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Effective Conductivity and Critical Properties of a Hexagonal Array of Superconducting Cylinders

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Contributions in Mathematics and Engineering

Abstract

Effective conductivity of a 2D composite corresponding to the regular hexagonal arrangement of superconducting disks is expressed in the form of a long series in the volume fraction of ideally conducting disks. According to our calculations based on various resummation techniques, both the threshold and critical index are obtained in good agreement with expected values. The critical amplitude is in the interval (5. 14, 5. 24) that is close to the theoretical estimation 5. 18. The next-order (constant) term in the high-concentration regime is calculated for the first time, and the best estimate is equal to − 6. 229. The final formula is derived for the effective conductivity for arbitrary volume fraction.

In Honor of Constantin Carathéodory

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Acknowledgements

The authors are grateful to Leonid Berlyand for stimulating discussion.

Author information

Authors and Affiliations

  1. Ekayna Vihara, Bathurst 3000, Apt 606, Toronto, ON, Canada, M6B 3B4

    Simon Gluzman

  2. Ryukoku University, Kyoto, Japan

    Simon Gluzman

  3. Department of Computer Sciences and Computer Methods, Pedagogical University, ul. Podchorazych 2, Krakow, 30-084, Poland

    Vladimir Mityushev & Wojciech Nawalaniec

  4. Dnipropetrovs’k Regional Institute of State Management of National Academy of State Management at the President of Ukraine Gogolya 29, UA-49631, Dnipropetrovs’k, Ukraine

    Galina Starushenko

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  1. Simon Gluzman
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  2. Vladimir Mityushev
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  3. Wojciech Nawalaniec
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  4. Galina Starushenko
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Correspondence to Vladimir Mityushev .

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Editors and Affiliations

  1. Industrial and Systems Engineering, University of Florida, Gainesville, Florida, USA

    Panos M. Pardalos

  2. Department of Mathematics, National Technical Unversity of Athens Department of Mathematics, Athens, Greece

    Themistocles M. Rassias

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Gluzman, S., Mityushev, V., Nawalaniec, W., Starushenko, G. (2016). Effective Conductivity and Critical Properties of a Hexagonal Array of Superconducting Cylinders. In: Pardalos, P., Rassias, T. (eds) Contributions in Mathematics and Engineering. Springer, Cham. https://doi.org/10.1007/978-3-319-31317-7_13

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