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Journal articles on the topic 'Discontinuous Function'

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Author: Grafiati

Published: 14 March 2023

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1

Pershyna, I. I. "RESTORATION OF DISCONTINUOUS FUNCTIONS BY DISCONTINUOUS INTERLINATION SPLINES." Radio Electronics, Computer Science, Control, no. 4 (December 4, 2022): 29. http://dx.doi.org/10.15588/1607-3274-2022-4-3.

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Context. The problem of development and research of methods for approximation of discontinuous functions by discontinuous interlination splines and its further application to problems of computed tomography. The object of the study was the modeling of objects with a discontinuous internal structure. Objective. The aim of this study is to develop a general method for constructing discontinuous interlining polynomial splines, which, as a special case, include discontinuous and continuously differentiated splines. Method. Modern methods of restoring functions are characterized by new approaches to obtaining, processing and analyzing information. There is a need to build mathematical models in which information can be represented not only by function values at points, but also in the form of a set of function traces on planes or straight lines. At the same time, practice shows that among the multidimensional objects that need to be investigated, more problems are described by a discontinuous functions. The paper develops a general method for constructing discontinuous interlining polynomial splines, which, as a special case, include discontinuous and continuously differentiable splines. It is considered that the domain of the definition of the required twodimensional function is divided into rectangular elements. Theorems on interlination and approximation properties of such discontinuous constructions are formulated and proved. The method is developed for approximating discontinuous functions of two variables based on the constructed discontinuous splines. The input data are the traces of an unknown function along a given system of mutually perpendicular straight lines. The proposed method has not only theoretical significance but also practical application in the IT domain, especially in computing tomography, allowing more accurately restore the internal structure of the body. Results. The discontinuous interlination operator from known traces of the function of two variables on a system of mutually perpendicular straight lines is researched. Conclusions. The functions of two variables that are discontinuous at some points or on some lines are better approximated by discontinuous spline interlinants. At the same time, equally high approximation estimates can be obtained. The results obtained have significant advantages over existing methods of interpolation and approximation of discontinuous functions. In further research, the authors plan to develop a theory of discontinuous splines on areas of complex shape bounded by arcs of known curves.

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2

Jarosz, Krzysztof, and Zbigniew Sawoń. "A discontinuous function does not operate on the real part of a function algebra." Časopis pro pěstování matematiky 110, no. 1 (1985): 58–59. http://dx.doi.org/10.21136/cpm.1985.118221.

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3

Hartman, James L. "A Terminally Discontinuous Function." College Mathematics Journal 27, no. 3 (May 1996): 211. http://dx.doi.org/10.2307/2687172.

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4

Hartman, James L. "A Terminally Discontinuous Function." College Mathematics Journal 27, no. 3 (May 1996): 211–12. http://dx.doi.org/10.1080/07468342.1996.11973781.

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5

Steprāns, Juris. "A very discontinuous Borel function." Journal of Symbolic Logic 58, no. 4 (December 1993): 1268–83. http://dx.doi.org/10.2307/2275142.

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AbstractIt is shown to be consistent that the reals are covered by ℵ1, meagre sets yet there is a Baire class 1 function which cannot be covered by fewer than ℵ2, continuous functions. A new cardinal invariant is introduced which corresponds to the least number of continuous functions required to cover a given function. This is characterized combinatorially. A forcing notion similar to, but not equivalent to, superperfect forcing is introduced.

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6

HARINI, LUH PUTU IDA, and KARTIKA SARI. "APLIKASI INTEGRAL DALAM BIDANG EKONOMI DAN FINANSIAL." E-Jurnal Matematika 9, no. 2 (June 1, 2020): 143. http://dx.doi.org/10.24843/mtk.2020.v09.i02.p291.

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The characteristics of a function are usually investigated by looking at the continuity of the function. But what happens if a function does not have continuous properties? To what extent can the characteristics of continuous function be maintained for discontinuous cases? The stochastic function that is widely involved in solving problems in the field of average financial mathematics is a discontinuous function. This is reflected by the acquisition of a smooth curve from the modeling drawing obtained. Today, the nature of continuous functions in [a, b] has been widely studied and developed. Some properties of the continuous function can be extended to the appropriate discontinuous function. In this paper, there will be some integral reviews for discontinuous functions which are closely related to stochastic functions.

