The discontinuous dynamical systems generated by the retarded functional equations has been defined in -. The dynamical systems with piecewiseconstant arguments has been studied in - and references therein. In this work we define the discontinuous dynamical systems generated by functional equations with piecewiseconstant arguments. The dynamical properties of the discontinuous dynamical system of the Riccati type equation will be discussed. Comparison with the corresponding discrete dynamical system of the Riccati type equation
In this paper, the numerical stability of a partial diﬀerential equation with piecewiseconstant arguments is considered. Firstly, the θ -methods are applied to approximate the original equation. Secondly, the numerical asymptotic stability conditions are given when the mesh ratio and the corresponding parameter satisfy certain conditions. Thirdly, the conditions under which the numerical stability region contains the analytic stability region are also established. Finally, some numerical examples are given to demonstrate the theoretical results.
 G. Papaschinopoulos, On asymptotic behavior of the solutions of a class of perturbed diﬀerential equations with piecewiseconstant argument and variable coeﬃcients, J. Math. Anal. Appl. 185 (1994), no. 2, 490–500. MR 95f:34064. Zbl 810.34079.  G. Seifert, Almost periodic solutions for delay-diﬀerential equations with inﬁnite delays, J.
was initiated in (,) where h (t) = [t] , [t − n] , [t + n] , etc. These types of equations have been intensively investigated for twenty five years. Systems de- scribed by DEP CA exist in a large area such as biomedicine, chemistry, physics and mechanical engineering. Busenberg and Cooke  first established a mathemati- cal model with a piecewiseconstant argument for analyzing vertically transmitted diseases. Examples in practice include machinery driven by servo units, charged particles moving in a piecewise constantly varying electric field, and elastic systems impelled by a Geneva wheel.
Diﬀerential equations with piecewiseconstant arguments (DEPCA) exist in a widely expanded areas such as biomedicine, chemistry, mechanical engineering, physics, and so on. To the best of our knowledge, the ﬁrst mathematical model that includes a piecewiseconstant argument was proposed by Busenberg and Cooke . They investigated the fol-
The study of diﬀerential equations with piecewiseconstant argument (EPCA) initiated in [, ]. These equations represent a hybrid of continuous and discrete dynamical systems and combine the properties of both diﬀerential and diﬀerence equations, hence, they are of importance in control theory and in certain biomedical models . In this paper the second order delay diﬀerential equation with a piecewiseconstant argument
Exploiting the piecewise constancy of TV-AR models, several methods are available to detect the changing instants of the AR coefficients, and thus facilitate what is often referred to as signal segmentation. The literature on signal segmentation is large since the topic is of interest in signal processing, applied statistics, and several other branches of science and engineering. Recent advances can be mainly divided in two categories. The first class adopts regularized LS criteria in order to impose piecewise- constantAR coefficients. To avoid “ oversegmentation, ” the LS cost is typically regularized with the total number of changes . The resulting estimator can be implemented via dynamic programming (DP), which incurs computa- tional burden that scales quadratically with the signal dimension. For large data sets, such as those considered in speech processing, this burden refrains practitioners from applying DP to segmentation, and heuristics are pursued instead based on the generalized likelihood ratio test
Recently, some properties of differential equations with piecewiseconstant arguments of various types as retarded, advanced and mixed types have been investigated, intensively in ,,,, (see also references therein). Many of investigations (existence, unique- ness, stability, oscillation, periodicity of solutions and ets.) are devoted to differential equations (linear and nonlinear) for different order (for example, see ,,). Dif- ferential equations with piecewiseconstant arguments are investigation subjects of many problems in life sciences such as physics, chemistry, biomedicine, mechanical engineering etc.
