6,849 research outputs found
Free Fibonacci Sequences
This paper describes a class of sequences that are in many ways similar to
Fibonacci sequences: given n, sum the previous two terms and divide them by the
largest possible power of n. The behavior of such sequences depends on n. We
analyze these sequences for small n: 2, 3, 4, and 5. Surprisingly these
behaviors are very different. We also talk about any n. Many statements about
these sequences are difficult or impossible to prove, but they can be supported
by probabilistic arguments, we have plenty of those in this paper.
We also introduce ten new sequences. Most of the new sequences are also
related to Fibonacci numbers proper, not just free Fibonacci numbers.Comment: 18 page
Finite sections of the Fibonacci Hamiltonian
We study finite but growing principal square submatrices of the one- or
two-sided infinite Fibonacci Hamiltonian . Our results show that such a
sequence , no matter how the points of truncation are chosen, is always
stable -- implying that is invertible for sufficiently large and
pointwise
Hidden dimers and the matrix maps: Fibonacci chains re-visited
The existence of cycles of the matrix maps in Fibonacci class of lattices is
well established. We show that such cycles are intimately connected with the
presence of interesting positional correlations among the constituent `atoms'
in a one dimensional quasiperiodic lattice. We particularly address the
transfer model of the classic golden mean Fibonacci chain where a six cycle of
the full matrix map exists at the centre of the spectrum [Kohmoto et al, Phys.
Rev. B 35, 1020 (1987)], and for which no simple physical picture has so far
been provided, to the best of our knowledge. In addition, we show that our
prescription leads to a determination of other energy values for a mixed model
of the Fibonacci chain, for which the full matrix map may have similar cyclic
behaviour. Apart from the standard transfer-model of a golden mean Fibonacci
chain, we address a variant of it and the silver mean lattice, where the
existence of four cycles of the matrix map is already known to exist. The
underlying positional correlations for all such cases are discussed in details.Comment: 14 pages, 2 figures. Submitted to Physical Review
The switch operators and push-the-button games: a sequential compound over rulesets
We study operators that combine combinatorial games. This field was initiated
by Sprague-Grundy (1930s), Milnor (1950s) and Berlekamp-Conway-Guy (1970-80s)
via the now classical disjunctive sum operator on (abstract) games. The new
class consists in operators for rulesets, dubbed the switch-operators. The
ordered pair of rulesets (R 1 , R 2) is compatible if, given any position in R
1 , there is a description of how to move in R 2. Given compatible (R 1 , R 2),
we build the push-the-button game R 1 R 2 , where players start by playing
according to the rules R 1 , but at some point during play, one of the players
must switch the rules to R 2 , by pushing the button ". Thus, the game ends
according to the terminal condition of ruleset R 2. We study the pairwise
combinations of the classical rulesets Nim, Wythoff and Euclid. In addition, we
prove that standard periodicity results for Subtraction games transfer to this
setting, and we give partial results for a variation of Domineering, where R 1
is the game where the players put the domino tiles horizontally and R 2 the
game where they play vertically (thus generalizing the octal game 0.07).Comment: Journal of Theoretical Computer Science (TCS), Elsevier, A
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