scholarly journals A geometric criterion for prescribing residues and some applications

2021 ◽  
pp. 1-56
Author(s):  
Hanlong Fang
Keyword(s):  
Geometric Criterion

scholarly journals Characteristic Cohomology I: Singularities of Given Type

2020 ◽  
Author(s):  
James Damon
Keyword(s):  
Geometric Criterion ◽  
Milnor Fiber ◽  
Compact Models ◽  
Singular Matrices

Abstract For a germ of a variety $\mathcal{V}, 0 \subset \mathbb C^N, 0$, a singularity $\mathcal{V}_0$ of ‘type $\mathcal{V}$’ is given by a germ $f_0 : \mathbb C^n, 0 \to \mathbb C^N, 0$ which is transverse to $\mathcal{V}$ in an appropriate sense so that $\mathcal{V}_0 = f_0^{\,-1}(\mathcal{V})$. If $\mathcal{V}$ is a hypersurface germ, then so is $\mathcal{V}_0 $, and by transversality ${\operatorname{codim}}_{\mathbb C} {\operatorname{sing}}(\mathcal{V}_0) = {\operatorname{codim}}_{\mathbb C} {\operatorname{sing}}(\mathcal{V})$ provided $n > {\operatorname{codim}}_{\mathbb C} {\operatorname{sing}}(\mathcal{V})$. So $\mathcal{V}_0, 0$ will exhibit singularities of $\mathcal{V}$ up to codimension n. For singularities $\mathcal{V}_0, 0$ of type $\mathcal{V}$, we introduce a method to capture the contribution of the topology of $\mathcal{V}$ to that of $\mathcal{V}_0$. It is via the ‘characteristic cohomology’ of the Milnor fiber (for $\mathcal{V}, 0$ a hypersurface), and complement and link of $\mathcal{V}_0$ (in the general case). The characteristic cohomology of the Milnor fiber $\mathcal{A}_{\mathcal{V}}(\,f_0; R)$, and respectively of the complement $\mathcal{C}_{\mathcal{V}}(\,f_0; R)$, are subalgebras of the cohomology of the Milnor fibers, respectively the complement, with coefficients R in the corresponding cohomology. For a fixed $\mathcal{V}$, they are functorial over the category of singularities of type $\mathcal{V}$. In addition, for the link of $\mathcal{V}_0$ there is a characteristic cohomology subgroup $\mathcal{B}_{\mathcal{V}}(\,f_0, \mathbf{k})$ of the cohomology of the link over a field $\mathbf{k}$ of characteristic 0. The cohomologies $\mathcal{C}_{\mathcal{V}}(\,f_0; R)$ and $\mathcal{B}_{\mathcal{V}}(\,f_0, \mathbf{k})$ are shown to be invariant under the $\mathcal{K}_{\mathcal{V}}$-equivalence of defining germs f0, and likewise $\mathcal{A}_{\mathcal{V}}(\,f_0; R)$ is shown to be invariant under the $\mathcal{K}_{H}$-equivalence of f0 for H the defining equation of $\mathcal{V}, 0$. We give a geometric criterion involving ‘vanishing compact models’ for both the Milnor fibers and complements which detect non-vanishing subalgebras of the characteristic cohomologies, and subgroups of the characteristic cohomology of the link. Also, we consider how in the hypersurface case the cohomology of the Milnor fiber is a module over the characteristic cohomology $\mathcal{A}_{\mathcal{V}}(\,f_0; R)$. We briefly consider the application of these results to a number of cases of singularities of a given type. In part II, we specialize to the case of matrix singularities and using results on the topology of the Milnor fibers, complements and links of the varieties of singular matrices obtained in another paper allow us to give precise results for the characteristic cohomology of all three types.

scholarly journals An application of a novel geometric criterion to global-stability problems of a nonlinear SEIVS epidemic model

