Search Results (151)

The scattering interaction between a circularly polarized Bessel pincer light-sheet beam and a chiral particle is investigated within the framework of generalized Lorenz–Mie theory (GLMT). The incident electric field distribution is rigorously derived via the vector angular spectrum decomposition method (VASDM), with subsequent determination of the beam-shape coefficients (BSCs) $p_{mn}^u$ and $q_{mn}^u$ through multipole expansion in the basis of vector spherical wave functions (VSWFs). The expansion coefficients for the scattered field ($A_{mn}^s$$B_{mn}^s$) and interior field ($A_{mn}$$B_{mn}$) are derived by imposing boundary conditions. Simulations highlight notable variations in the scattering field, near-surface field distribution, and far-field intensity, strongly influenced by the dimensionless size parameter $ka$, chirality $\kappa$, and beam parameters (beam order l and beam scaling parameter ${\alpha }_0$). These findings provide insights into the role of chirality in modulating scattering asymmetry and localization effects. The results are particularly relevant for applications in optical manipulation and super-resolution imaging in single-molecule microbiology. Full article

Traditional 4-dimensional variational data assimilation methods have limitations due to the Gaussian distribution assumption of observation errors, and the gradient of the objective functional is vulnerable to observation noise and outliers. To address these issues, this paper proposes a non-Gaussian nonlinear data assimilation method called α-4DVar, based on Rényi entropy and the α-generalized Gaussian distribution. By incorporating the heavy-tailed property of Rényi entropy, the objective function and its gradient suitable for non-Gaussian errors are derived, and numerical experiments are conducted using the Lorenz-63 model. Experiments are conducted with Gaussian and non-Gaussian errors as well as different initial guesses to compare the assimilation effects of traditional 4DVar and α-4DVar. The results show that α-4DVar performs as well as traditional method without observational errors. Its analysis field is closer to the truth, with RMSE rapidly dropping to a low level and remaining stable, particularly under non-Gaussian errors. Under different initial guesses, the RMSE of both the background and analysis fields decreases quickly and stabilizes. In conclusion, the α-4DVar method demonstrates significant advantages in handling non-Gaussian observational errors, robustness against noise, and adaptability to various observational conditions, thus offering a more reliable and effective solution for data assimilation. Full article

We establish a correspondence between Lorenz curves and Pickands dependence functions, thereby reframing the construction of any bivariate extreme-value copula as an inequality problem. We discuss the conditions under which a Lorenz curve generates a closed-form Pickands model, considerably expanding the small set of tractable parametrizations currently available. Furthermore, the Pickands measure-generating function M can be written explicitly in terms of the quantile function underlying the Lorenz curve, providing a constructive route to model specification. Finally, classical inequality indices like the Gini coincide with scale-free, rotation-invariant indices of global upper-tail dependence, thereby complementing local coefficients such as the upper tail dependence index ${\lambda }_U$. Full article

Accurate initial conditions are crucial for improving numerical weather prediction (NWP). Variational data assimilation relies on a static background error covariance matrix (B), yet its variance estimation is often inaccurate, affecting assimilation and forecast performance. This study introduces DRL-AST, a deep reinforcement learning-based adaptive variance rescaling strategy that dynamically adjusts the variances of B to optimize forecast skill through improved assimilation performance. By formulating variance rescaling as a Markov Decision Process and employing an actor–critic framework with Proximal Policy Optimization, DRL-AST autonomously selects spatio-temporal rescaling factors, enhancing assimilation and forecast accuracy without additional computational cost. As a new paradigm for adaptive variance tuning, DRL-AST demonstrates competitive improvements in forecast skill in experiments with the Lorenz-96 model by generating initial states that better conform to model dynamical consistency. Given its adaptability and efficiency, DRL-AST holds great potential for application in high-dimensional NWP models, where deep learning-based dimensionality reduction and reinforcement learning techniques could further enhance its feasibility and effectiveness in complex assimilation frameworks. Full article

