# The sum of squared logarithms inequality in arbitrary dimensions

@article{Borisov2015TheSO,
title={The sum of squared logarithms inequality in arbitrary dimensions},
author={Lev Borisov and Patrizio Neff and Suvrit Sra and Christian Thiel},
journal={PAMM},
year={2015},
volume={16}
}
• Published 17 August 2015
• Mathematics
• PAMM
We prove the sum of squared logarithms inequality (SSLI) which states that for nonnegative vectors x, y ∈ ℝn whose elementary symmetric polynomials satisfy ek(x) ≤ ek(y) (for 1 ≤ k < n) and en(x) = en(y) , the inequality ∑i(log xi)2 ≤ ∑i(log yi)2 holds. Our proof of this inequality follows by a suitable extension to the complex plane. In particular, we show that the function f : M ⊆ ℂn → ℝ with f(z) = ∑i(log zi)2 has nonnegative partial derivatives with respect to the elementary symmetric…
17 Citations
We consider a partial order on positive vectors induced by elementary symmetric polynomials. As a corollary we obtain a short proof of the SSLI inequality of Neff et al. (2012), which was first
• Mathematics
ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik
• 2019
We consider the problem to determine the optimal rotations R∈SO(n) which minimize W:SO(n)→R0+,W(R;D)≔sym(RD−1)2for a given diagonal matrix D≔diag(d1,…,dn)∈Rn×n with positive entries di>0 . The
• Mathematics, Computer Science
SIAM J. Appl. Algebra Geom.
• 2019
The optimal orthogonal matrices are determined which minimize the symmetrized Euclidean distance W(R,;D) =vert\vert{{sym}(R D - 1) .
• Mathematics
• 2017
We consider the two logarithmic strain measures $$\omega _{\mathrm {iso}}= ||{{\mathrm{dev}}}_n \log U ||$$ and $$\omega _{\mathrm {vol}}= |{{\mathrm{tr}}}(\log U) |$$, which are isotropic invariants
• Mathematics
• 2015
In any geometrically nonlinear quadratic Cosserat‐micropolar extended continuum model formulated in the deformation gradient field F:=∇φ:Ω→ GL +(n) and the microrotation field R:Ω→ SO (n) , the
• Mathematics, Computer Science
• 2016
This paper presents a simple method of deriving a posteriori error equalities and estimates for linear elliptic and parabolic partial differential equations using a combined norm taking into account both the primal and dual variables.
From a conceptual point of view, two-stage stochastic programs and bilevel problems under stochastic uncertainty are closely related. However, the step from the first to the latter mirrors the step
In this paper we prove the following theorem: Let $$\Omega \subset \mathbb {R}^{n}$$Ω⊂Rn be a bounded open set, $$\psi \in C_{c}^{2}(\mathbb {R}^{n})$$ψ∈Cc2(Rn), $$\psi > 0$$ψ>0 on \partial \Omega
• Mathematics
• 2016
This paper is concerned with the analysis and numerical investigations for the optimal control of first-order magneto-static equations. Necessary and sufficient optimality conditions are established

## References

SHOWING 1-10 OF 48 REFERENCES

• Mathematics
• 2013
The unitary polar factor Q = Up in the polar decomposition of Z = Up H is the minimizer over unitary matrices Q for both kLog(QZ)k 2 and its Hermitian part ksym * (Log(QZ))k 2 over both R and C for
• Mathematics
ArXiv
• 2014
It is seen that H and the intrinsically quantum informational quantity Q become surprisingly closely related in functional form, suggesting a special significance for the symmetric polynomials in quantum information theory.
• Mathematics
• 2014
We consider the sum of squared logarithms inequality and investigate possible connections with the theory of majorization. We also discuss alternative sufficient conditions on two sets of vectors
• Mathematics
• 2013
AbstractLet y1,y2,y3,a1,a2,a3∈(0,∞) be such that y1y2y3=a1a2a3 and y1+y2+y3≥a1+a2+a3,y1y2+y2y3+y1y3≥a1a2+a2a3+a1a3. Then
• Mathematics
• 2004
Let S be the von Neumann entropy of a finite ensemble E of pure quantum states. We show that S may be naturally viewed as a function of a set of geometrical volumes in Hilbert space defined by the
• Mathematics
• 2014
Assume n≥2$n\geq2$. Consider the elementary symmetric polynomials ek(y1,y2,…,yn)$e_{k}(y_{1},y_{2},\ldots, y_{n})$ and denote by E0,E1,…,En−1$E_{0},E_{1},\ldots,E_{n-1}$ the elementary symmetric
• Mathematics
SIAM J. Matrix Anal. Appl.
• 2006
This work defines the Log‐Euclidean mean from a Riemannian point of view, based on a lie group structure which is compatible with the usual algebraic properties of this matrix space and a new scalar multiplication that smoothly extends the Lie group structure into a vector space structure.