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Mallinson JB, Steel JK, Heywood ZE, Studholme SJ, Bones PJ, Brown SA. Experimental Demonstration of Reservoir Computing with Self-Assembled Percolating Networks of Nanoparticles. ADVANCED MATERIALS (DEERFIELD BEACH, FLA.) 2024; 36:e2402319. [PMID: 38558447 DOI: 10.1002/adma.202402319] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/13/2024] [Revised: 03/13/2024] [Indexed: 04/04/2024]
Abstract
The complex self-assembled network of neurons and synapses that comprises the biological brain enables natural information processing with remarkable efficiency. Percolating networks of nanoparticles (PNNs) are complex self-assembled nanoscale systems that have been shown to possess many promising brain-like attributes and which are therefore appealing systems for neuromorphic computation. Here experiments are performed that show that PNNs can be utilized as physical reservoirs within a nanoelectronic reservoir computing framework and demonstrate successful computation for several benchmark tasks (chaotic time series prediction, nonlinear transformation, and memory capacity). For each task, relevant literature results are compiled and it is shown that the performance of the PNNs compares favorably to that previously reported from nanoelectronic reservoirs. It is then demonstrated experimentally that PNNs can be used for spoken digit recognition with state-of-the-art accuracy. Finally, a parallel reservoir architecture is emulated, which increases the dimensionality and richness of the reservoir outputs and results in further improvements in performance across all tasks.
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Affiliation(s)
- Joshua B Mallinson
- The MacDiarmid Institute for Advanced Materials and Nanotechnology, School of Physical and Chemical Sciences, Te Kura Matū, University of Canterbury, Private Bag 4800, Christchurch, 8140, New Zealand
| | - Jamie K Steel
- The MacDiarmid Institute for Advanced Materials and Nanotechnology, School of Physical and Chemical Sciences, Te Kura Matū, University of Canterbury, Private Bag 4800, Christchurch, 8140, New Zealand
| | - Zachary E Heywood
- The MacDiarmid Institute for Advanced Materials and Nanotechnology, School of Physical and Chemical Sciences, Te Kura Matū, University of Canterbury, Private Bag 4800, Christchurch, 8140, New Zealand
- Electrical and Computer Engineering, University of Canterbury, Private Bag 4800, Christchurch, 8140, New Zealand
| | - Sofie J Studholme
- The MacDiarmid Institute for Advanced Materials and Nanotechnology, School of Physical and Chemical Sciences, Te Kura Matū, University of Canterbury, Private Bag 4800, Christchurch, 8140, New Zealand
| | - Philip J Bones
- Electrical and Computer Engineering, University of Canterbury, Private Bag 4800, Christchurch, 8140, New Zealand
| | - Simon A Brown
- The MacDiarmid Institute for Advanced Materials and Nanotechnology, School of Physical and Chemical Sciences, Te Kura Matū, University of Canterbury, Private Bag 4800, Christchurch, 8140, New Zealand
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2
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Tejo F, Fernandez-Roldan JA, Guslienko KY, Otxoa RM, Chubykalo-Fesenko O. Giant supermagnonic Bloch point velocities in cylindrical ferromagnetic nanowires. NANOSCALE 2024; 16:10737-10744. [PMID: 38721645 DOI: 10.1039/d3nr05013k] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/07/2024]
Abstract
Achieving high velocities of magnetic domain walls is a crucial factor for their use as information carriers in modern nanoelectronic applications. In nanomagnetism and spintronics, these velocities are often limited either by internal domain wall instabilities, known as the Walker breakdown phenomenon, or by spin wave emission, known as the magnonic regime. In the rigid domain wall model, the maximum magnon velocity acts as an effective "speed of light", providing a relativistic analogy for the domain wall speed limitation. Cylindrical magnetic nanowires are an example of systems without the Walker breakdown phenomenon. Here we demonstrate that the magnonic limit could be outstandingly surpassed in cylindrical nanowires with high magnetization, such as iron. Our numerical modeling shows the Bloch point domain wall velocities as high as 14 km s-1, well above the magnonic limit estimated in the interval 1.7-2.0 km s-1. The key ingredient is the three-dimensional conical shape of the domain wall, which elongates and breaks during the dynamics, expelling backwards pairs of Bloch points. This leads to domain wall acceleration, the effect, which resembles the "jet propulsion". This effect will be very important for three-dimensional networks based on cylindrical magnetic nanowires.
