By K Heiner Kamps, Timothy Porter

Summary homotopy concept relies at the statement that analogues of a lot of topological homotopy idea and easy homotopy idea exist in lots of different different types, akin to areas over a set base, groupoids, chain complexes and module different types. learning express models of homotopy constitution, equivalent to cylinders and course area buildings allows not just a unified improvement of many examples of identified homotopy theories, but additionally finds the interior operating of the classical spatial idea, truly indicating the logical interdependence of homes (in specific the lifestyles of yes Kan fillers in linked cubical units) and effects (Puppe sequences, Vogt's lemma, Dold's Theorem on fibre homotopy equivalences, and homotopy coherence idea)

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**Example 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.