Current issues in sustainable development concern research on comprehensiveness, coherence and practicality. Therefore, the objective was to develop and test a novelty approach to product sustainability assessment based on life cycle, quality, and costs. This approach extends the iterative design thinking process (DT), including overcoming the limitations of existing LCSA methods. We present a systematic process for obtaining and processing customer requirements with a survey and Pareto–Lorenz analysis. Then, using an algorithm developed in Matlab R2021a program, we generated product prototypes considering the key criteria presented in various dimensions of current and modified states. Next, we propose the modeling of prospective LCA for all prototypes in the OpenLCA program with Ecoinvent database. Finally, we aggregated the results considering the cost of prototypes in environmental–cost analysis to determine the direction of product sustainability. We tested this approach in detail with the example of vacuum cleaners for domestic and commercial use. After a literature review and survey research in customers, we developed 54 prototypes, where the modified key quality criteria were as follows: vacuum in the suction pipe, engine power, operating range, and length of the power cable. Using this approach, it was possible to select six prototypes that best meet customer requirements, are environmentally friendly, and cost-effective. Finally, we discuss contributions to DT and LCSA methodologies, and propose future directions for development within the application of artificial intelligence (AI). This approach can be a practical application in SMEs already in the early stages of product development (conceptualization), where access to detailed data is limited. Full article

A fractional-order Lorenz system is optimized to maximize its maximum Lyapunov exponent and Kaplan-York dimension using the Non-dominated Sorting Genetic Algorithm II (NSGA-II) algorithm. The fractional-order Lorenz system is integrated with a recent process called the “modified two-stage Runge-Kutta” (M2sFRK) method, which is very fast and efficient. A Pseudo-Random Number Generator (PRNG) was built using one of the optimized systems that was obtained. The M2sFRK method allows for obtaining a very fast optimization time and also designing a very efficient PRNG with linear complexity, $O\left(n\right)$. The designed PRNG generates 24 random bits at each iteration step, and the random sequences pass all the National Institute of Standards and Technology (NIST) and TestU01 statistical tests, making the PRNG suitable for cryptographic applications. The presented methodology could be extended to any other chaotic system. Full article

Learning high-dimensional chaos is a complex and challenging problem because of its initial value-sensitive dependence. Based on an echo state network (ESN), we introduce homotopy transformation in topological theory to learn high-dimensional chaos. On the premise of maintaining the basic topological properties, our model can obtain the key features of chaos for learning through the continuous transformation between different activation functions, achieving an optimal balance between nonlinearity and linearity to enhance the generalization capability of the model. In the experimental part, we choose the Lorenz system, Mackey–Glass (MG) system, and Kuramoto–Sivashinsky (KS) system as examples, and we verify the superiority of our model by comparing it with other models. For some systems, the prediction error can be reduced by two orders of magnitude. The results show that the addition of homotopy transformation can improve the modeling ability of complex spatiotemporal chaotic systems, and this demonstrates the potential application of the model in dynamic time series analysis. Full article

This study investigates the scattering dynamics of vector Bessel–Gaussian (BG) beams in winter haze environments, with a particular emphasis on the influence of ice-coated haze particles on light propagation. Employing the Generalized Lorenz–Mie Theory (GLMT), we analyze the scattering coefficients of particles transitioning from water to ice coatings under varying atmospheric conditions. Our results demonstrate that the presence of ice coatings significantly alters the scattering and extinction efficiencies of BG beams, revealing distinct differences compared to particles coated with water. Furthermore, the study examines the role of Orbital Angular Momentum (OAM) modes in shaping scattering behavior. We show that higher OAM modes, characterized by broader energy distributions and larger beam spot sizes, induce weaker localized interactions with individual particles, leading to diminished scattering and attenuation. In contrast, lower OAM modes, with energy concentrated in smaller regions, exhibit stronger interactions with particles, thereby enhancing scattering and attenuation. These findings align with the Beer–Lambert law in the single scattering regime, where beam intensity attenuation is influenced by the spatial distribution of radiation, while overall power attenuation follows the standard exponential decay with respect to propagation distance. The transmission attenuation of BG beams through haze-laden atmospheres is further explored, emphasizing the critical roles of particle concentration and humidity. This study provides valuable insights into the interactions between vector BG beams and atmospheric haze, advancing the understanding of optical communication and environmental monitoring in hazy conditions. Full article

