We present the theories used in solid-state nuclear magnetic resonance and the expansion schemes used as numerical integrators for solving the time dependent Schrodinger Equation. We highlight potential future theoret...We present the theories used in solid-state nuclear magnetic resonance and the expansion schemes used as numerical integrators for solving the time dependent Schrodinger Equation. We highlight potential future theoretical and numerical directions in solid-state nuclear magnetic resonancesuch as the Chebychev expansion and the transformation of Cayley.展开更多
We consider computing the minimal nonnegative solution of the nonsymmetric algebraic Riccati equation with M-matrix.It is well known that such equations can be efficiently solved via the structure-preserving doubling ...We consider computing the minimal nonnegative solution of the nonsymmetric algebraic Riccati equation with M-matrix.It is well known that such equations can be efficiently solved via the structure-preserving doubling algorithm(SDA)with the shift-and-shrink transformation or the generalized Cayley transformation.In this paper,we propose a more generalized transformation of which the shift-and-shrink transformation and the generalized Cayley transformation could be viewed as two special cases.Meanwhile,the doubling algorithm based on the proposed generalized transformation is presented and shown to be well-defined.Moreover,the convergence result and the comparison theorem on convergent rate are established.Preliminary numerical experiments show that the doubling algorithm with the generalized transformation is efficient to derive the minimal nonnegative solution of nonsymmetric algebraic Riccati equation with M-matrix.展开更多
Some mathematical aspects of the Lie groups SU (2) and in realization by two pairs of boson annihilation and creation operators and in the parametrization by the vector parameter instead of the Euler angles and ...Some mathematical aspects of the Lie groups SU (2) and in realization by two pairs of boson annihilation and creation operators and in the parametrization by the vector parameter instead of the Euler angles and the vector parameter c of Fyodorov are developed. The one-dimensional root scheme of SU (2) is embedded in two-dimensional root schemes of some higher Lie groups, in particular, in inhomogeneous Lie groups and is represented in text and figures. The two-dimensional fundamental representation of SU (2) is calculated and from it the composition law for the product of two transformations and the most important decompositions of general transformations in special ones are derived. Then the transition from representation of SU (2) to of is made where in addition to the parametrization by vector the convenient parametrization by vector c is considered and the connections are established. The measures for invariant integration are derived for and for SU (2) . The relations between 3D-rotations of a unit sphere to fractional linear transformations of a plane by stereographic projection are discussed. All derivations and representations are tried to make in coordinate-invariant way.展开更多
A theorem of transformation between two geometries is proved on the basis of Wu’s method of mechanical theorem proving.By using this theoremfor Cayley-Klein geometries of dimension two,we prove that the nine Cayley-K...A theorem of transformation between two geometries is proved on the basis of Wu’s method of mechanical theorem proving.By using this theoremfor Cayley-Klein geometries of dimension two,we prove that the nine Cayley-Klein geometries can be divided into three groups within which the geometries aremutually equivalent in the sense that certain geometry statements are correct inone geometry if and only if they are correct in the other geometries of the samegroup.This means that for each group we only need to choose a model geometryto study and the theorems of other geometries in the same group can be obtainedfrom the model geometry automatically.The three model geometries chosen forthe nine Cayley-Klein geometries are:Euclidean geometry,Riemann geometry,andGalilean geometry.展开更多
In an inner-product space, an invertible vector generates a reflection with respect to a hyperplane, and the Clifford product of several invertible vectors, called a versor in Clifford algebra, generates the compositi...In an inner-product space, an invertible vector generates a reflection with respect to a hyperplane, and the Clifford product of several invertible vectors, called a versor in Clifford algebra, generates the composition of the corresponding reflections, which is an orthogonal transformation. Given a versor in a Clifford algebra, finding another sequence of invertible vectors of strictly shorter length but whose Clifford product still equals the input versor, is called versor compression. Geometrically, versor compression is equivalent to decomposing an orthogonal transformation into a shorter sequence of reflections. This paper proposes a simple algorithm of compressing versors of symbolic form in Clifford algebra. The algorithm is based on computing the intersections of lines with planes in the corresponding Grassmann-Cayley algebra, and is complete in the case of Euclidean or Minkowski inner-product space.展开更多
文摘We present the theories used in solid-state nuclear magnetic resonance and the expansion schemes used as numerical integrators for solving the time dependent Schrodinger Equation. We highlight potential future theoretical and numerical directions in solid-state nuclear magnetic resonancesuch as the Chebychev expansion and the transformation of Cayley.
