Traffic flow modelling with junctions

Motivated by the modelling of a roundabout, we are led to study the traffic on a road with points of entry and exit. In this note, we would like to describe the modellisation of a junction and solve the Riemann problem for such a model. More precisely, between each point of discontinuity we use a multi-class extension of the LWR model to describe the evolution of the density of the vehicles, the ‘multi-class’ approach being used in order to distinguish the vehicles after their origin and destination. Then, we treat the points of entry and exit thanks to special boundary conditions that give bounds on the flows of the different types of vehicles. In the case of the one-T road we obtain a result of existence and uniqueness. This first step allows us to obtain a similar result for the n-T road. We describe these results and also some properties of the obtained solutions, in order to see how long this model is valid.     

Lu Qi-Keng's problem on some complex ellipsoids

In this note, we give an improvement on the Bergman kernel for the domain . As an application, we describe how the zeroes of the kernel depend on the defining parameters p,m,n. We also consider the domain .     

On preconditioned eigensolvers and Invert-Lanczos processes

This paper deals with the convergence analysis of various preconditioned iterations to compute the smallest eigenvalue of a discretized self-adjoint and elliptic partial differential operator. For these eigenproblems several preconditioned iterative solvers are known, but unfortunately, the convergence theory for some of these solvers is not very well understood.The aim is to show that preconditioned eigensolvers (like the preconditioned steepest descent iteration (PSD) and the locally optimal preconditioned conjugate gradient method (LOPCG)) can be interpreted as truncated approximate Krylov subspace iterations. In the limit of preconditioning with the exact inverse of the system matrix (such preconditioning can be approximated by multiple steps of a preconditioned linear solver) the iterations behave like Invert-Lanczos processes for which convergence estimates are derived.     

Unit Distances and Diameters in Euclidean Spaces

We show that the maximum number of unit distances or of diameters in a set of n points in d-dimensional Euclidean space is attained only by specific types of Lenz constructions, for all d≥4 and n sufficiently large depending on d. As a corollary, we determine the exact maximum number of unit distances for all even d≥6 and the exact maximum number of diameters for all d≥4 and all n sufficiently large depending on d. This material is based upon work supported by the South African National Research Foundation.     

The Diffie–Hellman problem and generalization of Verheul’s theorem

Bilinear pairings on elliptic curves have been of much interest in cryptography recently. Most of the protocols involving pairings rely on the hardness of the bilinear Diffie–Hellman problem. In contrast to the discrete log (or Diffie–Hellman) problem in a finite field, the difficulty of this problem has not yet been much studied. In 2001, Verheul (Advances in Cryptology—EUROCRYPT 2001, LNCS 2045, pp. 195–210, 2001) proved that on a certain class of curves, the discrete log and Diffie–Hellman problems are unlikely to be provably equivalent to the same problems in a corresponding finite field unless both Diffie–Hellman problems are easy. In this paper we generalize Verheul’s theorem and discuss the implications on the security of pairing based systems.      

二阶奇异微分方程组边值问题两个正解的存在性

利用锥拉伸和锥压缩不动点定理,给出了一类二阶微分方程组奇异边值问题两个正解的存在性.     

Model-theoretic complexity of automatic structures

We study the complexity of automatic structures via well-established concepts from both logic and model theory, including ordinal heights (of well-founded relations), Scott ranks of structures, and Cantor–Bendixson ranks (of trees). We prove the following results: (1) The ordinal height of any automatic well-founded partial order is bounded by ω^ω. (2) The ordinal heights of automatic well-founded relations are unbounded below , the first non-computable ordinal. (3) For any computable ordinal α, there is an automatic structure of Scott rank at least α. Moreover, there are automatic structures of Scott rank . (4) For any computable ordinal α, there is an automatic successor tree of Cantor–Bendixson rank α.     

Three points method for searching the best least absolute deviations plane

In this paper a new method for estimation of optimal parameters of a best least absolute deviations plane is proposed, which is based on the fact that there always exists a best least absolute deviations plane passing through at least three different data points. The proposed method leads to a solution in finitely many steps. Moreover, a modification of the aforementioned method is proposed that is especially adjusted to the case of a large number of data and the need to estimate parameters in real time. Both methods are illustrated by numerical examples on the basis of simulated data and by one practical example from the field of robotics.     

Computing eigenelements of boundary value problems with fractional derivatives

Variational iteration method has been successfully implemented to handle linear and nonlinear differential equations. The main property of the method is its flexibility and ability to solve nonlinear equations accurately and conveniently. In this paper, first, a general framework of the variational iteration method is presented for analytic treatment of differential equations of fractional order where the fractional derivatives are described in Caputo sense. Second, the new framework is used to compute approximate eigenvalues and the corresponding eigenfunctions for boundary value problems with fractional derivatives. Numerical examples are tested to show the pertinent features of this method. This approach provides a new way to investigate eigenvalue problems with fractional order derivatives.