A generalization of the Dotsenko-Fateev integral

The Dotsenko-Fateev integral, an analytic function of one complex variable arising in conformal field theory, is generalized in a natural way to an analytic function of two complex variables. A system of partial differential equations and a Pfaffian system of Fuchsian type are derived for this generalized Dotsenko- Fateev integral. The Fuchsian system permits to obtain local expansions of solutions in the neighborhoods of singularities of the system.     

Quasihomogene Singularitäten algebraischer Kurven

In this note we consider algebraic curves X over an algebraically closed field k of characteristic O. We give some characterizations in terms of differentials of quasihomogeneous curve singularities, similar to well-known characterizations of quasihomogeneous isolated singularities of hypersurfaces (see [2] and the literature quoted there). Moreover, we discuss the structure of the completion Ô_X of the local ring in such singularities. In case X has at most two analytic branches in a quasihomogeneous singularity a classification of the k-algebras Ô_X up to isomorphism can be given.     

Log canonical foliation singularities on surfaces

We give a classification of the dual graphs of the exceptional divisors on the minimal resolutions of log canonical foliation singularities on surfaces. As an application, we show the set of foliated minimal log discrepancies for foliated surface triples satisfies the ascending chain condition and a Grauert–Riemenschneider–type vanishing theorem for foliated surfaces with special log canonical foliation singularities.     

On analytic primals

Summary Cartain standard forms of equations of primals with singularities of a given type are investigated. Also the relation between the geometric and analytic case is given and some analytic invariants are found. The application of these methods to the resolution of singularities is indicated.     

Recurrence Criteria for Generalized Dirichlet Forms

We develop sufficient analytic conditions for recurrence and transience of non-sectorial perturbations of possibly non-symmetric Dirichlet forms on a general state space. These form an important subclass of generalized Dirichlet forms which were introduced in Stannat (Ann Scuola Norm Sup Pisa Cl Sci (4) 28(1):99–140, 1999). In case there exists an associated process, we show how the analytic conditions imply recurrence and transience in the classical probabilistic sense. As an application, we consider a generalized Dirichlet form given on a closed or open subset of \(\mathbb {R}^d\) which is given as a divergence free first-order perturbation of a non-symmetric energy form. Then, using volume growth conditions of the sectorial and non-sectorial first-order part, we derive an explicit criterion for recurrence. Moreover, we present concrete examples with applications to Muckenhoupt weights and counterexamples. The counterexamples show that the non-sectorial case differs qualitatively from the symmetric or non-symmetric sectorial case. Namely, we make the observation that one of the main criteria for recurrence in these cases fails to be true for generalized Dirichlet forms.     

Geometric bases and topological equivalence

There are many algebraic and topological invariants associated to a singular point of a complex analytic function. The intent here is to discuss some of these invariants and the topological classification of singularities. Specifically, we establish that the topological type is determined by the Lefschetz vanishing cycles obtained by unfolding the singularity and certain local monodromy operators defined by Gabrielov. In Brieskorn's terminology singularities with the same geometric bases are topologically indistinguishable. Thus the higher invariants in the hierarchy of Brieskorn are necessary to understand the geometry of higher singularities. As a corollary to our main theorem, we obtain the result of Lê-Ramanujam which states that the topological type is constant in a oneparameter family of singularities with constant Milnor number.     

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.     

Additive blending of local approximations into a globally-valid approximation with application to the dilogarithm

We show how local approximations, each accurate on a subinterval, can be blended together to form a global approximation which is accurate over the entire interval. The blending functions are smoothed approximations to a step function, constructed using the error function. The local approximations may be power series, asymptotic expansion, or other more exotic species. As an example, for the dilogarithm function, we construct a one-line analytic approximation which is accurate to one part in 700. This can be generalized to higher order merely by adding more terms in the local approximations. We also show the failure of the alternative strategy of subtracting singularities.     

The asymptotic blow-up of a surface in Euclidean 3-space

Given a smoothly immersed surface in Euclidean (or affine) 3-space, the asymptotic directions define a subset in the Grassmann bundle of unoriented one-dimensional subspaces over the surface. This links the Euler characteristic of the region where the Gauss curvature is nonpositive with the index of singularities in a natural line field defined on this subset. To apply this we need only identify mechanisms which restrict the index of the singularities. In Section 2.1 we show that specific configurations of nonpositive Gauss curvature cannot be realized by an immersed surface and that specific configurations in 2-sphere cannot be realized as Gauss images of surfaces. In Section 2.2 we prove an existence theorem for surfaces which satisfy regularity conditions and a Symplectic Monge Ampere PDE. In general, a PDE of this type will not restrict the indices of the singularities over a solution. However, we show that over a surface of nonzero constant mean curvature the indices are restricted and, hence, that specific configurations of nonpositive Gauss curvature cannot be realized by a constant mean curvature surface.     

Thom polynomials, symmetries and incidences of singularities

As an application of the generalized Pontryagin-Thom construction [RSz] here we introduce a new method to compute cohomological obstructions of removing singularities — i.e. Thom polynomials [T]. With the aid of this method we compute some sample results, such as the Thom polynomials associated to all stable singularities of codimension ≤8 between equal dimensional manifolds, and some other Thom polynomials associated to singularities of maps N ⁿ ?P ^n+k for k>0. We also give an application by reproving a weak form of the multiple point formulas of Herbert and Ronga ([H], [Ro2]). As a byproduct of the theory we define the incidence class of singularities, which — the author believes — may turn to be an interesting, useful and simple tool to study incidences of singularities. Oblatum 4-II-1999 & 19-VII-2000?Published online: 30 October 2000     

Space-Time Foam Differential Algebras of Generalized Functions and a Global Cauchy-Kovalevskaia Theorem

