Exclusion statistics,operator algebras and Fock space representations

We study exclusion statistics within the second quantized approach. We consider operator algebras with positive definite Fock space and restrict them in a such a way that certain state vectors in Fock space are forbidden ab initio.We describe three characteristic examples of such exclusion, namely exclusion on the base space which is characterized by states with specific constraint on quantum numbers belonging to base space M (e.g. Calogero-Sutherland type of exclusion statistics), exclusion in the single-oscillator Fock space, where some states in single oscillator Fock space are forbidden (e.g. the Gentile realization of exclusion statistics) and a combination of these two exclusions (e.g. Green's realization of para-Fermi statistics). For these types of exclusions we discuss extended Haldane statistics parameters g, recently introduced by two of us in Mod.Phys.Lett.A 11, 3081 (1996), and associated counting rules. Within these three types of exclusions in Fock space the original Haldane exclusion statistics cannot be realized.Comment: Latex,31 pages,no figures,to appear in J.Phys.A : Math.Ge

Supersymmetric Many-particle Quantum Systems with Inverse-square Interactions

The development in the study of supersymmetric many-particle quantum systems with inverse-square interactions is reviewed. The main emphasis is on quantum systems with dynamical OSp(2|2) supersymmetry. Several results related to exactly solved supersymmetric rational Calogero model, including shape invariance, equivalence to a system of free superoscillators and non-uniqueness in the construction of the Hamiltonian, are presented in some detail. This review also includes a formulation of pseudo-hermitian supersymmetric quantum systems with a special emphasis on rational Calogero model. There are quite a few number of many-particle quantum systems with inverse-square interactions which are not exactly solved for a complete set of states in spite of the construction of infinitely many exact eigen functions and eigenvalues. The Calogero-Marchioro model with dynamical SU(1,1|2) supersymmetry and a quantum system related to short-range Dyson model belong to this class and certain aspects of these models are reviewed. Several other related and important developments are briefly summarized.Comment: LateX, 65 pages, Added Acknowledgment, Discussions and References, Version to appear in Jouranl of Physics A: Mathematical and Theoretical (Commissioned Topical Review Article

Biomimetic rehabilitation engineering: the importance of somatosensory feedback for brain-machine interfaces.

Brain-machine interfaces (BMIs) re-establish communication channels between the nervous system and an external device. The use of BMI technology has generated significant developments in rehabilitative medicine, promising new ways to restore lost sensory-motor functions. However and despite high-caliber basic research, only a few prototypes have successfully left the laboratory and are currently home-deployed. The failure of this laboratory-to-user transfer likely relates to the absence of BMI solutions for providing naturalistic feedback about the consequences of the BMI's actions. To overcome this limitation, nowadays cutting-edge BMI advances are guided by the principle of biomimicry; i.e. the artificial reproduction of normal neural mechanisms. Here, we focus on the importance of somatosensory feedback in BMIs devoted to reproducing movements with the goal of serving as a reference framework for future research on innovative rehabilitation procedures. First, we address the correspondence between users' needs and BMI solutions. Then, we describe the main features of invasive and non-invasive BMIs, including their degree of biomimicry and respective advantages and drawbacks. Furthermore, we explore the prevalent approaches for providing quasi-natural sensory feedback in BMI settings. Finally, we cover special situations that can promote biomimicry and we present the future directions in basic research and clinical applications. The continued incorporation of biomimetic features into the design of BMIs will surely serve to further ameliorate the realism of BMIs, as well as tremendously improve their actuation, acceptance, and use