Types Of Singular Points Research Articles

Rotor type singular points of nonautonomous systems of differential equations and their role in the generation of singular attractors of nonlinear autonomous systems

Rotor type singular points of nonautonomous systems of differential equations and their role in the generation of singular attractors of nonlinear autonomous systems

Rotor type singular points of nonautonomous systems of differential equations and their role in the generation of singular attractors of nonlinear autonomous systems

Rotor type singular points of nonautonomous systems of differential equations and their role in the generation of singular attractors of nonlinear autonomous systems

Theory on the Stability of the Ferromagnetic Double Layer Structure and on the Peak Structure of the Magneto-Optical Spectra of CeSb

We propose the pf+pd mixing model for CeSb to explain the stability of the ferromagnetic double layer structure in the magnetic ordering. The pd mixing causes the saddle type singular points, neighboring the $\Delta$ axis, for the bands which gain energy through the pf hybridization with the occupied f state. The peak of the density of states due to this combined effect of the pf mixing and the pd mixing enhances the stability of the double layer structure. The same combined effect also causes the saddle type singular points in the joint density of states of the optical transition. The peak structure of the magneto-optical spectra which has been observed in experiments is explained by the present model.

Flows through Porous Media Containing a Shielded Spherical Inclusion

Compact formulas are obtained for constructing flow potentials in media containing a spherical inclusion shielded by a high- or low-permeability film (fracture or barrier) using the known potentials of steady-state incompressible fluid flows through homogeneous porous media. The formulas obtained can be extended both to inhomogeneous porous media in which the permeability functions have different constant factors inside and outside the inclusion and to bounded zones. The type of singular points of the potentials (sources, sinks, etc.) and the boundary conditions assigned in media without an inclusion are conserved for media containing a shielded inclusion. As an example, translational flow past a spherical shielded polluted zone is studied. This is of interest in connection with environmental problems.

Normal Forms in One-Parametric Optimization

We deal with one-parametric families of optimization problems in finite dimension with a finite number of (in-)equality constraints. Its set of generalized critical points can be classified according to five types (cf. ). In this paper the corresponding normal forms of the problem near each of these five types of singular points are presented.

Heat kernel coefficients and spectra of the vector Laplacians on spherical domains with conical singularities

The spherical domains with conical singularities are a convenient arena for studying the properties of tensor Laplacians on arbitrary manifolds with this type of singular point. In this paper the vector Laplacian on is considered and its spectrum is calculated exactly for any dimension d. This enables one to find the Schwinger - DeWitt coefficients of this operator by using the residues of the -function. In particular, the second coefficient, defining the conformal anomaly, is explicitly calculated on and its generalization to arbitrary manifolds is found. As an application of this result, the standard renormalization of the 1-loop effective action of gauge fields is demonstrated to be sufficient to remove the ultraviolet divergences up to first order in the conical deficit angle.

Water-surface slope at critical controls in open-channel flow

A general equation is proposed for the determination of water-surface slope of a steady, gradually varied openchannel flow at a critical section. Thereupon the different types of singular points and water-surface slopes are given at such a section caused by changes in (1) bed slope, (2) channel width, (3) surface roughness of the channel, and (4) discharge, or their combinations. Although these factors are very different they have similar hydraulic effects on the occurrence of a certain type of singular point and water-surface slope. It is shown that a saddle type singular point is the only one which can become a control. The water-surface slopes at a critical control are evaluated using the proposed equation and are verified by the experimental results of Wilkinson (1974) for cases (1) and (2).

N = 2 heterotic superstring and its dual theory in five dimensions

We study quantum effects in five dimensions in heterotic superstring theory compactified on K 3 × S 1 and analyze the conjecture that its dual effective theory is eleven-dimensional supergravity compactified on a Calabi-Yau threefold. This theory is also equivalent to type II superstring theory compactified on the same Calabi-Yau manifold, in an appropriate large volume limit. In this limit the conifold singularity disappears and is replaced by a singularity associated to enhanced gauge symmetries, as naïvely expected from the heterotic description. Furthermore, we exhibit the existence of additional massless states which appear in the strong coupling regime of the heterotic theory and are related to a different type of singular points on Calabi-Yau threefolds.

Analysis for an adhesively bonded finite strip repair to a cracked plate

A plate with a crack, repaired by an adhesively bonded overlay of a finite strip, is investigated. In the analysis model it is assumed that due to high stress concentration an internal debonding between the strip and the plate may generate from the crack surfaces or, in other words, it is assumed that the strip may be originally designed to be adhesively bonded partially instead of fully, to the cracked plate. With the aid of the Green functions for a pair of edge dislocations and for a concentrated body force, a singular integral equation method is presented, by which the stress singularities not only at the crack tips but also at the strip ends, and at the internal debonding fronts or the internal edges of the partly bonding areas, can be examined. The numerical results given show the effects of various geometric and material parameters, defining, respectively, the relative size and position of the strip to the crack, the distances of the internal debonding fronts or the internal bonding edges away from the crack surfaces, and the relative stiffness of the strip to the plate, on the stress intensity factors at all the three types of singular points.

Analysis of Stresses at Singular Points of Patterned Structures

AbstractAn iterative method describing the stress state at the vicinity of singularity point is presented here. This approach is based on concepts of rupture mechanics, uses classical solutions from finite element calculations and concerns plane problems. Through this approach, the stress state at the vicinity of several types of singular points in a patterned structure were analysed. The obtained results can be discussed according to failure mecanisms.

