Singular Points Research Articles

The very singular solution for the Anisotropic Fast Diffusion Equation and its consequences

We construct the Very Singular Solution (VSS) for the Anisotropic Fast Diffusion Equation (AFDE) in the suitably good exponent range. VSS is a solution that, starting from an infinite mass located at one point as initial datum, evolves as an admissible solution of the corresponding equation everywhere away from the point singularity. It is expected to represent important properties of the fundamental solutions when the initial mass is very big. Here we work in the whole Euclidean space.In this setting we show how the diffusion process distributes mass from the initial infinite singularity with different rates along the different space directions. Indeed, and up to constant factors, there is a simple partition formula for the anisotropic mass expansion, given approximately as the minimum of separate 1-D VSS solutions. This striking fact is a consequence of the improved scaling properties of the special solution, and it has strong consequences.If we consider the family of fundamental solutions for different masses, we prove that they all share the same universal tail behaviour (i.e., for large |x|) as the VSS. Namely, their tail is asymptotically convergent to the unique VSS tail. This means that the VSS partition formula holds also for the fundamental solutions at spatial infinity. With the help of this analysis we study the behaviour of the class of nonnegative finite-mass solutions of the Anisotropic FDE, and prove the Global Harnack Principle (GHP) and the Asymptotic Convergence in Relative Error (ACRE) under a natural assumption on the decay of the initial tail.

Analysis of long transients and detection of early warning signals of extinction in a class of predator-prey models exhibiting bistable behavior.

In this paper, we develop a method of analyzing long transient dynamics in a class of predator-prey models with two species of predators competing explicitly for their common prey, where the prey evolves on a faster timescale than the predators. In a parameter regime near a singular zero-Hopf bifurcation of the coexistence equilibrium state, we assume that the system under study exhibits bistability between a periodic attractor that bifurcates from the singular Hopf point and another attractor, which could be a periodic attractor or a point attractor, such that the invariant manifolds of the coexistence equilibrium point play central roles in organizing the dynamics. To find whether a solution that starts in a vicinity of the coexistence equilibrium approaches the periodic attractor or the other attractor, we reduce the equations to a suitable normal form, and examine the basin boundary near the singular Hopf point. A key component of our study includes an analysis of the long transient dynamics, characterized by their rapid oscillations with a slow variation in amplitude, by applying a moving average technique. We obtain a set of necessary and sufficient conditions on the initial values of a solution near the coexistence equilibrium to determine whether it lies in the basin of attraction of the periodic attractor. As a result of our analysis, we devise a method of identifying early warning signals, significantly in advance, of a future crisis that could lead to extinction of one of the predators. The analysis is applied to the predator-prey model considered in Sadhu (Discrete Contin Dyn Syst B 26:5251-5279, 2021) and we find that our theory is in good agreement with the numerical simulations carried out for this model.

On the singularity point in acoustic orthorhombic media

Abstract The acoustic orthorhombic model is widely used in seismic modeling and processing of P-wave data. However, the anisotropic acoustic models have so called S-wave artifacts (1 artifact in transversely isotropic acoustic medium and two artifacts in orthorhombic acoustic medium). I show that S-wave artifacts can have one singularity point that results in complications in polarization field and the group velocity surface. The conditions of the existence of this point are defined in terms of anellipticity parameters. This singularity point and its group velocity image are the objects of my analysis.

