Transcendental lattices of some K 3‐surfaces

In a previous paper, [12], we described six families of K 3‐surfaces (over ?) with Picard‐number 19, and we identified surfaces with Picard‐number 20. In these notes we classify some of the surfaces by computing their transcendental lattices. Moreover, we show that the surfaces with Picard‐number 19 are birational to a Kummer surface which is the quotient of a non‐product type abelian surface by an involution. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Bubbling of the heat flows for harmonic maps from surfaces

In this article we prove that any Palais-Smale sequence of the energy functional on surfaces with uniformly L²-bounded tension fields converges pointwise, by taking a subsequence if necessary, to a map from connected (possibly singular) surfaces, which consist of the original surfaces and finitely many bubble trees. We therefore get the corresponding results about how the solutions of heat flow for harmonic maps from surfaces form singularities at infinite time. © 1997 John Wiley & Sons, Inc.     

Biconservative surfaces in BCV‐spaces

Biconservative hypersurfaces are hypersurfaces with conservative stress‐energy tensor with respect to the bienergy functional, and form a geometrically interesting family which includes that of biharmonic hypersurfaces. In this paper we study biconservative surfaces in the 3‐dimensional Bianchi–Cartan–Vranceanu spaces, obtaining their characterization in the following cases: when they form a constant angle with the Hopf vector field; when they are SO(2)‐invariant.     

Deformation of a singularity of type E8 and Mordell‐Weil lattices in characteristic 2

In this paper, we study the Mordell‐Weil lattices of the family of elliptic surfaces which is arising from the E₈⁴ singularity, one of the ADE singularities in characteristic 2. And we construct a subfamily of the universal family of supersingular K 3 surfaces in characteristic 2 as an application (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the boundedness of some potential‐type operators with oscillating kernels

We consider a class of multidimensional potential‐type operators with kernels that have singularities at the origin and on the unit sphere and that are oscillating at infinity. We describe some convex sets in the (1/p, 1/q)‐plane for which these operators are bounded from L_p into L_q and indicate domains where they are not bounded. We also reveal some effects which show that oscillation and singularities of the kernels may strongly influence on the picture of boundedness of the operators under consideration. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

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.     

Complete algebraic vector fields on affine surfaces,Part I

In this work we study the singularities at infinity of algebraic vector fields in dimension 2.These singularities will be classified under a mild assumption. The general problem is also reduced to the study of the combinatorics of certain resolutions which will be developed in Part II. Our main results are local and therefore can be carried over more general surfaces. Whereas we deal with ℂ -complete vector fields, the results also apply to ℝ-complete ones thanks to a theorem of Forstneric [7].     

Sequential stability of weak solutions in compressible self‐gravitating fluids and stationary problem

In this paper, we prove the sequential stability of weak solutions over time, in relation to the Navier–Stokes system of compressible self‐gravitating fluids in a three‐dimensional domain. As a byproduct, we show that there exists at least one non‐negative solution to the stationary problem in any bounded domain with a given mass for the adiabatic constant γ > 3 ∕ 2. In particular, for the spherically symmetric case, these conclusions still hold for γ > 4 ∕ 3 or γ = 4 ∕ 3 with a small mass. Copyright © 2012 John Wiley & Sons, Ltd.     

Affine normal surfaces with simply-connected smooth locus

Given a normal affine surface V defined over \mathbbC{\mathbb{C}}, we look for algebraic and topological conditions on V which imply that V is smooth or has at most rational singularities. The surfaces under consideration are algebraic quotients \mathbbCⁿ/G{\mathbb{C}^n/G} with an algebraic group action of G and topologically contractible surfaces. Theorem 3.6 can be considered as a global version of the well-known result of Mumford giving a smoothness criterion for a germ of a normal surface in terms of the local fundamental group.     

Detection of C2‐singularities using uniform spline approximations

For the detection of C²‐singularities, we present lower estimates for the error in Schoenberg variation‐diminishing spline approximation with equidistant knots in terms of the classical second‐order modulus of smoothness. To this end, we investigate the behaviour of the iterates of the Schoenberg operator. In addition, we show an upper bound of the second‐order derivative of these iterative approximations. Finally, we provide an example of how to detect singularities based on the decay rate of the approximation error.     

