scholarly journals INTERSECTION COHOMOLOGY, MONODROMY AND THE MILNOR FIBER

2009 ◽  
Vol 20 (04) ◽  
pp. 491-507 ◽  
Author(s):  
DAVID B. MASSEY
Keyword(s):  
Analytic Function ◽  
Simple Relation ◽  
Intersection Cohomology ◽  
Critical Locus ◽  
One Dimensional ◽  
Milnor Fiber ◽  
Singular Sets ◽  
Vanishing Cycle ◽  
Thom Isomorphism ◽  
Complex Analytic Space

We say that a complex analytic space, X, is an intersection cohomology manifold if and only if the shifted constant sheaf on X is isomorphic to intersection cohomology; with field coefficients, this is quickly seen to be equivalent to X being a homology manifold. Given an analytic function f on an intersection cohomology manifold, we describe a simple relation between V(f) being an intersection cohomology manifold and the vanishing cycle Milnor monodromy of f. We then describe how the Sebastiani–Thom isomorphism allows us to easily produce intersection cohomology manifolds with arbitrary singular sets. Finally, as an easy application, we obtain restrictions on the cohomology of the Milnor fiber of a hypersurface with a special type of one-dimensional critical locus.

scholarly journals On the Topology of Isochronous Centers of Hamiltonian Differential Systems

2019 ◽  
Vol 29 (06) ◽  
pp. 1950099
Author(s):  
Guangfeng Dong ◽  
Changjian Liu ◽  
Jiazhong Yang
Keyword(s):  
Hamiltonian Systems ◽  
Isochronous Center ◽  
Level Curve ◽  
Topological Properties ◽  
Differential Systems ◽  
One Dimensional ◽  
Vanishing Cycle ◽  
Isochronous Centers ◽  
The One ◽  
Hamiltonian Functions

In this paper, we study the topology of isochronous centers of Hamiltonian differential systems with polynomial Hamiltonian functions [Formula: see text] such that the isochronous center lies on the level curve [Formula: see text]. We prove that, in the one-dimensional homology group of the Riemann surface (removing the points at infinity) of level curve [Formula: see text], the vanishing cycle of an isochronous center cannot belong to a subgroup generated by those small loops such that each of them is centered at a removed point at infinity of having one of the two special types described in the paper, where [Formula: see text] is sufficiently close to [Formula: see text]. Besides, we present some topological properties of isochronous centers for a large class of Hamiltonian systems of degree [Formula: see text], whose homogeneous parts of degree [Formula: see text] contain factors with multiplicity of no more than [Formula: see text]. As applications, we study the nonisochronicity for some Hamiltonian systems with quite complicated forms which are usually very hard to handle by the classical tools.

scholarly journals The Af condition and relative conormal spaces for functions with nonvanishing derivative

2019 ◽  
Vol 30 (10) ◽  
pp. 1950050
Author(s):  
Terence Gaffney ◽  
Antoni Rangachev
Keyword(s):  
Analytic Function ◽  
Analytic Space ◽  
Numerical Criterion ◽  
Join Construction ◽  
Complex Analytic Space ◽  
Text Condition

We introduce a join construction, as a way of completing the description of the relative conormal space of an analytic function on a complex analytic space that has a non-vanishing derivative at the origin. Then we show how to obtain a numerical criterion for Thom’s [Formula: see text] condition.

On the equations governing the perturbations of the Reissner–Nordström black hole

1979 ◽  
Vol 365 (1723) ◽  
pp. 453-465 ◽  
Keyword(s):  
Black Hole ◽  
Electromagnetic Waves ◽  
Wave Equations ◽  
Simple Relation ◽  
Opposite Parity ◽  
One Dimensional ◽  
Basic Equations ◽  
Dimensional Wave

By considering suitable combinations of the Weyl scalars and the spin coefficients, the basic equations governing the perturbations of the Reissner–Nordström black hole, in the Newman–Penrose formalism, are decoupled; a fundamental pair of decoupled equations are obtained. It is then shown how this pair of decoupled equations can be transformed into one dimensional wave equations which are appropriate for describing the perturbations of odd and of even parity. A simple relation is obtained which will allow derivation of a solution belonging to one parity from a solution belonging to the opposite parity. Finally, equations are derived in terms of which one can readily ascertain how an arbitrary superposition of gravitational and electromagnetic waves, incident on the black hole, will be reflected and absorbed.

