Classical foundations of quantum groups

The concept of classical r matrices is developed from a purely canonical standpoint. The final purpose of this work is to bring about a synthesis between recent developments in the theory of integrable systems and the general theory of quantization as a deformation of classical mechanics. The concept of quantization algebra is here dominant; in integrable systems this is the set of dynamical variables that appear in the Lax pair. The nature of this algebra, a solvable Lie algebra in such models as the Sine-Gordon and Toda field theories but semisimple in the case of spin systems, provides a useful scheme for the classification of integrable models. A completely different classification is obtained by the nature of the r matrix employed; there are three kinds: rational, trigonometric, and elliptic. All cases are studied in detail, with numerous examples. Some of the problems connected with quantization are discussed.This paper is dedicated to my friend Asim Barut.     

Type II quantum chaos

A classification of quantum systems into three categories, type I, II and III, is proposed. The classification is based on the degree of sensitivity upon initial conditions, and the appearance of chaos. The quantum dynamics of type I systems is quasi periodic displaying no exponential sensitivity. They arise, e.g., as the quantized versions of classical chaotic systems. Type II systems are obtained when classical and quantum degrees of freedom are coupled. Such systems arise naturally in a dynamic extension of the first step of the Born-Oppenheimer approximation, and are of particular importance to molecular and solid state physics. Type II systems can show exponential sensitivity in the quantum subsystem. Type III systems are fully quantized systems which show exponential sensitivity in the quantum dynamics. No example of a type III system is currently established. This paper presents a detailed discussion of a type II quantum chaotic system which models a coupled electronic-vibronic system. It is argued that type II systems are of importance for any field systems (not necessarily quantum) that couple to classical degrees of freedom.     

A DEGREE SELECTION METHOD OF MATRIX CONDENSATIONS FOR EIGENVALUE PROBLEMS

A new and automatic degree selection technique based on the approximate modal energy has been derived and developed for matrix condensations in this paper. The method is used to condense the number of degrees of the matrix when dealing with eigenvalue problems. By defining a new basis in the vicinity of the original space, the individual modal energy gradients can be evaluated. The primary degrees of freedom are then determined according to the variation of the energies in the neighborhood. In case the energy variation of a degree tends to increase in that neighborhood, the degree is classified as secondary since it relatively provides energy to the nodes nearby. On the other hand, if the energy variation is decreasing, then it is primary. All the classification criteria are finally mapped to one parameter, which is called the index of classification. That is, by examining the magnitude of the index of classification, one is able to determine the primary and secondary degrees. The new selection method is demonstrated and verified by a well-known cantilever beam problem in addition to the error bound estimation.     

超导故障限流器的研究现状及其应用

超导故障限流器(Superconducting Fault Current Limiter,SFCL)是利用超导体的基本特性,有效限制电力系统故障短路电流,提高电网安全性和稳定性的一种新型电力设备。文中在综合大量文献的基础上,对超导故障限流器进行了一种较为系统的分类。基于该种分类,结合目前国内外的研究现状,就电阻型、磁屏蔽型、饱和铁芯电抗器型、桥路型SFCL的工作原理作了详细的分析介绍,还给出了它们在电力系统中的安装位置。最后,对超导故障限流器的研究中存在的问题及其发展趋势做了说明。     

Stochastic flows,reaction — Diffusion processes,and morphogenesis

Recently, anexact procedure has been introduced [C. A. Walsh and J. J. Kozak,Phys. Rev. Lett. 47:1500 (1981)] for calculating the expected walk length 〈n〉 for a walker undergoing random displacements on a finite or infinite (periodic)d-dimensional lattice with traps (reactive sites). The method (which is based on a classification of the symmetry of the sites surrounding the central deep trap and a coding of the fate of the random walker as it encounters a site of given symmetry) is applied here to several problems in lattice statistics for each of whichexact results are presented. First, we assess the importance of lattice geometry in influencing the efficiency of reaction-diffusion processes in simple and multiple trap systems by reporting values of 〈n〉 for square (cubic) versus hexagonal lattices ind=2, 3. We then show how the method may be applied to variable-step (distance-dependent) walks for a single walker on a given lattice and also demonstrate the calculation of the expected walk length 〈n〉 for the case of multiple walkers. Finally, we make contact with recent discussions of “mixing” by showing that the degree of chaos associated with flows in certain lattice systems can be calibrated by monitoring the lattice walks induced by the Poincaré map of a certain parabolic function.     

