The Extended Fisher-Hartwig Conjecture for Symbols with Multiple Jump Discontinuities

1994 ◽  
pp. 16-28
Author(s):  
Estelle L. Basor ◽  
Kent E. Morrison
Keyword(s):  
Jump Discontinuities

scholarly journals Detecting Strength and Location of Jump Discontinuities in Numerical Data

2013 ◽  
Vol 04 (12) ◽  
pp. 1-14 ◽  
Author(s):  
Philipp Öffner ◽  
Thomas Sonar ◽  
Martina Wirz
Keyword(s):  
Numerical Data ◽  
Jump Discontinuities

scholarly journals TOPOLOGICAL EQUIVALENCE FOR DISCONTINUOUS RANDOM DYNAMICAL SYSTEMS AND APPLICATIONS

2013 ◽  
Vol 14 (01) ◽  
pp. 1350007 ◽  
Author(s):  
HUIJIE QIAO ◽  
JINQIAO DUAN
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Stochastic Differential Equation ◽  
Stochastic Differential Equations ◽  
Lévy Processes ◽  
Levy Processes ◽  
Topological Equivalence ◽  
Random Differential Equation ◽  
Jump Discontinuities ◽  
Non Gaussian

After defining non-Gaussian Lévy processes for two-sided time, stochastic differential equations with such Lévy processes are considered. Solution paths for these stochastic differential equations have countable jump discontinuities in time. Topological equivalence (or conjugacy) for such an Itô stochastic differential equation and its transformed random differential equation is established. Consequently, a stochastic Hartman–Grobman theorem is proved for the linearization of the Itô stochastic differential equation. Furthermore, for Marcus stochastic differential equations, this topological equivalence is used to prove the existence of global random attractors.

Solving two-point seismic-ray tracing problems in a heterogeneous medium

1980 ◽  
Vol 70 (1) ◽  
pp. 79-99 ◽  
Author(s):  
V. Pereyra ◽  
W. H. K. Lee ◽  
H. B. Keller
Keyword(s):  
Numerical Method ◽  
Ray Tracing ◽  
Finite Difference Methods ◽  
Three Dimensional ◽  
Nonlinear Boundary Conditions ◽  
Variable Order ◽  
Flexible Tool ◽  
Velocity Models ◽  
Jump Discontinuities ◽  
Variable Mesh

abstract A study of two-point seismic-ray tracing problems in a heterogeneous isotropic medium and how to solve them numerically will be presented in a series of papers. In this Part 1, it is shown how a variety of two-point seismic-ray tracing problems can be formulated mathematically as systems of first-order nonlinear ordinary differential equations subject to nonlinear boundary conditions. A general numerical method to solve such systems in general is presented and a computer program based upon it is described. High accuracy and efficiency are achieved by using variable order finite difference methods on nonuniform meshes which are selected automatically by the program as the computation proceeds. The variable mesh technique adapts itself to the particular problem at hand, producing more detailed computations where they are needed, as in tracing highly curved seismic rays. A complete package of programs has been produced which use this method to solve two- and three-dimensional ray-tracing problems for continuous or piecewise continuous media, with the velocity of propagation given either analytically or only at a finite number of points. These programs are all based on the same core program, PASVA3, and therefore provide a compact and flexible tool for attacking ray-tracing problems in seismology. In Part 2 of this work, the numerical method is applied to two- and three-dimensional velocity models, including models with jump discontinuities across interfaces.

Pre-treating GNSS time series using a recurrent neural network to improve the automated detection of jump discontinuities

2021 ◽  
Author(s):  
Luca Tavasci ◽  
Pasquale Cascarano ◽  
Stefano Gandolfi
Keyword(s):  
Neural Network ◽  
Time Series ◽  
Time Series Analysis ◽  
Ad Hoc ◽  
Short Term Memory ◽  
Learning Approaches ◽  
Jump Detection ◽  
Series Analysis ◽  
Jump Discontinuities ◽  
Gnss Time Series

