On the logarithms of matrices with central symmetry

2015 ◽  
Vol 23 (4) ◽  
Author(s):  
Chengbo Lu
Keyword(s):  
Central Symmetry ◽  
Nonsingular Matrix ◽  
Schur Decomposition

AbstractFor computing logarithms of a nonsingular matrix A, one well known algorithm named the inverse scaling and squaring method, was proposed by Kenney and Laub [14] for use with a Schur decomposition. In this paper we further consider (the computation of) the logarithms of matrices with central symmetry.We first investigate the structure of the logarithms of these matrices and then give the classifications of their logarithms. Then, we develop several algorithms for computing the logarithms. We show that our algorithms are four times cheaper than the standard inverse scaling and squaring method for matrices with central symmetry under certain conditions.

Tandem scanning reflected light microscopy: How it works and applications

1986 ◽  
Vol 44 ◽  
pp. 84-87
Author(s):  
Alan Boyde ◽  
Milan Hadravský ◽  
Mojmír Petran ◽  
Timothy F. Watson ◽  
Sheila J. Jones ◽  
...  
Keyword(s):  
Light Microscopy ◽  
Focal Plane ◽  
Central Symmetry ◽  
Single Line ◽  
Scanning Microscope ◽  
Reflected Light ◽  
Illuminating Light ◽  
Illuminating Beam

The principles of tandem scanning reflected light microscopy and the design of recent instruments are fully described elsewhere and here only briefly. The illuminating light is intercepted by a rotating aperture disc which lies in the intermediate focal plane of a standard LM objective. This device provides an array of separate scanning beams which light up corresponding patches in the plane of focus more intensely than out of focus layers. Reflected light from these patches is imaged on to a matching array of apertures on the opposite side of the same aperture disc and which are scanning in the focal plane of the eyepiece. An arrangement of mirrors converts the central symmetry of the disc into congruency, so that the array of apertures which chop the illuminating beam is identical with the array on the observation side. Thus both illumination and “detection” are scanned in tandem, giving rise to the name Tandem Scanning Microscope (TSM). The apertures are arranged on Archimedean spirals: each opposed pair scans a single line in the image.

Specific Features Pertinent to the Use of Planar Antenna Arrays with Central Symmetry in Interference Cancellers

Vestnik MEI ◽  
2018 ◽  
Vol 2 (2) ◽  
pp. 123-128
Author(s):  
Pavel S. Gribov ◽  
◽  
Maria A. Gribova ◽  
Aleksandr Yu. Shatilov ◽  
◽  
...  
Keyword(s):  
Antenna Arrays ◽  
Central Symmetry ◽  
Planar Antenna

Sign-Nonsingular Matrix Pairs

1992 ◽  
Vol 13 (1) ◽  
pp. 36-40 ◽  
Author(s):  
Richard A. Brualdi ◽  
Keith L. Chavey
Keyword(s):  
Nonsingular Matrix

Ball characterizations in spaces of constant curvature

2018 ◽  
Vol 55 (4) ◽  
pp. 421-478
Author(s):  
Jesus Jerónimo-Castro ◽  
Endre Makai, Jr.
Keyword(s):  
Convex Sets ◽  
Central Symmetry ◽  
Connected Components ◽  
Plane Convex ◽  
Converse Implication ◽  
Plane Convex Body ◽  
Dual Theorem ◽  
Centrally Symmetric ◽  
Whole Space ◽  
Straight Lines

High proved the following theorem. If the intersections of any two congruent copies of a plane convex body are centrally symmetric, then this body is a circle. In our paper we extend the theorem of High to spherical, Euclidean and hyperbolic spaces, under some regularity assumptions. Suppose that in any of these spaces there is a pair of closed convex sets of class C+2 with interior points, different from the whole space, and the intersections of any congruent copies of these sets are centrally symmetric (provided they have non-empty interiors). Then our sets are congruent balls. Under the same hypotheses, but if we require only central symmetry of small intersections, then our sets are either congruent balls, or paraballs, or have as connected components of their boundaries congruent hyperspheres (and the converse implication also holds). Under the same hypotheses, if we require central symmetry of all compact intersections, then either our sets are congruent balls or paraballs, or have as connected components of their boundaries congruent hyperspheres, and either d ≥ 3, or d = 2 and one of the sets is bounded by one hypercycle, or both sets are congruent parallel domains of straight lines, or there are no more compact intersections than those bounded by two finite hypercycle arcs (and the converse implication also holds). We also prove a dual theorem. If in any of these spaces there is a pair of smooth closed convex sets, such that both of them have supporting spheres at any of their boundary points Sd for Sd of radius less than π/2- and the closed convex hulls of any congruent copies of these sets are centrally symmetric, then our sets are congruent balls.

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