CR Runge Sets on Hypersurface Graphs

This work contains an improvement of earlier results of Boggess and Dwilewicz regarding global approximation of CR functions by entire functions in the case of hypersurface graphs. In this work, we show that if ω, an open subset of a real hypersurface in ℂ ⁿ , can be graphed over a convex subset in ℝ²ⁿ⁻¹, then ω is CR-Runge in the sense that continuous CR functions on ω can be approximated by entire functions on ℂ ⁿ in the compact open topology of ω. Examples are presented to show that this approximation result does not hold for graphed CR submanifolds in higher codimension. R. Dwilewicz is partially supported by the Polish Science Foundation (KBN) grant N201 019 32/805.     

Hypersurface Singularities with 2-Dimensional Critical Locus

Consider an analytic germ f:(C^m, 0)(C, 0) (m3) whose criticallocus is a 2-dimensional complete intersection with an isolatedsingularity (icis). We prove that the homotopy type of the Milnorfiber of f is a bouquet of spheres, provided that the extendedcodimension of the germ f is finite. This result generalizesthe cases when the dimension of the critical locus is zero [8],respectively one [12]. Notice that if the critical locus isnot an icis, then the Milnor fiber, in general, is not homotopicallyequivalent to a wedge of spheres. For example, the Milnor fiberof the germ f:(C⁴, 0)(C, 0), defined by f(x₁, x₂, x₃, x₄) =x₁x₂x₃x₄ has the homotopy type of S¹xS¹xS¹. On the other hand,the finiteness of the extended codimension seems to be the rightgeneralization of the isolated singularity condition; see forexample [9–12, 17, 18]. In the last few years different types of ‘bouquet theorems’have appeared. Some of them deal with germs f:(X, x)(C, 0) wheref defines an isolated singularity. In some cases, similarlyto the Milnor case [8], F has the homotopy type of a bouquetof (dim X–1)-spheres, for example when X is an icis [2],or X is a complete intersection [5]. Moreover, in [13] Siersmaproved that F has a bouquet decomposition FF₀Sⁿ...Sⁿ (whereF₀ is the complex link of (X, x)), provided that both (X, x)and f have an isolated singularity. Actually, Siersma conjecturedand Tibr proved [16] a more general bouquet theorem for thecase when (X, x) is a stratified space and f defines an isolatedsingularity (in the sense of the stratified spaces). In thiscase F_iF_i, where the F_i are repeated suspensions of complexlinks of strata of X. (If (X, x) has the ‘Milnor property’,then the result has been proved by Lê; for details see[6].) In our situation, the space-germ (X, x) is smooth, but f hasbig singular locus. Surprisingly, for dim Sing f^–1(0)2,the Milnor fiber is again a bouquet (actually, a bouquet ofspheres, maybe of different dimensions). This result is in thespirit of Siersma's paper [12], where dim Sing f^–1(0)= 1. In that case, there is only a rather small topologicalobstruction for the Milnor fiber to be homotopically equivalentto a bouquet of spheres (as explained in Corollary 2.4). Inthe present paper, we attack the dim Sing f^–1(0) = 2 case.In our investigation some results of Zaharia are crucial [17,18].     

Jet cohomology of Isolated Hypersurface Singularities and Spectral Sequences

We study jet cohomology of isolated hypersurface singularities defined by partial differential forms and prove formulas to compute jet cohomology groups by linear algebra.

     

Isolated Hypersurface Singularities and Special Polynomial Realizations of Affine Quadrics

Let V, $\tilde{V}$ be hypersurface germs in ? ^m , each having a quasi-homogeneous isolated singularity at the origin. We show that the biholomorphic equivalence problem for V, $\tilde{V}$ reduces to the linear equivalence problem for certain polynomials P, $\tilde{P}$ arising from the moduli algebras of V, $\tilde{V}$ . The polynomials P, $\tilde{P}$ are completely determined by their quadratic and cubic terms, hence the biholomorphic equivalence problem for V, $\tilde{V}$ in fact reduces to the linear equivalence problem for pairs of quadratic and cubic forms.     

Variation of the Milnor Fibration in Pencils of Hypersurface Singularities

Let = (f, g):(Cⁿ_{+ 1},0) (C₂, 0) be a pair of holomorphic germswith no blowing up in codimension 0. (Two examples are the following: defines an isolated complete intersection singularity; g =l^N where is a generic linear form with respect to f and N>0.) We study how the Milnor fibrations of the germs _(:ß)= g-ß f are related to each other when (:ß)varies in P¹. More precisely, we construct isotopic subfibrationsor subfibres of the Milnor fibrations of any two such germs.The proofs are based on the precise study of the subdiscs ofcomplex lines meeting a fixed complex plane curve germ transversally,generalizing Lê's work on the Cerf diagram. 2000 MathematicalSubject Classification: 32S55, 32S15, 32S30.     

