Piecewise Continuous Functions Research Articles

Adaptive exponential cluster synchronization in colored community networks via aperiodically intermittent pinning control

This paper investigates the problem of pinning cluster synchronization for colored community networks via adaptive aperiodically intermittent control. Firstly, a general colored community network model is proposed, where the isolated nodes can interact through different kinds of connections in different communities and the interactions between different pair of communities can also be different, and moreover, the nodes in different communities can have different state dimensions. Then, an adaptive aperiodically intermittent control strategy combined with pinning scheme is developed to realize cluster synchronization of such colored community network. By introducing a novel piecewise continuous auxiliary function, some globally exponential cluster synchronization criteria are rigorously derived according to Lyapunov stability theory and piecewise analysis approach. Based on the derived criteria, a guideline to illustrate which nodes in each community should be preferentially pinned is given. It is noted that the adaptive intermittent pinning control is aperiodic, in which both control width and control period are allowed to be variable. Finally, a numerical example is provided to show the effectiveness of the theoretical results obtained.

Adaptive exponential lag synchronization for neural networks with mixed delays via intermittent control

In this paper, the problem of exponential lag synchronization for a class of neural networks with mixed delays including discrete and distributed delays is investigated via adaptive intermittent control. Based on piecewise analytic method, some sufficient conditions for globally exponential lag synchronization are established through constructing a piecewise continuous auxiliary function. It is noted that both the control periods and the control widths in our adaptive intermittent control strategy are allowed to be nonidentical, which extends the scope of application of periodically intermittent control with fixed both control period and control width employed widely in previous works. Moreover, it is shown that the derived globally exponential lag synchronization criteria are related to the control rates rather than the control periods, which facilitates the choice of the control periods for practical problems. Finally, a numerical example is given to illustrate the correctness of the obtained theoretical results.

Singular Fourier–Padé series expansion of European option prices

We apply a new numerical method, the singular Fourier–Padé (SFP) method invented by Driscoll and Fornberg [Numer. Algorithms, 2001, 26, 77–92; The Gibbs Phenomenon in Various Representations and Applications, 2011], to price European-type options in Lévy and affine processes. The motivation behind this application is to reduce the inefficiency of current Fourier techniques when they are used to approximate piecewise continuous (non-smooth) probability density functions. When techniques such as fast Fourier transforms and Fourier series are applied to price and hedge options with non-smooth probability density functions, they cause the Gibbs phenomenon; accordingly, the techniques converge slowly for density functions with jumps in value or derivatives. This seriously adversely affects the efficiency and accuracy of these techniques. In this paper, we derive pricing formulae and their option Greeks using the SFP method to resolve the Gibbs phenomenon and restore the global spectral convergence rate. Moreover, we show that our method requires a small number of terms to yield fast error convergence, and it is able to accurately price any European-type option deep in/out of the money and with very long/short maturities. Furthermore, we conduct an error-bound analysis of the SFP method in option pricing. This new method performs favourably in numerical experiments compared with existing techniques.

Aplicación de primera derivada en solidificación de aleaciones

In this paper a numerical application of the derivative is performed, the algorithm based on the least squares method program is presented. The program is used for calculation of experimental melting temperature of an aluminum alloy, of about 660 °C, giving a relative error of 0.08%. This derivative is also used to evaluate its effectiveness in periodic and piecewise continuous functions, a correlation coefficient of 1.0 is obtained in all results. The work presented here is focused to undergraduate students in differential calculus and researchers that require the derivative with numerical analysis.

On Normal Modes of the Closed Waveguide with Discontinuous Filling

We consider a waveguide of a constant cross-section S with ideally conducting walls. We assume that the filling of waveguide doesn’tchange along its axis and is described by the piecewise continuous functions e and μ defined on waveguide cross-section. We show that it is possible to make substitutionwhich allows to work only with continuous functions. Instead of noncontinuous cross-components of an electromagnetic field E and H we offer to use four potentials ue,uh and ve,vh. We can prove as the generalization of Tikhonov—Samarskii theorem that any field in the waveguide allows representation in such form if we consider the potentials ue,uh as elements of Sobolev space Wo21(S) and the potentials ve,vh as elements of Sobolev space W21(S). If ϵ and m are the piecewise constant functions then Maxwell’s equations written in four potentialsreduce to a pair of independent systems. This statement give us new approach to theinvestigation of spectral properties of waveguides. First, we can prove the completenessof the system of the normal waves in closed waveguides using standard functionalspaces. Secondly, we can offer new technique for calculation of the normal waves usingstandard finite elements. FreeFem++ program for calculation of disperse lines ofwaveguides is presented. The question of calculation of modes at great values of k=ω/c is also considered.

