arXiv subjects

Rômulo Damasclin Chaves dos Santos

Publications and source records attributed to Rômulo Damasclin Chaves dos Santos.

At least 19 recordsLinked to original sources

arXiv2026

A Non-Commutative Voronovskaya Theorem for Quantum Neural Network Operators

We prove a complete asymptotic expansion for quantum neural network operators when they approximate arbitrary quantum channels. This is the non-commutative analogue of the classical Voronovskaya theorem. The expansion reveals that the approximation error splits into three fundamentally different parts: integer powers of $1/n$ involving ordinary Fréchet derivatives; fractional powers governed by Marchaud fractional derivatives, which capture the Hölder smoothness of the channel; and purely quantum commutator terms that have no classical counterpart. The remainder is bounded sharply by an explicit constant: \[ \norm{R_{m,n}(Φ,\bullet)}_\diamond \le C_{m,γ,d} \|Φ\|_{\cC^{m,γ}} \, n^{-(m+γ)} (\log n)^{3m/2}. \] We present a numerical test for a classical analogue that confirms the predicted convergence rate and the logarithmic correction, directly validating the asymptotic theory. Based on this expansion, we obtain three major advances: a quantum central limit theorem for the fluctuations of quantum neural network operators, a method to construct optimal interpolation geodesics between quantum channels via Kubo-Ando means, and a systematic understanding of how fractional smoothness limits the acceleration of quantum neural network approximations. The numerical test further demonstrates that the theoretical rates are sharp and that logarithmic enhancements are unavoidable. Altogether, our work builds a rigorous bridge between classical approximation theory, fractional calculus, and quantum machine learning, offering both theoretical insight and practical tools for designing and analyzing quantum neural networks in finite dimensions.

quant-ph↗

arXiv2026

Anisotropic Mixed Fractional Landau Inequalities for Rotating Compressible Flows

We develop a rigorous theory of anisotropic mixed fractional Landau inequalities for rotating compressible fluid flows at high Mach numbers, incorporating Coriolis and centrifugal forces. We introduce rotating fractional Sobolev spaces $\mathcal{W}^{ν,p}_{α,Ω}(\mathbb{R}^k)$, which encode directional scaling, fractional dissipation and rotational effects. We prove sharp fractional Landau inequalities with explicit dependence on the Mach number, the rotation rate $|Ω|$ and the anisotropy vector $α$. Key tools are a rotating Littlewood--Paley decomposition, anisotropic maximal estimates with rotational corrections, and commutator estimates for the Coriolis term. As applications, we establish stability bounds for neural operators approximating rotating compressible flows and derive optimal approximation rates of order $N^{-ν/d_{α,Ω}}$, where $d_{α,Ω}=\sum_iα_i^{-1}+κ|Ω|^{2/ν}$ is the effective anisotropic--rotational dimension. Our results provide a mathematical foundation for analysing high--Mach rotating flows and for designing physically informed neural architectures.

math.AP↗

arXiv2026

Asymptotic Expansions for Neural Network Approximations of Quantum Channels

This paper establishes the Quantum Voronovskaya--Damasclin (QVD) Theorem, providing a complete asymptotic characterization of Quantum Neural Network Operators in the approximation of arbitrary quantum channels. The result extends the classical Voronovskaya theorem from scalar approximation to the non-commutative operator framework of quantum information theory. We introduce rigorous quantum analogues of Sobolev and Hölder spaces defined through Fréchet differentiability in the Liouville representation and measured using the completely bounded (diamond) norm. Within this framework, we derive an explicit asymptotic expansion of the approximation error and identify the fundamental mechanisms governing convergence. The expansion separates integer-order differential contributions, fractional corrections associated with limited regularity, and intrinsically non-commutative effects arising from operator algebra structure. We also establish a sharp remainder estimate with explicit dependence on the regularity of the channel and the dimension of the underlying Hilbert space. Several applications demonstrate the scope of the theory. These include a quantum central limit theorem describing the fluctuation regime of quantum neural network operators, an optimal interpolation method based on operator geometric means, and a convergence acceleration procedure inspired by Richardson extrapolation. The results provide a rigorous mathematical foundation for the asymptotic analysis of quantum neural network models and establish a direct connection between classical approximation theory, operator algebras, and quantum information science, with implications for quantum algorithms and quantum machine learning.

