Search Results (371)

This paper presents a systematic methodology for identifying optimal scaling regions in segment-based box-counting fractal dimension calculations through a three-phase algorithmic framework combining grid offset optimization, boundary artifact detection, and sliding window optimization. Unlike traditional pixelated approaches that suffer from rasterization artifacts, the method used directly analyzes geometric line segments, providing superior accuracy for mathematical fractals and other computational applications. The three-phase optimization algorithm automatically determines optimal scaling regions and minimizes discretization bias without manual parameter tuning, achieving significant error reduction compared to traditional methods. Validation across the Koch curve, Sierpinski triangle, Minkowski sausage, Hilbert curve, and Dragon curve demonstrates substantial improvements: excellent accuracy for the Koch curve (0.11% error) and significant error reduction for the Hilbert curve. All optimized results achieve $R^2\ge 0.9988$. Iteration analysis establishes minimum requirements for reliable measurement, with convergence by level 6+ for the Koch curve and level 3+ for the Sierpinski triangle. Each fractal type exhibits optimal iteration ranges where authentic scaling behavior emerges before discretization artifacts dominate, challenging the assumption that higher iteration levels imply more accurate results. Application to a Rayleigh–Taylor instability interface (D = 1.835 ± 0.0037) demonstrates effectiveness for physical fractal systems where theoretical dimensions are unknown. This work provides objective, automated fractal dimension measurement with comprehensive validation establishing practical guidelines for mathematical and real-world fractal analysis. The sliding window approach eliminates subjective scaling region selection through systematic evaluation of all possible linear regression windows, enabling measurements suitable for automated analysis workflows. Full article

In special relativity, particle trajectories, whether mass-bearing or not, can be traced on the Minkowski spacetime manifold in (3+1)D. Meantime, in quantum mechanics, trajectories in the phase space are not strictly outlined because coordinate and linear momentum cannot be measured simultaneously with arbitrary precision since they do not commute within the Hilbert space formalism. However, from the density matrix representing a quantum system, the extracted information still produces an imperative description of its properties and, furthermore, by appropriately reordering the matrix entries, additional information can be obtained from the same content. Adhering to this line of work, the paper investigates the definition and the meaning of velocity and speed in a typical quantum phenomenon, the disentanglement for a bipartite system when dynamical evolution is displayed in a (3+1)D pseudo-spacetime whose coordinates are constructed from combinations of entries to the density matrix. The formalism is based on the definition of a Minkowski manifold with compact support, where trajectories are defined following the same reasoning and formalism present in the Minkowski manifold of special relativity. The space-like and time-like regions acquire different significations referred to entangled-like and separable-like, respectively. The definition and the sense of speed and velocities of disentanglement follow naturally from the formalism. Depending on the dynamics of the physical state of the system, trajectories may meander between regions of entanglement and separability in the space of new coordinates defined on the Minkowski manifold. Full article

The population activity of grid cells from a single module is topologically constrained to a toroidal manifold. Our work proposes an improved version of Gardner’s earlier model, which can account for both geometric properties and force field dynamics. Employing methods from Differential Geometry, we have derived Lagrangian densities that—under very general assumptions and avoiding dimensionful constants—provide a rationale for the trajectories associated with the synaptic spacetime as a global solution to the Einstein–Maxwell field equations. Then, we investigate the helical solutions to show that the synaptic toroidal topological space, as a locally flat Minkowski spacetime, with a Lorentzian metric is geodesically complete and, therefore, exhibits maximal stability. Finally, we consider a Lorentzian metric with curved spacetimes that give rise to Lorentzian tori admitting curvature spacetime singularities. Full article

