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Randomness certification is a foundational and practical aspect of quantum information science, essential for securing quantum communication protocols. Traditionally, these protocols have been implemented and validated with a single entanglement source, as in the paradigmatic Bell scenario. However, advancing these protocols to support more complex configurations involving multiple entanglement sources is key to building robust architectures and realizing large-scale quantum networks. In this work, we show how to certify randomness in an entanglement-teleportation experiment, the building block of a quantum repeater displaying two independent sources of entanglement. Utilizing the scalar extension method, we address the challenge posed by the non-convexity of the correlation set, providing effective bounds on an eavesdropper's knowledge of the shared secret bits. Our theoretical model characterizes the certifiable randomness within the network and is validated through the analysis of experimental data from a photonic quantum network.

By harnessing the unique properties of bilayer graphene, we present a flexible platform for achieving electrically tunable exciton polaritons within a microcavity. Using a semiclassical approach, we solve Maxwell's equations within the cavity, approximating the optical conductivity of bilayer graphene through its excitonic response as described by the Elliott formula. Transitioning to a quantum mechanical framework, we diagonalize the Hamiltonian governing excitons and cavity photons, revealing the resulting polariton dispersions, Hopfield coefficients and Rabi splittings. Our analysis predicts that, under realistic exciton lifetimes, the exciton-photon interaction reaches the strong coupling regime. Furthermore, we explore the integration of an epsilon-near-zero material within the cavity, demonstrating that such a configuration can further enhance the light-matter interaction., Comment: 13 pages, 5 figures, PRB Editor's suggestion

Classifying phase transitions is a fundamental and complex challenge in condensed matter physics. This work proposes a framework for identifying quantum phase transitions by combining classical shadows with unsupervised machine learning. We use the axial next-nearest neighbor Ising model as our benchmark and extend the analysis to the Kitaev-Heisenberg model on a two-leg ladder. Even with few qubits, we can effectively distinguish between the different phases of the Hamiltonian models. Moreover, given that we only rely on two-point correlator functions, the classical shadows protocol enables the cost of the analysis to scale logarithmically with the number of qubits, making our approach a scalable and efficient way to study phase transitions in many-body systems., Comment: 12 pages, 16 figures

Stellar feedback is an essential step in the baryon cycle of galaxies, but it remains unconstrained beyond Cosmic Noon. We study the ionized gas kinematics, dynamical mass and gas-flow properties of a sample of 16 sub-$L^{\star}$ star-forming galaxies at $4\leq z\leq7.6$, using high-resolution JWST/NIRSpec observations. The emission lines are resolved, with velocity dispersions ($\sigma_{\rm gas}{\rm~(km~s^{-1})}\simeq38-96$) comparable to more massive galaxies at Cosmic Noon. From $\sigma_{\rm gas}$ and the galaxy size ($r_e=400-960~$pc), we estimate the dynamical mass to be $\log M_{\rm dyn}/M_{\odot}=9.25-10.25$. Stellar-to-dynamical mass ratios are low ($\log M_{\star}/M_{\rm dyn}\in[-0.5,-2]$) and decrease with increasing SFR surface density ($\Sigma_{\rm SFR}$). We estimate the gas surface densities assuming a star-formation law, but the gas masses do not balance the baryon-to-dynamical mass ratios, which would require a decrease in the star-formation efficiency. We find evidence of ionized outflows in five out of the sixteen galaxies, based on the need of broad components to reproduce the emission-line wings. We only observe outflows from galaxies undergoing recent bursts of star formation ${\rm SFR_{10}/SFR_{100}\geq1}$, with elevated $\Sigma_{\rm SFR}$ and low $M_{\star}/M_{\rm dyn}$. This links high gas surface densities to increased outflow incidence and lower $M_{\star}/M_{\rm dyn}$. With moderate outflow velocities ($v_{\rm flow}{\rm~(km~s^{-1})}=150-250$) and mass outflow rates ($\dot{M}_{\rm flow}/{\rm M_{\odot} yr^{-1}}=0.2-5$), these high-redshift galaxies appear more efficient at removing baryons than low-redshift galaxies with similar $M_{\star}$, showing mass loading-factors of $\dot{M}_{\rm flow}/{\rm SFR}=0.04-0.4$. For their given dynamical mass, the outflow velocities exceed the escape velocities, meaning that they may eventually enrich the Circumgalactic Medium., Comment: 15 pages, 11 figures, 3 tables; submitted to MNRAS