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7

Górka, Przemysław, and Artur Słabuszewski. "A discontinuous Sobolev function exists." Proceedings of the American Mathematical Society 147, no. 2 (October 31, 2018): 637–39. http://dx.doi.org/10.1090/proc/14164.

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8

Yang, Xinsong, and Jinde Cao. "Synchronization of Discontinuous Neural Networks with Delays via Adaptive Control." Discrete Dynamics in Nature and Society 2013 (2013): 1–9. http://dx.doi.org/10.1155/2013/147164.

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The drive-response synchronization of delayed neural networks with discontinuous activation functions is investigated via adaptive control. The synchronization of this paper means that the synchronization error approaches to zero for almost all time as time goes to infinity. The discontinuous activation functions are assumed to be monotone increasing which can be unbounded. Due to the mild condition on the discontinuous activations, adaptive control technique is utilized to control the response system. Under the framework of Filippov solution, by using Lyapunov function and chain rule of differential inclusion, rigorous proofs are given to show that adaptive control can realize complete synchronization of the considered model. The results of this paper are also applicable to continuous neural networks, since continuous function is a special case of discontinuous function. Numerical simulations verify the effectiveness of the theoretical results. Moreover, when there are parameter mismatches between drive and response neural networks with discontinuous activations, numerical example is also presented to demonstrate the complete synchronization by using discontinuous adaptive control.

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9

Lytvyn, Oleg, Oleg Lytvyn, and Oleksandra Lytvyn. "Analysis of the results of a computational experiment to restore the discontinuous functions of two variables using projections." Physico-mathematical modelling and informational technologies, no. 33 (September 2, 2021): 12–17. http://dx.doi.org/10.15407/fmmit2021.33.012.

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This article presents the main statements of the method of approximation of discontinuous functions of two variables, describing an image of the surface of a 2D body or an image of the internal structure of a 3D body in a certain plane, using projections that come from a computer tomograph. The method is based on the use of discontinuous splines of two variables and finite Fourier sums, in which the Fourier coefficients are found using projection data. The method is based on the following idea: an approximated discontinuous function is replaced by the sum of two functions – a discontinuous spline and a continuous or differentiable function. A method is proposed for constructing a spline function, which has on the indicated lines the same discontinuities of the first kind as the approximated discontinuous function, and a method for finding the Fourier coefficients of the indicated continuous or differentiable function. That is, the difference between the function being approximated and the specified discontinuous spline is a function that can be approximated by finite Fourier sums without the Gibbs phenomenon. In the numerical experiment, it was assumed that the approximated function has discontinuities of the first kind on a given system of circles and ellipses nested into each other. The analysis of the calculation results showed their correspondence to the theoretical statements of the work. The proposed method makes it possible to obtain a given approximation accuracy with a smaller number of projections, that is, with less irradiation.

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10

Yang, Yi, Xiangguang Dai, Xianxiu Zhang, Yuelei Feng, Yashu Zhang, and Changcheng Xiang. "Stability for a Non-Smooth Filippov Ratio-Dependent Predator-Prey System through a Smooth Lyapunov Function." Mathematical Problems in Engineering 2022 (September 28, 2022): 1–6. http://dx.doi.org/10.1155/2022/6807336.

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For nonsmooth Filippov systems, the stability of the system is assumed to be proved by nonsmooth Lyapunov functions, such as piecewise smooth Lyapunov functions. This extension was based on the Filippov solution and Clarke generalized gradient. However, it is difficult to estimate the gradient of a non-smooth Lyapunov function. In some cases, the nonsmooth system can be divided into continuous and discontinuous components. If the Lebesgue measure of the discontinuous components is zero, the smooth Lyapunov function can be utilized to prove the stability of the system owing to the inner product of the gradient of the Lyapunov function of the discontinuous components being zero. In this paper, we apply the smooth Lyapunov function to prove the stability of the nonsmooth ratio-dependent predator-prey system. In contrast to the existing literature, in this paper, although the system is divided into continuous and discontinuous components, the inner product of the gradient of the Lyapunov function of the discontinuous part is not zero but negative. In the proof of stability, the negative value condition is stricter than the zero-value condition. This proof method only needs to construct a smooth Lyapunov function, which is simpler than a non-smooth Lyapunov function or other methods.

