We consider the problem of estimating the state of a diffusion process, based on discrete time observations in singular noise. We reduce the problem to a static problem, and we show that the solution is provided by the area or co–area formula of ge~ metric measure theory, provided the observed value is a regular value of the observation function. In order to address the case of singular values,...

We extend the perturbation theory of Vǐsik, Ljusternik and Lidskĭı for eigenvalues of matrices, using methods of min-plus algebra. We show that the asymptotics of the eigenvalues of a perturbed matrix is governed by certain discrete optimisation problems, from which we derive new perturbation formulæ, extending the classical ones and solving cases which where singular in previous approaches. Ou...

Perturbation bounds in numerical linear algebra are traditionally derived and expressed using norms. Norm bounds cannot reflect the scaling or sparsity of a problem and its perturbation, and so can be unduly weak. If the problem data and its perturbation are measured componentwise, much smaller and more revealing bounds can be obtained. A survey is given of componentwise perturbation theory in ...

in this paper an exponentially fitted finite difference method is presented for solving singularly perturbed two-point boundary value problems with the boundary layer. a fitting factor is introduced and the model equation is discretized by a finite difference scheme on an uniform mesh. thomas algorithm is used to solve the tri-diagonal system. the stability of the algorithm is investigated. it ...

First order perturbation theory for eigenvalues of arbitrary matrices is systematically developed in all its generality with the aid of the Newton diagram, an elementary geometric construction first proposed by Isaac Newton. In its simplest form, a square matrix A with known Jordan canonical form is linearly perturbed to A(ε) = A+ ε B for an arbitrary perturbation matrix B, and one is intereste...

The problem of computing the best rank-(p, q, r) approximation of a third order tensor is considered. First the problem is reformulated as a maximization problem on a product of three Grassmann manifolds. Then expressions for the gradient and the Hessian are derived in a local coordinate system at a stationary point, and conditions for a local maximum are given. A first order perturbation analy...

Perturbation bounds in numerical linear algebra are traditionally derived and expressed using norms. Norm bounds cannot reeect the scaling or sparsity of a problem and its perturbation, and so can be unduly weak. If the problem data and its perturbation are measured componentwise, much smaller and more revealing bounds can be obtained. A survey is given of componentwise perturbation theory in n...

The weakly nonlinear dynamics of packets of equatorial Kelvin waves is studied using singular perturbation theory applied to the shallow water wave equations. Within the limits of the perturbation theory, which is formally restricted to weak mean shear and weak nonlinearity, we derive a Nonlinear Schroedinger equation to describe the envelope of the wave packet. We find that nonlinearity has a ...

Triviality and Landau poles are often greeted as harbingers of new physics at 1 TeV. After briefly reviewing the ideas behind this, a model of singular quantum mechanics is introduced. Its ultraviolet structure, as well as some features of its vacuum, related to triviality, very much parallel λφ. The model is solvable, exactly and perturbatively, in any dimension. From its analysis we learn tha...