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Luis Vega

Publications and source records attributed to Luis Vega.

At least 55 records · Page 3Linked to original sources

A Hardy-type inequality and some spectral characterizations for the Dirac-Coulomb operator

We prove a sharp Hardy-type inequality for the Dirac operator. We exploit this inequality to obtain spectral properties of the Dirac operator perturbed with Hermitian matrix-valued potentials $\mathbf V$ of Coulomb type: we characterise its eigenvalues in terms of the Birman-Schwinger principle and we bound its discrete spectrum from below, showing that the \emph{ground-state energy} is reached if and only if $\mathbf V$ verifies some {rigidity} conditions. In the particular case of an electrostatic potential, these imply that $\mathbf V$ is the Coulomb potential.

math.AP↗

On the improvement of the Hardy inequality due to singular magnetic fields

We establish magnetic improvements upon the classical Hardy inequality for two specific choices of singular magnetic fields. First, we consider the Aharonov-Bohm field in all dimensions and establish a sharp Hardy-type inequality that takes into account both the dimensional as well as the magnetic flux contributions. Second, in the three-dimensional Euclidean space, we derive a non-trivial magnetic Hardy inequality for a magnetic field that vanishes at infinity and diverges along a plane.

math.SP↗

Asymptotics in Fourier space of self-similar solutions to the modified Korteweg-de Vries equation

We give the asymptotics of the Fourier transform of self-similar solutions to the modified Korteweg-de Vries equation, through a fixed point argument in weighted W^{1,\infty} around a carefully chosen, two term ansatz. Such knowledge is crucial in the study of stability properties of the self-similar solutions for the modified Korteweg-de Vries flow. In the defocusing case, the self-similar profiles are solutions to the Painlevé II equation. Although they were extensively studied in physical space, no result to our knowledge describe their behavior in Fourier space. We are able to relate the constants involved in the description in Fourier space with those involved in the description in physical space.

math.AP↗

Asymmetric emission of the [OIII]$λ$5007 profile in narrow-line Seyfert 1 galaxies

Many active galactic nuclei (AGN) and particularly narrow-line Seyfert 1 (NLS1) galaxies, usually exhibit blueshifts and blue wings in several emission lines, which are mainly associated with outflows and strong winds. In order to study the radial velocity difference between the narrow component of H$β$ and the core component of [OIII]$λ$5007 and the asymmetric emission of this forbidden line, we investigate a sample of NLS1 galaxies . One of the aims of this paper is to analyze the blue wings of the [OIII]$λ$5007 profiles and their relation with the central engine. We have obtained and studied medium-resolution spectra (190 km s$^{-1}$ FWHM at H$β$) of a sample of 28 NLS1 galaxies in the optical range 4300 - 5200Å. We performed Gaussian decomposition to the H$β$ and [OIII]$λλ$4959,5007 emission profiles in order to study the distinct components of these lines. A new blue outlier galaxy is found, in which the center of the core component of [OIII] is blueshifted by 405 km s$^{-1}$ relative to the center of the narrow component of H$β$ line. We confirmed a previously known correlation between the blueshift and the full width half maximum (FWHM) of the core component of [OIII]$λ$5007 line. We also corroborated the correlation between the latter and the velocity of the centroid of the blue wing. On the other hand, by studying the radial velocity difference between the blue end of the asymmetric emission and the centroid of the core component of [OIII], we found a correlation between it and the central black hole mass and, therefore, with the luminosity of the broad component of H$β$. Finally, we found a moderate correlation between the luminosity of the [OIII] blue wing and the black hole mass.

astro-ph.GA↗

Absence of eigenvalues of two-dimensional magnetic Schroedinger operators

By developing the method of multipliers, we establish sufficient conditions on the electric potential and magnetic field which guarantee that the corresponding two-dimensional Schroedinger operator possesses no point spectrum. The settings of complex-valued electric potentials and singular magnetic potentials of Aharonov-Bohm field are also covered.

math.SP↗

On the Relationship between the One-Corner Problem and the $M$-Corner Problem for the Vortex Filament Equation

In this paper, we give evidence that the evolution of the Vortex Filament Equation for a regular $M$-corner polygon as initial datum can be explained at infinitesimal times as the superposition of $M$ one-corner initial data. Therefore, and due to periodicity, the evolution at later times can be understood as the nonlinear interaction of infinitely many filaments, one for each corner. This interaction turns out to be some kind of nonlinear Talbot effect. We also give very strong numerical evidence of the transfer of energy and linear momentum for the $M$-corner case.

