Bi-Inner Dilations and Bi-Stable Passive Scattering Realizations of Schur Class Operator-Valued Functions

Let S(U; Y) be the class of all Schur functions (analytic contractive functions) whose values are bounded linear operators mapping one separable Hilbert space U into another separable Hilbert space Y , and which are defined on a domain , which is either the open unit disk or the open right half-plane . In the development of the Darlington method for passive linear time-invariant input/state/output systems (by Arov, Dewilde, Douglas and Helton) the following question arose: do there exist simple necessary and sufficient conditions under which a function has a bi-inner dilation mapping into ; here U ₁ and Y ₁ are two more separable Hilbert spaces, and the requirement that Θ is bi-inner means that Θ is analytic and contractive on Ω and has unitary nontangential limits a.e. on ∂Ω. There is an obvious well-known necessary condition: there must exist two functions and (namely and ) satisfying and for almost all . We prove that this necessary condition is also sufficient. Our proof is based on the following facts. 1) A solution ψ _r of the first factorization problem mentioned above exists if and only if the minimal optimal passive realization of θ is strongly stable. 2) A solution ψ _l of the second factorization problem exists if and only if the minimal *-optimal passive realization of θ is strongly co-stable (the adjoint is strongly stable). 3) The full problem has a solution if and only if the balanced minimal passive realization of θ is strongly bi-stable (both strongly stable and strongly co-stable). This result seems to be new even in the case where θ is scalar-valued.      

Weight Function for the Quantum Affine Algebra Uq( $$\widehat{\mathfrak{s}\mathfrak{l}}_3$$ )

We give a precise expression for the universal weight function of the quantum affine algebra U _q ( ). The calculations use the technique of projecting products of Drinfeld currents on the intersections of Borel subalgebras. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 145, No. 1, pp. 3–34, October, 2005.     

q-Difference Realization of Uq(sl(M|N)) and Its Application to Free Boson Realization of Uq $$(\widehat{{\text{sl}}}(2|1))$$

We present a q-difference realization of the quantum superalgebra U_q(sl(M|N)), which includes Grassmann even and odd coordinates and their derivatives. Based on this result, we obtain a free boson realization of the quantum affine superalgebra U_q of an arbitrary level k.     

Complicated dynamics of parabolic equations with simple gradient dependence

Let be a smooth bounded domain. Given positive integers , and , , ..., , consider the semilinear parabolic equation

where and are smooth functions. By refining and extending previous results of Polácik we show that arbitrary -jets of vector fields in can be realized in equations of the form (E). In particular, taking we see that very complicated (chaotic) behavior is possible for reaction-diffusion-convection equations with linear dependence on .

     

A new chaotic system and its circuit realization

Based on the Lü system, a new chaotic system is constructed, which can generate a Lorenz-like attractor, Chen-like attractor, Lü-like attractor and new attractor when its parameters are chosen appropriately. The detailed dynamical behaviours of this system are also investigated, including equilibria and stability, bifurcations, and Lyapunov exponent spectrum. Moreover, a novel analogue circuit diagram is designed for the verification of various attractors.     

EPR Entangled States for Bipartite Kinematics and New Bosonic Representation of SU（2） Algebra

We find that the Einstein-Podolsky-Rosen (EPR) entangled state representation descr/bing bipartite kinematics is closely related to a new Bose operator realization of SU(2) Lie algebra. By virtue of the new realization some ttamiltonian eigenfunction equation can be directly converted to the generalized confluent equation in the EPR entangled state representation and its solution is obtainable. This thus provides a new approach for studying dynamics of angular momentum systems.     

Spinor Field Realizations of W2,6 String and W6 String

In this paper the spinor field BRST charges of the W_2,6 string and W₆ string are constructed, where the BRST charges are graded.     

Realization of Configurations and the Löwner Ellipsoid

It is proved that any subset of an (m-1)-dimensional sphere of volume larger than l(m+ 1) of the volume of the entire sphere contains l+ 1 points forming a regular l-dimensional simplex. As a corollary, it is obtained that, if the exterior of a given m-dimensional filled ellipsoid contains no more than the 1/(m+ 1) fraction of some sphere, then the volume of the ellipsoid is no less than the volume of the corresponding ball. The existence of a pair of points a given spherical distance apart in a set of positive measure is examined.     

Performance optimization of queueing systems with perturbation realization

After the intensive studies of queueing theory in the past decades, many excellent results in performance analysis have been obtained, and successful examples abound. However, exploring special features of queueing systems directly in performance optimization still seems to be a territory not very well cultivated. Recent progresses of perturbation analysis (PA) and sensitivity-based optimization provide a new perspective of performance optimization of queueing systems. PA utilizes the structural information of queueing systems to efficiently extract the performance sensitivity information from a sample path of system. This paper gives a brief review of PA and performance optimization of queueing systems, focusing on a fundamental concept called perturbation realization factors, which captures the special dynamic feature of a queueing system. With the perturbation realization factors as building blocks, the performance derivative formula and performance difference formula can be obtained. With performance derivatives, gradient-based optimization can be derived, while with performance difference, policy iteration and optimality equations can be derived. These two fundamental formulas provide a foundation for performance optimization of queueing systems from a sensitivity-based point of view. We hope this survey may provide some inspirations on this promising research topic.     

Modeling and Analog Realization of the Fundamental Linear Fractional Order Differential Equation

This paper provides a rational function approximation of the irrational transfer function of the fundamental linear fra- ctional order differential equation, namely, whose transfer function is given by for 0<m<2. Simple methods, useful in system and control theory, which consists of approximating, for a given frequency band, the transfer function of this fractional order system by a rational function are presented. The impulse and step responses of this system are derived and simple analog circuit which can serve as fundamental analog fractional order system is also obtained. Illustrative examples are presented to show the exactitude and the usefulness of the approximation methods.