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The Gq/11 family of Gα subunits is necessary and sufficient for lower jaw development.
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1995
Abstract This is a foundational chapter, and everything in it (except perhaps Section 3.4 on contact structures) is needed to understand later chapters. The first section contains elementary definitions and first examples of symplectic manifolds. The second section is devoted to Darboux’s theorem.
Dusa McDuff, Dietmar Salamon
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Abstract This is a foundational chapter, and everything in it (except perhaps Section 3.4 on contact structures) is needed to understand later chapters. The first section contains elementary definitions and first examples of symplectic manifolds. The second section is devoted to Darboux’s theorem.
Dusa McDuff, Dietmar Salamon
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2017
The third chapter introduces the basic notions of symplectic topology, such as symplectic forms, symplectomorphisms, and Lagrangian submanifolds. A fundamental classical construction is Moser isotopy, with its various applications such as Darboux’s theorem and the Lagrangian neighbourhood theorem.
Dusa McDuff, Dietmar Salamon
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The third chapter introduces the basic notions of symplectic topology, such as symplectic forms, symplectomorphisms, and Lagrangian submanifolds. A fundamental classical construction is Moser isotopy, with its various applications such as Darboux’s theorem and the Lagrangian neighbourhood theorem.
Dusa McDuff, Dietmar Salamon
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Locally Lagrange-Symplectic Manifolds
Geometriae Dedicata, 1999Motivated by Lagrangian dynamics, several authors related the tangent structure of the tangent bundle \(TN\) of a manifold \(N\) with the symplectic form associated to a nondegenerate Lagrangian \({\mathcal L}: TN\to\mathbb{R}\). In this paper, the author gives a generalization of this relation for any symplectic manifold \((M,\omega)\) endowed with a ...
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Symplectic manifolds and Poisson manifolds
1987In this chapter we define the important notion of a symplectic manifold and some directly related notions: coisotropic, isotropic and Lagrangian submanifolds of a symplectic manifold, symplectomorphisms and locally or globally Hamiltonian vector fields.
Paulette Libermann, Charles-Michel Marle
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2004
In this section we recall some general facts about symplectic manifolds. Then we give a short discussion of Moser’s method, which is applied in the proof of numerous fundamental statements discussed in the text. The chapter concludes with a short review on what is known about the classification of symplectic 4-manifolds.
Burak Ozbagci, András I. Stipsicz
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In this section we recall some general facts about symplectic manifolds. Then we give a short discussion of Moser’s method, which is applied in the proof of numerous fundamental statements discussed in the text. The chapter concludes with a short review on what is known about the classification of symplectic 4-manifolds.
Burak Ozbagci, András I. Stipsicz
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Constructing Symplectic Manifolds
1995Abstract In this chapter we look at various ways to construct symplectic manifolds. It is also possible to do various kinds of surgery, though the possibilities have been explored more in the contact case. We begin by studying blowing up and down in both the complex and the symplectic contexts.
Dusa McDuff, Dietmar Salamon
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Constructing symplectic manifolds
2017This chapter examines various ways to construct symplectic manifolds and submanifolds. It begins by studying blowing up and down in both the complex and the symplectic contexts. The next section is devoted to a discussion of fibre connected sums and describes Gompf’s construction of symplectic four-manifolds with arbitrary fundamental group.
Dusa McDuff, Dietmar Salamon
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