Search Results (255)

Statistical data depth measures have been applied to density-based clustering techniques in an effort to achieve robustness in parameter selection via the affine invariant property of the depth measure. Specifically, the Mahalanobis depth measure is used in the application of DBSCAN. In this paper, we examine properties of the Mahalanobis depth measure that lead to instances where it fails to detect clusters in DBSCAN, whereas Euclidean distance is able to differentiate the clusters. We propose two solutions to the problems induced by these properties. The first re-examines clusters to determine if data shape is causing multiple clusters to be grouped into a single cluster. The second examines the use of a different measure as an alternate depth function. Experiments are provided. Full article

This work presents a comprehensive mathematical framework for symmetrized neural network operators operating under the paradigm of fractional calculus. By introducing a perturbed hyperbolic tangent activation, we construct a family of localized, symmetric, and positive kernel-like densities, which form the analytical backbone for three classes of multivariate operators: quasi-interpolation, Kantorovich-type, and quadrature-type. A central theoretical contribution is the derivation of the Voronovskaya–Santos–Sales Theorem, which extends classical asymptotic expansions to the fractional domain, providing rigorous error bounds and normalized remainder terms governed by Caputo derivatives. The operators exhibit key properties such as partition of unity, exponential decay, and scaling invariance, which are essential for stable and accurate approximations in high-dimensional settings and systems governed by nonlocal dynamics. The theoretical framework is thoroughly validated through applications in signal processing and fractional fluid dynamics, including the formulation of nonlocal viscous models and fractional Navier–Stokes equations with memory effects. Numerical experiments demonstrate a relative error reduction of up to 92.5% when compared to classical quasi-interpolation operators, with observed convergence rates reaching $\mathcal{O}\left(n^{-1.5}\right)$ under Caputo derivatives, using parameters $\lambda =3.5$, $q=1.8$, and $n=100$. This synergy between neural operator theory, asymptotic analysis, and fractional calculus not only advances the theoretical landscape of function approximation but also provides practical computational tools for addressing complex physical systems characterized by long-range interactions and anomalous diffusion. Full article

The acoustic scattering of targets in shallow-sea waveguides exhibits complex features such as multipath propagation and intricate echo components, with its acoustic field properties remaining incompletely understood. This study employs a hybrid method combining normal modes and scattering functions to numerically model the acoustic scattering of targets in waveguide channels. We analyze the coupling mechanisms of multipath acoustic waves and derive precise predictive formulas for the bright–dark interference fringe patterns in range–frequency spectra based on the physical mechanisms governing acoustic field interference. By tracking the peak trajectories of these interference fringes in range–frequency spectra, we investigate the variations of the waveguide invariant with frequency, range, and depth, revealing statistical patterns of the waveguide invariant in target–waveguide coupled scattering fields under different water depths. The results demonstrate that, under constant channel conditions, waveguide properties exhibit a weak correlation with target material characteristics. In shallow water environments, waveguide invariant values display broader distributions with higher probability density peaks, whereas increasing water depth progressively narrows the distribution range and monotonically reduces the peak magnitudes of the probability density function. Experimental validation via scaled elastic target echo testing confirms the observed trends of waveguide invariant variation with water depth. Full article

The twin propeller system can be powered by a motor with dual-rotating rotors, which generally necessitates that both rotors run at the same speed to prevent rolling. The motor with dual-rotating rotors is popular for applications that benefit from high torque density and an axially compact form factor. In order to minimize the effects of load disturbances and internal parameter perturbations on the motor performance, this paper proposes a control strategy combining disturbance observer and sliding mode control (SMC) technologies to realize the purpose of both rotors rotating at the same speed. There are issues with the conventional proportional-integral (PI) control for load disturbances and motor parameter variations, whereas the SMC method has its invariant properties. Meanwhile, the system disturbances obtained by a disturbance observer are estimated to be used as feed-forward compensation for the SMC control in order to reduce the undesired chattering during the SMC control process. The validity and practicability of the control strategies proposed in this paper are demonstrated by both simulations and experiments. Full article