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11

Pershyna, Iuliia. "RECONSTRUCTION OF THE TWO VARIABLES DISCONTINUOUS FUNCTION BY DIFFERENT INFORMATION OPERATORS USING TRIANGULAR ELEMENTS." Bulletin of the National Technical University "KhPI". Series: Mathematical modeling in engineering and technologies, no. 2 (November 30, 2021): 84–96. http://dx.doi.org/10.20998/2222-0631.2021.02.10.

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The paper examines methods for constructing mathematical models of two variables discontinuous functions using various information about them: one-sided values at points and one-sided traces along a given system of lines. The case is considered when the domain of the required function is triangulated by right-angled triangles. If interpolation or approximation methods are used, then for their construction the values of the function at given points must be given; if we use interlination methods, then traces of the desired function along a given system of lines. In this work, we construct a discontinuous interpolation and approximation splines for approximating a discontinuous function of two variables with given one-sided values in a given system of points (in our case, at the vertices of right-angled triangles), and prove theorems on the estimation of the approximation error by constructed discontinuous structures. In the paper a discontinuous interlination spline, which uses completely different information about the discontinuous function, namely one-sided traces along a given system of lines (in our case, along the sides of right-angled triangles) is also built. Interlination of functions can find wide application in the aircraft and automobile body design automation; when receiving and processing the results of sonar and radar, when solving problems of computed tomography, in digital signal processing and in many other areas. In the paper theorems on the integral form and an estimate of the approximation error by the constructed discontinuous interlination operator are also proved. Computational experiments that compare the results of the approximation of a discontinuous function of two variables by different information operators using triangular elements are presented. In the future, it is planned to apply the constructed operators of discontinuous approximation and interlination to solve a two-dimensional problem of computed tomography with a significant use of the inhomogeneity of the internal structure of the body, which must be reconstructed.

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12

Mezhuyev, V., O. M. Lytvyn, I. Pershyna, and O. Nechuiviter. "APPROXIMATION OF DISCONTINUOUS FUNCTIONS OF TWO VARIABLES BY DISCONTINUOUS INTERLINATION SPLINES USING TRIANGULAR ELEMENTS." Journal of the Serbian Society for Computational Mechanics 14, no. 1 (June 30, 2020): 75–89. http://dx.doi.org/10.24874/jsscm.2020.14.01.07.

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The paper develops a method for approximation of the discontinuous functions of two variables by discontinuous interlination splines using arbitrary triangular elements. Experimental data are one-sided traces of a function given along a system of lines (such data are commonly used in remote methods, in particular in tomography). The paper also proposes a method for approximating the discontinuous functions of two variables taking into account triangular elements having one curved side. The proposed methods improve approximation of the discontinuous functions, allowing an application to complex domains of definition and avoiding the Gibbs phenomenon.

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13

Hammond, Christopher N. B. "104.16 Another function with discontinuous derivative." Mathematical Gazette 104, no. 560 (June 18, 2020): 315–18. http://dx.doi.org/10.1017/mag.2020.58.

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14

Kaplun, Yuliya, Mykola Perestyuk, and Valeriy Samoylenko. "Implicit function equation with discontinuous trajectories." Miskolc Mathematical Notes 2, no. 2 (2001): 145. http://dx.doi.org/10.18514/mmn.2001.45.

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15

Liu, Xiaohua, Liu Liu, and Weinian Zhang. "Discontinuous function with continuous second iterate." Aequationes mathematicae 88, no. 3 (July 20, 2013): 243–66. http://dx.doi.org/10.1007/s00010-013-0220-z.

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16

Park, Dong-Ryeon. "Comparison of Nonparametric Function Estimation Methods for Discontinuous Regression Functions." Korean Journal of Applied Statistics 23, no. 6 (December 31, 2010): 1245–53. http://dx.doi.org/10.5351/kjas.2010.23.6.1245.

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17

Zhang, Jianming, Lei Han, Yudong Zhong, Yunqiao Dong, and Weicheng Lin. "Boundary element analysis for elasticity problems using expanding element interpolation method." Engineering Computations 37, no. 1 (November 14, 2019): 1–20. http://dx.doi.org/10.1108/ec-11-2018-0506.

