arXiv subjects

Juan M. Burgos

Publications and source records attributed to Juan M. Burgos.

5 recordsLinked to original sources

arXiv2022

Quantum network reliability with perfect nodes

We introduce the concept of quantum reliability as an extension of the concept of network reliability in the context of quantum networks. We show that this concept is intimately related to the concept of quantum reliability operator that we also introduce and show a Negami like splitting formula for it. Considering that the simple factorization formula for classical networks which is the basis of most of the calculation algorithms does not hold in the quantum context due to entanglement, a Negami like splitting for the quantum reliability operator becomes relevant.

quant-ph↗

arXiv2022

On the Lagrange-Dirichlet converse in dimension three

Consider a mechanical system with a real analytic potential. We prove that in dimension three, there is an open and dense subset of the set of non strict local minimums of the potential such that every one of its points is a Lyapunov unstable equilibrium point.

math.DS↗

arXiv2021

Adelic toric varieties and adelic loop groups

We study the proalgebraic space which is the inverse limit of all finite branched covers over a normal toric variety with branching set the invariant divisor under the algebraic torus action. These are completions (compactifications) of the adelic abelian algebraic group which is the profinite completion of the algebraic torus. We prove that the vector bundle category of the proalgebraic toric completion of a toric variety is the direct limit of the respective categories of the finite toric varieties coverings defining the completion. In the case of the complex projective line we obtain as proalgebraic completion the adelic projective line P. We define holomorphic vector bundles over P. We also introduce the smooth, Sobolev and Wiener adelic loop groups and the corresponding Grassmannans; we describe their properties and prove Birkhoff's factorization for these groups. We prove that the adelic Picard group of holomorphic line bundles is isomorphic to the rationals and prove the Birkhoff-Grothendieck splitting theorem for holomorphic bundles of higher rank over P.

math.AG↗

arXiv2019

Negami's like splitting formula for the Jones polynomial

Negami's splitting formula cannot be directly applied to get a Jones polynomial splitting formula for the contraction of certain planar graphs in the decomposition become non planar. Therefore, we build a Negami's like splitting formula from the scratch. Now, the new splitting matrix doesn't have the form of a Lindström matrix and it would be interesting to have similar results for it.

math.CO↗

arXiv2019

Adelic solenoid II: Ahlfors-Bers theory

We generalize the Ahlfors-Bers theory to the adelic Riemann sphere. In particular, after defining the appropriate notion of a Beltrami differential in the solenoidal context, we give a sufficient condition on it such that the corresponding Beltrami equation has a quasiconformal homeomorphism solution; i.e. The Ahlfors-Bers Theorem in the solenoidal case. This additional condition on the solenoidal Beltrami differentials can be written as a Banach norm in a subspace of solenoidal differentials. Moreover, this subspace is the completion under this norm of those solenoidal differentials locally constant at the fiber. As a toy example, we show how this technique works on a linear problem: We generalize the diophantine equation complex analytic extension problem to the respective solenoidal space.

math.CV↗