Singular Points Research Articles

Trace operators on bounded subanalytic manifolds

We prove that if M⊂Rn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$M\\subset {\\mathbb {R}}^n$$\\end{document} is a bounded subanalytic submanifold of Rn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {R}}^n$$\\end{document} such that B(x0,ε)∩M\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ extbf{B}}(x_0,\\varepsilon )\\cap M$$\\end{document} is connected for every x0∈M¯\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$x_0\\in {{\\overline{M}}}$$\\end{document} and ε>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varepsilon >0$$\\end{document} small, then, for p∈[1,∞)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p\\in [1,\\infty )$$\\end{document} sufficiently large, the space C∞(M¯)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathscr {C}}^\\infty ( {{\\overline{M}}})$$\\end{document} is dense in the Sobolev space W1,p(M)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$W^{1,p}(M)$$\\end{document}. We also show that for p large, if A⊂M¯\\M\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$A\\subset {{\\overline{M}}}\\setminus M$$\\end{document} is subanalytic then the restriction mapping C∞(M¯)∋u↦u|A∈Lp(A)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {\\mathscr {C}}^\\infty ( {{\\overline{M}}})\ i u\\mapsto u_{|A}\\in L^p(A)$$\\end{document} is continuous (if A is endowed with the Hausdorff measure), which makes it possible to define a trace operator, and then prove that compactly supported functions are dense in the kernel of this operator. We finally generalize these results to the case where our assumption of connectedness at singular points of M¯\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {{\\overline{M}}}$$\\end{document} is dropped.

Set theoretical pathologies in the problem of Lyapunov stability of singular points of vector fields

Set theoretical pathologies in the problem of Lyapunov stability of singular points of vector fields

Negatively enhanced thermopower near a Van Hove singularity in electron-doped Sr2RuO4

The layered perovskite Sr2RuO4 serves as a model material of the two-dimensional (2D) Fermi liquid but also exhibits various emergent phenomena including the non-Fermi-liquid (NFL) behavior under external perturbations such as uniaxial pressure and chemical substitutions. Here we present the thermoelectric transport of electron-doped system Sr2−yLayRuO4, in which a filling-induced Lifshitz transition occurs at the Van Hove singularity (VHS) point of y≈0.2. We find that the sign of the low-temperature thermopower becomes negative only near the VHS point, where the NFL behavior has been observed in the earlier study. This observation is incompatible with either a numerical calculation within a constant relaxation-time approximation or a toy-model calculation for the 2D Lifshitz transition adopting an elastic carrier scattering. As a promising origin of the observed negatively enhanced thermopower, we propose a skewed NFL state, in which an inelastic scattering with a considerable odd-frequency term plays a crucial role to negatively enhance the thermopower. Published by the American Physical Society 2024

Failure prediction of an open-hole laminate under compression

Failure prediction of open-hole laminates under compression still remains a great challenge. A number of unique mechanics theories for composites developed by the author are successfully applied to analyze in plane compression induced various failures of the notched laminates in this paper. A buckling mode must be incorporated into the analysis of the compression induced delamination, and the rules for selecting a scale factor for the buckling analysis through ABAQUS are established. To simulate delamination, the interlaminar matrix stress modification method is applied. A more pertinent criterion for delamination is proposed. When and where is delamination initiated, how can the initiated delamination be propagated and how much is the delaminated area to be attained can be easily reproduced. Intralaminar failures are estimated based on Bridging Model and the matrix true stress theory. The ultimate strength of a notched laminate is assumed when any primary layer element outside neighborhoods of the stress singularity and weak singularity points firstly attains an ultimate failure. A limited, if not the minimum, number of inputs are required all measurable independently and following existing standards with no data calibration. Except for the pre-buckling analysis, no iteration is needed for prediction of all the other failures. The predicted failure modes and ultimate compressive loads of several laminates with single or double holes agree well with our measured counterparts.