НЕКОТОРЫЕ СВОЙСТВА РЕШЕНИЙ ЗАДАЧИ КОШИ ДЛЯ ФУНКЦИОНАЛЬНО-ДИФФЕРЕНЦИАЛЬНОГО ВКЛЮЧЕНИЯ С МНОГОЗНАЧНЫМИ ИМПУЛЬСНЫМИ ВОЗДЕЙСТВИЯМИ

Deviation estimates in space of piecewise continuous functions of a set of the generalized decisions from beforehand given function are received. The continuous dependence of the generalized decisions on starting conditions is established.

Synthesis of intelligent fuzzy control in the space of bounded piecewise-continuous functions based on the combined maximum principle

It is shown that the solution of the problem of control synthesis using the Hamilton-Ostrogradskii principle leads to a variational inequality, from which the conditions for the maximum of the the generalized power function in the space of bounded piecewise continuous functions follow. It allows to find a feedback structure up to a synthesis function. Using the methods of the structure construction the nonlinear structures of the relay and continuous control laws are obtained. The proposed control method allows to avoid the mode with frequented switching. It consists of the following two stages. At the first stage, the control object is brought into the vicinity of the terminal state using the relay control law. In the second stage, the quasi-optimal continuous control is used. The uncertainty of the transition area size is resolved using fuzzy logic. The efficiency of the intelligent controls is demonstrated by the example of mathematical modeling of the system dynamics.

Model of the managment and control device for mobile radio systems

The article presents a mathematical model of a common panel for control and monitoring (PCM) of combined mobile radio engineering systems and its physical interpretation. The model describes management as a ramified process, the functions of which are entrusted on the PCM, the actual model of the PCM containing objects of different importance, among which the generator of orthogonal piecewise-continuous functions is distinguished, process of monitoring the state of objects. The physical interpretation of the above components of the model is determined. It involves the management of a mobile radio engineering system carried out from a remote control, which has in its composition a generator of orthogonal piecewise-continuous functions. The obtained results can be used for telecommunication platforms, which combined various types of communication, devices for monitoring objects and the state of data transmission paths, radar systems of detection, recognition, tracking and guidance. An example of a combined telecommunication platform is shown mobile digital troposcatter-radiorelay station.

General methods for converting impulsive fractional differential equations to integral equations and applications

AbstractIn this paper, we propose the concepts of Caputo fractional derivatives and Caputo type Hadamard fractional derivatives for piecewise continuous functions. We obtain general solutions of four classes of impulsive fractional differential equations (Theorem 3.1–Theorem 3.4) respectively. These results are applied to converting boundary value problems for impulsive fractional differential equations to integral equations. Some comments are made on recently published papers (see Section 4).

Impulsive control functional differential systems of fractional order: stability with respect to manifolds

In this paper, we consider nonlinear impulsive functional differential systems involving Caputo fractional derivatives. Using vector multivariable piecewise continuous Lyapunov functions and a new vector fractional comparison principle, stability criteria with respect to a manifold are established. Some examples are presented in order to illustrate our theoretical findings.

Exponential cluster synchronization in directed community networks via adaptive nonperiodically intermittent pinning control

In this paper, the problem of exponential cluster synchronization for a class of directed community networks is investigated via adaptive nonperiodically intermittent pinning control. By constructing a novel piecewise continuous Lyapunov function, some sufficient conditions to guarantee globally exponential cluster synchronization are derived. It is noted that the derived cluster synchronization criteria rely on the control rates, but not the control widths or the control periods, which facilitates the choice of the control periods in practical applications. A numerical example is finally presented to show the effectiveness of the obtained theoretical results.