The Shell of the Cosmos

The Big Bang theory, widely accepted by cosmologists, explains the origin, expansion, and eventual fate of the universe. It was further developed by inflation theory, which solves issues like the horizon and flatness problems. This theory states that the Cosmic Microwave Background (CMB), a remnant of the Big Bang, permeates the entire universe. Cosmologists believe these photons were emitted during the recombination epoch, approximately 380,000 years after the Big Bang when the cosmos had a temperature of around 3,000 Kelvin. Additionally, the Lambda-Cold Dark Matter (ΛCDM) model addresses questions arising from observations of the cosmos, such as the existence of light elements like hydrogen, helium, lithium, and the anisotropy in the CMB, and eventually, the continuous expansion of space. In this model, dark energy, represented as the cosmological constant lambda (Λ), exerts negative pressure on empty space, counteracting the effects of gravity with a repulsive force. While the Standard Model of Particle Physics (SMPP) posits that the universe's matter originated from fundamental particles (quarks and leptons), it cannot explain the origin of these particles and how they came into existence, because general relativity and the SMPP cannot be integrated to explain matter production at the singularity point. T-Consciousness Cosmology introduces the ‘Spherical Cosmos Model’ (SCM) to answer questions about the cause of the explosion, expansion, and shape of the universe, the nature of ordinary matter and energy, dark matter and energy, the fate of the universe, the reason for the high density of objects in the depths of space, etc. In this model, the spherical cosmos has a shell called the ‘Shell of the Cosmos,’ made of Taheri Absolute Matter (TAM) that not only isolates it but also produces dark matter and dark energy, ordinary matter and ordinary energy, and finally space mesh from the inner surface to the inside of the cosmos since the birth of the universe. The Shell of this isolated sphere is currently expanding at a speed faster than the speed of light. In this perspective, dark matter and energy are the same space mesh that have been compressed to varying degrees. Dark energy, which is constantly being released from the Shell into the cosmos, unlike the standard model of cosmology, is one of the factors in the expansion of the isolated cosmos by creating positive pressure in it. Also in this model, the recombination epoch will always be located spherically at a certain distance from the Shell until the ultimate stage of Rebound. In other words, not only is the origin of the CMB not related to the past of the cosmos, but it also exists now, and given the vastness of the sphere and the position of the Earth within this sphere, we detect it in the microwave wavelength with a delay of several billion years. Therefore, the observed distant galaxies that have been attributed to the early epochs of the cosmos in the Big Bang model are currently being created by the Shell according to the Spherical Cosmos Model, and we are surrounded by particles and objects that are constantly being produced.

Image Copyright Protection Based on Blockchain Technology Review

On a daily basis, a significant number of individuals distribute several photos and videos that have been marginally modified from the original material produced by copyright owners, such as photographers, graphic designers, and video producers. Individuals that infringe upon the rights of others, lacking the legal authority to access multimedia content, employ various digital image and picture manipulation techniques, it involves converting to gray scale, trimming, rotating, contracting the frame, and adjusting the background speed, to modify said content. Blockchain technology obviates the necessity of an intermediary, hence circumventing the possibility of a singular point of failure. Infractions to copyright poses a significant barrier to protecting commercial image and video information. The IPFS blockchain technology offers on-chain preservation for copyright information and off-chain storing for distinct multimedia files. The enhanced perceptual hashing algorithm significantly enhances the precision of identifying connections to identify digital image piracy. The photographers and designers that submit their photographs on websites are experiencing significant dissatisfaction due to a prevalent practice in which others attempt to claim credit and profit from the initial creator's effort.

Where is the orbital angular momentum in vortex superposition states?

In this paper, we explore the distribution of the orbital angular momentum (OAM) in the coaxial vortex superposition states based on the independent propagation principle of light in this interference process. We find that in this case, some specific singular points exist in the spatial intensity distribution. The first type of singular point is located at the center point of the spatial intensity distribution. The second type of specific singular point is at the critical location of the overlapping area in angular direction. By analogy with the angular momentum superposition of two axially rotating homogeneous disks with different radius in rigid body, We present a suggestion: the center point is located at the overlapping area of all the superposed components. Therefore, the topological charge value in the center point should be doubled by the actual number of superposition field components. The singular point at the critical location of the overlapping area in angular direction should also be co-owned by the superposition components outside the position of the ring (including the corresponding component of the ring). The total OAM is exactly equal to the sum of those two types contained in the superposition states, which is equal to the input OAM of the superposition state components. The conservation of the OAM in the coaxial interference process is demonstrated.

On singularity and normality of regular nilpotent Hessenberg varieties

Regular nilpotent Hessenberg varieties form an important family of subvarieties of the flag variety, which are often singular and sometimes not normal varieties. Like Schubert varieties, they contain distinguished points called permutation flags. In this paper, we give a combinatorial characterization for a permutation flag of a regular nilpotent Hessenberg variety to be a singular point. We also apply this result to characterize regular nilpotent Hessenberg varieties which are normal algebraic varieties.

