A DIRECT METHOD FOR THE UNIT CIRCLE PROBLEM

The unit circle problem is the problem of finding the number of eigenvalues of a non-Hermitian matrix inside and outside the unit circle . To reduce the cost of computing eigenvalues for the problem, a direct method, which is analogous to that given in [5], is proposed in this paper.     

Application of coupling methods to the Neumann eigenvalue problem

Summary By using coupling methods, some lower bounds are obtained for the first Neumann eigenvalue on Riemannian manifolds. This method is new and the results improve some known estimates. An example shows that our estimates can be sharp.Research supported in part by the Foundation of Beijing Normal University     

Observation of p(1×1)-type rows on TiO2(110)-p(1×2) surface by scanning tunneling microscope (STM)

The TiO₂(110) surfaces were observed by a Scanning Tunneling Microscope (STM). We found two types of bright p(1×1)-type rows on the p(1×2) surface. One p(1×1)-type formed independently and corresponds to the bridging oxygen rows. The second p(1×1)-type appeared in a bright grouping, forming narrow rows, and corresponds to the five-fold titanium rows. The above results suggest the following two conclusions. First, the density of state (DOS) on the bridging oxygen rows becomes higher than that on the five-fold titanium atom rows when a bridging oxygen row exists independently on the p(1×2) surface. Second, the bright rows on a TiO₂(110)-p(1×1) surface correspond to the five-fold titanium atom rows. The results further show the validity of DOS calculations on the TiO₂(110)-p(1×1) surface by Diebold et al. [Phys. Rev. Lett. 77 (1996) 1322]. The difference of width for Ti₂O₃ unit rows on the p(1×2) and p(1×3) surfaces in STM images are also discussed.     

Symmetries and conservation laws of Navier-Stokes equations

All the symmetries and conservation laws of Navier-Stokes equations are calculated.     

Literal‐paraconsistent and literal‐paracomplete matrices

We introduce a family of matrices that define logics in which paraconsistency and/or paracompleteness occurs only at the level of literals, that is, formulas that are propositional letters or their iterated negations. We give a sound and complete axiomatization for the logic defined by the class of all these matrices, we give conditions for the maximality of these logics and we study in detail several relevant examples. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Density estimation and adaptive control of markov processes: Average and discounted criteria

We consider a class of discrete-time Markov control processes with Borel state and action spaces, and ^d i.i.d. disturbances with unknown distribution . Under mild semi-continuity and compactness conditions, and assuming that is absolutely continuous with respect to Lebesgue measure, we establish the existence of adaptive control policies which are (1) optimal for the average-reward criterion, and (2) asymptotically optimal in the discounted case. Our results are obtained by taking advantage of some well-known facts in the theory of density estimation. This approach allows us to avoid restrictive conditions on the state space and/or on the system's transition law imposed in recent works, and on the other hand, it clearly shows the way to other applications of nonparametric (density) estimation to adaptive control.Research partially supported by The Third World Academy of Sciences under Research Grant No. MP 898-152.     

Partitioning multi-edge graphs

We introduce a new kind of graph called multi-edge graph which arises in many applications such as finite state Markov chains and broadcasting communication networks. A cluster in such a graph is a set of nodes such that for any ordered pair of nodes, there is a path of multi-edges from one to the other such that these edges remain within the same set. We give two algorithms to partition a multi-edge graph into maximal clusters. Both these algorithms are based on the depth-first search algorithm to find strongly connected components of the directed graph. We also discuss some applications of clustering in multi-edge graphs.     

On a problem proposed by H. Braß concerning the remainder term in quadrature for convex functions

Summary We prove that the error inn-point Gaussian quadrature, with respect to the standard weight functionw1, is of best possible orderO(n ^–2) for every bounded convex function. This result solves an open problem proposed by H. Braß and published in the problem section of the proceedings of the 2. Conference on Numerical Integration held in 1981 at the Mathematisches Forschungsinstitut Oberwolfach (Hämmerlin 1982; Problem 2). Furthermore, we investigate this problem for positive quadrature rules and for general product quadrature. In particular, for the special class of Jacobian weight functionsw _, (x)=(1–x)(1+x), we show that the above result for Gaussian quadrature is not valid precisely ifw _, is unbounded.Dedicated to Prof. H. Braß on the occasion of his 55th birthday     

Parallel and superfast algorithms for Hankel systems of equations

Summary Utilizing kernel structure properties a unified construction of Hankel matrix inversion algorithms is presented. Three types of algorithms are obtained: 1)O(n ²) complexity Levinson type, 2)O (n) parallel complexity Schur-type, and 3)O(n log² n) complexity asymptotically fast ones. All algorithms work without additional assumption (like strong nonsingularity).     

Nonlinear stability and convergence of finite-difference methods for the “good” Boussinesq equation

Summary The good Boussinesq equationu _tt =–u _xxxx +u _xx +(u ²) _xx has recently been found to possess an interesting soliton-interaction mechanism. In this paper we study the nonlinear stability and the convergence of some simple finite-difference schemes for the numerical solution of problems involving the good Boussinesq equation. Numerical experimentas are also reported.