Chris Bernhardt

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The tightly regulated expression patterns of structural cell wall proteins in several plant species indicate that they play a crucial role in determining the extracellular matrix structure for specific cell types. We demonstrate that AtPRP3, a proline-rich cell wall protein in Arabidopsis, is expressed in root-hair-bearing epidermal cells at the root/shoot(More)
We have characterized the molecular organization and expression of four proline-rich protein genes from Arabidopsis (AtPRPs). These genes predict two classes of cell wall proteins based on DNA sequence identity, repetitive motifs, and domain organization. AtPRP1 and AtPRP3 encode proteins containing an N-terminal PRP-like domain followed by a C-terminal(More)
Let T be a tree with n vertices. Let f : T → T be continuous and suppose that the n vertices form a periodic orbit under f . We show: (1) (a) If n is not a divisor of 2k then f has a periodic point with period 2k. (b) If n = 2pq, where q > 1 is odd and p ≥ 0, then f has a periodic point with period 2pr for any r ≥ q. (c) The map f also has periodic orbits(More)
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)
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 0, 3, 5, 6, 9, 10, 12, 15 are evil and that 1, 2, 4, 7, 8, 11, 13, 14 are odious. Next we say that if two consecutive integers are evil then this is a pair(More)
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)
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)
The forcing relation on n-modal cycles is studied. If α is an nmodal cycle then the n-modal cycles with block structure that force α form a 2n-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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