Singular Line Research Articles

Bifurcation of Limit Cycles from a Class of Piecewise Smooth Systems with Two Vertical Straight Lines of Singularity

Perturbing the following systems [Formula: see text] having a linear center with two vertical straight lines ([Formula: see text]) of singularity inside the class of all piecewise polynomials of degree [Formula: see text], we give the estimate of the maximum number [Formula: see text] of limit cycles bifurcating from the period annulus around the center based on the argument principle and the first order averaging theory. It shows that [Formula: see text] is at least [Formula: see text] bigger than the maximum number of limit cycles bifurcating from the period annulus around the linear center with two parallel straight lines of singularity in [Li & Liu, 2015].

Composite Dislocations in Smectic Liquid Crystals.

Smectic liquid crystals are characterized by layers that have a preferred uniform spacing and vanishing curvature in their ground state. Dislocations in smectics play an important role in phase nucleation, layer reorientation, and dynamics. Typically modeled as possessing one line singularity, the layer structure of a dislocation leads to a diverging compression strain as one approaches the defect center, suggesting a large, elastically determined melted core. However, it has been observed that for large charge dislocations, the defect breaks up into two disclinations [C. E. Williams, Philos. Mag. 32, 313 (1975)PHMAA40031-808610.1080/14786437508219956]. Here we investigate the topology of the composite core. Because the smectic cannot twist, transformations between different disclination geometries are highly constrained. We demonstrate the geometric route between them and show that despite enjoying precisely the topological rules of the three-dimensional nematic, the additional structure of line disclinations in three-dimensional smectics localizes transitions to higher-order point singularities.

Interaction of two singular Lissajous lines in free space.

The interaction of two singular Lissajous lines emergent from a polychromatic vector beam is studied. It is shown that singular Lissajous lines disappear with propagation; meanwhile Lissajous singularities take place. The handedness reversal, the changes in the shape of Lissajous figures, and the degree of polarization of Lissajous singularities, as well as the creation and annihilation of a single singularity, may appear by varying the control parameters. In addition, the transformation of the shape of line h=0, the creation and annihilation of pairs of Lissajous singularities not only with opposite topological charge and same handedness, but also with same degree of polarization, take place with propagation.

Analysis of Safety Protection Measures for Maintenance Work of 500 kV Double-Circuit Transmission Lines on Same Tower

Four transmission lines on the same tower are widely used because of their obvious economic and social benefits. But it also has high power supply reliability, so the choice of reasonable maintenance mode is particularly important. In this paper, we deducted the maintenance influence of the energized line to non-energized line, calculated and analyzed protection measures of non-energized singular line of 500kV double-circuit transmission line on the same tower with ATP software, and calculated field intensity distribution of typical operating position of the energized double-circuit transmission line with the finite element software. The calculation shows that when using the outage maintenance method, hanging both ground current and personal security line can reduce the current flowing through the operator’s body effectively. When using the live maintenance method, the field intensity of operator body strengths up to 383.69kV/m, The operator needs to wear shielding cloth with at least 43.08 dB shielding efficiency, in order to meet the security requirements.

A general extension theorem for cohomology classes on non reduced analytic subspaces

The main purpose of this paper is to generalize the celebrated L${}^2$ extension theorem of Ohsawa-Takegoshi in several directions : the holomorphic sections to extend are taken in a possibly singular hermitian line bundle, the subvariety from which the extension is performed may be non reduced, the ambient manifold is K{\"a}hler and holomorphically convex, but not necessarily compact.

Effects of Pendular Waist on Gecko's Climbing: Dynamic Gait, Analytical Model and Bio-inspired Robot

Most quadruped reptiles, such as lizards, salamanders and crocodiles, swing their waists while climbing on horizontal or vertical surfaces. Accompanied by body movement, the centroid trajectory also becomes more of a zigzag path rather than a straight line. Inspired by gecko's gait and posture on a vertical surface, a gecko inspired model with one pendular waist and four active axil legs, which is called GPL model, is proposed. Relationship between the waist position, dynamic gait, and driving forces on supporting feet is analyzed. As for waist trajectory planning, a singular line between the supporting feet is found and its effects on driving forces are discussed. Based on the GPL model, it is found that a sinusoidal waist trajectory, rather than a straight line, makes the driving forces on the supporting legs smaller. Also, a waist close to the pygal can reduce the driving forces compared to the one near middle vertebration, which is in accord with gecko's body bending in the process of climbing. The principles of configuration design and gait planning are proposed based on theoretical analyses. Finally, a bio-inspired robot DracoBot is developed and both of the driving force measurements and climbing experiments reinforce theoretical analysis and the rationality of gecko's dynamic gait.

