Qualitative Properties of alpha-Weighted Scheduling Policies

We consider a switched network, a fairly general constrained queueing network model that has been used successfully to model the detailed packet-level dynamics in communication networks, such as input-queued switches and wireless networks. The main operational issue in this model is that of deciding which queues to serve, subject to certain constraints. In this paper, we study qualitative performance properties of the well known $\alpha$-weighted scheduling policies. The stability, in the sense of positive recurrence, of these policies has been well understood. We establish exponential upper bounds on the tail of the steady-state distribution of the backlog. Along the way, we prove finiteness of the expected steady-state backlog when $\alpha<1$, a property that was known only for $\alpha\geq 1$. Finally, we analyze the excursions of the maximum backlog over a finite time horizon for $\alpha \geq 1$. As a consequence, for $\alpha \geq 1$, we establish the full state space collapse property.Comment: 13 page

Ground State Energy for Fermions in a 1D Harmonic Trap with Delta Function Interaction

Conjectures are made for the ground state energy of a large spin 1/2 Fermion system trapped in a 1D harmonic trap with delta function interaction. States with different spin J are separately studied. The Thomas-Fermi method is used as an effective test for the conjecture.Comment: 4 pages, 3 figure

Pointed Hopf Algebras with classical Weyl Groups

We prove that Nichols algebras of irreducible Yetter-Drinfeld modules over classical Weyl groups $A \rtimes \mathbb S_n$ supported by $\mathbb S_n$ are infinite dimensional, except in three cases. We give necessary and sufficient conditions for Nichols algebras of Yetter-Drinfeld modules over classical Weyl groups $A \rtimes \mathbb S_n$ supported by $A$ to be finite dimensional.Comment: Combined with arXiv:0902.4748 plus substantial changes. To appear International Journal of Mathematic

Exploration of Resonant Continuum and Giant Resonance in the Relativistic Approach

Single-particle resonant-states in the continuum are determined by solving scattering states of the Dirac equation with proper asymptotic conditions in the relativistic mean field theory (RMF). The regular and irregular solutions of the Dirac equation at a large radius where the nuclear potentials vanish are relativistic Coulomb wave functions, which are calculated numerically. Energies, widths and wave functions of single-particle resonance states in the continuum for ^{120}Sn are studied in the RMF with the parameter set of NL3. The isoscalar giant octupole resonance of ^{120}Sn is investigated in a fully consistent relativistic random phase approximation. Comparing the results with including full continuum states and only those single-particle resonances we find that the contributions from those resonant-states dominate in the nuclear giant resonant processes.Comment: 16 pages, 2 figure