О Топологии Волновых Фронтов В Пространствах Небольших Размерностей

Седых, Вячеслав Дмитриевич; Sedykh, V. D.

doi:10.4213/im4572

Izv. RAN. Ser. Mat.

2012

DOI: 10.4213/im4572

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О Топологии Волновых Фронтов В Пространствах Небольших Размерностей

Вячеслав Дмитриевич Седых¹,

V. D. Sedykh²

Abstract: О топологии волновых фронтов в пространствах небольших размерностей Вычислены индексы примыкания особенностей волновых фронтов общего положения в пространствах размерности n 6. В качестве следствия найдены новые условия сосуществования особенностей волновых фронтов. Библиография: 13 наименований. Ключевые слова: лежандровы отображения, волновые фронты, особенности (мультиособенности), индекс примыкания, эйлерова характеристика.

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Cited by 6 publications

References 11 publications

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The topology of adjacencies of type A and D Lagrangian singularities

Sedykh

2014

Funct Anal Its Appl

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Adjacencies of multisingularities of stable Lagrangian maps with singularities of types A ± µ and D ± µ are studied.We calculate the homotopy type of adjacencies of multisingularities in the image of a stable Lagrangian map with singularities of types A ± μ and D ± μ . Decompositions of real singularities have also been studied by other authors; see, e.g., [2] and [6].1. Singularities of Lagrangian maps. First, we recall the basic facts of the theory of Lagrangian singularities; see [1] for details.Let E be a smooth 2n-manifold with a symplectic structure ω. A smooth n-dimensional submanifold L in E is said to be Lagrangian if ω| L = 0. A smooth fibration ρ : E → V over a smooth n-manifold V is said to be Lagrangian if all of its fibers are Lagrangian submanifolds in E.Let ρ be a Lagrangian fibration, and let L be a Lagrangian submanifold in E. Then the map f = ρ| L : L → V is said to be Lagrangian. We consider only proper Lagrangian maps.According to Arnold's theorem, all simple stable singularities of Lagrangian maps are of typesTo be more precise, any simple stable germ of a Lagrangian map is Lagrange equivalent to the germ at zero of a Lagrangian map R n → R n , (t, q) → (−∂S(t, q)/∂t, q), where t = (t 1 , . . . , t k ), q = (q k+1 , . . . , q n ), and S = S(t, q) is a smooth function of one of the following forms (μ n + 1 is an integer):+ · · · + q 3 t 2 2 , μ 4, E ± 6 : S = t 3 1 ± t 4 2 + q 5 t 1 t 2 2 + q 4 t 1 t 2 + q 3 t 2 2 , μ = 6, E 7 : S = t 3 1 + t 1 t 3 2 + q 6 t 2 1 t 2 + q 5 t 2 1 + q 4 t 1 t 2 + q 3 t 2 2 , μ= 7,The number μ is called the degree of the singularity, and μ − 1 is its codimension. If μ is even or μ = 1, then all singularities of types A + μ and A − μ are Lagrange equivalent (and we denote these types by A μ ).Let S + be the free Abelian multiplicative semigroup generated by the symbols E 7 , and E 8 . We denote the identity element of this semigroup by 1.The elements of S + are used to index the types of multisingularities in the image a stable Lagrangian map with simple singularities. Namely, consider any element A = X 1 . . . X p ∈ S + \ {1}, where X 1 , . . . , X p is any set of generators of S + . A Lagrangian map f : L → V has a multisingularity of type A at a point y ∈ V if (i) f −1 (y) consists of p pairwise different points; and (ii) the points in f −1 (y) can be ordered as x 1 , . . . , x p so that f has singularities of types X 1 , . . . , X p , respectively, at these points. At the points of the complement V \ f (L), the map f has multisingularities of type 1.

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The topology of adjacencies of type A and D Lagrangian singularities

Sedykh

2014

Funct Anal Its Appl

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On Lagrangian and Legendrian Singularities

Sedykh

2020

Arnold Math J.

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On the topology of stable Lagrangian maps with singularities of types $ A$ and $ D$

Sedykh¹

2015

Izv. Math.

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We show that the 3D charged Banados-Teitelboim-Zanelli (BTZ) black hole solution interpolates between two different 2D AdS spacetimes: a near-extremal, nearhorizon AdS 2 geometry with constant dilaton and U(1) field and an asymptotic AdS 2 geometry with a linear dilaton. Thus, the charged BTZ black hole can be considered as interpolating between the two different formulations proposed until now for AdS 2 quantum gravity. In both cases the theory is the chiral half of a 2D CFT and describes, respectively, Brown-Hennaux-like boundary deformations and near-horizon excitations. The central charge c as of the asymptotic CFT is determined by 3D Newton constant G and the AdS length l, c as = 3l/G, whereas that of the near-horizon CFT also depends on the U(1) charge Q, c nh ∝ lQ/ √ G.

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О Топологии Волновых Фронтов В Пространствах Небольших Размерностей

Cited by 6 publications

References 11 publications

The topology of adjacencies of type A and D Lagrangian singularities

The topology of adjacencies of type A and D Lagrangian singularities

On Lagrangian and Legendrian Singularities

On the topology of stable Lagrangian maps with singularities of types $ A$ and $ D$

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