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., Comment: 17 pages

Detecting the first generation of stars, Population III (PopIII), has been a long-standing goal in astrophysics, yet they remain elusive even in the JWST era. Here we present a novel NIRCam-based selection method for PopIII galaxies, and carefully validate it through completeness and contamination simulations. We systematically search ~500 arcmin$^{2}$ across JWST legacy fields for PopIII candidates, including GLIMPSE which, assisted by gravitational lensing, has produced JWST's deepest NIRCam imaging thus far. We discover one promising PopIII galaxy candidate (GLIMPSE-16043) at $z=6.50^{+0.03}_{-0.24}$, a moderately lensed galaxy (mu=2.9) with an intrinsic UV magnitude of $M_{UV}$=-15.89. It exhibits key PopIII features: strong H$\alpha$ emission (rest-frame EW $2810\pm550$\AA); a Balmer jump; no dust (UV slope $\beta=-2.34\pm0.36$); and undetectable metal lines (e.g., [OIII]; [OIII]/H$\beta$<0.44) implying a gas-phase metallicity of Zgas/Zsun<0.5%. These properties indicate the presence of a nascent, metal-deficient young stellar population (<5Myr) with a stellar mass of $\simeq10^{5}M_{\odot}$. Intriguingly, this source deviates significantly from the extrapolated UV-metallicity relation derived from recent JWST observations at $z=4-10$, consistent with UV enhancement by a top-heavy PopIII initial mass function or the presence of an extremely metal-poor AGN. We also derive the first observational constraints on the PopIII UV luminosity function at z~6-7. The volume density of GLIMPSE-16043 ($\approx10^{-4}$ cMpc$^{-3}$) is in excellent agreement with theoretical predictions, independently reinforcing its plausibility. This study demonstrates the power of our novel NIRCam method to finally reveal distant galaxies even more pristine than the Milky Way's most metal-poor satellites, thereby promising to bring us closer to the first generation of stars than we have ever been before., Comment: 41 pages, 20 figures, and 10 Tables. Submitted to ApJ. Comments are welcome!

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 $\theta$-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., Comment: 13 pages

Efficient surgery room scheduling is essential for hospital efficiency, patient satisfaction, and resource utilization. This study addresses this challenge by introducing a novel concept of Random-Key Optimizer (RKO), rigorously tested on literature and new, real-world inspired instances. Our combinatorial optimization problem incorporates multi-room scheduling, equipment scheduling, and complex availability constraints for rooms, patients, and surgeons, facilitating rescheduling and enhancing operational flexibility. The RKO approach represents solutions as points in a continuous space, which are then mapped in the problem solution space via a deterministic function known as a decoder. The core idea is to operate metaheuristics and heuristics in the random-key space, unaware of the original solution space. We design the Biased Random-Key Genetic Algorithm with $Q$-Learning, Simulated Annealing, and Iterated Local Search for use within an RKO framework, employing a single decoder function. The proposed metaheuristics are complemented by lower-bound formulations, providing optimal gaps for evaluating the effectiveness of the heuristic results. Our results demonstrate significant lower and upper bounds improvements for the literature instances, notably proving one optimal result. Furthermore, the best-proposed metaheuristic efficiently generates schedules for the newly introduced instances, even in highly constrained scenarios. This research offers valuable insights and practical solutions for improving surgery scheduling processes, offering tangible benefits to hospitals by optimising resource allocation, reducing patient wait times, and enhancing overall operational efficiency., Comment: 38 pages, Preprint submitted to Applied Soft Computing

Let (F_n)_{n} be the classical Fibonacci sequence. It is well-known that it satisfies F_{n}^2 + F_{n+1}^2 = F_{2n+1}. In this study, we explore generalizations of this Diophantine equation in several directions. First, we solve the Diophantine equation (F_{n}^{(k)})^2 + (F_{n+d}^{(k)})^2 = F_{m}^{(k)} over the k-generalized Fibonacci numbers for every k \geq 2, generalizing Chaves and Marques. Next, we solve F_{n}^{s} + F_{n+d}^{s} = F_m over the Fibonacci numbers for every s \geq 2, generalizing Luca and Oyono. Finally, we solve the Diophantine equation F_{n}^s + \cdots + F_{n+d}^s = F_m for d+1 < n and s \geq 2., Comment: 27 pages, comments are very welcome!

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., Comment: 12 pages

Let (F_n^{(k)})_{n\geq -(k-2)} be the k-generalized Fibonacci sequence, defined as the linear recurrence sequence whose first k terms are \(0, 0, \ldots, 0, 1\), and whose subsequent terms are determined by the sum of the preceding k terms. This article is devoted to investigating when the sum of consecutive numbers in the k-generalized Fibonacci sequence belongs to the Fibonacci sequence. Namely, given d,k \in \N, with k \geq 3, our main theorem states that there are at most finitely many n \in \N such that F_n^{(k)} + \cdots + F_{n+d}^{(k)} is a Fibonacci number. In particular, the intersection between the Fibonacci sequence and the k-generalized Fibonacci sequence is finite., Comment: Comments are welcome !

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., Comment: 16 pages

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., Comment: 10 pages

Providing radio frequency (RF) signals to circuits working in cryogenic conditions requires bulky and expensive transmission cabling interfacing specialized RF electronics anchored at room temperature. Superconducting Josephson junction arrays (JJAs) can change this paradigm by placing the RF source and detector inside the chip. In this work, we demonstrate that DC-biased JJAs can emit signals in the C-band frequency spectrum and beyond. We fabricate reproducible JJAs comprised of amorphous MoGe or NbTiN superconducting islands and metallic Au weak links. Temperature, magnetic fields, applied currents, and device design are explored to control the operation of the RF sources, while we also identify important features that affect the ideal source behavior. Combined with the proven ability of these JJAs to detect microwave radiation, these sources allow us to propose a fully DC-operated cryogenic on-chip measurement platform that is a viable alternative to the high-frequency circuitry currently required for several quantum applications.