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Purpose This paper aims to propose a boundary element analysis of two-dimensional linear elasticity problems by a new expanding element interpolation method. Design/methodology/approach The expanding element is made up based on a traditional discontinuous element by adding virtual nodes along the perimeter of the element. The internal nodes of the original discontinuous element are referred to as source nodes and its shape function as raw shape function. The shape functions of the expanding element constructed on both source nodes and virtual nodes are referred as fine shape functions. Boundary variables are interpolated by the fine shape functions, while the boundary integral equations are collocated on source nodes. Findings The expanding element inherits the advantages of both the continuous and discontinuous elements while overcomes their disadvantages. The polynomial order of fine shape functions of the expanding elements increases by two compared with their corresponding raw shape functions, while the expanding elements still keep independence to each other as the original discontinuous elements. This feature makes the expanding elements able to naturally and accurately interpolate both continuous and discontinuous fields. Originality/value Numerical examples are presented to verify the proposed method. Results have demonstrated that the accuracy, efficiency and convergence rate of the expanding element method.

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18

Pershyna, Iuliia. "Approximation of discontinuous functions of three variables by discontinuous interpolation splines." Physico-mathematical modelling and informational technologies, no. 33 (September 4, 2021): 99–104. http://dx.doi.org/10.15407/fmmit2021.33.099.

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In this paper, discontinuous interpolation splines of three variables are constructed and a method for reconstructing of the discontinuous internal structure of a three-dimensional body by constructed splines is proposed. It is believed that a three-dimensional object, which is described by a function of three variables with discontinuities of the first kind on a given grid of nodes, is completely covered by a system of parallelepipeds. The experimental data are the one-sided value of the discontinuous function in a given grid of nodes. In the article, theorems on interpolation properties and the error of the constructed discontinuous structures are formulated and proved. Moreover, the constructed discontinuous interpolation splines include, as a special case, classical continuous splines. The developed approximation method can be applied in three-dimensional mathematical modeling of discontinuous processes, including in computed tomography.

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19

Burlakov, Evgenii O., and Ivan N. Malkov. "On connection between continuous and discontinuous neural ﬁeld models with microstructure: II. Radially symmetric stationary solutions in 2D (“bumps”)." Russian Universities Reports. Mathematics, no. 129 (2020): 6–17. http://dx.doi.org/10.20310/2686-9667-2020-25-129-6-17.

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We suggest a method allowing to investigate existence and the measure of proximity between the stationary solutions to continuous and discontinuous neural ﬁelds with microstructure. The present part involves results on proximity of the stationary solutions to speciﬁc homogenized neural ﬁeld equations with continuous and discontinuous activation functions. The results of numerical investigation of radially symmetric stationary solutions (bumps) to the neural ﬁeld with a discontinuous activation function and a given microstructure are presented.

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20

Ovan, A. Saputra, and N. Tasni. "Integral mean value theorem for discontinuous function." Journal of Physics: Conference Series 1918, no. 4 (June 1, 2021): 042033. http://dx.doi.org/10.1088/1742-6596/1918/4/042033.

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21

Chartrand, Rick. "Numerical Differentiation of Noisy, Nonsmooth Data." ISRN Applied Mathematics 2011 (May 11, 2011): 1–11. http://dx.doi.org/10.5402/2011/164564.

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We consider the problem of differentiating a function specified by noisy data. Regularizing the differentiation process avoids the noise amplification of finite-difference methods. We use total-variation regularization, which allows for discontinuous solutions. The resulting simple algorithm accurately differentiates noisy functions, including those which have a discontinuous derivative.

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22

Akhmet, Marat, Madina Tleubergenova, Mehmet Onur Fen, and Zakhira Nugayeva. "Unpredictable Solutions of Linear Impulsive Systems." Mathematics 8, no. 10 (October 16, 2020): 1798. http://dx.doi.org/10.3390/math8101798.

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We consider a new type of oscillations of discontinuous unpredictable solutions for linear impulsive nonhomogeneous systems. The models under investigation are with unpredictable perturbations. The definition of a piecewise continuous unpredictable function is provided. The moments of impulses constitute a newly determined unpredictable discrete set. Theoretical results on the existence, uniqueness, and stability of discontinuous unpredictable solutions for linear impulsive differential equations are provided. We benefit from the B-topology in the space of discontinuous functions on the purpose of proving the presence of unpredictable solutions. For constructive definitions of unpredictable components in examples, randomly determined unpredictable sequences are newly utilized. Namely, the construction of a discontinuous unpredictable function is based on an unpredictable sequence determined by a discrete random process, and the set of discontinuity moments is realized by the logistic map. Examples with numerical simulations are presented to illustrate the theoretical results.

