A Type of New Loop Algebra and a Generalized Tu Formula

A new Lie algebra, which is far different form the known An-1, is established, for which the corresponding loop algebra is given. From this, two isospectral problems are revealed, whose compatibility condition reads a kind of zero curvature equation, which permits Lax integrable hierarchies of soliton equations. To aim at generating Hamiltonian structures of such soliton-equation hierarchies, a beautiful Killing-Cartan form, a generalized trace functional of matrices, is given, for which a generalized Tu formula （GTF） is obtained, while the trace identity proposed by Tu Guizhang [J. Math. Phys. 30 （1989） 330] is a special case of the GTF. The computing formula on the constant γ to be determined appearing in the GTF is worked out, which ensures the exact and simple computation on it. Finally, we take two examples to reveal the applications of the theory presented in the article. In details, the first example reveals a new Liouville-integrable hierarchy of soliton equations along with two potential functions and Hamiltonian structure. To obtain the second integrable hierarchy of soliton equations, a higher-dimensional loop algebra is first constructed. Thus, the second example shows another new Liouville integrable hierarchy with 5-potential component functions and bi- Hamiltonian structure. The approach presented in the paper may be extensively used to generate other new integrable soliton-equation hierarchies with multi-Hamiltonian structures.     

A Complex Higher-Dimensional Lie Algebra with Real and Imaginary Structure Constants as Well as Its Decomposition

A new Lie algebra G of the Lie algebra sl（2） is constructed with complex entries whose structure constants are real and imaginary numbers. A loop algebra G corresponding to the Lie algebra G is constructed, for which it is devoted to generating a soliton hierarchy of evolution equations under the framework of generalized zero curvature equation which is derived from the compatibility of the isospectral problems expressed by Hirota operators. Finally, we decompose the Lie algebra G to obtain the subalgebras G1 and G2. Using the G2 and its one type of loop algebra G2, a Liouville integrable soliton hierarchy is obtained, furthermore, we obtain its bi-Hamiltonian structure by employing the quadratic-form identity.     

P-Twisted Affine Lie Algebras and Their Associated Vertex Algebras

For any finite-dimensional semisimple Lie algebra g, a Z＋-graded vertex algebra is construsted on the vacuum representation Vk（g[θ]of g[θ]）,which is a one-dimentionM central extension of 8-invariant subspace on the loop algebra Lg=g C（（t^1/p））.     

Characteristic Numbers of Matrix Lie Algebras

A notion of characteristic number of matrix Lie algebras is defined, which is devoted to distinguishing various Lie algebras that are used to generate integrable couplings of soliton equations. That is, the exact classification of the matrix Lie algebras by using computational formulas is given. Here the characteristic numbers also describe the relations between soliton solutions of the stationary zero curvature equations expressed by various Lie algebras.     

Induced Lie Algebras of a Six-Dimensional Matrix Lie Algebra

By using a six-dimensional matrix Lie algebra [Y.F. Zhang and Y. Wang, Phys. Lett. A 360 （2006） 92], three induced Lie algebras are constructed. One of them is obtained by extending Lie bracket, the others are higher-dimensional complex Lie algebras constructed by using linear transformations. The equivalent Lie algebras of the later two with multi-component forms are obtained as well. As their applications, we derive an integrable coupling and quasi-Hamiltonian structure of the modified TC hierarchy of soliton equations.     

A New Liouville Integrable Hamiltonian System

With the help of a Lie algebra, an isospectral Lax pair is introduced for which a new Liouville integrable hierarchy of evolution equations is generated. Its Hamiltonian structure is also worked out by use of the quadratic-form identity.     

Mutual Information and Relative Entropy of Sequential Effect Algebras

In this paper, we introduce and investigate the mutual information and relative entropy on the sequential effect algebra, we also give a comparison of these mutual information and relative entropy with the classical ones by the venn diagrams. Finally, a nice example shows that the entropies of sequential effect algebra depend extremely on the order of its sequential product.     

Two Integrable Couplings of Modified Tu Hierarchy

Two different integrable couplings of the modified Tu hierarchy are obtained under the zero curvature equation by using two higher dimension Lie algebras. Furthermore, a complex Hamiltonian structures of the second integrable couplings is presented by taking use of the variational identity.     

Realization of Vertex Operators of 7-Twisted Affine Lie Algebra sl（3,C）[θ]

In this paper, we construct a new algebra structure 7-twisted atone Lie algebra sl（3,C）[θ] and study its vertex operator representations.     

