Solution of Schrodinger Equation for Two-Dimensional Complex Ouartic Potentials

We investigate the quasi-exact solutions of the Schrodinger wave equation for two-dimensional non-hermitian complex Hamiltonian systems within the frame work of an extended complex phase space characterized by x = x1 ＋ ip3, y = x2 ＋ ip4, px= p1＋ ix3, py= p2 ＋ ix4. Explicit expressions of the energy eigenvalues and the eigenfunctions for ground and first excited states for a complex quartic potential are obtained. Eigenvalue spectra of some variants of the complex quartic potential, including PT-symmetrie one, are also worked out.     

Does the eigenratio synchronizability of λ2/λN represent the a complex network？

In the study of complex networks, it is commonly believed that the eigenratio λ2/λN of the Laplacian matrix of a network represents the network synchronizability, especially for symmetric networks. This paper gives two counterexamples to show that this is not true for the case where the network has a disconnected synchronized region. Consequently, a simple answer is presented to the question of when the eigenratio λ2/λN does represent the network synchronizability.     

Undetermination of the synchronizability and relation between network betweenness centrality

Betweenness centrality is taken as a sensible indicator of the synchronizability of complex networks. To test whether betweenness centrality is a proper measure of the synchronizability in specific realizations of random networks, this paper adds edges to the networks and then evaluates the changes of betweenness centrality and network synchronizability. It finds that the two quantities vary independently.     

Fusion,fission, and annihilation of complex waves for the （2＋l）-dimensional generalized Calogero-Bogoyavlenskii-Schiff system

With the help of the symbolic computation system, Maple and Riccati equation （ξ＇ = ao ＋ a1ξ＋ a2ξ2）, expansion method, and a linear variable separation approach, a new family of exact solutions with q = lx ＋ my ＋ nt ＋ Г（x,y, t） for the （2＋1）-dimensional generalized Calogero-Bogoyavlenskii-Schiff system （GCBS） are derived. Based on the derived solitary wave solution, some novel localized excitations such as fusion, fission, and annihilation of complex waves are investigated.     

Complex Wave Solutions for （2＋1）-Dimensional Modified Dispersive Water Wave System

The extended Riccati mapping approach^[1] is further improved by generalized Riccati equation, and combine it with variable separation method, abundant new exact complex solutions for the （2＋1）-dimensional modified dispersive water-wave （MDWW） system are obtained. Based on a derived periodic solitary wave solution and a rational solution, we study a type of phenomenon of complex wave.     

Numerical Complexiton Solutions of Complex KdV Equation

In this paper, we directly extend the applications of the Adomian decomposition method to investigate the complex KdV equation. By choosing different forms of wave functions as the initial values, three new types of realistic numerical solutions： numerical positon, negaton solution, and particularly the numerical analytical complexiton solution are obtained, which can rapidly converge to the exact ones obtained by Lou et al. Numerical simulation figures are used to illustrate the efficiency and accuracy of the proposed method.     

Higher-Dimensional Integrable Systems Arising from Motions of Curves on S^2（R） and S^3（R）

We show that higher-dimensional integrable systems including the （2＋1）-dimensional generalized sine-Gordon equation and the （2＋1）-dimensional complex mKdV equation are associated with motions of surfaces induced by endowing with an extra space variable to the motions of curves on S^2（R） and S^3（R）.     

Complex solutions and novel complex wave localized excitations for the(2+1)-dimensional Boiti–Leon–Pempinelli system

With the help of the symbolic computation system Maple, the Riccati equation mapping approach and a linear variable separation approach, a new family of complex solutions for the （2＋ 1）-dimensional Boiti-Leon-Pempinelli system （BLP） is derived. Based on the derived solitary wave solution, some novel complex wave localized excitations are obtained.     

Exponential Synchronization of Uncertain Complex Dynamical Networks with Delay Coupling

This paper studies the global exponential synchronization of uncertain complex delayed dynamical networks. The network model considered is general dynamical delay networks with unknown network structure and unknown coupling functions but bounded. Novel delay-dependent linear controllers are designed via the Lyapunov stability theory. Especially, it is shown that the controlled networks are globally exponentially synchronized with a given convergence rate. An example of typical dynamical network of this class, having the Lorenz system at each node, has been used to demonstrate and verify the novel design proposed. And, the numerical simulation results show the effectiveness of proposed synchronization approaches.     

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.     

Co-evolution of the brand effect and competitiveness in evolving networks

The principle that ‘the brand effect is attractive’ underlies the preferential attachment. Here we show that the brand effect is just one dimension of attractiveness. Another dimension is competitiveness. We firstly introduce a general framework that allows us to investigate the competitive aspect of real networks, instead of simply preferring popular nodes. Our model accurately describes the evolution of social and technological networks. The phenomenon that more competitive nodes become richer can help us to understand the evolution of many competitive systems in nature and society. In general,the paper provides an explicit analytical expression of degree distributions of the network. In particular, the model yields a nontrivial time evolution of nodes’ properties and the scale-free behavior with exponents depending on the microscopic parameters characterizing the competition rules. Secondly, through theoretical analyses and numerical simulations, we reveal that our model has not only the universality for the homogeneous weighted network, but also the character for the heterogeneous weighted network. Thirdly, we also develop a model based on the profit-driven mechanism. It can better describe the observed phenomenon in enterprise cooperation networks. We show that the standard preferential attachment,the growing random graph, the initial attractiveness model, the fitness model, and weighted networks can all be seen as degenerate cases of our model.     

