Abstract There exists rare integration formulas over quaternions due to the noncommutativity of quarternions multplication. Based on the class operator＇s formula of rotation group we derive a Gaussian integral formul...Abstract There exists rare integration formulas over quaternions due to the noncommutativity of quarternions multplication. Based on the class operator＇s formula of rotation group we derive a Gaussian integral formula for quaternions, which is similar in form to the integration for coherent state＇s completeness relation.展开更多
In this paper, we introduce new invariant sets, and the invariant sets and exact solutions to general reactiondiffusion equation are discussed. It is shown that there exist a class of exact solutions to the equations ...In this paper, we introduce new invariant sets, and the invariant sets and exact solutions to general reactiondiffusion equation are discussed. It is shown that there exist a class of exact solutions to the equations that belong to the invariant sets.展开更多
Geometric phases have natural manifestations in large deformations of geometrically exact rods. The primary concerns of this article are the physical implications and observable consequences of geometric phases arisin...Geometric phases have natural manifestations in large deformations of geometrically exact rods. The primary concerns of this article are the physical implications and observable consequences of geometric phases arising from the deformed patterns exhibited by a rod subjected to end moments. This mechanical problem is classical and has a long tradition dating back to Kirchhoff. However, the perspective from geometric phases seems to go more deeply into relations between local strain states and global geometry of shapes, and infuses genuinely new insights and better understanding, which enable one to describe this kind of deformation in a neat and elegant way. On the other hand, visual representations of these deformations provide beautiful illustrations of geometric phases and render the meaning of the abstract concept of holonomy more direct and transparent.展开更多
In this paper, we introduce a new invariant set Eo={u：ux=f＇（x）F（u）＋ε[g＇（x）-f＇（x）g（x）]F（u）×exp（-∫^u1/F（z）dz）}where f and g are some smooth functions of x, ε is a constant, and F is a smooth...In this paper, we introduce a new invariant set Eo={u：ux=f＇（x）F（u）＋ε[g＇（x）-f＇（x）g（x）]F（u）×exp（-∫^u1/F（z）dz）}where f and g are some smooth functions of x, ε is a constant, and F is a smooth function to be determined. The invariant sets and exact sohltions to nonlinear diffusion equation ut = （ D（u）ux）x ＋ Q（x, u）ux ＋ P（x, u）, are discussed. It is shown that there exist several classes of solutions to the equation that belong to the invariant set Eo.展开更多
Following Chen＇s method [Commun. Theor. Phys. 52 （2009） 549] to use ＆step line tilting to realize tip tilting, to achieve finer rotation, it is discovered that a 16-step line tilting method may realize a rotation ...Following Chen＇s method [Commun. Theor. Phys. 52 （2009） 549] to use ＆step line tilting to realize tip tilting, to achieve finer rotation, it is discovered that a 16-step line tilting method may realize a rotation two order smaller than that achieved by ＆step.展开更多
By direct calculation of rotation matrices of SO(3),we show how certain specific sequence of eight consecutiverotations of digital angles can yield a tilting of a facet mirror.We also design a detailed program specifi...By direct calculation of rotation matrices of SO(3),we show how certain specific sequence of eight consecutiverotations of digital angles can yield a tilting of a facet mirror.We also design a detailed program specifically to tiltan array of mirrors from planar orientation to the required focusing orientation.We describe how to use the 8-step torealize the focusing of the mirror array.We have found,in our designed program,an important feature of row-sharingduring the rotations for the columns and similarly the column-sharing during the rotations for the row.This feature cansave a lot of operating time during the actual realization of the mechanical movements.展开更多
In a recent paper the author constructed a continuous map from the configuration space of n distinct ordered points in 3-space to the flag manifold of the unitary group U(n), which is compatible with the action of the...In a recent paper the author constructed a continuous map from the configuration space of n distinct ordered points in 3-space to the flag manifold of the unitary group U(n), which is compatible with the action of the symmetric group. This map is also compatible with appropriate actions of the rotation group SO(3). In this paper the author studies the induced homomorphism in SO(3)-equivariant cohomology and shows that this contains much interesting information involving representations of the symmetric group.展开更多
The invariant sets and the solutions of the 1+2-dimensional generalized thin film equation are discussed. It is shown that there exists a class of solutions to the equations, which are invariant with respect to the se...The invariant sets and the solutions of the 1+2-dimensional generalized thin film equation are discussed. It is shown that there exists a class of solutions to the equations, which are invariant with respect to the set $$E_0 = \{ u:u_x = v_x F(u),u_y = v_y F(u)\} ,$$ where v is a smooth function of variables x, y and F is a smooth function of u. This extends the results of Galaktionov (2001) and for the 1+1-dimensional nonlinear evolution equations.展开更多
We investigate the orientably regular non-abelian coverings of regular maps.A complete classification of dihedral coverings of the Platonic maps for branching over faces(or,dually,vertices)is given.As a result,we gene...We investigate the orientably regular non-abelian coverings of regular maps.A complete classification of dihedral coverings of the Platonic maps for branching over faces(or,dually,vertices)is given.As a result,we generalise the results of Jones and Surowski on regular cyclic coverings of the Platonic maps.展开更多
Estimates of unemployment in the UK are based on data collected in the Labour Force Survey(LFS). The data is collected continuously and the survey design is structured in such a way asto provide quarterly estimates. T...Estimates of unemployment in the UK are based on data collected in the Labour Force Survey(LFS). The data is collected continuously and the survey design is structured in such a way asto provide quarterly estimates. These quarterly estimates are published each month as ‘rollingquarterly’ estimates. Currently the Office for National Statistics (ONS) publish rolling quarterlyestimates, and these have been assessed to be of sufficient quality to be badged as ‘NationalStatistics’. ONS also publish monthly estimates of a selection of labour force variables, but theseare designated ‘Experimental Statistics’ due to concerns over the quality of these data. Due to thesample design of the LFS, monthly estimates of change are volatile as there is no sample overlap.A state space model can be used to develop improved estimates of monthly change, accountingfor aspects of the survey design. An additional source of information related to unemployment isadministrative data on the number of people claiming unemployment related benefits. This datais more timely than survey data collected in the LFS and can be used to provide early estimatesof monthly unemployment.展开更多
基金National Natural Science Foundation of China under Grant No.10475056
文摘Abstract There exists rare integration formulas over quaternions due to the noncommutativity of quarternions multplication. Based on the class operator＇s formula of rotation group we derive a Gaussian integral formula for quaternions, which is similar in form to the integration for coherent state＇s completeness relation.