Large Péclet number forced convection from a circular wire in a uniform stream: hybrid approximations at small Reynolds numbers

Abstract We consider heat or mass transport from a circular cylinder under a uniform crossflow at small Reynolds numbers, $\mathrm{Re}\ll 1$ . This problem has been thwarted in the past by limitations inherent in the classical analyses of the singular flow problem, which have used asymptotic expansions in inverse powers of $\log \mathrm{Re}$ . We here make use of the hybrid approximation of Kropinski, Ward & Keller [(1995) SIAM J. Appl. Math.55, 1484], based upon a robust asymptotic expansion in powers of $\mathrm{Re}$ . In that approximation, the “inner” streamfunction is provided by the product of a pre-factor $S$ , a slowly varying function of $\mathrm{Re}$ , with a $\mathrm{Re}$ -independent “canonical” solution of a simple mathematical form. The pre-factor, in turn, is determined as an implicit function of $\log \mathrm{Re}$ via asymptotic matching with a numerical solution of the nonlinear single-scaled “outer” problem, where the cylinder appears as a point singularity. We exploit the hybrid approximation to analyse the transport problem in the limit of large Péclet number, $\mathrm{Pe}\gg 1$ . In that limit, transport is restricted to a narrow boundary layer about the cylinder surface – a province contained within the inner region of the flow problem. With $S$ appearing as a parameter, a similarity solution is readily constructed for the boundary-layer problem. It provides the Nusselt number as $0.5799(S\,\mathrm{Pe})^{1/3}$ . This asymptotic prediction is in remarkably close agreement with that of the numerical solution of the exact problem [Dennis, Hudson & Smith (1968) Phys. Fluids11, 933] even for moderate $\mathrm{Re}$ -values.

A Singular Tempered Sub-Diffusion Fractional Equation with Changing-Sign Perturbation

In this paper, we establish some new results on the existence of positive solutions for a singular tempered sub-diffusion fractional equation involving a changing-sign perturbation and a lower-order sub-diffusion term of the unknown function. By employing multiple transformations, we transform the changing-sign singular perturbation problem to a positive problem, then establish some sufficient conditions for the existence of positive solutions of the problem. The asymptotic properties of solutions are also derived. In deriving the results, we only require that the singular perturbation term satisfies the Carathéodory condition, which means that the disturbance influence is significant and may even achieve negative infinity near some time singular points.

Integration of a degenerate system of ODEs

The integrability of a two-dimensional autonomous polynomial system of ordinary differential equations (ODEs) with a degenerate singular point at the origin that depends on six parameters is investigated. The integrability condition for the first quasihomogeneous approximation allows one of these parameters to be fixed on a countable set of values. The further analysis is carried out for this value and five free parameters. Using the power geometry method, the system is reduced to a non-degenerate form through the blowup process. Then, the necessary conditions for its local integrability are calculated using the method of normal forms. In other words, the conditions for the parameters under which the original system is locally integrable near the degenerate stationary point are found. By resolving these conditions, we find seven twoparameter families in the five-dimensional parametric space. For parameter values from these families, the first integrals of the system are found. The cumbersome calculations that occur in the problem under consideration are carried out using computer algebra.

The Hilbert problem in a half-plane for generalized analytic functions with a singular point on the real axis

This article analyzes the inhomogeneous Hilbert boundary value problem for an upper half-plane with the finite index and boundary condition on the real axis for one generalized Cauchy–Riemann equation with a singular point on the real axis. A structural formula was obtained for the general solution of this equation under restrictions leading to an infinite index of the logarithmic order of the accompanying Hilbert boundary value problem for analytic functions. This formula and the solvability results of the Hilbert problem in the theory of analytic functions were applied to solve the set boundary value problem.

The analytical solutions of long waves over geometries with linear and nonlinear variations in the form of power-law nonlinearities with solid inclined wall

In this paper, we derive the exact analytical solutions for the long-wave equation in both linear and nonlinear power-law form depth and breadth geometries containing a solid inclined wall. Firstly, we give general information about the concept of partial reflection and its components, and formulate the solid inclined wall boundary condition. For these specific power-law forms of depth and breadth geometries, we show that in the presence of the solid inclined wall, the long-wave equation admits solutions in terms of Bessel-Z functions and the Cauchy–Euler series. Since the presence of solid vertical wall removes the singular point from the domain, the solution admits both the first and the second kind of the Bessel functions and Cauchy–Euler series terms. We derive results for the general case and also discuss their significance using six different geometries with solid inclined wall.

