The Permanent Functions of Tensors

Abstract

By a tensor we mean a multidimensional array (matrix) or hypermatrix over a number field. This article aims to set an account of the studies on the permanent functions of tensors. We formulate the definitions of 1-permanent, 2-permanent, and k-permanent of a tensor in terms of hyperplanes, planes, and k-planes of the tensor; we discuss the polytopes of stochastic tensors; at the end, we present an extension of the generalized matrix function for tensors.

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Acknowledgements

The work was done while the second author was visiting Shanghai University during his sabbatical leave from Nova Southeastern University. This expository article was written based on the second author’s presentation at ICMAA in Da Nang, Vietnam, June 14–18, 2017. The authors appreciate the communications with C. Bu, L. Cui, S. Hu, L. Qi, A. Taranenko, Y. Wei, and G. Yu during the preparation of the manuscript.

Funding

The work of Wang was partially supported by the Natural Science Foundation of China (11571220); the work of Zhang was partially supported by an NSU PFRDG Research Scholar grant.

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Correspondence to Fuzhen Zhang.

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Wang, Q., Zhang, F. The Permanent Functions of Tensors. Acta Math Vietnam 43, 701–713 (2018). https://doi.org/10.1007/s40306-018-0268-x

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Keywords

  • Birkhoff-von Neumann theorem
  • Doubly stochastic matrix
  • Hypermatrix
  • Matrix of higher order
  • Multidimensional array
  • Permanent
  • Polytope
  • Stochastic tensor
  • Tensor

Mathematics Subject Classification (2010)

  • 15A15
  • 15A02
  • 52B12