quant-ph↗

arXiv2026

Fractional Grönwall--Wendroff Inequalities for Implicit Systems with Distributed Memory

This work establishes a comprehensive analytical framework for studying implicit fractional differential systems with distributed memory and time delays. We develop novel fractional integral inequalities of Grönwall--Wendroff type that are specifically adapted to handle multivariate functions with singular kernels and implicit dependencies. These inequalities provide essential a priori estimates for analyzing complex memory-dependent systems. Building upon these results, we prove general existence and uniqueness theorems for implicit fractional differential equations using fixed-point theory in appropriately weighted Banach spaces. Furthermore, we establish Ulam--Hyers stability criteria, demonstrating that small perturbations in the governing equations lead to proportionally small deviations in solutions. The theoretical advances are applied to the fractional FitzHugh--Nagumo model with delay (FHN-$α$-$τ$), a neurodynamical system exhibiting both subdiffusive memory and discrete time delays. Our analysis yields rigorous conditions for the existence of limit cycles corresponding to action potentials, along with asymptotic stability criteria. The results reveal how fractional order $α$ and delay $τ$ jointly modulate neural excitability thresholds and dynamic regime transitions. This work bridges fundamental fractional calculus theory with applications in mathematical neuroscience, providing analytical tools for systems where present evolution depends non-trivially on historical states and intrinsic fractional rates of change.

math.DS↗

arXiv2025

Stochastic Fractional Neural Operators: A Symmetrized Approach to Modeling Turbulence in Complex Fluid Dynamics

In this work, we introduce a new class of neural network operators designed to handle problems where memory effects and randomness play a central role. In this work, we introduce a new class of neural network operators designed to handle problems where memory effects and randomness play a central role. These operators merge symmetrized activation functions, Caputo-type fractional derivatives, and stochastic perturbations introduced via Itô type noise. The result is a powerful framework capable of approximating functions that evolve over time with both long-term memory and uncertain dynamics. We develop the mathematical foundations of these operators, proving three key theorems of Voronovskaya type. These results describe the asymptotic behavior of the operators, their convergence in the mean-square sense, and their consistency under fractional regularity assumptions. All estimates explicitly account for the influence of the memory parameter $α$ and the noise level $σ$. As a practical application, we apply the proposed theory to the fractional Navier-Stokes equations with stochastic forcing, a model often used to describe turbulence in fluid flows with memory. Our approach provides theoretical guarantees for the approximation quality and suggests that these neural operators can serve as effective tools in the analysis and simulation of complex systems. By blending ideas from neural networks, fractional calculus, and stochastic analysis, this research opens new perspectives for modeling turbulent phenomena and other multiscale processes where memory and randomness are fundamental. The results lay the groundwork for hybrid learning-based methods with strong analytical backing.

cs.LG↗

arXiv2025

Revolutionizing Fractional Calculus with Neural Networks: Voronovskaya-Damasclin Theory for Next-Generation AI Systems

This work introduces rigorous convergence rates for neural network operators activated by symmetrized and perturbed hyperbolic tangent functions, utilizing novel Voronovskaya-Damasclin asymptotic expansions. We analyze basic, Kantorovich, and quadrature-type operators over infinite domains, extending classical approximation theory to fractional calculus via Caputo derivatives. Key innovations include parameterized activation functions with asymmetry control, symmetrized density operators, and fractional Taylor expansions for error analysis. The main theorem demonstrates that Kantorovich operators achieve $o(n^{-β(N-\varepsilon)})$ convergence rates, while basic operators exhibit $\mathcal{O}(n^{-βN})$ error decay. For deep networks, we prove $\mathcal{O}(L^{-β(N-\varepsilon)})$ approximation bounds. Stability results under parameter perturbations highlight operator robustness. By integrating neural approximation theory with fractional calculus, this work provides foundational mathematical insights and deployable engineering solutions, with potential applications in complex system modeling and signal processing.

cs.LG↗

arXiv2025

Generalized Neural Network Operators with Symmetrized Activations: Fractional Convergence and the Voronovskaya-Damasclin Theorem

