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Search: All articles in the CJM digital archive with keyword stability

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1. CJM Online first

Pérez-Chavela, Ernesto; Sánchez-Cerritos, Juan Manuel
 Euler-type relative equilibria in spaces of constant curvature and their stability We consider three point positive masses moving on $S^2$ and $H^2$. An Eulerian-relative equilibrium, is a relative equilibrium where the three masses are on the same geodesic, in this paper we analyze the spectral stability of these kind of orbits where the mass at the middle is arbitrary and the masses at the ends are equal and located at the same distance from the central mass. For the case of $S^2$, we found a positive measure set in the set of parameters where the relative equilibria are spectrally stable, and we give a complete classification of the spectral stability of these solutions, in the sense that, except on an algebraic curve in the space of parameters, we can determine if the corresponding relative equilibria is spectrally stable or unstable. On $H^2$, in the elliptic case, we prove that generically all Eulerian-relative equilibria are unstable; in the particular degenerate case when the two equal masses are negligible we get that the corresponding solutions are spectrally stable. For the hyperbolic case we consider the system where the mass in the middle is negligible, in this case the Eulerian-relative equilibria are unstable. Keywords:curved space, relative equilibrium, spectral stabilityCategories:70F07, 70G60

2. CJM 2016 (vol 68 pp. 1096)

Smith, Benjamin H.
 Singular $G$-Monopoles on $S^1\times \Sigma$ This article provides an account of the functorial correspondence between irreducible singular $G$-monopoles on $S^1\times \Sigma$ and $\vec{t}$-stable meromorphic pairs on $\Sigma$. A theorem of B. Charbonneau and J. Hurtubise is thus generalized here from unitary to arbitrary compact, connected gauge groups. The required distinctions and similarities for unitary versus arbitrary gauge are clearly outlined and many parallels are drawn for easy transition. Once the correspondence theorem is complete, the spectral decomposition is addressed. Keywords:connection, curvature, instanton, monopole, stability, Bogomolny equation, Sasakian geometry, cameral coversCategories:53C07, 14D20

3. CJM 2016 (vol 69 pp. 241)

 Finite Determinacy and Stability of Flatness of Analytic Mappings It is proved that flatness of an analytic mapping germ from a complete intersection is determined by its sufficiently high jet. As a consequence, one obtains finite determinacy of complete intersections. It is also shown that flatness and openness are stable under deformations. Keywords:finite determinacy, stability, flatness, openness, complete intersectionCategories:58K40, 58K25, 32S05, 58K20, 32S30, 32B99, 32C05, 13B40

4. CJM 2015 (vol 67 pp. 1247)

Barros, Carlos Braga; Rocha, Victor; Souza, Josiney
 Lyapunov Stability and Attraction Under Equivariant Maps Let $M$ and $N$ be admissible Hausdorff topological spaces endowed with admissible families of open coverings. Assume that $\mathcal{S}$ is a semigroup acting on both $M$ and $N$. In this paper we study the behavior of limit sets, prolongations, prolongational limit sets, attracting sets, attractors and Lyapunov stable sets (all concepts defined for the action of the semigroup $\mathcal{S}$) under equivariant maps and semiconjugations from $M$ to $N$. Keywords:Lyapunov stability, semigroup actions, generalized flows, equivariant maps, admissible topological spacesCategories:37B25, 37C75, 34C27, 34D05

5. CJM 2015 (vol 67 pp. 1270)

Carcamo, Cristian; Vidal, Claudio
 Stability of Equilibrium Solutions in Planar Hamiltonian Difference Systems In this paper, we study the stability in the Lyapunov sense of the equilibrium solutions of discrete or difference Hamiltonian systems in the plane. First, we perform a detailed study of linear Hamiltonian systems as a function of the parameters, in particular we analyze the regular and the degenerate cases. Next, we give a detailed study of the normal form associated with the linear Hamiltonian system. At the same time we obtain the conditions under which we can get stability (in linear approximation) of the equilibrium solution, classifying all the possible phase diagrams as a function of the parameters. After that, we study the stability of the equilibrium solutions of the first order difference system in the plane associated to mechanical Hamiltonian system and Hamiltonian system defined by cubic polynomials. Finally, important differences with the continuous case are pointed out. Keywords:difference equations, Hamiltonian systems, stability in the Lyapunov senseCategories:34D20, 34E10

6. CJM 2014 (vol 67 pp. 1024)

Ashraf, Samia; Azam, Haniya; Berceanu, Barbu
 Representation Stability of Power Sets and Square Free Polynomials The symmetric group $\mathcal{S}_n$ acts on the power set $\mathcal{P}(n)$ and also on the set of square free polynomials in $n$ variables. These two related representations are analyzed from the stability point of view. An application is given for the action of the symmetric group on the cohomology of the pure braid group. Keywords:symmetric group modules, square free polynomials, representation stability, Arnold algebraCategories:20C30, 13A50, 20F36, 55R80

7. CJM 2012 (vol 65 pp. 222)

Sauer, N. W.
 Distance Sets of Urysohn Metric Spaces A metric space $\mathrm{M}=(M;\operatorname{d})$ is {\em homogeneous} if for every isometry $f$ of a finite subspace of $\mathrm{M}$ to a subspace of $\mathrm{M}$ there exists an isometry of $\mathrm{M}$ onto $\mathrm{M}$ extending $f$. The space $\mathrm{M}$ is {\em universal} if it isometrically embeds every finite metric space $\mathrm{F}$ with $\operatorname{dist}(\mathrm{F})\subseteq \operatorname{dist}(\mathrm{M})$. (With $\operatorname{dist}(\mathrm{M})$ being the set of distances between points in $\mathrm{M}$.) A metric space $\boldsymbol{U}$ is an {\em Urysohn} metric space if it is homogeneous, universal, separable and complete. (It is not difficult to deduce that an Urysohn metric space $\boldsymbol{U}$ isometrically embeds every separable metric space $\mathrm{M}$ with $\operatorname{dist}(\mathrm{M})\subseteq \operatorname{dist}(\boldsymbol{U})$.) The main results are: (1) A characterization of the sets $\operatorname{dist}(\boldsymbol{U})$ for Urysohn metric spaces $\boldsymbol{U}$. (2) If $R$ is the distance set of an Urysohn metric space and $\mathrm{M}$ and $\mathrm{N}$ are two metric spaces, of any cardinality with distances in $R$, then they amalgamate disjointly to a metric space with distances in $R$. (3) The completion of every homogeneous, universal, separable metric space $\mathrm{M}$ is homogeneous. Keywords:partitions of metric spaces, Ramsey theory, metric geometry, Urysohn metric space, oscillation stabilityCategories:03E02, 22F05, 05C55, 05D10, 22A05, 51F99

8. CJM 2007 (vol 59 pp. 1245)

Chen, Qun; Zhou, Zhen-Rong
 On Gap Properties and Instabilities of $p$-Yang--Mills Fields We consider the $p$-Yang--Mills functional $(p\geq 2)$ defined as $\YM_p(\nabla):=\frac 1 p \int_M \|\rn\|^p$. We call critical points of $\YM_p(\cdot)$ the $p$-Yang--Mills connections, and the associated curvature $\rn$ the $p$-Yang--Mills fields. In this paper, we prove gap properties and instability theorems for $p$-Yang--Mills fields over submanifolds in $\mathbb{R}^{n+k}$ and $\mathbb{S}^{n+k}$. Keywords:$p$-Yang--Mills field, gap property, instability, submanifoldCategories:58E15, 53C05
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