Partial resolutions of quotient singularities

For quotient singularities the irreducible components of the (reduced) base space of the versal deformation are in one to one correspondence with certain partial resolutions, calledP-resolutions [1]. In [3] we found allP-resolutions for cyclic quotient singularities. In this note we determine theP-resolutions for the other quotient singularities. A simple lemma allows reduction to the cyclic case; the same technique was already used in [3, Sect. 7] to study the dihedral singularities, so we are mainly concerned with the exceptional cases.     

Minimal singularities for representations of Dynkin quivers

We develop some reduction techniques for the study of singularities in orbit closures of finite dimensional modules. This enables us to classify all singularities occurring in minimal degenerations of representations of Dynkin quivers. They are all smoothly equivalent to the singularity at the zero-matrix inside thep×q-matrices of rank at most one.     

Schwarz-Christoffel mapping of multiply connected domains

A Schwarz-Christoffel mapping formula is established for polygonal domains of finite connectivitym≥2 thereby extending the results of Christoffel (1867) and Schwarz (1869) form=1 and Komatu (1945),m=2. A formula forf, the conformal map of the exterior ofm bounded disks to the exterior ofm bounded disjoint polygons, is derived. The derivation characterizes the global preSchwarzianf″ (z)/f′ (z) on the Riemann sphere in terms of its singularities on the sphere and its values on them boundary circles via the reflection principle and then identifies a singularity function with the same boundary behavior. The singularity function is constructed by a “method of images” infinite sequence of iterations of reflecting prevertex singularities from them boundary circles to the whole sphere.     

Error estimates for the combinedh andp versions of the finite element method

Summary In theh-version of the finite element method, convergence is achieved by refining the mesh while keeping the degree of the elements fixed. On the other hand, thep-version keeps the mesh fixed and increases the degree of the elements. In this paper, we prove estimates showing the simultaneous dependence of the order of approximation on both the element degrees and the mesh. In addition, it is shown that a proper design of the mesh and distribution of element degrees lead to a better than polynomial rate of convergence with respect to the number of degrees of freedom, even in the presence of corner singularities. Numerical results comparing theh-version,p-version, and combinedh-p-version for a one dimensional problem are presented.     

Algebraicity of Global Real Analytic Hypersurfaces

Let X be an algebraic manifold without compact component and let V be a compact coherent analytic hypersurface in X, with finite singular set. We prove that V is diffeotopic (in X) to an algebraic hypersurface in X if and only if the homology class represented by V is algebraic and singularities are locally analytically equivalent to Nash singularities. This allows us to construct algebraic hypersurfaces in X with prescribed Nash singularities.     

Instability of the wet x soap film

For idealized, infinitely thin (“dry”) soap films, an Xis unstable, while for very thick (“wet”) soap films it is minimizing. We show that for soap films of relatively small but positive wetness, the Xis unstable. Full stability diagrams for the constant liquid fraction case and the constant pressure case are generated. Analogous questions about other singularities remain controversial.     

On the cotangent cohomology of rational surface singularities with almost reduced fundamental cycle

We prove dimension formulas for the cotangent spaces T ¹ and T ² for a class of rational surface singularities by calculating a correction term in the general dimension formulas. We get that it is zero if the dual graph of the rational surface singularity X does not contain a particular type of configurations, and this generalizes a result of Theo de Jong stating that the correction term c (X ) is zero for rational determinantal surface singularities. In particular our result implies that c (X ) is zero for Riemenschneiders quasi‐determinantal rational surface singularities, and this also generalizes results for quotient singularities. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

K-theory of blow-ups and vector bundles on the cone over a surface

TheK-theory of the derived categories as well as someK-theoretic invariants associated to the resolution of singularities are applied in order to compute theK ₀-groups of a variety with finitely many singular points. Explicit computations are given in order to determine theK ₀ and the relative CH* groups of the affine cone over a nonsingular surface of degreed in ^d+1 not contained in any hyperplane.     

Combinatorial aspects of sequences of point blowing ups

We study morphisms given by composition of a sequence of point blow ups of smoothd-dimensional varieties in terms of combinatorial information coming from thed-ary intersection form on divisors with exceptional support. Some application to the classification of space curve singularities are derived. Supported by D.G.I.C.Y.T. PB91-0210-C02-01     

Microlocal energy methods and pseudo-differential operators

A functionk(x, u) of the(n+n)-variables is said to be a positive Hermite kernel if , and the matrix(k(x _i, x_j))_i,j is positive semi-definite for every integerN and everyx ₁, ..., x_N. In this paper, we prove that this positive structure can be microlocalized in the category of microfunctions. Further we obtain a useful theorem concerning the positivity of pseudo-differential operators. This theory will play important roles in the study of analytic singularities of solutions of boundary value problems.     