<p>Ground motion monitoring is one of the main goals in the geoscientist community and at the time it is mainly performed by analyzing time series of data. Our capability of describing the most significant features characterizing the time evolution of a point-position is affected by the presence of undetected discontinuities in the time series. One of the most critical aspects in the automated time series analysis, which is quite necessary since the amount of data is increasing more and more, is still the detection of discontinuities and in particular the definition of their epoch. A number of algorithms have already been developed and proposed to the community in the last years, following different statistical approaches and different hypotheses on the coordinates behavior. In this work, we have chosen to analyze GNSS time series and to use an already published algorithm (STARS) for jump detection as a benchmark to test our approach, consisting of pre-treating the time series to be analyzed using a neural network. In particular, we chose a Long Short Term Memory (LSTM) neural network belonging to the class of the Recurrent Neural Networks (RNNs), ad hoc modified for the GNSS time series analysis. We focused both on the training algorithm and the testing one. The latter has been the object of a parametric test to find out the number of predicted data that mostly emphasize our capability of detecting jump discontinuities. Results will be presented considering several GNSS time series of daily positions. Finally, a discussion on the possible integration of machine learning approaches and classical deterministic approaches will be done.</p>

scholarly journals Integro-differential equations for the self-organisation of liver zones by competitive exclusion of cell-types

1987 ◽  
Vol 29 (2) ◽  
pp. 156-194 ◽  
Author(s):  
L. Bass ◽  
A. J. Bracken ◽  
K. Holmåker ◽  
B. R. F. Jefferies
Keyword(s):  
Differential Equations ◽  
Enzymatic Activity ◽  
Cell Types ◽  
Stationary Solutions ◽  
Stable Set ◽  
Model Parameters ◽  
Asymptotically Stable ◽  
Hepatic Sinusoid ◽  
Jump Discontinuities ◽  
Cell Densities

AbstractA model is developed for the seif-organisation of zones of enzymatic activity along a liver capillary (hepatic sinusoid) lined with cells of two types, which contain different enzymes and compete for sites on the wall of the sinusoid. An effectively non-local interaction between the cells arises from local consumption of oxygen from blood flowing throug1 the sinusoid, which gives rise to gradients of oxygen concentration in turn influencing rates of division and of death of the two cell-types. The process is modelled by a pair of coupled non-linear integro-differential equations for the cell-densities as functions of time and position along the sinusoid. Existence of a unique, bounded, non-negative solution of the equations is proved, for prescribed initial values. The equations admit infinitely many stationary solutions, but it is shown that all except one are unstable, for any given set of the model parameters. The remaining solution is shown to be asymptotically stable against a large class of perturbations. For certain ranges of the model parameters, the asymptotically stable stationaxy solution has a zonal structure, with cells of one type located entirely upstream of cells of the other type, and with jump discontinuities in the cell densities at a certain distance along the sinusoid. Such sinusoidal zones can account for zones of enzymatic activity observed in the intact liver. Exceptional cases are found for singular choices of model parameters, such that stationary cell-densities cannot be asymptotically stable individually, but together form an asymptotically stable set. Certain mathematical questions are left open, notably the behaviour of large deviations from stationary solutions, and the global stability of such solutions. Possible generalisations of the model are described.

scholarly journals DESIGN AND CONVERGENCE OF AFEM IN H(DIV)

2007 ◽  
Vol 17 (11) ◽  
pp. 1849-1881 ◽  
Author(s):  
J. MANUEL CASCON ◽  
RICARDO H. NOCHETTO ◽  
KUNIBERT G. SIEBERT
Keyword(s):  
Mixed Boundary ◽  
Total Error ◽  
Posteriori Error ◽  
Coefficient Matrix ◽  
Adaptive Finite Element Method ◽  
Adaptive Finite Element ◽  
A Posteriori Error ◽  
Energy Error ◽  
Jump Discontinuities ◽  
Posteriori Error Analysis

We design an adaptive finite element method (AFEM) for mixed boundary value problems associated with the differential operator A-∇div in H(div, Ω). For A being a variable coefficient matrix with possible jump discontinuities, we provide a complete a posteriori error analysis which applies to both Raviart–Thomas ℝ𝕋n and Brezzi–Douglas–Marini 𝔹𝔻𝕄n elements of any order n in dimensions d = 2, 3. We prove a strict reduction of the total error between consecutive iterates, namely a contraction property for the sum of energy error and oscillation, the latter being solution-dependent. We present numerical experiments for ℝ𝕋n with n = 0, 1 and 𝔹𝔻𝕄1 which document the performance of AFEM and corroborate as well as extend the theory.

Sign in / Sign up

Export Citation Format

Share Document