This paper explores some frame bundles and physical implications of Killing vector fields on the two-sphere ${\mathbb{S}}^2$, culminating in a novel application to Maxwell’s equations in free space. Initially, we investigate the Killing vector fields on ${\mathbb{S}}^2$ (represented by the unit sphere of ${\mathbb{R}}^3$), which generate the isometries of the sphere under the rotation group $SO\left(3\right)$. These fields, realized as functions $K_v:{\mathbb{S}}^2\to {\mathbb{R}}^3$, defined by $K_v\left(q\right)=v\times q$ for a fixed $v\in {\mathbb{R}}^3$ and any $q\in {\mathbb{S}}^2$, generate a three-dimensional Lie algebra isomorphic to $\mathfrak{so}\left(3\right)$. We establish an isomorphism $K:{\mathbb{R}}^3\to \mathcal{K}\left({\mathbb{S}}^2\right)$, mapping vectors $v=au$ (with $u\in {\mathbb{S}}^2$) to scaled Killing vector fields $a{K}_u$, and analyze its relationship with $SO\left(3\right)$ through the exponential map. Subsequently, at a fixed point $e\in {\mathbb{S}}^2$, we construct a smooth orthonormal right-handed tangent frame $f_e:{\mathbb{S}}^2\backslash \{e,-e\}\to T{\left({\mathbb{S}}^2\right)}^2$, defined as $f_e\left(u\right)=({\widehat{K}}_e\left(u\right),u\times {\widehat{K}}_e\left(u\right))$, where ${\widehat{K}}_e$ is the unit vector field of the Killing field $K_e$. We verify its smoothness, orthonormality, and right-handedness. We further prove that any smooth orthonormal right-handed frame on ${\mathbb{S}}^2\backslash \{e,-e\}$ is either $f_e$ or a rotation thereof by a smooth map $\rho :{\mathbb{S}}^2\backslash \{e,-e\}\to SO\left(3\right)$, reflecting the triviality of the frame bundle over the parallelizable domain. The paper then pivots to an innovative application, constructing solutions to Maxwell’s equations in free space by combining spherical symmetries with quantum mechanical de Broglie waves in tempered distribution wave space. The deeper scientific significance lies in bringing together differential geometry (via $SO\left(3\right)$ symmetries), quantum mechanics (de Broglie waves in Schwartz distribution theory), and electromagnetism (Maxwell’s solutions in Schwartz tempered complex fields on Minkowski space-time), in order to offer a unifying perspective on Maxwell’s electromagnetism and Schrödinger’s picture in relativistic quantum mechanics. Full article

Voronoi partitioning is a fundamental geometric concept with applications across computational geometry, robotics, optimization, and resource allocation. While Euclidean distance is the most commonly used metric, alternative distance functions can significantly influence the shape and properties of Voronoi cells. This paper presents a comprehensive mathematical analysis of various distance metrics used in Voronoi partitioning, including Euclidean, Manhattan, Minkowski, weighted, anisotropic, and geodesic metrics. We analyze their mathematical formulations, geometric properties, topological implications, and computational complexity. This work aims to provide a theoretical framework for selecting appropriate metrics for Voronoi-based modeling in diverse applications. Full article

By applying Maxwell’s equations to curved spacetimes, the Planck–Einstein energy–frequency relation for photons, originally formulated in Minkowski space, is generalized for application in Riemann space. According to this relation, photon energy depends not only on the photon frequency but also on the physical speed of photons, which may vary when locally measured in non-inertial static frames. In Minkowski space, the energy of free photons is conserved as neither frequency shifts nor changes in photon speed are observed. In Riemann space, energy of free photons also remains conserved as gravitational redshift is compensated by a corresponding variation in photon speed. The generalized Planck–Einstein relation may have significant astrophysical implications, particularly for gravitational lensing, observations of neutron star mergers, supernovae and quasars, the propagation of light near black holes, and expanding cosmologies. Full article

The optical force exerted on a dipole particle can be divided into gradient force, scattering force, and spin–curl force, all of which can be derived from Maxwell’s stress tensor with the dipole approximation. Here, we identify an additional spin–curl force for arbitrary objects beyond the dipole approximation, which is named the generalized spin–curl force in this paper. The generalized spin–curl force originates from the Minkowski force density and depends on the imaginary parts of the permittivity, permeability, and chirality of the object. However, it remains imperceptible in conventional optical force calculations due to its exact cancellation by a compensatory surface force during MST surface integration. The study of the generalized spin–curl force provides critical insights into elucidating the mechanisms underlying optical momentum transfer and internal force distribution within complex media. Furthermore, the generalized spin–curl force offers a novel mechanism for enhancing optical sensors, enabling highly sensitive detection of absorptive or chiral perturbations in systems such as microcavities and metasurfaces. Its ability to manipulate internal force distributions also provides new pathways for advancing optical force probes and chirality-selective sensing at the nanoscale. Full article

