On equivariant indices of 1-forms on varieties

Given a G-invariant holomorphic 1-form with an isolated singular point on a germ of a complex-analytic G-variety with an isolated singular point (G is a finite group), its equivariant homological index and (reduced) equivariant radial index are defined as elements of the ring of complex representations of the group. We show that these indices coincide on a germ of a smooth complex analytic G-variety. This makes it possible to consider the difference between them as a version of the equivariant Milnor number of a germ of a G-variety with an isolated singular point.     

Euler Obstruction and Defects of Functions on Singular Varieties

Several authors have proved Lefschetz type formulas for thelocal Euler obstruction. In particular, a result of this typehas been proved that turns out to be equivalent to saying thatthe local Euler obstruction, as a constructible function, satisfiesthe local Euler condition (in bivariant theory) with respectto general linear forms. The purpose of the paper is to determinewhat prevents the local Euler obstruction from satisfying thelocal Euler condition with respect to functions which are singularat the considered point. This is measured by an invariant (or‘defect’) of such functions. An interpretation ofthis defect is given in terms of vanishing cycles, which allowsit to be calculated algebraically. When the function has anisolated singularity, the invariant can be defined geometrically,via obstruction theory. This invariant unifies the usual conceptsof the Milnor number of a function and the local Euler obstructionof an analytic set.     

Milnor numbers and Euler obstruction*

Using a geometric approach, we determine the relations between the local Euler obstruction Eu_f of a holomorphic function f and several generalizations of the Milnor number for functions on singular spaces. *This work was partially supported by CNRS-CONACYT (12409) Cooperation Program. The first and third named authors partially supported by CONACYT grant G36357-E and DGPA (UNAM) grant IN 101 401.     

The Euler obstruction of a function on a determinantal variety and on a curve

Given an analytic function germ f: (X, 0) → C on an isolated determinantal singularity or on a reduced curve, we present formulas relating the local Euler obstruction of f to the vanishing Euler characteristic of the fiber X ∩ f^-1(0) and to the Milnor number of f. Restricting ourselves to the case where X is a complete intersection, we obtain an easy way to calculate the local Euler obstruction of f as the difference between the dimension of two algebras.     

高阶奇异积分的Hadamard主值

应用Ｅｕｌｅｒ径向微分算子:Ｄ＝ｚｌ＋…＋ｚｎ研究复ｎ维超球面Ｂ＝{ζ∈Ｃｎ|ζ＝(ζ１,…,ζｎ),|ζ１|２＋…＋|ζｎ|２＝１}上两类高阶奇异积分的Ｈａｄａｍａｒｄ主值．本文得到置换和合成公式并讨论了它们的拓广以及在偏微分奇异积分方程上的应用．     

Rough singular integrals on Triebel–Lizorkin space and Besov space

In this paper the authors prove that the homogeneous singular integral T_Ω with ΩH¹(Sⁿ⁻¹) is bounded on the Triebel–Lizorkin spaces and the Besov spaces. These results answer an open problem proposed by Chen and Zhang in [J. Chen, C. Zhang, Boundedness of rough singular integral on the Triebel–Lizorkin spaces, J. Math. Anal. Appl. 337 (2008) 1048–1052]. The same results hold also for the rough singular integral operators T_Ω,h with radial function kernels.     

Complex monodromy and the topology of real algebraic sets

A relation between the Euler characteristics of the Milnorfibres of a real analytic function is derived from a simple identity involvingcomplex monodromy and complex conjugation. A corollary is the result of Costeand Kurdyka that the Euler characteristic of the local link of an irreduciblealgebraic subset of a real algebraic set is generically constant modulo 4. Asimilar relation for iterated Milnor fibres of ordered sets of functions isused to define topological invariants of ordered collections of algebraicsubsets.     

Convergence Conditions for Interpolation Fractions at the Nodes Distinct from the Singular Points of a Function

We consider an interpolation process for the class of functions with finitely many singular points by means of the rational functions whose poles coincide with the singular points of the function under interpolation. The interpolation nodes constitute a triangular matrix and are distinct from the singular points of the function. We find a necessary and sufficient condition for uniform convergence of sequences of interpolation fractions to the function under interpolation on every compact set disjoint from the singular points of the function and other conditions for convergence.Original Russian Text Copyright © 2005 Lipchinskii A. G.__________Translated from Sibirskii Matematicheskii Zhurnal, Vol. 46, No. 4, pp. 822–833, July–August, 2005.     

