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Seth S, Bera A, Pakrashi V. Exploring chaos and ergodic behavior of an inductorless circuit driven by stochastic parameters. NONLINEAR DYNAMICS 2024; 112:19441-19462. [PMID: 39219722 PMCID: PMC11362311 DOI: 10.1007/s11071-024-10050-x] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 10/03/2023] [Accepted: 07/15/2024] [Indexed: 09/04/2024]
Abstract
There exist extensive studies on periodic and random perturbations of various smooth maps investigating their dynamics. Unlike smooth maps, non-smooth maps are yet to be studied extensively under a stochastic regime. This paper presents a stochastic piecewise-smooth map derived from a simple inductorless switching circuit. The stochasticity is introduced in parameter values. The distribution of the parameter values is bounded and randomly selected from uniform and triangular distributions and ranges between high and low bifurcation parameter values of the deterministic map. Due to this inherent stochasticity in parameter values, the time evolution of the state variable cannot be predicted at a specific time instant. We observe that the state variable exhibits completely ergodic behavior when the minimum value of the parameter is the same as the minimum bifurcation parameter of the deterministic system. However, the ensemble average of the state variable converges to a fixed value. The system demonstrates nonchaotic behavior for a particular range of parameter values but the deterministic map in that bifurcation range shows interplay between chaos and periodic orbits. The values of Lyapunov exponents decrease monotonically with increased asymmetry of the distribution from which the bifurcation parameter values are chosen. We determine the probability density function of the stochastic map and verify its invariance under initial conditions. The most noteworthy result is the disappearance of chaotic behavior when the lower range of the distribution is varied while maintaining a fixed upper threshold for a particular distribution, even though the deterministic map exhibits an array of periodic and chaotic behaviors within the range. As the period-incrementing cascade with chaotic inclusion only occurs in nonsmooth maps, this paper numerically shows the stochasticity of a piecewise-smooth map obtained from a practical system for the first time where randomness is introduced in the parameter space.
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Affiliation(s)
- Soumyajit Seth
- UCD Centre for Mechanics, Dynamical Systems and Risk Laboratory, School of Mechanical and Materials Engineering, University College Dublin, Dublin, Ireland
- School of Technology Management and Engineering, NMIMS Hyderabad, Jadcherla Campus, Raipalle, 509301 India
| | - Abhijit Bera
- Department of Physical Sciences, Indian Institute of Science Education and Research Kolkata, Mohanpur Campus, Kalyani, IN 741246 India
| | - Vikram Pakrashi
- UCD Centre for Mechanics, Dynamical Systems and Risk Laboratory, School of Mechanical and Materials Engineering, University College Dublin, Dublin, Ireland
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2
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Schöll E. Delayed feedback control of synchronization patterns: Comment on "Control of movement of underwater swimmers: Animals, simulated animates and swimming robots" by S.Yu. Gordleeva et al. Phys Life Rev 2024; 49:112-114. [PMID: 38574585 DOI: 10.1016/j.plrev.2024.03.010] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/20/2024] [Accepted: 03/26/2024] [Indexed: 04/06/2024]
Affiliation(s)
- Eckehard Schöll
- Institute for Theoretical Physics, Technical University of Berlin, Germany; Bernstein Center for Computational Neuroscience Berlin, Germany; Potsdam Institute for Climate Impact Research, Germany.
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Konishi K, Yoshida K, Sugitani Y, Hara N. Delay-induced amplitude death in multiplex oscillator network with frequency-mismatched layers. Phys Rev E 2024; 109:014220. [PMID: 38366515 DOI: 10.1103/physreve.109.014220] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/02/2023] [Accepted: 12/01/2023] [Indexed: 02/18/2024]
Abstract
The present paper analytically investigates the stability of amplitude death in a multiplex Stuart-Landau oscillator network with a delayed interlayer connection. The network consists of two frequency-mismatched layers, and all oscillators in each layer have identical frequencies. We show that, if the matrices describing the network topologies of each layer commute, then the characteristic equation governing the stability can be reduced to a simple form. This form reveals that the stability of amplitude death in the multiplex network is equally or more conservative than that in a pair of frequency-mismatched oscillators coupled by a delayed connection. In addition, we provide a procedure for designing the delayed interlayer connection such that amplitude death is stable for any commuting matrices and for any intralayer coupling strength. These analytical results are verified through numerical examples. Moreover, we numerically discuss the results for the case in which the commutative property does not hold.
