Time Singularity Research Articles

A Novel Stochastic Interacting Particle-Field Algorithm for 3D Parabolic-Parabolic Keller-Segel Chemotaxis System

We introduce an efficient stochastic interacting particle-field (SIPF) algorithm with no history dependence for computing aggregation patterns and near singular solutions of parabolic-parabolic Keller-Segel (KS) chemotaxis system in three-dimensional (3D) space. In our algorithm, the KS solutions are approximated as empirical measures of particles coupled with a smoother field (concentration of chemo-attractant) variable computed by a spectral method. Instead of using heat kernels that cause history dependence and high memory cost, we leverage the implicit Euler discretization to derive a one-step recursion in time for stochastic particle positions and the field variable based on the explicit Green’s function of an elliptic operator of the form Laplacian minus a positive constant. In numerical experiments, we observe that the resulting SIPF algorithm is convergent and self-adaptive to the high-gradient part of solutions. Despite the lack of analytical knowledge (such as a self-similar ansatz) of a blowup, the SIPF algorithm provides a low-cost approach to studying the emergence of finite-time blowup in 3D space using only dozens of Fourier modes and by varying the amount of initial mass and tracking the evolution of the field variable. Notably, the algorithm can handle multi-modal initial data and the subsequent complex evolution involving the merging of particle clusters and the formation of a finite time singularity with ease.

Shade watch: Mapping citywide shade dynamics through ray tracing and LiDAR data in Hong Kong's complex 3-D built environment

The importance of urban shade in city planning has been underscored by its various associations with thermal comfort, living environment quality, human health and well-being. However, existing methods for urban shade mapping are often limited to street-level scopes or singular sampling observational times, leaving citywide shade mapping, its spatiotemporal patterns, and controlling factors largely unexplored. To fill these knowledge gaps, this study proposed a novel protocol that enables seamless mapping of spatiotemporal urban shade patterns over the entire city by fusing look-up-table (LUT)-based ray-tracing approach with Google Earth Engine cloud computing technology, as well as high-resolution digital surface model (DSM) derived from LiDAR data. This protocol was applied to Hong Kong, a city characterized by a complex 3-D built environment, and then delved into the spatiotemporal patterns of urban shade and the associated controlling factors. Validation results indicate that the proposed method can accurately quantify urban shade compared to ground-based photo references and high-resolution satellite imagery. The analysis uncovers that: (1) Shade exhibits spatial heterogeneity across different 3-D built environments while maintaining a consistent "U-shape" diurnal shade fraction during the day; (2) Solar geometry, regardless of solar zenith and azimuth angle, is positively associated with aggregated metrics of shade landscape and negatively associated with fragmented metrics. Moreover, solar zenith angles exert stronger controls over the citywide shade landscape patterns; and (3) Persistent shade significantly reduces the sunlight hours (i.e., the accumulated sunshine hours per day) across administrative districts, local climate zones (LCZs), and seasons. Due to shade effects originating from 3-D urban structures, Hong Kong typically experiences an average of 4-8 sunlight hours per day within a one-year cycle, which is substantially lower than the 11-14 hours of natural daylength. This study offers a practical protocol for large-scale urban shade mapping, enhancing our understanding of urban shade, its spatiotemporal dynamics, and benefits in a broader spatiotemporal context. The associated datasets and findings from this study can inform urban planners and policymakers in developing effective strategies to create healthier and sustainable urban environments in Hong Kong.

Self-similar spirals for the generalized surface quasi-geostrophic equations

We construct a large class of nontrivial (nonradial) self-similar solutions of the generalized surface quasi-geostrophic equations (gSQG). To the best of our knowledge, this is the first rigorous construction of any self-similar solution for these equations. The solutions are of spiral type, locally integrable, and may have mixed sign. Moreover, they bear some resemblance to the finite time singularity scenario numerically proposed by Scott–Dritschel [J. Fluid Mechanics 863, article no. R2 (2019)] in the SQG patch setting.