2 n2-inequality for cA1 points and applications to birational rigidity

The $4 n^2$ -inequality for smooth points plays an important role in the proofs of birational (super)rigidity. The main aim of this paper is to generalize such an inequality to terminal singular points of type $cA_1$ , and obtain a $2 n^2$ -inequality for $cA_1$ points. As applications, we prove birational (super)rigidity of sextic double solids, many other prime Fano 3-fold weighted complete intersections, and del Pezzo fibrations of degree $1$ over $\mathbb {P}^1$ satisfying the $K^2$ -condition, all of which have at most terminal $cA_1$ singularities and terminal quotient singularities. These give first examples of birationally (super)rigid Fano 3-folds and del Pezzo fibrations admitting a $cA_1$ point which is not an ordinary double point.

Noise filtering options for conically scanning Doppler lidar measurements with low pulse accumulation

Abstract. Doppler lidar (DL) applications with a focus on turbulence measurements sometimes require measurement settings with a relatively small number of accumulated pulses per ray in order to achieve high sampling rates. Low pulse accumulation comes at the cost of the quality of DL radial velocity estimates and increases the probability of outliers, also referred to as “bad” estimates or noise. Careful filtering is therefore the first important step in a data processing chain that begins with radial velocity measurements as DL output variables and ends with turbulence variables as the target variable after applying an appropriate retrieval method. It is shown that commonly applied filtering techniques have weaknesses in distinguishing between “good” and “bad” estimates with the sensitivity needed for a turbulence retrieval. For that reason, new ways of noise filtering have been explored, taking into account that the DL background noise can differ from generally assumed white noise. It is shown that the introduction of a new coordinate frame for a graphical representation of DL radial velocities from conical scans offers a different perspective on the data when compared to the well-known velocity–azimuth display (VAD) and thus opens up new possibilities for data analysis and filtering. This new way of displaying DL radial velocities builds on the use of a phase-space perspective. Following the mathematical formalism used to explain a harmonic oscillator, the VAD’s sinusoidal representation of the DL radial velocities is transformed into a circular arrangement. Using this kind of representation of DL measurements, bad estimates can be identified in two different ways: either in a direct way by singular point detection in subsets of radial velocity data grouped in circular rings or indirectly by localizing circular rings with mostly good radial velocity estimates by means of the autocorrelation function. The improved performance of the new filter techniques compared to conventional approaches is demonstrated through both a direct comparison of unfiltered with filtered datasets and a comparison of retrieved turbulence variables with independent measurements.

Holonomy of the Planar Brownian Motion in a Poisson Punctured Plane

We define a family of diffeomorphism-invariant models of random connections on principal G-bundles over the plane, whose curvatures are concentrated on singular points. We study the limit when the number of points grows to infinity whilst the singular curvature on each point diminishes, and prove that the holonomy along a Brownian trajectory converges towards an explicit limit.

Use of GIS tools, enhanced by FFT filtering methods, to detect blurred craters in Synthetic Digital Elevation Models, to improve their location and morphological characterisation

Filtered Geomorphic Reference Models (FGRMs), extracted from High-Pass Filters based on the Fast Fourier Transform (HPF-FFT), provide a semi-automated and objective detection of a wide range of geoforms and geomorphic situations (flat areas, slopes, depressions, or valley bottoms). The method is sensitive enough to reveal numerous inflection points that enable the identification of subtle changes in relief polarity, resulting in accurate representations of real land depressions. The effectiveness of this method has been demonstrated in lunar landscapes, utilizing altimetry extracted from high-precision Planetary Digital Elevation Models (PDEMs) to construct crater inventories with well-defined FGRMs. However, when analysing lunar environments with closely-spaced geomorphic features, the FGRMs display blurring and generalization effects, reducing their capability to identify all existing crater depressions. This limitation controls the ability of FGRMs, to generate reliable crater inventories. This study examines such effects in three experiments. Using Synthetic Digital Relief Models (SDRMs) constructed by means of a MATLAB script two experiments are performed. In Experiment_1 different scenarios of proximity between aligned craters are analysed. The marks, or footprints, formed by craters of identical size and shape (i.e., circular bowl-shaped depressions with central peaks) are considered, resulting in simplified PDEMs. Various scenarios of crater centre separation (proximity) and the presence of internal features within craters are also analysed. Experiment_2 validates the method with different proximities, locations and crater sizes. Experiment_3 validates the approach in a sampled area of LRO lunar PDEM corresponding to Zelinskiy crater (Mare Ingenii area). To address these blurring effects, a hybrid methodology is employed, generating FGRMs extracted from HPF-FFT, in conjunction with other GIS and remote sensing tools, such as singular point zoning, aspect-slope zoning, and edge-limits zoning. As a result, the correspondence between the FGRMs and the synthetic forms is examined, aiding the comprehension of the graphical response provided by filtering in these cratered areas. The methodology demonstrates that FGRMs derived from HPF-FFT are sensitive to capturing connections between shapes as soon as the crater features start to merge, while blurring becomes evident when the craters are located at a proximity of <20 model units. Additionally, the results highlight the method's utility in accurately locating crater footprints and morphologically characterizing inner crater features, even when crater centres are closely spaced. This approach effectively mitigates blurring effects and enables the precise localization of crater marks and their internal features. The validations carried out in complex relief and high precise DEMs provide a use in real situation.

