Showing papers on "Singularity published in 1974"

Single-particle states, Kohn anomaly, and pairing fluctuations in one dimension

Abstract: We compute the single-particle spectral density, susceptibility near the Kohn anomaly, and pair propagator for a one-dimensional interacting-electron gas. With an attractive interaction, the pair propagator is divergent in the zero-temperature limit and the Kohn singularity is removed. For repulsive interactions, the Kohn singularity is stronger than the free-particle case and the pair propagator is finite. The low-temperature behavior of the interacting system is not consistent with the usual Ginzburg-Landau functional because the frequency, temperature, and momentum dependences are characterized by power-law behavior with the exponent dependent on the interaction strength. Similarly, the enrgy dependence of the single-particle spectral density obeys a power law whose exponent depends on the interaction and exhibits no quasiparticle character. Our calculations are exact for the Luttinger or Tomonage model of the one-dimensional interacting system.

Fundamental singularities of viscous flow

Abstract: The image system for the fundamental singularities of viscous (including potential) flow are obtained in the vicinity of an infinite stationary no-slip plane boundary. The image system for a: stokeslet, the fundamental singularity of Stokes flow; rotlet (also called a stresslet), the fundamental singularity of rotational motion; a source, the fundamental singularity of potential flow and also the image system for a source-doublet are discussed in terms of illustrative diagrams. Their far-fields are obtained and interpreted in terms of singularities. Both the stokeslet and rotlet have similar far field characteristics: for force or rotational components parallel to the wall a far-field of a stresslet typeO(r −2) is obtained, whereas normal components are of higher orderO(r −3).

Singular points in the magnetization curve of a polycrystalline ferromagnet

Abstract: The reversible magnetization curve M(H) of a polycrystalline ferromagnet has been investigated with the objective of examining the properties of the singularities located at H = −HA, the anisotropy field, along the hard direction. Using a simple model that approximates the system as a continuous assembly of noninteracting particles, we show that the singular point becomes apparent in the successive derivatives dnM/dHn plotted as functions of H. Indeed, differentiation accentuates more and more the singularity hidden in the magnetization curve, and the minimum order n* at which it can be detected is the order for which the function dn*M/dHn* has a discontinuity in the slope at the singular point. The general formula for the dependence of n* on the symmetry properties of the hard direction is given together with the analytical expression for the shape and amplitude of the singularity in the most important cases. The same sort of phenomenon is shown to be present in the reversible transverse susceptibility, ...

Small-Scale Yielding Analysis of Mixed Mode Plane-Strain Crack Problems

Abstract: The small-scale yielding analysis of an elastic-plastic body with a line crack under plane-strain conditions subject to combinations of Mode I and II loadings is examined. The analysis of the near-tip field follows the works of Hutchinson and Rice and Rosengren. Dominant singularity solutions governing the asymptotic behavior of the stresses and strains at the crack tip are obtained for the complete range of loadings between Mode I and II. The results of an accurate finite element technique, which imbeds the dominant singularity solutions, directly relates the near-tip behavior to the elastic stress intensity factors K(I) and K(II). Implications of this study to mixed mode fracture mechanics is also discussed, particularly with respect to the direction of crack initiation and the relation of fracture toughness under mixed modes to that in Mode I. Details of the mixed mode plastic zone sizes and shapes are also given.

Application of quadratic isoparametric finite elements in linear fracture mechanics