Determining triple half-twist of optical Möbius strips in electromagnetic fields with a non-planar structure.

Polarization ellipses in a nonparaxial field form twisted strips when traced along closed contours around a circular polarization singularity line. We found an analytical expression for the twist number of the strip when the contour is coplanar with the polarization ellipse in its center. Necessary and sufficient conditions for strips having one or three half-twists are found. A set of five parameters of electromagnetic field at the polarization singularity point is found which definitely determines the value of the twist number of the strip.

Energy Resolved Mass Spectrometry for Interoperable Non-resonant Collisional Spectra in Metabolomics.

In untargeted metabolomics, the unambiguous identification of metabolites remains a major challenge. This requires high-quality spectral libraries for reliable metabolite identification, which is essential for translating metabolomics data into meaningful biological information. Several attempts have been made to generate reproducible product ion spectra (PIS) under a low collision energy (ELab) regime and nonresonant collisional conditions but have not fully succeeded. We examined the ERMS (energy-resolved mass spectrometry) breakdown curves of two lipo-amino acids and showed the possibility to highlight "singular points", called descriptors hereafter (linked to respective ELab depending on the instrument), for each of the monomodal product ion profiles. Using several instruments based on different technologies, the PIS recorded at these specific ELab sites shows remarkable similarities. The descriptors appeared as being independent of the fragmentation mechanisms and can be used to overcome the main instrumental effects that limit the interoperability of spectral libraries. This proof-of-concept study, performed on two particular lipo-amino acids, demonstrates the high potential of ERMS-derived information to determine the instrument-specific ELab at which PIS recorded in nonresonant conditions become highly similar and instrument-independent, thus comparable across platforms. This innovative but straightforward approach could help remove some of the obstacles to metabolite identification in nontargeted metabolomics, putting an end to a challenging chimera.

A novel approach to cosmological particle production

In this work we present a novel approach to the study of cosmological particle production in asymptotically Minkowski spacetimes. We emphasize that it is possible to determine the amount of particle production by focusing on the mathematical properties of the mode function equations, i.e. their singularities and monodromies, sidestepping the need to solve those equations. We consider in detail creation of scalar and spin 1/2 particles in four dimensional asymptotically Minkowski flat FLRW spacetimes. We explain that when the mode function equation for scalar fields has only regular singular points, the corresponding scale factors are asymptotically Minkowski. For Dirac spin 1/2 fields, the requirement of mode function equations with only regular points is more restrictive, and picks up a subset of the aforementioned scale factors. For the scalar case, we argue that there are two different regimes of particle production; while most of the literature has focused on only one of these regimes, the other regime presents enhanced particle production. On the other hand, for Dirac fermions we find a single regime of particle production. Finally, we very briefly comment on the possibility of studying particle production in spacetimes that don't asymptote to Minkowski, by considering mode function equations with irregular singular points.

Nonlinear stochastic behavior of soft-core sandwich panels

The nonlinear stochastic behavior of soft-core sandwich panels and the evolution of uncertainty along the nonlinear path are investigated. The paper focuses on the way the geometrically nonlinear and unstable behavior is affected by uncertainty and considers, for the first time, the evolution of uncertainty along the unstable structural response. The structural modeling relies on the EHSAPT and its integration into the geometrically nonlinear stochastic regime. A tailored nonlinear FE model that is based on a variational formulation and an arc-length solution scheme are developed. The stochastic analysis adopts the Perturbation-based SFEM and its generalization for the evolution of uncertainty along the process axis. A numerical study demonstrates the stochastic methodology and explores the impact of uncertainty on the nonlinear structural behavior. The evolution of the stochastic stress resultants at the displaced end is studied along the two axes of evolution. The differences between the two representations and their physical and engineering significance are highlighted revealing significant changes to the uncertainty evolution pattern with the shift from global buckling to localized wrinkling instabilities. The representation of the stochastic results along the process axis also quantifies the level of uncertainty near critical and singular points where conventional analyses diverge.