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Affiliation(s)
- Keiji Konishi
- Department of Electrical and Electronic Systems Engineering, Osaka Metropolitan University, 1-1 Gakuen-cho, Naka-ku, Sakai, Osaka 599-8531, Japan
| | - Koki Yoshida
- National Institute of Technology, Toyama College, 13 Hongo-machi, Toyama city, Toyama 939-8630, Japan
| | - Yoshiki Sugitani
- Department of Electrical and Electronic Systems Engineering, Osaka Metropolitan University, 1-1 Gakuen-cho, Naka-ku, Sakai, Osaka 599-8531, Japan
| | - Naoyuki Hara
- Department of Electrical and Electronic Systems Engineering, Osaka Metropolitan University, 1-1 Gakuen-cho, Naka-ku, Sakai, Osaka 599-8531, Japan
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Sathiyadevi K, Chandrasekar VK, Lakshmanan M. Emerging chimera states under nonidentical counter-rotating oscillators. Phys Rev E 2022; 105:034211. [PMID: 35428132 DOI: 10.1103/physreve.105.034211] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/05/2021] [Accepted: 03/04/2022] [Indexed: 06/14/2023]
Abstract
Frequency plays a crucial role in exhibiting various collective dynamics in the coexisting corotating and counter-rotating systems. To illustrate the impact of counter-rotating frequencies, we consider a network of nonidentical and globally coupled Stuart-Landau oscillators with additional perturbation. Primarily, we investigate the dynamical transitions in the absence of perturbation, demonstrating that the transition from desynchronized state to cluster oscillatory state occurs through an interesting partial synchronization state in the oscillatory regime. Following this, the system dynamics transits to amplitude death and oscillation death states. Importantly, we find that the observed dynamical states do not preserve the parity (P) symmetry in the absence of perturbation. When the perturbation is increased one can note that the system dynamics exhibits a kind of transition which corresponds to a change from incoherent mixed synchronization to coherent mixed synchronization through a chimera state. In particular, incoherent mixed synchronization and coherent mixed synchronization states completely preserve the P symmetry, whereas the chimera state preserves the P symmetry only partially. To demonstrate the occurrence of such partial symmetry-breaking (chimera) state, we use basin stability analysis and discover that partial symmetry breaking exists as a result of the coexistence of symmetry-preserving and symmetry-breaking behavior in the initial state space. Further, a measure of the strength of P symmetry is established to quantify the P symmetry in the observed dynamical states. Subsequently, the dynamical transitions are investigated in the parametric spaces. Finally, by increasing the network size, the robustness of the chimera state is also inspected, and we find that the chimera state is robust even in networks of larger sizes. We also show the generality of the above results in the related reduced phase. model as well as in other coupled models such as the globally coupled van der Pol and Rössler oscillators.
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Affiliation(s)
- K Sathiyadevi
- Department of Nonlinear Dynamics, School of Physics, Bharathidasan University, Tiruchirappalli 620 024, Tamil Nadu, India
| | - V K Chandrasekar
- Department of Physics, Centre for Nonlinear Science & Engineering, School of Electrical & Electronics Engineering, SASTRA Deemed University, Thanjavur 613 401, Tamil Nadu, India
| | - M Lakshmanan
- Department of Nonlinear Dynamics, School of Physics, Bharathidasan University, Tiruchirappalli 620 024, Tamil Nadu, India
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Sawicki J, Berner R, Löser T, Schöll E. Modeling Tumor Disease and Sepsis by Networks of Adaptively Coupled Phase Oscillators. FRONTIERS IN NETWORK PHYSIOLOGY 2022; 1:730385. [PMID: 36925568 PMCID: PMC10013027 DOI: 10.3389/fnetp.2021.730385] [Citation(s) in RCA: 6] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/24/2021] [Accepted: 11/19/2021] [Indexed: 06/18/2023]
Abstract
In this study, we provide a dynamical systems perspective to the modelling of pathological states induced by tumors or infection. A unified disease model is established using the innate immune system as the reference point. We propose a two-layer network model for carcinogenesis and sepsis based upon the interaction of parenchymal cells and immune cells via cytokines, and the co-evolutionary dynamics of parenchymal, immune cells, and cytokines. Our aim is to show that the complex cellular cooperation between parenchyma and stroma (immune layer) in the physiological and pathological case can be qualitatively and functionally described by a simple paradigmatic model of phase oscillators. By this, we explain carcinogenesis, tumor progression, and sepsis by destabilization of the healthy homeostatic state (frequency synchronized), and emergence of a pathological state (desynchronized or multifrequency cluster). The coupled dynamics of parenchymal cells (metabolism) and nonspecific immune cells (reaction of innate immune system) are represented by nodes of a duplex layer. The cytokine interaction is modeled by adaptive coupling weights between the nodes representing the immune cells (with fast adaptation time scale) and the parenchymal cells (slow adaptation time scale) and between the pairs of parenchymal and immune cells in the duplex network (fixed bidirectional coupling). Thereby, carcinogenesis, organ dysfunction in sepsis, and recurrence risk can be described in a correct functional context.