math.NA↗

Singularity formation for the 1-D cubic NLS and the Schrödinger map on $\mathbb S^2$

In this note we consider the 1-D cubic Schrödinger equation with data given as small perturbations of a Dirac-$δ$ function and some other related equations. We first recall that although the problem for this type of data is ill-posed one can use the geometric framework of the Schrödinger map to define the solution beyond the singularity time. Then, we find some natural and well defined geometric quantities that are not regular at time zero. Finally, we make a link between these results and some known phenomena in fluid mechanics that inspired this note.

math.AP↗

A strategy for self-adjointness of Dirac operators: Applications to the MIT bag model and delta-shell interactions

We develop an approach to prove self-adjointness of Dirac operators with boundary or transmission conditions at a $\mathcal{C}^2$-compact surface without boundary. To do so we are lead to study the layer potential induced by the Dirac system as well as to define traces in a weak sense for functions in the appropriate Sobolev space. Finally, we introduce Calderón projectors associated with the problem and illustrate the method in two special cases: the well-known MIT bag model and an electrostatic $δ$-shell interaction.

math-ph↗

Spectral stability of Schroedinger operators with subordinated complex potentials

We prove that the spectrum of Schroedinger operators in three dimensions is purely continuous and coincides with the non-negative semiaxis for all potentials satisfying a form-subordinate smallness condition. By developing the method of multipliers, we also establish the absence of point spectrum for Schroedinger operators in all dimensions under various alternative hypotheses, still allowing complex-valued potentials with critical singularities.

math.SP↗

On the regularity of solutions to the $k$-generalized Korteweg-de Vries equation

This work is concerned with special regularity properties of solutions to the $k$-generalized Korteweg-de Vries equation. In \cite{IsazaLinaresPonce} it was established that if the initial datun $u_0\in H^l((b,\infty))$ for some $l\in\mathbb Z^+$ and $b\in \mathbb R$, then the corresponding solution $u(\cdot,t)$ belongs to $H^l((β,\infty))$ for any $β\in \mathbb R$ and any $t\in (0,T)$. Our goal here is to extend this result to the case where $\,l\in \mathbb R^+$.

math.AP↗

Optimal Staged Self-Assembly of General Shapes

We analyze the number of tile types $t$, bins $b$, and stages necessary to assemble $n \times n$ squares and scaled shapes in the staged tile assembly model. For $n \times n$ squares, we prove $\mathcal{O}(\frac{\log{n} - tb - t\log t}{b^2} + \frac{\log \log b}{\log t})$ stages suffice and $Ω(\frac{\log{n} - tb - t\log t}{b^2})$ are necessary for almost all $n$. For shapes $S$ with Kolmogorov complexity $K(S)$, we prove $\mathcal{O}(\frac{K(S) - tb - t\log t}{b^2} + \frac{\log \log b}{\log t})$ stages suffice and $Ω(\frac{K(S) - tb - t\log t}{b^2})$ are necessary to assemble a scaled version of $S$, for almost all $S$. We obtain similarly tight bounds when the more powerful flexible glues are permitted.

cs.CG↗

Spectral nuclear properties of NLS1 galaxies

It is not yet well known whether narrow line Seyfert 1 (NLS1) galaxies follow the $M_{BH} - σ_{\star}$ relation found for normal galaxies. Emission lines, such as [SII] or [OIII]$λ$5007, have been used as a surrogate of the stellar velocity dispersion and various results have been obtained. On the other hand, some active galactic nuclei (AGNs) have shown Balmer emission with an additional intermediate component (IC) besides the well-known narrow and broad lines. In order to re-examine the location of NLS1 in the $M_{BH} - σ_{\star}$ relation, we test some emission lines, such as the narrow component (NC) of H$α$ and the forbidden [NII]$λλ$6548,6584 and [SII]$λλ$6716,6731 lines, as replacements for $σ_{\star}$. On the other hand, we study the properties of the IC of H$α$ found in 14 galaxies of the sample to find a link between this component, the central engine, and the remaining lines. We also compare the emission among the broad component (BC) of H$α$ and those emitted at the narrow line region (NLR). We have obtained and studied medium-resolution spectra (170 km s$^{-1}$ FWHM at H$α$) of 36 NLS1 galaxies in the optical range $\sim$5800 - 6800Å. We found that, in general, most of the galaxies lie below the $M_{BH} - σ_{\star}$ relation when the NC of H$α$, [SII], and [NII] lines are used as a surrogate of $σ_{\star}$. Besides this, we found that 13 galaxies show an IC, most of which are in the velocity range $\sim$ 700 $-$ 1500 km s$^{- 1}$. This is same range as in AGN types and is well correlated with the FWHM of BC and, therefore, with the BH mass. On the other hand, we found that there is a correlation between the luminosity of the BC of H$α$ and NC, IC, [NII]$λ$6584, and [SII] lines.

