On deformation of associative algebras and graph homology

Deformation theory of associative algebras and in particular of Poisson algebras is reviewed. The role of an “almost contraction” leading to a canonical solution of the corresponding Maurer–Cartan equation is noted. This role is reminiscent of the Homotopical Perturbation Lemma, with the infinitesimal deformation cocycle as “initiator.”Applied to star-products, we show how Moyal's formula can be obtained using such an almost contraction and conjecture that the “merger operation” provides a canonical solution at least in the case of linear Poisson structures.     

Coorbits of Quantum Homogeneous Spaces

The notion of coorbits for spaces with quantum group actions is introduced. A space with a quantum group action is given by a pair of algebras: an associative algebra which is the analog of a classical topological space, and a Hopf algebra which is the analog of a classical topological group. The Hopf algebra acts on the associative algebra via a comodule structure mapping which is also an algebra homomorphism. For a space with a quantum group action, a coorbit is a pair of spaces given by the image and the kernel of an algebra homomorphism from the associative algebra to the Hopf algebra. The coorbits of several types of quantum homogeneous spaces are discussed. In the case when the associative algebra is the group algebra of a group and the Hopf algebra is a quotient of the group algebra, the connection between the set of coorbits and the character group is established.     

A quantum mechanical approach to a fuzzy theory

Recently, we proposed a general measurement theory for classical and quantum systems (i.e., “objective fuzzy measurement theory”). In this paper, we propose “subjective fuzzy measurement theory”, which is characterized as the statistical method of the objective fuzzy measurement theory. Our proposal of course has a lot of advantages. For example, we can directly see “membership functions” (= “fuzzy sets”) in this theory. Therefore, we can propose the objective and the subjective methods of membership functions. As one of the consequences, we assert the objective (i.e., individualistic) aspect of Zadeh's theory. Also, as a quantum application, we clarify Heisenberg's uncertainty relation.     

Exponential p-stability of delayed Cohen–Grossberg-type BAM neural networks with impulses

An impulsive Cohen–Grossberg-type bidirectional associative memory (BAM) neural networks with distributed delays is studied. Some new sufficient conditions are established for the existence and global exponential stability of a unique equilibrium without strict conditions imposed on self regulation functions. The approaches are based on Laypunov–Kravsovskii functional and homeomorphism theory. When our results are applied to the BAM neural networks, our results generalize some previously known results. It is believed that these results are significant and useful for the design and applications of Cohen–Grossberg-type bidirectional associative memory networks.     

Vertex-IRF transformations, dynamical quantum groups and harmonic analysis

It is shown that a dynamical quantum group arising from a vertex-IRF transformation has a second realization with untwisted dynamical multiplication but nontrivial bigrading. Applied to the SL 2; ) dynamical quantum group, the second realization is naturally described in terms of Koornwinder's twisted primitive elements. This leads to an intrinsic explanation why harmonic analysis on the “classical” SL(2; ) quantum group with respect to twisted primitive elements, as initiated by Koornwinder, is the same as harmonic analysis on the SL(2; C) dynamical quantum group.     

EXPONENTIAL STABILITY AND PERIODIC SOLUTION OF HYBRID BIDIRECTIONAL ASSOCIATIVE MEMORY NEURAL NETWORKS WITH DISCRETE DELAYS

In this paper, we study the existence, uniqueness, and the global exponential stability of the periodic solution and equilibrium of hybrid bidirectional associative memory neural networks with discrete delays. By ingeniously importing real parameters di > 0 (i = 1,2, …, n) which can be adjusted, making use of the Lyapunov functional method and some analysis techniques, some new sufficient conditions are established. Our results generalize and improve the related results in [9]. These conditions can be used both to design globally exponentially stable and periodical oscillatory hybrid bidirectional associative neural networks with discrete delays, and to enlarge the area of designing neural networks. Our work has important significance in related theory and its application.     

Gravitational instanton in Hilbert space and the mass of high energy elementary particles

While the theory of relativity was formulated in real spacetime geometry, the exact formulation of quantum mechanics is in a mathematical construction called Hilbert space. For this reason transferring a solution of Einstein’s field equation to a quantum gravity Hilbert space is far of being a trivial problem.

On the other hand ^(∞) spacetime which is assumed to be real is applicable to both, relativity theory and quantum mechanics. Consequently, one may expect that a solution of Einstein’s equation could be interpreted more smoothly at the quantum resolution using the Cantorian ^(∞) theory.

