Position Of Singular Points Research Articles

A Rapid Numerical Method for the Mullins–Sekerka Flow with Application to Contact Angle Problems

The Mullins–Sekerka problem is numerically solved in R2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {R}}^2$$\\end{document} with the aid of the charge simulation method. This is an expansion of the numerical scheme by which Sakakibara and Yazaki computed the Hele–Shaw flow. We investigate a relationship among a time step, the number of collocation points and the position of singular points to ensure that the length of the generated approximate polygonal curves gradually decreases. We propose a new benchmark function for the Mullins–Sekerka flow to confirm that the scheme works well. Moreover, by changing the fundamental solutions of the charge simulation method, we are successful to establish a numerical scheme that can be used to treat the Mullins–Sekerka problem with the contact angle condition.

Analytic structure of three-point functions from contour deformations

We explore the analytic structure of three-point functions using contour deformations. This method allows continuing calculations analytically from the spacelike to the timelike regime. We first elucidate the case of two-point functions with explicit explanations how to deform the integration contour and the cuts in the integrand to obtain the known cut structure of the integral. This is then applied to one-loop three-point integrals. We explicate individual conditions of the corresponding Landau analysis in terms of contour deformations. In particular, the emergence and position of singular points in the complex integration plane are relevant to determine the physical thresholds. As an exploratory demonstration of this method's numerical implementation we apply it to a coupled system of functional equations for the propagator and the three-point vertex of $\phi^3$ theory. We demonstrate that under generic circumstances the three-point vertex function displays cuts which can be determined from modified Landau conditions.

Some exact solutions that describe an ideal fluid flow with a free boundary

The study is aimed at constructing exact solutions that describe unsteady flows of an ideal fluid with a free boundary. Currently existing examples of solutions of this kind have a linear or quadratic initial field of velocities. The object of the present study is flows with a cubic initial velocity field. Several exact solutions are obtained by using a method based on the complex Hopf equation for the complex velocity. One of the main problems in solving the Euler equations in the domain with a free boundary is to study the positions of singular points because these points can reach the free boundary with time, which may lead to solution destruction. In the present case, singularities can be found explicitly. However, as the initial velocity field is defined by a third-power polynomial, one has to study the Riemann surfaces for the function of complex velocity in order to determine the mutual positions of the singularities and flow domain. It is shown that any initial velocity field defined by a cubic polynomial yields one of seven flows described in the paper with time. It is of interest that there are examples of flows that exist for a finite time and break down when the singularities reach the free boundary. Using a certain modification of the Hopf equation, we obtain a two-parameter family of exact solutions that describe fluid flows with a free boundary. Two examples of this family are considered. In one of them, the solution breaks down within a finite time. The second example is of interest because the motion of a singular point in the fluid leads to the formation of a narrow channel with free boundaries.

Kinematic Singularity and Bifurcation Analysis of Sidestay Landing Gear Locking Mechanisms

A dual-sidestay landing gear is prone to locking failure in the deployed state due to the restriction of movement between two sidestays. However, the principle of its locking movement still remains unclear. The present study is aimed at investigating the synchronous locking performance of the dual-sidestay landing gear based on the singularity and bifurcation theory. From the perspective of the kinematic mechanism, the reason for high sensitivity to structural dimensions in the locking process is explained, and the locked position is investigated by employing the numerical continuation method in the case of a single-sidestay landing gear. Afterwards, the reason for the locking failure of the dual-sidestay landing gear is analyzed, and a kinematic optimization method for the synchronous locking is proposed. The results reveal that the lock links must reach the lower overcenter singular point to fully lock the landing gear, and the singular point is sensitively affected by structural parameters. Owing to the different positions of singular points, the movements of fore and aft sidestays seriously restrict each other, causing locking failure of the dual-sidestay landing gear. The singular points of two sidestays can be optimized to be approximately identical, making their movements more coordinated.

Eight-connected contour method for accurate position detection of optical vortices using Shack–Hartmann wavefront sensor

We propose a simple method of realizing an accurate position detection of phase singularities in an optical vortex (OV) beam using a Shack–Hartmann wavefront sensor (SH-WFS). The method calculates circulations which are the discrete contour integrals of phase slope vectors measured by the SH-WFS and then determines the accurate positions of the singular points by calculating the centers-of-gravity with a fixed window size around the local peak of the circulation distribution. We use closed paths that connect the centers of eight-connected, instead of 2×2-neighboring lenslet apertures for calculating the circulations. Both the numerical analysis and proof-of-principle experiment were performed to confirm the measurement accuracy. In experiments, the positions of singular points in OV beams generated by a liquid-crystal-on-silicon spatial light modulator were measured. The root-mean-square error of the position measurement was approximately 0.09 in units of the lens size of the lenslet array used in the SH-WFS. We also estimated the topological charges of the singular points being detected based on the peak circulations, and the results agreed well with theoretical ones. The method achieves both rapid implementation and sublens-size spatial resolution detection and is suitable for applications that require real-time control of OV beams.

