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Fernando Chamizo

Publications and source records attributed to Fernando Chamizo.

At least 19 recordsLinked to original sources
arXiv2025

Nonparametric estimation of the multivariate Spearman's footrule: a further discussion

In this paper, we propose two new estimators of the multivariate rank correlation coefficient Spearman's footrule which are based on two general estimators for Average Orthant Dependence measures. We compare the new proposals with a previous estimator existing in the literature and show that the three estimators are asymptotically equivalent, but, in small samples, one of the proposed estimators outperforms the others. We also analyse Pitman efficiency of these indices to test for multivariate independence as compared to multivariate versions of Kendall's tau and Spearman's rho.

math.ST↗
arXiv2024

Plateaux of probability for the expanded quantum infinite well

If the standard 1D quantum infinite potential well initially in its ground state suffers a sudden expansion, it turns out that in the evolution of the system they may appear plateaux of probability for some fractional times, as noticed by C. Aslangul in 2008. We introduce a mathematical framework to explain this phenomenon. Remarkably, the characterization of these plateaux depends on nontrivial number theoretical considerations.

quant-ph↗
arXiv2024

On sets with missing differences in compact abelian groups

A much-studied problem posed by Motzkin asks to determine, given a finite set $D$ of integers, the so-called Motzkin density for $D$, i.e., the supremum of upper densities of sets of integers whose difference set avoids $D$. We study the natural analogue of this problem in compact abelian groups. Using ergodic-theoretic tools, this is shown to be equivalent to the following discrete problem: given a lattice $\Lambda\subset \mathbb{Z}^r$, letting $D$ be the image in $\mathbb{Z}^r/\Lambda$ of the standard basis, determine the Motzkin density for $D$ in $\mathbb{Z}^r/\Lambda$. We study in particular the periodicity question: is there a periodic $D$-avoiding set of maximal density in $\mathbb{Z}^r/\Lambda$? The Greenfeld--Tao counterexample to the periodic tiling conjecture implies that the answer can be negative. On the other hand, we prove that the answer is positive in several cases, including the case rank$(\Lambda)=1$ (in which we give a formula for the Motzkin density), the case rank$(\Lambda)=r-1$, and hence also the case $r\leq 3$. It follows that, for up to three missing differences, the Motzkin density in a compact abelian group is always a rational number.

math.CO↗
arXiv2024

Regular Polygonal Vortex Filament Evolution and Exponential Sums

In this paper, we give a rigorous proof for the expression of the angle between adjacent sides in the skew polygons appearing at rational times in the evolution of regular polygons of $M$ sides under the vortex filament equation. The proof depends on showing that some exponential sums with arithmetic content are purely imaginary.

math.NT↗
arXiv2023

Exact quantum revivals for the Dirac equation

In the present work, the results obtained in [1] about the revivals of a relativistic fermion wave function on a torus are considerably enlarged. In fact, all the possible quantum states exhibiting revivals are fully characterized. The revivals are exact, that is, are true revivals without taking any particular limit such as the non relativistic one. The present results are of interest since they generalize the Talbot effect and the revivals of the Schr\"odinger equation to a relativistic situation with non zero mass. This makes the problem nontrivial, as the dispersion relation is modified and is not linear. The present results are obtained by the use of arithmetic tools which are described in certain detail. In addition, several plots of the revivals are presented, which are useful for exemplifying the procedure proposed along the text.

quant-ph↗
arXiv2023

Correlation and lower bounds of arithmetic expressions

We explore the use of correlation with simple functions to get lower bounds for arithmetic quantities. In particular, we apply this idea to the power moments of the error term when counting visible lattice points in large spheres.

math.NT↗
arXiv2023

About the quantum Talbot effect on the sphere

The Schr\"odinger equation on a circle with an initially localized profile of the wave function is known to give rise to revivals or replications, where the probability density of the particle is partially reproduced at rational times. As a consequence of the convolutional form of the general solution it is deduced that a piecewise constant initial wave function remains piecewise constant at rational times as well. For a sphere instead, it is known that this piecewise revival does not necessarily occur, indeed the wave function becomes singular at some specific locations at rational times. It may be desirable to study the same problem, but with an initial condition being a localized Dirac delta instead of a piecewise constant function, and this is the purpose of the present work. By use of certain summation formulas for the Legendre polynomials together with properties of Gaussian sums, it is found that revivals on the sphere occur at rational times for some specific locations, and the structure of singularities of the resulting wave function is characterized in detail. In addition, a partial study of the regions where the density vanishes, named before valley of shadows in the context of the circle, is initiated here. It is suggested that, differently from the circle case, these regions are not lines but instead some specific set of points along the sphere. A conjecture about the precise form of this set is stated and the intuition behind it is clarified.