The certification of randomness is essential for both fundamental science and information technologies. Unlike traditional random number generators, randomness obtained from nonlocal correlations is fundamentally guaranteed to be unpredictable. However, it is also highly susceptible to noise. Here, we show that extending the conventional bipartite Bell scenario to hybrid quantum networks -- which incorporate both quantum channels and entanglement sources -- enhances the robustness of certifiable randomness. Our protocol even enables randomness to be certified from Bell-local states, broadening the range of quantum states useful for this task. Through both theoretical analysis and experimental validation in a photonic network, we demonstrate enhanced performance and improved noise resilience., Comment: 14 pages, 6 figures

Observational and/or astrophysical systematics modulating the observed number of luminous tracers can constitute a major limitation in the cosmological exploitation of surveys of the large scale structure of the universe. Part of this limitation arises on top of our ignorance on how such systematics actually impact the observed galaxy/quasar fields. In this work we develop a generic, hybrid model for an arbitrary number of systematics that may modulate observations in both an additive and a multiplicative way. This model allows us devising a novel algorithm that addresses the identification and correction for either additive and/or multiplicative contaminants. We test this model on galaxy mocks and systematics templates inspired from data of the third data release of the {\it Javalambre Photometric Local Universe Survey} (J-PLUS). We find that our method clearly outperforms standard methods that assume either an additive or multiplicative character for all contaminants in scenarios where both characters are actually acting on the observed data. In simpler scenarios where only an additive or multiplicative imprint on observations is considered, our hybrid method does not lie far behind the corresponding simplified, additive/multiplicative methods. Nonetheless, in scenarios of mild/low impact of systematics, we find that our hybrid approach converges towards the standard method that assumes additive contamination, as predicted by our model describing systematics. Our methodology also allows for the estimation of biases induced by systematics residuals on different angular scales and under different observational configurations, although these predictions necessarily restrict to the subset of {\em known/identified} potential systematics, and say nothing about ``unknown unknowns" possibly impacting the data., Comment: 17 pages, 9 figures, to be submitted to the Open Journal for Astrophysics. Comments welcome!

The {\it Javalambre Photometric Local Universe Survey} (J-PLUS) is a {\it spectro-photometric} survey covering about 3,000~deg$^2$ in its third data release (DR3), and containing about 300,000 galaxies with high quality ({\it odds}$>0.8$) photometric redshifts (hereafter photo-$z$s). We use this galaxy sample to conduct a tomographic study of the counts and redshift angular fluctuations under Gaussian shells sampling the redshift range $z\in[0.05,0.25]$. We confront the angular power spectra of these observables measured under shells centered on 11 different redshifts with theoretical expectations derived from a linear Boltzmann code ({\tt ARFCAMB}). Overall we find that J-PLUS DR3 data are well reproduced by our linear, simplistic model. We obtain that counts (or density) angular fluctuations (hereafter ADF) are very sensitive to the linear galaxy bias $b_g(z)$, although weakly sensitive to radial peculiar velocities of the galaxy field, while suffering from systematics residuals for $z>0.15$. Angular redshift fluctuations (ARF), instead, show higher sensitivity to radial peculiar velocities and also higher sensitivity to the average uncertainty in photo-$z$s ($\sigma_{\rm Err}$), with no obvious impact from systematics. For $z<0.15$ both ADF and ARF agree on measuring a monotonically increasing linear bias varying from $b_g(z=0.05)\simeq 0.9\pm 0.06$ up to $b_g(z=0.15)\simeq 1.5\pm 0.05$, while, by first time, providing consistent measurements of $\sigma_{\rm Err}(z)\sim 0.014$ that are $\sim 40~\%$ higher than estimates from the photo-$z$ code {\tt LePhare}, ($\sigma_{\rm Err}^{\rm LePhare}=0.010$). As expected, this photo-$z$ uncertainty level prevents the detection of radial peculiar velocities in the modest volume sampled by J-PLUS DR3, although prospects for larger galaxy surveys of similar (and higher) photo-$z$ precision are promising., Comment: About 20 pages and 14 figures, to be submitted to the Open Journal for Astrophysics. Comments welcome!

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 $H^s(\Omega(t))$ 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 $L^2(\Omega)$ 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., Comment: 11 pages

We report the discovery of an ultra-massive grand-design red spiral galaxy, named Zh\'ul\'ong (Torch Dragon), at $z_{\rm phot} = 5.2^{+0.3}_{-0.2}$ in the JWST PANORAMIC survey, identified as the most distant bulge+disk galaxy with spiral arms known to date. Zh\'ul\'ong displays an extraordinary combination of properties: 1) a classical bulge centered in a large, face-on exponential stellar disk (half-light radius of $R_{\rm e} = 3.7 \pm 0.1 \, \mathrm{kpc}$), with spiral arms extending across 19 kpc; 2) a clear transition from the red, quiescent core ($F150W-F444W=3.1$ mag) with high stellar mass surface density ($\log(\Sigma M_{\star}/M_{\odot} \, \mathrm{kpc}^{-2}) = 9.91_{-0.09}^{+0.11}$) to the star-forming outer regions, as revealed by spatially resolved SED analysis, which indicates significant inside-out galaxy growth; 3) an extremely high stellar mass at its redshift, with $\log (M_{\star}/M_{\odot})=11.03_{-0.08}^{+0.10}$ comparable to the Milky Way, and an implied baryon-to-star conversion efficiency ($\epsilon \sim 0.3$) that is 1.5 times higher than even the most efficient galaxies at later epochs; 4) despite an active disk, a relatively modest overall star formation rate ($\mathrm{SFR} =66_{-46}^{+89} ~M_{\odot} \, \mathrm{yr}^{-1}$), which is $>$0.5 dex below the star formation main sequence at $z \sim 5.2$ and $>$10 times lower than ultra-massive dusty galaxies at $z=5-6$. Altogether, Zh\'ul\'ong shows that mature galaxies emerged much earlier than expected in the first billion years after the Big Bang through rapid galaxy formation and morphological evolution. Our finding offers key constraints for models of massive galaxy formation and the origin of spiral structures in the early universe., Comment: 11 pages, 5 main figures, 2 extended figures, 1 table; submitted to A&A