Turbulence. 2. Pfaff’s systems in turbulence models

Based on the classification of dynamic coordinates presented in Part 1 of this work and analogy with the classical systems constrained by nonholonomic coupling, Pfaff’s systems of turbulent dynamics are constructed. A method of constructing trajectory bundles for particles forming a vortex sheet is described. Thermodynamic interpretation of Pfaff’s coefficients is suggested.     

Linked by loops: Network structure and switch integration in complex dynamical systems

Simple nonlinear dynamical systems with multiple stable stationary states are often taken as models for switchlike biological systems. This paper considers the interaction of multiple such simple multistable systems when they are embedded together into a larger dynamical “supersystem.” Attention is focused on the network structure of the resulting set of coupled differential equations, and the consequences of this structure on the propensity of the embedded switches to act independently versus cooperatively. Specifically, it is argued that both larger average and larger variance of the node degree distribution lead to increased switch independence. Given the frequency of empirical observations of high variance degree distributions (e.g., power-law) in biological networks, it is suggested that the results presented here may aid in identifying switch-integrating subnetworks as comparatively homogenous, low-degree, substructures. Potential applications to ecological problems such as the relationship of stability and complexity are also briefly discussed.     

Group classification of a class of coupled equations

A complete group classification of a class of coupled equations that appear in many physical problems is presented by developing the method of preliminary group classification of Ibragimovet al. We give a symmetry group analysis for an interesting example.     

Classification of “Real” Bloch-bundles: Topological quantum systems of type AI

We provide a classification of type AI topological quantum systems in dimension $d=1,2,3,4$ which is based on the equivariant homotopy properties of “Real” vector bundles. This allows us to produce a fine classification able to take care also of the non stable regime which is usually not accessible via $K$-theoretic techniques. We prove the absence of non-trivial phases for one-band AI free or periodic quantum particle systems in each spatial dimension by inspecting the second equivariant cohomology group which classifies “Real” line bundles. We also show that the classification of “Real” line bundles suffices for the complete classification of AI topological quantum systems in dimension $d⩽3$. In dimension $d=4$ the determination of different topological phases (for free or periodic systems) is fixed by the second “Real” Chern class which provides an even labeling identifiable with the degree of a suitable map. Finally, we provide explicit realizations of non trivial 4-dimensional free models for each given topological degree.     

Conformal Geodesics on Vacuum Space-times

 A classification of discrete integrable systems on quad–graphs, i.e. on surface cell decompositions with quadrilateral faces, is given. The notion of integrability laid in the basis of the classification is the three–dimensional consistency. This property yields, among other features, the existence of the discrete zero curvature representation with a spectral parameter. For all integrable systems of the obtained exhaustive list, the so called three–leg forms are found. This establishes Lagrangian and symplectic structures for these systems, and the connection to discrete systems of the Toda type on arbitrary graphs. Generalizations of these ideas to the three–dimensional integrable systems and to the quantum context are also discussed. Received: 14 February 2002 / Accepted: 22 September 2002 Published online: 8 January 2003 Acknowledgements. This research was partly supported by DFG (Deutsche Forschungsgemeinschaft) in the frame of SFB 288 ``Differential Geometry and Quantum Physics'. V.A. was also supported by the RFBR grant 02-01-00144. He thanks TU Berlin for warm hospitality during the visit when part of this work has been fulfilled. Communicated by L. Takhtajan     

An Orientation‐Independent Imaging Technique for the Classification of Blood Cells

This study presents an orientation independent imaging technique for the classification and recognition of blood cells with relevant applications to related problems in cytometry and medical diagnosis. The proposed method integrates three important aspects towards its practical implementation: 1. The use of the principal component (PC) transform to reorient in an optimal fashion the image data and to make of the feature matching an orientation independent process; 2. The establishment of similarity measures to quantify the matching process of the shape of the blood cell populations with any chosen degree of certainty; and 3. The application of the recognition and classification processes once all the similarity measurements are gathered. An extension of this two‐dimensional (2‐D) method to a three‐dimensional (3‐D) base is proposed. Results of this technique using real‐world image data of blood cell populations are given. It is appropriate to note that although the images used here are real‐world samples of blood cell populations, other samples containing different biological specimens could have been used just as well.     

