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Another note on focus-focus singularities  [PDF]
Nguyen Tien Zung
Mathematics , 2001,
Abstract: We show a natural relation between the monodromy formula for focus-focus singularities of integrable Hamiltonian systems and a formula of Duistermaat-Heckman, and extend the main results of our previous note on focus-focus singularities ($\bbS^1$-action, monodromy, and topological classification) to the degenerate case. We also consider the non-Hamiltonian case, local normal forms, etc.
Focus-focus singularities in classical mechanics  [PDF]
Gleb Smirnov
Mathematics , 2013,
Abstract: In this paper the local singularities of integrable Hamiltonian systems with two degrees of freedom are studied. The topological obstruction to the existence of focus-focus singularity with given complexity was found. It has been showed that only simple focus-focus singularities can appear in a typical mechanical system. The model examples of mechanical systems with complex focus-focus singularity are given.
A note on graded hypersurface singularities  [PDF]
Patricio Gallardo
Mathematics , 2013,
Abstract: For a weighted quasihomogeneous two dimensional hypersurface singularity, we define a smoothing with unipotent monodromy and an isolated graded normal singularity. We study the natural weighted blow up of both the smoothing and the surface. In particular, we describe our construction for the quasihomogeneous singularities of type I, the 14 unimodal exceptional singularities and we relate it to their stable replacement.
A note on toroidalization: the problem of resolution of singularities of morphisms in the logarithmic category  [PDF]
Kenji Matsuki
Mathematics , 2003,
Abstract: This is a note on toroidalization, formulated as the problem of resolution of singularities of morphisms in the logarithmic category. It is submitted to the proceedings of the Barrett conference held at the University of Tennessee at Knoxville in April 2002.
Semiclassical inverse spectral theory for singularities of focus-focus type  [PDF]
álvaro Pelayo,San V? Ngoc
Mathematics , 2013, DOI: 10.1007/s00220-014-1998-9
Abstract: We prove, assuming that the Bohr-Sommerfeld rules hold, that the joint spectrum near a focus-focus critical value of a quantum integrable system determines the classical Lagrangian foliation around the full focus-focus leaf. The result applies, for instance, to h-pseudodifferential operators, and to Berezin-Toeplitz operators on prequantizable compact symplectic manifolds.
On semi-global invariants for focus-focus singularities  [PDF]
San Vu Ngoc
Mathematics , 2002,
Abstract: This article gives a classification, up to symplectic equivalence, of singular Lagrangian foliations given by a completely integrable system of a 4-dimensional symplectic manifold, in a full neighbourhood of a singular leaf of focus-focus (=nodal) type.
Higher index focus-focus singularities in the Jayne-Cummings-Gaudin model : symplectic invariants and monodromy  [PDF]
Olivier Babelon,Benoit Doucot
Physics , 2013,
Abstract: We study the symplectic geometry of the Jaynes-Cummings-Gaudin model with $n=2m-1$ spins. We show that there are focus-focus singularities of maximal Williamson type $(0,0,m)$. We construct the linearized normal flows in the vicinity of such a point and show that soliton type solutions extend them globally on the critical torus. This allows us to compute the leading term in the Taylor expansion of the symplectic invariants and the monodromy associated to this singularity.
A note on bimodal singularities and mirror symmetry  [PDF]
Makiko Mase,Kazushi Ueda
Mathematics , 2014,
Abstract: We discuss the relation between transposition mirror symmetry of Berlund and H\"ubsch for bimodal singularities and polar duality of Batyrev for associated toric K3 hypersurfaces. We also show that homological mirror symmetry for singularities implies the geometric construction of Coxeter-Dynkin diagrams of bimodal singularities by Ebeling and Ploog.
A note on exceptional unimodal singularities and K3 surfaces  [PDF]
Masanori Kobayashi,Makiko Mase,Kazushi Ueda
Mathematics , 2011,
Abstract: This is a short note on the relation between the graded stable derived categories of 14 exceptional unimodal singularities and the derived category of K3 surfaces obtained as compactifications of the Milnor fibers. As a corollary, we obtain a basis of the numerical Grothendieck group similar to the one given by Ebeling and Ploog (arXiv:0809.2738).
A note on the $δ$-singularities of the static electric and magnetic fields  [PDF]
Constantin Vrejoiu,Roxana Zus
Physics , 2010,
Abstract: The $\delta-$singularities of the electric and magnetic fields in the static case are established based on the regularized $\delta_\eps(\rvec)$ function introduced by Jackson.
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