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23

Mattiola, Simone, and Francesca Masini. "Discontinuous reduplication: a typological sketch." STUF - Language Typology and Universals 75, no. 2 (July 1, 2022): 271–316. http://dx.doi.org/10.1515/stuf-2022-1055.

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Abstract The paper investigates discontinuous reduplication (DR), a pattern where reduplicant and base are separated by other material, by annotating a 214-example dataset collected from a 99-language sample. Several items turned out to serve as interposing elements, although their nature does not seem to correlate with function, unlike the category of the base. DR’s functions are a subset of those associated with reduplication cross-linguistically. All languages displaying DR also present contiguous reduplication, suggesting a contiguous reduplication > discontinuous reduplication hierarchy. Finally, a corpus-based analysis of Italian (lacking DR according to grammars) unveiled a wealth of DR patterns, suggesting that corpora are essential for the typological enterprise.

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24

den Boer, Arnoud V., and N. Bora Keskin. "Discontinuous Demand Functions: Estimation and Pricing." Management Science 66, no. 10 (October 2020): 4516–34. http://dx.doi.org/10.1287/mnsc.2019.3446.

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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.

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25

Wang, Yonglong, Bill Samson, David Ellison, and Louis Natanson. "MIQR Active Learning on a Continuous Function and a Discontinuous Function." Neural Computing & Applications 10, no. 3 (December 1, 2001): 253–63. http://dx.doi.org/10.1007/s521-001-8053-6.

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26

Avrutin, Viktor, Bernd Eckstein, and Michael Schanz. "On detection of multi-band chaotic attractors." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 463, no. 2081 (March 6, 2007): 1339–58. http://dx.doi.org/10.1098/rspa.2007.1826.

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In this work, we present two numerical methods for the detection of the number of bands of a multi-band chaotic attractor. The first method is more efficient but can be applied only for dynamical systems with a continuous system function, whereas the second one is applicable for dynamical systems with a discontinuous system function as well. Using the developed methods, we investigate a one-dimensional piecewise-linear map and report for both cases of a continuous and a discontinuous system functions some new bifurcation scenarios involving multi-band chaotic attractors.

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27

Gavaldà, Ricard, and Hava T. Siegelmann. "Discontinuities in Recurrent Neural Networks." Neural Computation 11, no. 3 (April 1, 1999): 715–45. http://dx.doi.org/10.1162/089976699300016638.

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This article studies the computational power of various discontinuous real computational models that are based on the classical analog recurrent neural network (ARNN). This ARNN consists of finite number of neurons; each neuron computes a polynomial net function and a sigmoid-like continuous activation function. We introduce arithmetic networks as ARNN augmented with a few simple discontinuous (e.g., threshold or zero test) neurons. We argue that even with weights restricted to polynomial time computable reals, arithmetic networks are able to compute arbitrarily complex recursive functions. We identify many types of neural networks that are at least as powerful as arithmetic nets, some of which are not in fact discontinuous, but they boost other arithmetic operations in the net function (e.g., neurons that can use divisions and polynomial net functions inside sigmoid-like continuous activation functions). These arithmetic networks are equivalent to the Blum-Shub-Smale model, when the latter is restricted to a bounded number of registers. With respect to implementation on digital computers, we show that arithmetic networks with rational weights can be simulated with exponential precision, but even with polynomial-time computable real weights, arithmetic networks are not subject to any fixed precision bounds. This is in contrast with the ARNN that are known to demand precision that is linear in the computation time. When nontrivial periodic functions (e.g., fractional part, sine, tangent) are added to arithmetic networks, the resulting networks are computationally equivalent to a massively parallel machine. Thus, these highly discontinuous networks can solve the presumably intractable class of PSPACE-complete problems in polynomial time.

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Yun, Beong In. "Improving Fourier Partial Sum Approximation for Discontinuous Functions Using a Weight Function." Abstract and Applied Analysis 2017 (2017): 1–7. http://dx.doi.org/10.1155/2017/1364914.