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Affiliation(s)
- Felipe Tejo
- Universidad Central de Chile, Escuela de Ingeniería, Santiago de Chile, 8330601, Chile
| | - Jose Angel Fernandez-Roldan
- Institute of Ion Beam Physics and Materials Research, Helmholtz-Zentrum Dresden-Rossendorf e.V., Bautzner Landstrasse 400, Dresden, 01328, Germany
| | - Konstantin Y Guslienko
- Depto. Polímeros y Materiales Avanzados: Física, Química y Tecnología, Universidad del País Vasco, UPV/EHU, San Sebastian, 20018, Spain
- IKERBASQUE, the Basque Foundation for Science, Bilbao, 48009, Spain
| | - Rubén M Otxoa
- Hitachi Cambridge Laboratory, J. J. Thomson Avenue, Cambridge, CB3 0HE, UK
- Donostia International Physics Center, San Sebastian, 20018, Spain
| | - Oksana Chubykalo-Fesenko
- Instituto de Ciencias de Materiales de Madrid, CSIC, Sor Juana Ines de la Cruz, 3, Madrid, 28049, Spain.
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3
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Yang C, Huang Y, Pei K, Long X, Yang L, Luo Y, Lai Y, Zhang J, Cao G, Che R. Current-Controllable and Reversible Multi-Resistance-State Based on Domain Wall Number Transition in 2D Ferromagnet Fe 3GeTe 2. ADVANCED MATERIALS (DEERFIELD BEACH, FLA.) 2024; 36:e2311831. [PMID: 38253422 DOI: 10.1002/adma.202311831] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/08/2023] [Revised: 01/15/2024] [Indexed: 01/24/2024]
Abstract
Controlling the multi-state switching is significantly essential for the extensive utilization of 2D ferromagnet in magnetic racetrack memories, topological devices, and neuromorphic computing devices. The development of all-electric functional nanodevices with multi-state switching and a rapid reset remains challenging. Herein, to imitate the potentiation and depression process of biological synapses, a full-current strategy is unprecedently established by the controllable resistance-state switching originating from the spin configuration rearrangement by domain wall number modulation in Fe3GeTe2. In particular, a strong correlation is uncovered in the reduction of domain wall number with the corresponding resistance decreasing by in-situ Lorentz transmission electron microscopy. Interestingly, the magnetic state is reversed instantly to the multi-domain wall state under a single pulse current with a higher amplitude, attributed to the rapid thermal demagnetization by simulation. Based on the neuromorphic computing system with full-current-driven artificial Fe3GeTe2 synapses with multi-state switching, a high accuracy of ≈91% is achieved in the handwriting image recognition pattern. The results identify 2D ferromagnet as an intriguing candidate for future advanced neuromorphic spintronics.