Implicitly enriched Galerkin methods for numerical solutions of fourth‐order partial differential equations containing singularities

Highlights are the following:

• For any integer , we construct ‐continuous partition of unity (PU) functions with flat‐top from B‐spline functions to have numerical solutions of fourth‐order equations with singularities. B‐spline functions are modified to satisfy clamped boundary conditions.
• To handle singularity arising in fourth‐order elliptic differential equations, these modified B‐spline functions are enriched either by introducing enrichment basis functions implicitly through particular geometric mappings or by adding singular basis functions explicitly.
• To show the effectiveness of the proposed implicit enrichment methods (mapping method), the accuracy, the number of degrees of freedom (DOF), and matrix condition numbers are computed and compared in the h‐refinement, the p‐refinement, and the k‐refinement of the approximation space of B‐spline basis functions.

Using Partition of unity (PU) functions with flat‐top, B‐spline functions are modified to satisfy boundary conditions of the fourth‐order equations. Since the standard isogeometric analysis (IGA) as well as the conventional FEM have limitations in handling fourth‐order differential equations containing singularities, we consider two enrichment methods (explicit and implicit) in the framework of the p‐, the k, and the h‐refinements of IGA. We demonstrate that both enrichment methods yield good approximate solutions, but explicit enrichment methods give large (almost singular) matrix condition numbers and face integrating singular functions. Because of these limitations of external enrichment methods, we extensively investigate implicit enrichment methods (mapping methods) that virtually convert fourth‐order elliptic problems with singularities to problems with no influence of the singularities. Effectiveness of the proposed mapping method extensively tested to one‐dimensional fourth‐order equation with singularities. The implicit enrichment (mapping) method is extended to the two‐dimensional cases and test it to fourth‐order partial differential equations on cracked domains.     

Regularity of area-minimizing surfaces in 3D polytopes and of invariant hypersurfaces inR n

In (the surface of) a convex polytope P ³ inR ⁴,an areaminimizing surface avoids the vertices of P and crosses the edges orthogonally. In a smooth Riemannian manifold M with a group of isometries G, an areaminimizing G-invariant oriented hypersurface is smooth (except for a very small singular set in high dimensions). Already in 3D, area-minimizing G-invariant unoriented surfaces can have certain singularities, such as three orthogonal sheets meeting at a point. We also treat other categories of surfaces such as rectifiable currents modulo v and soap films.     

Strengthened Cauchy–Bunyakowski–Schwarz inequality for a three‐dimensional elasticity system

The constant γ of the strengthened Cauchy–Bunyakowski–Schwarz (CBS) inequality plays a fundamental role in the convergence rate of multilevel iterative methods. The main purpose of this work is to give an estimate of the constant γ for a three‐dimensional elasticity system. The theoretical results obtained are practically important for the successful implementation of the finite element method to large‐scale modelling of complicated structures as they allow us to construct optimal order algebraic multilevel iterative solvers for a wide class of real‐life elasticity problems. Copyright © 2001 John Wiley & Sons, Ltd.     

Integral Menger curvature for surfaces

We develop the concept of integral Menger curvature for a large class of nonsmooth surfaces. We prove uniform Ahlfors regularity and a C1,λ-a priori bound for surfaces for which this functional is finite. In fact, it turns out that there is an explicit length scale R>0 which depends only on an upper bound E for the integral Menger curvature Mp(Σ) and the integrability exponent p, and not on the surface Σ itself; below that scale, each surface with energy smaller than E looks like a nearly flat disc with the amount of bending controlled by the (local) Mp-energy. Moreover, integral Menger curvature can be defined a priori for surfaces with self-intersections or branch points; we prove that a posteriori all such singularities are excluded for surfaces with finite integral Menger curvature. By means of slicing and iterative arguments we bootstrap the Hölder exponent λ up to the optimal one, λ=1−(8/p), thus establishing a new geometric ‘Morrey–Sobolev’ imbedding theorem.As two of the various possible variational applications we prove the existence of surfaces in given isotopy classes minimizing integral Menger curvature with a uniform bound on area, and of area minimizing surfaces subjected to a uniform bound on integral Menger curvature.     