On the Coordinate of a Singular Point of the Time Correlation Function for a Spin System on a Simple Hypercubic Lattice at High Temperatures

Using the d ^–1 expansion method (d is the space dimension), we estimate the coordinate of the time-dependent autocorrelation function singular point on the imaginary time axis for the spin 1/2 Heisenberg model on a simple hypercubic lattice at high temperatures. We represent the coefficients of the time expansion (the spectral moments) for the autocorrelation function as the sums of the weighted lattice figures in which the trees constructed from the double bonds give the leading contributions with respect to d ^–1 and the same trees with the built-in squares from six bonds or diagrams with the fourfold bonds give the contribution of the next-to-leading order. We find the corrections to the coordinate of the autocorrelation function singular point that are due to the latter contributions.     

Maximum principle for optimal control of distributed-parameter systems with singular mass matrix

A maximum principle for the open-loop optimal control of a vibrating system relative to a given convex index of performance is investigated. Though maximum principles have been studied by many people (see, e.g., Refs. 1–5), the principle derived in this paper is of particular use for control problems involving mechanical structures. The state variable satisfies general initial conditions as well as a self-adjoint system of partial differential equations together with a homogeneous system of boundary conditions. The mass matrix is diagonal, constant, and singular, and the viscous damping matrix is diagonal. The maximum principle relates the optimal control with the solution of the homogeneous adjoint equation in which terminal conditions are prescribed in terms of the terminal values of the optimal state variable. An application of this theory to a structural vibrating system is given in a companion paper (Ref. 6).     

Lê cycles and Milnor classes

The purpose of this work is to establish a link between the theory of Chern classes for singular varieties and the geometry of the varieties in question. Namely, we show that if Z is a hypersurface in a compact complex manifold, defined by the complex analytic space of zeroes of a reduced non-zero holomorphic section of a very ample line bundle, then its Milnor classes, regarded as elements in the Chow group of Z, determine the global Lê cycles of Z; and vice versa: The Lê cycles determine the Milnor classes. Morally this implies, among other things, that the Milnor classes determine the topology of the local Milnor fibres at each point of Z, and the geometry of the local Milnor fibres determines the corresponding Milnor classes.     

The differential equations for the feynman amplitude of a single-loop graph with four vertices

A system of second-order partial differential equations for the Feynman amplitude of a single-loop graph with four vertices is obtained. It is proved that the symbol of differential operators of this system is singular (in the sense of I. N. Bernshtein) on the Landau manifold of the Feynman amplitude under consideration. The derived system of differential equations is a multidimensional generalization of the system of differential equations for the hypergeometric function of two variables of Appell and Kampé de Fériet.Translated from Matematicheskie Zametki, Vol. 23, No. 1, pp. 113–119, January, 1978.     

The Poincaré index and the χy-characteristic of Hirzebruch

In the paper, we discuss several methods for computing the homology of contravariant and covariant versions of the classical De Rham complex on analytic spaces. Our approach is based on the theory of holomorphic and regular meromorphic differential forms, and is applicable in different settings depending on concrete types of varieties. Among other things, we describe how to compute by elementary calculations the homological index of vector fields and differential forms given on Cohen–Macaulay curves, graded normal surfaces, complete intersections and some others. In these situations, making use of ideas of X. Gómez-Mont, we derive explicit expressions for the local topological index of Poincaré and its generalizations. Furthermore, applying similar methods to the study of certain other complexes, we investigate some challenges, relating to the computation of classical topological–analytical invariants, such as the Euler characteristic of the Milnor fibre of an isolated singularity, the multiplicity of the discriminant of the versal deformation, the dimension of torsion and cotorsion modules, and so on.     