An Online Sequential Learning Non-parametric Value-at-Risk Model for High-Dimensional Time Series

Online Value-at-Risk (VaR) analysis in high-dimensional space remains a challenge in the era of big data. In this paper, we propose an online sequential learning non-parametric VaR model called OS-GELM which is an autonomous cognitive system. This model uses a Generalized Autoregressive Conditional Heteroskedasticity (GARCH) process and an online sequential extreme learning machine (OS-ELM) to cognitively calculate VaR, which can be used for online risk analysis. The proposed model not only learns the data one-by-one or chunk-by-chunk but also calculates VaR in real time by extending OS-ELM from machine learning to the non-parametric GARCH process. The GARCH process is also extended to one-by-one and chunk-by-chunk mode. In OS-GELM, the parameters of hidden nodes are randomly selected. The output weights are analytically determined based on the sequentially arriving data. In addition, the generalization performance of the OS-GELM model attains a small training error and generates the smallest norm of weights. Experimentally obtained VaRs are compared with those given by GARCH-type models and conventional OS-ELM. The computational results demonstrate that the OS-GELM model obtains more accurate results and is better at forecasting the online VaR. OS-GELM model is an autonomous cognitive system to dynamically calculate Value-at-Risk, which can be used for online financial risk assessment about human being’s behavior. The OS-GELM model can calculate VaR in real time, which can be used as a tool for online risk management. OS-GELM can handle any bounded, non-constant, piecewise-continuous membership function to realize real-time VaR monitoring.

Linear extensions of some Baire-one functions

Let X be a Hausdorff topological space, and let {mathscr {B}}_1(X) denote the space of all real Baire-one functions defined on X. Let A be a nonempty subset of X endowed with the topology induced from X, and let {mathscr {F}}(A) be the set of functions Arightarrow {mathbb R} with a property {mathscr {F}} making {mathscr {F}}(A) a linear subspace of {mathscr {B}}_1(A). We give a sufficient condition for the existence of a linear extension operator T_A:{mathscr {F}}(A)rightarrow {mathscr {F}}(X), where {mathscr {F}} means to be piecewise continuous on a sequence of closed andG_delta subsets ofX and is denoted by {mathscr {P}_0}. We show that T_A restricted to bounded elements of {mathscr {F}}(A) endowed with the supremum norm is an isometry. As a consequence of our main theorem, we formulate the conclusion about existence of a linear extension operator for the classes of Baire-one-star and piecewise continuous functions.

Analytical representation of switching current impulses for study of metal-oxide surge arrester models

The object of research is an analytical expression for representing the switching current impulse of a surge arrester. Any current impulse (both lightning and switching) is characterized by such parameters as the virtual front time and the virtual time to half-value on the tail. According to the standard IEC 60099-4:2014, switching current impulse has a virtual time to half-value on the tail of roughly twice the virtual front time. This requirement is one of the most problematic places in this task. The existing approaches used to represent lightning current impulses are not suitable in this case, since these impulses have virtual time to half-value on the tail of two and a half times the virtual front time.This problem can be solved with a help of analytical piecewise continuous functions.It is shown how to describe switching current impulses of the surge arresters with a help of analytical piecewise continuous functions. In contrast to other expressions, the resulting expressions for the switching impulse have only one parameter (angular frequency). Instead of approximate calculation, the front time of the resulting impulse is calculated by an analytically exact formula. Hence, the tolerance of virtual front time is equal to zero. The time to half-value on the tail of the resulting impulses is determined with some error that can be reduced by some complication of the original expression.The proposed functions satisfy the requirements of the IEC 60099-4:2014 standard regarding switching current impulses of surge arresters. These functions allow representing current impulses having virtual time to half-value on the tail of roughly twice the virtual front time (30/60 or 45/90 microseconds). In such cases, minimal tolerance of time to half-value on the tail is +3.78 %. Additional study shows that one of proposed functions allows representing current impulses having virtual time to half-value on the tail of two and a half times the virtual front time (8/20 or 4/10 microseconds). Tolerance of time to half-value on the tail for such impulses is 0.55 %. The obtained functions are intended for study of various models of metal-oxide surge arresters on personal computers.

