117,292 research outputs found
The 99th Fibonacci Identity
We provide elementary combinatorial proofs of several Fibonacci and Lucas number identities left open in the book Proofs That Really Count [1], and generalize these to Gibonacci sequences Gn that satisfy the Fibonacci recurrence, but with arbitrary real initial conditions. We offer several new identities as well.
[1] A. T. Benjamin and J. J. Quinn, Proofs That Really Count: The Art of Combinatorial Proof, The Dolciani Mathematical Expositions, 27, Mathematical Association of America, Washington, DC, 200
Tiling approach to obtain identities for generalized Fibonacci and Lucas numbers
In Proofs that Really Count [2], Benjamin and Quinn have used “square
and domino tiling” interpretation to provide tiling proofs of many Fibonacci
and Lucas formulas. We explore this approach in order to provide tiling
proofs of some generalized Fibonacci and Lucas identities.
Keywords: Generalized Fibonacci and Lucas numbers; Tiling proofs
A note on coherent orientations for exact Lagrangian cobordisms
Let be a spin, exact Lagrangian cobordism
in the symplectization of the 1-jet space of a smooth manifold . Assume that
has cylindrical Legendrian ends . It is well
known that the Legendrian contact homology of can be defined with
integer coefficients, via a signed count of pseudo-holomorphic disks in the
cotangent bundle of . It is also known that this count can be lifted to a
mod 2 count of pseudo-holomorphic disks in the symplectization , and that induces a morphism between the -valued DGA:s of the ends in a functorial way. We prove that
this hold with integer coefficients as well. The proofs are built on the
technique of orienting the moduli spaces of pseudo-holomorphic disks using
capping operators at the Reeb chords. We give an expression for how the DGA:s
change if we change the capping operators.Comment: 41 pages, final version, accepted for publication in Quantum
Topology. More details have been added to some of the proof
- …