The discontinuous dynamical systems generated by the retarded functional equations have been defined in -. The dynamical systems with piecewiseconstant arguments have been studied in - and the references therein. In this work we define the discontinuous dynamical systems generated by functional equations with piecewiseconstant arguments. The dynamic properties of two discontinuous dynamical systems of the Logistic equation will be discussed. Comparison with the corresponding discrete dynamical systems of the Logistic equation
where a and b have the same meaning as in the logistic equation (), τ is a positive con- stant. Equation () is known as Hutchinson’s equation or (autonomous) delayed logistic equation. There exist many studies about logistic equation with or without time delay [– ]. In general, delay diﬀerential equations exhibit much more complicated dynamics than ordinary diﬀerential equations since a time delay could cause a stable equilibrium to be- come unstable and lead the populations to ﬂuctuate. There are also several papers [–] that study the logistic equation with piecewiseconstant argument of the following form:
PHAVerLite PHAVerLite is a variant of the stand-alone verification tool PHAVer, sharing the same capabilities and formal soundness guarantees. It is worth stressing that PHAVerLite, being a stand-alone tool, differs from the PHAVer-lite SpaceEx plugin that participated in the friendly competition in 2018. For instance, while PHAVer-lite was able to accept input specified using the SpaceEx syntax for hybrid automata, at present PHAVerLite can only accept input specified using the PHAVer syntax. The main difference with respect to PHAVer is the adoption of the new polyhedra library PPLite : thanks to a novel representation and conversion algorithm  for NNC (Not Necessarily Closed) polyhedra, PPLite is able to obtain significant efficiency improvements with respect to the classical polyhedra implementation used in PHAVer (which is based on the Parma Polyhedra Library ). The development of PHAVerLite was motivated by the desire to go beyond the main change above and also revisit many of the key design and implementation choices of the original PHAVer: this allowed to experiment with novel algorithms or design tradeoffs, also exploiting some of the more recent advances in the implementation of operators on the polyhedral domains. At present, PHAVerLite has only been used to analyze systems characterized by piecewiseconstant dynamics; also note that a few of the PHAVer functionalities (e.g., the computation of simulation relations) have been deliberately removed.
The present choice of piecewiseconstant functions limits the optimal convergence rate to classes of low or no pointwise regularity. While the practical extension of our method to higher order piecewise polynomial approximations is almost straightforward, its analysis in this more general context becomes significantly more difficult and will be given in a forthcoming paper. This is so far a weakness of our approach from the theoretical perspective, compared to the complexity regularization approach for which optimal convergence results could be obtained in the piecewise polynomial context (using for instance Gy¨orfy et al., 2002, Theorem 12.1).
quotes arrive in a scarce way, it is more realistic to assume that conditional expec- tations move in a more piecewiseconstant fashion. Such type of processes didn’t receive attention so far, and our paper aims at filling this gap. We focused on the construction of piecewiseconstant martingales that is, martingales whose trajecto- ries are piecewiseconstant. Such processes are indeed good candidates to model the dynamics of conditional expectations of random variables under partial (punctual) information. The time-changed approach proves to be quite powerful: starting with a martingale in a given range, we obtain a PWC martingale by using a piecewiseconstant time-change process. Among those time-change processes, lazy clocks are specifically appealing: these are time-change processes staying always in arrears to the real clock, and that synchronizes to the calendar time at some random times. This ensures that θ t ≤ t which is a convenient feature when one needs to sample trajec-
problem, BACH conducts the bounded checking in a “path-oriented” layered style. It finds potential paths which can reach the target location on the graph structure first, then encodes the feasibility of such path into a linear programming problem and solve it afterwards. In this way, as the number of paths in the discrete graph structure of an LHA under a given bound is finite, all candidate paths can be enumerated and checked one by one to tackle the bounded reachability analysis of LHA. Furthermore, the memory usage is well controlled as it only encodes and solves one path at a time. Meanwhile, BACH provides an efficient way to locate the infeasible path segment core when a path is reported as infeasible to guide the backtracking in the graph structure traversing to achieve good performance . Such infeasible path segments can also be used to derive complete state arguments under certain conditions . Lyse Lyse is a tool for the reachability analysis of convex hybrid automata, namely hybrid automata with piecewiseconstant dynamics, whose constraints are possibly non-linear but re- quired to be convex. In this class are HPWC whose flow is contrained in rectangles, polyhedra, but also ellipses and parabolae. Linear hybrid automata are a special case. Lyse performs for- ward reachability analysis by means of template-polyhedra, whose directions are incrementally extracted from spurious counterexamples. The extraction is performed by a novel technique that generates interpolants by means of convex programming .
Variational methods for image segmentation have had great success, which are characterized by deriving an en- ergy functional from some a priori mathematical model and minimizing this energy functional over all possible partitions. Among them, the Mumford-Shah (M-S) model  is one of the most widely studied mathemat- ical models for image analysis. The M-S functional con- tains a data fidelity term and two/a regularity terms imposing a piecewise smooth/constant representation of an image and penalizing the Hausdorff measure of the set of discontinuities, resulting in simultaneous restoration
In this chapter, we develop a fast approach for elliptic interface equations with piecewiseconstant but discontinuous coefficients. The idea is to precondition the original elliptic equation before using the immersed interface method IIM. In order to take advantage of existing fast Poisson solvers on cubic domains, we introduce a new problem by rewriting the PDE and introducing an intermediate unknown jump in 𝑢𝑢 . The new problem looks like a Poisson equation, which can be discretized by using the standard seven point stencil with some modification to the right-hand side. Then some existing fast Poisson solver can be called directly. Basically, this approach is equivalent to using a second order finite difference scheme to approximate the Poisson equation in Ω and Ω , and a second order discretization