This paper presents a deep MIRI/JWST medium resolution spectroscopy (MRS) covering the rest-frame optical spectrum of the GN-z11 galaxy. The [OIII]5008 and H$\alpha$ emission lines are detected and spectroscopically resolved. The line profiles are well-modeled by a narrow Gaussian component with intrinsic FWHMs of 189$\pm$25 and 231$\pm$52 kms$^{-1}$, respectively. We do not find any evidence of a dominant broad H$\alpha$ emission line component tracing a Broad Line Region in a type 1 active galactic nuclei (AGN). However, a broad ($\sim$430-470 kms$^{-1}$) and weak ($<$ 20-30%) H$\alpha$ line component, tracing a minor AGN contribution in the optical, cannot be ruled out completely with the sensitivity of the present data. The physical and excitation properties of the ionized gas are consistent with a low-metallicity starburst forming stars at a rate of SFR(H$\alpha$)$=$24 $\pm$3$M_{\odot}$yr$^{-1}$. The electron temperature of the ionized gas is $T_{\mathrm{e}}$(O$^{++}$)$=$14000$\pm$2100K, while the direct-$T_{\mathrm{e}}$ gas-phase metallicity is 12+$\log$(O/H)$=$7.91$\pm$0.07 (Z=0.17$\pm$0.03Z$_{\odot}$). The optical line ratios locate GN-z11 in the starburst or AGN region but more consistent with those of local low-metallicity starbursts and high-$z$ luminous galaxies detected at redshifts similar to GN-z11. We conclude that the MRS optical spectrum of GN-z11 is consistent with that of a massive, compact, and low-metallicity starburst galaxy. Due to its high SFR and stellar mass surface densities, close to that of the densest stellar clusters, we speculate that GN-z11 could be undergoing a feedback-free, highly efficient starburst phase. Additional JWST data are needed to validate this scenario, and other recently proposed alternatives, to explain the existence of bright compact galaxies in the early Universe., Comment: accepted for publication in A&A (17 pages, 6 figures, 2 tables)

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 \( a(x, \xi) \in S^m(\mathbb{R}^n) \), 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 \( p_0(x, \xi) \), with energy dissipation controlled by the imaginary part of \( p_0(x, \xi) \). 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., Comment: 28 pages

This paper explores the application of Madelung hydrodynamic models to study two-dimensional electron gases, with a focus on nonlocal plasmonics and nonlinear optics. We begin by reviewing the derivation of the Madelung equations. Using the Madelung equations in conjunction with Poisson's equation, we calculate the spectrum of magnetoplasmons and the magneto-optical conductivity in the electrostatic regime, incorporating nonlocal corrections due to the Fermi pressure. In the absence of a magnetic field, we analyze nonlinear and nonlocal second-harmonic generation, demonstrating how plasmon excitation enhances this process. We further discuss the emergence of self-modulation phenomena driven by nonlinearity, leading to the renormalization of the plasmon dispersion. Notably, we show that nonlinearity amplifies nonlocal effects and, leveraging the hydrodynamic formalism, derive a simple analytic expression for the renormalized spectra., Comment: 14 pages and 4 figures

Twistronics has received much attention as a new method to manipulate the properties of 2D van der Waals structures by introducing moir\'e patterns through a relative rotation between two layers. Here we begin a theoretical exploration of twistronics beyond the realm of van der Waals materials by developing a first-principles description of the electronic structure and interlayer interactions of ultrathin perovskite bilayers. We construct both an ab initio tight-binding model as well as a minimal 3-band effective model for the valence bands of monolayers and bilayers of oxides derived from the Ruddlesden-Popper phase of perovskites, which is amenable to thin-layer formation. We illustrate the approach with the specific example of Sr$_2$TiO$_4$ layers but also provide model parameters for Ca$_2$TiO$_4$ and Ba$_2$TiO$_4$ ., Comment: 14 pages, 7 figures, supplementary material

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 $\rho(t, x)$, 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., Comment: 15 pages

The FASTSUM Collaboration has developed a comprehensive research programme in thermal lattice QCD using 2+1 flavour ensembles. We review our recent hadron spectrum analyses of open charm mesons and charm baryons at non-zero temperature. We also detail our determination of the interquark potential in the bottomonium system using NRQCD quarks. All of our work uses anisotropic lattices where the temporal lattice spacing is considerably finer than the spatial one allowing better resolution of temporal correlation functions., Comment: 6 pages. Contribution to 42nd International Conference on High Energy Physics (ICHEP2024), 18-24 July 2024, Prague, Czech Republic

