Physical Complexity of Classical and Quantum Objects and Their Dynamical Evolution From an Information-Theoretic Viewpoint

Charles Bennett's measure of physical complexity for classical objects, namely logical-depth, is used to prove that a chaotic classical dynamical system is not physically complex. The natural measure of physical complexity for quantum objects, quantum logical-depth, is then introduced to prove that a chaotic quantum dynamical system too is not physically complex.     

The minimal polynomial of a matrix

Let Rbe a finite dimensional central simple algebra over a field FA be any n× n matrix over R. By using the method of matrix representation, this paper obtains the structure formula of the minimal polynomial q_A(λ) of A over F. By using q_A(λ), this paper discusses the structure of right (left) eigenvalues set of A, and obtains the necessary and sufficient condition that a matrix over a finite dimensional central division algebra is similar to a diagonal matrix.     

再扩充模糊逻辑中逻辑程序的模型论语义和不动点语义研究

模糊逻辑及其扩充是处理不确定性与模糊性信息的重要数学工具,在近似推理、人工智能等领域有着广泛的应用。而逻辑程序也已经成为人工智能的研究热点之一。本文是在再扩充模糊逻辑中,对逻辑程序进行了语义的研究。给出了其语法和语义描述,并且将逻辑程序的许多主要结论推广到再扩充模糊逻辑中。首先,得到了关于模糊逻辑推论的充分必要条件以及模型交与强模型交性质。然后,通过引入一个H erbrand解释算子Pτ:LBp→LBp,给出了确定性程序P的H erbrand模型的充分条件和强H erbrand模型的充分必要条件。最后,建立了确定性程序的最小强H erbrand模型的不动点刻画定理。     

A stability concept for matrix game optimal strategies and its application to linear programming sensitivity analysis

This paper studies a class of perturbations of a game matrix that alters each row by a different amount. We find that completely mixed optimal strategies are stable under these perturbations provided the norm of the vector of additive amounts is sufficiently small. Using this concept we give a new characterization of completely mixed grames. We also obtain a sensitivity result for a class of perturbations of the technological coefficient matrix of positive linear programs. The stability of an optimal strategy holds throughout at least a spherical neighborhood of the zero perturbation. We give a computational formula and equivalent programming formulations for the radius of this neighborhood.     

Quantized Gromov-Hausdorff distance

A quantized metric space is a matrix order unit space equipped with an operator space version of Rieffel's Lip-norm. We develop for quantized metric spaces an operator space version of quantum Gromov-Hausdorff distance. We show that two quantized metric spaces are completely isometric if and only if their quantized Gromov-Hausdorff distance is zero. We establish a completeness theorem. As applications, we show that a quantized metric space with 1-exact underlying matrix order unit space is a limit of matrix algebras with respect to quantized Gromov-Hausdorff distance, and that matrix algebras converge naturally to the sphere for quantized Gromov-Hausdorff distance.     

On solving discrete two-stage stochastic programs having mixed-integer first- and second-stage variables

In this paper, we propose a decomposition-based branch-and-bound (DBAB) algorithm for solving two-stage stochastic programs having mixed-integer first- and second-stage variables. A modified Benders' decomposition method is developed, where the Benders' subproblems define lower bounding second-stage value functions of the first-stage variables that are derived by constructing a certain partial convex hull representation of the two-stage solution space. This partial convex hull is sequentially generated using a convexification scheme such as the Reformulation-Linearization Technique (RLT) or lift-and-project process, which yields valid inequalities that are reusable in the subsequent subproblems by updating the values of the first-stage variables. A branch-and-bound algorithm is designed based on a hyperrectangular partitioning process, using the established property that any resulting lower bounding Benders' master problem defined over a hyperrectangle yields the same objective value as the original stochastic program over that region if the first-stage variable solution is an extreme point of the defining hyperrectangle or the second-stage solution satisfies the binary restrictions. We prove that this algorithm converges to a global optimal solution. Some numerical examples and computational results are presented to demonstrate the efficacy of this approach.     