This paper explores the asymptotic behavior of univariate neural network operators, with an emphasis on both classical and fractional differentiation over infinite domains. The analysis leverages symmetrized and perturbed hyperbolic tangent activation functions to investigate basic, Kantorovich, and quadrature-type operators. Voronovskaya-type expansions, along with the novel Vonorovskaya-Damasclin theorem, are derived to obtain precise error estimates and establish convergence rates, thereby extending classical results to fractional calculus via Caputo derivatives. The study delves into the intricate interplay between operator parameters and approximation accuracy, providing a comprehensive framework for future research in multidimensional and stochastic settings. This work lays the groundwork for a deeper understanding of neural network operators in complex mathematical.

math.GM↗

arXiv2025

Extension of Symmetrized Neural Network Operators with Fractional and Mixed Activation Functions

We propose a novel extension to symmetrized neural network operators by incorporating fractional and mixed activation functions. This study addresses the limitations of existing models in approximating higher-order smooth functions, particularly in complex and high-dimensional spaces. Our framework introduces a fractional exponent in the activation functions, allowing adaptive non-linear approximations with improved accuracy. We define new density functions based on $q$-deformed and $θ$-parametrized logistic models and derive advanced Jackson-type inequalities that establish uniform convergence rates. Additionally, we provide a rigorous mathematical foundation for the proposed operators, supported by numerical validations demonstrating their efficiency in handling oscillatory and fractional components. The results extend the applicability of neural network approximation theory to broader functional spaces, paving the way for applications in solving partial differential equations and modeling complex systems.

stat.ML↗

arXiv2025

Non-Linear Interactions in Neural Network Operators: New Theorems on Symmetry-Preserving Transformations

This paper advances the study of multivariate function approximation using neural network operators activated by symmetrized and perturbed hyperbolic tangent functions. We propose new non-linear operators that preserve dynamic symmetries within Euclidean spaces, extending current results on Voronovskaya-type asymptotic expansions. The developed operators utilize parameterized deformations to model higher-order interactions in multivariate settings, achieving improved accuracy and robustness. Fundamental theorems demonstrate convergence properties and quantify error bounds, highlighting the operators' ability to approximate functions and derivatives in Sobolev spaces. These results provide a rigorous foundation for theoretical studies and further applications in symmetry-driven modeling and regularization strategies in machine learning and data analysis.

math.GM↗

arXiv2024

Voronovskaya-Type Asymptotic Expansions and Convergence Analysis for Neural Network Operators in Complex Domains

This paper extends the classical theory of Voronovskaya-type asymptotic expansions to generalized neural network operators defined on non-Euclidean and fractal domains. We introduce and analyze smooth operators activated by modified and generalized hyperbolic tangent functions, extending their applicability to manifold and fractal geometries. Key theoretical results include the preservation of density properties, detailed convergence rates, and asymptotic expansions. Additionally, we explore the role of fractional derivatives in defining neural network operators, which capture non-local behavior and are particularly useful for modeling systems with long-range dependencies or fractal-like structures. Our findings contribute to a deeper understanding of neural network operators in complex, structured spaces, offering robust mathematical tools for applications in signal processing on manifolds and solving partial differential equations on fractals.

math.GM↗

arXiv2024

Generalized Lagrangian Coherent Structures in Finsler Manifolds

This paper introduces a novel theoretical framework for identifying Lagrangian Coherent Structures (LCS) in manifolds with non-constant curvature, extending the theory to Finsler manifolds. By leveraging Riemannian and Finsler geometry, we generalize the deformation tensor to account for geodesic stretching in these complex spaces. The main result demonstrates the existence of invariant surfaces acting as LCS, characterized by dominant eigenvalues of the generalized deformation tensor. We discuss potential applications in astrophysics, relativistic fluid dynamics, and planetary science. This work paves the way for exploring LCS in intricate geometrical settings, offering new tools for dynamical system analysis.

math.GM↗

arXiv2024

A Sophisticated Analytical Methodology for Refining the Smagorinsky Model in Turbulent Flows