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., Comment: 27 pages

We report the discovery of five galaxy candidates at redshifts between $15.9

We present cosmological results from the measurement of clustering of galaxy, quasar and Lyman-$\alpha$ forest tracers from the first year of observations with the Dark Energy Spectroscopic Instrument (DESI Data Release 1). We adopt the full-shape (FS) modeling of the power spectrum, including the effects of redshift-space distortions, in an analysis which has been validated in a series of supporting papers. In the flat $\Lambda$CDM cosmological model, DESI (FS+BAO), combined with a baryon density prior from Big Bang Nucleosynthesis and a weak prior on the scalar spectral index, determines matter density to $\Omega_\mathrm{m}=0.2962\pm 0.0095$, and the amplitude of mass fluctuations to $\sigma_8=0.842\pm 0.034$. The addition of the cosmic microwave background (CMB) data tightens these constraints to $\Omega_\mathrm{m}=0.3056\pm 0.0049$ and $\sigma_8=0.8121\pm 0.0053$, while further addition of the the joint clustering and lensing analysis from the Dark Energy Survey Year-3 (DESY3) data leads to a 0.4% determination of the Hubble constant, $H_0 = (68.40\pm 0.27)\,{\rm km\,s^{-1}\,Mpc^{-1}}$. In models with a time-varying dark energy equation of state, combinations of DESI (FS+BAO) with CMB and type Ia supernovae continue to show the preference, previously found in the DESI DR1 BAO analysis, for $w_0>-1$ and $w_a<0$ with similar levels of significance. DESI data, in combination with the CMB, impose the upper limits on the sum of the neutrino masses of $\sum m_\nu < 0.071\,{\rm eV}$ at 95% confidence. DESI data alone measure the modified-gravity parameter that controls the clustering of massive particles, $\mu_0=0.11^{+0.45}_{-0.54}$, while the combination of DESI with the CMB and the clustering and lensing analysis from DESY3 constrains both modified-gravity parameters, giving $\mu_0 = 0.04\pm 0.22$ and $\Sigma_0 = 0.044\pm 0.047$, in agreement with general relativity. [Abridged.], Comment: This DESI Collaboration Key Publication is part of the 2024 publication series using the first year of observations (see https://data.desi.lbl.gov/doc/papers/). 55 pages, 10 figures

We present the samples of galaxies and quasars used for DESI 2024 cosmological analyses, drawn from the DESI Data Release 1 (DR1). We describe the construction of large-scale structure (LSS) catalogs from these samples, which include matched sets of synthetic reference `randoms' and weights that account for variations in the observed density of the samples due to experimental design and varying instrument performance. We detail how we correct for variations in observational completeness, the input `target' densities due to imaging systematics, and the ability to confidently measure redshifts from DESI spectra. We then summarize how remaining uncertainties in the corrections can be translated to systematic uncertainties for particular analyses. We describe the weights added to maximize the signal-to-noise of DESI DR1 2-point clustering measurements. We detail measurement pipelines applied to the LSS catalogs that obtain 2-point clustering measurements in configuration and Fourier space. The resulting 2-point measurements depend on window functions and normalization constraints particular to each sample, and we present the corrections required to match models to the data. We compare the configuration- and Fourier-space 2-point clustering of the data samples to that recovered from simulations of DESI DR1 and find they are, generally, in statistical agreement to within 2\% in the inferred real-space over-density field. The LSS catalogs, 2-point measurements, and their covariance matrices will be released publicly with DESI DR1., Comment: This DESI Collaboration Key Publication is part of the 2024 publication series using the first year of observations (see https://data.desi.lbl.gov/doc/papers/)

We present the measurements and cosmological implications of the galaxy two-point clustering using over 4.7 million unique galaxy and quasar redshifts in the range $0.1

The Dark Energy Spectroscopic Instrument (DESI) survey is mapping the large-scale distribution of millions of Emission Line Galaxies (ELGs) over vast cosmic volumes to measure the growth history of the Universe. However, compared to Luminous Red Galaxies (LRGs), very little is known about the connection of ELGs with the underlying matter field. In this paper, we employ a novel theoretical model, SHAMe-SF, to infer the connection between ELGs and their host dark matter subhaloes. SHAMe-SF is a version of subhalo abundance matching that incorporates prescriptions for multiple processes, including star formation, tidal stripping, environmental correlations, and quenching. We analyse the public measurements of the projected and redshift-space ELGs correlation functions at $z=1.0$ and $z=1.3$ from DESI One Percent data release, which we fit over a broad range of scales $r \in [0.1, 30]/h^{-1}$Mpc to within the statistical uncertainties of the data. We also validate the inference pipeline using two mock DESI ELG catalogues built from hydrodynamical (TNG300) and semi-analytical galaxy formation models (\texttt{L-Galaxies}). SHAMe-SF is able to reproduce the clustering of DESI-ELGs and the mock DESI samples within statistical uncertainties. We infer that DESI ELGs typically reside in haloes of $\sim 10^{11.8}h^{-1}$M$_{\odot}$ when they are central, and $\sim 10^{12.5}h^{-1}$M$_{\odot}$ when they are a satellite, which occurs in $\sim$30 \% of the cases. In addition, compared to the distribution of dark matter within halos, satellite ELGs preferentially reside both in the outskirts and inside haloes, and have a net infall velocity towards the centre. Finally, our results show evidence of assembly bias and conformity., Comment: 23 pages, 17 figures. To be submitted to A&A

