Uniform boundary Harnack principle and generalized triangle property

Let D be a bounded open subset in R^d, d?2, and let G denote the Green function for D with respect to (-Δ)^α/2, 0<α?2, α<d. If α<2, assume that D satisfies the interior corkscrew condition; if α=2, i.e., if G is the classical Green function on D, assume—more restrictively—that D is a uniform domain. Let g=G(·,y₀)∧1 for some y₀∈D. Based on the uniform boundary Harnack principle, it is shown that G has the generalized triangle property which states that when d(z,x)?d(z,y). An intermediate step is the approximation G(x,y)≈|x-y|^α-dg(x)g(y)/g(A)², where A is an arbitrary point in a certain set B(x,y).This is discussed in a general setting where D is a dense open subset of a compact metric space satisfying the interior corkscrew condition and G is a quasi-symmetric positive numerical function on D×D which has locally polynomial decay and satisfies Harnack's inequality. Under these assumptions, the uniform boundary Harnack principle, the approximation for G, and the generalized triangle property turn out to be equivalent.     

Exact uncertainty approach in quantum mechanics and quantum gravity

The assumption that an ensemble of classical particles is subject to nonclassical momentum fluctuations, with the fluctuation uncertainty fully determined by the position uncertainty, has been shown to lead from the classical equations of motion to the Schrödinger equation. This ‘exact uncertainty’ approach may be generalised to ensembles of gravitational fields, where nonclassical fluctuations are added to the field momentum densities, of a magnitude determined by the uncertainty in the metric tensor components. In this way one obtains the Wheeler-DeWitt equation of quantum gravity, with the added bonus of a uniquely specified operator ordering. No a priori assumptions are required concerning the existence of wave functions, Hilbert spaces, Planck's constant, linear operators, etc. Thus this approach has greater transparency than the usual canonical approach, particularly in regard to the connections between quantum and classical ensembles. Conceptual foundations and advantages are emphasised.     

Uncomplementability of spaces of compact operators in larger spaces of operators

In the first part of the paper we prove some new result improving all those already known about the equivalence of the nonexistence of a projection (of any norm) onto the space of compact operators and the containment of c ₀ in the same space of compact operators. Then we show several results implying that the space of compact operators is uncomplemented by norm one projections in larger spaces of operators. The paper ends with a list of questions naturally rising from old results and the results in the paper.     

A Stochastic Algorithm for Fault Inverse Problems in Elastic Half Space with Proof of Convergence

A general stochastic algorithm for solving mixed linear and nonlinear problems was introduced in [11]. We show in this paper how it can be used to solve the fault inverse problem, where a planar fault in elastic half-space and a slip on that fault have to be reconstructed from noisy surface displacement measurements. With the parameter giving the plane containing the fault denoted by $m$ and the regularization parameter for the linear part of the inverse problem denoted by $C$,both modeled as random variables, we derive a formula for the posterior marginal of $m.$ Modeling $C$ as a random variable allows to sweep through a wide range of possible values which was shown to be superior to selecting a fixed value [11]. We prove that this posterior marginal of $m$ is convergent as the number of measurement points and the dimension of the space for discretizing slips increase. Simply put, our proof only assumes that the regularized discrete error functional for processing measurements relates to an order 1 quadrature rule and that the union of the finite-dimensional spaces for discretizing slips is dense. Our proof relies on trace class operator theory to show that an adequate sequence of determinants is uniformly bounded. We also explain how our proof can be extended to a whole class of inverse problems, as long as some basic requirements are met. Finally, we show numerical simulations that illustrate the numerical convergence of our algorithm.     

Estimation variances for estimators of product densities and pair correlation functions of planar point processes

Approximations of the estimation variances of kernel estimators of the pair correlation function and the product density of a planar Poisson process are given. Furthermore, a heuristic approximation of the estimation variance of an estimator of the pair correlation function of a general planar point process is suggested. All formulae have been tested by simulation experiments.     

An approximation algorithm for the m-machine permutation flow shop scheduling problem with controllable processing times

The paper deals with the m-machine permutation flow shop scheduling problem in which job processing times, along with a processing order, are decision variables. It is assumed that the cost of processing a job on each machine is a linear function of its processing time and the overall schedule cost to be minimized is the total processing cost plus maximum completion time cost. A algorithm for the problem with m = 2 is provided; the best approximation algorithm until now has a worst-case performance ratio equal to . An extension to the m-machine (m ≥2) permutation flow shop problem yields an approximation algorithm with a worst-case bound equal to

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, where is the worst-case performance ratio of a procedure used, in the proposed algorithm, for solving the (pure) sequencing problem. Moreover, examples which achieve this bound for = 1 are also presented.     

A SECOND ORDER CONTROL-VOLUME FINITE-ELEMENT LEAST-SQUARES STRATEGY FOR SIMULATING DIFFUSION IN STRONGLY ANISOTROPIC MEDIA

An unstructured mesh finite volume discretisation method for simulating diffusion in anisotropic media in two-dimensional space is discussed. This technique is considered as an extension of the fully implicit hybrid control-volume finite-element method and it retains the local continuity of the flux at the control volume faces. A least squares function recon-struction technique together with a new flux decomposition strategy is used to obtain an accurate flux approximation at the control volume face, ensuring that the overall accuracy of the spatial discretisation maintains second order. This paper highlights that the new technique coincides with the traditional shape function technique when the correction term is neglected and that it significantly increases the accuracy of the previous linear scheme on coarse meshes when applied to media that exhibit very strong to extreme anisotropy ratios. It is concluded that the method can be used on both regular and irregular meshes,and appears independent of the mesh quality.     

Approximations for Steiner Trees with Minimum Number of Steiner Points

Given n terminals in the Euclidean plane and a positive constant, find a Steiner tree interconnecting all terminals with the minimum number of Steiner points such that the Euclidean length of each edge is no more than the given positive constant. This problem is NP-hard with applications in VLSI design, WDM optical networks and wireless communications. In this paper, we show that (a) the Steiner ratio is 1/ 4, that is, the minimum spanning tree yields a polynomial-time approximation with performance ratio exactly 4, (b) there exists a polynomial-time approximation with performance ratio 3, and (c) there exists a polynomial-time approxi-mation scheme under certain conditions.     

Explicit solution of the operator equation

In this paper we find the explicit solution of the equation

A^*X+X^*A=B