In the present paper we will attempt to implement the above strategy to study the Eguchi–Hanson gravitational instanton solution and its interpretation by ‘t Hooft in the context of quantum gravity Hilbert space as an event and a possible solitonic “extended” particle. Subsequently we do not only reproduce the result of ‘t Hooft but also find the mass of a fundamental “exotic” symplictic-transfinite particle m1.8 MeV as well as the mass M_x and M (Planck) which are believed to determine the GUT and the total unification of all fundamental interactions respectively. This may be seen as a further confirmation to an argument which we put forward in various previous publications in favour of an alternative mass acquisition mechanism based on unification and duality considerations. Thus even in case that we never find the Higgs particle experimentally, the standard model would remain substantially intact as we can appeal to tunnelling and unification arguments to explain the mass. In fact a minority opinion at present is that finding the Higgs particle is not a final conclusive argument since one could ask further how the Higgs particle came to its mass which necessitates a second Higgs field. By contrast the present argument could be viewed as an ultimate theory based on the existence of a “super” force, beyond which nothing else exists.     

Introducing groups into quantum theory (1926–1930)

In the second half of the 1920s, physicists and mathematicians introduced group-theoretic methods into the recently invented “new” quantum mechanics. Group representations turned out to be a highly useful tool in spectroscopy and in giving quantum-mechanical explanations of chemical bonds. H. Weyl explored the possibilities of a group-theoretic approach to quantization. In his second version of a gauge theory for electromagnetism, he even started to build a bridge between quantum theoretic symmetries and differential geometry. Until the early 1930s, an active group of young quantum physicists and mathematicians contributed to this new challenging field. But around the turn to the 1930s, opposition to the new methods in physics grew. This article focuses on the work of those physicists and mathematicians who introduced group-theoretic methods into quantum physics.     

Isomorphism of a Ring to the Endomorphism Ring of an Abelian Group

This paper presents necessary and sufficient conditions under which isomorphism of endomorphism rings of additive groups of arbitrary associative rings with 1 implies isomorphism of these rings. For a certain class of Abelian groups, we present a criterion which shows when isomorphism of their endomorphism rings implies isomorphism of these groups. We demonstrate necessary and sufficient conditions under which an arbitrary ring is the endomorphism ring of an Abelian group. This solves Problem 84 in L. Fuchs’ “Infinite Abelian Groups.”__________Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 9, No. 1, pp. 231–234, 2003.     

Using the wigner function for quantum transport in device simulation

The Wigner function was introduced as a generalization of the concept of distribution function for quantum statistics. The aim of this work is pushing further the formal analogy between quantum and classical approaches. The Wigner function is defined as an ensemble average, i.e., in terms of a mixture of pure states. From the point of view of basic physics, it would be very appealing to be able to define a Wigner function also for pure states and the associated expectation values for quantum observables, in strict analogy with the definition of mean value of a physical quantity in classical mechanics; then correct results for any quantum system should be recovered as appropriate superpositions of such “pure-state” quantities. We will show that this is actually possible, at the cost of dealing with generalized functions in place of proper functions.     

Quiver varieties and Lusztig's algebra

We study preprojective algebras of graphs and their relationship to module categories over representations of quantum SL(2). As an application, ADE quiver varieties of Nakajima are shown to be subvarieties of the variety of representations of a certain associative algebra introduced by Lusztig.     

Experimental study of quantum chaos with cold atoms

We investigate experimentally the quantum behavior of laser-cooled atoms in a pulsed standing wave, a system that is an atomic analog of the quantum kicked rotor. In particular, it may display the well-known phenomenon of “dynamical localization”, when the standing wave is driven periodically. Furthermore, we study some interesting properties of a quasi-periodically driven kicked rotor, which presents resonances that are shown to be sharper than the inverse of the driven–excitation duration, thus presenting a sub-Fourier character.     

Dynamics of bidirectional associative memory networks with distributed delays and reaction–diffusion terms

In this paper, the periodic oscillatory solution and stability are investigated for a class of bidirectional associative memory neural networks with distributed delays and reaction–diffusion terms. By constructing a new Lyapunov functional, applying M-matrix theory and inequality technique, several novel sufficient conditions are derived to ensure the existence and uniqueness of periodic oscillatory solutions for bidirectional associative memory neural networks with distributed delays and reaction–diffusion terms, and all other solutions of this network converge exponentially to the unique periodic oscillatory solution. Moreover, the exponential convergence rate is estimated, which depends on the delay kernel functions and the system parameters. Two numerical examples are given to show the effectiveness of the obtained results. The results extend and improve the previously known results.     

