Complex Cauchy Problems with Intersecting Singularities

We consider linear Cauchy problems of order two in a complex domain. We assume that the initial values have singularities along a family of hypersurfaces, which cross pairwise transversally along a single intersection. We study the propagation of the singularities of the solution. We show that the solution may have anomalous singularities, and study the monodromy of the solution.     

Heat flow for p-harmonic maps with small initial data

We study developing singularities for surfaces of rotation with free boundaries and evolving under volume-preserving mean curvature flow. We show that singularities form a finite, discrete set along the axis of rotation. We prove a monotonicity formula and conclude that type I singularities are asymtotically cylindrical.     

Timelike Constant Mean Curvature Surfaces with Singularities

We use integrable systems techniques to study the singularities of timelike non-minimal constant mean curvature (CMC) surfaces in the Lorentz–Minkowski 3-space. The singularities arise at the boundary of the Birkhoff big cell of the loop group involved. We examine the behavior of the surfaces at the big cell boundary, generalize the definition of CMC surfaces to include those with finite, generic singularities, and show how to construct surfaces with prescribed singularities by solving a singular geometric Cauchy problem. The solution shows that the generic singularities of the generalized surfaces are cuspidal edges, swallowtails, and cuspidal cross caps.     

Singularity spectrum of multifractal functions involving oscillating singularities

We give general mathematical results concerning oscillating singularities and we study examples of functions composed only of oscillating singularities. These functions are defined by explicit coefficients on an orthonormal wavelet basis. We compute their Hölder regularity and oscillation at every point and we deduce their spectrum of oscillating singularities.     

The heat equation and reflected Brownian motion in time-dependent domains.: II. Singularities of solutions

We study the space-time Brownian motion and the heat equation in non-cylindrical domains. The paper is mostly devoted to singularities of the heat equation near rough points of the boundary. Two types of singularities are identified—heat atoms and heat singularities. A number of explicit geometric conditions are given for the existence of singularities. Other properties of the heat equation solutions are analyzed as well.     

Maximal modifications and Auslander–Reiten duality for non-isolated singularities

We first generalize classical Auslander–Reiten duality for isolated singularities to cover singularities with a one-dimensional singular locus. We then define the notion of CT modules for non-isolated singularities and we show that these are intimately related to noncommutative crepant resolutions (NCCRs). When R has isolated singularities, CT modules recover the classical notion of cluster tilting modules but in general the two concepts differ. Then, wanting to generalize the notion of NCCRs to cover partial resolutions of \(\operatorname{Spec}R\) , in the main body of this paper we introduce a theory of modifying and maximal modifying modules. Under mild assumptions all the corresponding endomorphism algebras of the maximal modifying modules for three-dimensional Gorenstein rings are shown to be derived equivalent. We then develop a theory of mutation for modifying modules which is similar but different to mutations arising in cluster tilting theory. Our mutation works in arbitrary dimension, and in dimension three the behavior of our mutation strongly depends on whether a certain factor algebra is artinian.     

RECOVERY OF THE SINGULARITIES OF A POTENTIAL FROM FIXED ANGLE SCATTERING DATA

We study the inverse scattering problem for Schroedinger equation. We prove that for non-smooth potential the main singularities of the potential are contained in the Born approximation which can be obtained from measurement of the scattering amplitude in a single outgoing direction. We measure singularities in the scale of Sobolev spaces.     

A Sub-Density Theorem of Sturm-Liouville Eigenvalue Problem with Finitely Many Singularities

We study the distribution of the Sturm-Liouville eigenvalues of a potential with finitely many singularities. There is an asymptotically periodical structure on this class of eigenvalues as described by the entire function theory. We describe the singularities of its potential function explicitly in its eigenvalue asymptotics.     

On weakly rational singularities in complex analytic geometry

Summary We study particular singularities of complex analytic spaces that we call weakly rational and that contain rational singularities. In fact, a weakly rational singularity is rational if and only if it is Cohen-Macauley. Invariance under morphisms and deformations of weakly rational singularities is also studied.Partially supported by C.N.R.     