基金National Natural Science Foundation of China under Grant Nos.10472091,10332030 and 10502042the Natural Science Foundation of Shanxi Province under Grant No.2003A03
文摘In this paper, we introduce new invariant sets, and the invariant sets and exact solutions to general reactiondiffusion equation are discussed. It is shown that there exist a class of exact solutions to the equations that belong to the invariant sets.
文摘Geometric phases have natural manifestations in large deformations of geometrically exact rods. The primary concerns of this article are the physical implications and observable consequences of geometric phases arising from the deformed patterns exhibited by a rod subjected to end moments. This mechanical problem is classical and has a long tradition dating back to Kirchhoff. However, the perspective from geometric phases seems to go more deeply into relations between local strain states and global geometry of shapes, and infuses genuinely new insights and better understanding, which enable one to describe this kind of deformation in a neat and elegant way. On the other hand, visual representations of these deformations provide beautiful illustrations of geometric phases and render the meaning of the abstract concept of holonomy more direct and transparent.
基金National Natural Science Foundation of China under Grant Nos.10472091,10332030,and 10502042the Natural Science Foundation of Shaanxi Province under Grant No.2003A03
文摘In this paper, we introduce a new invariant set Eo={u：ux=f＇（x）F（u）＋ε[g＇（x）-f＇（x）g（x）]F（u）×exp（-∫^u1/F（z）dz）}where f and g are some smooth functions of x, ε is a constant, and F is a smooth function to be determined. The invariant sets and exact sohltions to nonlinear diffusion equation ut = （ D（u）ux）x ＋ Q（x, u）ux ＋ P（x, u）, are discussed. It is shown that there exist several classes of solutions to the equation that belong to the invariant set Eo.
基金Supported by NKBRPC under Grant No.2006CB805905National Natural Science Foundation of China under Grant Nos.10901153 and 10771203CNSF under Grant No.10773009,SRFDP,and CAS
文摘Following Chen＇s method [Commun. Theor. Phys. 52 （2009） 549] to use ＆step line tilting to realize tip tilting, to achieve finer rotation, it is discovered that a 16-step line tilting method may realize a rotation two order smaller than that achieved by ＆step.
文摘By direct calculation of rotation matrices of SO(3),we show how certain specific sequence of eight consecutiverotations of digital angles can yield a tilting of a facet mirror.We also design a detailed program specifically to tiltan array of mirrors from planar orientation to the required focusing orientation.We describe how to use the 8-step torealize the focusing of the mirror array.We have found,in our designed program,an important feature of row-sharingduring the rotations for the columns and similarly the column-sharing during the rotations for the row.This feature cansave a lot of operating time during the actual realization of the mechanical movements.
文摘In a recent paper the author constructed a continuous map from the configuration space of n distinct ordered points in 3-space to the flag manifold of the unitary group U(n), which is compatible with the action of the symmetric group. This map is also compatible with appropriate actions of the rotation group SO(3). In this paper the author studies the induced homomorphism in SO(3)-equivariant cohomology and shows that this contains much interesting information involving representations of the symmetric group.
基金This work was supported by the National Natural Science Foundation of China (Grant No. 10671156)the Program for New Century Excellent Talents in Universities (Grant No. NCET-04-0968)
文摘The invariant sets and the solutions of the 1+2-dimensional generalized thin film equation are discussed. It is shown that there exists a class of solutions to the equations, which are invariant with respect to the set $$E_0 = \{ u:u_x = v_x F(u),u_y = v_y F(u)\} ,$$ where v is a smooth function of variables x, y and F is a smooth function of u. This extends the results of Galaktionov (2001) and for the 1+1-dimensional nonlinear evolution equations.
文摘We investigate the orientably regular non-abelian coverings of regular maps.A complete classification of dihedral coverings of the Platonic maps for branching over faces(or,dually,vertices)is given.As a result,we generalise the results of Jones and Surowski on regular cyclic coverings of the Platonic maps.
文摘Estimates of unemployment in the UK are based on data collected in the Labour Force Survey(LFS). The data is collected continuously and the survey design is structured in such a way asto provide quarterly estimates. These quarterly estimates are published each month as ‘rollingquarterly’ estimates. Currently the Office for National Statistics (ONS) publish rolling quarterlyestimates, and these have been assessed to be of sufficient quality to be badged as ‘NationalStatistics’. ONS also publish monthly estimates of a selection of labour force variables, but theseare designated ‘Experimental Statistics’ due to concerns over the quality of these data. Due to thesample design of the LFS, monthly estimates of change are volatile as there is no sample overlap.A state space model can be used to develop improved estimates of monthly change, accountingfor aspects of the survey design. An additional source of information related to unemployment isadministrative data on the number of people claiming unemployment related benefits. This datais more timely than survey data collected in the LFS and can be used to provide early estimatesof monthly unemployment.
