Symmetries of Type D Pure Radiation Fields

The geometrical symmetries corresponding to the continuous groups of collineations and motions generated by a null vector l are considered. These symmetries have been translated into the language of Newman-Penrose formalism for pure radiation (PR) type D fields. It is seen that for such fields, conformal, special conformal and homothetic motions degenerate to motion. The concept of free curvature, matter curvature and matter affine collineations have been introduced and the conditions under which PR type D fields admit such collineations have been obtained. Moreover, it is shown that the projective collineation degenerate to matter affine, special projective, conformal, special conformal, null geodesic and special null geodesic collineations. It is also seen that type D pure radiation fields admit Maxwell collineation along the propagation vector l.     

Lag projective synchronization in fractional-order chaotic (hyperchaotic) systems

In this Letter, a new lag projective synchronization for fractional-order chaotic (hyperchaotic) systems is proposed, which includes complete synchronization, anti-synchronization, lag synchronization, generalized projective synchronization. It is shown that the slave system synchronizes the past state of the driver up to a scaling factor. A suitable controller for achieving the lag projective synchronization is designed based on the stability theory of linear fractional-order systems and the pole placement technique. Two examples are given to illustrate effectiveness of the scheme, in which the lag projective synchronizations between fractional-order chaotic Rössler system and fractional-order chaotic Lü system, between fractional-order hyperchaotic Lorenz system and fractional-order hyperchaotic Chen system, respectively, are successfully achieved. Corresponding numerical simulations are also given to verify the analytical results.     

Projective Synchronization Between Two Nonidentical Variable Time Delayed Systems

In this paper,we propose a method for the projective synchronization between two different chaotic systems with variable time delays.Using active control approach,the suitable controller is constructed to make the states of two different diverse time delayed systems asymptotically synchronize up to the desired scaling factor.Based on the Lyapunov stability theory,the sufficient condition for the projective synchronization is calculated theoretically.Numerical simulations of the projective synchronization between Mackey-Glass system and Ikeda system with variable time delays are shown to validate the effectiveness of the proposed algorithm.     

Integrable System and Motion of Curves in Projective and Similarity Geometries

Based on the natural frame in the projective geometry, motions of curves in projective geometry are studied. It is shown that several integrable equations including Sawada-Kotera and KK equations arise from motion of plane curves in projective geometries. Motion of space curves described by acceleration field and governed by endowing an extra space variable in similarity geometry P³ is also studied.     

Function projective synchronization in partially linear drive-response chaotic systems

This paper gives the definition of function projective synchronization with less conservative demand for a scaling function,and investigates the function projective synchronization in partially linear drive-response chaotic systems.Based on the Lyapunov stability theory,it has been shown that the function projective synchronization with desired scaling function can be realized by simple control law.Moreover it does not need scaling function to be differentiable,bounded and non-vanished.The numerical simulations are provided to verify the theoretical result.     

Geometrical aspects of Schlesinger''s equation

The equation (Schlesinger's equation) for the isomonodromic deformations of an (SL (2, C) connection with four simple poles on the projective line is shown to describe a holomorphic projective structure on a surface. The space of geodesics of this structure is, by a primitive version of twistor theory, a two-dimensional complex Poisson manifold containing complete rational curves. The Poisson structure degenerates on a divisor and it is shown that the complement of the divisor is a symplectic manifold which can be identified with the quotient of the moduli space of representations of a free group on three generators in SL (2, ) by the action of a braid group.     

Hybrid projective synchronization of chaotic fractional order systems with different dimensions

The hybrid projective synchronization of different dimensional fractional order chaotic systems is investigated in this paper. It is shown that the slave system can be synchronized with the projection of the master system generated through state transformation. Based on the stability theorem of linear fractional order systems, a suitable controller for achieving the synchronization is given. The hybrid projective synchronization between the fractional order chaotic system and hyperchaotic system is successfully achieved in both reduced order and increased order. The corresponding numerical results verify the effectiveness of the proposed method.     

Function projective synchronization in drive-response dynamical network

In this Letter, the function projective synchronization in the drive-response dynamical network is investigated, where the response dynamical network is affected not only by the drive system, but also coupled via a linearly feedback scheme. Based on Lyapunov stability theory, it is shown that the function projective synchronization with desired scaling function can be realized in the drive-response dynamical network by a simple control law. Moreover it is no need for the scaling function to be differentiable, bounded and nonzero all the time. The numerical simulations are provided to verify the theoretical result.     

Scheme for probabilistic remotely preparing a multi-particle entangled GHZ state

This paper presents a scheme for probabilistic remote preparation of a three-particle entangled Greenberger-Horne-Zeilinger （GHZ） state via three-particle orthonormal basis projective measurement, and then directly generalize the scheme to multi-particle case. It is shown that by using N pairs of bipartite non-maximally entangled states as the quantum channel and N-particle orthonormal basis projective measurement, the multi-particle remote preparation can be successfully realized with a certain probability.     

Integrable motions of curves in projective geometries

In this paper, integrable systems satisfied by the curvatures of curves under inextensible motions in projective geometries are identified. It is shown that motions of curves in two-, three- and four-dimensional projective geometries are respectively related to the Kaup–Kupershmidt hierarchy, a coupled Kaup–Kupershmidt–KdV system and their extension.     