Abstract: Quadratic isoparametri c elements are shown to embody 1//r singularity f or c alculating s tress i ntensity f actors of elastic f racture mechanics. The singularity is obtained by placing the mid-side node on any side at the quarter p oint. Figure 1 shows the 2-dimensional, 8-noded quadrilateral (a) and 6-noded triangle (b), isoparametric e lements with the mid-side nodes near the crack tip at the quarter nodes. Figure 2 shows the S-dimensional elements w ith the mid-side node near the crack edge at the quarter p oints. The local s trains in these e lements vary as 1//r throughout the element. In the 3-D case, the strains along the crack edge are non-singular. A very important feature of these elements is that they satisfy the necessary requirements for convergence [I] in their singular form as well as in their non-singular form. They, therefore, pass the patch test [2], possess rigid body motion (R.B.M.), constant strain modes, interelement compatibility, and continuity of displacements. In contrast, other special crack tip elements [3,4], do not possess rigid body motion modes and do not pass the patch test, thus making their use in the problems cited below questionable. The existence of rigid body motion and constant strain modes in the proposed isoparametric elements allows the calculation of stress intensity factors for thermal gradients in 2- and 3-D problems and in problems where symmetry about the crack cannot be invoked (R.B.M. exists). In addition, since these elements are part of the element library of most general purpose programs, their use in linear fracture mechanics is very tractable. The element formulation in its non-singular form is well documented ([I], pp. 103-154). The element in the singular form is formulated exactly in the same manner except for a restriction on the location of the nodal points. In summary, the element is formulated by mapping its geometry from the cartesian space into a unit curvilinear space using special quadratic functions [i]. The same functions, in the curvilinear space, are used to interpolate the displacements within the element, hence the name isoparametric. In order to achieve the required singularity, the Jacobian of transformation [J], from the cartesian to the curvilinear space, is made singular by placing the mid-side nodes near the crack tip at the quarter points. The singularity occurs only at the crack tip point. It can be easily shown, for example, that for the rectangular form of the case in Figure la, the strain in the local x-direction along the line i-2, is given by

Local Contributions to Global Deformations of Surfaces

Abstract: In [23], Schiffer and Spencer prove that all small deformations of complex structure on a compact Riemann surface may be realized by altering the complex structure only within an arbitrarily small neighborhood of a point on the surface. It seems interesting in general to consider whether it is possible to construct deformations of an algebraic variety or complex manifold from deformations of neighborhoods of certain subvarieties. Further motivation for trying to understand the role of subvarieties in deformations is suggested by the instability under deformation of the Neron-Severi group, i.e., the group of divisors modulo numerical equivalence. As an example, one may consider the family of affine surfaces Vt: x 2 + y 2 + z 2 = t z (t is a parameter; V o has a nodal singularity at the origin). This family admits a resolution {Xt} --, {V~}, with {X,} a smooth family of non-singular surfaces, and each X~ is a minimal resolution of V t ([4] or [5]). The exceptional curve E in X o is a IP 1 with self-intersection 2 which does not appear in any Xt, for t#:0. One may ask whether every smooth surface X with such a curve in it admits a one-parameter family of deformations arising from this local model. Furthermore, if X contains several disjoint such curves, does each one independently contribute one dimension to the moduli of X? The Hartogs ' theorem of [14] says that one cannot simply plumb in the local deformation, leaving the structure of X unchanged outside a small neighborhood of E. Moreover, old examples of Segre [26] show that the nodes on certain hypersurfaces V in IP 3 are not "independent", i.e., there aren't enough deformations of the resolution X of V to allow for a one-dimensional contribution from each node. Theorem (3.7) of this paper says that the regularity of the Kuranishi variety of X is sufficient for the deformations of X to realize independently

Equisingular deformations of plane algebroid curves

Abstract: We construct a formal versal equisingular deformation of a plane algebroid curve (in characteristic zero), and show it is smoothly embedded in the whole deformation space of the singularity. Closer analysis relates equisingular deformations of the curve to locally trivial deformations of a certain (nonreduced) projective curve. Finally, we prove that algebraic r1 of the complement of a plane algebroid curve remains constant during formal equisingular deformation. Introduction. In a series of papers ([10], [11], [121), Zariski has studied the concept of equisingularity of plane algebroid curves. Two curves are equisingular if one can simultaneously resolve their singularities; this equivalence relation is weaker than analytic equivalence, but stronger than equimultiplicity. Using topological techniques, Zariski proves that two equisingular curves over C have locally the same topological embedding in C2; in particular, the characteristic pairs of their branches are the same, whence they yield knots of the same knot type in R3 (cf. [4]). Utilizing techniques developed by M. Schlessinger [61, we study infinitesimal equisingular families of curves. Our deformation theory takes place over the category C of artin local C-algebras. Recall that if f e C[[X, Y]] is reduced, and if g1, *.*. * gm e C[[X, Y]] induce a basis of the artin ring C[[X, Y]/(jf fx' fy), then the formal family f + t1g1 + * .*+ tmgm C[[X, Y, t1,., till induces a formal versal (or semiuniversal) deformation of the singularity defined by (f ). Thus, in a weak sense, the family represents the functor on C of infinitesimal deformations of the singularity. To define equisingular deformation, we emulate Zariski's original definition. Recall that every plane algebroid singularity can be reduced to a number of ordinary double points by a finite number of quadratic transforms. We say a deformation Received by the editors December 19, 1972. AMS (MOS) subject classifications (1970). Primary 14D15, 14H20; Secondary 14B10, 32C40.