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)., Comment: 13 pages

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 Schr\"odinger-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., Comment: 12 pages

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., Comment: 22 pages

This article presents an innovative extension of the Smagorinsky model incorporating dynamic boundary conditions and advanced regularity methods. We formulate the modified Navier-Stokes equations with the Smagorinsky term to model dissipation in turbulence and prove theorems concerning the existence, uniqueness, and asymptotic behavior of solutions. The first theorem establishes the existence and uniqueness of solutions in higher Sobolev spaces, considering the effect of the nonlinear Smagorinsky term and dynamic boundary conditions. The proof employs the Galerkin method and energy estimates, culminating in the application of Gr\"onwall's theorem. The second theorem investigates the asymptotic behavior of solutions, focusing on anomalous dissipation in high-turbulence regimes. We demonstrate that the dissipated energy does not decrease with vanishing viscosity, indicating the occurrence of anomalous dissipation. The third theorem explores advanced regularity in higher Sobolev spaces, allowing for more rigorous control of nonlinear terms and ensuring improved stability conditions. The proof utilizes the energy method combined with Sobolev estimates and Gr\"onwall's inequality. These mathematical results are fundamental for the analysis of dissipation in turbulent flows and can inspire new approaches in numerical simulations of fluids., Comment: 12 pages

The innermost region of the Milky Way harbors the central molecular zone (CMZ). This region contains a large amount of molecular gas but a poor star formation rate considering the densities achieved by the gas in this region. We used the arepo code to perform a hydrodynamic and star formation simulation of the Galaxy, where a Ferrers bar was adiabatically introduced. During the stage of bar imposition, the bar strength excites density waves close to the inner Lindblad resonance guiding material toward the inner Galaxy, driving the formation of a ring that we qualitatively associate with the CMZ. During the simulation, we identified that the ring passes three main phases, namely: formation, instability, and quasi-stationary stages. During the whole evolution, and particularly in the quasi-stationary stage, we observe that the ring is associated with the x2 family of periodic orbits. Additionally, we found that most of the star formation occurs during the ring formation stage, while it drastically decreases in the instability stage. Finally, we found that when the gas has settled in a stable x2 orbit, the star formation takes place mostly after the dense gas passes the apocenter, triggering the conveyor-belt mechanism described in previous studies., Comment: Accepted for publication in PASA

This paper enhances the classic Smagorinsky model by introducing an innovative, adaptive dissipation term that adjusts dynamically with distance from boundary regions. This modification addresses a known limitation of the standard model over dissipation near boundaries thereby improving accuracy in turbulent flow simulations in confined or wall-adjacent areas. We present a rigorous theoretical framework for this adaptive model, including two foundational theorems. The first theorem guarantees existence and uniqueness of solutions, ensuring that the model is mathematically well-posed within the adaptive context. The second theorem provides a precise bound on the energy dissipation rate, demonstrating that dissipation remains controlled and realistic even as boundary effects vary spatially. By allowing the dissipation coefficient to decrease near boundary layers, this approach preserves the finer turbulent structures without excessive smoothing, yielding a more physically accurate representation of the flow. Future work will focus on implementing this adaptive model in computational simulations to empirically verify the theoretical predictions and assess performance in scenarios with complex boundary geometries., Comment: 9 pages

This study presents an extension of the corrected Smagorinsky model, incorporating advanced techniques for error estimation and regularity analysis of far-from-equilibrium turbulent flows. A new formulation that increases the model's ability to explain complex dissipative processes in turbulence is presented, using higher-order Sobolev spaces to address incompressible and compressible Navier-Stokes equations. Specifically, a refined energy dissipation mechanism that provides a more accurate representation of turbulence is introduced, particularly in the context of multifractal flow regimes. Furthermore, we derive new theoretical results on energy regularization in multifractal turbulence, contributing to the understanding of anomalous dissipation and vortex stretching in turbulent flows. The work also explores the numerical implementation of the model in the presence of challenging boundary conditions, particularly in dynamically evolving domains, where traditional methods struggle to maintain accuracy and stability. Theoretical demonstrations and analytical results are provided to validate the proposed framework, with implications for theoretical advances and practical applications in computational fluid dynamics. This approach provides a basis for more accurate simulations of turbulence, with potential applications ranging from atmospheric modeling to industrial fluid dynamics., Comment: 25 pages

