Journal articles: 'Stochastic Differential Geometry'

1

Kendall, Wilfrid S. "Stochastic differential geometry: An introduction." Acta Applicandae Mathematicae 9, no. 1-2 (1987): 29–60. http://dx.doi.org/10.1007/bf00580820.

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Catuogno, Pedro, and Paulo Ruffino. "Geometry of Stochastic Delay Differential Equations." Electronic Communications in Probability 10 (2005): 190–95. http://dx.doi.org/10.1214/ecp.v10-1151.

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Thiruthummal, Abhiram Anand, and Eun-jin Kim. "Monte Carlo Simulation of Stochastic Differential Equation to Study Information Geometry." Entropy 24, no. 8 (August 12, 2022): 1113. http://dx.doi.org/10.3390/e24081113.

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Information Geometry is a useful tool to study and compare the solutions of a Stochastic Differential Equations (SDEs) for non-equilibrium systems. As an alternative method to solving the Fokker–Planck equation, we propose a new method to calculate time-dependent probability density functions (PDFs) and to study Information Geometry using Monte Carlo (MC) simulation of SDEs. Specifically, we develop a new MC SDE method to overcome the challenges in calculating a time-dependent PDF and information geometric diagnostics and to speed up simulations by utilizing GPU computing. Using MC SDE simulations, we reproduce Information Geometric scaling relations found from the Fokker–Planck method for the case of a stochastic process with linear and cubic damping terms. We showcase the advantage of MC SDE simulation over FPE solvers by calculating unequal time joint PDFs. For the linear process with a linear damping force, joint PDF is found to be a Gaussian. In contrast, for the cubic process with a cubic damping force, joint PDF exhibits a bimodal structure, even in a stationary state. This suggests a finite memory time induced by a nonlinear force. Furthermore, several power-law scalings in the characteristics of bimodal PDFs are identified and investigated.

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Kendall, Wilfrid S. "Stochastic Differential Geometry, a Coupling Property, and Harmonic Maps." Journal of the London Mathematical Society s2-33, no. 3 (June 1986): 554–66. http://dx.doi.org/10.1112/jlms/s2-33.3.554.

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Dimakis, Aristophanes, and Folkert M�ller-Hoissen. "Stochastic differential calculus, the Moyal *-product, and noncommutative geometry." Letters in Mathematical Physics 28, no. 2 (June 1993): 123–37. http://dx.doi.org/10.1007/bf00750305.

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Manton, Jonathan H. "A Primer on Stochastic Differential Geometry for Signal Processing." IEEE Journal of Selected Topics in Signal Processing 7, no. 4 (August 2013): 681–99. http://dx.doi.org/10.1109/jstsp.2013.2264798.

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Kühnel, Line, Stefan Sommer, and Alexis Arnaudon. "Differential geometry and stochastic dynamics with deep learning numerics." Applied Mathematics and Computation 356 (September 2019): 411–37. http://dx.doi.org/10.1016/j.amc.2019.03.044.

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Agrachev, Andrei, Ugo Boscain, Robert Neel, and Luca Rizzi. "Intrinsic random walks in Riemannian and sub-Riemannian geometry via volume sampling." ESAIM: Control, Optimisation and Calculus of Variations 24, no. 3 (2018): 1075–105. http://dx.doi.org/10.1051/cocv/2017037.

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We relate some constructions of stochastic analysis to differential geometry, via random walk approximations. We consider walks on both Riemannian and sub-Riemannian manifolds in which the steps consist of travel along either geodesics or integral curves associated to orthonormal frames, and we give particular attention to walks where the choice of step is influenced by a volume on the manifold. A primary motivation is to explore how one can pass, in the parabolic scaling limit, from geodesics, orthonormal frames, and/or volumes to diffusions, and hence their infinitesimal generators, on sub-Riemannian manifolds, which is interesting in light of the fact that there is no completely canonical notion of sub-Laplacian on a general sub-Riemannian manifold. However, even in the Riemannian case, this random walk approach illuminates the geometric significance of Ito and Stratonovich stochastic differential equations as well as the role played by the volume.

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Ashyralyev, Allaberen, and Ülker Okur. "Stability of Stochastic Partial Differential Equations." Axioms 12, no. 7 (July 24, 2023): 718. http://dx.doi.org/10.3390/axioms12070718.

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In this paper, we study the stability of the stochastic parabolic differential equation with dependent coefficients. We consider the stability of an abstract Cauchy problem for the solution of certain stochastic parabolic differential equations in a Hilbert space. For the solution of the initial-boundary value problems (IBVPs), we obtain the stability estimates for stochastic parabolic equations with dependent coefficients in specific applications.

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Andrianantenainarinoro, T. R. H., R. A. Randrianomenjanahary, and T. J. Rabeherimanana. "AMPLITUDE ADJUSTMENT WITH FIWASVJ MODEL." Advances in Mathematics: Scientific Journal 11, no. 4 (April 28, 2022): 383–413. http://dx.doi.org/10.37418/amsj.11.4.7.

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Andrianantenainarinoro\cite{maReference101} remarked that the price amplitudes of financial models may not correspond to the reality and we propose here a model in continuous time Fractionally Integrated WASC Stochastic Volatility Jump. To do this, we introduce a fractal index in the WASC Stochastic Volatility Jump model and we have two others characteristics: amplitude adjustment and memory of process. We present also several theories in stochastic calculus, algebraic, differential geometry, numerical method and estimating method which can use to financial such us: sense of a fractional integral, relationship between trace and determinant operator, Euler's approximation for an unresolved differential equation and convergence speed.