Leveraging Singular Spectrum Analysis and Time Delay Neural Network for Improved Potato Price Forecasting

Leveraging Singular Spectrum Analysis and Time Delay Neural Network for Improved Potato Price Forecasting

Singularity formation for the relativistic Euler equations of Chaplygin gases in Schwarzschild spacetime

AbstractWe study the formation of singularities of smooth solutions to the relativistic Euler equations of Chaplygin gases in Schwarzschild spacetime. The system is in the spherically symmetric form, and its coefficients and nonhomogeneous terms contain a parameter reflecting the mass of the black hole, which makes it highly nonlinear and complicated. To overcome the influence of the mass parameter of black hole, we introduce a pair of suitable auxiliary variables related to it and derive their characteristic decompositions to establish the estimates of the smooth solution. We show that, for a kind of initial data, the smooth solution develops singularity in finite time and the mass‐energy density itself approaches infinity at the blowup point.

Singular times and multiple temporalities of the future in tourism planning

Critical thinking about the future needs to be at the centre of tourism planning scholarship. By using complexity theory as a framework to understand the shaping power of time, we review understandings about the future in tourism planning literature concerning three interrelated aspects: How the future is known in tourism planning, how the future has been planned in tourism, and what futures have been planned. Changing, diverse and mutually intertwined views of understanding time and approaching the future are identified and gathered into two interwoven bundles. In the first bundle, the future in tourism planning is singular, foreseeable, and technical, with strong linear and cause-effect assumptions. In the second grouping, as cause-effect assumptions are questioned, futures in tourism planning become multiple, nested, simultaneous and emerge from non-linear, relational, value-laden and time-dependent socio-political processes. A characterisation of singular time and multiple temporalities in tourism planning is presented, and their coexistence in the literature is acknowledged for its potential to stimulate ethical future-making through planning and tourism.

Canonical surgeries in rotationally invariant Ricci flow

We construct a rotationally invariant Ricci flow through surgery starting at any closed rotationally invariant Riemannian manifold. We demonstrate that a sequence of such Ricci flows with surgery converges to a Ricci flow spacetime in the sense of Kleiner and Lott [Acta Math. 219 (2017), pp. 65–134]. Results of Bamler-Kleiner [Acta Math. 228 (2022), pp. 1–215] and Haslhofer [Proc. Amer. Math. Soc. 150 (2022), pp. 5433–5437] then guarantee the uniqueness and stability of these spacetimes given initial data. We simplify aspects of this proof in our setting, and show that for rotationally invariant Ricci flows, the closeness of spacetimes can be measured by equivariant comparison maps. Finally we show that the blowup rate of the curvature near a singular time for these Ricci flows is bounded by the inverse of remaining time squared.

Predicting mine water inflow volumes using a decomposition-optimization algorithm-machine learning approach

Disasters caused by mine water inflows significantly threaten the safety of coal mining operations. Deep mining complicates the acquisition of hydrogeological parameters, the mechanics of water inrush, and the prediction of sudden changes in mine water inflow. Traditional models and singular machine learning approaches often fail to accurately forecast abrupt shifts in mine water inflows. This study introduces a novel coupled decomposition-optimization-deep learning model that integrates Complete Ensemble Empirical Mode Decomposition with Adaptive Noise (CEEMDAN), Northern Goshawk Optimization (NGO), and Long Short-Term Memory (LSTM) networks. We evaluate three types of mine water inflow forecasting methods: a singular time series prediction model, a decomposition-prediction coupled model, and a decomposition-optimization-prediction coupled model, assessing their ability to capture sudden changes in data trends and their prediction accuracy. Results show that the singular prediction model is optimal with a sliding input step of 3 and a maximum of 400 epochs. Compared to the CEEMDAN-LSTM model, the CEEMDAN-NGO-LSTM model demonstrates superior performance in predicting local extreme shifts in mine water inflow volumes. Specifically, the CEEMDAN-NGO-LSTM model achieves scores of 96.578 in MAE, 1.471% in MAPE, 122.143 in RMSE, and 0.958 in NSE, representing average performance improvements of 44.950% and 19.400% over the LSTM model and CEEMDAN-LSTM model, respectively. Additionally, this model provides the most accurate predictions of mine water inflow volumes over the next five days. Therefore, the decomposition-optimization-prediction coupled model presents a novel technical solution for the safety monitoring of smart mines, offering significant theoretical and practical value for ensuring safe mining operations.