The fractional differential quadrature method (FDQM) with generalized Caputo derivatives is used in this paper to show a new numerical way to solve fractional Riccati equations and fractional Lorenz systems. Unlike previous FDQM applications that have primarily focused on linear problems, our work pioneers the use of this method for nonlinear fractional initial value problems. By combining Lagrange interpolation polynomials and discrete singular convolution (DSC) shape functions with the generalized Caputo operator, we effectively transform nonlinear fractional equations into algebraic systems. An iterative method is then utilized to address the nonlinearity. Our numerical results, obtained using MATLAB, demonstrate the exceptional accuracy and efficiency of this approach, with convergence rates reaching 10⁻⁸. Comparative analysis with existing methods highlights the superior performance of the DSC shape function in terms of accuracy, convergence speed, and reliability. Our results highlight the versatility of our approach in tackling a wider variety of intricate nonlinear fractional differential equations. Full article

A new family of inequality indices based on the deviation between the expected maximum and the expected minimum of random samples, called the nth Gini index is presented. These indices generalize the Gini index. At the same time, this family of indices and the S-Gini index are generalized by proposing the $uv-$Gini index, which turns out to be a convex combination of the S-Gini index and the Lorenz family of inequality measures. This family of Gini indices provides a methodology for achieving perfect equality in a given distribution of incomes. This is achieved through a series of successive and equal increases in the incomes of each individual. Full article

This paper examines the socio-economic significance of forest visits and the collection of forest berries and mushrooms (FBMs) in the Czech Republic, emphasising their role in enhancing human well-being and contributing to regional economies. Over a 30-year period, data were collected on the quantities and economic values of FBMs, alongside the intensity of forest visits by the Czech population. This study incorporates a detailed analysis of time series data on FBM collection, exploring trends and fluctuations in the harvested quantities and their economic value. A Lorenz curve analysis reveals significant disparities in the distribution of economic benefits, with a small segment of the population accounting for the majority of the FBM-derived value. Additionally, the research investigates the impact of forest visitation on well-being at the regional level, highlighting the relationship between forest access, visitation intensity, and public health benefits. This study also examines visitors’ expectations, motivations, and perceptions regarding an ideal forest for visitation, providing recommendations for effective marketing strategies. Furthermore, the study explores the contribution of FBMs to net income across different regions, demonstrating substantial regional variation in their economic importance. Notably, the analysis shows that the value of FBMs represents approximately 37% of the net income generated by traditional forestry activities, underscoring its significant economic potential. The findings emphasise the potential of territorial marketing strategies to enhance well-being, particularly in economically disadvantaged regions, and advocate for sustainable forest management practices to protect these valuable resources and ensure equitable access to the benefits provided by forest ecosystems. Full article

This paper investigates the feasibility of downscaling within high-dimensional Lorenz models through the use of machine learning (ML) techniques. This study integrates atmospheric sciences, nonlinear dynamics, and machine learning, focusing on using large-scale atmospheric data to predict small-scale phenomena through ML-based empirical models. The high-dimensional generalized Lorenz model (GLM) was utilized to generate chaotic data across multiple scales, which was subsequently used to train three types of machine learning models: a linear regression model, a feedforward neural network (FFNN)-based model, and a transformer-based model. The linear regression model uses large-scale variables to predict small-scale variables, serving as a foundational approach. The FFNN and transformer-based models add complexity, incorporating multiple hidden layers and self-attention mechanisms, respectively, to enhance prediction accuracy. All three models demonstrated robust performance, with correlation coefficients between the predicted and actual small-scale variables exceeding 0.9. Notably, the transformer-based model, which yielded better results than the others, exhibited strong performance in both control and parallel runs, where sensitive dependence on initial conditions (SDIC) occurs during the validation period. This study highlights several key findings and areas for future research: (1) a set of large-scale variables, analogous to multivariate analysis, which retain memory of their connections to smaller scales, can be effectively leveraged by trained empirical models to estimate irregular, chaotic small-scale variables; (2) modern machine learning techniques, such as FFNN and transformer models, are effective in capturing these downscaling processes; and (3) future research could explore both downscaling and upscaling processes within a triple-scale system (e.g., large-scale tropical waves, medium-scale hurricanes, and small-scale convection processes) to enhance the prediction of multiscale weather and climate systems. Full article

