Integrating Singular Functions on the Sphere

We obtain rigorous results concerning the evaluation of integrals on the two sphere using complex methods. It is shown that for regular as well as singular functions which admit poles, the integral can be reduced to the calculation of residues through a limiting procedure.Comment: 15 pages, revte

Fuzzy spectral and spatial feature integration for classification of nonferrous materials in hyperspectral data

Hyperspectral data allows the construction of more elaborate models to sample the properties of the nonferrous materials than the standard RGB color representation. In this paper, the nonferrous waste materials are studied as they cannot be sorted by classical procedures due to their color, weight and shape similarities. The experimental results presented in this paper reveal that factors such as the various levels of oxidization of the waste materials and the slight differences in their chemical composition preclude the use of the spectral features in a simplistic manner for robust material classification. To address these problems, the proposed FUSSER (fuzzy spectral and spatial classifier) algorithm detailed in this paper merges the spectral and spatial features to obtain a combined feature vector that is able to better sample the properties of the nonferrous materials than the single pixel spectral features when applied to the construction of multivariate Gaussian distributions. This approach allows the implementation of statistical region merging techniques in order to increase the performance of the classification process. To achieve an efficient implementation, the dimensionality of the hyperspectral data is reduced by constructing bio-inspired spectral fuzzy sets that minimize the amount of redundant information contained in adjacent hyperspectral bands. The experimental results indicate that the proposed algorithm increased the overall classification rate from 44% using RGB data up to 98% when the spectral-spatial features are used for nonferrous material classification

Understanding singularities in Cartan's and NSF geometric structures

In this work we establish a relationship between Cartan's geometric approach to third order ODEs and the 3-dim Null Surface Formulation (NSF). We then generalize both constructions to allow for caustics and singularities that necessarily arise in these formalisms.Comment: 22 pages, 2 figure

A framework for randomized time-splitting in linear-quadratic optimal control

Inspired by the successes of stochastic algorithms in the training of deep neural networks and the simulation of interacting particle systems, we propose and analyze a framework for randomized time-splitting in linear-quadratic optimal control. In our proposed framework, the linear dynamics of the original problem is replaced by a randomized dynamics. To obtain the randomized dynamics, the system matrix is split into simpler submatrices and the time interval of interest is split into subintervals. The randomized dynamics is then found by selecting randomly one or more submatrices in each subinterval. We show that the dynamics, the minimal values of the cost functional, and the optimal control obtained with the proposed randomized time-splitting method converge in expectation to their analogues in the original problem when the time grid is refined. The derived convergence rates are validated in several numerical experiments. Our numerical results also indicate that the proposed method can lead to a reduction in computational cost for the simulation and optimal control of large-scale linear dynamical systemsThis project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (Grant agreement No: 694126- DyCon), the Alexander von Humboldt-Professorship program, the European Unions Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie Grant agreement No.765579-ConFlex and the Transregio 154 Project “Mathematical Modelling, Simulation and Optimization Using the Example of Gas Networks”, project C08, of the German DFG, the grant PID2020-112617GB-C22, “Kinetic equations and learning control” of the Spanish MINECO, and the COST Action grant CA18232, “Mathematical models for interacting dynamics on networks” (MAT-DYN-NET

Schwarzschild Tests of the Wahlquist-Estabrook-Buchman-Bardeen Tetrad Formulation for Numerical Relativity

A first order symmetric hyperbolic tetrad formulation of the Einstein equations developed by Estabrook and Wahlquist and put into a form suitable for numerical relativity by Buchman and Bardeen (the WEBB formulation) is adapted to explicit spherical symmetry and tested for accuracy and stability in the evolution of spherically symmetric black holes (the Schwarzschild geometry). The lapse and shift which specify the evolution of the coordinates relative to the tetrad congruence are reset at frequent time intervals to keep the constant-time hypersurfaces nearly orthogonal to the tetrad congruence and the spatial coordinate satisfying a kind of minimal rate of strain condition. By arranging through initial conditions that the constant-time hypersurfaces are asymptotically hyperbolic, we simplify the boundary value problem and improve stability of the evolution. Results are obtained for both tetrad gauges (``Nester'' and ``Lorentz'') of the WEBB formalism using finite difference numerical methods. We are able to obtain stable unconstrained evolution with the Nester gauge for certain initial conditions, but not with the Lorentz gauge.Comment: (accepted by Phys. Rev. D) minor changes; typos correcte