Singular Fourier-Padd Series Expansion of European Option Prices

We apply a new numerical method, the singular Fourier-Pad e (SFP) method invented by Driscoll and Fornberg (2001, 2011), to price European-type options in L evy and affine processes. The motivation behind this application is to reduce the inefficiency of current Fourier techniques when they are used to approximate piecewise continuous (non-smooth) probability density functions. When techniques such as fast Fourier transforms and Fourier series are applied to price and hedge options with non-smooth prob- ability density functions, they cause the Gibbs phenomenon; accordingly, the techniques converge slowly for density functions with jumps in value or derivatives. This seriously adversely affects the efficiency and accuracy of these techniques. In this paper, we derive pricing formulae and their option Greeks using the SFP method to resolve the Gibbs phenomenon and restore the global spectral convergence rate. More- over, we show that our method requires a small number of terms to yield fast error convergence, and it is able to accurately price any European-type option deep in/out of the money and with very long/short maturities. Furthermore, we conduct an error-bound analysis of the SFP method in option pricing. This new method performs favourably in numerical experiments compared with existing techniques.

Consensus in Networks of Agents With Unknown High-Frequency Gain Signs and Switching Topology

The agreement control problem of single and double-integrator agents with unknown and nonidentical control directions is addressed in this note under switching network topology. Distributed nonlinear PI control laws are proposed which ensure asymptotic consensus among the agents based on a new boundedness lemma and a generalized version of Barbalat's lemma for uniformly piecewise right continuous functions. Theoretical results are verified by simulation studies.

Fractional order differential switched systems with coupled nonlocal initial and impulsive conditions

In this paper, we discuss a new class of fractional order differential switched systems with coupled nonlocal initial and impulsive conditions in Rn. We firstly derive a solution formula for this system. Secondly, we utilize three well-known fixed point methods to present the existence results. Moreover, we use Schauder topological degree theory to show a new existence result for resonant case: Landesman–Lazer conditions. Finally, we introduce the concepts of Ulam's type stability and present new stability results in the space of fractional version piecewise continuous functions.

A problem of unilateral contact of two hanging chains

The unilateral contact of two closely hanging chains (heavy inextensible strings) under the action of gravity and small horizontal loading is considered. Each chain has one end fixed and the other free. The chains have different lengths; the linear densities of the chains may be variable. The horizontal loading slightly deflects the chains from the vertical line, so that the system is geometrically linear. The basic problem is to find the shapes of the chains. This problem is reduced to the problem of finding the density of the forces of interaction between the chains. The accurate formulation of the latter contact problem is propounded. The unknown density of the contact forces is assumed to be the sum of piecewise continuous (and one-sided continuous) function and delta functions describing the concentrated forces. The uniqueness of the solution of the problem is proved. The analytical solution is constructed in some special cases. The following contact patterns are found out: contact at one point, contact along the full length of the shorter chain, contact along a part of the shorter chain, contact at a point and along a part of the shorter chain.

Hyers-Ulam Stability of First-Order Non-Linear Delay Differential Equations with Fractional Integrable Impulses

This paper proves the Hyers-Ulam stability and Hyers-Ulam-Rassias stability of first-order non-linear delay differential equations with fractional integrable impulses. Our approach uses abstract Gronwall lemma together with integral inequality of Gronwall type for piecewise continuous functions

Discontinuous Demand Functions: Estimation and Pricing

We consider a dynamic pricing problem with an unknown and discontinuous demand function. There is a seller who dynamically sets the price of a product over a multiperiod time horizon. The expected demand for the product is a piecewise continuous and parametric function of the charged price, allowing for possibly multiple discontinuity points. The seller initially knows neither the locations of the discontinuity points nor the parameters of the demand function but can infer them by observing stochastic demand realizations over time. We measure the seller’s performance by the revenue loss relative to a clairvoyant who knows the underlying demand function with certainty. We construct a dynamic estimation-and-pricing policy that accounts for demand discontinuities, derive the convergence rates of discontinuity- and parameter-estimation errors under this policy, and prove that it achieves near-optimal revenue performance. We also extend our analysis to the cases of time-varying demand discontinuities and inventory constraints. This paper was accepted by Noah Gans, stochastic models and simulation.