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Alejandro J. Castro

Publications and source records attributed to Alejandro J. Castro.

At least 19 recordsLinked to original sources
arXiv2025

Solving wave equation problems on D-Wave quantum annealers

We solve the one-dimensional Helmholtz equation in several scenarios using the quantum annealer provided by D-Wave Systems within a pseudospectral framework, where the solution is encoded in an appropriate set of basis functions. We evaluate the performance of different encoding strategies based on algebraic conditioning and adiabatic considerations, and benchmark their performance against the classical simulated annealing algorithm. In particular, we analyze the minimum energy gap, the so-called dynamic range, and the mean squared error to assess the numerical stability, consistency, and accuracy of the solutions returned by each strategy. Our work highlights the importance of developing custom embedding techniques ensuring well-conditioned algebraic systems. In particular, we show that encoding strategies producing full-rank matrices with reduced dynamic ranges enhance the performance of the quantum annealer even under polychromatic driving and nontrivial initial conditions. We further discuss the potential of developing hybrid quantum-classical schemes designed to simultaneously satisfy algebraic conditioning and adiabatic requirements.

quant-ph↗
arXiv2023

Topological magnetoelectric response in passive magnetic devices

Despite the prospect of next-generation electronic technologies has spurred the investigation of the remarkable topological magnetoelectric response, it remains largely unexplored its potential in the application of basic electronic devices. In this paper, we undertake this task at the theoretical level by addressing the $θ$-electrodynamics and examine electromagnetic properties (e.g. tunable inductance, operating frequency range, and power consumption) of three fundamental passive magnetic devices endowed with this effect: the primitive transformer, the bilayer solenoid inductor, and the solenoid actuator. We further exploit the methodology of magnetic circuits to obtain an extended Hopkinson's law that is valid for both topological and ordinary magnetoelectric responses (provided it is uniform in the bulk). Under low-power conditions, we find out that the functionally passive part of the topological-magnetoelectric transformer, solenoid inductor as well as solenoid actuator is indistinguishable from the conventional situation up to second-order in the magnetoelectric susceptibility; and argue that the main benefit of using topological insulators essentially relies on a lower power consumption. Our theoretical framework is also convenient to analyse magnetoelectric inductors endowed with a relatively large magnetoelectric susceptibility, they display a broad inductance tunability of over 200% up to 100 GHz in the millimeter length scale. Conversely, our treatment predicts that the operating frequency range could be restricted below the ultra low frequency by a significantly strong magnetoelectric response (e.g. retrieved by certain multiferroic heterostructures).

physics.app-ph↗
arXiv2020

Wigner instability analysis of the damped Hirota equation

We address the modulation instability of the Hirota equation in the presence of stochastic spatial incoherence and linear time-dependent amplification/attenuation processes via the Wigner function approach. We show that the modulation instability remains baseband type, though the damping mechanisms substantially reduce the unstable spectrum independent of the higher-order contributions (e.g. the higher-order nonlinear interaction and the third-order dispersion). Additionally, we find out that the unstable structure due to the Kerr interaction exhibits a significant resilience to the third-order-dispersion stabilizing effects in comparison with the higher-order nonlinearity, as well as a moderate Lorentzian spectrum damping may assist the rising of instability. Finally, we also discuss the relevance of our results in the context of current experiments exploring extreme wave events driven by the modulation instability (e.g. the generation of the so-called rogue waves).

nlin.PS↗
arXiv2020

Regularity of Fourier integral operators with amplitudes in general Hörmander classes

We prove the global $L^p$-boundedness of Fourier integral operators that model the parametrices for hyperbolic partial differential equations, with amplitudes in classical Hörmander classes $S^{m}_{ρ, δ}(\mathbb{R}^n)$ for parameters $0<ρ\leq 1$, $0\leq δ<1$. We also consider the regularity of operators with amplitudes in the exotic class $S^{m}_{0, δ}(\mathbb{R}^n)$, $0\leq δ< 1$ and the forbidden class $S^{m}_{ρ, 1}(\mathbb{R}^n)$, $0\leqρ\leq 1.$ Furthermore we show that despite the failure of the $L^2$-boundedness of operators with amplitudes in the forbidden class $S^{0}_{1, 1}(\mathbb{R}^n)$, the operators in question are bounded on Sobolev spaces $H^s(\mathbb{R}^n)$ with $s>0.$ This result extends those of Y. Meyer and E. M. Stein to the setting of Fourier integral operators.

math.AP↗
arXiv2019

Regularity properties of Schrödinger integral operators and general oscillatory integrals