Sharp non-uniqueness for the 3D hyperdissipative Navier-Stokes equations: Beyond the Lions exponent

We study the 3D hyperdissipative Navier-Stokes equations on the torus, where the viscosity exponent α can be larger than the Lions exponent 5/4. It is well-known that, due to Lions [1], for any L2 divergence-free initial data, there exist unique smooth Leray-Hopf solutions when α≥5/4. We prove that even in this high dissipative regime, the uniqueness would fail in the supercritical spaces LtγWxs,p, in view of the Ladyženskaja-Prodi-Serrin criteria. The non-uniqueness is proved in the strong sense and, in particular, yields the sharpness at two endpoints (3/p+1−2α,∞,p) and (2α/γ+1−2α,γ,∞). Moreover, the constructed solutions are allowed to coincide with the unique Leray-Hopf solutions near the initial time and, more delicately, admit the partial regularity outside a fractal set of singular times with zero Hausdorff Hη⁎ measure, where η⁎>0 is any given small positive constant. These results also provide the sharp non-uniqueness in the supercritical Lebesgue and Besov spaces. Furthermore, we prove the strong vanishing viscosity result for the hyperdissipative Navier-Stokes equations.

Necessary and sufficient conditions for Kolmogorov’s flux laws on T2 and T3

Necessary and sufficient conditions for the third order Kolmogorov universal scaling flux laws are derived for the stochastically forced incompressible Navier Stokes equations on the torus in 2D and 3D. This paper rigorously generalises the result of (Bedrossian 2019 Commun. Math. Phys. 367 1045–75) to functions which are heavy-tailed in Fourier space or have local finite time singularities in the inviscid limit. In other words, we have rigorously derived the existence of the well known physical relationships, the direct and inverse cascades. Furthermore we show that the rate of the direct cascade is proportional to the amount of energy ‘escaping to infinity’ in spectral space as well as a measure of the total singularities within the solution. Similarly, an inverse cascade is proportional to the amount of energy that moves towards the k = 0 Fourier mode in the invisicid limit.

Formation of Finite Time Singularity for Axially Symmetric Magnetohydrodynamic Waves in 3-D

Formation of Finite Time Singularity for Axially Symmetric Magnetohydrodynamic Waves in 3-D

Singularities of the network flow with symmetric initial data

We study the formation of singularities for the curvature flow of networks when the initial data is symmetric with respect to a pair of perpendicular axes and has two triple junctions. We show that, in this case, the set of singular times is finite.

The dielectric and radiation shielding features of Zn0.9Ni0.1Co2-xFexO4 nanopowder

Nano Zn0.9Ni0.1Co2-xFexO4 (x = 0, 0.05, 0.1, 0.15) specimens were synthesized utilizing the hydrothermal method. An extensive assessment was conducted on the structure and dielectric characteristics of the fabricated specimens. The synchrotron x-ray diffraction and scanning electron microscopy techniques were employed to analyze the formed phases and the morphological characteristics of the specimens. Rietveld refinement was utilized for determining the structural and microstructural parameters of all specimens. The impact of temperature and frequency on the dielectric properties of the material is thoroughly investigated. Except for the specimen with x = 0.15, all samples exhibit ferroelectric characteristics. The electric modulus corroborated the existence of the non-Debye relaxation phenomenon and the presence of relaxation times distributed at a specific frequency. Each specimen demonstrates a singular relaxation time, which was modified by the introduction of Fe ions. Through the utilization of the Phy-X/PSD software, the radiation shielding parameters for the examined specimens were computed across a wide energy spectrum ranging from 15 KeV to 15 MeV. These parameters encompass the linear attenuation coefficients (LAC), mean free path (MFP), mass attenuation coefficient (MAC), half value length (HVL), effective nuclear number (Z eff), and fast neutron removal cross-section (FNRCS). The specimens of Zn0.9Ni0.1Co2-xFexO4 exhibit elevated FNRCS values compared to RS-253-G18, RS-360, and RS-520 commercial shielding glasses.

Gravitational lensing by transparent Janis–Newman–Winicour naked singularities

The Janis–Newman–Winicour (JNW) spacetime can describe a naked singularity with a photon sphere that smoothly transforms into a Schwarzschild black hole. Our analysis reveals that photons, upon entering the photon sphere, converge to the singularity in a finite coordinate time. Furthermore, if the singularity is subjected to some regularization, these photons can traverse the regularized singularity. Subsequently, we investigate the gravitational lensing of distant sources and show that new images emerge within the critical curve formed by light rays escaping from the photon sphere. These newfound images offer a powerful tool for the detection and study of JNW naked singularities.