scholarly journals Real canonical cycle and asymptotics of oscillating integrals

2003 ◽  
Vol 171 ◽  
pp. 187-196
Author(s):  
Daniel Barlet
Keyword(s):  
Analytic Function ◽  
Real Analytic Function ◽  
Isolated Singularity ◽  
Connected Component ◽  
Analytic Set ◽  
Milnor Fiber ◽  
Real Analytic ◽  
Topological Construction

AbstractLet Xℝ ⊂ ℝN a real analytic set such that its complexification Xℂ ⊂ ℂN is normal with an isolated singularity at 0. Let fℝ : Xℝ → ℝ a real analytic function such that its complexification fℂ : Xℂ → ℂ has an isolated singularity at 0 in Xℂ. Assuming an orientation given on to a connected component A of we associate a compact cycle Γ(A) in the Milnor fiber of fℂ which determines completely the poles of the meromorphic extension of or equivalently the asymptotics when T → ±∞ of the oscillating integrals . A topological construction of Γ(A) is given. This completes the results of [BM] paragraph 6.

TOPOLOGY OF 1-PARAMETER DEFORMATIONS OF NON-ISOLATED REAL SINGULARITIES

2021 ◽  
pp. 1-15
Author(s):  
NICOLAS DUTERTRE ◽  
JUAN ANTONIO MOYA PÉREZ
Keyword(s):  
Analytic Function ◽  
Euler Characteristic ◽  
Topological Degree ◽  
Isolated Singularities ◽  
Function Germ ◽  
Milnor Fiber ◽  
Real Singularities ◽  
Parameter Deformation

Abstract Let $f\,{:}\,(\mathbb R^n,0)\to (\mathbb R,0)$ be an analytic function germ with non-isolated singularities and let $F\,{:}\, (\mathbb{R}^{1+n},0) \to (\mathbb{R},0)$ be a 1-parameter deformation of f. Let $ f_t ^{-1}(0) \cap B_\epsilon^n$ , $0 < \vert t \vert \ll \epsilon$ , be the “generalized” Milnor fiber of the deformation F. Under some conditions on F, we give a topological degree formula for the Euler characteristic of this fiber. This generalizes a result of Fukui.

Interaction among n Parallel Dislocations in One-Dimensional Hexagonal Quasicrystals

2015 ◽  
Vol 775 ◽  
pp. 133-137
Author(s):  
Guan Ting Liu ◽  
Li Ying Yang
Keyword(s):  
Analytic Function ◽  
Complex Variable ◽  
Function Theory ◽  
Reference Value ◽  
Interaction Force ◽  
Complex Variable Function ◽  
One Dimensional ◽  
Action Point ◽  
Variable Function ◽  
Equivalent Action

By means of analytic function theory, the problems of interaction amongparallel dislocations in one-dimensional hexagonal quasicrystals are investigated. The interaction force of parallel dislocations in the material is obtained in forms of complex variable function firstly, which is the versions of well-known Peach-Koehler formula in one-dimensional hexagonal quasicrystals on parallel dislocations. These results are development of the corresponding parts of quasicrystals. Meanwhile, in this paper, we firstly give the equivalent action point of parallel dislocations in one-dimensional hexagonal quasicrystals, which be of important reference value to researching the interaction problems of many dislocations in fracture mechanics of quasicrystals.

A one-dimensional self-gravitating stellar gas

1966 ◽  
Vol 25 ◽  
pp. 46-48 ◽  
Author(s):  
M. Lecar
Keyword(s):  
Gravitational Field ◽  
Phase Space ◽  
Motion Picture ◽  
Space Distribution ◽  
Two Dimensional ◽  
One Dimensional ◽  
Phase Space Distribution ◽  
Dimensional Phase Space ◽  
The Mean

“Dynamical mixing”, i.e. relaxation of a stellar phase space distribution through interaction with the mean gravitational field, is numerically investigated for a one-dimensional self-gravitating stellar gas. Qualitative results are presented in the form of a motion picture of the flow of phase points (representing homogeneous slabs of stars) in two-dimensional phase space.

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