Parseval Theory of Complex Wavelet Transform for Wavelet Family Including Rotational Parameters

A rotational parameter Rθ has been introduced to complex wavelet transform （CWT）. The rotational CWT （RCWT） corresponds to a matrix element 〈φ｜U2（θ;μ;κ）[F〉 in the context of quantum mechanics, where U2（θ;μ;κ） is a two-mode rotational displacing-squeezing operator in the 〈η｜ representation. Based on this, the Parseval theorem and the inversion formula of RCWT have been proved. The concise proof not only manifestly shows the merit of Dirac＇s representation theory but also leads to a new orthogonal property of complex mother wavelets in parameter space.     

Robust H∞ observer-based of a class of complex control for synchronization dynamical networks

This paper is concerned with the robust Hoo synchronization problem for a class of complex dynamical networks by applying the observer-based control. The proposed feedback control scheme is developed to ensure the asymptotic stability of the augmented system, to reconstruct the non-measurable state variables of each node and to improve the H∞ performance related to the synchronization error and observation error despite the external disturbance. Based on the Lyapunov stability theory, a synchronization criterion is obtained under which the controlled network can be robustly stabilized onto a desired state with a guaranteed H∞ performance. The controller and the observer gains can be given by the feasible solutions of a set of linear matrix inequalities （LMIs）. The effectiveness of the proposed control scheme is demonstrated by a numerical example through simulation.     

A Formula of Average Path Length for Unweighted Networks

In this paper, based on the adjacency matrix of the network and its powers, the formulas are derived for the shortest path and the average path length, and an effective algorithm is presented. Furthermore, an example is provided to demonstrate the proposed method.     

Transmission time in the reflectionless complex potential

A phase time definition directly obtained from the Schr6dinger equation is used to investigate the time delay of a particle scattered by complex reflectionless potential. The artifacts introduced by truncating in the numerical simulation are clarified. The time delay of the transmitted wave packet is found to be equal to the reflection time of the truncated potential. Both time delays are the same as the traversal time in the free space, but shorter than the time taken by a classical particle to pass the same potential.     

Inconsistence of Non-Singular Spherically Symmetric Solution and a Satisfactory Energy-Momentum Complex

Two spherically symmetric non-singular black hole solutions in Moiler tetrad theory of gravitation have been obtained. Although the two solutions have the same form of metric （spherically symmetric nonsingular black hole）, their energy contents are different. We use another method given by Gibbons and Hawking to calculate the energy content of these solutions. We also obtained different value of energy. Study the requirements of a satisfactory energymomentum complex given by Moiler we find that the second solution, which behaves as 1/√r, is not transformed as a four-vector under Lorentz transformation.     

Some Topological Properties of Anyons

In this paper, starting with the well known U（1） Chern-Simons Lagrangian and the covariant derivative of a complex scalar matter field, we give a detailed discussion of some topological properties of anyons. We show that the ＂basic＂ charge carried by anyons has an inner structure and can be decomposed in terms of the Chern-Simons coupling and the gauge coupling constants of the theory. Also some incorrect results obtained in the literature are revised.     

1D配合物[Tb(3,5-pdc)(3,5-Hpdc)(H2O)(phen)]n的合成、晶体结构及发光性质

用硝酸铽与3,5-吡啶二甲酸、邻菲罗啉在水热条件下进行反应,合成了配合物[Tb(3,5-pdc)(3,5-Hpdc)(H₂O)(phen)]_n(3,5-H₂pdc =3,5-吡啶二甲酸, phen=1,10-邻菲罗啉)。单晶X射线衍射测试表明,该配合物属单斜晶系,C2/c空间群,晶胞参数分别为:a=2.734 9(3) nm,b=1.099 48(12) nm,c=2.200 0(3) nm,α=γ= 90°,β=132.147 3(12)°,Z= 8,V=4 904.8(10) nm³。配合物中Tb³⁺与配体3,5-pdc^2-和3,5-Hpdc^-交替连接形成了一维链结构,链与链之间通过邻菲罗啉芳香环的π…π 堆积弱相互作用形成了三维结构。在波长为352 nm的紫外光激发下,配合物发射出Tb³⁺的特征荧光。     

Remarks on interpretations of the Eotvos experiment and misinterpretation of E=mc^2

The Eotvos experiment on the verification of equivalence between inertial mass and gravitational mass of a body is famous for its accuracy. A question is, however, can these experimental results be applied to the case of a physical space in general relativity, where the space coordinates could be arbitrary？ It is pointed out that it can be validly applied because it has been proven that Einstein＇s equivalence principle for a physical space must have a frame of reference with the Euclidean-like structure. Will claimed further that such an overall accuracy can be translated into an accuracy of the equivalence between inertial mass and each type of energy. It is shown that, according to general relativity, such a claim is incorrect. The root of this problem is due to an inadequate understanding of special relativity that produced the famous equation E=mc^2, which must be understood in terms of energy conservation. Concurrently, it is pointed out that this error is a problem in Will＇s book, ‘Theory and Experiment in Gravitational Physics＇.     

First-Principles Calculations of Atomic and Electronic Properties of T1 and In on Si（111）

The atomic and electronic structures of T1 and In on Si（111） surfaces are investigated using the firstprinciples total energy calculations. Total energy optimizations show that the energetically favored structure is 1/3 ML T1 adsorbed at the T4 sites on Si（111） surfaces. The adsorption energy difference of one T1 adatom between （√3 × √3） and （1 × 1） is less than that of each In adatom. The DOS indicates that TI 6p and Si 3p electrons play a very important role in the formation of the surface states. It is concluded that the bonding of TI adatoms on Si（111） surfaces is mainly polar covalent, which is weaker than that of In on Si（111）. So T1 atom is more easy to be migrated than In atom in the same external electric field and the structures of T1 on Si（111） is prone to switch between （√3 × √3） and （1 × 1）.