In this work, we present three important theorems related to the corrected Smagorinsky model for turbulence in time-dependent domains. The first theorem establishes an improved regularity criterion for the solution of the corrected Smagorinsky model in Sobolev spaces $H^s(Ω(t))$ with smooth and evolving boundaries. The result provides a bound on the Sobolev norm of the solution, ensuring that the solution remains regular over time. The second theorem quantifies the approximation error between the corrected Smagorinsky model and the true Navier-Stokes solution. Taking advantage of high-order Sobolev spaces and energy methods, we derive an explicit error estimate for the velocity fields, showing the relationship between the error and the external force term. The third theorem focuses on the asymptotic convergence of the corrected Smagorinsky model to the solution of the Navier-Stokes equations as time progresses. We provide an upper bound for the error in the $L^2(Ω)$ norm, demonstrating that the error decreases as time increases, especially as the external force term vanishes. This result highlights the long-term convergence of the corrected model to the true solution, with explicit dependences on the initial conditions, viscosity, and external forces.

math.AP↗

arXiv2024

Semiclassical and Microlocal Analysis of Energy Dissipation and Cascades in Turbulent Flows

This work presents a comprehensive study of the microlocal energy decomposition and propagation of singularities for semiclassically adjusted dissipative pseudodifferential operators. The analysis focuses on the behavior of energy dissipation in turbulent flows modeled by operators $ P_h $ with symbols $ a(x, ξ) \in S^m(\mathbb{R}^n) $, where $ m < 0 $. Using microlocal partitions of unity, we derive an expression for the energy dissipation rate $ \varepsilon_h $ in both spatial and spectral regions, showing its asymptotic equivalence to a two-dimensional integral over phase space. This framework is then applied to the study of the propagation of singularities in solutions to the equation $ P_h u_h = 0 $, where the wavefront set $ \operatorname{WF}_h(u_h) $ evolves along generalized bicharacteristics of the principal symbol $ p_0(x, ξ) $, with energy dissipation controlled by the imaginary part of $ p_0(x, ξ) $. The results combine microlocal analysis, semiclassical techniques, and symbolic calculus to offer insights into the complex dynamics of wave propagation and energy dissipation in turbulent systems, with implications for both theoretical and applied studies in mathematical physics.

math.GM↗

arXiv2024

Curvature-Enhanced Dynamics and Exponential Decay of the Non-Cutoff Boltzmann Equation on Riemannian Manifolds

In this work, we investigate the long-time behavior of solutions to the non-cutoff Boltzmann equation on compact Riemannian manifolds with bounded Ricci curvature. The paper introduces new results on the exponential decay of hydrodynamic quantities, such as density, momentum, and energy fields, influenced by both the curvature of the manifold and singularities in the collision kernel. We demonstrate that for initial data in $H^s_x \times L^p_v$, the solutions exhibit sharp exponential decay rates in Sobolev norms, with the decay rate determined by the manifold's geometry and the regularity of the kernel. Specifically, we prove that the density $ρ(t, x)$, momentum $\mathbf{m}(t, x)$, and energy field $E(t, x)$ all decay exponentially in time, with decay rates that depend on the manifold's curvature and the nature of the collision kernel's singularity. Additionally, we address the case of angular singularities in the collision kernel, providing conditions under which the exponential decay persists. The analysis combines energy methods, Fourier analysis, and coercivity estimates for the collision operator, extended to curved geometries. These results extend the understanding of dissipation mechanisms in kinetic theory, especially in curved settings, and offer valuable insights into the behavior of rarefied gases and plasma flows in non-Euclidean environments. The findings have applications in plasma physics, astrophysics, and the study of rarefied gases, opening new directions for future research in kinetic theory and geometric analysis.

math.AP↗

arXiv2024

Mathematical Analysis of Regularity, Bifurcations, and Turbulence in Fluid Dynamics via Sobolev, Besov, and Triebel-Lizorkin Spaces

