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Some geometrical aspects of binary, ternary, quaternary, quinary and senary structures in physics

100%

Suzuki O. , Ławrynowicz J. , Nowak-Kępczyk M. , Zubert M.

nr 2

109-122

https://doi.org/10.26485/0459-6854/2018/68.2/11 Obserwujemy, że struktury kwinarne i senarne, zarówno w przypadku pentacenu, jak i innych polimerów, można utworzyć ze struktur binarnych i senarnych w sensie równań różniczkowych i opisu geometrycznego. Liście pentacenu umieszczone na silikonowym podłożu mają postać pięciu połączonych węglowo-wodorowych sześciokątów; w całości nie tworzą dokładnie struktury planarnej lecz lekko falującą, która minimalizuje energię całkowitą. W przypadku struktury kwinarnej liście tworzą odosobnione, niemal periodyczne zygzaki i meandry.

https://doi.org/10.26485/0459-6854/2018/68.2/11 It is observed that quinary and senary structures like in pentacene and several other polymers may be composed from binary and ternary structures in the sense of differentialequational and geometrical description. In the case of pentacene its leaves are attached to the silicon background and have the form of five connected carbon-hydrogen hexagons; in total they do not form the precisely planar structure but a slightly wavy structure which minimizes total energy. In the case of a quinary structure the leaves form solitary, nearly periodical zigzags and meanders.

Binary and ternary structures in physics III. Galois-type theory for binary and ternary structures

100%

Suzuki O. , Ławrynowicz J. , Nowak-Kępczyk M.

nr 1

https://doi.org/10.26485/0459-6854/2018/68.1/8 Przedstawiona jest teoria typu Galois dla maszyny Turinga wraz z jej odpowiednikiem dla struktur binarnych i ternarnych w fizyce. Ponadto zaprezentowana idea binarnoternarnej dekompozycji struktur kwinarnych i senarnych w zastosowaniu do badań nad polimerami.

https://doi.org/10.26485/0459-6854/2018/68.1/8 A Galois-type theory for Turing machine is presented as well as its counterpart for binary and ternary structures in physics. In addition an idea of binary-ternary decomposition of quinary and senary structures is indicated with application to polymer research.

Differential and integral calculus for a Schauder basis on a fractal set (I) (Schauder basis 80 years after)

100%

Ławrynowicz J. , Ogata T. , Suzuki O.

Banach Center Publications

2009

tom 87

nr 1

115-140

In this paper we introduce a concept of Schauder basis on a self-similar fractal set and develop differential and integral calculus for them. We give the following results: (1) We introduce a Schauder/Haar basis on a self-similar fractal set (Theorems I and I'). (2) We obtain a wavelet expansion for the L²-space with respect to the Hausdorff measure on a self-similar fractal set (Theorems II and II'). (3) We introduce a product structure and derivation on a self-similar fractal set (Theorem III). (4) We give the Taylor expansion theorem on a fractal set (Theorem IV and IV'). (5) By use of the Taylor expansion for wavelet functions, we introduce basic functions, for example, exponential and trigonometrical functions, and discuss the relationship between the usual and introduced corresponding special functions (Theorem V). (6) Finally we discuss the relationship between the wavelet functions and the generating functions of the dynamical systems on a fractal set and show that our wavelet expansions can describe several fluctuations observed in nature.

Supercomplex structures, surface soliton equations, and quasiconformal mappings

100%

Ławrynowicz J. , Kędzia K. , Suzuki O.

nr 1

245-268

Hurwitz pairs and triples are discussed in connection with algebra, complex analysis, and field theory. The following results are obtained: (i) A field operator of Dirac type, which is called a Hurwitz operator, is introduced by use of a Hurwitz pair and its characterization is given (Theorem 1). (ii) A field equation of the elliptic Neveu-Schwarz model of superstring theory is obtained from the Hurwitz pair (𝔼⁴,𝔼³) (Theorem 2), and its counterpart connected with the Hurwitz triple $(𝔼^{11},𝔼^{11},𝔼^{26})$ is mentioned. (iii) Isospectral deformations of the Hurwitz operator of the Hurwitz pair (𝔼²,𝔼²) induce various soliton equations (Theorem 3). (iv) A special complex structure, which is called a supercomplex structure, is introduced on separable Hilbert spaces (Definition 10). A correspondence between such structures and reduction solutions of Sato's version of Kadomtsev-Petviashvili system is established (Theorem 4). (v) The general class of quasiconformal mappings in the plane is obtained from generalized Hurwitz pairs (Theorem 5). From these results we conclude that Hurwitz pairs and triples give rise to several interesting applications.