Tilting at cosmological singularities

Abstract: A detailed investigation is made of the simplest type of general relativistic perfect fluid cosmological models that possess a singularity at which all physical quantities are well-behaved. These models are spatially homogeneous, axisymmetric generalisations of the open (k/it=−1) Robertson-Walker universes. A pictorial description of the evolution of the models is obtained by using the qualitative theory of differential equations. The most surprising feature that emerges is that for some (non-empty) models the matter density may become zero, within a finite time, on a null hypersurface which acts as a Cauchy horizon for the models. This result is generalized to most other types of spatially homogeneous models. It is also discovered that the behaviour of the models varies dramatically with the type of matter content. This casts some doubts on the validity of assuming definite equations of state in general relativity, and suggests an investigation of the structural stability of Einstein's field equations.

On laminar boundary-layer separation

Abstract: Iterative finite-difference techniques are developed for integrating the boundary-layer equations, without approximation, through a region of reversed flow. The numerical procedures are used to calculate incompressible laminar separated flows and to investigate the conditions for regular behavior at the point of separation. Regular flows are shown to be characterized by an integrable saddle-type singularity that makes it difficult to obtain numerical solutions which pass continuously into the separated region. The singularity is removed and continuous solutions ensured by specifying the wall shear distribution and computing the pressure gradient as part of the solution. Calculated results are presented for a number of separated flows and the accuracy of the method is verified.

A note on the stress singularity at a non-uniformly moving crack tip

Abstract: The stress-singularity at a crack tip moving arbitrarily in an elastic plate under plane strain conditions is investigated. By formulating the wave-equations in a polar coordinate system attached to the crack-tip, it is found by an asymptotic analysis that the angle-dependence of the singularity is only dependent on the instantaneous cracktip velocity. This result is used to derive a relation between the dynamic stress-intensity factor and the energyrelease rate.

Virtual Singularities in the Artificial Satellite Theory

Abstract: When the coordinate system used in perturbation theory presents a geometrical singularity and when the perturbation technique fails to take account of this, the solution developed may present singularities which are no longer easily explained by purely geometrical means. These singularities have been calledvirtual singularities by Deprit and Rom (1970). We propose to demonstrate that virtual singularities can in general be avoided by the use of Lie transforms. In general, it is sufficient to recognize that the original Hamiltonian function presents the d'Alembert characteristic with respect to pairs of action-angle variables and that the averaging operations preserve this characteristic. We then apply this criterion to the artificial satellite theory (for small to moderate eccentricity) showing that all of three possible virtual singularities can be avoided at the same time. A new set of elliptic elements, well suited to the problem at hand, is introduced.

A bound for the fixed-point index of an area-preserving map with applications to mechanics