Some new simple stability criteria of linear neutral systems with a single delay

This article mainly considers the linear neutral delay-differential systems with a single delay. Using the characteristic equation of the system, new simple delay-independent asymptotic and exponential stability criteria are derived in terms of the matrix measure, the spectral norm and the spectral radius of the corresponding matrices. Numerical examples demonstrate that our criteria are less conservative than those of previous corresponding results [L.M. Li, Stability of linear neutral delay-differential systems, Bull. Aust. Math. Soc. 38 (1988) 339–344; G.D. Hu, G.D. Hu. Some simple criteria for stability of neutral delay-differential systems, Appl. Math. Comput. 80 (1996) 257–271; D.Q. Cao, Ping He, Sufficient conditions for stability of linear neutral systems with a single delay, Appl. Math. Lett. 17 (2004) 139–144; G.D. Hu, G.D. Hu, B. Cahlon, Algebraic criteria for stability of linear neutral systems with a single delay, J. Comput. Appl. Math. 135 (2001) 125–130].     

Higher-order harmonics in bleached silver halide holograms

A large number of plane wave holograms were recorded in Agfa-Gevaert 8E75HD holographic plates, at a wide range of bias exposures and fringe visibilities. The plates were processed by various combinations of developers (AAC, Pyrogallol and Catechol) and bleaching agents (R-9 and EDTA). The phase gratings were studied by phase-contrast microscopy, using a high-power immersion (100×) objective. The phase-contrast photomicrographs were Fourier analysed. Thus, first-, second-, and third-order modulations of the refractive index as a function of bias exposure and visibility of the recording interference pattern could be determined. The ratio of the amplitudes of higher-order modulations to that of the first-order can serve as a measure of the nonlinearity of the holographic recording.     

Approximation of the joint spectral radius using sum of squares

We provide an asymptotically tight, computationally efficient approximation of the joint spectral radius of a set of matrices using sum of squares (SOS) programming. The approach is based on a search for an SOS polynomial that proves simultaneous contractibility of a finite set of matrices. We provide a bound on the quality of the approximation that unifies several earlier results and is independent of the number of matrices. Additionally, we present a comparison between our approximation scheme and earlier techniques, including the use of common quadratic Lyapunov functions and a method based on matrix liftings. Theoretical results and numerical investigations show that our approach yields tighter approximations.     

Matrix-Based Analytical Model of Critical Coupling Length of Titanium In-Diffused Integrated-Optic Directional Coupler on Lithium Niobate Substrate

The critical coupling length Lc is the most important design parameter of an integrated-optic directional coupler, which is the basic building block of a variety of optical integrated circuits. The existing methods of estimating Lc of a Ti in-diffused waveguide directional coupler on LiNbO3 substrate are based on the beam propaga tion method (BPM), which requires huge computing time. A considerably simpler and computationally faster analytical method involving only multiplication of 2 x 2 matrices with no iterations is described in this article. A staircase type step-index profile is generated from the one-dimensional effective index profile in the lateral direction by partitioning the latter into a large number of thin sections of different refractive indices. The overall transfer matrix of the step-index layered structure so obtained may be computed by the progressive multiplication of individual transfer matrices (2 x 2) relating the field components in adjacent layers. Hence the wave amplitude in any layer may be computed as a function of the input wave amplitude for different angles of incidence. This method has been successfully employed to compute the propagation constants for symmetric and asymmetric guided modes in the coupled region of the directional coupler and hence compute the coupling length. The values of Lc computed for a variety of devices agree closely with the published experimental results reported by several research groups. The computer program implementing the model runs reasonably fast in an ordinary PC and is versatile enough to consider arbitrary waveguide dimensions. Ti film thickness, and diffusion parameters for any wavelength of input light and for both transverse electric and transverse magnetic polarizations.