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Christakos, George, Dionissios T. Hristopulos, and Xinyang Li. "Multiphase flow in heterogeneous porous media from a stochastic differential geometry viewpoint." Water Resources Research 34, no. 1 (January 1998): 93–102. http://dx.doi.org/10.1029/97wr02715.

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DIAZ-GUILERA, ALBERT. "NONLINEAR STOCHASTIC DIFFERENTIAL EQUATIONS AND SELF-ORGANIZED CRITICALITY." Fractals 01, no. 04 (December 1993): 963–67. http://dx.doi.org/10.1142/s0218348x93001039.

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Several nonlinear stochastic differential equations have been proposed in connection with self-organized critical phenomena. Due to the threshold condition involved in its dynamic evolution, an infinite number of nonlinearities arise in a hydrodynamic description. We study two models with different noise correlations which make all nonlinear contributions to be equally relevant below the upper critical dimension. The asymptotic values of the critical exponents are estimated from a systematic expansion in the number of coupling constants by means of the dynamic renormalization group.

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Luo, Mei, Michal Fečkan, Jin-Rong Wang, and Donal O’Regan. "g-Expectation for Conformable Backward Stochastic Differential Equations." Axioms 11, no. 2 (February 14, 2022): 75. http://dx.doi.org/10.3390/axioms11020075.

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In this paper, we study the applications of conformable backward stochastic differential equations driven by Brownian motion and compensated random measure in nonlinear expectation. From the comparison theorem, we introduce the concept of g-expectation and give related properties of g-expectation. In addition, we find that the properties of conformable backward stochastic differential equations can be deduced from the properties of the generator g. Finally, we extend the nonlinear Doob–Meyer decomposition theorem to more general cases.

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Crisan, Dan. "The stochastic filtering problem: a brief historical account." Journal of Applied Probability 51, A (December 2014): 13–22. http://dx.doi.org/10.1239/jap/1417528463.

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Onwards from the mid-twentieth century, the stochastic filtering problem has caught the attention of thousands of mathematicians, engineers, statisticians, and computer scientists. Its applications span the whole spectrum of human endeavour, including satellite tracking, credit risk estimation, human genome analysis, and speech recognition. Stochastic filtering has engendered a surprising number of mathematical techniques for its treatment and has played an important role in the development of new research areas, including stochastic partial differential equations, stochastic geometry, rough paths theory, and Malliavin calculus. It also spearheaded research in areas of classical mathematics, such as Lie algebras, control theory, and information theory. The aim of this paper is to give a brief historical account of the subject concentrating on the continuous-time framework.

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Crisan, Dan. "The stochastic filtering problem: a brief historical account." Journal of Applied Probability 51, A (December 2014): 13–22. http://dx.doi.org/10.1017/s002190020002115x.

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Onwards from the mid-twentieth century, the stochastic filtering problem has caught the attention of thousands of mathematicians, engineers, statisticians, and computer scientists. Its applications span the whole spectrum of human endeavour, including satellite tracking, credit risk estimation, human genome analysis, and speech recognition. Stochastic filtering has engendered a surprising number of mathematical techniques for its treatment and has played an important role in the development of new research areas, including stochastic partial differential equations, stochastic geometry, rough paths theory, and Malliavin calculus. It also spearheaded research in areas of classical mathematics, such as Lie algebras, control theory, and information theory. The aim of this paper is to give a brief historical account of the subject concentrating on the continuous-time framework.

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Nayak, Sukanta, Tshilidzi Marwala, and Snehashish Chakraverty. "Stochastic differential equations with imprecisely defined parameters in market analysis." Soft Computing 23, no. 17 (July 27, 2018): 7715–24. http://dx.doi.org/10.1007/s00500-018-3396-2.

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Batiha, Iqbal M., Ahmad A. Abubaker, Iqbal H. Jebril, Suha B. Al-Shaikh, and Khaled Matarneh. "A Numerical Approach of Handling Fractional Stochastic Differential Equations." Axioms 12, no. 4 (April 17, 2023): 388. http://dx.doi.org/10.3390/axioms12040388.

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This work proposes a new numerical approach for dealing with fractional stochastic differential equations. In particular, a novel three-point fractional formula for approximating the Riemann–Liouville integrator is established, and then it is applied to generate approximate solutions for fractional stochastic differential equations. Such a formula is derived with the use of the generalized Taylor theorem coupled with a recent definition of the definite fractional integral. Our approach is compared with the approximate solution generated by the Euler–Maruyama method and the exact solution for the purpose of verifying our findings.

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Muratore-Ginanneschi, Paolo. "On the use of stochastic differential geometry for non-equilibrium thermodynamic modeling and control." Journal of Physics A: Mathematical and Theoretical 46, no. 27 (June 18, 2013): 275002. http://dx.doi.org/10.1088/1751-8113/46/27/275002.

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Wu, Meng, Nan-jing Huang, and Chang-Wen Zhao. "Fixed Points and Stability in Neutral Stochastic Differential Equations with Variable Delays." Fixed Point Theory and Applications 2008 (2008): 1–12. http://dx.doi.org/10.1155/2008/407352.

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CHAI, RUISHUAI. "FRACTAL DIMENSION OF FRACTIONAL BROWNIAN MOTION BASED ON RANDOM SETS." Fractals 28, no. 08 (July 10, 2020): 2040020. http://dx.doi.org/10.1142/s0218348x20400204.