On singularities of the Galilean spherical darboux ruled surface of a space curve in <Emphasis Type="Italic">G</Emphasis><Subscript>3</Subscript>

We study the singularities of Galilean height functions intrinsically related to the Frenet frame along a curve embedded into the Galilean space. We establish the relationships between the singularities of the discriminant and the sets of bifurcations of the function and geometric invariants of curves in the Galilean space.     

Contact points and fractional singularities for semigroups of holomorphic self-maps of the unit disc

We study boundary singularities which can appear for infinitesimal generators of one-parameter semigroups of holomorphic self-maps of the unit disc. We introduce “regular” fractional singularities and characterize them in terms of the behavior of the associated semigroups and K?nigs functions. We also provide necessary and sufficient geometric criteria on the shape of the image of the K?nigs function for having such singularities. In order to do this, we study contact points of semigroups and prove that any contact (not fixed) point of a one-parameter semigroup corresponds to a maximal arc on the boundary to which the associated infinitesimal generator extends holomorphically as a vector field tangent to this arc.     

Finite time singularities in a 1D model of the quasi-geostrophic equation

In this paper we study 1D equations with nonlocal flux. These models have resemblance of the 2D quasi-geostrophic equation. We show the existence of singularities in finite time and construct explicit solutions to the equations where the singularities formed are shocks. For the critical viscosity case we show formation of singularities and global existence of solutions for small initial data.     

Generalized surfaces with constant <Emphasis Type="Italic">H</Emphasis>/<Emphasis Type="Italic">K</Emphasis> in Euclidean three-space

Surfaces in Euclidean three-space with constant ratio of mean curvature to Gauss curvature arise naturally as the parallel surfaces to minimal surfaces. They might possess singularities which occur naturally as focal points of minimal surfaces. We study geometric properties and the singularities of such surfaces, prove some global results about them, and provide a Björling formula to construct such surfaces with prescribed point or curve singularities.     

Singularités de solutions d'équations aux dérivées partielles

We develop the general mathematical setting necessary to study the singularities of local solutions of the quasi-linear first-order systems of PDEs with a free initial condition. In the good cases, it is possible to describe these singularities as a function of the free initial conditions satisfied by the solutions. Using the transversality theorems, it is then possible to describe the singularities of generic solutions, and of generic families of solutions under deformation of the initial conditions. We apply this study by giving classifications of an important classe of hyperbolic quasi-linear first-order systems in the plane, the reducible systems, and of an almost general class of hyperbolic quasi-linear second-order equations in the plane.     

Some properties of Malgrange isomonodromic deformations of linear 2 × 2 systems

We study movable singularities of the Malgrange isomonodromic deformation of a linear differential 2 × 2 system with two irregular singularities of Poincaré rank 1 and with an arbitrary number of Fuchsian singular points.     

Semiconcave solutions of partial differential inclusions

We obtain new results on the propagation of singularities for semiconcave solutions of partial differential inclusions. These results will be used to study the behavior of singularities of the value function for a reflected control problem.     

Equisingular Deformations of Sandwiched Singularities

We relate the equisingular deformation theory of plane curve singularities and sandwiched surface singularities. We show the existence of a smooth map between the two corresponding deformation functors and study the kernel of this map. In particular we show that the map is an isomorphism when a certain invariant is large enough.     

Generic singular configurations of linkages

We study the topological and differentiable singularities of the configuration space C(Γ) of a mechanical linkage Γ in Rd, defining an inductive sufficient condition to determine when a configuration is singular. We show that this condition holds for generic singularities, provide a mechanical interpretation, and give an example of a type of mechanism for which this criterion identifies all singularities.     

On Auslander modules of normal surface singularities

Abstract We study thefundamental sequences of normal surface singularities. Our main result asserts that for rational singularities (with a technical side-condition) and for minimally elliptic singularities the middle termA, theAuslander module, is isomorphic to the module of Zariski differentials if and only if the singularity is quasihomogeneous.