Kaloshin Vadim and Zhang Ke
- Published in print:
- 2020
- Published Online:
- May 2021
- ISBN:
- 9780691202525
- eISBN:
- 9780691204932
- Item type:
- chapter
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691202525.003.0004
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics
This chapter examines the geometrical structure for the system at double resonance. After describing the normal form near a double resonance, it reduces the system to the slow mechanical system with ...
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This chapter examines the geometrical structure for the system at double resonance. After describing the normal form near a double resonance, it reduces the system to the slow mechanical system with perturbation. The system is conjugate to a perturbation of a two degrees of freedom mechanical system after a coordinate change and an energy reduction. The chapter then formulates the non-degeneracy conditions and theorems about their genericity. It also considers the normally hyperbolic invariant cylinders, and sketches the proof using local and global maps. The periodic orbits obtained in Theorem 4.4 correspond to the fixed points of compositions of local and global maps, when restricted to the suitable energy surfaces.Less
This chapter examines the geometrical structure for the system at double resonance. After describing the normal form near a double resonance, it reduces the system to the slow mechanical system with perturbation. The system is conjugate to a perturbation of a two degrees of freedom mechanical system after a coordinate change and an energy reduction. The chapter then formulates the non-degeneracy conditions and theorems about their genericity. It also considers the normally hyperbolic invariant cylinders, and sketches the proof using local and global maps. The periodic orbits obtained in Theorem 4.4 correspond to the fixed points of compositions of local and global maps, when restricted to the suitable energy surfaces.
Kaloshin Vadim and Zhang Ke
- Published in print:
- 2020
- Published Online:
- May 2021
- ISBN:
- 9780691202525
- eISBN:
- 9780691204932
- Item type:
- chapter
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691202525.003.0010
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics
This chapter proves the geometric picture of double resonance described in Chapter 4. There are two cases. In the simple critical homology case, the chapter shows the homoclinic orbit can be extended ...
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This chapter proves the geometric picture of double resonance described in Chapter 4. There are two cases. In the simple critical homology case, the chapter shows the homoclinic orbit can be extended to periodic orbits both in positive and negative energy. The union of these periodic orbits forms a normally hyperbolic invariant manifold (which is homotopic to a cylinder with a puncture). In the non-simple homology case, the chapter demonstrates that for positive energy, there exist periodic orbits. The strategy is to prove the existence of these periodic orbits as hyperbolic fixed points of composition of local and global maps. A main technical tool to prove the existence and uniqueness of these fixed points is the Conley-McGehee isolation block.Less
This chapter proves the geometric picture of double resonance described in Chapter 4. There are two cases. In the simple critical homology case, the chapter shows the homoclinic orbit can be extended to periodic orbits both in positive and negative energy. The union of these periodic orbits forms a normally hyperbolic invariant manifold (which is homotopic to a cylinder with a puncture). In the non-simple homology case, the chapter demonstrates that for positive energy, there exist periodic orbits. The strategy is to prove the existence of these periodic orbits as hyperbolic fixed points of composition of local and global maps. A main technical tool to prove the existence and uniqueness of these fixed points is the Conley-McGehee isolation block.
Marco Abate
Araceli Bonifant, Mikhail Lyubich, and Scott Sutherland (eds)
- Published in print:
- 2014
- Published Online:
- October 2017
- ISBN:
- 9780691159294
- eISBN:
- 9781400851317
- Item type:
- chapter
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691159294.003.0017
- Subject:
- Mathematics, Combinatorics / Graph Theory / Discrete Mathematics
This chapter uses techniques from the theory of local dynamics of holomorphic germs tangent to the identity to prove three index theorems for global meromorphic maps of projective space. More ...
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This chapter uses techniques from the theory of local dynamics of holomorphic germs tangent to the identity to prove three index theorems for global meromorphic maps of projective space. More precisely, the chapter seeks to prove a particular index theorem: Let f : ℙⁿ ⇢ ℙⁿ be a meromorphic self-map of degree ν + 1 ≥ 2 of the complex n-dimensional projective space. Let Σ(f) = Fix(f) ∪ I(f) be the union of the indeterminacy set I(f) of f and the fixed points set Fix(f) of f. Let Σ(f) = ⊔subscript Greek Small Letter AlphaΣsubscript Greek Small Letter Alpha be the decomposition of Σ in connected components, and denote by N the tautological line bundle of ℙⁿ. After laying out the statements under this theorem, the chapter discusses the proofs.Less
This chapter uses techniques from the theory of local dynamics of holomorphic germs tangent to the identity to prove three index theorems for global meromorphic maps of projective space. More precisely, the chapter seeks to prove a particular index theorem: Let f : ℙⁿ ⇢ ℙⁿ be a meromorphic self-map of degree ν + 1 ≥ 2 of the complex n-dimensional projective space. Let Σ(f) = Fix(f) ∪ I(f) be the union of the indeterminacy set I(f) of f and the fixed points set Fix(f) of f. Let Σ(f) = ⊔subscript Greek Small Letter AlphaΣsubscript Greek Small Letter Alpha be the decomposition of Σ in connected components, and denote by N the tautological line bundle of ℙⁿ. After laying out the statements under this theorem, the chapter discusses the proofs.