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The fractal dimension of fractional Brownian motion can effectively describe random sets, reflecting the regularity implicit in complex random sets. Data mining algorithms based on fractal theory usually follow the calculation of the fractal dimension of fractional Brownian motion. However, the existing fractal dimension calculation methods of fractal Brownian motion have high time complexity and space complexity, which greatly reduces the efficiency of the algorithm and makes it difficult for the algorithm to adapt to high-speed and massive data flow environments. Therefore, several existing fractal dimension calculation methods of fractional Brownian motion are summarized and analyzed, and a random method is proposed, which uses a fixed memory space to quickly estimate the associated dimension of the data stream. Finally, a comparison experiment with existing algorithms proves the effectiveness of this random algorithm. Second, in the sense of two different measures, based on the principle of stochastic comparison, the stability of the stochastic fuzzy differential equations is derived using the stability of the comparison equations, and the practical stability criterion of two measures according to probability is obtained. Then, the stochastic fuzzy differential equations are discussed. The definition of stochastic exponential stability is given and the stochastic exponential stability criterion is proved.

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Antonyuk, A. Val, and A. Vik Antonyuk. "Regularity of nonlinear flows on noncompact Riemannian manifolds: Differential geometry versus stochastic geometry or what kind of variations is natural?" Ukrainian Mathematical Journal 58, no. 8 (August 2006): 1145–70. http://dx.doi.org/10.1007/s11253-006-0126-1.

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Farinelli, Simone, and Hideyuki Takada. "Geometry and Spectral Theory Applied to Credit Bubbles in Arbitrage Markets: The Geometric Arbitrage Approach to Credit Risk." Symmetry 14, no. 7 (June 27, 2022): 1330. http://dx.doi.org/10.3390/sym14071330.

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We apply Geometric Arbitrage Theory (GAT) to obtain results in mathematical finance for credit markets, which do not need stochastic differential geometry in their formulation. The remarkable aspect of the GAT is the gauge symmetry, which can be translated to the financial context, by packaging all of the asset model information into a (stochastic) principal fiber bundle. We obtain closed-form equations involving default intensities and loss-given defaults characterizing the no-free-lunch-with-vanishing-risk condition for government and corporate bond markets while relying on the spread-term structure with default intensity and loss-given default. Moreover, we provide a sufficient condition equivalent to the Novikov condition implying the absence of arbitrage. Furthermore, the generic dynamics for an isolated credit market allowing for arbitrage possibilities (and minimizing the total quantity of potential arbitrage) are derived, and arbitrage credit bubbles for both base credit assets and credit derivatives are explicitly computed. The existence of an approximated risk-neutral measure allowing the definition of fundamental values for the assets is inferred through spectral theory. We show that instantaneous bond returns are serially uncorrelated and centered, that the expected value of credit bubbles remains constant for future times where no coupons are paid, and that the variance of the market portfolio nominals is concurrent with that of the corresponding bond deflators.

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Drăgan, Vasile, Ivan Ganchev Ivanov, and Ioan-Lucian Popa. "A Game — Theoretic Model for a Stochastic Linear Quadratic Tracking Problem." Axioms 12, no. 1 (January 11, 2023): 76. http://dx.doi.org/10.3390/axioms12010076.

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In this paper, we solve a stochastic linear quadratic tracking problem. The controlled dynamical system is modeled by a system of linear Itô differential equations subject to jump Markov perturbations. We consider the case when there are two decision-makers and each of them wants to minimize the deviation of a preferential output of the controlled dynamical system from a given reference signal. We assume that the two decision-makers do not cooperate. Under these conditions, we state the considered tracking problem as a problem of finding a Nash equilibrium strategy for a stochastic differential game. Explicit formulae of a Nash equilibrium strategy are provided. To this end, we use the solutions of two given terminal value problems (TVPs). The first TVP is associated with a hybrid system formed by two backward nonlinear differential equations coupled by two algebraic nonlinear equations. The second TVP is associated with a hybrid system formed by two backward linear differential equations coupled by two algebraic linear equations.

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Krikštolaitis, Ričardas, Gintautas Mozgeris, Edmundas Petrauskas, and Petras Rupšys. "A Statistical Dependence Framework Based on a Multivariate Normal Copula Function and Stochastic Differential Equations for Multivariate Data in Forestry." Axioms 12, no. 5 (May 8, 2023): 457. http://dx.doi.org/10.3390/axioms12050457.

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Stochastic differential equations and Copula theories are important topics that have many advantages for applications in almost every discipline. Many studies in forestry collect longitudinal, multi-dimensional, and discrete data for which the amount of measurement of individual variables does not match. For example, during sampling experiments, the diameters of all trees, the heights of approximately 10% of the trees, and the tree crown base height and crown width for a significantly smaller number of trees are measured. In this study, for estimating five-dimensional dependencies, we used a normal copula approach, where the dynamics of individual tree variables (diameter, potentially available area, height, crown base height, and crown width) are described by a stochastic differential equation with mixed-effect parameters. The approximate maximum likelihood method was used to obtain parameter estimates of the presented stochastic differential equations, and the normal copula dependence parameters were estimated using the pseudo-maximum likelihood method. This study introduced the normalized multi-dimensional interaction information index based on differential entropy to capture dependencies between state variables. Using conditional copula-type probability density functions, the exact form equations defining the links among the diameter, potentially available area, height, crown base height, and crown width were derived. All results were implemented in the symbolic algebra system MAPLE.