The Guangdong-Hong Kong-Macao Greater Bay Area (GBA) is one of China’s three major urban agglomerations. Over the past thirty years, the region has undergone intensive economic development and urban expansion, resulting in significant changes in its ecological conditions. Due to the region’s humid and rainy climate, traditional remote sensing ecological indexes (RSEIs) struggle to ensure consistency in long-term ecological quality assessments. To address this, this study developed a unified RSEI (URSEI) model, incorporating optimized data selection, composite index construction, normalization using invariant regions, and multi-temporal principal component analysis. Using Landsat imagery from 1990 to 2020, this study examined the spatiotemporal evolution of ecological quality in the GBA. Building on this, spatial autocorrelation analysis was applied to explore the distribution characteristics of the URSEI, followed by geodetector analysis to investigate its driving factors, including temperature, precipitation, elevation, slope, land use, population density, GDP, and nighttime light. The results indicate that (1) the URSEI effectively mitigates the impact of cloudy and rainy conditions on data consistency, producing seamless ecological quality maps that accurately reflect the region’s ecological evolution; (2) ecological quality showed a “decline-then-improvement” trend during the study period, with the URSEI mean dropping from 0.65 in 1990 to 0.60 in 2000, then rising to 0.63 by 2020. Spatially, ecological quality was higher in the northwest and northeast, and poorer in the central urbanized areas; and (3) in terms of driving mechanisms, nighttime light, GDP, and temperature were the most influential, with the combined effect of “nighttime light + land use” being the primary driver of URSEI spatial heterogeneity. Human-activity-related factors showed the most notable variation in influence over time. Full article

Most high-order Euler-type methods have been proposed to solve one-dimensional scalar hyperbolic conservational law. These methods resolve smooth variations in flow parameters accurately and simultaneously identify the discontinuities. A disadvantage of Euler-type methods is the parameter change stretching in the shock over a few mesh cells. In reality, in the shock, the flow properties change abruptly at once for the computational mesh. In our considerations, the mean free path of a flow particle is much smaller than the mesh cell size. This paper describes a modification of the semi-Lagrangian Godunov-type method, which was proposed by the author in the previously published paper. The modified method also does not have numerical viscosity for shocks. In the previous article, a linear law for the distribution of flow parameters was employed for a rarefaction wave when modeling the Shu-Osher problem with the aim of reducing parasitic oscillations. Additionally, the nonlinear law derived from the Riemann invariants was used for the remaining test problems. This article proposes an advanced method, namely, a unified formula for the density distribution of rarefaction waves and modification of the scheme for modeling moderately strong shock waves. The obtained results of numerical analysis, including the standard problem of Sod, the Riemann problem of Lax, the Shu–Osher shock-tube problem and a few author’s test cases are compared with the exact solution, the data of the previous method and the Total Variation Deminishing (TVD) scheme results. This article delineates the further advancement of the numerical scheme of the proposed method, specifically presenting a unified mathematical formulation for an expanded set of test problems. Full article

With the rapid development of information technology, the global data volume has been continuously expanding, placing unprecedented demands on communication networks to accommodate precipitously increasing throughput. Thin-film lithium niobate (TFLN) modulators, characterized by their large theoretical bandwidth, low half-wave voltage, and suitability for high-density integration, show great application potential in high-speed optical modules and optical interconnection networks. However, the persistent issue of velocity mismatch between radio frequency (RF) signals and optical carriers invariably hinders the utilization of higher-frequency bands, which restricts the modulation speed of the fabricated devices. In this paper, an electrode co-loaded with square serrations and T-shaped stubs was utilized to achieve precise velocity matching and excellent impedance matching. Leveraging this approach, a TFLN modulator chip with an electro-optic bandwidth far exceeding 67 GHz and a return loss of greater than 12 dB was successfully fabricated on a silicon substrate. The velocity of RF signals can be tuned by altering the lengths of the slow-wave structures, which provides guidance for the design and optimization of broadband modulators. Full article