Abstract: Carl P. Simon (Ann Arbor) Area-preserving maps and flows play an essential role in the study of motions of mechanical systems, especially in celestial mechanics (see [1, 14]). Since one is often interested in the behavior of an area-preserving map around a fixed point and in the number and type of critical points and periodic orbits of an area-preserving flow, the ./i'xed-point index of a map and the index oj" a singularity or a closed orbit of a flow can yield much information about the map or flow. For example, the fact that the index of an isolated singularity of an area-preserving flow can never be greater than +1 has aided in setting a lower bound for the number of stationary points of certain area-preserving flows. It has also been a useful necessary condition for a flow to be area-preserving. For these reasons, the conjecture that the fixed-point index of an area-preserving homeomorphism of a 2- manifold is always less than or equal to + 1 has drawn attention. In this paper, we answer this conjecture in the affirmative for smooth maps and then put this bound to work to show that certain maps must have at least two fixed points and certain flows at least two periodic orbits. An important application is the following generalization of a famous theorem of Liapunov: a Hamiltonian vectorfield on M 4 must have two distinct one-parameter families of periodic orbits around a non-degenerate minimum (or maximum) of the Hamiltonian, even when the pure imaginary characteristic exponents are in resonance.

A separation of the logarithmic singularity in the exact kernel of the cylinderical antenna integral equation

Abstract: The well-known logarithmic behavior of the singular "exact" kernel of cylindrical antenna integral equations is explicitly separated. The elliptic integral partitioning of the kernel by Schelkunoff is the point of departure in the development. The result is an expression for the kernel comprising the logarithmic term and a well-behaved economically computable residual term. This result is readily amenable to numerical solutions of integral equations of cylindrical geometries. Numerical results obtained from method-of-moments solutions to cylindrical antennas are given to verify the result.

The inviscid nonlinear instability of parallel shear flows

Abstract: In this paper we assume the existence of a nonlinear boundary layer centred on the critical point, and explore its effect on the development of unstable parallel shear flows. A velocity matching condition derived in a qualitative discussion suggests a growth of harmonics which differs from that predicted by previous theories; however, the prediction is in excellent agreement with experimental data. A hyperbolic-tangent velocity profile, subjected to perturbations with wavenumbers and frequencies close to marginal values, is then chosen as a mathematical model of the nonlinear development, both temporal and spatial instability growth being considered.A singularity in the analysis which has been treated in previous theories by the introduction of viscosity is dealt with in the present work by the introduction of a growth boundary layer. The asymptotics are non-uniform and the time-dependent solution does not resemble the steady viscous solutions, even as the growth rate tends to zero. The theory suggests that the instability will develop as a series of temporally growing spiral vortices, a description differing from that of a cat's-eye pattern predicted by existing theories, but in accord with experimental and field observations.

On the Connectedness Structure of the Coulomb S-Matrix*

Abstract: The forward direction singularity of the non-relativistic Coulomb S-matrix is examined and discussed. The relativistic Coulomb S-matrix to order e is shown to have a similar singularity.

Oscillatory supersonic kernel function method for interfering surfaces

Abstract: In the method presented in this paper, a collocation technique is used with the nonplanar supersonic kernel function to solve multiple lifting surface problems with interference in steady or oscillatory flow. The pressure functions used are based on conical flow theory solutions and provide faster solution convergence than is possible with conventional functions. In the application of the nonplanar supersonic kernel function, an improper integral of a 3/2 power singularity along the Mach hyperbola is described and treated. The method is compared with other theories and experiment for two wing-tail configurations in steady and oscillatory flow.

Singularities in Weyl gravitational fields

Abstract: The peculiarities of the scalarS ≡RijklRijkl are exhibited for two axially-symmetric static (Weyl) gravitational fields. By examiningS along curved families of trajectories to the Weyl singularities, examples are found which contradict previous claims by Gautreau and Anderson regarding ‘directional singularities’. Proper circumferences about the Bach and Weyl line-mass singularity are also examined. There is no apparent correlation between the source structure and the behaviour ofS from this analysis.

Tachyon singularity: A spacelike counterpart of the Schwarzschild black hole

Abstract: A Schwarzschild-like metric is presented for the region around a singularity associated with a spacelike world-line (tachyon singularity). Kruskal-type extensions of the metric are provided. The form of the metric suggests a qualitative explanation of tachyon motion (v>c) that does not involve transport of energy or information at speeds greater than the velocity of light. The extended metrics show both normal and time-reversed gravitational Cerenkov cones of high space curvature. The solutions correspond to half-advanced, half-retarded fields in accord with the Wheeler-Feynman absorber theory.