About singularities at angular points of a trailing edge under the Joukowskii–Kutta Condition

The linear problem for the velocity potential around a slightly curved thin finite wing is considered under the Joukowskii–Kutta hypothesis. The exponents of possible singularities of solutions at angular points on wing's trailing edge are expressed in terms of eigenvalues of mixed boundary value problems for the Beltrami–Laplace operator on the hemisphere and the semicircle. These singularities have a structure such that the circulation function turns out to be continuous in interior angular points of the trailing edge. In the case of trapezoidal shape of the wing ends there occur square-root singularities of the velocity field at the trailing edge endpoints and the same singularities, of course, are extended along the lateral sides of the wake behind the wing. It is proved that for any angular point on the trailing edge the exponents of all above-mentioned singularities form a countable set in the upper complex half-plane with the only accumulation point at infinity. © 1998 B. G. Teubner Stuttgart—John Wiley & Sons, Ltd.     

Formation of singularities of solutions of the equations of motion of compressible fluids subjected to external forces in the case of several spatial variables

One considers a class of solutions with finite total energy and moment of inertia for the equations of motion of compressible fluids. It is shown that for a wide class of right-hand sides, including the viscosity term, initially smooth solutions may acquire singularities on a finite time interval. A sufficient condition for the appearance of singularities is found. This condition may be called “the best possible sufficient condition” in the sense that one can explicitly construct a time-global smooth solution for which this condition does not hold to within arbitrary infinitely small quantities. For a nontrivial constant state, perturbations with compact support are considered. A generalization is proved for the known theorem on the initial conditions for which the solution acquires singularities on a finite time interval. The effect of dry friction and rotation on the formation of singularities of smooth solutions is examined. __________ Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 26, pp. 274–308, 2007.     

On the search of supersingularities,more severe than a common crack singularity

Stress singularities at two-dimensional multi-material-junctions, consisting of dissimilar, homogeneous, isotropic and linear-elastic wedges under a plane strain state are considered. The stresses formed at the vertex of this multi-material composite situation are analyzed by complex variable method, based on a choice of the Kolosov-potentials which are applicable in the vicinity of the vertex. The characteristic equation for the order of the stress singularities is available in a closed-form analytical manner and investigated numerically. By varying parameters, like the stiffness ratio of one material with respect to the remaining stiffnesses or the angle between the involved sectors of the junction, useful information about the variation of the order of the stress singularities is provided. A particular outcome of the performed investigations is that there are many situations where supersingularities occur, i.e. singularities more severe than those due to a crack inside a homogeneous material. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the Singularities of Fractionalized Potential Functions

The singularities of harmonic functions generated under the action of an integral transform with a fractionalized kernel are studied. A simple relationship arises connecting pairs of singularities.     

Infinitesimally stable and unstable singularities of 2-degrees of freedom completely integrable systems

In this article we give a list of 10 rank zero and 6 rank one singularities of 2-degrees of freedom completely integrable systems. Among such singularities, 14 are the singularities that satisfy a non-vanishing condition on the quadratic part, the remaining 2 are rank 1 singularities that play a role in the geometry of completely integrable systems with fractional monodromy. We describe which of them are stable and which are unstable under infinitesimal completely integrable deformations of the system.      

Singularities of solutions to linear,second order analytic elliptic equations in two independent variables I. The completely regular boundary

Two methods are described for the a priori location of singularities of solutions to exterior boundary value problems. One uses an expansion for the solution in a circle centered on a regular exterior point P. A singularity lies on the circle of convergence. The envelope of these circles, generated as P makes a circuit about the closed boundary, circumscribes the singularities. The radius of convergence depends on singularities of the solution u(s) and its normal derivative v(s) on the boundary. The second method employs complex characteristics to relate singularities of the boundary data to real singularities of the solution. Integral equations connecting (y), v(s) and the analytic boundary condition are used to continue the data into the complex s-plane and to locate their singularities. Explicit solution of the integral equations is unnecessary; some nonlinear boundary conditions can be handled.     

