scholarly journals Minimal Height Companion Matrices for Euclid Polynomials

2018 ◽  
Vol 13 (1-2) ◽  
pp. 41-56 ◽  
Author(s):  
E. Y. S. Chan ◽  
R. M. Corless
Keyword(s):  
Minimal Height ◽  
Companion Matrices

scholarly journals Toeplitz nonnegative realization of spectra via companion matrices

2019 ◽  
Vol 7 (1) ◽  
pp. 230-245
Author(s):  
Macarena Collao ◽  
Mario Salas ◽  
Ricardo L. Soto
Keyword(s):  
Eigenvalue Problem ◽  
Half Plane ◽  
Toeplitz Matrix ◽  
Toeplitz Matrices ◽  
Nonnegative Matrix ◽  
Inverse Eigenvalue Problem ◽  
Complex Numbers ◽  
Inverse Eigenvalue ◽  
Companion Matrices ◽  
Solution Matrix

Abstract The nonnegative inverse eigenvalue problem (NIEP) is the problem of finding conditions for the existence of an n × n entrywise nonnegative matrix A with prescribed spectrum Λ = {λ1, . . ., λn}. If the problem has a solution, we say that Λ is realizable and that A is a realizing matrix. In this paper we consider the NIEP for a Toeplitz realizing matrix A, and as far as we know, this is the first work which addresses the Toeplitz nonnegative realization of spectra. We show that nonnegative companion matrices are similar to nonnegative Toeplitz ones. We note that, as a consequence, a realizable list Λ= {λ1, . . ., λn} of complex numbers in the left-half plane, that is, with Re λi≤ 0, i = 2, . . ., n, is in particular realizable by a Toeplitz matrix. Moreover, we show how to construct symmetric nonnegative block Toeplitz matrices with prescribed spectrum and we explore the universal realizability of lists, which are realizable by this kind of matrices. We also propose a Matlab Toeplitz routine to compute a Toeplitz solution matrix.

scholarly journals On the Number of Quadratic Twists with a Rational Point of Almost Minimal Height

2020 ◽  
Author(s):  
Joachim Petit
Keyword(s):  
Asymptotic Formula ◽  
Elliptic Curve ◽  
Asymptotic Estimate ◽  
Rational Point ◽  
Quadratic Fields ◽  
Real Quadratic Fields ◽  
Quadratic Twists ◽  
The Family ◽  
Minimal Height ◽  
Real Quadratic

Abstract We investigate the number of curves having a rational point of almost minimal height in the family of quadratic twists of a given elliptic curve. This problem takes its origin in the work of Hooley, who asked this question in the setting of real quadratic fields. In particular, he showed an asymptotic estimate for the number of such fields with almost minimal fundamental unit. Our main result establishes the analogue asymptotic formula in the setting of quadratic twists of a fixed elliptic curve.

scholarly journals Bounds on polynomial roots using intercyclic companion matrices

2018 ◽  
Vol 539 ◽  
pp. 94-116
Author(s):  
Kevin N. Vander Meulen ◽  
Trevor Vanderwoerd
Keyword(s):  
Polynomial Roots ◽  
Companion Matrices

scholarly journals Influence of misalignments on the performance of externally occulted solar coronagraphs

2018 ◽  
Vol 612 ◽  
pp. A82 ◽  
Author(s):  
S. V. Shestov ◽  
A. N. Zhukov
Keyword(s):  
European Space Agency ◽  
Numerical Approach ◽  
Primary Objective ◽  
Optical Elements ◽  
Trade Off ◽  
Space Agency ◽  
Minimal Height ◽  
Different Types ◽  
Points Of View ◽  
Quantitative Influence

Context. The ASPIICS instrument is a novel externally occulted coronagraph that will be launched on board the PROBA-3 mission of the European Space Agency. The external occulter will be placed on one satellite ~150 m ahead of the second satellite that will carry an optical instrument. During 6 h out of 19.38 h of orbit, the satellites will fly in a precise (accuracy around a few millimeters) formation, constituting a giant externally occulted coronagraph. The large distance between the external occulter and the primary objective will allow observations of the white-light solar corona starting from extremely low heights ~ 1.1R⊙. Aims. We intend to analyze influence of shifts of the satellites and misalignments of optical elements on the ASPIICS performance in terms of diffracted light. Based on the quantitative influence of misalignments on diffracted light, we provide a recipe for choosing the size of the internal occulter (IO) to achieve a trade-off between the minimal height of observations and sustainability to possible misalignments. Methods. We considered different types of misalignments and analyzed their influence from optical and computational points of view. We implemented a numerical model of the diffracted light and its propagation through the optical system and computed intensities of diffracted light throughout the instrument. Our numerical approach is based on a model from the literature that considered the axisymmetrical case. Here we extend the model to include nonsymmetrical cases and possible misalignments. Results. The numerical computations fully confirm the main properties of the diffracted light that we obtained from semi-analytical consideration. We obtain that relative influences of various misalignments are significantly different. We show that the internal occulter with RIO = 1.694 mm = 1.1R⊙ is large enough to compensate possible misalignments expected to occur in PROBA-3/ASPIICS. Besides that we show that apodizing the edge of the internal occulter leads to additional suppression of the diffracted light. Conclusions. We conclude that the most important misalignment is the tilt of the telescope with respect to the line connecting the center of the external occulter and the entrance aperture. Special care should be taken to co-align the external occulter and the coronagraph, which means co-aligning the diffraction fringe from the external occulter and the internal occulter. We suggest that the best orientation strategy is to point the coronagraph to the center of the external occulter.

scholarly journals Linear fractional transforms of companion matrices

1979 ◽  
Vol 20 (2) ◽  
pp. 129-132 ◽  
Author(s):  
N. J. Young
Keyword(s):  
Characteristic Polynomial ◽  
Canonical Form ◽  
Stability Theory ◽  
Companion Matrix ◽  
Arbitrary Field ◽  
Companion Matrices ◽  
Image Position ◽  
Fractional Transforms

Questions about polynomials can be turned into questions about matrices by associating with the polynomial(over an arbitrary field) its companion matrixwhich has p/an as its characteristic polynomial. This technique is often used in stability theory, as indicated in [1]; companion matrices also occur in the theory of the rational canonical form.

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