We introduce the notion of Schrödinger integral operators and prove sharp local and global regularity results for these (including propagators for the quantum mechanical harmonic oscillator). Furthermore we introduce general classes of oscillatory integral operators with inhomogeneous phase functions, whose local and global regularity are also established in classical function spaces (both in the Banach and quasi-Banach scales). The results are then applied to obtain optimal (local in time) estimates for the solution to the Cauchy problem for variable-coefficient Schrödinger equations as well as other evolutionary partial differential equations.

math.AP↗
arXiv2019

Fourier Neural Networks: A Comparative Study

We review neural network architectures which were motivated by Fourier series and integrals and which are referred to as Fourier neural networks. These networks are empirically evaluated in synthetic and real-world tasks. Neither of them outperforms the standard neural network with sigmoid activation function in the real-world tasks. All neural networks, both Fourier and the standard one, empirically demonstrate lower approximation error than the truncated Fourier series when it comes to an approximation of a known function of multiple variables.

cs.NE↗
arXiv2018

Riesz transforms, Cauchy-Riemann systems and amalgam Hardy spaces

In this paper we study Hardy spaces $\mathcal{H}^{p,q}(\mathbb{R}^d)$, $0<p,q<\infty$, modeled over amalgam spaces $(L^p,\ell^q)(\mathbb{R}^d)$. We characterize $\mathcal{H}^{p,q}(\mathbb{R}^d)$ by using first order classical Riesz transforms and compositions of first order Riesz transforms depending on the values of the exponents $p$ and $q$. Also, we describe the distributions in $\mathcal{H}^{p,q}(\mathbb{R}^d)$ as the boundary values of solutions of harmonic and caloric Cauchy-Riemann systems. We remark that caloric Cauchy-Riemann systems involve fractional derivative in the time variable. Finally we characterize the functions in $L^2(\mathbb{R}^d) \cap \mathcal{H}^{p,q}(\mathbb{R}^d)$ by means of Fourier multipliers $m_θ$ with symbol $θ(\cdot/|\cdot|)$, where $θ\in C^\infty(\mathbb{S}^{d-1})$ and $\mathbb{S}^{d-1}$ denotes the unit sphere in $\mathbb{R}^d$.

math.CA↗
arXiv2018

BMO functions and Balayage of Carleson measures in the Bessel setting

By $BMO_o(R)$ we denote the space consisting of all those odd and bounded mean oscillation functions on R. In this paper we characterize the functions in $BMO_o(R)$ with bounded support as those ones that can be written as a sum of a bounded function on $(0,\infty )$ plus the balayage of a Carleson measure on $(0,\infty )\times (0,\infty )$ with respect to the Poisson semigroup associated with the Bessel operator $B_λ=-x^{-λ}Dx^{2λ}Dx^{-λ}$, $λ>0$. This result can be seen as an extension to Bessel setting of a classical result due to Carleson.

math.CA↗
arXiv2017

Transference of local to global $L^2$ maximal estimates for dispersive partial differential equations

In this paper we give an elementary proof for transference of local to global maximal estimates for dispersive PDEs. This is done by transferring local $L^2$ estimates for certain oscillatory integrals with rough phase functions, to the corresponding global estimates. The elementary feature of our approach is that it entirely avoids the use of the wave packet techniques which are quite common in this context, and instead is based on scalings and classical oscillatory integral estimates.

math.AP↗
arXiv2017

Variable exponent Hardy spaces associated with discrete Laplacians on graphs

In this paper we develop the theory of variable exponent Hardy spaces associated with discrete Laplacians on infinite graphs. Our Hardy spaces are defined by square integrals, atomic and molecular decompositions. Also we study boundedness properties of Littlewood-Paley functions, Riesz transforms, and spectral multipliers for discrete Laplacians on variable exponent Hardy spaces.

math.CA↗
arXiv2016

Homogenization of a parabolic Dirichlet problem by a method of Dahlberg

Consider the linear parabolic operator in divergence form $$\mathcal{H} u =\partial_t u(X,t)-\text{div}(A(X)\nabla u(X,t)).$$ We employ a method of Dahlberg to show that the Dirichlet problem for $\mathcal{H}$ in the upper half plane is well-posed for boundary data in $L^p$, for any elliptic matrix of coefficients $A$ which is periodic and satisfies a Dini-type condition. This result allows us to treat a homogenization problem for the equation $\partial_t u_\varepsilon(X,t)-\text{div}(A(X/\varepsilon)\nabla u_\varepsilon(X,t))$ in Lipschitz domains with $L^p$-boundary data.

math.AP↗
arXiv2016

Oscillation of generalized differences of Hölder and Zygmund functions

In this paper we analyze the oscillation of functions having derivatives in the Hölder or Zygmund class in terms of generalized differences and prove that its growth is governed by a version of the classical Kolmogorov's Law of the Iterated Logarithm. A better behavior is obtained for functions in the Lipschitz class via an interesting connection with Calderón-Zygmund operators.

math.CA↗