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Affiliation(s)
- Chendi Yang
- Laboratory of Advanced Materials, Shanghai Key Lab of Molecular Catalysis and Innovative Materials, Academy for Engineering & Technology, Fudan University, Shanghai, 200438, China
| | - Yalei Huang
- Materials Genome Institute, Shanghai University, Shanghai, 200444, China
| | - Ke Pei
- Laboratory of Advanced Materials, Shanghai Key Lab of Molecular Catalysis and Innovative Materials, Academy for Engineering & Technology, Fudan University, Shanghai, 200438, China
| | - Xiumin Long
- Materials Genome Institute, Shanghai University, Shanghai, 200444, China
| | - Liting Yang
- Laboratory of Advanced Materials, Shanghai Key Lab of Molecular Catalysis and Innovative Materials, Academy for Engineering & Technology, Fudan University, Shanghai, 200438, China
| | - Yongming Luo
- School of Electronics and Information Engineering, Hangzhou Dianzi University, Hangzhou, 310018, China
| | - Yuxiang Lai
- Pico Electron Microscopy Center, Innovation Institute for Ocean Materials Characterization, Center for Advanced Studies in Precision Instruments, Hainan University, Haikou, 570228, China
| | | | - Guixin Cao
- Materials Genome Institute, Shanghai University, Shanghai, 200444, China
- Zhejiang Laboratory, Hangzhou, 311100, China
| | - Renchao Che
- Laboratory of Advanced Materials, Shanghai Key Lab of Molecular Catalysis and Innovative Materials, Academy for Engineering & Technology, Fudan University, Shanghai, 200438, China
- Zhejiang Laboratory, Hangzhou, 311100, China
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4
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Bittencourt GHR, Carvalho-Santos VL, Altbir D, Chubykalo-Fesenko O, Moreno R. Tuning domain wall oscillation frequency in bent nanowires through a mechanical analogy. NANOTECHNOLOGY 2023; 35:065709. [PMID: 38009501 DOI: 10.1088/1361-6528/ad0a4b] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/19/2023] [Accepted: 11/07/2023] [Indexed: 11/29/2023]
Abstract
In this work, we present a theoretical model for domain wall (DW) oscillations in a curved magnetic nanowire with a constant curvature under the action of a uniaxial magnetic field. Our results show that the DW dynamics can be described as that of the mechanical pendulum, and both the NW curvature and the external magnetic field influence its oscillatory frequency. A comparison between our theoretical approach and experimental data in the literature shows an excellent agreement. The results presented here can be used to design devices demanding the proper control of the DW oscillatory motion in NWs.
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Affiliation(s)
- G H R Bittencourt
- Departamento de Física, Universidade Federal de Viçosa, Av. PH Rolfs s/n, 36570-900, Viçosa, Brazil
| | - V L Carvalho-Santos
- Departamento de Física, Universidade Federal de Viçosa, Av. PH Rolfs s/n, 36570-900, Viçosa, Brazil
| | - D Altbir
- Universidad de Santiago de Chile, CEDENNA, 9170124, Santiago, Chile
- Universidad Diego Portales, Ejército 441, Santiago, Chile
| | - O Chubykalo-Fesenko
- Instituto de Ciencia de Materiales de Madrid, CSIC, Cantoblanco, E-28049 Madrid, Spain
| | - R Moreno
- Instituto de Física Enrique Gaviola, IFEG (UNC-CONICET), Medina Allende s/n, Ciudad Universitaria, 5000 Córdoba, Argentina
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5
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Ross A, Leroux N, De Riz A, Marković D, Sanz-Hernández D, Trastoy J, Bortolotti P, Querlioz D, Martins L, Benetti L, Claro MS, Anacleto P, Schulman A, Taris T, Begueret JB, Saïghi S, Jenkins AS, Ferreira R, Vincent AF, Mizrahi FA, Grollier J. Multilayer spintronic neural networks with radiofrequency connections. NATURE NANOTECHNOLOGY 2023; 18:1273-1280. [PMID: 37500772 DOI: 10.1038/s41565-023-01452-w] [Citation(s) in RCA: 7] [Impact Index Per Article: 7.0] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/10/2022] [Accepted: 06/12/2023] [Indexed: 07/29/2023]
Abstract
Spintronic nano-synapses and nano-neurons perform neural network operations with high accuracy thanks to their rich, reproducible and controllable magnetization dynamics. These dynamical nanodevices could transform artificial intelligence hardware, provided they implement state-of-the-art deep neural networks. However, there is today no scalable way to connect them in multilayers. Here we show that the flagship nano-components of spintronics, magnetic tunnel junctions, can be connected into multilayer neural networks where they implement both synapses and neurons thanks to their magnetization dynamics, and communicate by processing, transmitting and receiving radiofrequency signals. We build a hardware spintronic neural network composed of nine magnetic tunnel junctions connected in two layers, and show that it natively classifies nonlinearly separable radiofrequency inputs with an accuracy of 97.7%. Using physical simulations, we demonstrate that a large network of nanoscale junctions can achieve state-of-the-art identification of drones from their radiofrequency transmissions, without digitization and consuming only a few milliwatts, which constitutes a gain of several orders of magnitude in power consumption compared to currently used techniques. This study lays the foundation for deep, dynamical, spintronic neural networks.
