AMS classification: 15A48 47H07 47H09 47H10 Keywords: Perron-Frobenius theorem for nonnegative tensors Convergence of the power algorithmWe prove an analog of Perron-Frobenius theorem for multilinear forms with nonnegative coefficients, and more generally, for polynomial maps with nonnegative coefficients. We determine the geometric convergence rate of the power algorithm to the unique normalized eigenvector.

Abstract. This note studies the dynamical behavior of polynomial mappings with polynomial inverse from the real or complex plane to itself.

Abstract. We establish several mathematical and computational properties of the nuclear norm for higher-order tensors. We show that like tensor rank, tensor nuclear norm is dependent on the choice of base field -the value of the nuclear norm of a real 3-tensor depends on whether we regard it as a real 3-tensor or a complex 3-tensor with real entries. We show that every tensor has a nuclear norm attaining decomposition and every symmetric tensor has a symmetric nuclear norm attaining decomposition. There is a corresponding notion of nuclear rank that, unlike tensor rank, is upper semicontinuous. We establish an analogue of Banach's theorem for tensor spectral norm and Comon's conjecture for tensor rank -for a symmetric tensor, its symmetric nuclear norm always equals its nuclear norm. We show that computing tensor nuclear norm is NP-hard in several sense. Deciding weak membership in the nuclear norm unit ball of 3-tensors is NP-hard, as is finding an ε-approximation of nuclear norm for 3-tensors. In addition, the problem of computing spectral or nuclear norm of a 4-tensor is NP-hard, even if we restrict the 4-tensor to be bi-Hermitian, bisymmetric, positive semidefinite, nonnegative valued, or all of the above. We discuss some simple polynomial-time approximation bounds. As an aside, we show that the nuclear (p, q)-norm of a matrix is NP-hard in general but can be computed in polynomial-time if p = 1, q = 1, or p = q = 2, with closed-form expressions for the nuclear (1, q)-and (p, 1)-norms.

Uncertainty relations are a distinctive characteristic of quantum theory that impose intrinsic limitations on the precision with which physical properties can be simultaneously determined. The modern work on uncertainty relations employs entropic measures to quantify the lack of knowledge associated with measuring non-commuting observables. However, there is no fundamental reason for using entropies as quantifiers; any functional relation that characterizes the uncertainty of the measurement outcomes defines an uncertainty relation. Starting from a very reasonable assumption of invariance under mere relabelling of the measurement outcomes, we show that Schur-concave functions are the most general uncertainty quantifiers. We then discover a fine-grained uncertainty relation that is given in terms of the majorization order between two probability vectors, significantly extending a majorization-based uncertainty relation first introduced in [M. H. Partovi, Phys. Rev. A 84, 052117 (2011)]. Such a vector-type uncertainty relation generates an infinite family of distinct scalar uncertainty relations via the application of arbitrary uncertainty quantifiers. Our relation is therefore universal and captures the essence of uncertainty in quantum theory.Uncertainty relations lie at the core of quantum mechanics and are a direct manifestation of the noncommutative structure of the theory. In contrast to classical physics, where in principle any observable can be measured with arbitrary precision, quantum mechanics introduces severe restrictions on the allowed measurement results of two or more non-commuting observables. Uncertainty relations are not a manifestation of the experimentalists' (in)ability of performing precise measurements, but are inherently determined by the incompatibility of the measured observables.The first formulation of the uncertainty principle was provided by Heisenberg [1], who noted that more knowledge about the position of a single quantum particle implies less certainty about its momentum and vice-versa. He expressed the principle in terms of standard deviations of the momentum and position operators ∆X · ∆P 2 .Robertson [2] generalized Heisenberg's uncertainty principle to any two arbitrary observables A and B asA major drawback of Robertson's uncertainty principle is that it depends on the state |ψ of the system. In particular, when |ψ belongs to the null-space of the * Electronic address: friedlan@uic. Here H(A) is the Shannon entropy [5] of the probability distribution induced by measuring the state |ψ of the system in the eigenbasis {|a j } of the oservable A (and similarly for B). The bound on the right hand side c(A, B) := max m,n | a m |b n | represents the maximum overlap between the bases elements, and is independent of the state |ψ . Recently the study of uncertainty relations intensified [6,7] (see also [1,9] for recent surveys), and as a result various important applications have been discovered, ranging from security proofs for quantum cryptography [10][11][12], information locking ...

Abstract. We consider the formulation and local analysis of various quadratically convergent methods for solving the symmetric matrix inverse eigenvalue problem. One of these methods is new. We study the case where multiple eigenvalues are given: we show how to state the problem so that it is not overdetermined, and describe how to modify the numerical methods to retain quadratic convergence on the modified problem. We give a general convergence analysis, which covers both the distinct and the multiple eigenvalue cases. We also present numerical experiments which illustrate our results.

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Consider all colorings of a finite box in a multidimensional grid with a given number of colors subject to given local constraints. We outline the most recent theory for the computation of the exponential growth rate of the number of such colorings as a function of the dimensions of the box. As an application we compute the monomer-dimer constant for the 2-dimensional grid to 9 decimal digits, agreeing with the heuristic computations of Baxter, and for the 3-dimensional grid with an error smaller than 1.35%.  2004 Elsevier Inc. All rights reserved.

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