Free system

Abstract: We discuss the possibility of interpreting a q-deformed non-interacting system as incorporating the effects of interactions among its particles. This can be accomplished, for instance, in an ensemble of q-Boson by means of the virial expansion of a real gas in powers of the deformed parameter. The lowest-order virial coefficient reduces to the case of the standard, non-interacting Bose gas, while the higher-order virial coefficients contain effects arising from the interaction. The same picture can be drawn in a quantum-mechanical system where it is shown that the q-deformed momentum can be expanded in a series containing high-order powers of the standard quantum phase-space variables. Motivated by this result, we introduce, in the classical framework, a transformation relating the momentum of a free system to the momentum of an interacting system. It is shown that the canonical quantization applied to the interacting system implies a q-deformed quantization for the free system.

Abstract: Interactions of massless fields of all spins in four dimensions with currents of any spin is shown to result from a solution of the linear problem that describes a gluing between rank-one (massless) system and rank-two (current) system in the unfolded dynamics approach. Since the rank-two system is dual to a free rank-one higher-dimensional system, that effectively describes conformal fields in six space-time dimensions, the constructed system can be interpreted as describing a mixture between linear conformal fields in four and six dimensions. Interpretation of the obtained results in spirit of AdS/CFT correspondence is discussed.

Abstract: In this article, we are concerned with the boundary stabilisation of the Euler–Bernoulli beam equation for which all eigenvalues of the (control) free system are located on the imaginary axis of the complex plane. The fourth-order system in spacial variable is transformed into a coupled heat-like system. This enables us to make a natural backstepping transformation in vector form to transform the system into a target system which has arbitrary decay rate. The state feedback is thus designed. It is shown that the original closed-loop system is exponentially stable with the given arbitrary decay rate.

Abstract: The existence of closed bounded balls in the complex plane the union of which contains all the poles of a system subject to delayed dynamics with constant internal point delays is proved. Each ball is centred at each pole of the associated delay-free system and it may contain either a finite or infinite number of poles of the delayed system. As a result, the stability of the delay free system is preserved with a sufficiently small contribution from delayed dynamics.