Aiming to explore the subtle connection between the number of nonlinear terms in Lorenz-like systems and hidden attractors, this paper introduces a new simple sub-quadratic four-thirds-degree Lorenz-like system, where $\dot{x}=a(y-x)$, $\dot{y}=cx-\sqrt[3]{x}z$, $\dot{z}=-bz+\sqrt[3]{x}y$, and uncovers the following property of these systems: decreasing the powers of the nonlinear terms in a quadratic Lorenz-like system where $\dot{x}=a(y-x)$, $\dot{y}=cx-xz$, $\dot{z}=-bz+xy$, may narrow, or even eliminate the range of the parameter c for hidden attractors, but enlarge it for self-excited attractors. By combining numerical simulation, stability and bifurcation theory, most of the important dynamics of the Lorenz system family are revealed, including self-excited Lorenz-like attractors, Hopf bifurcation and generic pitchfork bifurcation at the origin, singularly degenerate heteroclinic cycles, degenerate pitchfork bifurcation at non-isolated equilibria, invariant algebraic surface, heteroclinic orbits and so on. The obtained results may verify the generalization of the second part of the celebrated Hilbert’s sixteenth problem to some degree, showing that the number and mutual disposition of attractors and repellers may depend on the degree of chaotic multidimensional dynamical systems. Full article

Misunderstandings in dyadic interactions often persist despite our best efforts, particularly between native and non-native speakers, resembling a broken duet that refuses to harmonise. This paper delves into the computational mechanisms underpinning these misunderstandings through the lens of the broken Lorenz system—a continuous dynamical model. By manipulating a specific parameter regime, we induce bistability within the Lorenz equations, thereby confining trajectories to distinct attractors based on initial conditions. This mirrors the persistence of divergent interpretations that often result in misunderstandings. Our simulations reveal that differing prior beliefs between interlocutors result in misaligned generative models, leading to stable yet divergent states of understanding when exposed to the same percept. Specifically, native speakers equipped with precise (i.e., overconfident) priors expect inputs to align closely with their internal models, thus struggling with unexpected variations. Conversely, non-native speakers with imprecise (i.e., less confident) priors exhibit a greater capacity to adjust and accommodate unforeseen inputs. Our results underscore the important role of generative models in facilitating mutual understanding (i.e., establishing a shared narrative) and highlight the necessity of accounting for multistable dynamics in dyadic interactions. Full article

Electromagnetic scattering is a routine tool for rapid, non-contact characterization of particle media. In previous work, the interaction targets of scattering intensity, scattering efficiency, and extinction efficiency of Bessel pincer light-sheet beams were all aimed at dielectric spheres. However, most particles in nature are charged. Considering the boundary condition on a charged sphere, the beam shape coefficients (BSCs) ($p_{mn},q_{mn}$) of the charged spherical particle illuminated by a Bessel pincer light-sheet beam are obtained. The extinction, scattering, and absorption efficiencies are derived under the generalized Lorenz–Mie theory (GLMT) framework. This study reveals the significant differences in scattering characteristics of Bessel pincer light-sheet beams on a charged particle compared to traditional beams. The simulations show a few apparent differences in the far-field scattering intensity and efficiencies between charged and natural spheres under the influence of dimensionless size parameters. As dimensionless parameters increase, the difference between the charged and neutral spheres decreases. The effects of refractive index and beam parameters on scattering, extinction, and absorption coefficients are different but tend to converge with increasing dimensionless parameters. When applied to charged spheres with different refractive indices, the scattering, extinction, and absorption efficiencies of Bessel pincer light-sheet beams change with variations in surface charge. However, once the surface charge reaches saturation, these efficiencies become stable. This study is significant for understanding optical manipulation and super-resolution imaging in single-molecule microbiology. Full article