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We introduce a generalized sigmoidal transformation wm(r;x) on a given interval [a,b] with a threshold at x=r∈(a,b). Using wm(r;x), we develop a weighted averaging method in order to improve Fourier partial sum approximation for a function having a jump-discontinuity. The method is based on the decomposition of the target function into the left-hand and the right-hand part extensions. The resultant approximate function is composed of the Fourier partial sums of each part extension. The pointwise convergence of the presented method and its availability for resolving Gibbs phenomenon are proved. The efficiency of the method is shown by some numerical examples.

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29

Wolf, Christian. "A shift map with a discontinuous entropy function." Discrete & Continuous Dynamical Systems - A 40, no. 1 (2020): 319–29. http://dx.doi.org/10.3934/dcds.2020012.

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Begmatov, Akram H., and Z. H. Ochilov. "Integral geometry problem with a discontinuous weight function." Doklady Mathematics 80, no. 3 (December 2009): 823–25. http://dx.doi.org/10.1134/s1064562409060106.

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Huang, Gan, and Jinde Cao. "Multistability of neural networks with discontinuous activation function." Communications in Nonlinear Science and Numerical Simulation 13, no. 10 (December 2008): 2279–89. http://dx.doi.org/10.1016/j.cnsns.2007.07.005.

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Jia, Xiao, Jin-Song Hong, Ya-Chun Gao, Hong-Chun Yang, Chun Yang, Chuan-Ji Fu, and Jian-Quan Hu. "Percolation phase transition of static and growing networks under a weighted function." International Journal of Modern Physics C 27, no. 07 (May 24, 2016): 1650082. http://dx.doi.org/10.1142/s0129183116500820.

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We investigate the percolation phase transitions in both the static and growing networks where the nodes are sampled according to a weighted function with a tunable parameter [Formula: see text]. For the static network, i.e. the number of nodes is constant during the percolation process, the percolation phase transition can evolve from continuous to discontinuous as the value of [Formula: see text] is tuned. Based on the properties of the weighted function, three typical values of [Formula: see text] are analyzed. The model becomes the classical Erdös–Rényi (ER) network model at [Formula: see text]. When [Formula: see text], it is shown that the percolation process generates a weakly discontinuous phase transition where the order parameter exhibits an extremely abrupt transition with a significant jump in large but finite system. For [Formula: see text], the cluster size distribution at the lower pseudo-transition point does not obey the power-law behavior, indicating a strongly discontinuous phase transition. In the case of growing network, in which the collection of nodes is increasing, a smoother continuous phase transition emerges at [Formula: see text], in contrast to the weakly discontinuous phase transition of the static network. At [Formula: see text], on the other hand, probability modulation effect shows that the nature of strongly discontinuous phase transition remains the same with the static network despite the node arrival even in the thermodynamic limit. These percolation properties of the growing networks could provide useful reference for network intervention and control in practical applications in consideration of the increasing size of most actual networks.

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Tian, R. L., Z. J. Zhao, X. W. Yang, and Y. F. Zhou. "Subharmonic Bifurcation for a Nonsmooth Oscillator." International Journal of Bifurcation and Chaos 27, no. 10 (September 2017): 1750163. http://dx.doi.org/10.1142/s0218127417501632.

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A nonsmooth pendulum model with multiple impulse effect is constructed to detect the bifurcation of a periodic orbit with multiple jump discontinuous points. Subharmonic Melnikov function of this kind of nonsmooth systems is studied. Differences of subharmonic Melnikov function between the nonsmooth system with multiple jump discontinuities and the smooth system are analyzed by using the Hamiltonian function and piecewise integral method. Applying the recursive method and perturbation principle, the effects of the jump discontinuous points on the subharmonic Melnikov function are converted to integral items which can be easily calculated. Hence, the subharmonic Melnikov function for the subharmonic orbit with multiple jump discontinuous points is obtained. Finally, the existence conditions for periodic motion of the subharmonic orbit are derived and the efficiency of the conclusions is verified via numerical simulations.

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34

Turner, J. D. "On the Simulation of Discontinuous Functions." Journal of Applied Mechanics 68, no. 5 (April 16, 2001): 751–57. http://dx.doi.org/10.1115/1.1387022.