This study investigates the dynamics of incompressible fluid flows through quaternionic variables integrated within Sobolev-Besov spaces. Traditional mathematical models for fluid dynamics often employ Sobolev spaces to analyze the regularity of the solution to the Navier-Stokes equations. However, with the unique ability of Besov spaces to provide localized frequency analysis and handle high-frequency behaviors, these spaces offer a refined approach to address complex fluid phenomena such as turbulence and bifurcation. Quaternionic analysis further enhances this approach by representing three-dimensional rotations directly within the mathematical framework. The author presents two new theorems to advance the study of regularity and energy dissipation in fluid systems. The first theorem demonstrates that energy dissipation in quaternionic fluid systems is dominated by the higher-frequency component in Besov spaces, with contributions decaying at a rate proportional to the frequency of the quaternionic component. The second theorem provides conditions for regularity and existence of solutions in quaternionic fluid systems with external forces. By integrating these hypercomplex structures with Sobolev-Besov spaces, our work offers a new mathematically rigorous framework capable of addressing frequency-specific dissipation patterns and rotational symmetries in turbulent flows. The findings contribute to fundamental questions in fluid dynamics, particularly by improving our understanding of high Reynolds number flows, energy cascade behaviors, and quaternionic bifurcation. This framework therefore paves the way for future research on regularity in complex fluid dynamics., Comment: 13 pages

This paper advances the stochastic regularity theory for the Navier-Stokes equations by introducing a variable-intensity noise model within the Sobolev and Besov spaces. Traditional models usually assume constant-intensity noise, but many real-world turbulent systems exhibit fluctuations of varying intensities, which can critically affect flow regularity and energy dynamics. This work addresses this gap by formulating a new regularity theorem that quantifies the impact of stochastic perturbations with bounded variance on the energy dissipation and smoothness properties of solutions. The author employs techniques such as the Littlewood-Paley decomposition and interpolation theory, deriving rigorous bounds, and we demonstrate how variable noise intensities influence the behavior of the solution over time. This study contributes theoretically by improving the understanding of energy dissipation in the presence of stochastic perturbations, particularly under conditions relevant to turbulent flows where randomness cannot be assumed to be uniform. The findings have practical implications for more accurate modeling and prediction of turbulent systems, allowing potential adjustments in simulation parameters to better reflect the observed physical phenomena. This refined model therefore provides a fundamental basis for future work in fluid dynamics, particularly in fields where variable stochastic factors are prevalent, including meteorology, oceanography, and engineering applications involving fluid turbulence. The present approach not only extends current theoretical frameworks but also paves the way for more sophisticated computational tools in the analysis of complex and stochastic fluid systems., Comment: 11 pages

This paper presents the Random-Key Optimizer (RKO), a versatile and efficient stochastic local search method tailored for combinatorial optimization problems. Using the random-key concept, RKO encodes solutions as vectors of random keys that are subsequently decoded into feasible solutions via problem-specific decoders. The RKO framework is able to combine a plethora of classic metaheuristics, each capable of operating independently or in parallel, with solution sharing facilitated through an elite solution pool. This modular approach allows for the adaptation of various metaheuristics, including simulated annealing, iterated local search, and greedy randomized adaptive search procedures, among others. The efficacy of the RKO framework, implemented in C++, is demonstrated through its application to three NP-hard combinatorial optimization problems: the alpha-neighborhood p-median problem, the tree of hubs location problem, and the node-capacitated graph partitioning problem. The results highlight the framework's ability to produce high-quality solutions across diverse problem domains, underscoring its potential as a robust tool for combinatorial optimization., Comment: 54 pages, 16 figures, 8 tables

We use the Dirac continuum model to study the propagation of electronic wave packets in graphene with periodically arranged circular potential steps. The time propagation of the wave packets are calculated using the split-operator method for different size, height and separation of the barriers. The time propagation of the wave packets is calculated using the split-operator method for various barrier sizes, heights, and separations. We found that, despite the pronounced Klein tunneling effect in graphene, the presence of a lattice of defects significantly impacts the propagation properties of the wave packets. For example, depending on the height and size of the incident wave packet, the transmission probability can decrease by more than 20\%. The alteration of the polarity of the potential barriers also contributes to the transmission probabilities of the wave packet in graphene. The obtained results could provide valuable insights into the fundamental understanding of charge carrier dynamics in graphene-based nanodevices.

Mathematical modeling of fluid dynamics for computer graphics requires high levels of theoretical rigor to ensure visually plausible and computationally efficient simulations. This paper presents an in-depth theoretical framework analyzing the mathematical properties, such as stability, convergence, and error bounds, of numerical schemes used in fluid simulation. Conditions for stability in semi-Lagrangian and particle-based methods were derived, demonstrating that these methods remain stable under certain conditions. Furthermore, convergence rates for Navier-Stokes discretizations were obtained, showing that numerical solutions converge to analytical solutions as spatial resolution and time step decrease. Furthermore, new theoretical results were introduced on the maintenance of incompressibility and conservation of vorticity, which are crucial for the physical accuracy of simulations. The findings serve as a mathematical foundation for future research in adaptive fluid simulation, guiding the development of robust simulation techniques for real-time graphics applications., Comment: 6 pages