In this work, we study field theoretic systems of a single axion-like field with linear potentials modulated by cosine terms, allegedly induced by non-perturbative instanton configurations. These systems are considered in expanding-Universe spacetime backgrounds (of Friedmann–Lema$\widehat{\mathrm{i}}$tre–Robertson–Walker-type). Using a dynamical system approach, we classify the various de Sitter-like (inflationary) vacua from the point of view of their stability, which depend on the values of the model parameters. In this respect, bifurcation points are found to be present for the various models under consideration. Part of the parameter space of the systems under consideration includes the running-vacuum (approximately) linear axion monodromy potentials, considered in previous works by some of the authors, where inflation is induced by primordial gravitational wave condensates. A particularly interesting case, corresponding to another part of the parameter space of the models, includes a series of stable de Sitter vacua, which physically may correspond to a series of successive tunnelings of the system, via say non-perturbative effects, with a decreasing effective cosmological constant. Under certain values of the parameters, these successive tunnelings can reach a Minkowski spacetime, with zero value of the minimum of the axion potential. The situation is not dissimilar to the one of discrete inflation that arguably characterizes some minimal non-critical-string (Liouville) models of cosmology. Finally, for comparison, we also include in this article a dynamical system study of standard axion monodromy-modulated potentials characterizing some string/brane compactification models of inflation. Full article

Demand-side flexible resources in building energy systems hold significant potential for enhancing grid reliability and operational efficiency. However, their effective coordination remains challenging due to the complexity of modeling and aggregating heterogeneous loads. To address this, this paper proposes a feasible region aggregation and coordination method for load aggregators based on a GNMTL-LSTM-Zonotope framework. A Gradient Normalized Multi-Task Learning Long Short-Term Memory (GNMTL-LSTM) model is developed to forecast the power trajectories of diverse flexible resources, including air-conditioning systems, energy storage units, and diesel generators. Using these predictions and associated uncertainty bounds, dynamic feasible regions for individual resources are constructed with Zonotope structures. To enable scalable aggregation, a Minkowski sum-based method is applied to merge the feasible regions of multiple resources efficiently. Additionally, a directionally weighted Zonotope refinement strategy is introduced, leveraging time-varying flexibility revenues from energy and reserve markets to enhance approximation accuracy during high-value periods. Case studies based on real-world office building data from Shandong Province validate the effectiveness, modeling precision, and economic responsiveness of the proposed method. The results demonstrate that the framework enables fine-grained coordination of flexible loads and enhances their adaptability to market signals. This study is the first to integrate GNMTL-LSTM forecasting with market-oriented Zonotope modeling for heterogeneous demand-side resources, enabling simultaneous improvements in dynamic accuracy, computational scalability, and economic responsiveness. Full article

This study evaluates hierarchical clustering methodologies to identify patterns associated with timeout requests for EuroLeague basketball games. Using play-by-play data from 3743 games spanning the 2008–2023 seasons (over 1.9 million instances), we applied Principal Component Analysis to reduce dimensionality and tested multiple agglomerative and divisive clustering techniques (e.g., Ward and DIANA) with different distance metrics (Euclidean, Manhattan, and Minkowski). Clustering quality was assessed using internal validation indices such as Silhouette, Dunn, Calinski–Harabasz, Davies–Bouldin, and Gap statistics. The results show that Ward.D and Ward.D2 methods using Euclidean distance generate well-balanced and clearly defined clusters. Two clusters offer the best overall quality, while four clusters allow for meaningful segmentation of game situations. The analysis revealed that teams that did not request timeouts often exhibited better scoring efficiency, particularly in the advanced game phases. These findings offer data-driven insights into timeout dynamics and contribute to strategic decision-making in professional basketball. Full article

In this study, We explore for Minkowski 3-space ${\mathbb{E}}_1^3$ harmonic surfaces’ geometric features by employing a common tangent vector field along a curve situated on the surface. Our analysis is grounded in the rotation minimizing (RM) Darboux frame, which offers a robust alternative to the classical Frenet frame particularly valuable in the Lorentzian setting, where singularities frequently arise. The RM Darboux frame, tailored to curves lying on surfaces, enables the expression of fundamental invariants such as geodesic curvature, normal curvature, and geodesic torsion. We derive specific conditions that characterize harmonic surfaces based on these invariants. We also clarify the connection between the components of the RM Darboux frame and thesurface’s mean curvature vector. This formulation provides fresh perspectives on the classification and intrinsic structure of harmonic surfaces within Minkowski geometry. To support our findings, we present several illustrative examples that demonstrate the applicability and strength of the RM Darboux approach in Lorentzian differential geometry. Full article