The hp-Version of the Boundary Element Method in ℝ3 The Basic Approximation Results

This paper deals with the basic approximation properties of the h–p version of the boundary element method (BEM) in ℝ³. We extend the results on the exponential convergence of the h–p version of the boundary element method on geometric meshes from problems in polygonal domains to problems in polyhedral domains. In 2D elliptic boundary value problems the solutions have only corner singularities whereas in 3D problems they contain additional edge and corner-edge singularities. The solutions of the corresponding boundary integral equations inherit those singularities. The detailed investigations in our analysis take care of the various types of those singularities. While edge singularities can be analysed using standard one-dimensional approximation results the corner-edge singularities demand a new analysis. © 1997 by B. G. Teubner Stuttgart–John Wiley & Sons Ltd.     

Space-Time Foam Dense Singularities and de Rham Cohomology

In an earlier paper of the authors, it was shown that the sheaf theoretically based recently developed abstract differential geometry of the first author can, in an easy and natural manner, incorporate singularities on arbitrary closed nowhere dense sets in Euclidean spaces, singularities which therefore can have arbitrary large positive Lebesgue measure. As also shown, one can construct in such a singular context a de Rham cohomology, as well as a short exponential sequence, both of which are fundamental in differential geometry. In this paper, these results are significantly strengthened, motivated by the so-called space-time foam structures in general relativity, where singularities can be dense. In fact, this time one can deal with singularities on arbitrary sets, provided that their complementaries are dense, as well. In particular, the cardinal of the set of singularities can be larger than that of the nonsingular points.     

On corner singularities in Reissner-Mindlin's theory of elastic plates

In the framework of linear elasticity, singularities occur in domains with non-smooth boundaries. Particularly in Fracture Mechanics, the local stress field near stress concentrations is of interest. In this work, singularities at re-entrant corners or sharp notches in Reissner-Mindlin plates are studied. Therefore, an asymptotic solution of the governing system of partial differential equations is obtained by using a complex potential approach which allows for an efficient calculation of the singularity exponent λ. The effect of the notch opening angle and the boundary conditions on the singularity exponent is discussed. The results show, that it can be distinguished between singularities for symmetric and antisymmetric loading and between singularities of the bending moments and the transverse shear forces. Also, stronger singularities than the classical crack tip singularity with free crack faces are observed. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Invariance of the cone algebra without asymptotics

Let B be a manifold with conical singularities, and denote by the smooth bounded manifold with cylindrical ends obtained by blowing up near the singularities.B.-W. Schulze has developed a framework for a pseudodifferential calculus on B by defining various classes of distribution spaces and operator algebras, working in fixed coordinates on the manifold . I am showing here that the Mellin Sobolev spaces without asymptotics, the cone algebra without asymptotics, and its ideal of smoothing operators are independent of the choice of coordinates and therefore may be considered intrinsic objects for manifolds with conical singularities.     

Resolution of Corank 1 Singularities of a Generic Front

We construct a resolution of singularities for wave fronts having only stable singularities of corank 1. It is based on a transformation that takes a given front to a new front with singularities of the same type in a space of smaller dimension. This transformation is defined by the class A_µ of Legendre singularities. The front and the ambient space obtained by the A_µ-transformation inherit topological information on the closure of the manifold of singularities A_µ of the original front. The resolution of every (reducible) singularity of a front is determined by a suitable iteration of A_µ-transformations. As a corollary, we obtain new conditions for the coexistence of singularities of generic fronts.     

Boundary singularities with a simple decomposition

We define the decomposition of a boundary singularity as a pair (a singularity in the ambient space together with a singularity of the restriction to the boundary). We prove that the Lagrange transform is an involution on the set of boundary singularities that interchanges the singularities that occur in the decomposition of a boundary singularity. We classify the boundary singularities for which both of these singularities are simple. Bibliography: 8 titles.Translated fromTrudy Seminara imeni I. G. Petrovskogo, No. 15, pp. 55–69, 1991.     