This article presents a comprehensive mathematical framework for the study of regularity, bifurcations, and turbulence in fluid dynamics, leveraging the power of Sobolev and Besov function spaces. We delve into the detailed definitions, properties, and notations of these spaces, illustrating their relevance in the context of partial differential equations governing fluid flow. The work emphasizes the intricate connections between Sobolev, Besov, and Triebel-Lizorkin spaces, highlighting their interplay in the analysis of fluid systems. We propose new regularity criteria for solutions to the Navier-Stokes equations, based on the interaction of low and high-frequency modes in turbulent regimes. These criteria offer a novel perspective on the conditions under which singularities may form, providing critical insights into the structure of turbulent flows. The article further explores the applications of these function spaces to the analysis of bifurcations in fluid systems, offering a deeper understanding of the mechanisms that lead to complex flow phenomena such as turbulence. Through the development of rigorous theorems and proofs, the paper aims to bridge the gap between abstract mathematical theory and practical fluid dynamics. In particular, the results contribute to ongoing efforts in solving the Navier-Stokes existence and smoothness problem, a key challenge in the field, and have potential implications for the Millennium Prize Problem. The conclusion underscores the significance of these findings, offering a pathway for future research in the analysis of fluid behavior at both small and large scales.

math.AP↗

arXiv2024

Quantum-Inspired Stochastic Modeling and Regularity Analysis in Turbulent Flows

This paper introduces a novel mathematical framework for examining the regularity and energy dissipation properties of solutions to the stochastic Navier-Stokes equations. By integrating Sobolev-Besov hybrid spaces, fractional differential operators, and quantum-inspired modeling techniques, we provide a comprehensive analysis that captures the multiscale and chaotic dynamics inherent in turbulent flows. Central to this framework is a Schrödinger-type operator adapted for fluid dynamics, which encapsulates quantum-scale turbulence effects, thereby elucidating the mechanisms of energy redistribution across scales. Additionally, we develop anisotropic stochastic models with direction-dependent viscosity, characterized by a pseudo-differential operator and a covariance matrix governing directional diffusion. These models more accurately reflect real-world turbulence, where viscosity varies with flow orientation, enhancing both theoretical insights and practical simulation capabilities. Our main contributions include new regularity theorems and rigorous a priori estimates for solutions in Sobolev-Besov spaces, alongside proofs of energy dissipation properties in anisotropic contexts. These findings advance the understanding of fluid turbulence by offering a refined approach to studying scale interactions, stochastic effects, and anisotropy in turbulent flows.

math.AP↗

arXiv2024

Theoretical Refinements of the Smagorinsky Model for Turbulence Simulations

This paper presents a rigorous theoretical extension of the Smagorinsky model for turbulence simulations. The author builds on its fundamental framework, addressing known limitations, and making new mathematical advances. Specifically, this work introduces new theorems on the existence and uniqueness of weak solutions and establishes the stability of the model under small perturbations. The approach leverages advanced tools in functional analysis, particularly Sobolev and Banach spaces, increasing the theoretical robustness of the model. Each theorem is accompanied by detailed proofs employing variational techniques and energy estimations. Furthermore, the author proposes a variational formulation that demonstrates improvements in the mathematical behavior of the model's subgrid-scale stress representation. This work aims to provide a deeper understanding of the Smagorinsky model, setting the stage for further theoretical and computational developments. The obtained results are expected to impact turbulence modeling by offering a more robust theoretical basis for Large Eddy Simulations (LES).

physics.flu-dyn↗

arXiv2024

Mathematical Analysis of Axisymmetrization and Enhanced Inviscid Damping in 2D Linearized Euler Flow

This paper extends the mathematical theory of axisymmetrization and vorticity depletion within the two-dimensional (2D) Euler equations, with an emphasis on the dynamics of radially symmetric, monotonic vorticity profiles. By analyzing inviscid damping, we establish new optimal decay rates for radial and angular velocity components in weighted Sobolev spaces, showing that vorticity depletion enhances damping effects beyond those observed in passive scalar dynamics. Our methodology involves the construction of advanced Greens functions and the use of spectral techniques to achieve precise asymptotic expansions, providing a comprehensive framework for analyzing long-term stability. These results mark the first rigorous confirmation of enhanced inviscid damping rates in axisymmetric vortices, substantially advancing the theoretical understanding of coherent vortex structures in high Reynolds number flows. This work has potential applications in fluid stability theory, turbulence modeling, and other fields that involve the study of inviscid flows.

physics.flu-dyn↗