Search results for "Mathematica"
showing 10 items of 7971 documents
One-dimensional families of projections
2008
Let m and n be integers with 0 < m < n. We consider the question of how much the Hausdorff dimension of a measure may decrease under typical orthogonal projections from onto m-planes provided that the dimension of the parameter space is one. We verify the best possible lower bound for the dimension drop and illustrate the sharpness of our results by examples. The question stems naturally from the study of measures which are invariant under the geodesic flow.
Computation of a few smallest eigenvalues of elliptic operators using fast elliptic solvers
2001
The computation of a few smallest eigenvalues of generalized algebraic eigenvalue problems is studied. The considered problems are obtained by discretizing self-adjoint second-order elliptic partial differential eigenvalue problems in two- or three-dimensional domains. The standard Lanczos algorithm with the complete orthogonalization is used to compute some eigenvalues of the inverted eigenvalue problem. Under suitable assumptions, the number of Lanczos iterations is shown to be independent of the problem size. The arising linear problems are solved using some standard fast elliptic solver. Numerical experiments demonstrate that the inverted problem is much easier to solve with the Lanczos…
A regularized Newton method for locating thin tubular conductivity inhomogeneities
2011
We consider the inverse problem of determining the position and shape of a thin tubular object, such as for instance a wire, a thin channel or a curve-like crack, embedded in some three-dimensional homogeneous body from a single measurement of electrostatic currents and potentials on the boundary of the body. Using an asymptotic model describing perturbations of electrostatic potentials caused by such thin objects, we reformulate the inverse problem as a nonlinear operator equation. We establish Frechet differentiability of the corresponding operator, compute its Frechet derivative and set up a regularized Newton scheme to solve the inverse problem numerically. We discuss our implementation…
Comparison of discretization strategies for the model-free information-theoretic assessment of short-term physiological interactions
2023
This work presents a comparison between different approaches for the model-free estimation of information-theoretic measures of the dynamic coupling between short realizations of random processes. The measures considered are the mutual information rate (MIR) between two random processes [Formula: see text] and [Formula: see text] and the terms of its decomposition evidencing either the individual entropy rates of [Formula: see text] and [Formula: see text] and their joint entropy rate, or the transfer entropies from [Formula: see text] to [Formula: see text] and from [Formula: see text] to [Formula: see text] and the instantaneous information shared by [Formula: see text] and [Formula: see…
The Calderón Problem for a Space-Time Fractional Parabolic Equation
2020
In this article we study an inverse problem for the space-time fractional parabolic operator $(\partial_t-\Delta)^s+Q$ with $0<s<1$ in any space dimension. We uniquely determine the unknown bounded...
Jacobian of solutions to the conductivity equation in limited view
2022
Abstract The aim of hybrid inverse problems such as Acousto-Electric Tomography or Current Density Imaging is the reconstruction of the electrical conductivity in a domain that can only be accessed from its exterior. In the inversion procedure, the solutions to the conductivity equation play a central role. In particular, it is important that the Jacobian of the solutions is non-vanishing. In the present paper we address a two-dimensional limited view setting, where only a part of the boundary of the domain can be controlled by a non-zero Dirichlet condition, while on the remaining boundary there is a zero Dirichlet condition. For this setting, we propose sufficient conditions on the bounda…
Towards a local approach to fatigue, for the calculation of structures, applied to continuous fibre reinforced composite materials and structures
2018
The original concept of mechanical fatigue was linked to the failure of structures and was treated at first within the framework of Fracture Mechanics. Models developed to explain this phenomenon must therefore be able to be applied to structures: changing the structure means changes to the model and its identification. It is therefore judicious to develop models capable of treating both the damage processes and also able to be used within a local framework: to this end a method based on Damage Mechanics seems appropriate. This approach has long been employed and requires only the identification of damage processes at the level of the RVE (Representative Volume Element) to be used for any s…
The Calderón problem for the fractional Schrödinger equation
2020
We show global uniqueness in an inverse problem for the fractional Schr\"odinger equation: an unknown potential in a bounded domain is uniquely determined by exterior measurements of solutions. We also show global uniqueness in the partial data problem where the measurements are taken in arbitrary open, possibly disjoint, subsets of the exterior. The results apply in any dimension $\geq 2$ and are based on a strong approximation property of the fractional equation that extends earlier work. This special feature of the nonlocal equation renders the analysis of related inverse problems radically different from the traditional Calder\'on problem.
On Erlang B-formula and ERT method extension
2010
The key result of the paper is the theorem on traffic splitting and the ERT method extension for estimation of the throughput for schemes with traffic splitting. The excellent accuracy (relative error is less than 1%) is shown in numerical example. The paper also contains new Erlang-B formula algorithm for non-integer number of channels based on parabolic approximation.
On spline methods of approximation under L-fuzzy information
2011
This work is closely related to our previous papers on algorithms of approximation under L-fuzzy information. In the classical theory of approximation central algorithms were worked out on the basis of usual, that is crisp splines. We describe central methods for solution of linear problems with balanced L-fuzzy information and develop the concept of L-fuzzy splines.