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Affiliation(s)
- Andrew Ross
- Unité Mixte de Physique CNRS/Thales, CNRS, Thales, Université Paris-Saclay, Palaiseau, France
| | - Nathan Leroux
- Unité Mixte de Physique CNRS/Thales, CNRS, Thales, Université Paris-Saclay, Palaiseau, France
| | - Arnaud De Riz
- Unité Mixte de Physique CNRS/Thales, CNRS, Thales, Université Paris-Saclay, Palaiseau, France
| | - Danijela Marković
- Unité Mixte de Physique CNRS/Thales, CNRS, Thales, Université Paris-Saclay, Palaiseau, France
| | - Dédalo Sanz-Hernández
- Unité Mixte de Physique CNRS/Thales, CNRS, Thales, Université Paris-Saclay, Palaiseau, France
| | - Juan Trastoy
- Unité Mixte de Physique CNRS/Thales, CNRS, Thales, Université Paris-Saclay, Palaiseau, France
| | - Paolo Bortolotti
- Unité Mixte de Physique CNRS/Thales, CNRS, Thales, Université Paris-Saclay, Palaiseau, France
| | - Damien Querlioz
- Centre de Nanosciences et de Nanotechnologies, Université Paris-Saclay, CNRS, Palaiseau, France
| | - Leandro Martins
- International Iberian Nanotechnology Laboratory (INL), Braga, Portugal
| | - Luana Benetti
- International Iberian Nanotechnology Laboratory (INL), Braga, Portugal
| | - Marcel S Claro
- International Iberian Nanotechnology Laboratory (INL), Braga, Portugal
| | - Pedro Anacleto
- International Iberian Nanotechnology Laboratory (INL), Braga, Portugal
| | | | - Thierry Taris
- Laboratoire de l'Intégration du Matériau au Système (IMS; UMR 5218), Univ. Bordeaux, CNRS, Bordeaux INP, Talence, France
| | - Jean-Baptiste Begueret
- Laboratoire de l'Intégration du Matériau au Système (IMS; UMR 5218), Univ. Bordeaux, CNRS, Bordeaux INP, Talence, France
| | - Sylvain Saïghi
- Laboratoire de l'Intégration du Matériau au Système (IMS; UMR 5218), Univ. Bordeaux, CNRS, Bordeaux INP, Talence, France
| | - Alex S Jenkins
- International Iberian Nanotechnology Laboratory (INL), Braga, Portugal
| | - Ricardo Ferreira
- International Iberian Nanotechnology Laboratory (INL), Braga, Portugal
| | - Adrien F Vincent
- Laboratoire de l'Intégration du Matériau au Système (IMS; UMR 5218), Univ. Bordeaux, CNRS, Bordeaux INP, Talence, France
| | - Frank Alice Mizrahi
- Unité Mixte de Physique CNRS/Thales, CNRS, Thales, Université Paris-Saclay, Palaiseau, France.
| | - Julie Grollier
- Unité Mixte de Physique CNRS/Thales, CNRS, Thales, Université Paris-Saclay, Palaiseau, France.
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6
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Körber L, Heins C, Hula T, Kim JV, Thlang S, Schultheiss H, Fassbender J, Schultheiss K. Pattern recognition in reciprocal space with a magnon-scattering reservoir. Nat Commun 2023; 14:3954. [PMID: 37402733 DOI: 10.1038/s41467-023-39452-y] [Citation(s) in RCA: 2] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/07/2022] [Accepted: 06/13/2023] [Indexed: 07/06/2023] Open
Abstract
Magnons are elementary excitations in magnetic materials and undergo nonlinear multimode scattering processes at large input powers. In experiments and simulations, we show that the interaction between magnon modes of a confined magnetic vortex can be harnessed for pattern recognition. We study the magnetic response to signals comprising sine wave pulses with frequencies corresponding to radial mode excitations. Three-magnon scattering results in the excitation of different azimuthal modes, whose amplitudes depend strongly on the input sequences. We show that recognition rates as high as 99.4% can be attained for four-symbol sequences using the scattered modes, with strong performance maintained with the presence of amplitude noise in the inputs.
