An Identity Conjectured by Lacasse via the Tree Function

Prodinger, Helmut

doi:10.37236/3309

Cited by 7 publications

(4 citation statements)

References 5 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Proofs of Lacasse's conjecture were given by Chen et al [11], Prodinger [59], Sun [73], and Younsi [80]. We will prove (3.2.6) by showing that both sides are equal to T (x)/(1 − T (x)) 3 .…”

Section: A Rational Function Expansionmentioning

confidence: 79%

Lagrange inversion

Gessel

2016

Journal of Combinatorial Theory, Series A

View full text Add to dashboard Cite

show abstract

Section: A Rational Function Expansionmentioning

confidence: 79%

Lagrange inversion

Gessel

2016

Journal of Combinatorial Theory, Series A

View full text Add to dashboard Cite

show abstract

“…3). To this end, it is useful to consider the generating function F (z; q, r) of the weighted Riordan paths, which can be obtained by the so called "Kernel method" [35]. The autocorrelator in terms of F (z; q, r) reads…”

Section: Autocorrelator At T = ∞mentioning

confidence: 99%

Dissipative spin chain as a non-Hermitian Kitaev ladder

2019

View full text Add to dashboard Cite

We derive exact results for the Lindblad equation for a quantum spin chain (one-dimensional quantum compass model) with dephasing noise. The system possesses doubly degenerate nonequilibrium steady states due to the presence of a conserved charge commuting with the Hamiltonian and Lindblad operators. We show that the system can be mapped to a non-Hermitian Kitaev model on a two-leg ladder, which is solvable by representing the spins in terms of Majorana fermions. This allows us to study the Liouvillian gap, the inverse of relaxation time, in detail. We find that the Liouvillian gap increases monotonically when the dissipation strength γ is small, while it decreases monotonically for large γ, implying a kind of phase transition in the first decay mode. The Liouvillian gap and the transition point are obtained in closed form in the case where the spin chain is critical. We also obtain the explicit expression for the autocorrelator of the edge spin. The result implies the suppression of decoherence when the spin chain is in the topological regime.

show abstract

“…[2], [5], [20], [23]), the proof we present remains, as far as we know, the only one providing equivalent expressions for the functions ξ and ξ 2 that are simpler and more convenient from a numerical perspective. Furthermore, none of the aforementioned papers [2], [5], [20], [23] discuss the context and relevance of Conjecture 2.1 in the theory of machine learning.…”

mentioning

confidence: 91%

A combinatorial conjecture from PAC-Bayesian machine learning

Younsi,

Lacasse

2020

Preprint

View full text Add to dashboard Cite

show abstract

An Identity Conjectured by Lacasse via the Tree Function

Abstract: A. Lacasse conjectured a combinatorial identity in his study of learning theory. Various people found independent proofs. Here is another one that is based on the study of the tree function, with links to Lamberts W -function and Ramanujan's Q-function. It is particularly short.

Cited by 7 publications

References 5 publications

Lagrange inversion

Lagrange inversion

Dissipative spin chain as a non-Hermitian Kitaev ladder

A combinatorial conjecture from PAC-Bayesian machine learning

Contact Info

Product

Resources

About