A hybrid path-following approach for constrained nonlinear-buckling optimization of variable stiffness composite shells with shape imperfections

Nonlinear buckling optimization of variable stiffness (VS) composite shells is computationally expensive due to frequent nonlinear analyses. Although the generalized path-following method can serve to reduce the redundant computation, it is weakened by two kinds of discontinuity. First, the design variables may not always be searched continuously in the constrained optimization. Second, the critical buckling states may not be continuous at singular points on the search path. To address this issue, a hybrid path-following approach is proposed to trace the critical buckling states in the nonlinear-buckling optimization with constraints on the realizability of the lamination parameters and the amplitude range of geometric imperfections. Firstly, the optimal search direction is determined with the sensitivity analysis of the critical buckling equations in the isogeometric continuum shell model. Then, a continuous following strategy is adopted to provide a proper initial estimate for the Newton iterations at a disconnected point. Moreover, an investigation into the singularity of the critical buckling equations is conducted to identify the discontinuous points of the critical states in a line-search. Upon detection of the singularity, a segment of path in state space is inserted to recover the continuity. Numerical experiments show that the continuity of the critical states in the optimization is well maintained, which leads to a more efficient optimization for the VS shell with shape imperfections.

Analytic auxiliary mass flow to compute master integrals in singular kinematics

The computation of master integrals from their differential equations requires boundary values to be supplied by an independent method. These boundary values are often desired at singular kinematical points. We demonstrate how the auxiliary mass flow technique can be extended to compute the expansion coefficients of master integrals in a singular limit in an analytical manner, thereby providing these boundary conditions. To illustrate the application of the method, we re-compute the phase space integrals relevant to initial-final antenna functions at NNLO, now including higher-order terms in their ϵ-expansion in view of their application in third-order QCD corrections.

NEWTON’S METHOD IN CONSTRUCTING A SINGULAR SET OF A MINIMAX SOLUTION IN A CLASS OF BOUNDARY VALUE PROBLEMS FOR THE HAMILTON - JACOBI EQUATIONS

The non-smooth features of the minimax (generalized) solution of the considered class of Dirichlet problems for equations of Hamiltonian type are due to the existence of pseudo-vertices - singular points of the boundary of the boundary set. The paper develops analytical and numerical methods for constructing pseudo-vertices and their accompanying constructive elements, which include local diffeomorphisms generating pseudo-vertices, as well as markers - numerical characteristics of these points. For markers, an equation with a characteristic structure inherent in equations for fixed points is obtained. An iterative procedure based on Newton’s method for the numerical construction of its solution is proposed. The convergence of the procedure to the pseudovertex marker is proved. An example of the numerical-analytical construction of a minimax solution is given, illustrating the effectiveness of the developed approaches for constructing nonsmooth solutions of boundary value problems.

Research on the Permeability and Pore Structure Distribution Characteristics of High-Performance Mortar for Surface Treatment of Bridge Piers and Columns

In marine environments, bridge piers and columns are prone to corrosion caused by harmful media, particularly chloride ions. This corrosion can lead to cracking of the steel bars in the protective layer of the bridge piers. To enhance the corrosion resistance of concrete in bridge piers, this article introduces the use of nanoclay-modified cement mortar. This innovative material offers high-performance surface treatment options that can effectively slow down the erosion process of harmful media and reduce the risk of bridge pier column cracking. To evaluate the ion erosion resistance of this nanoclay-modified cement mortar, we conducted detailed experiments on the pore structure of cement paste. The pore structure of cement paste with different dosages of nano-kaolinite clay and the dispersion method was studied using mercury intrusion porosimetry (MIP). The fractal dimension of the pore surface area of the net cement paste was calculated from the fractal model based on thermodynamic relationships of the pore structure-related parameters obtained with mercury pressure experiments. The relationship among the multiple fractal dimensions, pore structure parameters, dispersion mode, and permeability is explored. The results show that the addition of nano-kaolinite clay particles can improve the internal pore structure of cement materials. When 1.5% nano-kaolinite clay is mechanical dispersed, the total specific pore volume and the most probable pore size are reduced by 47.83% and 56.87%, respectively, compared with the control group. The fractal dimension image of cement-based materials with nano-kaolinite clay has a range of singular points and does not have fractal characteristics in this range. Nano-kaolinite clay has a significant effect on the fractal dimension of pore size range I. The fractal dimension of the whole pore size range is not suitable for the analysis of permeability, and the fractal dimension calculated by selecting less than the critical pore size range has a good correlation with permeability.