In one of our previous papers, we obtained a general class of potentials inside the nucleus, such that the singular solution of the Dirac equation for the S-states of hydrogen atoms outside the nucleus can be matched with the corresponding regular solution inside the nucleus (the proton) at the boundary. The experimental charge density distribution inside the proton generates a particular case of such potentials inside the proton. In this way, the existence of the second kind of hydrogen atom was predicted: atoms having only S-states. This theoretical prediction was then evidenced by several different types of atomic experiments and by astrophysical observations. In the present paper we provide the following new results. First, we show that the solution of the Dirac equation inside the proton can be (and is) found within the class of functions that are non-analytic at r = 0—in distinction to the traditional practice of limiting the search for the solution by the class of analytic functions. We demonstrate that this non-analytic solution inside the proton can be matched at the proton boundary R with the corresponding singular solution outside the proton regardless of the particular value of R. Second, we show that the interior and exterior solutions are scale-invariant with respect to the change of the boundary R between these two regions. Such invariance is the manifestation of a new symmetry—in addition to the previously discussed symmetries of the Dirac equation in the literature. Third, based on the new, more accurate results for the wave function inside and outside the proton, we revisit the resolution of the neutron lifetime puzzle initially outlined in our previous papers. On the basis of the more accurate calculations, we reconfirm that (A) the 2-body decay of neutrons produces overwhelmingly the SFHA (rather than the usual hydrogen atoms) and (B) the strengthened-in-this-way branching ratio for the 2-body decay of neutrons (compared to the 3-body decay) is in excellent agreement with the branching ratio required for reconciling the neutron lifetime values measured in the trap and beam experiments, so that the neutron lifetime puzzle seems to be indeed resolved in this way. Full article

An asymmetric non-singular bouncing cosmological model is proposed in the framework of a locally scale-invariant scalar-tensor version of the standard model of particle physics and gravitation. The scalar field $\varphi$ is complex. In addition to local scale invariance, the theory is U(1)-symmetric and has a conserved global charge associated with time variations of the phase of $\varphi$. An interplay between the positive energy density contributions of relativistic and non-relativistic matter and that of the negative kinetic energy associated with the phase of $\varphi$ results in a classical non-singular stable bouncing dynamics deep in the radiation-dominated era. This encompasses the observed redshifting era, which is preceded by a blueshifting era. The proposed model potentially avoids the flatness and horizon problems, as well as allowing for the generation of a scale-invariant spectrum of metric perturbations of the scalar type during a matter-dominated-like pre-bounce phase, with no recourse to an inflationary era. Full article

An essential aspect of any government in a smart city is to examine the issues of internal and external migration. Migration is a complex phenomenon. In order to effectively manage it, it is not only necessary to be able to accurately predict migration patterns but also to understand which factors influence these patterns. Current approaches to the development of migration models rely on macroeconomic indicators without considering the specificities of intraregional interactions among individuals. In this paper, we propose a method for determining the dynamics of migration balance based on Lagrangian mechanics. We derive and interpret the potential energy of a migration network by introducing specific functions that determine migration patterns. The solution of the migration equations and selection of parameters, as well as external forces, are achieved through the use of physics-informed neural networks. We also use external factors to explain the non-homogeneity in the dynamic equation through the use of a regression model. We analyze settlement priorities using transfer operator theory and invariant density. The findings obtained enable the assessment of migration flows and analysis of external migration factors. Full article

Assimilating complex systems to multifractal-type objects reveals continuous and non-differentiable curve dynamics, aligning with the Multifractal Theory of Motion. Two scenarios, a Schrödinger-type and a Madelung-type multifractal scenario, are possible in this setting. If the Madelung scenario employs maximized information entropy for a distribution density, then Newtonian and oscillator-type forces can be determined. In the presence of these forces and a matter background, we analyze the two-body problem. The obtained results are as follows: a generalized Hubble-type law, a dependence of Newton’s constant on the epoch and background density, a generalization of Lorentz transform (involving the Hubble constant, Newton’s constant, the speed of light, and cosmic matter density), etc. Moreover, in the same scenario, the functionality of a diffusion-type equation implies instabilities, such as period doubling, through an SL(2R) invariance. Thus, multiple infragalactic and extragalactic instabilities are exemplified. Full article