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Zhang, Pei, Adriana Irawati Nur Ibrahim, and Nur Anisah Mohamed. "Backward Stochastic Differential Equations (BSDEs) Using Infinite-Dimensional Martingales with Subdifferential Operator." Axioms 11, no. 10 (October 8, 2022): 536. http://dx.doi.org/10.3390/axioms11100536.

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In this paper, we focus on a family of backward stochastic differential equations (BSDEs) with subdifferential operators that are driven by infinite-dimensional martingales. We shall show that the solution to such infinite-dimensional BSDEs exists and is unique. The existence and uniqueness of the solution are established using Yosida approximations. Furthermore, as an application of the main result, we shall show that the backward stochastic partial differential equation driven by infinite-dimensional martingales with a continuous linear operator has a unique solution under the special condition that the Ft-progressively measurable generator F of the model we proposed in this paper equals zero.

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Doubbakh, Salima, Nabil Khelfallah, Mhamed Eddahbi, and Anwar Almualim. "Malliavin Regularity of Non-Markovian Quadratic BSDEs and Their Numerical Schemes." Axioms 12, no. 4 (April 10, 2023): 366. http://dx.doi.org/10.3390/axioms12040366.

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We study both Malliavin regularity and numerical approximation schemes for a class of quadratic backward stochastic differential equations (QBSDEs for short) in cases where the terminal data need not be a function of a forward diffusion. By using the connection between the QBSDE under study and some backward stochastic differential equations (BSDEs) with global Lipschitz coefficients, we firstly prove Lq, (q≥2) existence and uniqueness results for QBSDE. Secondly, the Lp-Hölder continuity of the solutions is established for (q>4 and 2≤p<q2). Then, we analyze some numerical schemes for our systems and establish their rates of convergence. Moreover, our results are illustrated with three examples.

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Bayro-Corrochano, Eduardo. "Editorial." Robotica 26, no. 4 (July 2008): 415–16. http://dx.doi.org/10.1017/s0263574708004785.

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Robotic sensing is a relatively new field of activity compared with the design and control of robot mechanisms. In both areas the role of geometry is natural and necessary for the development of devices, their control and use in challenging environments. At the very beginning odometry, tactile and touch sensors dominated robot sensing. More recently, due to the fall in the price of laser devices, they have become more attractive to the community. On the other hand, progress in photogrametry, particularly during the nineties as the n-view geometry in projective geometry matured, boot-strapped the use of computer vision as an extra powerful sensor technique for robot guidance. Cameras were used in monocular or stereoscopic fashion, catadioptric systems for ominidirectional vision, fish-eye cameras and camera networks made the use of computer vision even more diverse. Researchers started to combine sensors for 2D and 3D sensing by fusing sensor data in a projective framework. Thanks to the continuous progress in mechatronics, the low prices of fast computers and increasing accuracy of sensor systems, one can build a robot to perceive its surroundings, reconstruct, plan and ultimately act intelligently. In these perception-action systems there is of course, the urgent need for a geometric stochastic framework to deal with uncertainty in the sensing, planning and action in a robust manner. Here geometry can play a central role for the representation and computing in higher dimensions using projective geometry and differential geometry on Lie groups manifolds with a pseudo Euclidean metric. Let us review briefly the developments towards modern geometry that have been often overlooked by the robotic researchers and practitioners.

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GROTHAUS, MARTIN, and PATRIK STILGENBAUER. "GEOMETRIC LANGEVIN EQUATIONS ON SUBMANIFOLDS AND APPLICATIONS TO THE STOCHASTIC MELT-SPINNING PROCESS OF NONWOVENS AND BIOLOGY." Stochastics and Dynamics 13, no. 04 (October 7, 2013): 1350001. http://dx.doi.org/10.1142/s0219493713500019.

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In this paper we develop geometric versions of the classical Langevin equation on regular submanifolds in Euclidean space in an easy, natural way and combine them with a bunch of applications. The equations are formulated as Stratonovich stochastic differential equations on manifolds. The first version of the geometric Langevin equation has already been detected before by Lelièvre, Rousset and Stoltz with a different derivation. We propose an additional extension of the models, the geometric Langevin equations with velocity of constant Euclidean norm. The latters are seemingly new and provide a galaxy of new, beautiful and powerful mathematical models. Up to the authors best knowledge there are not many mathematical papers available dealing with geometric Langevin processes. We connect the first version of the geometric Langevin equation via proving that its generator coincides with the generalized Langevin operator proposed by Soloveitchik, Jørgensen or Kolokoltsov. All our studies are strongly motivated by industrial applications in modeling the fiber lay-down dynamics in the production process of nonwovens. We light up the geometry occurring in these models and show up the connection with the spherical velocity version of the geometric Langevin process. Moreover, as a main point, we construct new smooth industrial relevant three-dimensional fiber lay-down models involving the spherical Langevin process. Finally, relations to a class of swarming models are presented and further applications of the geometric Langevin equations are given.

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Zulfiqar, Hina, Shenglan Yuan, and Muhammad Shoaib Saleem. "Slow Manifolds for Stochastic Koper Models with Stable Lévy Noises." Axioms 12, no. 3 (March 3, 2023): 261. http://dx.doi.org/10.3390/axioms12030261.

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The Koper model is a vector field in which the differential equations describe the electrochemical oscillations appearing in diffusion processes. This work focuses on the understanding of the slow dynamics of a stochastic Koper model perturbed by stable Lévy noise. We establish the slow manifold for a stochastic Koper model with stable Lévy noise and verify exponential tracking properties. We also present two practical examples to demonstrate the analytical results with numerical simulations.