Enhanced diffusion, which describes the accelerated spread of passive scalars due to the interaction between advection and molecular diffusion, has been extensively studied in simplified geometries, such as uniform shear and radial flows. However, many real-world applications occur in complex, anisotropic domains where standard assumptions do not hold. This paper extends the theory of enhanced diffusion to anisotropic and non-uniform geometries, where scaling in the \(x\)- and \(y\)-directions follows distinct, power-law relationships, and the velocity field exhibits spatially varying regularity. We define a generalized framework for anisotropic diffusion enhancement and rigorously derive new scaling laws for diffusion rates in these settings. Specifically, we show that in a domain with anisotropic scaling functions \( f(x) \sim |x|^p \) and \( g(y) \sim |y|^q \), the enhanced diffusion rate \( r(\kappa) \) for a passive scalar satisfies \( r(\kappa) = C \kappa^{\frac{pq}{p+q+2}} \), where \( C \) depends on the regularity and scaling properties of the domain but remains independent of the diffusivity \( \kappa \). This theoretical result is validated through detailed proofs leveraging stochastic process techniques and variance estimation. Our findings offer a new perspective on the role of domain geometry and anisotropy in diffusion processes, with potential applications in environmental science, engineering, and beyond. This framework lays the groundwork for future studies in both theoretical and applied settings, enabling a deeper understanding of transport phenomena in complex domains., Comment: 9 pages

This work presents a comprehensive framework for enhanced diffusion modeling in fluid-structure interactions by combining the Immersed Boundary Method (IBM) with stochastic trajectories and high-order spectral boundary conditions. Using semi-Lagrangian schemes, this approach captures complex diffusion dynamics at moving interfaces, integrating probabilistic methods that reflect multi-scale fluctuations. In addition to a rigorous mathematical foundation that includes stability proofs, this model exhibits reduced numerical diffusion errors and improved stability in long-term simulations. Comparative studies highlight its effectiveness in multi-scale scenarios that require precision in interface dynamics. Focusing on various shear and circular flows, including those with H\"older and Lipschitz regularities and critical points, we establish sharp bounds on effective diffusion rates using specific initial data examples. This dual exploration in enhanced diffusion highlights how flow regularity and critical points influence dissipation. These findings advance both the theoretical understanding and practical applications of enhanced diffusion in fluid dynamics, offering new insights into diffusion rate optimization through interface dynamics and flow structure regularities. Future research can further refine the IBM framework by exploring alternative probabilistic methods to improve interface accuracy, opening up the potential for improved modeling in applications that require precise control over mixing rates and dissipation processes., Comment: 16 pages

This article proposes a hybrid adaptive numerical method based on the Dual Reciprocity Method (DRM) to solve problems with non-linear boundary conditions and large-scale problems, named Hybrid Adaptive Dual Reciprocity Method (H-DRM). The method uses a combination of DRM to handle non-homogeneous terms, iterative techniques to deal with non-linear boundary conditions, and an adaptive multiscale approach for large-scale problems. Additionally, the H-DRM incorporates local finite elements in critical regions of the domain. This method aims to improve computational efficiency and accuracy for problems involving complex geometry and non-linearities at the boundary, offering a robust solution for physical and engineering problems. Demonstrations and computational results are presented, validating the effectiveness of the method compared to other known methods through an iterative process of 7 million iterations., Comment: 15 pages

This article presents a comprehensive analysis of the formation and dissipation of vortices within chaotic fluid flows, leveraging the framework of Sobolev and Besov spaces on Riemannian manifolds. Building upon the Navier-Stokes equations, we introduce a hypercomplex bifurcation approach to characterize the regularity and critical thresholds at which vortices emerge and dissipate in chaotic settings. We explore the role of differential geometry and bifurcation theory in vortex dynamics, providing a rigorous mathematical foundation for understanding these phenomena. Our approach addresses spectral decomposition, asymptotic stability, and dissipation thresholds, offering critical insights into the mechanisms of vortex formation and dissipation. Additionally, we introduce two new theorems that further elucidate the role of geometric stability and bifurcations in vortex dynamics. The first theorem demonstrates the geometric stability of vortices on manifolds with positive Ricci curvature, while the second theorem analyzes the bifurcation points that lead to the formation or dissipation of vortex structures. This research contributes to the existing literature by providing a more complete mathematical picture of the underlying mechanisms of vortex dynamics in turbulent fluid flows., Comment: 15 pages

This research explores the development and application of the High-Order Dynamic Integration Method for solving integro-differential equations, with a specific focus on turbulent fluid dynamics. Traditional numerical methods, such as the Finite Difference Method and the Finite Volume Method, have been widely employed in fluid dynamics but struggle to accurately capture the complexities of turbulence, particularly in high Reynolds number regimes. These methods often require significant computational resources and are prone to errors in nonlinear dynamic systems. The High-Order Dynamic Integration Method addresses these challenges by integrating higher-order interpolation techniques with dynamic adaptation strategies, significantly enhancing accuracy and computational efficiency. Through rigorous numerical analysis, this method demonstrates superior performance over the Finite Difference Method and the Finite Volume Method in handling the nonlinear behaviors characteristic of turbulent flows. Furthermore, the High-Order Dynamic Integration Method achieves this without a substantial increase in computational cost, making it a highly efficient tool for simulations in computational fluid dynamics. The research validates the capabilities of the High-Order Dynamic Integration Method through a series of benchmark tests and case studies. Results indicate a marked improvement in both accuracy and stability, particularly in simulations of high-Reynolds-number flows, where traditional methods often falter. This innovative approach offers a robust and efficient alternative for solving complex fluid dynamics problems, contributing to advances in the field of numerical methods and computational fluid dynamics., Comment: 7 pages