Study on fast integration method for Bessho form translating–pulsating source Green’s function distributing on a panel

The singularities, oscillatory performance of the panel source functions, which are derived by using a single integral expression of Bessho form translating–pulsating source Green’s function and represent the velocity potentials due to the translating–pulsating sources distributing on a panel, are analyzed. Relative fast numerical integration methods, such as the “LOBATTO Rule”, the “Variable Substitution Method (VSM)” and the “Steepest Descent Integration Method (SDIM)”, are used to evaluate this type of Green’s function. For SDIM, complex domain is restricted only on the θ-plane; meanwhile, the integral along the real axis is computed by use of the VSM to avoid the complication of numerical search of the steepest descent line. Auxiliary techniques, such as local refinement of integral steps in narrow zone near inflection points when using VSM, are applied so as to improve accuracy. Contour integral technique is adopted, and a simple numerical method for justifying the relative positions of singular points and integral contour is established based on Green’s theorem. The numerical method is validated by comparison to other existing results, and is shown to be efficient and reliable.

Number of Jordan blocks of the maximal size for local monodromies

AbstractWe prove formulas for the number of Jordan blocks of the maximal size for local monodromies of one-parameter degenerations of complex algebraic varieties where the bound of the size comes from the monodromy theorem. In the case when the general fibers are smooth and compact, the proof calculates some part of the weight spectral sequence of the limit mixed Hodge structure of Steenbrink. In the singular case, we can prove a similar formula for the monodromy on the cohomology with compact supports, but not on the usual cohomology. We also show that the number can really depend on the position of singular points in the embedded resolution, even in the isolated singularity case, and hence there are no simple combinatorial formulas using the embedded resolution in general.

Solubility of aluminum, gallium, and indium perchlorates in water

Solubility polytherms of the system M(ClO4)3-H2O (M3+ = Al3+, Ga3+, In3+) were measured. Positions of singular points and vitrification ranges were determined. Values of eutectic concentrations were compared with earlier known data on the inversion concentrations measured by the viscometry method. Compositions of found crystalline hydrates were calculated from the solubility polytherms. The concept of the ability of the studied aqua cations to show properties of aqua acids, of the structure of solutions of posteutectic concentrations (structurally forced processes and microheterogeneity), of the electronic structure of M3+ cations, and of the localized hydrolysis phenomenon were used to understand the existence of the polyaqua compounds.

Singular Problem and Improved Plane Interpolation Algorithm

In the 5-axis machining, when the tool orientation approaches the one of the rotation axis, the sharp changing of the rotation angles obtained by the transforming of tool orientations causes a large over cutting or under cutting for the limit of the machine’s dynamic characteristic, which is called singular problem. After analyzing the causing of the singular problem, the position of singular point, the singular value and the nonlinear error, etc, an improved plane interpolation algorithm based on plane interpolation of tool orientation algorithm is given out in the paper following with the simulation and test of the circumference milling of the tilt plane (tilt angle: 1°) separately with the plane interpolation and improved plane interpolation based on a 5-axis machining center with two swiveling heads (rotation axes: C/A). The results prove that the new interpolation is valid and correct.

Topology and vortex structures of a curving turbine cascade with tip clearance (II)—Topological flow pattern and vortex structure in the transverse section of a blade cascade

By means of ink trace visualization of the flows in conventional straight, positively curved and negatively curved cascades with tip clearance, and measurement of the aerodynamic parameters in transverse section, and by appling topology theory, the topological structures and vortex structure in the transverse section of a blade cascade were analyzed. Compared with conventional straight cascade, blade positive curving eliminates the separation line of the upper passage vortex, and leads the secondary vortex to change from close separation to open separation, while blade negative curving effects merely the positions of singular points and the intensities and scales of vortex.

Topology and vortex structures of a curving turbine cascade with tip clearance (I)—Experimental model and topological flow patterns on both endwalls and blade surfaces

By means of ink trace visualization of the flows in conventional straight, positively curved and negatively curved cascades with tip clearance, and measurement of the aerodynamic parameters in the transverse section, and by appling topology theory, the structures on both endwalls and blade surfaces were analyzed. Compared with conventional straight cascade, blade positive curving eliminates the separation line of the upper passage vortex and leads the secondary vortex to change from close separation to open separation, while blade negative curving effects merely the positions of singular points and the intensities and scales of vortex.