On the Cauchy Principal Value of the Khenkin-Ramirez Singular Integral in Strictly Pseudoconvex Domains of ℂ <Superscript>n</Superscript>

We show that in the multidimensional case (unlike the complex plane) the Cauchy principal value of the Khenkin-Ramirez singular integral in strictly pseudoconvex domains is equal to the limit value of this integral inside the domain.Original Russian Text Copyright © 2005 Kytmanov A. M. and Myslivets S. G.The first author was supported by a grant of the President of the Russian Federation and the State Maintenance Program for the Leading Scientific Schools of the Russian Federation (Grant NSh-1212.2003.1); the second author was supported by the Krasnoyarsk Region Science Foundation (Grant 12F0063C).__________Translated from Sibirskii Matematicheskii Zhurnal, Vol. 46, No. 3, pp. 625–633, May–June, 2005.     

Note on: N.E. Aguilera, M.S. Escalante, G.L. Nasini, “The disjunctive procedure and blocker duality”

Aguilera et al. [Discrete Appl. Math. 121 (2002) 1–13] give a generalization of a theorem of Lehman through an extension of the disjunctive procedure defined by Balas, Ceria and Cornuéjols. This generalization can be formulated as(A) For every clutter , the disjunctive index of its set covering polyhedron coincides with the disjunctive index of the set covering polyhedron of its blocker, .In Aguilera et al. [Discrete Appl. Math. 121 (2002) 1–3], (A) is indeed a corollary of the stronger result(B) .Motivated by the work of Gerards et al. [Math. Oper. Res. 28 (2003) 884–885] we propose a simpler proof of (B) as well as an alternative proof of (A), independent of (B). Both of them are based on the relationship between the “disjunctive relaxations” obtained by and the set covering polyhedra associated with some particular minors of .     

Riemannian Geometry of Conical Singular Sets

In this paper, we study a class of singular Riemannian manifolds. The singular set itself is a smooth manifold with a cone-like neighborhood. By imposing a reasonable convergence condition on the metric, we can determine the local geometrical structure near the singular set. In general, the curvature near the singular set is unbounded. We prove that a bounded curvature assumption would have a strong implication on the geometrical and topological structures near the singular set. We also establish the Gauss–Bonnet–Chern formula, which can be applied to the study of singular Eistein 4-manifolds.     

Asymptotic analysis of a singularly perturbed linear system with a singular limit pencil of matrices

With the help of perturbation methods and Newton diagrams, an asymptotic analysis is conducted of the general solution of a linear singularly perturbed system of ordinary differential equations in the case of degeneracy of a matrix multiplying the derivative in the approach of a small parameter to zero. It is assumed that the pencil of limit matrices of the system is singular and possesses a minimal index for rows and columns.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 1, pp. 106–122, January, 1992.     

Local Constraints on Einstein–Weyl Geometries: The 3-Dimensional Case

Following an earlier study [3], we consider the Einstein–Weyl equations on a fixed (complex) background metric as an equation for a 1-form and its first few derivatives. If the background is flat then we conclude that the only solutions are conformal rescalings of constant curvature metrics. If the background is a homogeneous 3-geometry in Bianchi class A (i.e., with unimodular isometry group), we find necessary and sufficient conditions on the 3-geometry for solutions of the Einstein–Weyl equations to exist. The solutions we find are complexifications of known ones. In particular, we find that the general left-invariant metric on S³ and the metric 'Sol' admit no local solutions of the Einstein–Weyl equations.     

Motivic Milnor fibre of cyclic <Emphasis Type="Italic">L</Emphasis><Subscript>∞</Subscript>-algebras

We define the motivic Milnor fiber of cyclic L _∞-algebras of dimension three using the method of Denef and Loeser of motivic integration. It is proved by Nicaise and Sebag that the topological Euler characteristic of the motivic Milnor fiber is equal to the Euler characteristic of the étale cohomology of the analytic Milnor fiber. We prove that the value of Behrend function on the germ moduli space determined by a cyclic L _∞-algebra L is equal to the Euler characteristic of the analytic Milnor fiber. Thus we prove that the Behrend function depends only on the formal neighborhood of the moduli space.     

On the Index of Differential Operators on Manifolds with Conical Singularities

The paper contains the proof of the index formula for manifolds with conical points. For operators subject to an additional condition of spectral symmetry, the index is expressed as the sum of multiplicities of spectral points of the conormal symbol (indicial family) and the integral from the Atiyah–Singer form over the smooth part of the manifold. The obtained formula is illustrated by the example of the Euler operator on a two-dimensional manifold with conical singular point.