By V. G. Turaev, A. M. Vershik, V. A. Rokhlin
This booklet is devoted to the reminiscence of the phenomenal Russian mathematician, V. A. Rokhlin (1919--1984). it's a selection of examine papers written by way of his former scholars and fans, who're now specialists of their fields. the subjects during this quantity comprise topology (the Morse-Novikov concept, spin bordisms in size 6, and skein modules of links), genuine algebraic geometry (real algebraic curves, airplane algebraic surfaces, algebraic hyperlinks, and complicated orientations), dynamics (ergodicity, amenability, and random package transformations), geometry of Riemannian manifolds, concept of Teichmüller areas, degree idea, and so on. The publication additionally contains a biography of Rokhlin by means of Vershik and articles of old curiosity.
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Extra info for Topology, ergodic theory, real algebraic geometry: Rokhlin's memorial
Sample text
We conclude that d is a divisor of r3, ... , r n-1, and finally, of r n. Thus, we just proved that the Euclidean algorithm, when applied to two natural numbers a and b, does yield their greatest common divisor. The greatest common divisor of the numbers a and b will be denoted in what follows by (a, b). Clearly, a is divisible by b if and only if (a, b) = b. As an example, let us find (U20, U15) = (6765,610). 3) is 6765 = 610 . 11 + 55, 610 = 55· 11 + 5, 55 = 5·11. Thus, The fact that the greatest common divisor of two Fibonacci numbers turns out to be another Fibonacci number is not accidental.
Chapter 2 N umber-Theoretic Properties of Fibonacci Numbers 1. We are going to study some properties of Fibonacci numbers related to their divisibility by other numbers. The first result addresses the divisibility of a Fibonacci number by another Fibonacci number. Theorem. If n is divisible by m, then Un is divisible by Urn. Proof. Assume n is divisible by m and set n = mk. We will carry out the proof by induction on k. If k = 1, then n = m, and in this case it is obvious that Un is divisible by Urn.
However, the above mentioned drawback of the Fibonacci numeration sys- tem regarding the bigger capacity provided by the binary system turns out to be a positive feature of the system when it comes to what is called antijamming. Let us point this out as follows. In the binary system, any string of digits represents a certain number. Therefore, an error of any kind - omission or typo - in a string, produces a valid representation of a different number and, therefore, such an error would easily pass undetected.