Multichannel algorithm based on generalized positional numeration system

This report is devoted to introduction in multichannel algorithm based on generalized numeration notations (GPN). The internal, external and mixed account are entered. The concept of the GPN and its classification as decomposition of an integer on composed of integers is discussed. Realization of multichannel algorithm on the basis of GPN is introduced. In particular, some properties of Fibonacci multichannel algorithm are discussed.Comment: 7 pages, 7 tables, report at the conference "European Economy: Present And Future

Morava motives of projective quadrics

The present Ph. D. thesis is devoted to Morava motives of projective quadrics, meaning that we replace the Chow theory by another oriented cohomology theory. We consider arbitrary oriented cohomology theories as we wish to obtain invariants that are simpler than Chow motives. In fact, there exists a series of theories, more precisely, Morava K-theories K(n)*, which starts from K^0 and tends to CH*. The most important and interesting results are the following ones: Theorem (Theorem 1.3.9) Let Q be a generic quadric of dimension D > 0, and n > 1; we denote N = 2^n for D = 2d even, or N = 2^n-1 for D = 2d+1 odd. Then K(n)-motive of Q has an indecomposable summand of rank min(N, 2d + 2), and max(0, 2d + 2 - N) summands isomorphic to Tate motives. Theorem (Theorem 2.0.1) For a group G_m = Spin_m or G_m = SO_m with m > 2^(n+1), n > 1, the canonical map K(n)*(G_m; F_2) → K(n)*(G_(m+2); F_2) is an isomorphism. We also describe several algorithms useful for computer computations of K(n)-motives of small-dimensional varieties.In der vorliegenden Doktorarbeit betrachten wir die Motive von projektiven Quadriken in Morava K-Theorie, d.h. wir ersetzen die Chow-Theorie durch eine andere orientierte Kohomologietheorie. Indem wir unseren Focus auf beliebige orientierte Kohomologietheorie erweiten, hoffen wir Invariantanten zu finden, die einfacher sind als Chow-Motive. Genauer betrachten wir eine Reihe von Theorien, die Morava K-Theorien K(n)*, welche von K^0 ausgehend gegen CH* “konvergieren”. Als Hauptergebnisse erhalten wir: Theorem (Theorem 1.3.9) Es sei Q eine generische Quadrik von der Dimension D > 0 und es sei n > 1. Bezeichne N = 2^n für D = 2d gerade oder N = 2^n - 1 für D = 2d + 1 ungerade. Dann hat das K(n)-Motiv von Q einen unzerlegbaren Summanden vom Rang min(N, 2d + 2) und max(0, 2d + 2 - N) Summanden, die isomorph zu Tate Motiven sind. Theorem (Theorem 2.0.1) Für G_m = Spin_m oder G_m = SO_m ist der kanonische Homomorphismus K(n)*(G_m; F_2) → K(n)*(G_(m+2); F_2 ) ein Isomorphismus, für m> 2^(n + 1), n > 1. Außerdem präsentieren wir auch verschiedene Algorithmen für die Berechnungen von K(n)-Motiven von Varietaten kleiner Dimensionen