Topologically equisingular deformations of homogeneous hypersurfaces with line singularities are equimultiple

We prove that if [Formula: see text] is a family of line singularities with constant Lê numbers and such that [Formula: see text] is a homogeneous polynomial, then [Formula: see text] is equimultiple. This extends to line singularities a well-known theorem of Gabrièlov and Kušnirenko concerning isolated singularities. As an application, we show that if [Formula: see text] is a topologically [Formula: see text]-equisingular family of line singularities, with [Formula: see text] homogeneous, then [Formula: see text] is equimultiple. This provides a new partial positive answer to the famous Zariski multiplicity conjecture for a special class of non-isolated hypersurface singularities.

Equidistribution for sequences of line bundles on normal Kähler spaces

We study the asymptotics of Fubini-Study currents and zeros of random holomorphic sections associated to a sequence of singular Hermitian line bundles on a compact normal Kaehler complex space.

Dynamics of Traveling Wave Solutions to a New Highly Nonlinear Shallow Water Wave Equation

In this paper, the dynamics of traveling wave solutions in a shallow water wave model with a regime for large-amplitude is studied. The corresponding traveling wave system is a singular planar dynamical system with one or two singular straight lines. By using the method of dynamical systems, bifurcation diagrams are presented. The existence of solitary wave solutions, periodic wave solutions, quasi-peakon solution, periodic peakon solutions and compacton solutions under different parameter conditions are determined.

Geometric localization of thermal fluctuations in red blood cells

The thermal fluctuations of membranes and nanoscale shells affect their mechanical characteristics. Whereas these fluctuations are well understood for flat membranes, curved shells show anomalous behavior due to the geometric coupling between in-plane elasticity and out-of-plane bending. Using conventional shallow shell theory in combination with equilibrium statistical physics we theoretically demonstrate that thermalized shells containing regions of negative Gaussian curvature naturally develop anomalously large fluctuations. Moreover, the existence of special curves, "singular lines," leads to a breakdown of linear membrane theory. As a result, these geometric curves effectively partition the cell into regions whose fluctuations are only weakly coupled. We validate these predictions using high-resolution microscopy of human red blood cells (RBCs) as a case study. Our observations show geometry-dependent localization of thermal fluctuations consistent with our theoretical modeling, demonstrating the efficacy in combining shell theory with equilibrium statistical physics for describing the thermalized morphology of cellular membranes.

Connected hidden singularities and toward successive state flipping in degenerate optical microcavities

Using scattering matrix formalism, we report the formation of a special hidden singular line connecting multiple second-order hidden singular points, in a non-uniformly pumped degenerate optical microcavity. Such singularities are known as exceptional points (EPs), and the line is proposed as an exceptional line. Exploring the unconventional behavior of cavity resonances created by a spatially imbalanced gain–loss profile, we have established the adiabatic state-flipping mechanism of coupled resonances encountering such EPs. Various interesting encircling situations, incorporating smooth as well as fluctuating variations of the control parameters, have been analyzed. We exploit the above scheme for the first time, to the best of our knowledge, to analyze the optical performance and stability of the cascaded flip-of-state phenomenon assisted by successive encirclement of either single or multiple singularities following the exceptional line in the context of optical mode converters.

The spectrum of solar relativistic cosmic ray measurements and numerical simulation

The solar relativistic protons are measured with the worldwide network of neutron monitors. The big pulses of relativistic protons appeared after the flares occurring in the West side of the Sun disk. They arrive to the Earth along Archimedean magnetic lines without collisions in ∼15 min after a flare. This prompt anisotropic flux contains information about the exponential ∼exp(−E/E0) spectrum of protons ejected from the solar cosmic ray source. After delay of 15 - 20 min the proton flux becomes isotropic with power spectrum E−γ where γ ∼ 5. Apparently, beam instability is developed. The protons accelerated in eastern flares can reach the neutron monitor due to diffusion across the magnetic lines. The magnetic field energy accumulation in the current sheet in the solar corona above the active region is proved by MHD simulations and the position of observed flare thermal X-ray source. During a flare the magnetic field energy is transferred into the particle energy. Proton acceleration up to relativistic energy can occur in the electric field applied along the singular line in a current sheet. The electric field E = -V×B/c is created due to the fast rate of reconnection V. At typical V = 2×107 cm/s the measured spectrum coincides with the calculated spectrum.