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Affiliation(s)
- Lukas Körber
- Institut für Ionenstrahlphysik und Materialforschung, Helmholtz-Zentrum Dresden - Rossendorf, Bautzner Landstr. 400, Dresden, D-01328, Germany.
- Fakultät Physik, Technische Universität Dresden, Dresden, D-01062, Germany.
| | - Christopher Heins
- Institut für Ionenstrahlphysik und Materialforschung, Helmholtz-Zentrum Dresden - Rossendorf, Bautzner Landstr. 400, Dresden, D-01328, Germany
- Fakultät Physik, Technische Universität Dresden, Dresden, D-01062, Germany
| | - Tobias Hula
- Institut für Ionenstrahlphysik und Materialforschung, Helmholtz-Zentrum Dresden - Rossendorf, Bautzner Landstr. 400, Dresden, D-01328, Germany
- Institut für Physik, Technische Universität Chemnitz, Chemnitz, D-09107, Germany
| | - Joo-Von Kim
- Centre de Nanosciences et de Nanotechnologies, CNRS, Université Paris-Saclay, 91120, Palaiseau, France
| | - Sonia Thlang
- Centre de Nanosciences et de Nanotechnologies, CNRS, Université Paris-Saclay, 91120, Palaiseau, France
| | - Helmut Schultheiss
- Institut für Ionenstrahlphysik und Materialforschung, Helmholtz-Zentrum Dresden - Rossendorf, Bautzner Landstr. 400, Dresden, D-01328, Germany
- Fakultät Physik, Technische Universität Dresden, Dresden, D-01062, Germany
| | - Jürgen Fassbender
- Institut für Ionenstrahlphysik und Materialforschung, Helmholtz-Zentrum Dresden - Rossendorf, Bautzner Landstr. 400, Dresden, D-01328, Germany
- Fakultät Physik, Technische Universität Dresden, Dresden, D-01062, Germany
| | - Katrin Schultheiss
- Institut für Ionenstrahlphysik und Materialforschung, Helmholtz-Zentrum Dresden - Rossendorf, Bautzner Landstr. 400, Dresden, D-01328, Germany.
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7
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Vidamour IT, Ellis MOA, Griffin D, Venkat G, Swindells C, Dawidek RWS, Broomhall TJ, Steinke NJ, Cooper JFK, Maccherozzi F, Dhesi SS, Stepney S, Vasilaki E, Allwood DA, Hayward TJ. Quantifying the computational capability of a nanomagnetic reservoir computing platform with emergent magnetisation dynamics. NANOTECHNOLOGY 2022; 33:485203. [PMID: 35940063 DOI: 10.1088/1361-6528/ac87b5] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/30/2022] [Accepted: 08/08/2022] [Indexed: 06/15/2023]
Abstract
Devices based on arrays of interconnected magnetic nano-rings with emergent magnetization dynamics have recently been proposed for use in reservoir computing applications, but for them to be computationally useful it must be possible to optimise their dynamical responses. Here, we use a phenomenological model to demonstrate that such reservoirs can be optimised for classification tasks by tuning hyperparameters that control the scaling and input-rate of data into the system using rotating magnetic fields. We use task-independent metrics to assess the rings' computational capabilities at each set of these hyperparameters and show how these metrics correlate directly to performance in spoken and written digit recognition tasks. We then show that these metrics, and performance in tasks, can be further improved by expanding the reservoir's output to include multiple, concurrent measures of the ring arrays' magnetic states.