基金The work of B.Tang was supported partly by Hunan Provincial Innovation Foundation for Postgraduate(No.CX2016B249)Hunan Provincial Natural Science Foundation of China(No.2018JJ3019)+1 种基金The work of N.Dong was supported partly by the Hunan Provincial Natural Science Foundation of China(Nos.14JJ2114,2017JJ2071)the Excellent Youth Foundation and General Foundation of Hunan Educational Department(Nos.17B071,17C0466).
文摘We consider computing the minimal nonnegative solution of the nonsymmetric algebraic Riccati equation with M-matrix.It is well known that such equations can be efficiently solved via the structure-preserving doubling algorithm(SDA)with the shift-and-shrink transformation or the generalized Cayley transformation.In this paper,we propose a more generalized transformation of which the shift-and-shrink transformation and the generalized Cayley transformation could be viewed as two special cases.Meanwhile,the doubling algorithm based on the proposed generalized transformation is presented and shown to be well-defined.Moreover,the convergence result and the comparison theorem on convergent rate are established.Preliminary numerical experiments show that the doubling algorithm with the generalized transformation is efficient to derive the minimal nonnegative solution of nonsymmetric algebraic Riccati equation with M-matrix.
文摘Some mathematical aspects of the Lie groups SU (2) and in realization by two pairs of boson annihilation and creation operators and in the parametrization by the vector parameter instead of the Euler angles and the vector parameter c of Fyodorov are developed. The one-dimensional root scheme of SU (2) is embedded in two-dimensional root schemes of some higher Lie groups, in particular, in inhomogeneous Lie groups and is represented in text and figures. The two-dimensional fundamental representation of SU (2) is calculated and from it the composition law for the product of two transformations and the most important decompositions of general transformations in special ones are derived. Then the transition from representation of SU (2) to of is made where in addition to the parametrization by vector the convenient parametrization by vector c is considered and the connections are established. The measures for invariant integration are derived for and for SU (2) . The relations between 3D-rotations of a unit sphere to fractional linear transformations of a plane by stereographic projection are discussed. All derivations and representations are tried to make in coordinate-invariant way.
文摘A theorem of transformation between two geometries is proved on the basis of Wu’s method of mechanical theorem proving.By using this theoremfor Cayley-Klein geometries of dimension two,we prove that the nine Cayley-Klein geometries can be divided into three groups within which the geometries aremutually equivalent in the sense that certain geometry statements are correct inone geometry if and only if they are correct in the other geometries of the samegroup.This means that for each group we only need to choose a model geometryto study and the theorems of other geometries in the same group can be obtainedfrom the model geometry automatically.The three model geometries chosen forthe nine Cayley-Klein geometries are:Euclidean geometry,Riemann geometry,andGalilean geometry.
文摘In an inner-product space, an invertible vector generates a reflection with respect to a hyperplane, and the Clifford product of several invertible vectors, called a versor in Clifford algebra, generates the composition of the corresponding reflections, which is an orthogonal transformation. Given a versor in a Clifford algebra, finding another sequence of invertible vectors of strictly shorter length but whose Clifford product still equals the input versor, is called versor compression. Geometrically, versor compression is equivalent to decomposing an orthogonal transformation into a shorter sequence of reflections. This paper proposes a simple algorithm of compressing versors of symbolic form in Clifford algebra. The algorithm is based on computing the intersections of lines with planes in the corresponding Grassmann-Cayley algebra, and is complete in the case of Euclidean or Minkowski inner-product space.