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Affiliation(s)
- Jakub Sawicki
- Potsdam Institute for Climate Impact Research, Potsdam, Germany
| | - Rico Berner
- Institut für Mathematik, Technische Universität Berlin, Berlin, Germany
- Institut für Theoretische Physik, Technische Universität Berlin, Berlin, Germany
| | | | - Eckehard Schöll
- Potsdam Institute for Climate Impact Research, Potsdam, Germany
- Institut für Theoretische Physik, Technische Universität Berlin, Berlin, Germany
- Bernstein Center for Computational Neuroscience Berlin, Humboldt-Universität, Berlin, Germany
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Chouzouris T, Roth N, Cakan C, Obermayer K. Applications of optimal nonlinear control to a whole-brain network of FitzHugh-Nagumo oscillators. Phys Rev E 2021; 104:024213. [PMID: 34525550 DOI: 10.1103/physreve.104.024213] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/05/2021] [Accepted: 07/26/2021] [Indexed: 01/21/2023]
Abstract
We apply the framework of optimal nonlinear control to steer the dynamics of a whole-brain network of FitzHugh-Nagumo oscillators. Its nodes correspond to the cortical areas of an atlas-based segmentation of the human cerebral cortex, and the internode coupling strengths are derived from diffusion tensor imaging data of the connectome of the human brain. Nodes are coupled using an additive scheme without delays and are driven by background inputs with fixed mean and additive Gaussian noise. Optimal control inputs to nodes are determined by minimizing a cost functional that penalizes the deviations from a desired network dynamic, the control energy, and spatially nonsparse control inputs. Using the strength of the background input and the overall coupling strength as order parameters, the network's state-space decomposes into regions of low- and high-activity fixed points separated by a high-amplitude limit cycle, all of which qualitatively correspond to the states of an isolated network node. Along the borders, however, additional limit cycles, asynchronous states, and multistability can be observed. Optimal control is applied to several state-switching and network synchronization tasks, and the results are compared to controllability measures from linear control theory for the same connectome. We find that intuitions from the latter about the roles of nodes in steering the network dynamics, which are solely based on connectome features, do not generally carry over to nonlinear systems, as had been previously implied. Instead, the role of nodes under optimal nonlinear control critically depends on the specified task and the system's location in state space. Our results shed new light on the controllability of brain network states and may serve as an inspiration for the design of new paradigms for noninvasive brain stimulation.
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Affiliation(s)
- Teresa Chouzouris
- Institut für Softwaretechnik und Theoretische Informatik, Technische Universität Berlin, Marchstraße 23, 10587 Berlin, Germany
| | - Nicolas Roth
- Institut für Softwaretechnik und Theoretische Informatik, Technische Universität Berlin, Marchstraße 23, 10587 Berlin, Germany
| | - Caglar Cakan
- Institut für Softwaretechnik und Theoretische Informatik, Technische Universität Berlin, Marchstraße 23, 10587 Berlin, Germany.,Bernstein Center for Computational Neuroscience Berlin, Philippstraße 13, 10115 Berlin, Germany
| | - Klaus Obermayer
- Institut für Softwaretechnik und Theoretische Informatik, Technische Universität Berlin, Marchstraße 23, 10587 Berlin, Germany.,Bernstein Center for Computational Neuroscience Berlin, Philippstraße 13, 10115 Berlin, Germany
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Mao X, Li X, Ding W, Wang S, Zhou X, Qiao L. Dynamics of a multiplex neural network with delayed couplings. APPLIED MATHEMATICS AND MECHANICS 2021; 42:441-456. [PMID: 33551533 PMCID: PMC7852048 DOI: 10.1007/s10483-021-2709-6] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/06/2020] [Revised: 12/16/2020] [Indexed: 06/12/2023]
Abstract
Multiplex networks have drawn much attention since they have been observed in many systems, e.g., brain, transport, and social relationships. In this paper, the nonlinear dynamics of a multiplex network with three neural groups and delayed interactions is studied. The stability and bifurcation of the network equilibrium are discussed, and interesting neural activities of the network are explored. Based on the neuron circuit, transfer function circuit, and time delay circuit, a circuit platform of the network is constructed. It is shown that delayed couplings play crucial roles in the network dynamics, e.g., the enhancement and suppression of the stability, the patterns of the synchronization between networks, and the generation of complicated attractors and multi-stability coexistence.