Corrections to fully developed turbulent spectra due to compressibility of the fluid

We construct the regular expansion at small compressibilities for the theory of fully developed turbulence of an isotropic homogeneous compressible fluid with MSR-type action. The parameter of the expansion is the Mach numberMa. For the inertial range of a compressible fluid, we study the infrared singularities determined by the transverse fields, which are used in the theory of incompressible fluids. These singularities are connected with the composite operators of transverse fields that are investigated by the quantum field renormalization group method. As a result, it is shown that the transverse fields induce scaling behavior with theMa scaling dimension equal to 1/3 (i.e.,Ma k^–1/3 is the dimensionless scaling parameter of the correlation functions in the inertial range).Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 106, No. 3, pp. 375–389, March, 1996.Translated by L. O. Chekhov.     

Nonfattening of Mean Curvature Flow at Singularities of Mean Convex Type

We show that a mean curvature flow starting from a compact, smoothly embedded hypersurface M ⊆ ℝ^{n + 1} remains unique past singularities, provided the singularities are of mean convex type, i.e., if around each singular point, the surface moves in one direction. Specifically, the level set flow of M does not fatten if all singularities are of mean convex type. We further show that assumptions of the theorem hold provided all blowup flows are of the kind appearing in a mean convex flow, i.e., smooth, multiplicity 1 , and convex. Our results generalize the well-known fact that the level set flow of a mean convex initial hypersurface M does not fatten. They also provide the first instance where nonfattening is concluded from local information around the singular set or from information about the singularity profiles of a flow. © 2019 Wiley Periodicals, Inc.     

Singular integral operators on complicated contours and pseudodifferential operators

We apply the pseudodifferential operator technique to the study of algebras of singular operators on complicated contours. This technique is used to construct a symbolic calculus for theC ^*-algebra generated by singular integral operators whose coefficients may have singularities of the second kind on complicated contours; the curves forming a node are not required to have a tangent at the node. Translated fromMatematicheskie Zametki, Vol. 58, No. 1, pp. 67–85, July, 1995. This work was partially supported by the Russian Foundation for Basic Research under grant No. 93-04-691.     

Weakly-exceptional quotient singularities

A singularity is said to be weakly-exceptional if it has a unique purely log terminal blow-up. In dimension 2, V. Shokurov proved that weakly-exceptional quotient singularities are exactly those of types D _n , E ₆, E ₇, E ₈. This paper classifies the weakly-exceptional quotient singularities in dimensions 3 and 4.     

Negative powers of a singular Schrödinger operator and convergence of spectral decompositions

We compare theL ₂( ^N )-norms of negative powers of various Laplace and Schrödinger operators possessing a singular potential whose singularities lie on some manifolds. We write out sufficient conditions for uniform convergence and localization of spectral decompositions of functions from the Liouville class.Translated fromMatematicheskie Zametki, Vol. 59, No. 3, pp. 428–436, March, 1996.The author wishes to express deep gratitude to Prof. Sh. A. Alimov for his attention to this work.     

On Morse Conjecture for Flows on Closed Surfaces

Let M be a closed orientable surface and let ϕ be a C¹‐flow on M with set of singularities compact countable. In this paper, we prove the Morse conjecture for ϕ: if ϕ is topologically transitive then it is metrically transitive.     

Ehresmann connection for the canonical foliation on a manifold over a local algebra

For a canonical foliation on a manifoldM ^A over a local algebra, theA-affine horizontal distribution complementary to the leaves, similar to the horizontal distribution of a higher order connection on the fiber bundle ofA-jets, is defined. In the case of a complete manifoldM ^A, theA-affine horizontal distribution is proved to be an Ehresmann connection in the sense of Blumental-Hebda. It is shown that theA-affine horizontal distribution onM ^A exists if and only if the Atiyah class of a certain foliated principal bundle vanishes.Translated fromMatematicheskie Zametki, Vol. 59, No. 2, pp. 303–310, February, 1996.     

Extension of Fréchet valued real analytic functions from subvarieties of ℝd

It is shown that for an algebraic subvariety X of ℝ^d every Fréchet valued real analytic function on X can be extended to a real analytic function on ℝ^d if and only if X is of type (PL), i.e. all of its singularities are of a certain type. Necessity of this condition is shown for any real analytic variety. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A. S. Convergence of two-parameter banach space valued martingales and the radon-nikodym property of banach spaces

In this paper, we prove that under theF ₄ conditions, anyL log⁺ L bounded two-parameter Banach spece valued martingale converges almost surely to an integrable Banach space valued random variable if and only if the Banach space has the Radon-Nikodym property. We further prove that the above conclusion remains true if theF ₄ condition is replaced by the weaker localF ₄ condition. Project supported by the National Natural Science Foundation of China and the State Education Commission Ph. D. Station Foundation