable by UV-Radiation Curing

Interpenetrating Polymer Networks (IPNs) composed of polystyrene (PS) and poly(methyl methacrylate) (PMMA) were synthesized from a precursor mixture by using dissimilar photo-cross-link reactions. When the reation yields exceeded a certain threshold, the mixture was quenched from one-phase region into two-phase region, leading to phase separation. Upon irradiation with strong UV-light, an intensity gradient was formed along the propagating direction of the exciting light, generating a gradient of quench depth via the spatial inhomogeneity of the cross-link reactions. As a consequence, a gradient of the characteristic length scales was continuously generated from the top to the bottom of the mixture. The resulting three-dimensional (3-D) morphology was in-situ observed at different depths of the mixture by using a laser-scanning confocal microscope (LSCM). From this 3-D observation, it was found that phase separation was accelerated at the bottom of the mixture and proceeded in an autocatalytic fashion. The mechanism for the formation of the graded morphology was discussed in conjunction with the kinetics of the autocatalytic phase separation.     

Thermal waves scattering by a subsurface sphere in a semi-infinite exponentially graded material using non-Fourier's model

In this study, the multiple scattering of thermal waves and temperature distribution resulting from a subsurface sphere in a semi-infinite exponentially graded material are investigated, and the analytical expression of the temperature at the surface of the graded material is obtained. Non-Fourier heat conduction equation is applied to solve the temperature at the surface, and the image method is used to satisfy the semi-infinite boundary condition of graded material. The thermal wave fields are expressed using wave function expansion method, and the expanded mode coefficients are determined by satisfying the boundary condition of the sphere. According to the wave equation of heat conduction, a general solution of scattered thermal waves is presented for the first time. The temperature distribution and phase difference at the surface of the semi-infinite material with different parameters are graphically presented. Analyses show that the hyperbolic heat conduction equation cannot be regarded as a continuation of the parabolic heat conduction equation at very short time scale. The effects of the incident wave number, the structural and physical parameters on the distribution of temperature and phase difference in the semi-infinite material are also examined.     

On Orderable Poly Engel Groups

It is known that an orderable n-Engel group is nilpotent. We show that an orderable group that is an extension of an n-Engel group by an n-Engel group is nilpotent-by-nilpotent. However, a finitely generated orderable poly n-Engel group need not be solvable in general.