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Discontinuous function constraints arise during the calculation of surface contact, stiction, and friction effects in studies of the behavior of complex systems. These nonlinear effects are mathematically defined by inequality constraints of the form 0⩾gxt,t. The unknown in the problem is the time, t*, when the equality condition is reached. This paper presents an exact solution for t*, which is obtained by introducing a slack variable that replaces time as the independent variable, leading to an extended state-space that is noniteratively integrated to the constraint surface. Several applications are presented to demonstrate the method.

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Ullrich, P. A. "A global finite-element shallow-water model supporting continuous and discontinuous elements." Geoscientific Model Development 7, no. 6 (December 17, 2014): 3017–35. http://dx.doi.org/10.5194/gmd-7-3017-2014.

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Abstract. This paper presents a novel nodal finite-element method for either continuous and discontinuous elements, as applied to the 2-D shallow-water equations on the cubed sphere. The cornerstone of this method is the construction of a robust derivative operator that can be applied to compute discrete derivatives even over a discontinuous function space. A key advantage of the robust derivative is that it can be applied to partial differential equations in either a conservative or a non-conservative form. However, it is also shown that discontinuous penalization is required to recover the correct order of accuracy for discontinuous elements. Two versions with discontinuous elements are examined, using either the g1 and g2 flux correction function for distribution of boundary fluxes and penalty across nodal points. Scalar and vector hyperviscosity (HV) operators valid for both continuous and discontinuous elements are also derived for stabilization and removal of grid-scale noise. This method is validated using four standard shallow-water test cases, including geostrophically balanced flow, a mountain-induced Rossby wave train, the Rossby–Haurwitz wave and a barotropic instability. The results show that although the discontinuous basis requires a smaller time step size than that required for continuous elements, the method exhibits better stability and accuracy properties in the absence of hyperviscosity.

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36

Ullrich, P. A. "A global finite-element shallow-water model supporting continuous and discontinuous elements." Geoscientific Model Development Discussions 7, no. 4 (August 8, 2014): 5141–82. http://dx.doi.org/10.5194/gmdd-7-5141-2014.

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Abstract. This paper presents a novel nodal finite element method for either continuous and discontinuous elements, as applied to the 2-D shallow-water equations on the cubed-sphere. The cornerstone of this method is the construction of a robust derivative operator which can be applied to compute discrete derivatives even over a discontinuous function space. A key advantage of the robust derivative is that it can be applied to partial differential equations in either conservative or non-conservative form. However, it is also shown that discontinuous penalization is required to recover the correct order of accuracy for discontinuous elements. Two versions with discontinuous elements are examined, using either the g1 and g2 flux correction function for distribution of boundary fluxes and penalty across nodal points. Scalar and vector hyperviscosity operators valid for both continuous and discontinuous elements are also derived for stabilization and removal of grid-scale noise. This method is validated using three standard shallow-water test cases, including geostrophically balanced flow, a mountain-induced Rossby wave train and a barotropic instability. The results show that although the discontinuous basis requires a smaller time step size than that required for continuous elements, the method exhibits better stability and accuracy properties in the absence of hyperviscosity.

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37

Smolyanova, M. O. "A continuously differentiable discontinuous function on the space D." Izvestiya: Mathematics 59, no. 5 (October 31, 1995): 1077–82. http://dx.doi.org/10.1070/im1995v059n05abeh000048.

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Hui, Chi-Wai. "Optimizing chemical processes with discontinuous function — A novel formulation." Computers & Chemical Engineering 23 (June 1999): S479—S482. http://dx.doi.org/10.1016/s0098-1354(99)80118-5.

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Cheung, Kwok-Yuen, and Chi-Wai Hui. "Heat exchanger network optimization with discontinuous exchanger cost function." Applied Thermal Engineering 21, no. 13-14 (October 2001): 1397–405. http://dx.doi.org/10.1016/s1359-4311(01)00029-1.

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Chen, Zhen-Qing, and Renming Song. "Drift transforms and Green function estimates for discontinuous processes." Journal of Functional Analysis 201, no. 1 (June 2003): 262–81. http://dx.doi.org/10.1016/s0022-1236(03)00087-9.