In-vivo mapping of human polymorphic ventricular tachycardia in Brugada syndrome

Abstract Background/introduction Brugada syndrome (BrS) is associated with ventricular fibrillation (VF) and polymorphic ventricular tachycardia (PMVT). Different mechanisms have been described for human PMVT, including epicardial drifting of rotational activity before stabilization on the apex. Purpose The aim of this study is mapping human PMVT mechanisms in BrS with ECG imaging (ECGI) in-vivo. Methods All BrS patients, were prospectively enrolled between 1992 and 2022. Inclusion criteria for the study were: 1) BrS diagnosis; 2) PMVT mapped with ECGI during thoracoscopic epicardial right ventricle outflow tract ablation. PMVT was adjudicated if a changing QRS pattern was present. Phase mapping was performed for each PMVT. All maps were imported on a dedicated software and analyzed offline. Left anterior descending coronary artery was added, based on CT scan reconstruction as a reference for the interventricular septum (in grey in Figure 1). Phase maps were performed using the Hilbert transform on detrended signals and the phases of wavefront propagation were color-coded. Rotational activity was defined as a continuous progression of phase from -π to +π around a phase singularity point. Focal activity was defined as a wavefront that appears de novo, without arising from another wavefront and spreading from its origin. Results A total of 21 BrS patients with VF were analyzed and 2 patients (9.5%) were found with PMVT. A total of 2 episodes of PMVT were mapped (one for each BrS patient) with a duration of 2.9 s and 1.3 s, respectively. Both PMVT had spontaneous initiation and termination. The mean number of rotational activities was 2.45 ± 0.07, with a mean of 2.9 ± 0.71 rotations on the apex and 0.8 ± 1.13 rotations on the septum. No other rotational activity was observed. There were 2 and 1 focal activities on the septum, respectively. No other focal activity was observed. The phase mapping of both PMVT episodes was consistent with the mechanism shown in Figure 1. In the first phase of PMVT (negative QRS) a stable rotor (white arrow) is found over ventricular apex (Figure 1, Panels A-D). In Figure 1, Panel E a septal activity appears (yellow arrow). The reduction of QRS voltage is secondary to a fusion between septal (yellow arrow) and apical (white arrow) rotational activities (Figure 1, Panels F-G). In the second phase of PMVT (positive QRS) a stable focal activity (yellow arrow) is found over the ventricular septum (Figure 1, Panels H-J). Conclusion In patients with BrS, change in the QRS pattern during PMVT is secondary to a fusion between rotational septal and apical activities.Figure 1

The graded ring database for Fano 3-folds and the Bogomolov stability bound

My paper (Suzuki 2003) produced some computer routines in Magma (Bosma et al. J Symb Comp 24:235–265, 1997) for the numerical invariants of Fano 3-folds, and used them in particular to determine the maximum value f=19\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f=19$$\\end{document} of the Fano index. As a byproduct of the research, extensive data associated with all possible sets of singular points of Fano 3-folds with Fano indices greater than or equal to 2 was obtained. Collaborative research with Gavin Brown developed an improved version of the Magma program. The data discussed above was added to the Graded Ring Data Base (Brown et al. 2015) at the University of Kent. Subsequently, GRDB, now located to the University of Warwick, recently modified its interface to accommodate additional conditions, facilitating a more refined selection of Fano manifolds. In this context, we focus on the inequality known as the Bogomolov stability bound. We present a list of candidates for Fano 3-folds that do not satisfy these conditions and propose the conjecture that they do not exist.This result has been independently obtained in Liu and Liu (2023).