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Diop, Amadou, and Wei-Shih Du. "Existence of Mild Solutions for Multi-Term Time-Fractional Random Integro-Differential Equations with Random Carathéodory Conditions." Axioms 10, no. 4 (October 12, 2021): 252. http://dx.doi.org/10.3390/axioms10040252.

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In this paper, we investigate the existence of mild solutions to a multi-term fractional integro-differential equation with random effects. Our results are mainly relied upon stochastic analysis, Mönch’s fixed point theorem combined with a random fixed point theorem with stochastic domain, measure of noncompactness and resolvent family theory. Under the condition that the nonlinear term is of Carathéodory type and satisfies some weakly compactness condition, we establish the existence of random mild solutions. A nontrivial example illustrating our main result is also given.

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Xu, Xing, Long Chen, Liqin Sun, and Xiaodong Sun. "Dynamic Ride Height Adjusting Controller of ECAS Vehicle with Random Road Disturbances." Mathematical Problems in Engineering 2013 (2013): 1–9. http://dx.doi.org/10.1155/2013/439515.

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The ride height control system is greatly affected by the random road excitation during the ride height adjusting of the driving condition. The structure of ride height adjusting system is first analyzed, and then the mathematical model of the ride height adjusting system with the random disturbance is established as a stochastic nonlinear system. This system is decoupled using the differential geometry theory and stabilized using the Variable Structure Control (VSC) technique. The designed ride height control system converges in probability to be asymptotically stable in the sliding motion band, and the desired control law is solved to ensure the stable adjustment of the ride height system. Simulation results show that the proposed stochastic VSC method is effective for the dynamic adjusting of the ride height. Finally, the semiphysical rig test illustrates the applicability of the proposed scheme.

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Zakarya, Mohammed, Mahmoud A. Abd-Rabo, and Ghada AlNemer. "Hypercomplex Systems and Non-Gaussian Stochastic Solutions with Some Numerical Simulation of χ-Wick-Type (2 + 1)-D C-KdV Equations." Axioms 11, no. 11 (November 21, 2022): 658. http://dx.doi.org/10.3390/axioms11110658.

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In this article, we discuss the (2 + 1)-D coupled Korteweg–De Vries (KdV) equations whose coefficients are variables, and stochastic (2 + 1)-D C-KdV (C-KdV) equations with the χ-Wick-type product. White noise functional solutions (WNFS) are presented with the homogeneous equilibrium principle, Hermite transform (HT), and technicality via the F-expansion procedure. By means of the direct connection between the theory of hypercomplex systems (HCS) and white noise analysis (WNA), we establish non-Gaussian white noise (NGWN) by studying stochastic partial differential equations (PDEs) with NG-parameters. So, by using the F-expansion method we present multiples of exact and stochastic families from variable coefficients of travelling wave and stochastic NG-functional solutions of (2 + 1)-D C-KdV equations. These solutions are Jacobi elliptic functions (JEF), trigonometric, and hyperbolic forms, respectively.

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Mahrouf, Marouane, Adnane Boukhouima, Houssine Zine, El Mehdi Lotfi, Delfim F. M. Torres, and Noura Yousfi. "Modeling and Forecasting of COVID-19 Spreading by Delayed Stochastic Differential Equations." Axioms 10, no. 1 (February 7, 2021): 18. http://dx.doi.org/10.3390/axioms10010018.

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The novel coronavirus disease (COVID-19) pneumonia has posed a great threat to the world recent months by causing many deaths and enormous economic damage worldwide. The first case of COVID-19 in Morocco was reported on 2 March 2020, and the number of reported cases has increased day by day. In this work, we extend the well-known SIR compartmental model to deterministic and stochastic time-delayed models in order to predict the epidemiological trend of COVID-19 in Morocco and to assess the potential role of multiple preventive measures and strategies imposed by Moroccan authorities. The main features of the work include the well-posedness of the models and conditions under which the COVID-19 may become extinct or persist in the population. Parameter values have been estimated from real data and numerical simulations are presented for forecasting the COVID-19 spreading as well as verification of theoretical results.

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Osada, Hirofumi. "Stochastic geometry and dynamics of infinitely many particle systems—random matrices and interacting Brownian motions in infinite dimensions." Sugaku Expositions 34, no. 2 (October 12, 2021): 141–73. http://dx.doi.org/10.1090/suga/461.