2‐ and 3‐factors of graphs on surfaces

It has been conjectured that any 5‐connected graph embedded in a surface Σ with sufficiently large face‐width is hamiltonian. This conjecture was verified by Yu for the triangulation case, but it is still open in general. The conjecture is not true for 4‐connected graphs. In this article, we shall study the existence of 2‐ and 3‐factors in a graph embedded in a surface Σ. A hamiltonian cycle is a special case of a 2‐factor. Thus, it is quite natural to consider the existence of these factors. We give an evidence to the conjecture in a sense of the existence of a 2‐factor. In fact, we only need the 4‐connectivity with minimum degree at least 5. In addition, our face‐width condition is not huge. Specifically, we prove the following two results. Let G be a graph embedded in a surface Σ of Euler genus g.

• (1) If G is 4‐connected and minimum degree of G is at least 5, and furthermore, face‐width of G is at least 4g?12, then G has a 2‐factor.
• (2) If G is 5‐connected and face‐width of G is at least max{44g?117, 5}, then G has a 3‐factor.

The connectivity condition for both results are best possible. In addition, the face‐width conditions are necessary too. Copyright © 2010 Wiley Periodicals, Inc. J Graph Theory 67:306‐315, 2011     

On the normal bundle of curves on nodal quartic surfaces

Let k be an algebraically closed field, char k = 0. Let C be an irreducible nonsingular curve such that 2C = S ? F, where S and F are two surfaces and all the singularities of F are rational double points (if any). We prove that C can never pass through rational singularities of types A _2n n∈N, E₆ and E₈. We give conditions for C to pass through rational singularities of types. A _2k+1 k∈Z⁺ D_n n≥4 and E₇, (0.8).     

Inner metric geometry of complex algebraic surfaces with isolated singularities

We produce examples of complex algebraic surfaces with isolated singularities such that these singularities are not metrically conical, i.e., the germs of the surfaces near singular points are not bi‐Lipschitz equivalent, with respect to the inner metric, to cones. The technique used to prove the nonexistence of the metric conical structure is related to a development of metric homology. The class of the examples is rather large and includes some Kleinian singularities. © 2008 Wiley Periodicals, Inc.     

Natural hp‐BEM for the electric field integral equation with singular solutions

We apply the hp ‐version of the boundary element method (BEM) for the numerical solution of the electric field integral equation (EFIE) on a Lipschitz polyhedral surface Γ. The underlying meshes are supposed to be quasi‐uniform triangulations of Γ, and the approximations are based on either Raviart‐Thomas or Brezzi‐Douglas‐Marini families of surface elements. Nonsmoothness of Γ leads to singularities in the solution of the EFIE, severely affecting convergence rates of the BEM. However, the singular behavior of the solution can be explicitly specified using a finite set of functions (vertex‐, edge‐, and vertex‐edge singularities), which are the products of power functions and poly‐logarithmic terms. In this article, we use this fact to perform an a priori error analysis of the hp ‐BEM on quasi‐uniform meshes. We prove precise error estimates in terms of the polynomial degree p, the mesh size h, and the singularity exponents. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2012     

Singularities of Solutions to Quadratic Vector Equations on the Complex Upper Half‐Plane

Let S be a positivity‐preserving symmetric linear operator acting on bounded functions. The nonlinear equation with a parameter z in the complex upper half‐plane ? has a unique solution m with values in ?. We show that the z‐dependence of this solution can be represented as the Stieltjes transforms of a family of probability measures v on ?. Under suitable conditions on S , we show that v has a real analytic density apart from finitely many algebraic singularities of degree at most 3. Our motivation comes from large random matrices. The solution m determines the density of eigenvalues of two prominent matrix ensembles: (i) matrices with centered independent entries whose variances are given by S and (ii) matrices with correlated entries with a translation‐invariant correlation structure. Our analysis shows that the limiting eigenvalue density has only square root singularities or cubic root cusps; no other singularities occur.© 2016 Wiley Periodicals, Inc.