2021 ◽  
Author(s):  
Xingyu Wang ◽  
Zhijun Liu ◽  
Lianwen Wang ◽  
Caihong Guo ◽  
Huili Xiang
Keyword(s):  
Global Stability ◽  
Epidemic Model ◽  
Geometric Criterion ◽  
Stability Problems

scholarly journals Geometric criterion for solvability of lattice spin systems

2020 ◽  
Vol 102 (24) ◽  
Author(s):  
Masahiro Ogura ◽  
Yukihisa Imamura ◽  
Naruhiko Kameyama ◽  
Kazuhiko Minami ◽  
Masatoshi Sato
Keyword(s):  
Spin Systems ◽  
Geometric Criterion ◽  
Lattice Spin ◽  
Lattice Spin Systems

Geometric Criterion for Controllability under Arbitrary Constraints on the Control

2007 ◽  
Vol 134 (2) ◽  
pp. 161-176 ◽  
Author(s):  
V. I. Korobov
Keyword(s):  
Geometric Criterion

Geometric criterion for accurate conductance quantization in a lateral constriction

1992 ◽  
Vol 4 (11) ◽  
pp. L185-L190
Author(s):  
V Polyanovsky
Keyword(s):  
Geometric Criterion ◽  
Conductance Quantization

The new geometric criterion of polynomial stability

1988 ◽  
Vol 9 (5) ◽  
pp. 485-488
Author(s):  
Wang Xiao-jun
Keyword(s):  
Polynomial Stability ◽  
Geometric Criterion

A Geometric Criterion for Positive Topological Entropy¶II: Homoclinic Tangencies

1999 ◽  
Vol 208 (2) ◽  
pp. 267-273 ◽  
Author(s):  
Ale Jan Homburg ◽  
Howard Weiss
Keyword(s):  
Topological Entropy ◽  
Geometric Criterion ◽  
Positive Topological Entropy ◽  
Homoclinic Tangencies

Stability and Hopf Bifurcation of a Delayed Density-Dependent Predator–Prey System with Beddington–DeAngelis Functional Response

2016 ◽  
Vol 26 (10) ◽  
pp. 1650165 ◽  
Author(s):  
Haiyin Li ◽  
Gang Meng ◽  
Zhikun She
Keyword(s):  
Hopf Bifurcation ◽  
Density Dependence ◽  
Functional Response ◽  
Positive Equilibrium ◽  
Stability Switches ◽  
Predator Prey ◽  
Geometric Criterion ◽  
Density Dependent ◽  
Prey System ◽  
The Stability

In this paper, we investigate the stability and Hopf bifurcation of a delayed density-dependent predator–prey system with Beddington–DeAngelis functional response, where not only the prey density dependence but also the predator density dependence are considered such that the studied predator–prey system conforms to the realistically biological environment. We start with the geometric criterion introduced by Beretta and Kuang [2002] and then investigate the stability of the positive equilibrium and the stability switches of the system with respect to the delay parameter [Formula: see text]. Especially, we generalize the geometric criterion in [Beretta & Kuang, 2002] by introducing the condition [Formula: see text] which can be assured by the condition [Formula: see text], and adopting the technique of lifting to define the function [Formula: see text] for alternatively determining stability switches at the zeroes of [Formula: see text]s. Afterwards, by the Poincaré normal form for Hopf bifurcation in [Kuznetsov, 1998] and the bifurcation formulae in [Hassard et al., 1981], we qualitatively analyze the properties for the occurring Hopf bifurcations of the system (3). Finally, an example with numerical simulations is given to illustrate the obtained results.

Comment on ‘Geometric criterion for the absence of limit cycles in nonlinear control systems’

1973 ◽  
Vol 9 (24) ◽  
pp. 576 ◽  
Author(s):  
P.C. Byrne ◽  
M. Sheeran
Keyword(s):  
Nonlinear Control ◽  
Control Systems ◽  
Limit Cycles ◽  
Nonlinear Control Systems ◽  
Geometric Criterion
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