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We explain the general theories involved in solving an infinite-dimensional stochastic differential equation (ISDE) for interacting Brownian motions in infinite dimensions related to random matrices. Typical examples are the stochastic dynamics of infinite particle systems with logarithmic interaction potentials such as the sine, Airy, Bessel, and also for the Ginibre interacting Brownian motions. The first three are infinite-dimensional stochastic dynamics in one-dimensional space related to random matrices called Gaussian ensembles. They are the stationary distributions of interacting Brownian motions and given by the limit point processes of the distributions of eigenvalues of these random matrices. The sine, Airy, and Bessel point processes and interacting Brownian motions are thought to be geometrically and dynamically universal as the limits of bulk, soft edge, and hard edge scaling. The Ginibre point process is a rotation- and translation-invariant point process on R 2 \mathbb {R}^2 , and an equilibrium state of the Ginibre interacting Brownian motions. It is the bulk limit of the distributions of eigenvalues of non-Hermitian Gaussian random matrices. When the interacting Brownian motions constitute a one-dimensional system interacting with each other through the logarithmic potential with inverse temperature β = 2 \beta = 2 , an algebraic construction is known in which the stochastic dynamics are defined by the space-time correlation function. The approach based on the stochastic analysis (called the analytic approach) can be applied to an extremely wide class. If we apply the analytic approach to this system, we see that these two constructions give the same stochastic dynamics. From the algebraic construction, despite being an infinite interacting particle system, it is possible to represent and calculate various quantities such as moments by the correlation functions. We can thus obtain quantitative information. From the analytic construction, it is possible to represent the dynamics as a solution of an ISDE. We can obtain qualitative information such as semi-martingale properties, continuity, and non-collision properties of each particle, and the strong Markov property of the infinite particle system as a whole. Ginibre interacting Brownian motions constitute a two-dimensional infinite particle system related to non-Hermitian Gaussian random matrices. It has a logarithmic interaction potential with β = 2 \beta = 2 , but no algebraic configurations are known.The present result is the only construction.

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Alnafisah, Yousef, and Moustafa El-Shahed. "Deterministic and Stochastic Prey–Predator Model for Three Predators and a Single Prey." Axioms 11, no. 4 (March 28, 2022): 156. http://dx.doi.org/10.3390/axioms11040156.

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In this paper, a deterministic prey–predator model is proposed and analyzed. The interaction between three predators and a single prey was investigated. The impact of harvesting on the three predators was studied, and we concluded that the dynamics of the population can be controlled by harvesting. Some sufficient conditions were obtained to ensure the local and global stability of equilibrium points. The transcritical bifurcation was investigated using Sotomayor’s theorem. We performed a stochastic extension of the deterministic model to study the fluctuation environmental factors. The existence of a unique global positive solution for the stochastic model was investigated. The exponential–mean–squared stability of the resulting stochastic differential equation model was examined, and it was found to be dependent on the harvesting effort. Theoretical results are illustrated using numerical simulations.

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Toscano, Gregorio, Ricardo Landa, Giomara Lárraga, and Guillermo Leguizamón. "On the use of stochastic ranking for parent selection in differential evolution for constrained optimization." Soft Computing 21, no. 16 (February 18, 2016): 4617–33. http://dx.doi.org/10.1007/s00500-016-2073-6.

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Bakare, Emmanuel A., Snehashish Chakraverty, and Radovan Potucek. "Numerical Solution of an Interval-Based Uncertain SIR (Susceptible–Infected–Recovered) Epidemic Model by Homotopy Analysis Method." Axioms 10, no. 2 (June 6, 2021): 114. http://dx.doi.org/10.3390/axioms10020114.

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This work proposes an interval-based uncertain Susceptible–Infected–Recovered (SIR) epidemic model. The interval model has been numerically solved by the homotopy analysis method (HAM). The SIR epidemic model is proposed and solved under different uncertain intervals by the HAM to obtain the numerical solution of the model. Furthermore, the SIR ODE model was transformed into a stochastic differential equation (SDE) model and the results of the stochastic and deterministic models were compared using numerical simulations. The results obtained were compared with the numerical solution and found to be in good agreement. Finally, various simulations were done to discuss the solution.

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Gao, Yuan, Yong Xia, Xu Wang, and Wu Zhang. "Control System Design of Missile Based on Back-Stepping and Variable Structure Control Method." Applied Mechanics and Materials 644-650 (September 2014): 516–22. http://dx.doi.org/10.4028/www.scientific.net/amm.644-650.516.

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A nonlinear decoupling control method based on back-stepping design and sliding variable structure control theory was proposed in this paper according to the nonlinear and strong coupling characteristics of missile at hingh angle of attack. The affine nonlinear model of missile was firstly established, the exact linearization and input/output decoupling of system were then realized using differential geometry theory. On the basis of this, the robust controller was designed using back-stepping design and variable structure control method. Simulation results indicate that the influences caused by missile parameter uncertainties, unsuited uncertainties with known upper boundaries and stochastic interference can be effectively restrained.

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Freiling, G., and A. Hochhaus. "On a class of rational matrix differential equations arising in stochastic control." Linear Algebra and its Applications 379 (March 2004): 43–68. http://dx.doi.org/10.1016/s0024-3795(02)00651-1.

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Fragoso, Marcelo D., and Jack Baczynski. "On an infinite dimensional perturbed Riccati differential equation arising in stochastic control." Linear Algebra and its Applications 406 (September 2005): 165–76. http://dx.doi.org/10.1016/j.laa.2005.04.003.

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Maqbol, Sahar M. A., R. S. Jain, and B. S. Reddy. "On stability of nonlocal neutral stochastic integro differential equations with random impulses and Poisson jumps." Cubo (Temuco) 25, no. 2 (August 4, 2023): 211–29. http://dx.doi.org/10.56754/0719-0646.2502.211.

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This article aims to examine the existence and Hyers-Ulam stability of non-local random impulsive neutral stochastic integrodifferential delayed equations with Poisson jumps. Initially, we prove the existence of mild solutions to the equations by using the Banach fixed point theorem. Then, we investigate stability via the continuous dependence of solutions on the initial value. Next, we study the Hyers-Ulam stability results under the Lipschitz condition on a bounded and closed interval. Finally, we give an illustrative example of our main result.