A Direct Proof that Solutions of the Six Painlevé Equations Have No Movable Singularities Except Poles

The Painlevé property of an nth-order differential equation is that no solution has any movable singularities other than poles. This property is strongly indicative of complete integrability (the existence of n ? 1 integrals). However, the usual technique employed to test for the Painlevé property seeks only movable algebraic (or logarithmic) singularities. More general singularities are ignored. But, the six standard Painlevé equations are known to have no such singularities. Painlevé's proof of this is long and laborious; we give here a direct proof.     

A complete stability theorem for foliations with singularities

We study codimension one smooth foliations with singularities on closed manifolds. We assume that the singularities are nondegenerate (of Bott-Morse type) in the sense of Scárdua and Seade (2009) [9] and prove a version of Thurston-Reeb stability theorem in terms of a component of the singular set: If all singularities are of center type and the foliation exhibits a compact leaf with trivial Cohomology group of degree one or a codimension ?3 component of the singular set with trivial Cohomology group of degree one then the foliation is compact and stable.     

Detection of the singularities of a complex function by numerical approximations of its Laurent coefficients

For several applications, it is important to know the location of the singularities of a complex function: just for example, the rightmost singularity of a Laplace Transform is related to the exponential order of its inverse function. We discuss a numerical method to approximate, within an input accuracy tolerance, a finite sequence of Laurent coefficients of a function by means of the Discrete Fourier Transform (DFT) of its samples along an input circle. The circle may also enclose some singularities, since the method works with the Laurent expansion. The DFT is computed by the FFT algorithm so that, from a computational point of view, the efficiency is guaranteed. The function samples may be obtained by solving a numerical problem such as, for example, a differential problem. We derive, as consequences of the method, some new outcomes able to detect those singularities which are close to the circle and to discover if the singularities are all external or internal to the circle so that the Laurent expansion reduces to its regular or singular part, respectively. Other singularities may be located by means of a repeated application of the method, as well as an analytic continuation. Some examples and results, obtained by a first implementation, are reported.     

The Hilbert scheme of points for supersingular abelian surfaces

We study the geometry of Hilbert schemes of points on abelian surfaces and Beauville’s generalized Kummer varieties in positive characteristics. The main result is that, in characteristic two, the addition map from the Hilbert scheme of two points to the abelian surface is a quasifibration such that all fibers are nonsmooth. In particular, the corresponding generalized Kummer surface is nonsmooth, and minimally elliptic singularities occur in the supersingular case. We unravel the structure of the singularities in dependence of p-rank and a-number of the abelian surface. To do so, we establish a McKay Correspondence for Artin’s wild involutions on surfaces. Along the line, we find examples of canonical singularities that are not rational singularities.     

Saddle singularities of complexity 1 of integrable Hamiltonian systems

Properties of saddle singularities of rank 0 and complexity 1 for integrable Hamiltonian systems are studied. An invariant (f _n -graph) solving the problem of semi-local classification of saddle singularities of rank 0 for an arbitrary complexity was constructed earlier by the author. In this paper, a more simple form of the invariant for singularities of complexity 1 is obtained and some properties of such singularities are described in algebraic terms. In addition, the paper contains a list of saddle singularities of complexity 1 for systems with three degrees of freedom.     

Singularities of the Minkowski set and affine equidistants for a curve and a surface

Generic singularities of envelopes of families of chords and bifurcations of affine equidistants defined by a pair of a curve and a surface in R³ are classified. The chords join pairs of points of the curve and the surface such that the tangent line to the curve is parallel to the tangent plane to the surface. The classification contains singularities of stable Lagrange and Legendre projections, boundary singularities and some less known classes appearing at the points of the surface and the curve themselves.     

A wavelet method for solving singular integral equation of MHD

This paper presents Haar wavelet approximation to solve a singular integral equation which has singularities on a diagonal of the domain R=[-1,1]×[-1,1]. The singularities arise basically due to modified Bessel function K₀ which appears as a part of the kernel. Thus the integral equation is weakly (logarithmically) singular only. The problem is solved considering all the singularities of the kernel and results are examined for approximations of different orders. Our interest to solve the problem using Haar wavelet basis is due to its simplicity and efficiency in numerical approximation. The results show convergence trend as mesh is refined. Comparison is made with some available results obtained earlier by partial consideration of the singularities.