A note on the semiclassical measure at singular points of the boundary of the Bunimovich stadium

A note on the semiclassical measure at singular points of the boundary of the Bunimovich stadium

Fringe-based depth segmentation via minimum-fringe-period-based singular points extraction

In the field of machine vision, depth segmentation plays a crucial role in dividing targets into different regions based on abrupt changes in depth. Phase-shifting depth segmentation is a technique that extracts singular points to form segmentation lines by leveraging the phase-shifting invariance of singular points in different wrapped phase maps. This makes it immune to color, texture, and camera exposure. However, current phase-shifting depth segmentation techniques face challenges in the precision of segmentation. To overcome this issue, this paper proposes a singular points extraction technique by constructing a more comprehensive threshold with the help of the minimum period of the phase map. Taking full advantage of the proposed technique, mean-value points and order singular points are accurately filtered out, and the integrity of segmentation lines in high-curvature regions can be guaranteed. During optimization processing, the precision of segmentation is improved by employing a low-cost morphology-based optimization model. Simulation results demonstrate the segmentation accuracy reaches up to 98.58% even in a noisy condition. Experimental results on different objects indicate that the proposed method exhibits good generalization and robustness.

Stability condition for nilpotent singularities by its complex separatrices

This work is focused in the center problem for nilpotent singularities of differential systems in the plane. Although there are involved methods to approach the center problem in this work, we present a new stability criteria based on the existence of the complex separatrices passing through the singular point, and independent of the normal form. The algebraic method needs only the computation of such complex separatrices up to certain order, which can be done with a computer algebra package.

"Utilizing Layer-2 Blockchain and IPFS Innovations for Establishing a Robust System for Secure Document Signing and Management"

Data holds immense significance within any information system. In traditional setups, data typically funnels into a central server for processing and storage, rendering the entire system reliant on this singular point. The ongoing digitalization trend, particularly accelerated by the global pandemic, has ushered in many transformative shifts in various operational aspects, including the validation, storage, and electronic signing of many documents. This surge in digital transactions has amplified both the data storage requirements and the volume of data traversing the Internet, thereby heightening the potential risks associated with fraud and document tampering. Consequently, an imperative need for a robust and secure document management system arises. Blockchain technology emerges as an innovative solution poised to address these challenges. In this study, we explore the compelling advantages that Blockchain technology offers to the realm of digital document storage, culminating in the development of a secure, web-based solution for document storage and processing.

Remarks on the smoothness of the C1,α asymptotically self-similar singularity in the 3D Euler and 2D Boussinesq equations

We show that the constructions of C1,α asymptotically self-similar singularities for the three-dimensional (3D) Euler equations by Elgindi, and for the 3D Euler equations with large swirl and 2D Boussinesq equations with boundary by Chen-Hou can be extended to construct singularity with velocity u∈C1,α that is not smooth at only one point. The proof is based on a carefully designed small initial perturbation to the blowup profile, and a BKM-type continuation criterion for the one-point nonsmoothness. We establish the criterion using weighted Hölder estimates with weights vanishing near the singular point. Our results are inspired by the recent work of Cordoba, Martinez-Zoroa and Zheng that it is possible to construct a C1,α singularity for the 3D axisymmetric Euler equations without swirl and with velocity u∈C∞(R3∖{0}) .

Deformation in the wrinkle–crease transformation

We investigate deformation in the wrinkle–crease transformation to reveal the missing link between the wrinkling and creasing instabilities, which occur on the surface of elastomers under compression. Plane-strain finite element (FE) analysis is performed by combining a nonlinear perturbation approach to find two bifurcated solutions for the flat surface in a metastable state. The first solution is the stable deformation path for crease evolution, and the second solution is the previously unknown unstable deformation path from wrinkling to creasing. The two deformation paths are not independent, and they are connected at the critical strain for creasing εC. The resulting deformation at εC acts as a precursive deformation to cause creasing. Although the unstable path is initiated by the wrinkling instability as a bifurcation, the creasing instability (i.e., εC) is interpreted as the singular point at which the unstable path switches to the stable path. A chain reaction model is developed to describe the mechanism and deformation of the wrinkle–crease transformation, and it shows good agreement with the FE prediction within an inevitable limitation of FEs.