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Affiliation(s)
- I T Vidamour
- Department of Materials Science and Engineering, University of Sheffield, Sheffield S1 3JD, United Kingdom
| | - M O A Ellis
- Department of Computer Science, University of Sheffield, Sheffield S1 4DP, United Kingdom
| | - D Griffin
- Department of Computer Science, University of York, York YO10 5GH, United Kingdom
| | - G Venkat
- Department of Materials Science and Engineering, University of Sheffield, Sheffield S1 3JD, United Kingdom
| | - C Swindells
- Department of Materials Science and Engineering, University of Sheffield, Sheffield S1 3JD, United Kingdom
| | - R W S Dawidek
- Department of Materials Science and Engineering, University of Sheffield, Sheffield S1 3JD, United Kingdom
| | - T J Broomhall
- Department of Materials Science and Engineering, University of Sheffield, Sheffield S1 3JD, United Kingdom
| | - N J Steinke
- ISIS Neutron and Muon Source, Rutherford Appleton Lab, Didcot, OX11 0QX, United Kingdom
| | - J F K Cooper
- ISIS Neutron and Muon Source, Rutherford Appleton Lab, Didcot, OX11 0QX, United Kingdom
| | - F Maccherozzi
- Diamond Light Source, Harwell Science and Innovation Campus, Didcot, Oxfordshire OX11 0DE, United Kingdom
| | - S S Dhesi
- Diamond Light Source, Harwell Science and Innovation Campus, Didcot, Oxfordshire OX11 0DE, United Kingdom
| | - S Stepney
- Department of Computer Science, University of York, York YO10 5GH, United Kingdom
| | - E Vasilaki
- Department of Computer Science, University of Sheffield, Sheffield S1 4DP, United Kingdom
- Institute of Neuroinformatics, University of Zurich and ETH Zurich, 8057 Zürich, Switzerland
| | - D A Allwood
- Department of Materials Science and Engineering, University of Sheffield, Sheffield S1 3JD, United Kingdom
| | - T J Hayward
- Department of Materials Science and Engineering, University of Sheffield, Sheffield S1 3JD, United Kingdom
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8
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Gaurav A, Song X, Manhas S, Gilra A, Vasilaki E, Roy P, De Souza MM. Reservoir Computing for Temporal Data Classification Using a Dynamic Solid Electrolyte ZnO Thin Film Transistor. FRONTIERS IN ELECTRONICS 2022. [DOI: 10.3389/felec.2022.869013] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/13/2022] Open
Abstract
The processing of sequential and temporal data is essential to computer vision and speech recognition, two of the most common applications of artificial intelligence (AI). Reservoir computing (RC) is a branch of AI that offers a highly efficient framework for processing temporal inputs at a low training cost compared to conventional Recurrent Neural Networks (RNNs). However, despite extensive effort, two-terminal memristor-based reservoirs have, until now, been implemented to process sequential data by reading their conductance states only once, at the end of the entire sequence. This method reduces the dimensionality, related to the number of signals from the reservoir and thereby lowers the overall performance of reservoir systems. Higher dimensionality facilitates the separation of originally inseparable inputs by reading out from a larger set of spatiotemporal features of inputs. Moreover, memristor-based reservoirs either use multiple pulse rates, fast or slow read (immediately or with a delay introduced after the end of the sequence), or excitatory pulses to enhance the dimensionality of reservoir states. This adds to the complexity of the reservoir system and reduces power efficiency. In this paper, we demonstrate the first reservoir computing system based on a dynamic three terminal solid electrolyte ZnO/Ta2O5 Thin-film Transistor fabricated at less than 100°C. The inherent nonlinearity and dynamic memory of the device lead to a rich separation property of reservoir states that results in, to our knowledge, the highest accuracy of 94.44%, using electronic charge-based system, for the classification of hand-written digits. This improvement is attributed to an increase in the dimensionality of the reservoir by reading the reservoir states after each pulse rather than at the end of the sequence. The third terminal enables a read operation in the off state, that is when no pulse is applied at the gate terminal, via a small read pulse at the drain. This fundamentally allows multiple read operations without increasing energy consumption, which is not possible in the conventional two-terminal memristor counterpart. Further, we have also shown that devices do not saturate even after multiple write pulses which demonstrates the device’s ability to process longer sequences.
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