Integrable Evolution Equations on Associative Algebras

This paper surveys the classification of integrable evolution equations whose field variables take values in an associative algebra, which includes matrix, Clifford, and group algebra valued systems. A variety of new examples of integrable systems possessing higher order symmetries are presented. Symmetry reductions lead to an associative algebra-valued version of the Painlevé transcendent equations. The basic theory of Hamiltonian structures for associative algebra-valued systems is developed and the biHamiltonian structures for several examples are found. Received: 12 March 1997 / Accepted: 27 August 1997     

Photorefraction images analysis through neural networks

The importance of an early evaluation of infants’ visual system condition is long time recognized. Non-corrected visual disorders may lead to major vision and developmental non-reversible limitations in the future. Among the objective methods of refraction, photorefractive techniques are specifically designed for screening young children. Over the years a number of photorefraction systems with different grades of complexity and automation were developed. A critical problem that one needs to deal with in any approach to these systems is the interpretation and classification of the photorefraction images. In digital photorefraction conventional image processing operators and Fourier techniques were currently used. In this communication we will report on the use of Neural Networks for automated classification of digital photorefraction images.     

A one-dimension convolutional neural network based interference classification method

Interference is a common problem in wireless communication, navigation and radar systems. A wide variety of interferences are used to degrade the communication quality especially in electronic warfare environment. In modern military communication systems, interference classification is an important module for its ability to obtain prior interference information before adopting related anti-interference method. This paper proposes a deep learning based interference classification method, which applies one-dimension convolutional neural networks to automatically extract interference features for classification. Computer simulations show better classification performance and lower computational complexity. Meanwhile, this proposed method is implied on software defined radios (SDR) hardware, more than 99% correct classification probability can be achieved with limited samples of the received signal, which verifies the robustness of this proposed method.     

Polynomial integrability of the Hamiltonian systems with homogeneous potential of degree − 3

In this paper, we study the polynomial integrability of natural Hamiltonian systems with two degrees of freedom having a homogeneous potential of degree k given either by a polynomial, or by an inverse of a polynomial. For k=−2,−1,…,3,4, their polynomial integrability has been characterized. Here, we have two main results. First, we characterize the polynomial integrability of those Hamiltonian systems with homogeneous potential of degree −3. Second, we extend a relation between the nontrivial eigenvalues of the Hessian of the potential calculated at a Darboux point to a family of Hamiltonian systems with potentials given by an inverse of a homogeneous polynomial. This relation was known for such Hamiltonian systems with homogeneous polynomial potentials. Finally, we present three open problems related with the polynomial integrability of Hamiltonian systems with a rational potential.     

Diabetic Retinal Grading Using Attention-Based Bilinear Convolutional Neural Network and Complement Cross Entropy

Diabetic retinopathy (DR) is a common complication of diabetes mellitus (DM), and it is necessary to diagnose DR in the early stages of treatment. With the rapid development of convolutional neural networks in the field of image processing, deep learning methods have achieved great success in the field of medical image processing. Various medical lesion detection systems have been proposed to detect fundus lesions. At present, in the image classification process of diabetic retinopathy, the fine-grained properties of the diseased image are ignored and most of the retinopathy image data sets have serious uneven distribution problems, which limits the ability of the network to predict the classification of lesions to a large extent. We propose a new non-homologous bilinear pooling convolutional neural network model and combine it with the attention mechanism to further improve the network’s ability to extract specific features of the image. The experimental results show that, compared with the most popular fundus image classification models, the network model we proposed can greatly improve the prediction accuracy of the network while maintaining computational efficiency.     

The K-Orbits, Identities, and Classification of Four-Dimensional Homogeneous Spaces with the Group of Poincaré and de Sitter Transforms

The method of orbits traditionally applied to geometric quantization problems is used to study homogeneous spaces. Based on the proposed classification of the orbits of co-adjoint representation (K-orbits), a classification of homogeneous spaces is constructed. This classification allows one, in particular, to point out the explicit form of identities – functional relations between the transform-group generators – which are of great importance in applied problems (e.g., in the theory of separation of variables). All four-dimensional homogeneous spaces with the group of Poincaré and de Sitter transforms are classified and all independent identities on these spaces are given in explicit form.     

Integrability of coupled conformal field theories

The massive phase of two-layer integrable systems is studied by means of RSOS restrictions of affine Toda theories. A general classification of all possible integrable perturbations of coupled minimal models is pursued by an analysis of the (extended) Dynkin diagrams. The models considered in most detail are coupled minimal models which interpolate between magnetically coupled Ising models and Heisenberg spin ladders along the c < 1 discrete series.