quant-ph↗
arXiv2022

Current induction and macroscopic forces for superconducting strings

Vortons are extended superconducting rings, which hypothetically may play a role in cosmology and even may have significance in connection with cosmic rays of high energy. Some of these objects are able to confine fermions which consequently become massless in the core of the object \cite{witten}, \cite{vorton3}. These fermions travel at light speed in the core and may generate a large current without dissipation. This raises interest about the generation mechanisms for these currents inside the defect. This question is analyzed here by studying the inverse photoelectric effect for these objects namely, the absorption of a fermion with the consequent emission of a photon or a massive boson by the extended defect. Another motivation for the present work is that there exists a discussion in condensed matter about the role of the bound spectrum in the macroscopic Magnus force that the vortex experiences in certain type of superfluids or superconductors. The discussion is about wether the main force comes from scattering of these fermions by the object or by the effect of the environment on the bound states in the object, which may induce a spectral flow leading to an effective macroscopic force \cite{pelea1}, \cite{pelea2}, \cite{pelea3}. Without claiming that the results described here are conclusive in the context of condensed matter, this work presents a comparison between these two effects for vortons interacting with a plasma of fermions.

hep-th↗
arXiv2022

Extendable orthogonal sets of integral vectors

Motivated by a model in quantum computation we study orthogonal sets of integral vectors of the same norm that can be extended with new vectors keeping the norm and the orthogonality. Our approach involves some arithmetic properties of the quaternions and other hypercomplex numbers.

math.NT↗
arXiv2021

A simple evaluation of a theta value and the Kronecker limit formula

We evaluate the classic sum $\sum_{n\in\mathbb{Z}} e^{-\pi n^2}$. The novelty of our approach is that it does not require any prior knowledge about modular forms, elliptic functions or analytic continuations. Even the $\Gamma$ function, in terms of which the result is expressed, only appears as a complex function in the computation of a real integral by the residue theorem. Another contribution of this note is to provide a very simple proof of the Kronecker limit formula.

math.NT↗
arXiv2020

On an integral identity

We give three elementary proofs of a nice equality of definite integrals, which arises from the theory of bivariate hypergeometric functions, and has connections with irrationality proofs in number theory. We furthermore provide a generalization together with an equally elementary proof and discuss some consequences.

math.CA↗
arXiv2019

Pointwise monotonicity of heat kernels

In this paper the authors present a proof of a pointwise radial monotonicity property of heat kernels that is shared by the euclidean spaces, spheres and hyperbolic spaces. The main result deals with monotonicity from special points on revolution hypersurfaces from which the aforementioned are deduced. The proof relies on a non straightforward but elementary application of the parabolic maximum principle.

math.CA↗
arXiv2018

Fourier series in BMO with number theoretical implications

We introduce an elementary argument to bound the $\textrm{BMO}$ seminorm of Fourier series with gaps giving in particular a sufficient condition for them to be in this space. Using finer techniques we carry out a detailed study of the series $\sum n^{-1}e^{2πi n^2 x}$ providing some insight into how much this $\text{BMO}$ Fourier series differs from defining an $L^\infty$ function.

math.CA↗
arXiv2018

Invariant subspaces for Bishop operators and beyond

Bishop operators $T_α$ acting on $L^2[0,1)$ were proposed by E. Bishop in the fifties as possible operators which might entail counterexamples for the Invariant Subspace Problem. We prove that all the Bishop operators are biquasitriangular and, derive as a consequence that they are norm limits of nilpotent operators. Moreover, by means of arithmetical techniques along with a theorem of Atzmon, the set of irrationals $α\in (0,1)$ for which $T_α$ is known to possess non-trivial closed invariant subspaces is considerably enlarged, extending previous results by Davie, MacDonald and Flattot. Furthermore, we essentially show that when our approach fails to produce invariant subspaces it is actually because Atzmon Theorem cannot be applied. Finally, upon applying arithmetical bounds obtained, we deduce local spectral properties of Bishop operators proving, in particular, that neither of them satisfy the Dunford property $(C)$.

math.FA↗
arXiv2018

A short proof of sharp Weyl's law for the special orthogonal group

We give a short proof of a strong form of Weyl's law for $\text{SO}(N)$ using well known facts of the theory of modular forms. The exponent of the error term is sharp when the rank is at least~$4$. We also discuss the cases with smaller rank improving previous results.

math.AP↗