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Let M (n, n) be the set of all n × n matrices over a commutative ring with identity. Then the Cayley Hamilton Theorem states: Theorem. Let A ∈ M (n, n) with characteristic polynomial det(tI − A) = c 0 t n + c 1 t n−1 + c 2 t n−2 + · · · + c n. Then c 0 A n + c 1 A n−1 + c 2 A n−2 + · · · + c n I = 0. In this note we give a variation on a standard proof (see… (More)

- C. BERNHARDT, I. MULVEY
- 1992

Let n and 6 be cyclic permutations of finite ordered sets. We say that n forces 6 if every continuous map of the interval which has a representative of n also has one of 6. We give a geometric version of Jungreis' combinatorial algorithm for deciding in certain cases whether n forces 9 .

- Chris Bernhardt
- 2011

An integer is called evil if the number of ones in its binary expansion is even and odious if the number of ones in the binary expansion is odd. If we look at the integers between 0 and 15 we find that Next we say that if two consecutive integers are evil then this is a pair of evil twins and that if two consecutive integers are odious then this is a pair… (More)

- CHRIS BERNHARDT
- 2002

Time-symmetric cycles are defined. Unimodal time-symmetric cycles are classified. A one-parameter family of maps of the closed unit interval that exhibits this classification is given. 1. Introduction. In one-dimensional dynamics, the maps that are studied are usually noninvertible. The initial reason for studying these maps was that they had complicated… (More)

- CHRIS BERNHARDT
- 2002

The main result of this note is that given any two time-symmetric cycles, one can find a time-symmetric extension of one by the other. This means that given a time-symmetric cycle, both time-symmetric doubles and square roots can be found. 1. Introduction. In [4], the idea of time-symmetric cycles was introduced. The basic idea underlying the definition of… (More)

- CHRIS BERNHARDT
- 1996

The forcing relation on n-modal cycles is studied. If α is an n-modal cycle then the n-modal cycles with block structure that force α form a 2 n-horseshoe above α. If n-modal β forces α, and β does not have a block structure over α, then β forces a 2-horseshoe of simple extensions of α.

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