保存積分を用いた3次元ピエゾ接合体の界面角部近傍の応力特異線上の特異応力の強さの評価

保存積分を用いた3次元ピエゾ接合体の界面角部近傍の応力特異線上の特異応力の強さの評価

Structure of current and plasma in current sheets depending on the conditions of sheet formation

The structure of current sheets created under laboratory conditions is characterized by a large variety, which depends substantially on the conditions under which the sheet if formed. In this work, we present the results of an experimental study of the structure and evolution of current sheets that were formed in magnetic configurations with a singular line of the X type. It has been shown that the change in the transverse magnetic field gradient, the strength of the longitudinal magnetic field, and the mass of the ions in plasma makes it possible to significantly vary the main parameters of the current sheets. This offers the challenges of using laboratory experimental results for analyzing and simulating the cosmophysical processes.

Understanding Peakons, Periodic Peakons and Compactons via a Shallow Water Wave Equation

In this paper, a shallow water wave model is used to introduce the concepts of peakon, periodic peakon and compacton. Traveling wave solutions of the shallow water equation are presented. The corresponding traveling wave system is a singular planar dynamical system with one singular straight line. By using the method of dynamical systems, bifurcation diagrams and explicit exact parametric representations of the solutions are given, including solitary wave solution, periodic wave solution, peakon solution, periodic peakon solution and compacton solution under different parameter conditions.

Evaluating Multiscale Transform Based Image Compression Using Encoding Techniques

Image compression is the emerging field to transmit the multimedia products like image, video and audio. Image compression is used to reduce the storage quantity as much as possible. The objective of this paper is to compare the multiscale transform based image compression encoding techniques. The multiscale transforms involved in this paper are wavelet transform, bandelet transform, curvelet transform, ridgelet transform and contourlet transform. Wavelet transform allows good localization both in time and frequency domain. Bandelet transform takes geometric regularity of the natural images to improve the efficiency of representation. Curvelet transform handles curve discontinuities well. Curvelets are the good tool for the analysis and the computation of partial differential equations. Curvelets also have micro local features which make them especially adapted to certain reconstruction problems with missing data. The ridgelet transform is the core idea behind curvelet transform. It is used to represent objects with line singularities. The contourlet transform gets smooth contours and edges at any orientation. It filters noise as well. The Encoding techniques involved in this paper are spatial orientation tree wavelet (STW), set partitioned embedded block (SPECK) and compression with reversible embedded wavelet (CREW). The performance parameters such as peak signal to noise ratio (PSNR), image quality index and structural similarity index (SSIM) are used for the purpose of evaluation. It is found that bandelet transform with all the encoding techniques work well.

Equidistribution and convergence speed for zeros of holomorphic sections of singular Hermitian line bundles

We establish the equidistribution of zeros of random holomorphic sections of powers of a semipositive singular Hermitian line bundle, with an estimate of the convergence speed.

Equidistribution on Big Line Bundles with Singular Metrics for Moderate Measures

We establish an equidistribution theorem for the common zeros of random sections of high powers of several singular Hermitian big line bundles associated to moderate measures.

Interaction of the Lissajous singularity and singular Lissajous line carried by non-paraxial vector beams

The interaction of the Lissajous singularity and singular Lissajous line carried by non-paraxial polychromatic beams is studied. It has been found that with propagation the singular Lissajous line breaks into more than two Lissajous singularities, pairs creation and annihilation of Lissajous singularities, the handedness reversal and change in the degree of polarization of Lissajous singularities, as well as the change in the shape of Lissajous figures may take place.

Exact Traveling Wave Solutions and Bifurcations for a Shallow Water Equation Modeling Surface Waves of Moderate Amplitude

In this paper, we consider the traveling wave solutions for a shallow water equation. The corresponding traveling wave system is a singular planar dynamical system with one singular straight line. On the basis of the theory of the singular traveling wave systems, we obtain the bifurcations of phase portraits and explicit exact parametric representations for solitary wave solutions and smooth periodic wave solutions, as well as periodic peakon solutions. We show the existence of compacton solutions of the equation under different parameter conditions.