Homology and cohomology of hypersurfaces with quadratic singular points in generic position

We calculate the homology groups of hypersurfaces in ℂPn+1, n≥3, with fixed number and, maybe, position of singular points and sufficiently high degree. In the case of quadratic singularities, we use the results of the calculations to give a topological description (as specific as possible) of such a hypersurface by means of decomposing it into a connected sum. In this case the topological type of the hypersurface is determined by its dimension, degree, and the number of singular points. Bibliography: 7 titles.

The topological structure of complex hypersurfaces with quadractic singular points

Hypersurfaces of sufficiently high degree in ℂPn+1, n≥3, with fixed number and possibly fixed positions of singular points are studied. In the case where all singularities are quadratic, a topological description of such a hypersurface is given bymeans of decomposing it into a connected sum of special form. In this case, the diffeomorphism type of the hypersurface is determined by its dimension, degree, and the number of singular points. Bibliography: 6 titles.

INDUCED GRAVITY ANOMALIES AND ROCK‐BURST RISK IN COAL MINES: A CASE HISTORY1

AbstractThe use of the gravity method to predict rock bursts in mines is based on the relationship between the development of a dilatancy process in the exploited rock mass and the time‐dependent gravity anomalies induced by this phenomenon. The differences between successive observations of anomalies and the time behaviour of their trend amplitudes as precursors of preceding changes of rock stability are interpreted. The centres of zones of induced rock density variation are determined by computing the position of singular points of the differences between anomalies.Two gravity surveys have been carried out in the Radbod coal mine (Germany). The first survey took place at the level of the Dickebank seam (depth 1030 m), the second in the Sonnenschein seam (depth 1090 m). The observations were made with Worden and LaCoste‐Romberg (D‐type) gravimeters. The differences between successive anomalies were less than 100 μGal.In the case of the Dickebank seam, the position of singular points demonstrates the effect of two approaching longwalls on a previously mined‐out seam and on the gallery in which the gravity observations were made. In the case of the Sonnenschein seam, the trend amplitudes show distinct variations in the formation of the approaching longwall below the edges of all previously mined‐out seams. In particular, the effect of a remnant pillar has caused the largest gravity gradients. This result corresponds to the existence of a zone of rock‐burst hazard known from test drilling. The computed singular points are grouped together under the remnant pillar indicating two local hazard zones.Both results, the observed development of rock instability with time and the information about the position of the disturbed rock mass relative to the mine workings, are of importance, subsurface gravity surveying can therefore be a valuable tool for predicting rock‐bursts.

A bifurcation study of laminar flow in a curved tube of rectangular cross-section

The spiralling flow within a curved tube of rectangular cross-section normally forms two distinct cells in the plane of the cross-section, but a transition to a different, four-cell flow is known to occur at a critical value of the axial pressure gradient. This paper shows that, for a square cross-section, the transition is a result of a complex structure of multiple, symmetric and asymmetric solutions. The structure is revealed by solving extended systems of equations for steady-state, fully developed, laminar flow, in a finite-element approximation, to locate exactly the positions of singular points. Continuation methods are used to trace the paths of these points as the aspect ratio and radius of curvature vary. For a square cross-section, multiple, symmetric flows of both two and four cells are found in two distinct ranges of axial pressure gradient q. In addition, a pair of asymmetric solutions arise from a symmetry-breaking bifurcation point on the primary flow branch. The paths of limit points and symmetry-breaking bifurcation points are obtained as the aspect ratio γ varies, and a transcritical bifurcation point is found at γ = 1.43. For larger aspect ratios the secondary four-cell branch is disconnected from that of the primary two-cell flow. The bifurcation set in the two parameters q and γ has a complex structure, with a number of higher-order singularities; in particular, the path of limit points is found to cross in the manner of an unfolded swallowtail catastrophe. A stability analysis shows that all multiple solutions except the two-cell type are unstable with respect to either symmetric or antisymmetric perturbations. For values of the aspect ratio less than 1.43 there is a range of axial pressure gradients for which there is no stable flow. The critical Dean numbers of the singular points are found to vary strongly at small values of radius of curvature β, but the bifurcation set remains qualitatively the same. Previous work is interpreted in the light of the present results.

Synthesis of Planar Four-Link Path Generator Planar Four-Link Path Generator : On the Singular Points

Four-link coupler curve is discontinuous at the change point, and is separable into two or four curves by reversing the driven link about the fixed link. Nevertheless, in former algebraic analysis of singular points, it was not able to distinguish between the singular points on one curve and those on two curves. For practical uses of mechanisms it is required to analyze the singular points on one curve and to synthesize a mechanism having such singular points. In the present paper, the authors first obtain the number of singular points on one coupler curve of planar four-link crank-and-rocker mechanism, and derive the conditions for syntheses. Second, calculating the inclinations of tangents at the singular points, they synthesize the mechanisms satisfying the number and the positions of singular points and the inclinations of tangents at the singular points. As a result, it became possible to apply the coupler curve to automatic assembling and conveying machines.