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41

Kato, Junji, and Pan Jaqi. "Stability via the Liapunov function with a discontinuous derivative." Journal of Mathematical Analysis and Applications 152, no. 1 (October 1990): 229–39. http://dx.doi.org/10.1016/0022-247x(90)90100-t.

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42

Basu, Amitabh, Michele Conforti, and Marco Di Summa. "An extreme function which is nonnegative and discontinuous everywhere." Mathematical Programming 179, no. 1-2 (September 4, 2018): 447–53. http://dx.doi.org/10.1007/s10107-018-1322-0.

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43

Brownstein, K. R. "Bound state variational principle employing a discontinuous trial function." Journal of Mathematical Physics 36, no. 1 (January 1995): 76–85. http://dx.doi.org/10.1063/1.531327.

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Van Cranenburgh, Andreas, Remko Scha, and Rens Bod. "Data-Oriented Parsing with Discontinuous Constituents and Function Tags." Journal of Language Modelling 4, no. 1 (April 13, 2016): 57. http://dx.doi.org/10.15398/jlm.v4i1.100.

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Jaroszewska, Joanna. "A note on iterated function systems with discontinuous probabilities." Chaos, Solitons & Fractals 49 (April 2013): 28–31. http://dx.doi.org/10.1016/j.chaos.2013.01.012.

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Sánchez-Borrego, I. R., M. D. Martínez-Miranda, and A. González-Carmona. "Local linear kernel estimation of the discontinuous regression function." Computational Statistics 21, no. 3-4 (November 15, 2006): 557–69. http://dx.doi.org/10.1007/s00180-006-0014-z.

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47

Liu, Zhong-yan, and Huan-zhen Chen. "Discontinuous Galerkin Immersed Finite Volume Element Method for Anisotropic Flow Models in Porous Medium." Abstract and Applied Analysis 2014 (2014): 1–10. http://dx.doi.org/10.1155/2014/520404.

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By choosing the trial function space to the immersed finite element space and the test function space to be piecewise constant function space, we develop a discontinuous Galerkin immersed finite volume element method to solve numerically a kind of anisotropic diffusion models governed by the elliptic interface problems with discontinuous tensor-conductivity. The existence and uniqueness of the discrete scheme are proved, and an optimal-order energy-norm estimate andL2-norm estimate for the numerical solution are derived.

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48

ZHANG, Zhiqian, and Hirohisa NOGUCHI. "155 h-Adaptivity Analysis of Nonlinear Multi-material Structures using Meshfree Methods with Discontinuous Derivative Basis Function." Proceedings of The Computational Mechanics Conference 2006.19 (2006): 563–64. http://dx.doi.org/10.1299/jsmecmd.2006.19.563.

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Karapınar, Erdal, Andreea Fulga, and Poom Kumam. "Revisiting the Meir-Keeler contraction via simulation function." Filomat 34, no. 5 (2020): 1645–57. http://dx.doi.org/10.2298/fil2005645k.

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In this paper, we aim to obtain a fixed point theorem which guarantee the existence of a fixed point for both the continuous and discontinuous mappings that fullfill certain conditions in the context of metric space. We also consider some examples to illustrate our results.

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50

Cheraghi-Shami, Farideh, Ali-Akbar Gharaveisi, Malihe M. Farsangi, and Mohsen Mohammadian. "Discontinuous Lyapunov functions for a class of piecewise affine systems." Transactions of the Institute of Measurement and Control 41, no. 3 (May 24, 2018): 729–36. http://dx.doi.org/10.1177/0142331218771138.

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In this paper, a Lyapunov-based method is provided to study the local asymptotic stability of planar piecewise affine systems with continuous vector fields. For such systems, the state space is supposed to be partitioned into several bounded convex polytopes. A piecewise affine function, not necessarily continuous on the boundaries of the polytopic partitions, is proposed as a candidate Lyapunov function. Then, sufficient conditions for the local asymptotic stability of the system, including a monotonicity condition at switching instants, are formulated as a linear programming problem. In addition, when the problem does not have a feasible solution based on the original partitions of the system, a new partition refinement algorithm is presented. In this way, more flexibility can be provided in searching for the Lyapunov function. Owing to relaxation of the continuity condition imposed on the system boundaries, the proposed method reaches to less conservative results, compared with the previous methods based on continuous piecewise affine Lyapunov functions. Simulation results illustrate the effectiveness of the proposed method.

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