D21 Proyectos de Investigaciónhttps://hdl.handle.net/10902/1522024-03-01T06:47:33Z2024-03-01T06:47:33ZLow-energy points on the sphere and the real projective planeBeltrán Álvarez, CarlosEtayo Rodríguez, UjuéLópez Gómez, Pedro Ramónhttps://hdl.handle.net/10902/319282024-02-26T15:28:35Z2023-06-01T00:00:00ZLow-energy points on the sphere and the real projective plane
Beltrán Álvarez, Carlos; Etayo Rodríguez, Ujué; López Gómez, Pedro Ramón
We present a generalization of a family of points on S², the Diamond ensemble, containing collections of N points on S² with very small logarithmic energy for all N∈N. We extend this construction to the real projective plane RP² and we obtain upper and lower bounds with explicit constants for the Green and logarithmic energy on this last space.
2023-06-01T00:00:00ZA Lower Bound for the Logarithmic Energy on S² and for the Green Energy on SⁿBeltrán Álvarez, CarlosLizarte López, Fátimahttps://hdl.handle.net/10902/319272024-02-26T15:18:04Z2023-12-01T00:00:00ZA Lower Bound for the Logarithmic Energy on S² and for the Green Energy on Sⁿ
Beltrán Álvarez, Carlos; Lizarte López, Fátima
We show an alternative proof of the sharpest known lower bound for the logarithmic energy on the unit sphere S². We then generalize this proof to get new lower bounds for the Green energy on the unit n-sphere Sⁿ.
2023-12-01T00:00:00ZOn gegenbauer point processes on the unit intervalBeltrán Álvarez, CarlosDelgado, AntoniaFernández, LidiaSánchez-Lara, Joaquínhttps://hdl.handle.net/10902/319262024-02-27T07:34:40Z2024-01-01T00:00:00ZOn gegenbauer point processes on the unit interval
Beltrán Álvarez, Carlos; Delgado, Antonia; Fernández, Lidia; Sánchez-Lara, Joaquín
In this paper we compute the logarithmic energy of points in the unit interval [-1,1] chosen from different Gegenbauer Determinantal Point Processes. We check that all the different families of Gegenbauer polynomials yield the same asymptotic result to third order, we compute exactly the value for Chebyshev polynomials and we give a closed expression for the minimal possible logarithmic energy. The comparison suggests that DPPs cannot match the value of the minimum beyond the third asymptotic term.
2024-01-01T00:00:00ZAnalytic semiroots for plane branches and singular foliationsCano Torres, FelipeCorral Pérez, NuriaSenovilla Sanz, Davidhttps://hdl.handle.net/10902/318342024-02-21T11:27:34Z2023-05-01T00:00:00ZAnalytic semiroots for plane branches and singular foliations
Cano Torres, Felipe; Corral Pérez, Nuria; Senovilla Sanz, David
The analytic moduli of equisingular plane branches has the semimodule of differential values as the most relevant system of discrete invariants. Focusing in the case of cusps, the minimal system of generators of this semimodule is reached by the differential values attached to the differential 1-forms of the so-called standard bases. We can complete a standard basis to an extended one by adding a last differential 1-form that has the considered cusp as invariant branch and the "correct" divisorial order. The elements of such extended standard bases have the "cuspidal" divisor as a "totally dicritical divisor" and hence they define packages of plane branches that are equisingular to the initial one. These are the analytic semiroots. In this paper we prove
that the extended standard bases are well structured from this geometrical and foliated viewpoint, in the sense that the semimodules of differential values of the branches in the dicritical packages are described just by a truncation of the list of generators of the initial semimodule at the corresponding differential value. In particular they have all the same semimodule of differential values.
2023-05-01T00:00:00Z