An Exact Riemann Solver for a Fluidized Bed Model

We study a 2 x 2 hyperbolic system of conservation laws withsource term arising in a fluidized bed model. The system issolved numerically and results are presented to demonstratethe occurrence of ‘slugging’ in the full model equations.The numerical procedure is based on operator splitting and Godunov'smethod, for which we derive the exact solution of the Riemannproblem. A second-order improvement due to Davis (1988) mayproduce small oscillations near shocks and these can be reducedif the underlying flux limiter of the Davis method is replacedby the minmod limiter.     

Certain series involving products of laguerre polynomials

The author discusses here certain infinite sums of products of generalized Laguerre polynomials.     

Symmetric function means of positive operators

Many authors have studied means involving ratios of the elementary symmetric functions. This work involves generalizing these means to an operator theoretic setting, and extending many of the classical scalar inequalities to operators. It is shown that these operator means are related to electrical networks.     

On completeness of quadratic systems

A dynamical system is called complete if every solution of it exists for all t∈R. Let K be the dimension of the vector space of quadratic systems. The set of complete quadratic systems is shown to contain a vector subspace of dimension 2K/3. We provide two proofs, one by the Gronwall lemma and the second by compactification that is capable of showing incompleteness as well. Characterization of a vector subspace of complete quadratic systems is provided. The celebrated Lorenz system for all real ranges of its parameters is shown to belong to this subspace. We also provide a sufficient condition for a system to be incomplete.     

Eigenvalues of the difference and product of projections

There is an isomorphism between the matrices over the Boolean algebra of subsets of a k-element set and the k-tuples of Boolean binary (i.e. zero-one) matrices. This isomorphism allows many problems concerning nonbinary Boolean matrices to the referred to the binary ease. However, there are some features of the general (i.e. nonbinary) case that have not been mentioned, although they differ somewhat from the binary case. We exhibit characterizations of the linear operators that preserve several invariants of matrices over finite Boolean algebras to illustrate the differences and similarities of the general vs. the binary cases. We employ a canonical form that is useful in applying the isomorphism.     

Two functionals connected to the Laplacian in a class of doubly connected domains on rank one symmetric spaces of non-compact type

Let B ₁ be a ball in a non-compact rank-one symmetric space and let B ₀ be a smaller ball inside it. It is shown that if y is the solution of the problem ?Δu = 1 in ${B_1 \setminus \bar{B_0}}$ vanishing on the boundary, then the Dirichlet-energy of y is minimal if and only if the balls are concentric. It is also shown that the first Dirichlet eigenvalue of the Laplacian on ${B_1 \setminus \bar{B_0}}$ is maximal if and only if the two balls are concentric. The formalism of Damek-Ricci harmonic spaces is used.