Cut-threshold graphs

We study the structure of the networks in which connectedness and disconnectedness can be expressed by a threshold system. This means that the elements of the network have a certain “destruction cost” and that the enemy can disconnect the network if and only if they pay a large enough price. We give polynomial algorithms for the recognition of such networks, and for the determination of the appropriate costs and threshold value.     

连续反馈联想记忆的吸引域和指数收敛速度的估计

利用一些实分析技巧和Liapunov方法,重新估计了连续反馈联想记忆模式的吸引域及其中每一点趋向记忆模式的指数收敛速度,得到一些新的结果,这些结果可用于连续反馈联想记忆网络的容错性能评价及其综合过程.     

Entangled Markov chains

Motivated by the problem of finding a satisfactory quantum generalization of the classical random walks, we construct a new class of quantum Markov chains which are at the same time purely generated and uniquely determined by a corresponding classical Markov chain. We argue that this construction yields as a corollary, a solution to the problem of constructing quantum analogues of classical random walks which are “entangled” in a sense specified in the paper.The formula giving the joint correlations of these quantum chains is obtained from the corresponding classical formula by replacing the usual matrix multiplication by Schur multiplication.The connection between Schur multiplication and entanglement is clarified by showing that these quantum chains are the limits of vector states whose amplitudes, in a given basis (e.g. the computational basis of quantum information), are complex square roots of the joint probabilities of the corresponding classical chains. In particular, when restricted to the projectors on this basis, the quantum chain reduces to the classical one. In this sense we speak of entangled lifting, to the quantum case, of a classical Markov chain. Since random walks are particular Markov chains, our general construction also gives a solution to the problem that motivated our study.In view of possible applications to quantum statistical mechanics too, we prove that the ergodic type of an entangled Markov chain with finite state space (thus excluding random walks) is completely determined by the corresponding ergodic type of the underlying classical chain. Mathematics Subject Classification (2000) Primary 46L53, 60J99; Secondary 46L60, 60G50, 62B10     

Global asymptotic stability of BAM neural networks with distributed delays and reaction–diffusion terms

The global asymptotic stability of bi-directional associative memory neural networks with distributed delays and reaction–diffusion terms are studied by using the analysis technique and Lyapunov functional. A sufficient condition is proposed. Two numerical examples are given to show the correctness of our analysis.     

Quantum Backreaction (Casimir) Effect I. What are Admissible Idealizations?

Casimir effect, in a broad interpretation which we adopt here, consists in a backreaction of a quantum system to adiabatically changing external conditions. Although the system is usually taken to be a quantum field, we show that this restriction rather blurs than helps to clarify the statement of the problem. We discuss the problem from the point of view of algebraic structure of quantum theory, which is most appropriate in this context. The system in question may be any quantum system, among others both finite- as infinite-dimensional canonical systems are allowed. A simple finite-dimensional model is discussed. We identify precisely the source of difficulties and infinities in most of traditional treatments of the problem for infinite-dimensional systems (such as quantum fields), which is incompatibility of algebras of observables or their representations. We formulate conditions on model idealizations which are acceptable for the discussion of the adiabatic backreaction problem. In the case of quantum field models in that class we find that the normal ordered energy density is a well-defined distribution, yielding global energy in the limit of a unit test function. Although we see the “zero point” expressions as inappropriate, we show how they can arise in the quantum field theory context as a result of uncontrollable manipulations.Communicated by Klaus Fredenhagensubmitted 13/04/04, accepted 24/11/04     

Glimpses of the Octonions and Quaternions History and Today’s Applications in Quantum Physics

Before we dive in this essay into the accessibility stream of nowadays indicatory applications of octonions and quaternions to computer and other sciences and to quantum physics (see for example [50-53], [41], [33]) and to Clifford algebras (see for example [17,16], 18) let us focus for a while on the crucially relevant events for today’s revival on interest to nonassociativities while the role of associative quaternions in eight periodicity constructive classification of associative Clifford algebras is now a text-book knowledge.     

具有时滞的双向联想记忆神经网络的指数稳定性和周期解的存在性

通过构造Lyapunov泛函和运用不等式分析技巧，得到了判断一类时滞双向联想记忆神经网络平衡点的指数稳定性和周期解的存在性的充分条件，为综合设计具有时滞的双向联想记忆神经网络提供了依据。