2,987 research outputs found
Characterizing Real-Valued Multivariate Complex Polynomials and Their Symmetric Tensor Representations
In this paper we study multivariate polynomial functions in complex variables
and the corresponding associated symmetric tensor representations. The focus is
on finding conditions under which such complex polynomials/tensors always take
real values. We introduce the notion of symmetric conjugate forms and general
conjugate forms, and present characteristic conditions for such complex
polynomials to be real-valued. As applications of our results, we discuss the
relation between nonnegative polynomials and sums of squares in the context of
complex polynomials. Moreover, new notions of eigenvalues/eigenvectors for
complex tensors are introduced, extending properties from the Hermitian
matrices. Finally, we discuss an important property for symmetric tensors,
which states that the largest absolute value of eigenvalue of a symmetric real
tensor is equal to its largest singular value; the result is known as Banach's
theorem. We show that a similar result holds in the complex case as well
Scientific Endeavors of A.M. Mathai: An Appraisal on the Occasion of his Eightieth Birthday, April 2015
A.M. Mathai is Emeritus Professor of Mathematics and Statistics at McGill
University, Canada, and Director of the Centre for Mathematical and Statistical
Sciences, India. He has published over 300 research papers and more than 25
books on topics in mathematics, statistics, physics, astrophysics, chemistry,
and biology. He is a Fellow of the Institute of Mathematical Statistics,
National Academy of Sciences of India, President of the Mathematical Society of
India, and a Member of the International Statistical Institute. He is the
founder of the Canadian Journal of Statistics and the Statistical Society of
Canada. He is instrumental in the implementation of the United Nations Basic
Space Science Initiative. The paper is an attempt to capture the broad spectrum
of scientific endeavors of Professor A.M. Mathai at the occasion of his
anniversary.Comment: 21 pages, LaTe
Gaussian operator bases for correlated fermions
We formulate a general multi-mode Gaussian operator basis for fermions, to
enable a positive phase-space representation of correlated Fermi states. The
Gaussian basis extends existing bosonic phase-space methods to Fermi systems
and thus enables first-principles dynamical or equilibrium calculations in
quantum many-body Fermi systems. We prove the completeness and positivity of
the basis, and derive differential forms for products with one- and two-body
operators. Because the basis satisfies fermionic superselection rules, the
resulting phase space involves only c-numbers, without requiring anti-commuting
Grassmann variables
Random matrices with external source and KP functions
In this paper we prove that the partition function in the random matrix model
with external source is a KP function.Comment: 12 pages, title change
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