Tag Archives: Complex Variables

Constrained Least Squares for Extended Complex Factor Analysis [CL]

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http://arxiv.org/abs/1804.00430


For subspace estimation with an unknown colored noise, Factor Analysis (FA) is a good candidate for replacing the popular eigenvalue decomposition (EVD). Finding the unknowns in factor analysis can be done by solving a non-linear least square problem. For this type of optimization problems, the Gauss-Newton (GN) algorithm is a powerful and simple method. The most expensive part of the GN algorithm is finding the direction of descent by solving a system of equations at each iteration. In this paper we show that for FA, the matrices involved in solving these systems of equations can be diagonalized in a closed form fashion and the solution can be found in a computationally efficient way. We show how the unknown parameters can be updated without actually constructing these matrices. The convergence performance of the algorithm is studied via numerical simulations.

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A. Sardarabadi, A. Veen and L. Koopmans
Tue, 3 Apr 18
48/57

Comments: N/A

How constant shifts affect the zeros of certain rational harmonic functions [CL]

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http://arxiv.org/abs/1702.07593


We study the effect of constant shifts on the zeros of rational harmomic functions $f(z) = r(z) – \conj{z}$. In particular, we characterize how shifting through the caustics of $f$ changes the number of zeros and their respective orientations. This also yields insight into the nature of the singular zeros of $f$. Our results have applications in gravitational lensing theory, where certain such functions $f$ represent gravitational point-mass lenses, and a constant shift can be interpreted as the position of the light source of the lens.

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J. Liesen and J. Zur
Mon, 27 Feb 17
25/49

Comments: 26 pages, 9 figures

Creating images by adding masses to gravitational point lenses [GA]

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http://arxiv.org/abs/1405.2785


A well-studied maximal gravitational point lens construction of S.H. Rhie produces $5n$ images of a light source using $n+1$ deflector masses. The construction arises from a circular, symmetric deflector configuration on $n$ masses (producing only $3n+1$ images) by adding a tiny mass in the center of the other mass positions (and reducing all the other masses a little bit).
In a recent paper we studied this “image creating effect” from a purely mathematical point of view (S\`ete, Luce & Liesen, ArXiv eprints 2014). Here we discuss a few consequences of our findings for gravitational microlensing models. We present a complete characterization of the effect of adding small masses to these point lens models, with respect to the number of images. In particular, we give several examples of maximal lensing models that are different from Rhie’s construction and that do not share its highly symmetric appearance. We give generally applicable conditions that allow the construction of maximal point lenses on $n+1$ masses from maximal lenses on $n$ masses.

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O. Sete, R. Luce and J. Liesen
Tue, 13 May 14
37/58

Comments: N/A

Perturbing rational harmonic functions by poles [CL]

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http://arxiv.org/abs/1403.0906


We study how adding certain poles to rational harmonic functions of the form $R(z)-\bar{z}$, with $R(z)$ rational and of degree $d\geq 2$, affects the number of zeros of the resulting functions. Our results are motivated by and generalize a construction of Rhie derived in the context of gravitational microlensing (ArXiv e-print 2003). Of particular interest is the construction and the behavior of rational functions $R(z)$ that are {\em extremal} in the sense that $R(z)-\bar{z}$ has the maximal possible number of $5(d-1)$ zeros.

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O. Sete, R. Luce and J. Liesen
Wed, 5 Mar 14
68/75