Singularities of Centre Symmetry Sets

The center symmetry set (CSS) of a smooth hypersurface S inan affine space Rⁿ is the envelope of lines joining pairs ofpoints where S has parallel tangent hyperplanes. The idea stemsfrom a definition of Janeczko, in an alternative version dueto Giblin and Holtom. For n = 2 the envelope is always real,while for n > 3 the existence of a real envelope dependson the geometry of the hypersurface. In this paper we make alocal study of the CSS, some results applying to n 5 and othersto the cases n = 2,3. The method is to construct a generatingfunction whose bifurcation set contains the CSS and possiblysome other redundant components. Focal sets of smooth hypersurfacesare a special case of the construction, but the CSS is an affineand not a euclidean invariant. Besides the familiar local formsof focal sets there are other local forms corresponding to boundarysingularities, and yet others which do not appear to have arisenelsewhere in a geometrical context. There are connections withFinsler geometry. This paper concentrates on the theory andthe proof of the local normal forms for the CSS. 2000 MathematicsSubject Classification 57R45, 58K40, 32S25, 58B20.     

Separating Sets as Envelopes

The set which forms the boundary of the regions of interferencewhen a family of curves is superposed on a parallel gratingis related to the envelopes of two ‘interference sets’in two fixed directions.     

四维Minkowski空间中类空超曲面的双曲高斯映射的奇点

介绍了四维Minkowski空间中类空超曲面的局部理论,定义了类空超曲面上的双曲高斯映射,双曲高度函数及距离平方函数,给出了一些定理的详细证明.介绍了一种证明高度函数是Morse族的新方法并应用Arnold等建立的Lagrange奇点理论对类空超曲面的双曲高斯映射的奇点进行了分类.     

齐次均匀Moran集的拟Lipschitz等价

本文证明了两个正则齐次均匀Moran集拟Lipschitz等价当且仅当它们的Hausdorff维数相等.     

拟对称极小的齐次完全集

本文用质量分布原理,证明了由有界正整数序列定义的Hausdorff维数为1的齐次完全集是一维拟对称极小的.     

On the Generalized Cartan Matrices Arising from k-th Yau Algebras of Isolated Hypersurface Singularities

Algebras and Representation Theory - Let (V,0) be an isolated hypersurface singularity defined by the holomorphic function $f: (mathbb {C}^{n}, 0)rightarrow (mathbb {C}, 0)$ . The k-th Yau...     

Nonexistence of Solutions for Nonlinear Differential Inequalities with Singularities on Unbounded Sets

Mathematical Notes - We study some new Liouville theorems for nonlinear differential inequalities with gradient terms and singular variable coefficients that have singularities on unbounded sets....     

Sets of Finite Perimeter and the Gauss-Green Theorem with Singularities

We establish the n-dimensional divergence theorem in a formreaching the limits of generality with respect to the geometricaland the analytical assumptions.     

Distinct Distances in Homogeneous Sets in Euclidean Space

It is shown that every homogeneous set of n points in d-dimensional Euclidean space determines at least distinct distances for a constant c(d) > 0. In three-space the above general bound is slightly improved and it is shown that every homogeneous set of n points determines at least distinct distances.     

裁元齐次Moran集的Hausdorff维数

将齐次Moran集迭代过程中的k项序列集Dk={(i1,...,ik):1≤ij≤nj,1≤j≤k}裁减为Dk={(i1,...,ik):1≤ij≤nj, ij≠2 unless ij-1=1, 2≤j≤k},相应的集合称为裁元齐次Moran集．本文确定了一类裁元齐次Moran集的Hausdorff维数．     

齐次完全集的拟对称极小性

作为Cantor型集的推广,文志英和吴军引入了齐次完全集的概念,并基于齐次完全集的基本区间的长度以及基本区间之间的间隔的长度,得到了齐次完全集的Hausdorff维数.本文研究齐次完全集的拟对称极小性,证明在某些条件下Hausdorff维数为1的齐次完全集是1维拟对称极小的.     

R~d中齐次Moran集的Hausdorff维数

本文利用位势理论给出了Ｒｄ中齐次Ｍｏｒａｎ集的Ｈａｕｓｄｏｒｆｆ维数公式,从而回答了山中的问题．     

Allowable Interval Sequences and Separating Convex Sets in the Plane

In 2005, Goodman and Pollack introduced the concept of an allowable interval sequence, a combinatorial object which encodes properties of a family of pairwise disjoint convex sets in the plane. They, Dhandapani, and Holmsen used this concept to address Tverberg’s (1,k)-separation problem: How many pairwise disjoint compact convex sets in the plane are required to guarantee that one can be separated by a line from k others? (Denote this number by f _k .) A new proof was provided that f ₂=5, a result originally obtained by Tverberg himself, and the application of allowable interval sequences to the case of general k was left as an open problem. Hope and Katchalski, using other methods, proved in 1990 that 3k?1≤f _k ≤12(k?1). In this paper, we apply the method of allowable interval sequences to give an upper bound on f _k of under 7.2(k?1), shrinking the range given by Hope and Katchalski by more than half. For a family of translates we obtain a tighter upper bound of approximately 5.8(k?1).