A nonlinear superposition principle admitted by coupled Riccati equations of the projective type

A definite theorem due to Lie which group theoretically characterizes those systems of ordinary differential equations which possess nonlinear superposition principles is employed along with an observation by Lie on the exponentiated form of a fibered Lie algebra to obtain an explicit expression for the Vessiot-Guldberg-Lie nonlinear superposition principle admitted by n-coupled Riccati equations of the projective type. This also, immediately, yields an explicit expression for the generalized cross-ratio for the projective group in n-dimensions.Reported at the Georgia Workshop in Mathematical Physics, November 26–28, 1979, UGA, Athens, Georgia.     

Notes on Entropic Uncertainty Relations Beyond the Scope of Riesz’s Theorem

We discuss those relations for entropies of two quantum measurements that do not follow from the Riesz theorem and its varieties. As measures of an uncertainty, both the Rényi entropies and Tsallis entropies are utilized. It is assumed that values of entropic parameters do not share the usual relation which is required for application of Riesz’s theorem. As is shown, the question is quite reduced to estimating from below the sum or the product of norm-like functions of two generated probability distributions. The considered approach is developed in details within the two examples. In the first example, entropic uncertainty relations are given for a pair of spin-1/2 components along two non-orthogonal axes. The second example deals with two measurements for state discrimination. The former projective measurement is assigned to the Helstrom scheme, the second POVM is related to the B92 protocol of quantum key distribution.     

On unitary/antiunitary projective representations of groups

Unitary/antiunitary projective representations of groups (i.e., projective representations of groups where unitary as well as antiunitary operators in a separable complex Hilbert space are considered) are studied in a systematic way. Particular emphasis is put on continuous unitary/antiunitary projective representations of a Polish group G. It is shown that every continuous unitary/antiunitary projective representation of G can be lifted to a Borel unitary/antiunitary multiplier representation of G (namely, to a representation “up to a factor” which is a Borel mapping) and that this, in turn, can be derived from a continuous unitary/antiunitary (ordinary) representation of a Polish group obtained from an extension of G by the multiplicative group of all complex numbers of absolute value 1.     

On the representations of deformed Galilei group

The projective representations of k-Galilei group G _k are found by contracting the relevant representations of –Poincare group. The projective multiplier is found. It is shown that it is not possible to replace the projective representations of G _k by vector representations of some its extension.     

Yukawa couplings between (2, 1)-forms

The compactification of superstrings leads to an effective field theory for which the space-time manifold is the product of a four-dimensional Minkowski space with a six-dimensional Calabi-Yau space. The particles that are massless in the four-dimensional world correspond to differential forms of type (1, 1) and of type (2, 1) on the Calabi-Yau space. The Yukawa couplings between the families correspond to certain integrals involving three differential forms. For an important class of Calabi-Yau manifolds, which includes the cases for which the manifold may be realized as a complete intersection of polynomial equations in a projective space, the families correspond to (2, 1)-forms. The relation between (2, 1)-forms and the geometrical deformations of the Calabi-Yau space is explained and it is shown, for those cases for which the manifold may be realized as the complete intersection of polynomial equations in a single projective space or for many cases when the manifold may be realized as the transverse intersection of polynomial equations in a product of projective spaces, that the calculation of the Yukawa coupling reduces to a purely algebraic problem involving the defining polynomials. The generalization of this process is presented for a general Calabi-Yau manifold.     

Measurement of proper time in a Weyl space

In a Weyl space it is possible to construct geometrical clocks along timelike geodesics using projective and conformal techniques. We present a precise definition of the limiting processes involved and prove analytically that the clock readings coincide with an affine parameter of the Weyl connection.     

基于改进的主动控制法实现混沌系统广义投影同步

通过引进特殊矩阵构造并基于Lyapunov稳定性理论,提出了一种改进的主动控制法来实现混沌系统的广义投影同步.改进后的主动控制不依赖于罗斯-霍维兹判据,与未改进的主动控制相比,简化了相关运算步骤和复杂度.通过对混沌能源系统和Nuclear Spin Generator系统的研究,并与其他同步方法进行比较,说明了该方法具有简单,直观,稳健,高效等优点,且对混沌系统的自结构和异结构广义投影同步均适用.数值模拟的结果进一步表明了该方法的有效性和理论分析的正确性. 关键词： 改进的主动控制 广义投影同步 特殊矩阵构造     

Projective geometry and special relativity

Some concepts of real and complex projective geometry are applied to the fundamental physical notions that relate to Minkowski space and the Lorentz group. In particular, it is shown that the transition from an infinite speed of propagation for light waves to a finite one entails the replacement of a hyperplane at infinity with a light cone and the replacement of an affine hyperplane – or rest space – with a proper time hyperboloid. The transition from the metric theory of electromagnetism to the pre‐metric theory is discussed in the context of complex projective geometry, and ultimately, it is proposed that the geometrical issues are more general than electromagnetism, namely, they pertain to the transition from point mechanics to wave mechanics.     

Local obstructions to projective surfaces admitting skew-symmetric Ricci tensor

The equation determining whether a projective structure admits a connection in its given projective class that has skew-symmetric Ricci tensor is an overdetermined system of semi-linear partial differential equations which we call the projective Einstein–Weyl (pEW) equation. In 2-dimensions, we give local obstructions for projective surfaces to admit such a connection in its projective class. The obstructions are the resultants of polynomial equations that have to be satisfied for there to admit any pEW solution.