The new global version of the Cauchy-Kovalevskaia theorem presented here is a strengthening and extension of the regularity of similar global solutions obtained earlier by the author. Recently the space-time foam differential algebras of generalized functions with dense singularities were introduced. A main motivation for these algebras comes from the so called space-time foam structures in General Relativity, where the set of singularities can be dense. A variety of applications of these algebras have been presented elsewhere, including in de Rham cohomology, Abstract Differential Geometry, Quantum Gravity, etc. Here a global Cauchy-Kovalevskaia theorem is presented for arbitrary analytic nonlinear systems of PDEs. The respective global generalized solutions are analytic on the whole of the domain of the equations considered, except for singularity sets which are closed and nowhere dense, and which upon convenience can be chosen to have zero Lebesgue measure. In view of the severe limitations due to the polynomial type growth conditions in the definition of Colombeau algebras, the class of singularities such algebras can deal with is considerably limited. Consequently, in such algebras one cannot even formulate, let alone obtain, the global version of the Cauchy-Kovalevskaia theorem presented in this paper.     

A new approach to center conditions for simple analytic monodromic singularities

In this paper we present an alternative algorithm for computing Poincaré-Lyapunov constants of simple monodromic singularities of planar analytic vector fields based on the concept of inverse integrating factor. Simple monodromic singular points are those for which after performing the first (generalized) polar blow-up, there appear no singular points. In other words, the associated Poincaré return map is analytic. An improvement of the method determines a priori the minimum number of Poincaré-Lyapunov constants which must cancel to ensure that the monodromic singularity is in fact a center when the explicit Laurent series of an inverse integrating factor is known in (generalized) polar coordinates. Several examples show the usefulness of the method.     

Focus-center problem of planar degenerate system

In this paper we discuss how to determine a degenerate equilibrium of planar analytic systems to be focus-center type. A method of generalized normal sectors is used to determine orbits in exceptional directions near high degenerate equilibria. We obtain a sufficient and necessary condition for the existence of orbits going to origin in a generalized normal sector in class III. Thus, together with some criterions of orbits going to origin in a generic quasi-sector, we can characterize whether the degenerate equilibrium is of focus-center type in every case. The effectiveness of our methods is shown in an example which has a high degenerate equilibrium.     

Removable singularities of CR functions on singular boundaries

The problem of analytic representation of integrable CR functions on hypersurfaces with singularities is treated. The nature of singularities does not matter while the set of singularities has surface measure zero. For simple singularities like cuspidal points, edges, corners, etc., also the behaviour of representing analytic functions near singular points is studied. Received: 8 December 2000; in final form: 24 June 2001/Published online: 1 February 2002     

Isometry theorem for the Segal-Bargmann transform on a noncompact symmetric space of the complex type

We consider the Segal-Bargmann transform on a noncompact symmetric space of the complex type. We establish isometry and surjectivity theorems for the transform, in a form as parallel as possible to the results in the dual compact case. The isometry theorem involves integration over a tube of radius R in the complexification, followed by analytic continuation with respect to R. A cancellation of singularities allows the relevant integral to have a nonsingular extension to large R, even though the function being integrated has singularities.     

Stability in systems with aftereffect when there are singularities in the integral kernels

Systems with aftereffect are considered. The state of these systems is described by integrodifferential equations of the Volterra type, which depend on functionals in integral form and, in particular, on analytic functionals which are represented by Frechet series. The integral kernels can allow of singularities of Abel kernel singularities. The total stability (i.e. stability under persistent disturbances) is investigated, and the structure of the general solution is investigated in the neighbourhood of zero for an equation with a holomorphic non-linearity assuming asymptotic stability of the trivial solution of the linearized unperturbed equation. The conditions for instability are given in the critical case of a single zero root, which generalise results obtained previously.     

Maxwell’s equations,the Euler index,and Morse theory

We show that the singularities of the Fresnel surface for Maxwell’s equation on an anisotrpic material can be accounted from purely topological considerations. The importance of these singularities is that they explain the phenomenon of conical refraction predicted by Hamilton. We show how to desingularise the Fresnel surface, which will allow us to use Morse theory to find lower bounds for the number of critical wave velocities inside the material under consideration. Finally, we propose a program to generalise the results obtained to the general case of hyperbolic differential operators on differentiable bundles.     

On a method of calculation of hypersingular integrals

We propose a method for calculation of one-dimensional and two-dimensional hypersingular integrals, which allows to obtain a solution in the analytic form in a series of cases. Based on this method, we construct an approximate method for calculation of one-dimensional and twodimensional hypersingular integrals having singularities on rather complicated manifolds. We give estimates of errors of suggested algorithms.     

On analytic solutions to the generalized Cauchy problem with data on three surfaces for a quasilinear system

The generalized Cauchy problem with data on three surfaces is under consideration for a quasilinear analytic system of the third order. Under some simplifying assumption, we find necessary and sufficient conditions for existence of a solution in the form of triple series in the powers of the independent variables. We obtain convenient sufficient conditions under which the data of the generalized Cauchy problem has a unique locally analytic solution. We give counterexamples demonstrating that in the case we study it is impossible to state necessary and sufficient conditions for analytic solvability of the generalized Cauchy problem. We also show that the analytic solution can fail to exist even if the generalized Cauchy problem with data on three surfaces has a formal solution since the series converge only at a sole point, the origin.     

Differential equations with meromorphic coefficients

The following problems of the analytic theory of differential equations are considered: Hilbert’s 21st problem for Fuchsian systems of linear differential equations, the Birkhoff normal form problem for systems of linear differential equations with irregular singularities, and the classification problem for isomonodromic deformations of Fuchsian systems.