Cluster Sets and Periodicity in Some Structure Fractals

Ławrynowicz, Julian; Marchiafava, S.; Nowak-Kȩpczyk, Małgorzata

doi:10.1142/9789812709806_0020

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“…The proof is analogous to that of the corresponding theorem given in [4][5][6]. However, in order to understand two important differences:…”

Section: We Havementioning

confidence: 83%

“…Noncommutativity and phase-space theorem referring to the well--known Gelfand-Naǐmark theorem (1943): Let A be a commutative C * -algebra and M denote the set of maximal ideals of A. Then, equipped with a natural topology, M is a locally compact topological space, and A = C 0 (M ), where C 0 denotes the C * -algebra of continuous functions of M vanishing at infinity [5,16].…”

Section: Proof Of the Periodicity Theorem Formentioning

confidence: 99%

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Structure fractals and para-quaternionic geometry

Ławrynowicz¹,

Vaccaro²

2011

Annales UMCS, Mathematica

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Abstract. It is well known that starting with real structure, the CayleyDickson process gives complex, quaternionic, and octonionic (Cayley) structures related to the Adolf Hurwitz composition formula for dimensions p = 2, 4 and 8, respectively, but the procedure fails for p = 16 in the sense that the composition formula involves no more a triple of quadratic forms of the same dimension; the other two dimensions are n = 2 7 . Instead, Ławrynowicz and Suzuki (2001) have considered graded fractal bundles of the flower type related to complex and Pauli structures and, in relation to the iteration process p → p + 2 → p + 4 → . . . , they have constructed 2 4 -dimensional "bipetals" for p = 9 and 2 7 -dimensional "bisepals" for p = 13. The objects constructed appear to have an interesting property of periodicity related to the gradating function on the fractal diagonal interpreted as the "pistil" and a family of pairs of segments parallel to the diagonal and equidistant from it, interpreted as the "stamens". The first named author, M. Nowak-Kępczyk, and S. Marchiafava (2006, 2009a, b) gave an effective, explicit determination of the periods and expressed them in terms of complex and quaternionic structures, thus showing the quaternionic background of that periodicity. In contrast to earlier results, the fractal bundle flower structure, in particular petals, sepals, pistils, and stamens are not introduced ab initio; they are quoted a posteriori, when they are fully motivated. Physical concepts of dual and conjugate objects as well as of antiparticles led us to extend the periodicity theorem to structure fractals in para-quaternionic formulation, applying some results in this direction by the second named author. The paper is concluded by outlining some applications.2000 Mathematics Subject Classification. 81R25, 32L25, 53A50, 15A66.

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“…The proof is analogous to that of the corresponding theorem given in [4][5][6]. However, in order to understand two important differences:…”

Section: We Havementioning

confidence: 83%

Section: Proof Of the Periodicity Theorem Formentioning

confidence: 99%