Zhuolin Feng and Luyang Gao
- Published in print:
- 2020
- Published Online:
- November 2020
- ISBN:
- 9780198853022
- eISBN:
- 9780191887420
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198853022.003.0006
- Subject:
- Business and Management, Knowledge Management, Political Economy
International collaborations among higher education institutions have existed along the Silk Road for a long time. With the initiative of the Belt and Road, it is believed that cooperation among ...
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International collaborations among higher education institutions have existed along the Silk Road for a long time. With the initiative of the Belt and Road, it is believed that cooperation among universities and colleges may be influenced in various respects. In this study, seventeen university consortia that have been formed in the last thirty years along the Silk Road were investigated, including the membership structure, their starting point, goals and developments of these consortia, and the roles of world-class universities. The findings showed: 1) these consortia share the similar goals of multinational collaboration to foster solutions to new global challenges; 2) the thirty-year development of these consortia was divided into three phases, and most of the member universities are from the EU, China, and the US; 3) world-class universities were, are and will be playing the leading roles in these consortia continuously.Less
International collaborations among higher education institutions have existed along the Silk Road for a long time. With the initiative of the Belt and Road, it is believed that cooperation among universities and colleges may be influenced in various respects. In this study, seventeen university consortia that have been formed in the last thirty years along the Silk Road were investigated, including the membership structure, their starting point, goals and developments of these consortia, and the roles of world-class universities. The findings showed: 1) these consortia share the similar goals of multinational collaboration to foster solutions to new global challenges; 2) the thirty-year development of these consortia was divided into three phases, and most of the member universities are from the EU, China, and the US; 3) world-class universities were, are and will be playing the leading roles in these consortia continuously.
Yulia Frumer
- Published in print:
- 2018
- Published Online:
- September 2018
- ISBN:
- 9780226516448
- eISBN:
- 9780226524719
- Item type:
- chapter
- Publisher:
- University of Chicago Press
- DOI:
- 10.7208/chicago/9780226524719.003.0007
- Subject:
- History, History of Science, Technology, and Medicine
Chapter 6 demonstrates how ideas about time measurement that originated in astronomical practice and spread in early nineteenth century Japanese society through surveying and navigation manifested in ...
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Chapter 6 demonstrates how ideas about time measurement that originated in astronomical practice and spread in early nineteenth century Japanese society through surveying and navigation manifested in individuals’ perceptions of time. The chapter does so by focusing on the work of one individual—Endō Takanori. Takanori was born into a high-ranking samurai family, in line to become an infrastructure minister of the Kaga domain, which roughly overlaps with present-day Kanazawa. In preparation for his vocation, he studied astronomy and surveying, and indirectly learned from another Kaga-born scholar: Honda Toshiaki. Takanori absorbed the notion of time as rooted in celestial motion and geographic location, and concluded that human measurement of time had to correspond with physical reality. Later in life he became interested in Western drawing methods and developed an aesthetic theory, according to which Western-style depiction offered a superior representation of physical reality because it was based on the same principles as astronomy and surveying. And finally, his interrogation of castaways who brought him a Russian clock led him to contemplate Japan’s place on the global map.Less
Chapter 6 demonstrates how ideas about time measurement that originated in astronomical practice and spread in early nineteenth century Japanese society through surveying and navigation manifested in individuals’ perceptions of time. The chapter does so by focusing on the work of one individual—Endō Takanori. Takanori was born into a high-ranking samurai family, in line to become an infrastructure minister of the Kaga domain, which roughly overlaps with present-day Kanazawa. In preparation for his vocation, he studied astronomy and surveying, and indirectly learned from another Kaga-born scholar: Honda Toshiaki. Takanori absorbed the notion of time as rooted in celestial motion and geographic location, and concluded that human measurement of time had to correspond with physical reality. Later in life he became interested in Western drawing methods and developed an aesthetic theory, according to which Western-style depiction offered a superior representation of physical reality because it was based on the same principles as astronomy and surveying. And finally, his interrogation of castaways who brought him a Russian clock led him to contemplate Japan’s place on the global map.
