A commonly used mathematical model for the dynamic behaviour of power systems is that of a diierential/algebraic system. Such a system consists of a mixture of diierential equations and algebraic constraints. In this paper the behaviour of solutions to such systems which display certain jump discontinuities are investigated. To analyse stability properties of these solutions Lyapunov theory is ...

An applied compact introductory survey of Markov stochastic processes and control in continuous time is presented. The presentation is in tutorial stages, beginning with deterministic dynamical systems for contrast and continuing on to perturbing the deterministic model with diffusions using Wiener processes. Then jump perturbations are added using simple Poisson processes constructing the theo...

In this paper, we systematically derive jump conditions for the immersed interface method [SIAM J. Numer. Anal., 31 (1994), pp. 1019–1044; SIAM J. Sci. Comput., 18 (1997), pp. 709–735] to simulate three-dimensional incompressible viscous flows subject to moving surfaces. The surfaces are represented as singular forces in the Navier–Stokes equations, which give rise to discontinuities of flow qu...

For an area-preserving twist map F of a finite annulus, we obtain “quasi-foliations” of the annulus by one-sided-F -invariant graphs of functions that are either continuous (in which case the graph is a fully invariant curve) or have countably many jump discontinuities. For each of the one-sided-invariant graphs there is a well defined (one-sided) rotation number and all the numbers between the...

A ve-equation model for compressible twouid ow is proposed, that is based on physical ow equations only. The model is conservative and pressure-oscillation free. Equations for continuous ow and jump conditions for discontinuities are given, as well as a discretization of the equations and an adaptation of the HLL Riemann solver to twouid ow. Numerical tests in 1D and 2D show the accuracy of the...

Phase-field modeling—a continuous approach to discontinuities—is gaining popularity for simulating rock fractures due its ability handle complex, discontinuous geometry without an explicit surface tracking algorithm. None of the existing phase-field models, however, incorporates impact roughness on mechanical response fractures—such as elastic deformability and shear-induced dilation—despite im...

We study the orthogonal polynomials and Hankel determinants associated with Gaussian weight two jump discontinuities. When degree n is finite, are shown to be connected coupled Painlevé IV system. Using this connection, we obtain a sequence of special function solutions In double scaling limit as discontinuities tend edge spectrum grows infinity, establish asymptotic expansions for polynomials,...

In this paper, we derive a local Carleman estimate for the complex second order elliptic operator with Lipschitz coefficients having jump discontinuities. Combing result by M. Bellassoued and J. Le Rousseau (2018) arguments Di Cristo, E. Francini, C.-L. Lin, S. Vessella, J.-N. Wang (2017), present an elementary method to under optimal regularity assumption on coefficients.

We introduce a nonconforming finite-element method for second order elliptic interface problems. Our approach applies to problems in which discontinuous coefficients and singular sources on the interface may give rise to jump discontinuities in either the solution or its normal derivative. Given a standard background mesh and an interface that passes between elements, the key idea is to constru...

In this paper, we consider a one-dimensional inverse problem of estimating an input signal that a priori is known to be smooth functions with a few jump discontinuities. A popular method of regularizing the inverse problem with such prior information is to use a total variation (TV) penalty as a regularizing functional. In this article, we adopt a different approach, considering the problem fro...