A gas-kinetic scheme for shallow-water equations with source terms

In this paper, the Kinetic Flux Vector Splitting (KFVS) scheme is extended to solving the shallow water equations with source terms. To develop a well-balanced scheme between the source term and the flow convection, the source term effect is accounted in the flux evaluation across cell interfaces. This leads to a modified gas-kinetic scheme with particular application to the shallow water equations with bottom topography. Numerical experiments show better resolution of the unsteady solution than conventional finite difference method and KFVS method with little additional cost. Moreover, some positivity properties of the gas-kinetic scheme is established.     

On Fraenkel’s <Emphasis Type="Italic">N</Emphasis>-Heap Wythoff’s Conjectures

The N-heap Wythoffs game is a two-player impartial game with N piles of tokens of sizes Players take turns removing any number of tokens from a single pile, or removing (a₁,..., a_N) from all piles - a_i tokens from the i-th pile, providing that where is the nim addition. The first player that cannot make a move loses. Denote all the P-positions (i.e., losing positions) by Two conjectures were proposed on the game by Fraenkel [7]. When are fixed, i) there exists an integer N₁ such that when . ii) there exist integers N₂ and _2 such that when , the golden section.In this paper, we provide a sufficient condition for the conjectures to hold, and subsequently prove them for the three-heap Wythoffs game with the first piles having up to 10 tokens.AMS Subject Classification: 91A46, 68R05.     

On the quantifier complexity of Δ<Subscript> n+1</Subscript> (T)– induction

In this paper we continue the study of the theories I _n+1 (T), initiated in [7]. We focus on the quantifier complexity of these fragments and theirs (non)finite axiomatization. A characterization is obtained for the class of theories such that I _n+1 (T) is _n+2 –axiomatizable. In particular, I _n+1 (I _n+1 ) gives an axiomatization of Th _n+2 (I _n+1 ) and is not finitely axiomatizable. This fact relates the fragment I _n+1 (I _n+1 ) to induction rule for _n+1 –formulas. Our arguments, involving a construction due to R. Kaye (see [9]), provide proofs of Parsons conservativeness theorem (see [16]) and (a weak version) of a result of L.D. Beklemishev on unnested applications of induction rules for _n+2 and _n+1 formulas (see [2]).Research partially supported by grant PB96–1345 (Spanish Goverment)Mathematics Subject Classification (2000): 03F30, 03H15     

Robust H ∞ Filtering for a Class of Nonlinear Discrete-Time Markovian Jump Systems

This paper considers the problem of the robust H filtering for a class of nonlinear discrete-time Markovian jump systems with real time-varying norm-bounded parameter uncertainty. For each mode, the nonlinearity is assumed to satisfy the global Lipschitz conditions and appears in both the state and measured output equations. The problem that we address is the design of a nonlinear filter which ensures robust stochastic stability and a prescribed H performance level of the filtering error system for all admissible uncertainties. A sufficient condition for the solvability of this problem is obtained in terms of a set of linear matrix inequalities; an explicit expression of a desired nonlinear H filter is also given. Finally, an example is provided to demonstrate the effectiveness of the proposed approach.     

On the Green Function for a Complement of a Finite Number of Real Intervals

We give a new proof (and a generalization) of recent remarkable result by V. Totik concerning the smoothness properties of the equilibrium measure for compact subsets of the real line and the Green function for complement of these subsets taken with respect to the complex plane.     

Stabilization and H{infty} control of two-dimensional Markovian jump systems

This paper extends the results obtained for one-dimensionalMarkovian jump systems, to investigate the problems of stochasticstabilization and H control of two-dimensional (2D) systemswith Markovian jump parameters. The mathematical model of 2Djump systems is established upon the well-known Roesser model,and sufficient conditions are obtained for the existence ofdesired controllers in terms of linear matrix inequalities,which can be readily solved by available numerical software.These obtained results are further extended to more generalcases whose system matrices also contain parameter uncertaintiesrepresented by either polytopic or norm-bounded approaches.A numerical example is provided to show the applicability ofthe proposed theories.     

On the Irreducibility of Commuting Varieties Associated with Involutions of Simple Lie Algebras

Let be a reductive Lie algebra over an algebraically closed field of characteristic zero and an arbitrary -grading. We consider the variety , which is called the commuting variety associated with the -grading. Earlier it was proved by the author that is irreducible, if the -grading is of maximal rank. Now we show that is irreducible for and (E₆,F₄). In the case of symmetric pairs of rank one, we show that the number of irreducible components of is equal to that of nonzero non--regular nilpotent G ₀-orbits in . We also discuss a general problem of the irreducibility of commuting varieties.     

A note on the fundamental solution for the Tricomi-type equation in the hyperbolic domain

Barros-Neto and Gelfand (Duke Math. J. 98 (3) (1999) 465; Duke Math. J. 117 (2) (2003) 561) constructed for the Tricomi operator on the plane the fundamental solutions with the supports in the regions related to the geometry of the characteristics of the Tricomi operator. We give for the Tricomi-type operator a fundamental solution relative to an arbitrary point of Rⁿ⁺¹ with the support in the region t?0, where the operator is hyperbolic. Our key observation is that the fundamental solution for the Tricomi-type operator can be written like an integral of the distributions generated by the fundamental solution of the Cauchy problem for the wave equation. The application of that fundamental solution to the L^p-L^q estimate for the forced Tricomi-type equation is given as well.     

Commutators, C0-semigroups and resolvent estimates

We study the existence and the continuity properties of the boundary values on the real axis of the resolvent of a self-adjoint operator H in the framework of the conjugate operator method initiated by Mourre. We allow the conjugate operator A to be the generator of a C₀-semigroup (finer estimates require A to be maximal symmetric) and we consider situations where the first commutator [H,iA] is not comparable to H. The applications include the spectral theory of zero mass quantum field models.