Background: Traditional neonatal assessments rely on anthropometric measures such as weight, body size, and head circumference. However, recent studies suggest that objective movement quantification may serve as a complementary clinical indicator of development in preterm infants. Methods: This study evaluates non-invasive computer vision-based quantification of neonatal movement using contactless pose tracking based on computer vision. We analyzed approximately 800,000 postural data points from ten preterm infants to identify reliable algorithms, optimal recording duration, and whether whole-body or regional tracking is sufficient. Results: Our findings show that 30 s video segments are adequate for consistent motion quantification. Optical flow methods produced inconsistent results, while distance-based algorithms—particularly Chebyshev and Minkowski—offered greater stability, with coefficients of variation of 5.46% and 6.40% in whole-body analysis. Additionally, Minkowski and Mahalanobis metrics applied to the lower body yielded results similar to full-body tracking, with minimal differences of 0.89% and 1%. Conclusions: The results demonstrate that neonatal movement can be quantified objectively and without physical contact using computer vision techniques and reliable computational methods. This approach may serve as a complementary clinical indicator of neonatal progression, alongside conventional measures such as weight and size, with applications in continuous monitoring and early clinical decision-making for preterm infants. Full article

In this article, a high-gain axisymmetric Minkowski fractal reflectarray is designed and fabricated for Ku-Band satellite internet communications. High gain is achieved here by carefully optimising the number of unit cells, their shape modifier, focal length, feed position and scan angle. The space-filling properties of Minkowski fractals help in miniaturising the fractal. The scan angle of the reflectarray varied by adjusting the fractal scaling factor for each unit cell in the array. The reflectarray is symmetric along the X-axis in its design and configuration. Initially, a Minkowski fractal unit cell is designed using iteration-1 in the simulation software. Then, its design parameters are optimised to achieve high gain, a narrow beam, and beam scan capabilities. The sensitivity of design parameters is examined individually using the array synthesis method to achieve these performance parameters. It helps to establish the maximum range of design and performance parameters for this design. The proposed reflectarray resonates at 12 GHz, achieving a gain of over 20 dB and a narrow beamwidth of less than 15 degrees. Finally, the designed fractal reflectarray is tested in real-time simulation environments using MATLAB R2023b, and its performance is evaluated in an interference scenario involving LEO and MEO satellites, as well as a ground station, under various time conditions. For real-world applicability, it is necessary to identify, analyse, and mitigate the unwanted interference signals that degrade the desired satellite signal. The proposed reflectarray, with its performance characteristics and beam scanning capabilities, is found to be an excellent choice for Ku-band satellite internet communications. Full article

In this paper, we introduce and develop the concept of osculating curve pairs in the three-dimensional Minkowski space. By defining a vector lying in the intersection of osculating planes of two non-lightlike curves, we characterize osculating mates based on their Frenet frames. We then derive the transformation matrix between these frames and investigate the curvature and torsion relations under varying causal characterizations of the curves—timelike and spacelike. Furthermore, we determine the conditions under which these generalized osculating pairs reduce to well-known curve pairs such as Bertrand, Mannheim, and Bäcklund pairs. Our results extend existing theories by unifying several known curve pair classifications under a single geometric framework in Lorentzian space. Full article

The increasing penetration of distributed photovoltaic (PV) and energy storage (ES) systems in power grids, while advancing the transition to clean energy and enhancing grid flexibility, poses resource dispersion, uncertainty, and scheduling challenges. Consequently, it becomes crucial to manage and optimize these resources. In this paper, we innovatively propose a distribution network day-ahead optimal scheduling model that takes into account distributed resource aggregation and uncertainty. Firstly, distributed PV aggregation (PVA) is performed using the Minkowski summation method, and distributed ES aggregation (ESA) is performed using the polytope inner approximation method. Then, in order to deal with the uncertainty, the supply–demand balance of flexibility is modeled using kernel density estimation (KDE). Finally, the aggregation model and the flexibility supply–demand balance model are incorporated into the distribution network day-ahead optimization. The simulation study of the IEEE 69-node distribution system shows that the aggregate feasible region (AFR) is improved by about 90% and the active flexibility is improved by about 10% compared to box inner approximate aggregation methods, demonstrating their effectiveness in managing operational uncertainties and optimizing the utilization of distributed energy resources in day-ahead scheduling. Full article