The point equation of the associated curve of the indefinite numerical range is derived, following Fiedler's approach for definite inner product spaces. The classification of the associated curve is presented in the 3 × 3 indefinite case, using Newton's classification of cubic curves. Illustrative examples of all the different possibilities are given. The results obtained extend to Krein spaces results of Kippenhahn on the classical numerical range. Keywords: indefinite numerical range, indefinite inner product space, plane algebraic curve MSC 2000 : 15A60, 15A63, 46C20
Part of the beauty of the Azores lies under the feet of anyone who walks in the archipela- go. The traditional Portuguese pavement decorates public spaces with a very typical aesthetic. These examples illustrate different types of rosettes, friezes, and patterns. The rosettes are limited planar configurations, usually presented in circular frames. The friezes, due to their unidimensional nature, are found on sidewalks. The patterns, which can cover plane surfaces, show up in plazas. All the seven friezes, cyclic rosettes, dihedral rosettes, and some of the seventeen types of wallpa- pers were detected in the Azores islands. In this work, a card deck about the subject is presented. This deck suggests a walk through the 9 islands of the Azores and the symmetries of its pavements.
In 1704 the French priest Sébastien Truchet published a paper where he explored and counted patterns made up from a square divided by a diagonal line into two colored parts, , now known as a Truchet tile. A few years later, Father Dominique Doüat continued Truchet's work and published a book in 1722 containing many more patterns and further counts of configurations.In this paper, we extend the work introduced by Truchet and Doüat by considering all possible rosettes made up of an m × n array of square or non-square Truchet tiles, or . We then classify the rosettes according to their symmetry group and count all the distinct rosettes in each group, for all possible sizes. The results are summarized in a separate section where we further analyze the asymptotic behavior of the counts for square arrays. Finally, some applications are shown using two types of square flexagons.
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