Underwater acoustic channel (UWAC) is characterized by significant multipath effects, strong time-varying properties and complex noise environments, which make achieving high-rate and reliable underwater communication a formidable task. To address the above adverse challenges, this study first presents a novel, robust and efficient polar code construction (NREPCC) method using the base-adversarial polarization weight (BPW) algorithm tailored for typical ocean channel models, including invariable sound velocity gradient (ISVG) channels, negative sound velocity gradient (NSVG) channels, and positive sound velocity gradient (PSVG) channels. Subsequently, a robust and reliable polar-coded UWAC system model based on the orthogonal frequency division multiplexing (OFDM) technique is designed using the $t$-distribution noise model in conjunction with real sea noise data fitting. Then, an enhanced time synchronization and packet detection algorithm based on $t$-distribution is proposed for the performance optimization of the polar-coded UWAC OFDM system. Finally, extensive numerical simulation results confirm the excellent performance of the proposed NREPCC method and polar-coded UWAC OFDM system under a variety of channel conditions. Specifically, the NREPCC method outperforms low-density parity-check (LDPC) codes by approximately 0.5~1 dB in PSVG and ISVG channels while maintaining lower encoding and decoding complexity. Moreover, the robustness of the NREPCC method under $t$-distribution noise with varying degrees of freedom is rigorously validated, which renders vital technical support for the design of high-precision and high-robustness UWAC systems. Full article

The enigmatic phenomenon of dark energy (DE) is the elusive entity driving the accelerated expansion of our Universe. A plausible candidate for DE is the non-zero Einstein Cosmological Constant ${\Lambda }_E$ manifested as a constant energy density of the vacuum, yet it seemingly defies gravitational effects. In this work, we interpret the non-zero ${\Lambda }_E$ through the lens of scale-invariant cosmology. We revisit the conformal scale factor $\lambda$ and its defining equations within the Scale-Invariant Vacuum (SIV) paradigm. Furthermore, we address the profound problem of the missing mass across galactic and extragalactic scales by deriving an MOND-like relation, $g\sim \sqrt{a_0{g}_N}$, within the SIV context. Remarkably, the values obtained for ${\Lambda }_E$ and the MOND fundamental acceleration, $a_0$, align with observed magnitudes, specifically, $a_0\approx {10}^{-10}\mathrm{m}{\mathrm{s}}^{-2}$ and ${\Lambda }_E\approx 1.8\times {10}^{-52}{\mathrm{m}}^{-2}$. Moreover, we propose a novel early dark energy term, ${\tilde{T}}_{\mu \nu }\sim \kappa H$, within the SIV paradigm, which holds potential relevance for addressing the Hubble tension. Full article

According to general relativity, the cosmological redshift may be caused by other mechanisms than the source moving away from the observer. It can occur on a global scale, similar to the gravitational redshift near massive stars. In principle, these are differences in the time-dependent global metric field between the source in the past and the observer in the present. In this paper we attempt a new interpretation of the simple solution of Einstein’s equations within a fully conformal metric for the case of a time-independent energy-momentum tensor. The scaling factor here acts identically on all four space-time coordinates and the speed of light is independent of the conformal time. The fully conformal metric is interpreted here as a universal geometric background which is scale invariant and acts universally on all objects, including gauges and clocks, regardless of their dimensions and internal interactions. The associated scale invariant exponential expansion is thus only relative and all observers at different times are completely equal. The model introduces the concept of the appearent age of the universe, which is the limiting consequence of time dilation into the past, and corresponds to the present value of the age of the universe H⁻¹ according to the standard model. This appearent age is the same for all observers, and the Hubble constant is thus a true universal constant, invariant to time translations. The motivation of this work was to test the possibility of the above cosmological redshift mechanism in confrontation with astrophysical data. Probably the most important consequence is the generalized formulation and interpretation of the Hubble-Lemaître law z(r) = (e^Hr/c − 1), which shows good agreement with astrophysical data even for the most distant supernovae. Confronting the conformal metric model with some astrophysical data shows an interesting agreement with the observed spatial distribution of astrophysical sources such as γ-ray bursts and quasars. On a cosmological scale, the above fully conformal metric naturally determines the global energy density, spatial flatness, and solves the horizon problem and Olbers’ paradox in infinite spacetime. Full article

Let G be a random variable of functionals of an isonormal Gaussian process X defined on some probability space. Studies have been conducted to determine the exact form of the density function of the random variable G. In this paper, unlike previous studies, we will use the Stein’s method for invariant measures of diffusions to obtain the density formula of G. By comparing the density function obtained in this paper with that of the diffusion invariant measure, we find that the diffusion coefficient of an Itô diffusion with an invariant measure having a density can be expressed as in terms of operators in Malliavin calculus. Full article