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By using the derivative method, we obtained the same result with that of the previous work of Chen et al. in 2006. Different from the integral form, the derivative form of the surface expression published in this paper is derived from differential equation and based on the theory of non-imaging focusing heliostat proposed by Chen et al. in 2001. Th...
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... angle θ fo- cusing on one point in the position of the target. Denote * E-mail address: leeeee@mail.ustc.edu.cn † E-mail address: ytchen@ustc.edu.cn ‡ E-mail address: shu@ustc.edu.cn the distance between the target T and origin O by L, then the coordinate of the target is (2L sin θ, 0, L cos θ). Firstly, focus on the cross section plane y = 0 ( Fig. 2). Denote the truncated curve of the mirror surface by˜zby˜ by˜z(x), which is a function of the coordinate x. The normal di- rection is perpendicular to the tangent at every point x. Compared with the supporting plane of the heliostat, the tangent at position x changed an angle arctañ z (x), there- fore the normal at position x changed ...
Citations
Among others, the removal of boron is the key issue to effectively purify metallurgical silicon into solar grade silicon. An innovative slagging process is proposed in this article after reviewing many previous studies. In the process, strong radiation is employed as catalysis to promote the chemical reaction. The authors have for the first time unveiled the secrets including the newly developed high temperature tools, the materials used, the method of mixing materials before purification, the process for separating silicon from residuals after purification, etc.. In the meantime, the authors also propose a postulate to explain the mechanism of the catalysis process. Except the removal of boron, the processes to remove other impurities, such as phosphorous and residual metals using electron beam and magnetic floating directional solidification are also discussed for the completion of a new industrial process to fabricate pure silicon. In the article, the authors also extend their discussion to the possibility of using the proposed non-imaging heliostat in other fields; to the possibility to further manufacture electronic grade silicon using the same process; to the possibility to obtain more experimental evidences for the postulate of strong radiation catalysis and to other subjects related to authors' claimed creations. It is sincerely hoped that the publication of the innovations reported in this article may provide an alternative way in the route map for the use of renewable energy of low cost and low carbon emission.
We consider the generalised dispersive modified Benjamin–Bona–Mahony equation, which describes an approximation status for long surface wave existed in the non-linear dispersive media. By employing the truncated Painlevé expansion method, we derive its non-local symmetry and Bäcklund transformation. The non-local symmetry is localised by a new variable, which provides the corresponding non-local symmetry group and similarity reductions. Moreover, a direct method can be provided to construct a kind of finite symmetry transformation via the classic Lie point symmetry of the normal prolonged system. Finally, we find that the equation is a consistent Riccati expansion solvable system. With the help of the Jacobi elliptic function, we get its interaction solutions between solitary waves and cnoidal periodic waves.
With the help of symbolic computation, two types of complete scalar classification for dark Burgers’ equations are derived by requiring the existence of higher order differential polynomial symmetries. There are some free parameters for every class of dark Burgers’ systems; so some special equations including symmetry equation and dual symmetry equation are obtained by selecting the free parameter. Furthermore, two kinds of recursion operators for these dark Burgers’ equations are constructed by two direct assumption methods.
Under investigation in this paper is the variant Boussinesq system, which describes the propagation of surface long wave towards two directions in a certain deep trough. With the help of the truncated Painlevé expansion, we construct its nonlocal symmetry, Bäcklund transformation, and Schwarzian form, respectively. The nonlocal symmetries can be localised to provide the corresponding nonlocal group, and finite symmetry transformations and similarity reductions are computed. Furthermore, we verify that the variant Boussinesq system is solvable via the consistent Riccati expansion (CRE). By considering the consistent tan-function expansion (CTE), which is a special form of CRE, the interaction solutions between soliton and cnoidal periodic wave are explicitly studied.
In this article, the (2+1)-dimensional dispersive long wave equation (DLWE) is investigated, which is derived in the context of a water wave propagating in narrow infinitely long channels of finite constant depth. By using of the truncated Painlevé expansion, we construct its nonlocal symmetry and Bäcklund transformation. After implanting the equation into an enlarged one, then the residual symmetry is localised. Meanwhile, the symmetry group transformation can be computed from the prolonged system. Furthermore, the equation is verified to be consistent Riccati expansion (CRE) solvable. Outing from the CRE, the soliton-cnoidal wave interaction solution in terms of Jacobi elliptic functions and the third type of incomplete elliptic integral are studied, respectively.
Among others, the removal of boron is the key issue to effectively purify metallurgical silicon into solar grade silicon. An innovative slagging process is proposed in this article after reviewing many previous studies. In the process, strong radiation is employed as catalysis to promote the chemical reaction. The authors have for the first time unveiled the secrets including the newly developed high temperature tools, the materials used, the method of mixing materials before purification, the process for separating silicon from residuals after purification, etc.. In the meantime, the authors also propose a postulate to explain the mechanism of the catalysis process. Except the removal of boron, the processes to remove other impurities, such as phosphorous and residual metals using electron beam and magnetic floating directional solidification are also discussed for the completion of a new industrial process to fabricate pure silicon. In the article, the authors also extend their discussion to the possibility of using the proposed non-imaging heliostat in other fields; to the possibility to further manufacture electronic grade silicon using the same process; to the possibility to obtain more experimental evidences for the postulate of strong radiation catalysis and to other subjects related to authors' claimed creations. It is sincerely hoped that the publication of the innovations reported in this article may provide an alternative way in the route map for the use of renewable energy of low cost and low carbon emission.