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Affiliation(s)
- Xiaochen Mao
- Department of Engineering Mechanics, College of Mechanics and Materials, Hohai University, Nanjing, 211100 China
| | - Xingyong Li
- Department of Engineering Mechanics, College of Mechanics and Materials, Hohai University, Nanjing, 211100 China
| | - Weijie Ding
- Department of Engineering Mechanics, College of Mechanics and Materials, Hohai University, Nanjing, 211100 China
| | - Song Wang
- Department of Engineering Mechanics, College of Mechanics and Materials, Hohai University, Nanjing, 211100 China
| | - Xiangyu Zhou
- Department of Engineering Mechanics, College of Mechanics and Materials, Hohai University, Nanjing, 211100 China
| | - Lei Qiao
- Department of Engineering Mechanics, College of Mechanics and Materials, Hohai University, Nanjing, 211100 China
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Gerster M, Berner R, Sawicki J, Zakharova A, Škoch A, Hlinka J, Lehnertz K, Schöll E. FitzHugh-Nagumo oscillators on complex networks mimic epileptic-seizure-related synchronization phenomena. CHAOS (WOODBURY, N.Y.) 2020; 30:123130. [PMID: 33380049 DOI: 10.1063/5.0021420] [Citation(s) in RCA: 39] [Impact Index Per Article: 9.8] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/10/2020] [Accepted: 11/23/2020] [Indexed: 06/12/2023]
Abstract
We study patterns of partial synchronization in a network of FitzHugh-Nagumo oscillators with empirical structural connectivity measured in human subjects. We report the spontaneous occurrence of synchronization phenomena that closely resemble the ones seen during epileptic seizures in humans. In order to obtain deeper insights into the interplay between dynamics and network topology, we perform long-term simulations of oscillatory dynamics on different paradigmatic network structures: random networks, regular nonlocally coupled ring networks, ring networks with fractal connectivities, and small-world networks with various rewiring probability. Among these networks, a small-world network with intermediate rewiring probability best mimics the findings achieved with the simulations using the empirical structural connectivity. For the other network topologies, either no spontaneously occurring epileptic-seizure-related synchronization phenomena can be observed in the simulated dynamics, or the overall degree of synchronization remains high throughout the simulation. This indicates that a topology with some balance between regularity and randomness favors the self-initiation and self-termination of episodes of seizure-like strong synchronization.
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Affiliation(s)
- Moritz Gerster
- Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstr. 36, 10623 Berlin, Germany
| | - Rico Berner
- Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstr. 36, 10623 Berlin, Germany
| | - Jakub Sawicki
- Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstr. 36, 10623 Berlin, Germany
| | - Anna Zakharova
- Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstr. 36, 10623 Berlin, Germany
| | - Antonín Škoch
- National Institute of Mental Health, Topolová 748, 250 67 Klecany, Czech Republic
| | - Jaroslav Hlinka
- National Institute of Mental Health, Topolová 748, 250 67 Klecany, Czech Republic
| | - Klaus Lehnertz
- Department of Epileptology, University of Bonn Medical Centre, Venusberg Campus 1, 53127 Bonn, Germany
| | - Eckehard Schöll
- Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstr. 36, 10623 Berlin, Germany
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Ruzzene G, Omelchenko I, Sawicki J, Zakharova A, Schöll E, Andrzejak RG. Remote pacemaker control of chimera states in multilayer networks of neurons. Phys Rev E 2020; 102:052216. [PMID: 33327161 DOI: 10.1103/physreve.102.052216] [Citation(s) in RCA: 12] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/16/2020] [Accepted: 10/30/2020] [Indexed: 06/12/2023]
Abstract
Networks of coupled nonlinear oscillators allow for the formation of nontrivial partially synchronized spatiotemporal patterns, such as chimera states, in which there are coexisting coherent (synchronized) and incoherent (desynchronized) domains. These complementary domains form spontaneously, and it is impossible to predict where the synchronized group will be positioned within the network. Therefore, possible ways to control the spatial position of the coherent and incoherent groups forming the chimera states are of high current interest. In this work we investigate how to control chimera patterns in multiplex networks of FitzHugh-Nagumo neurons, and in particular we want to prove that it is possible to remotely control chimera states exploiting the multiplex structure. We introduce a pacemaker oscillator within the network: this is an oscillator that does not receive input from the rest of the network but is sending out information to its neighbors. The pacemakers can be positioned in one or both layers. Their presence breaks the spatial symmetry of the layer in which they are introduced and allows us to control the position of the incoherent domain. We demonstrate how the remote control is possible for both uni- and bidirectional coupling between the layers. Furthermore we show which are the limitations of our control mechanisms when it is generalized from single-layer to multilayer networks.
