Fernando Chamizo
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View article: Regular Polygonal Vortex Filament Evolution and Exponential Sums
Regular Polygonal Vortex Filament Evolution and Exponential Sums Open
When taking a regular planar polygon of $M$ sides and length $2\pi $ as the initial datum of the vortex filament equation, $\mathbf{X}_{t}= \mathbf{X}_{s}\wedge \mathbf{X}_{ss}$ , the solution becomes polygonal at times …
View article: Plateaux of probability for the expanded quantum infinite well
Plateaux of probability for the expanded quantum infinite well Open
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 …
View article: On sets with missing differences in compact abelian groups
On sets with missing differences in compact abelian groups Open
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 n…
View article: Regular Polygonal Vortex Filament Evolution and Exponential Sums
Regular Polygonal Vortex Filament Evolution and Exponential Sums Open
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…
View article: Correlation and lower bounds of arithmetic expressions
Correlation and lower bounds of arithmetic expressions Open
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.
View article: Exact quantum revivals for the Dirac equation
Exact quantum revivals for the Dirac equation Open
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 rev…
View article: Correlation and lower bounds of arithmetic expressions
Correlation and lower bounds of arithmetic expressions Open
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.
View article: About the quantum Talbot effect on the sphere
About the quantum Talbot effect on the sphere Open
The Schrödinger 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 …
View article: Current induction and macroscopic forces for superconducting strings
Current induction and macroscopic forces for superconducting strings Open
Superconducting strings are topological defects appearing in cosmological early stage models, in tentative explanations of the high energy cosmic rays, galaxy formation and even in condensed matter to deal with some kind of superconductors…
View article: About the quantum Talbot effect on the sphere
About the quantum Talbot effect on the sphere Open
The Schrödinger 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 …
View article: Current induction and macroscopic forces for superconducting strings
Current induction and macroscopic forces for superconducting strings Open
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 consequent…
View article: Extendable orthogonal sets of integral vectors
Extendable orthogonal sets of integral vectors Open
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 th…
View article: Extendable orthogonal sets of integral vectors
Extendable orthogonal sets of integral vectors Open
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…
View article: On an Integral Identity
On an Integral Identity Open
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 generali…
View article: A simple evaluation of a theta value and the Kronecker limit formula
A simple evaluation of a theta value and the Kronecker limit formula Open
We evaluate the classic sum $\sum_{n\in\mathbb{Z}} e^{-πn^2}$. The novelty of our approach is that it does not require any prior knowledge about modular forms, elliptic functions or analytic continuations. Even the $Γ$ function, in terms o…
View article: The additive problem for the number of representations as a sum of two squares
The additive problem for the number of representations as a sum of two squares Open
We improve a previous unconditional result about the asymptotic behavior of $\sum_{n\le x} r(n)r(n+m)$ with $r(n)$ the number of representations of $n$ as a sum of two squares when $m$ may vary with $x$.
View article: Lattice points in bodies of revolution II
Lattice points in bodies of revolution II Open
In [3] it was shown that when a three‐dimensional smooth convex body has rotational symmetry around a coordinate axis one can find better bounds for the lattice point discrepancy than what is known for more general convex bodies. To accomp…
View article: Morphisms and period matrices
Morphisms and period matrices Open
This Accepted Manuscript will be available for reuse under a CC BY-NC-ND licence after 24 months of embargo period
View article: Fourier series in BMO with number theoretical implications
Fourier series in BMO with number theoretical implications Open
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 seri…
View article: Pointwise monotonicity of heat kernels
Pointwise monotonicity of heat kernels Open
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 rev…
View article: A short proof of sharp Weyl's law for the special orthogonal group
A short proof of sharp Weyl's law for the special orthogonal group Open
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 ra…
View article: Tachyonic instabilities in 2 + 1 dimensional Yang–Mills theory and its connection to number theory
Tachyonic instabilities in 2 + 1 dimensional Yang–Mills theory and its connection to number theory Open
We consider the $2+1$ dimensional Yang-Mills theory with gauge group\n$\\text{SU}(N)$ on a flat 2-torus under twisted boundary conditions. We study\nthe possibility of phase transitions (tachyonic instabilities) when $N$ and the\nvolume va…
View article: Lattice points in elliptic paraboloids
Lattice points in elliptic paraboloids Open
We consider the lattice point problem corresponding to a family of elliptic paraboloids in $\mathbb{R}^d$ with $d\ge3$ and we prove the expected to be optimal exponent, improving previous results. This is especially noticeable for $d=3$ be…