arXiv subjects

Primitivo Acosta-Humánez

Publications and source records attributed to Primitivo Acosta-Humánez.

5 recordsLinked to original sources

arXiv2024

Non-integrability of the axisymmetric Bianchi IX cosmological model via Differential Galois Theory

We investigate the integrability of an anisotropic universe with matter and cosmological constant formulated as Bianchi IX models. The presence of the cosmological constant causes the existence of a critical point in the finite part of the phase space. The separatrix associated to this Einstein's static universe is entirely contained in an invariant isotropic plane forgetting the singularity at the origin. This invariant plane of isotropy is an integrable sub-space of the Taub type. In this paper we analyse the differential Galois group of the second order variational equations to this plane in order to apply the integrability theorem of the second author with Ramis and Sim\' o. The main result is that the model is non-integrable by meromorphic functions.

math.DS↗

arXiv2023

Darboux Transformations for Orthogonal Differential Systems and Differential Galois Theory

Darboux developed an ingenious algebraic mechanism to construct infinite chains of ''integrable'' second-order differential equations as well as their solutions. After a surprisingly long time, Darboux's results were rediscovered and applied in many frameworks, for instance in quantum mechanics (where they provide useful tools for supersymmetric quantum mechanics), in soliton theory, Lax pairs and many other fields involving hierarchies of equations. In this paper, we propose a method which allows us to generalize the Darboux transformations algorithmically for tensor product constructions on linear differential equations or systems. We obtain explicit Darboux transformations for third-order orthogonal systems ($\mathfrak{so}(3, C_K)$ systems) as well as a framework to extend Darboux transformations to any symmetric power of $\mathrm{SL}(2,\mathbb{C})$-systems. We introduce SUSY toy models for these tensor products, giving as an illustration the analysis of some shape invariant potentials. All results in this paper have been implemented and tested in the computer algebra system Maple.

math.CA↗

arXiv2016

Differential Galois Groups and Representation of Quivers for Seismic Models with Constant Hessian of Square of Slowness

The trajectory of energy is modeled by the solution of the Eikonal equation, which can be solved by solving a Hamiltonian system. This system is amenable of treatment from the point view of the theory of Differential Algebra. In particular, by Morales-Ramis theory it is possible to analyze integrable Hamiltonian systems through the abelian structure of their variational equations. In this paper we obtain the abelian differential Galois groups and the representation of the quiver, that allow us to obtain such abelian differential Galois groups, for some seismic models with constant Hessian of square of slowness, proposed in Cerveny 2001, which are equivalent to linear Hamiltonian systems with three uncoupled harmonic oscillators.

math-ph↗

arXiv2013

Frames of subspaces in Hilbert spaces with $W$-metrics

If $\left(\h,\langle\cdot,\cdot\rangle\right)$ is a Hilbert space and on it we consider the sesquilinear form $\langle\,W\cdot,\cdot\rangle$ so-called $W$-metric, where $W^{*}=W\in\BH$, and $\ker\,W=\{0\}$, then the space $\left(\h,\langle\,W\cdot,\cdot\rangle\right)$ is called Hilbert space with $W$-metric or simply $W$-space. In this paper we investigate the dynamic of frames of subspace on these spaces, where the sense of dynamics refers to the behavior of frames of subspace in $\h_{W}$ (the completion of $\left(\h,\langle\,W\cdot,\cdot\rangle\right)$) comparing with $\h$ and vice versa. This work is based on the study made in \cite{KEFER,GMMM} on frames in Krein spaces. In a similar way, Casazza and Kutyniok obtained some results in the context of Hilbert spaces, see \cite{CG}. We take tools of theory of $C^{*}$-algebra, and properties of $\BH$, to show that every Hilbert space with $W$-metric $\h_{W}$ with $0\inσ(W)$ has a decomposition $$\h_{W}=\bigoplus_{n\in\N\cup\{\infty\}}\h_{ψ_{n}}^{W},$$ where $\h_{ψ_{n}}^{W}\simeq \Ele(σ(W),x\,dμ_{n}(x))$ are Krein spaces, for every $n\in\N\cup\{\infty\}$. Moreover, we investigate the dynamics of frames of subspace when the self-adjoint operator $W$ is unbounded.

math.FA↗

arXiv2013

Pasting and Reversing Operations over some Vector Spaces

Pasting and Reversing operations have been used successfully over the set of integer numbers, simple permutations, rings and recently over a generalized vector product. In this paper, these operations are defined from a natural way to be applied over vector spaces. In particular we study Pasting and Reversing over vectors, matrices and we rewrite some properties for polynomials as vector space. Finally we study some properties of Reversing through linear transformations as for example an analysis of palindromic and antipalindromic vector subspaces.

math.HO↗