A Singular Tempered Sub-Diffusion Fractional Model Involving a Non-Symmetrically Quasi-Homogeneous Operator

In this paper, we focus on the existence of positive solutions for a singular tempered sub-diffusion fractional model involving a quasi-homogeneous nonlinear operator. By using the spectrum theory and computing the fixed point index, some new sufficient conditions for the existence of positive solutions are derived. It is worth pointing out that the nonlinearity of the equation contains a tempered fractional sub-diffusion term, and is allowed to possess strong singularities in time and space variables. In particular, the quasi-homogeneous operator is a nonlinear and non-symmetrical operator.

Polytropic Dynamical Systems with Time Singularity

In this paper we consider a class of second order singular homogeneous differential equations called the Lane-Emden-type with time singularity in the drift coefficient. Lane-Emden equations are singular initial value problems that model phenomena in astrophysics such as stellar structure and are governed by polytropics with applications in isothermal gas spheres. A hybrid method that combines two simple methods; Euler's method and shooting method, is proposed to approximate the solution of this type of dynamic equations. We adopt the shooting method to reduce the boundary value problem, then we apply Euler's algorithm to the resulted initial value problem to get approximations for the solution of the Lane-Emden equation. Finally, numerical examples and simulation are provided to show the validity and efficiency of the proposed technique, as well as the convergence and error estimation are analyzed.

Singularity formation for the cylindrically symmetric rotating relativistic Euler equations of Chaplygin gases

This paper studies the formation of singularities in smooth solutions of the relativistic Euler equations of Chaplygin gases with cylindrically symmetric rotating structures. This is a nonhomogeneous hyperbolic system with highly nonlinear structures and fully linearly degenerating characteristic fields. We introduce a pair of auxiliary functions and use the characteristic decomposition technique to overcome the influence of the rotating structures in the system. It is verified that smooth solutions develop into a singularity in finite time and the mass-energy density tends to infinity at the blowup point for a type of rotating initial data.

On the Solvability of a Singular Time Fractional Parabolic Equation with Non Classical Boundary Conditions

This paper deals with a singular two dimensional initial boundary value problem for a Caputo time fractional parabolic equation supplemented by Neumann and non-local boundary conditions. The well posedness of the posed problem is demonstrated in a fractional weighted Sobolev space. The used method based on some functional analysis tools has been successfully showed its efficiency in proving the existence, uniqueness and continuous dependence of the solution upon the given data of the considered problem. More precisely, for proving the uniqueness of the solution of the posed problem, we established an energy inequality for the solution from which we deduce the uniqueness. For the existence, we proved that the range of the operator generated by the considered problem is dense.

Homogenization of a Multivariate Diffusion with Semipermeable Interfaces

AbstractWe study the homogenization problem for a system of stochastic differential equations with local time terms that models a multivariate diffusion in the presence of semipermeable hyperplane interfaces with oblique penetration. We show that this system has a unique weak solution and determine its weak limit as the distances between the interfaces converge to zero. In the limit, the singular local times terms vanish and give rise to an additional regular interface-induced drift.

Stability of nonsingular cosmologies in Galileon models with torsion: A no-go theorem for eternal subluminality

Generic models in Galileons or Horndeski theory do not have cosmological solutions that are free of instabilities and singularities in the entire time of evolution. We extend this no-go theorem to a spacetime with torsion. On this more general geometry the no-go argument now holds provided the additional hypothesis that the graviton is also subluminal throughout the entire evolution. Thus, critically different for Galileons’ stability on a torsionful spacetime, an arguably unphysical although arbitrarily short (deep UV) phase occurring at an arbitrary time, when the speed of gravity (cg) is slightly higher than luminal (c), and by at least an amount cg≥2c, can lead to an all-time linearly stable and nonsingular cosmology. As a proof of principle we build a stable model for a cosmological bounce that is almost always subluminal, where the short-lived superluminal phase occurs before the bounce, and that transits to general relativity in the asymptotic past and future. Published by the American Physical Society 2024