Flow near the critical line in the gas-dynamic model of stellar wind with thermal conductivity

A one-dimensional, time-independent, spherically symmetric model of stellar wind is studied within the framework of gas dynamics. It is assumed that the energy transfer in the external stellar atmosphere occurs by electron thermal conduction. The form of the critical line on which the transition from subsonic to supersonic flow occurs and the types of singular points and their connection with the directions of separatrices is studied in detail. The stability of the solutions with respect to perturbations in the neighbourhood of singular points is considered, using the method suggested in [1].

Analysis of Stresses at Singular Points of Patterned Structures.

ABSTRACTAn iterative method describing the stress state at the vicinity of singularity point is presented here. This approach is based on concepts of rupture mechanics, uses classical solutions from finite element calculations and concerns plane problems. Through this approach, the stress state at the vicinity of several types of singular points in a patterned structure were analysed. The obtained results can be discussed according to failure mecanisms.

Theory of singular points of ordinary differential equations in complex domain

In this paper, the topological of integral surfaces near certain of Lyapunov type singular points and certain type of nodes of ordinary differential equations in complex domain are studied. We introduce Briot-Bouquet transformation, in order to study the topological structure of integral surfaces near higher order singular points. At last we give an estimate of the maximum number of isolated limit integral surfaces passing through certain type of higher order singular points.

Note on the model oscillators of irregular stellar variability

A model oscillator for irregular stellar variability presented by Tanaka and Takeuti has only one singularity in the original form. We show it has three different types of singular points when damping terms are ignored. The nonlinear pulsation of Baker's one-zone stellar models is discussed in comparison with the properties of the Tanaka-Takeuti oscillator.

A characterization of curves of cyclic order four

Differentiable curves of cyclic order 4 are well known as regards the number and type of singular points. This paper characterizes curves of cyclic order 4 in the real conformal plane when no differentiability conditions are assumed by introducing multiplicities for singular points. It is shown that the sum of the multiplicities of the singular points is exactly four.

Trajectories and singular points in steady-state models of two-phase flows

The purpose of this paper is to adapt the methodology of dynamical systems to the study of a wide class of mathematical models currently used to solve problems of steady 1-D two-phase flows. A detailed study of the geometrical features of the ensemble of solutions is then used for two purposes. First, it makes it possible to understand the physical characteristics of such flows without the need to produce complete solutions. Secondly, the methodology gives valuable indications as to how to supplement computer codes which become inadequate in the neighborhood of singular points. This leads to avoidable numerical difficulties and incorrect interpretations. This methodology is particularly useful in the study of the phenomena of cooking. The important contribution of this general analysis is to show that, regardless of the number of equations in the model, the generic solutions involve only three types of singular points: saddle points, spirals or nodes. By way of example, the method is applied to the mathematically simplest case, the homogeneous flow model with adiabatic boundary conditions. The channel consists of a convergent portion with transition to a divergent portion through a smooth throat. The divergent portion possesses an inflection point after which the rate of area divergence decreases. The fluid is a mixture of water and steam in thermodynamic equilibrium. The expected saddle point, located just downstream from the throat, is followed by an unexpected spiral. The phase space consisting of pressure P, enthalpy h and spatial coordinate z divides itself into four distinct areas. In area A of figure 13 all solutions are single-valued, possess a pressure minimum and correspond to physically acceptable subcritical flows. In area B the solutions of the differential equations (all possessing a turning point) are irrelevant for the physical problem at hand. Through the saddle point there passes a trajectory which is unique in the subcritical portion and continues with two branches, one subcritical ending at pressure P′, the other supercritical, ending at pressure P″. Area C between those branches contains states which are not described by the postulated model. It is conjectured that flows against a back-pressure, P″ < P a < P′, evidently observed in nature, must presumably be described by an extension to the model involving enhanced entropy production. The remaining area D contains supercritical state points and is of no present interest.

Structures of steady states of an enzyme reaction in a CSTR

Structures at steady states have been investigated for an enzyme reaction in a continuous stirred tank reactor that has a random bi-uni reaction mechanism, using the imperfect bifurcation theory via singularities. The analysis has shown that two types of singular points exist. One of these points has the two types of transition states characterized by the hysteresis and double-limit points. It is obtained when the derivative of the steady-state equation with respect to the bifurcation parameter does not vanish. When the derivative vanishes, the other type of singular point is obtained. This point has the two transition states of hysteresis and bifurcation points.

単一ジンバルＣＭＧの特異点について

A single gimbal CMG system is a promising candidate for an actuator of attitude control systems for large space structures. One of its problems is the existence of singular states. Various steering laws assume redundancy in the system and avoid singular states by maximizing a certain criterion function value by a gradient method. But it is not theoretically clear whether these steering laws can avoid all singular states actually. In this paper, a quadratic form is defined for each singular state, which enables to classify singular states into three types. In the vicinity of the two types of singular states, escape from the vicinity is not guaranteed locally. Also it is diffcult to guarantee that the system does not go into the vicinity of any two types of singular points by using the gradient method only. Examples of three type singular states are shown for three configurations.

Invariance properties in Hermite-Padé approximation theory

The Padé approximant is invariant under both linear fractional transformations of the function value and linear fractional transformations (origin fixed) of the argument. The paper reports on an investigation of these invariance properties to the class of Hermite-Padé approximants based on the class of general ordinary nonlinear differential equations. Further, the paper characterizes the class with these invariance properties. Uniqueness is discussed. Examples are given, such as the Padé-Riccati approximants, and allowed types of singular points are discussed.

Meron solutions of the sigma model and singular harmonic maps

We study the meron solutions of theO(3) sigma model in two-dimensions using the singular harmonic maps. We find new solutions of the topological charge (the Brouwer degree) zero having one dimensional singularities of the fold and collapse types. In general any singularity of a solution of the sigma model is locally equivalent with one of the four types of singular points of a harmonic map.