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Sabir, Zulqurnain, Tareq Saeed, Juan L. G. Guirao, Juan M. Sánchez, and Adrián Valverde. "A Swarming Meyer Wavelet Computing Approach to Solve the Transport System of Goods." Axioms 12, no. 5 (May 8, 2023): 456. http://dx.doi.org/10.3390/axioms12050456.

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The motive of this work is to provide the numerical performances of the reactive transport model that carries trucks with goods on roads by exploiting the stochastic procedures based on the Meyer wavelet (MW) neural network. An objective function is constructed by using the differential model and its boundary conditions. The optimization of the objective function is performed through the hybridization of the global and local search procedures, i.e., swarming and interior point algorithms. Three different cases of the model have been obtained, and the exactness of the stochastic procedure is observed by using the comparison of the obtained and Adams solutions. The negligible absolute error enhances the exactness of the proposed MW neural networks along with the hybridization of the global and local search schemes. Moreover, statistical interpretations based on different operators, histograms, and boxplots are provided to validate the constancy of the designed stochastic structure.

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Bandyopadhyay, Abhirup, and Samarjit Kar. "On fuzzy type-1 and type-2 stochastic ordinary and partial differential equations and numerical solution." Soft Computing 23, no. 11 (February 12, 2018): 3803–21. http://dx.doi.org/10.1007/s00500-018-3043-y.

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Berkolaiko, Gregory, Evelyn Buckwar, Cónall Kelly, and Alexandra Rodkina. "Almost sure asymptotic stability analysis of the θ-Maruyama method applied to a test system with stabilising and destabilising stochastic perturbations." LMS Journal of Computation and Mathematics 15 (April 1, 2012): 71–83. http://dx.doi.org/10.1112/s1461157012000010.

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AbstractWe perform an almost sure linear stability analysis of the θ-Maruyama method, selecting as our test equation a two-dimensional system of Itô differential equations with diagonal drift coefficient and two independent stochastic perturbations which capture the stabilising and destabilising roles of feedback geometry in the almost sure asymptotic stability of the equilibrium solution. For small values of the constant step-size parameter, we derive close-to-sharp conditions for the almost sure asymptotic stability and instability of the equilibrium solution of the discretisation that match those of the original test system. Our investigation demonstrates the use of a discrete form of the Itô formula in the context of an almost sure linear stability analysis.

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Jackson, Christopher S., and Carlton M. Caves. "Simultaneous Measurements of Noncommuting Observables: Positive Transformations and Instrumental Lie Groups." Entropy 25, no. 9 (August 23, 2023): 1254. http://dx.doi.org/10.3390/e25091254.

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We formulate a general program for describing and analyzing continuous, differential weak, simultaneous measurements of noncommuting observables, which focuses on describing the measuring instrument autonomously, without states. The Kraus operators of such measuring processes are time-ordered products of fundamental differential positive transformations, which generate nonunitary transformation groups that we call instrumental Lie groups. The temporal evolution of the instrument is equivalent to the diffusion of a Kraus-operator distribution function, defined relative to the invariant measure of the instrumental Lie group. This diffusion can be analyzed using Wiener path integration, stochastic differential equations, or a Fokker-Planck-Kolmogorov equation. This way of considering instrument evolution we call the Instrument Manifold Program. We relate the Instrument Manifold Program to state-based stochastic master equations. We then explain how the Instrument Manifold Program can be used to describe instrument evolution in terms of a universal cover that we call the universal instrumental Lie group, which is independent not just of states, but also of Hilbert space. The universal instrument is generically infinite dimensional, in which case the instrument’s evolution is chaotic. Special simultaneous measurements have a finite-dimensional universal instrument, in which case the instrument is considered principal, and it can be analyzed within the differential geometry of the universal instrumental Lie group. Principal instruments belong at the foundation of quantum mechanics. We consider the three most fundamental examples: measurement of a single observable, position and momentum, and the three components of angular momentum. As these measurements are performed continuously, they converge to strong simultaneous measurements. For a single observable, this results in the standard decay of coherence between inequivalent irreducible representations. For the latter two cases, it leads to a collapse within each irreducible representation onto the classical or spherical phase space, with the phase space located at the boundary of these instrumental Lie groups.

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Williams, P. D., T. W. N. Haine, P. L. Read, S. R. Lewis, and Y. H. Yamazaki. "QUAGMIRE v1.3: a quasi-geostrophic model for investigating rotating fluids experiments." Geoscientific Model Development Discussions 1, no. 1 (September 5, 2008): 187–241. http://dx.doi.org/10.5194/gmdd-1-187-2008.

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Abstract. QUAGMIRE is a quasi-geostrophic numerical model for performing fast, high-resolution simulations of multi-layer rotating annulus laboratory experiments on a desktop personal computer. The model uses a hybrid finite-difference/spectral approach to numerically integrate the coupled nonlinear partial differential equations of motion in cylindrical geometry in each layer. Version 1.3 implements the special case of two fluid layers of equal resting depths. The flow is forced either by a differentially rotating lid, or by relaxation to specified streamfunction or potential vorticity fields, or both. Dissipation is achieved through Ekman layer pumping and suction at the horizontal boundaries, including the internal interface. The effects of weak interfacial tension are included, as well as the linear topographic beta-effect and the quadratic centripetal beta-effect. Stochastic forcing may optionally be activated, to represent approximately the effects of random unresolved features. A leapfrog time stepping scheme is used, with a Robert filter. Flows simulated by the model agree well with those observed in the corresponding laboratory experiments.