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Affiliation(s)
- Giulia Ruzzene
- Department of Information and Communication Technologies, Universitat Pompeu Fabra, Carrer Roc Boronat 138, 08018 Barcelona, Catalonia, Spain
| | - Iryna Omelchenko
- Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstrasse 36, 10623 Berlin, Germany
| | - Jakub Sawicki
- Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstrasse 36, 10623 Berlin, Germany
| | - Anna Zakharova
- Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstrasse 36, 10623 Berlin, Germany
| | - Eckehard Schöll
- Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstrasse 36, 10623 Berlin, Germany
| | - Ralph G Andrzejak
- Department of Information and Communication Technologies, Universitat Pompeu Fabra, Carrer Roc Boronat 138, 08018 Barcelona, Catalonia, Spain
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Berner R, Sawicki J, Schöll E. Birth and Stabilization of Phase Clusters by Multiplexing of Adaptive Networks. PHYSICAL REVIEW LETTERS 2020; 124:088301. [PMID: 32167358 DOI: 10.1103/physrevlett.124.088301] [Citation(s) in RCA: 35] [Impact Index Per Article: 8.8] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/27/2019] [Revised: 12/05/2019] [Accepted: 01/16/2020] [Indexed: 06/10/2023]
Abstract
We propose a concept to generate and stabilize diverse partial synchronization patterns (phase clusters) in adaptive networks which are widespread in neuroscience and social sciences, as well as biology, engineering, and other disciplines. We show by theoretical analysis and computer simulations that multiplexing in a multilayer network with symmetry can induce various stable phase cluster states in a situation where they are not stable or do not even exist in the single layer. Further, we develop a method for the analysis of Laplacian matrices of multiplex networks which allows for insight into the spectral structure of these networks enabling a reduction to the stability problem of single layers. We employ the multiplex decomposition to provide analytic results for the stability of the multilayer patterns. As local dynamics we use the paradigmatic Kuramoto phase oscillator, which is a simple generic model and has been successfully applied in the modeling of synchronization phenomena in a wide range of natural and technological systems.
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Affiliation(s)
- Rico Berner
- Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstrasse 36, 10623 Berlin, Germany
- Institut für Mathematik, Technische Universität Berlin, Hardenbergstrasse 36, 10623 Berlin, Germany
| | - Jakub Sawicki
- Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstrasse 36, 10623 Berlin, Germany
| | - Eckehard Schöll
- Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstrasse 36, 10623 Berlin, Germany
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Otto A, Just W, Radons G. Nonlinear dynamics of delay systems: an overview. PHILOSOPHICAL TRANSACTIONS. SERIES A, MATHEMATICAL, PHYSICAL, AND ENGINEERING SCIENCES 2019; 377:20180389. [PMID: 31329061 PMCID: PMC6661329 DOI: 10.1098/rsta.2018.0389] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/12/2023]
Abstract
Time delays play an important role in many fields such as engineering, physics or biology. Delays occur due to finite velocities of signal propagation or processing delays leading to memory effects and, in general, infinite-dimensional systems. Time delay systems can be described by delay differential equations and often include non-negligible nonlinear effects. This overview article introduces the theme issue 'Nonlinear dynamics of delay systems', which contains new fundamental results in this interdisciplinary field as well as recent developments in applications. Fundamentally, new results were obtained especially for systems with time-varying delay and state-dependent delay and for delay system with noise, which do often appear in real systems in engineering and nature. The applications range from climate modelling over network dynamics and laser systems with feedback to human balancing and machine tool chatter. This article is part of the theme issue 'Nonlinear dynamics of delay systems'.
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Affiliation(s)
- A. Otto
- Institute of Physics, Chemnitz University of Technology, 09107 Chemnitz, Germany
- e-mail:
| | - W. Just
- School of Mathematical Sciences, Queen Mary University of London, London, UK
| | - G. Radons
- Institute of Physics, Chemnitz University of Technology, 09107 Chemnitz, Germany
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