We study the classification problem of singularities of function-germs with harmonic leading terms of two variables under the right-equivalence. We observe that the multiple actions of Laplacian appear for the classifications of such class of function-germs.
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13 results for “function-germ”
Study empty polar varieties' impact on singular function-germs.
problem Topology of singular function-germs with nonisolated singularities.
method Analysis of empty polar varieties.
result Topology implications of nonempty polar varieties.
We study the classification problem of singularities of function-germs with harmonic leading terms of two variables under the right-equivalence. We study the classification in the cases that the order of function-germs is at most 7. Moreover, we observe that the multiple actions of Laplacian appear for the classificati…
Study local topology of a function-germ deformation with a one-dimensional critical set.
problem Analyze the local topology of a deformation of a function-germ with a one-dimensional critical set.
method Use the Brasselet number to study the local topology of a deformation of a function-germ.
result Present a new proof of the Lê-Iomdin formula for the Brasselet number.
The paper examines bi-Lipschitz triviality of function germs on singular varieties.
problem Analyzing the bi-Lipschitz triviality of deformations of function germs on singular varieties.
method Introducing strongly rational -bi-Lipschitz trivial families and providing an infinitesimal criterion for bi-Lipschitz triviality.
result Bi-Lipschitz triviality of deformations of on when and are homogeneous of the same degree.
The Brasselet number helps calculate function germs with one-dimensional critical sets.
problem Calculating topological information of function germs with nonisolated singularities.
method Using the Brasselet number, the paper presents formulas for function germs with a one-dimensional critical locus.
result Formulas for function germs with a one-dimensional critical locus.
Tête-à-tête graphs were introduced by N. A'Campo in 2010 with the goal of modeling the monodromy of isolated plane curves. Mixed tête-à-tête graphs provide a generalization which define mixed tête-à-tête twists, which are pseudo-periodic automorphisms on surfaces. We characterize the mixed tête-à-tête twists as those p…
Study oscillatory integrals with degenerate singular points in multivariable phase functions.
problem Analyzing oscillatory integrals with degenerate singular points in phase functions.
method Using asymptotic expansions and results from one variable, the study examines multivariable phase functions.
result Asymptotic expansions of oscillatory integrals for multivariable phase functions with degenerate singular points.
The level curves of an analytic function germ almost always have bumps at unexpected points near the singularity. This profound discovery of N. A'Campo is fully explored in this paper for $f(z,w)\in \C\{z,w\}$, using the Newton-Puiseux infinitesimals and the notion of gradient canyon. Equally unexpected is the Dirac ph…
Khimshiashvili proved a topological degree formula for the Eu-ler characteristic of the Milnor fibres of a real function-germ with an isolated singularity. We give two generalizations of this result for non-isolated singularities. As corollaries we obtain an algebraic formula for the Euler characteristic of the fibres …
Paper proves families of singularities can be topologically trivialized.
problem Understanding behavior of singularities under small perturbations.
method Establishes sufficient conditions for embedded topological trivialization.
result New instances of topological stability, including -constant deformations.
The paper classifies symplectic invariants of specific singularities in integrable Hamiltonian systems.
problem Classifying symplectic invariants of singularities in integrable Hamiltonian systems.
method Smooth symplectic classification of Lagrangian fibrations near singularities.
result Action variables form complete symplectic invariants for parabolic orbits and cuspidal tori.
Symplectic classification for a specific type of singularity in integrable systems.
problem Symplectic classification of integrable systems near singular points of type .
method Real-analytic symplectic normal forms and classification of Lagrangian foliations.
result All integrable systems are symplectically equivalent near singular points of this type.