New Matrix Loop Algebra and Its Application

A new matrix Lie algebra and its corresponding Loop algebra are constructed firstly, as its application, the multi-component TC equation hierarchy is obtained, then by use of trace identity the Hamiltonian structure of the above system is presented. Finally, the integrable couplings of the obtained system is worked out by the expanding matrix Loop algebra.     

A New Lie Algebra and Its Related Liouville Integrable Hierarchies

A new Lie algebra G and its two types of loop algebras G1 and G2 are constructed. Basing on G1 and G2, two different isospectral problems are designed, furthermore, two Liouville integrable soliton hierarchies are obtained respectively under the framework of zero curvature equation, which is derived from the compatibility of the isospectral problems expressed by Hirota operators. At the same time, we obtain the Hamiltonian structure of the first hierarchy and the bi-Hamiltonian structure of the second one with the help of the quadratic-form identity.     

Two Types of Expanding Lie Algebra and New Expanding Integrable Systems

From a new Lie algebra proposed by Zhang, two expanding Lie algebras and its corresponding loop algebras are obtained. Two expanding integrable systems are produced with the help of the generalized zero curvature equation. One of them has complex Hamiltion structure with the help of generalized Tu formula （GTM）.     

Kac-Moody-Virasoro Symmetry Algebra of （2＋1）-Dimensional Dispersive Long-Wave Equation with Arbitrary Order Invariant

By Lie symmetry method, the Lie point symmetries and its Kac Moody-Virasoro （KMV） symmetry algebra of （2＋1）-dimensional dispersive long-wave equation （DLWE） are obtained, and the finite transformation of DLWE is given by symmetry group direct method, which can recover Lie point symmetries. Then KMV symmetry algebra of DLWE with arbitrary order invariant is also obtained. On basis of this algebra the group invariant solutions and similarity reductions are also derived.     

kth-order antibunching effect for a new kind of excited even and odd q-coherent states

A new kind of excited even q-coherent states （aq^-1）^m｜α〉q^e and excited odd q-coherent states （aq^-1）^m｜α〉q^o is constructed by acting with inverse boson operators on the even and odd q-coherent states. The m dependence of the kth-order antibunching effect is numerically studied for k = 2, 3, 4, 5. It is shown that the kth-order antibunching effect enhances as m increases. The larger k, the quicker the antibunching effect enhances.     

Hamiltonian Forms for a Hierarchy of Discrete Integrable Coupling Systems

A semi-direct sum of two Lie algebras of four-by-four matrices is presented, and a discrete four-by-four matrix spectral problem is introduced. A hierarchy of discrete integrable coupling systems is derived. The obtained integrable coupling systems are all written in their Hamiltonian forms by the discrete variational identity. Finally, we prove that the lattice equations in the obtained integrable coupling systems are all Liouville integrable discrete Hamiltonian systems.     

An ART iterative reconstruction algorithm for computed tomography of diffraction enhanced imaging

X-ray diffraction enhanced imaging （DEI） has extremely high sensitivity for weakly absorbing low- Z samples in medical and biological fields. In this paper, we propose an Algebra Reconstruction Technique （ART） iterative reconstruction algorithm for computed tomography of diffraction enhanced imaging （DEI-CT）. An Ordered Subsets （OS） technique is used to accelerate the ART reconstruction. Few-view reconstruction is also studied, and a partial differential equation （PDE） type filter which has the ability of edge-preserving and denoising is used to improve the image quality and eliminate the artifacts. The proposed algorithm is validated with both the numerical simulations and the experiment at the Beijing synchrotron radiation facility （BSRF）.     

Applications of Polynomial Algebras to 2-Dimensional Deformed Oscillators

The polynomial algebra is a deformed su(2) algebra. Here, we use polynomial algebra as a method to solve a series of deformed oscillators. Thus, we find a series of physics systems corresponding with polynomial algebra with different highest orders.     

多项式角动量代数的代数表示及实现

本文利用三参数李群求代数表示的方法求出多项式角动量代数的代数表示及其酉表示，找到一个能同时承载李代数及相对应的多项式角动量代数的基底，并在该基底下求出两种代数之间的联系，利用该联系则也可求出多项式角动量代数的代数表示.最后求出多项式角动量代数的单玻色实现及其在有限维多项式函数空间的微分实现. 关键词： 多项式角动量代数 Higgs代数 su(2)代数     

P-Twisted Affine Lie Algebras and Their Associated Vertex Algebras

For any finite-dimensional semisimple Lie algebra g, a Z₊-graded vertex algebra is construsted on the vacuum representation V_k(\hat{g}[ θ]) of \hat{g}[θ], which is a one-dimentional central extension of θ-invariant subspace on the loop algebra Lg=g\otimes C((t^1/p)).