Polar Bloch points in strained ferroelectric films

Topological domain structures have drawn great attention as they have potential applications in future electronic devices. As an important concept linking the quantum and classical magnetism, a magnetic Bloch point, predicted in 1960s but not observed directly so far, is a singular point around which magnetization vectors orient to nearly all directions. Here we show polar Bloch points in tensile-strained ultrathin ferroelectric PbTiO3 films, which are alternatively visualized by phase-field simulations and aberration-corrected scanning transmission electron microscopic imaging. The phase-field simulations indicate local steady-state negative capacitance around the Bloch points. The observation of polar Bloch points and their emergent properties consequently implies novel applications in future integrated circuits and low power electronic devices.

Unified framework for the separation property in binary phase-segregation processes with singular entropy densities

Abstract This paper investigates the separation property in binary phase-segregation processes modelled by Cahn-Hilliard type equations with constant mobility, singular entropy densities and different particle interactions. Under general assumptions on the entropy potential, we prove the strict separation property in both two and three-space dimensions. Namely, in 2D, we notably extend the minimal assumptions on the potential adopted so far in the literature, by only requiring a mild growth condition of its first derivative near the singular points $\pm 1$ , without any pointwise additional assumption on its second derivative. For all cases, we provide a compact proof using De Giorgi’s iterations. In 3D, we also extend the validity of the asymptotic strict separation property to the case of fractional Cahn-Hilliard equation, as well as show the validity of the separation when the initial datum is close to an ‘energy minimizer’. Our framework offers insights into statistical factors like particle interactions, entropy choices and correlations governing separation, with broad applicability.

Solutions with moving singularities for a one-dimensional nonlinear diffusion equation

The aim of this paper is to study singular solutions for a one-dimensional nonlinear diffusion equation. Due to slow diffusion near singular points, there exists a solution with a singularity at a prescribed position depending on time. To study properties of such singular solutions, we define a minimal singular solution as a limit of a sequence of approximate solutions with large Dirichlet data. Applying the comparison principle and the intersection number argument, we discuss the existence and uniqueness of a singular solution for an initial-value problem, the profile near singular points and large-time behavior of solutions. We also give some results concerning the appearance of a burning core, convergence to traveling waves and the existence of an entire solution.

A replaceable-component method to construct single-degree-of-freedom multi-mode planar mechanisms with up to eight links

Abstract. The multi-mode planar mechanisms (MMPMs) are excellent-performance reconfigurable mechanisms, which not only inherit structural characteristics of planar mechanisms but also have the multi-task, multi-working-condition application advantages of multi-mode mechanisms. However, lacking common bifurcation analysis and construction methods, their industrial application and development are seriously hindered. This paper presents a replaceable-component method to construct a set of single-degree-of-freedom (single-DOF) MMPMs based on the branch graphs of the corresponding planar mechanisms and the proposed multi-mode modules (MMMs). First, according to the established loop equations, all the kinematic information of the original planar mechanism is obtained by the branch graphs and singularity points using Maple. Then, compared to the relationship between the concepts of the branch and motion mode, the number and continuity of branches are taken as the index to identify the potential bifurcation and mode conversion ability for the corresponding planar mechanisms. Subsequently, the MMM is presented to help the planar mechanisms break the singularity positions to form the corresponding MMPMs, and the steps of constructing single-DOF MMPMs are summarized. Finally, a single-DOF Stephenson six-bar three-mode planar mechanism, a Watt six-bar three-mode planar mechanism, and an eight-bar four-mode planar mechanism are constructed for the first time, and the corresponding multi-mode motion analyses are made. The results can give the available configuration for the design of corresponding MMPMs. The proposed method will provide strong guidance for the configuration design of MMPMs.