Fractional Diffusion Limit for Collisional Kinetic Equations

This paper is devoted to diffusion limits of linear Boltzmann equations. When the equilibrium distribution function is a Maxwellian distribution, it is well known that for an appropriate time scale, the small mean free path limit gives rise to a diffusion equation. In this paper, we consider situations in which the equilibrium distribution function is a heavy-tailed distribution with infinite variance. We then show that for an appropriate time scale, the small mean free path limit gives rise to a fractional diffusion equation.     

Recognizing binet matrices

Binet matrices generalize network matrices and play an important role in combinatorial optimization. A first polynomial-time algorithm for recognizing binet matrices appeared in the author’s doctoral thesis. In this paper, we present some key ideas and results involved in the design of this algorithm. We show how we can find a Camion basis of the input matrix, whenever this one is binet, and then reduce the recognition problem to that of special binet matrices called bicyclic and cyclic.     

Analysis of diffuse-layer effects on time-dependent interfacial kinetics

We investigate the subtle effects of the diffuse charged layer on interfacial kinetics by solving the governing equations for ion transport (Nernst–Planck) with realistic boundary conditions representing reaction kinetics (Butler–Volmer) and compact-layer capacitance (Stern) in the asymptotic limit =λ_D/L→0, where λ_D is the Debye screening length and L is the distance between the working and counter electrodes. Using the methods of singular perturbation theory, we derive the leading-order steady-state response to a nonzero applied current in the case of the oxidation of a neutral species into cations, without any supporting electrolyte. In certain parameter regimes, the theory predicts a reaction-limited current smaller than the classical diffusion-limited current; this over potential effect is not due to ohmic drop effects in the bulk of the cell but rather to antagonist processes involved in the surface charge transfer and diffuse layer charging respectively. We demonstrate that the charging of diffuse charge, since it is intimately coupled to the surface reaction and cannot be considered independently, plays a fundamental role in nonequilibrium surface reactions when the transport of one of the reacting species is coupled to the total interfacial response of the compact and diffuse layers.     

Intermediary rings in normal pairs

We establish in this paper a result that gives the number of intermediary rings between R and S where (R,S) is a normal pair of rings. This result answers in particular a question which was left open in [A. Jaballah, Finiteness of the set of intermediary rings in a normal pair, Saintama Math. J. 17 (1999) 59-61]. Further applications are also given.     

On the solitaire cone and its relationship to multi-commodity flows

The classical game of Peg Solitaire has uncertain origins, but was certainly popular by the time of Louis XIV, and was described by Leibniz in 1710. The modern mathematical study of the game dates to the 1960s, when the solitaire cone was first described by Boardman and Conway. Valid inequalities over this cone, known as pagoda functions, were used to show the infeasibility of various peg games. In this paper we study the extremal structure of solitaire cones for a variety of boards, and relate their structure to the well studied metric cone. In particular we give:?1. an equivalence between the multicommodity flow problem with associated dual metric cone and a generalized peg game with associated solitaire cone;?2. a related NP-completeness result;?3. a method of generating large classes of facets;?4. a complete characterization of 0-1 facets;?5. exponential upper and lower bounds (in the dimension) on the number of facets;?6. results on the number of facets, incidence and adjacency relationships and diameter for small rectangular, toric and triangular boards;?7. a complete characterization of the adjacency of extreme rays, diameter, number of 2-faces and edge connectivity for rectangular toric boards. Received: July 1996 / Accepted: February 2000?Published online February 22, 2001