Superintegrability in Two Dimensions and the Racah–Wilson Algebra

The analysis of the most general second-order superintegrable system in two dimensions: the generic 3-parameter model on the 2-sphere is cast in the framework of the Racah problem for the \({\mathfrak{su}(1,1)}\) algebra. The Hamiltonian of the 3-parameter system and the generators of its quadratic symmetry algebra are seen to correspond to the total and intermediate Casimir operators of the combination of three \({\mathfrak{su}(1,1)}\) algebras, respectively. The construction makes explicit the isomorphism between the Racah–Wilson algebra, which is the fundamental algebraic structure behind the Racah problem for \({\mathfrak{su}(1, 1)}\) , and the invariance algebra of the generic 3-parameter system. It also provides an explanation for the occurrence of the Racah polynomials as overlap coefficients in this context. The irreducible representations of the Racah–Wilson algebra are reviewed as well as their connection with the Askey scheme of classical orthogonal polynomials.     

Statistics of a Flux in Burgers Turbulence¶with One-Sided Brownian Initial Data

We study the statistics of the flux of particles crossing the origin, which is induced by the dynamics of ballistic aggregation in dimension 1, under certain random initial conditions for the system. More precisely, we consider the cases when particles are uniformly distributed on ℝ at the initial time, and if u(x,t) denotes the velocity of the particle located at x at time t, then u(x,0)= 0 for x<0 and (u(x,0), x≥ 0) is either a white noise or a Brownian motion. Received: 18 April 2001 / Accepted: 16 July 2001     

A variance-based method to rank input variables of the Mesh Adaptive Direct Search algorithm

The Mesh Adaptive Direct Search algorithm (Mads) algorithm is designed for nonsmooth blackbox optimization problems in which the evaluation of the functions defining the problems are expensive to compute. The Mads algorithm is not designed for problems with a large number of variables. The present paper uses a statistical tool based on variance decomposition to rank the relative importance of the variables. This statistical method is then coupled with the Mads algorithm so that the optimization is performed either in the entire space of variables or in subspaces associated with statistically important variables. The resulting algorithm is called Stats-Mads and is tested on bound constrained test problems having up to 500 variables. The numerical results show a significant improvement in the objective function value after a fixed budget of function evaluations.     

The electrical conductivity of binary disordered systems,percolation clusters,fractals and related models

We review theoretical and experimental studies of the AC dielectric response of inhomogeneous materials, modelled as bond percolation networks, with a binary (conductor-dielectric) distribution of bond conductances. We first summarize the key results of percolation theory, concerning mostly geometrical and static (DC) transport properties, with emphasis on the scaling properties of the critical region around the percolation threshold. The frequency-dependent (AC) response of a general binary model is then studied by means of various approaches, including the effective-medium approximation, a scaling theory of the critical region, numerical computations using the transfer-matrix algorithm, and several exactly solvable deterministic fractal models. Transient regimes, related to singularities in the complex-frequency plane, are also investigated. Theoretical predictions are made more explicit in two specific cases, namely R-C and RL-C networks, and compared with a broad variety of experimental results, concerning, for example, granular composites, thin films, powders, microemulsions, cermets, porous ceramics and the viscoelastic properties of gels.     

Controllability cost of conservative systems: resolvent condition and transmutation

This article concerns the exact controllability of unitary groups on Hilbert spaces with unbounded control operator. It provides a necessary and sufficient condition not involving time which blends a resolvent estimate and an observability inequality. By the transmutation of controls in some time L for the corresponding second-order conservative system, it is proved that the cost of controls in time T for the unitary group grows at most like exp(αL²/T) as T tends to 0. In the application to the cost of fast controls for the Schrödinger equation, L is the length of the longest ray of geometric optics which does not intersect the control region. This article also provides observability resolvent estimates implying fast smoothing effect controllability at low cost, and underscores that the controllability cost of a system is not changed by taking its tensor product with a conservative system.     

Genetic analysis of the GRIK2 modifier effect in Huntington's disease

Background  

In Huntington's disease (HD), age at neurological onset is inversely correlated with the length of the CAG trinucleotide repeat mutation, but can be modified by genetic factors beyond the HD gene. Association of a relatively infrequent 16 TAA allele of a trinucleotide repeat polymorphism in the GRIK2 3'UTR with earlier than expected age at neurological onset has been suggested to reflect linkage disequilibrium with a functional polymorphism in GRIK2 or an adjacent gene.     

Scalarization of vector optimization problems

In this paper, we investigate the scalar representation of vector optimization problems in close connection with monotonic functions. We show that it is possible to construct linear, convex, and quasiconvex representations for linear, convex, and quasiconvex vector problems, respectively. Moreover, for finding all the optimal solutions of a vector problem, it suffices to solve certain scalar representations only. The question of the continuous dependence of the solution set upon the initial vector problems and monotonic functions is also discussed.The author is grateful to the two referees for many valuable comments and suggestions which led to major imporvements of the paper.     

On the height of random m-ary search trees

A random m-ary seach tree is constructed from a random permutation of 1,…, n. A law of large numbers is obtained for the height H_n of these trees by applying the theory of branching random walks. in particular, it is shown that H_n/log n→γ in probability as n→∞ where γ = γ(m) is a constant depending upon m only. Interestingly, as m→∞, γ(m) is asymptotic to 1/log m, the coefficient of log n in the asymptotic expression for the height of the complete m-ary search tree. This proves that for large m, random m-ary search trees behave virtually like complete m-ary trees.