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Williams, P. D., T. W. N. Haine, P. L. Read, S. R. Lewis, and Y. H. Yamazaki. "QUAGMIRE v1.3: a quasi-geostrophic model for investigating rotating fluids experiments." Geoscientific Model Development 2, no. 1 (February 17, 2009): 13–32. http://dx.doi.org/10.5194/gmd-2-13-2009.

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Abstract. QUAGMIRE is a quasi-geostrophic numerical model for performing fast, high-resolution simulations of multi-layer rotating annulus laboratory experiments on a desktop personal computer. The model uses a hybrid finite-difference/spectral approach to numerically integrate the coupled nonlinear partial differential equations of motion in cylindrical geometry in each layer. Version 1.3 implements the special case of two fluid layers of equal resting depths. The flow is forced either by a differentially rotating lid, or by relaxation to specified streamfunction or potential vorticity fields, or both. Dissipation is achieved through Ekman layer pumping and suction at the horizontal boundaries, including the internal interface. The effects of weak interfacial tension are included, as well as the linear topographic beta-effect and the quadratic centripetal beta-effect. Stochastic forcing may optionally be activated, to represent approximately the effects of random unresolved features. A leapfrog time stepping scheme is used, with a Robert filter. Flows simulated by the model agree well with those observed in the corresponding laboratory experiments.

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Goda, Katsuichiro, and Parva Shoaeifar. "Prospective Fault Displacement Hazard Assessment for Leech River Valley Fault Using Stochastic Source Modeling and Okada Fault Displacement Equations." GeoHazards 3, no. 2 (May 21, 2022): 277–93. http://dx.doi.org/10.3390/geohazards3020015.

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In this study, an alternative method for conducting probabilistic fault displacement hazard analysis is developed based on stochastic source modeling and analytical formulae for evaluating the elastic dislocation due to an earthquake rupture. It characterizes the uncertainty of fault-rupture occurrence in terms of its position, geometry, and slip distribution and adopts so-called Okada equations for the calculation of fault displacement on the ground surface. The method is compatible with fault-source-based probabilistic seismic hazard analysis and can be implemented via Monte Carlo simulations. The new method is useful for evaluating the differential displacements caused by the fault rupture at multiple locations simultaneously. The proposed method is applied to the Leech River Valley Fault located in the vicinity of Victoria, British Columbia, Canada. Site-specific fault displacement and differential fault displacement hazard curves are assessed for multiple sites within the fault-rupture zone. The hazard results indicate that relatively large displacements (∼0.5 m vertical uplift) can be expected at low probability levels of 10−4. For critical infrastructures, such as bridges and pipelines, quantifying the uncertainty of fault displacement hazard is essential to manage potential damage and loss effectively.

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Arif, Muhammad Shoaib, Kamaleldin Abodayeh, and Yasir Nawaz. "A Reliable Computational Scheme for Stochastic Reaction–Diffusion Nonlinear Chemical Model." Axioms 12, no. 5 (May 9, 2023): 460. http://dx.doi.org/10.3390/axioms12050460.

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The main aim of this contribution is to construct a numerical scheme for solving stochastic time-dependent partial differential equations (PDEs). This has the advantage of solving problems with positive solutions. The scheme provides conditions for obtaining positive solutions, which the existing Euler–Maruyama method cannot do. In addition, it is more accurate than the existing stochastic non-standard finite difference (NSFD) method. Theoretically, the suggested scheme is more accurate than the current NSFD method, and its stability and consistency analysis are also shown. The scheme is applied to the linear scalar stochastic time-dependent parabolic equation and the nonlinear auto-catalytic Brusselator model. The deficiency of the NSFD in terms of accuracy is also shown by providing different graphs. Many observable occurrences in the physical world can be traced back to certain chemical concentrations. Examining and understanding the inter-diffusion between chemical concentrations is important, especially when they coincide. The Brusselator model is the gold standard for describing the relationship between chemical concentrations and other variables in chemical systems. A computational code for the proposed model scheme may be made available to readers upon request for convenience.

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DURIN, GIANFRANCO, GIORGIO BERTOTTI, and ALESSANDRO MAGNI. "FRACTALS, SCALING AND THE QUESTION OF SELF-ORGANIZED CRITICALITY IN MAGNETIZATION PROCESSES." Fractals 03, no. 02 (June 1995): 351–70. http://dx.doi.org/10.1142/s0218348x95000278.

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The main physical aspects and the theoretical description of stochastic domain wall dynamics in soft magnetic materials are reviewed. The intrinsically random nature of domain wall motion results in the Barkhausen effect, which exibits scaling properties at low magnetization rates and 1/f power spectra. It is shown that the Barkhausen signal ν, as well as the size Δx and the duration Δu of jumps follow distributions of the form ν−α, Δx−β, Δu−γ, with α=1−c, β=3/2−c/2, γ=2–c, where c is a dimensionless parameter proportional to the applied field rate. These results are analytically calculated by means of a stochastic differential equation for the domain wall dynamics in a random perturbed medium with brownian properties and then compared to experiments. The Barkhausen signal is found to be related to a random Cantor dust with fractal dimension D=1−c, from which the scaling exponents are calculated using simple properties of fractal geometry. Fractal dimension Δ of the signal v is also studied using four different methods of calculation, giving Δ≈1.5, independent of the method used and of the parameter c. The stochastic model is analyzed in detail in order to clarify if the shown properties can be interpreted as manifestations of self-organized criticality in magnetic systems.

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Journal articles on the topic 'Stochastic Differential Geometry'

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