astro-ph.GA↗

Universal Shape Replicators via Self-Assembly with Attractive and Repulsive Forces

We show how to design a universal shape replicator in a self-assembly system with both attractive and repulsive forces. More precisely, we show that there is a universal set of constant-size objects that, when added to any unknown hole-free polyomino shape, produces an unbounded number of copies of that shape (plus constant-size garbage objects). The constant-size objects can be easily constructed from a constant number of individual tile types using a constant number of preprocessing self-assembly steps. Our construction uses the well-studied 2-Handed Assembly Model (2HAM) of tile self-assembly, in the simple model where glues interact only with identical glues, allowing glue strengths that are either positive (attractive) or negative (repulsive), and constant temperature (required glue strength for parts to hold together). We also require that the given shape has specified glue types on its surface, and that the feature size (smallest distance between nonincident edges) is bounded below by a constant. Shape replication necessarily requires a self-assembly model where parts can both attach and detach, and this construction is the first to do so using the natural model of negative/repulsive glues (also studied before for other problems such as fuel-efficient computation); previous replication constructions require more powerful global operations such as an "enzyme" that destroys a subset of the tile types.

cs.CG↗

Power Side Channels in Security ICs: Hardware Countermeasures

Power side-channel attacks are a very effective cryptanalysis technique that can infer secret keys of security ICs by monitoring the power consumption. Since the emergence of practical attacks in the late 90s, they have been a major threat to many cryptographic-equipped devices including smart cards, encrypted FPGA designs, and mobile phones. Designers and manufacturers of cryptographic devices have in response developed various countermeasures for protection. Attacking methods have also evolved to counteract resistant implementations. This paper reviews foundational power analysis attack techniques and examines a variety of hardware design mitigations. The aim is to highlight exposed vulnerabilities in hardware-based countermeasures for future more secure implementations.

cs.CR↗

An isoperimetric-type inequality for electrostatic shell interactions for Dirac operators

In this article we investigate spectral properties of the coupling $H+V_λ$, where $H=-iα\cdot\nabla +mβ$ is the free Dirac operator in $\mathbb R^3$, $m>0$ and $V_λ$ is an electrostatic shell potential (which depends on a parameter $λ\in\mathbb R$) located on the boundary of a smooth domain in $\mathbb R^3$. Our main result is an isoperimetric-type inequality for the admissible range of $λ$'s for which the coupling $H+V_λ$ generates pure point spectrum in $(-m,m)$. That the ball is the unique optimizer of this inequality is also shown. Regarding some ingredients of the proof, we make use of the Birman-Schwinger principle adapted to our setting in order to prove some monotonicity property of the admissible $λ$'s, and we use this to relate the endpoints of the admissible range of $λ$'s to the sharp constant of a quadratic form inequality, from which the isoperimetric-type inequality is derived.

math-ph↗

The vortex filament equation as a pseudorandom generator

In this paper, we consider the evolution of the so-called vortex filament equation (VFE), \begin{equation*} \mathbf X_t = \mathbf X_s\wedge\mathbf X_{ss}, \end{equation*} taking a planar regular polygon of $M$ sides as initial datum. We study VFE from a completely novel point of view: that of an evolution equation which yields a very good generator of pseudorandom numbers in a completely natural way. This essential randomness of VFE is in agreement with the randomness of the physical phenomena upon which it is based.

math.NT↗

Vortex Filament Equation for a Regular Polygon

In this paper, we study the evolution of the vortex filament equation (VFE), $$\mathbf X_t = \mathbf X_s \wedge \mathbf X_{ss},$$ with $\mathbf X(s, 0)$ being a regular planar polygon. Using algebraic techniques, supported by full numerical simulations, we give strong evidence that $\mathbf X(s, t)$ is also a polygon at any rational time; moreover, it can be fully characterized, up to a rigid movement, by a generalized quadratic Gauß sum. We also study the fractal behavior of $\mathbf X(0, t)$, relating it with the so-called Riemann's non-differentiable function, that was proved by Jaffard to be a multifractal.

math.AP↗