• 제목/요약/키워드: symplectic geometry

검색결과 8건 처리시간 0.017초

A NEW QUARTERNIONIC DIRAC OPERATOR ON SYMPLECTIC SUBMANIFOLD OF A PRODUCT SYMPLECTIC MANIFOLD

  • Rashmirekha Patra;Nihar Ranjan Satapathy
    • Korean Journal of Mathematics
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    • 제32권1호
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    • pp.83-95
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    • 2024
  • The Quaternionic Dirac operator proves instrumental in tackling various challenges within spectral geometry processing and shape analysis. This work involves the introduction of the quaternionic Dirac operator on a symplectic submanifold of an exact symplectic product manifold. The self adjointness of the symplectic quaternionic Dirac operator is observed. This operator is verified for spin ${\frac{1}{2}}$ particles. It factorizes the Hodge Laplace operator on the symplectic submanifold of an exact symplectic product manifold. For achieving this a new complex structure and an almost quaternionic structure are formulated on this exact symplectic product manifold.

EQUIVARIANT EMBEDDING OF TWO-TORUS INTO SYMPLECTIC MANIFOLD

  • Kim, Min Kyu
    • 충청수학회지
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    • 제20권2호
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    • pp.157-161
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    • 2007
  • We show that there is an equivariant symplectic embedding of a two-torus with a nontrivial action into a symplectic manifold with a symplectic circle action if and only if the circle action on the manifold is non-Hamiltonian. This is a new equivalent condition for non-Hamiltonian action and gives us a new insight to solve the famous conjecture by Frankel and McDuff.

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HOMOGENEOUS POLYNOMIAL HYPERSURFACE ISOLATED SINGULARITIES

  • Akahori, Takao
    • 대한수학회지
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    • 제40권4호
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    • pp.667-680
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    • 2003
  • The mirror conjecture means originally the deep relation between complex and symplectic geometry in Calabi-Yau manifolds. Recently, this conjecture is posed beyond Calabi-Yau, and even to, open manifolds (e.g. $A_{n}$ singularities and its resolution) is discussed. While if we treat open manifolds, we can't avoid the boundary (in our case, CR manifolds). Therefore we pose the more precise conjecture (mirror symmetry with boundaries). Namely, in mirror symmetry, for boundaries, what kind of structure should correspond\ulcorner For this problem, the $A_{n}$ case is studied.

A SURVEY ON SYMPLECTIC GEOMETRY

  • Nam, Jeong-Koo
    • 한국수학교육학회지시리즈B:순수및응용수학
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    • 제3권1호
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    • pp.19-32
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    • 1996
  • A symplectic manifold is a pair (M, $\omega$) consisting of a smooth manifold M and a non-degenerate closed 2-form $\omega$ on M. Locally, $\omega$ = (equation omitted) and d$\omega$ = 0, when n = dimM. The condition d$\omega$ = 0 implies that locally $\omega$ = d${\alpha}$ with ${\alpha}$ = (equation omitted). There are three main sources of symplectic manifolds.(omitted)

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Modal parameter identification of civil structures using symplectic geometry mode decomposition

  • Feng Hu;Lunhai Zhi;Zhixiang Hu;Bo Chen
    • Wind and Structures
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    • 제36권1호
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    • pp.61-73
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    • 2023
  • In this article, a novel structural modal parameters identification methodology is developed to determine the natural frequencies and damping ratios of civil structures based on the symplectic geometry mode decomposition (SGMD) approach. The SGMD approach is a new decomposition algorithm that can decompose the complex response signals with better decomposition performance and robustness. The novel method firstly decomposes the measured structural vibration response signals into individual mode components using the SGMD approach. The natural excitation technique (NExT) method is then used to obtain the free vibration response of each individual mode component. Finally, modal natural frequencies and damping ratios are identified using the direct interpolating (DI) method and a curve fitting function. The effectiveness of the proposed method is demonstrated based on numerical simulation and field measurement. The structural modal parameters are identified utilizing the simulated non-stationary responses of a frame structure and the field measured non-stationary responses of a supertall building during a typhoon. The results demonstrate that the developed method can identify the natural frequencies and damping ratios of civil structures efficiently and accurately.

A NOTE ON LOCAL CALIBRATIONS OF ALMOST COMPLEX STRUCTURES

  • Kim, Hyeseon
    • 호남수학학술지
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    • 제44권3호
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    • pp.384-390
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    • 2022
  • In this paper, we study the obstruction on the jets of an almost complex structure J to the existence of a symplectic form ω such that J is compatible with ω. We describe some almost complex structures on ℝ4 and on ℝ6, respectively, that cannot be calibrated by any symplectic forms. In particular, these examples pertain to the model almost complex structure on ℝ4 in [3], and the simple model structure on ℝ6 in [7].

RECENT DEVELOPMENTS IN DIFERENTIAL GEOMETRY AND MATHEMATICAL PHYSICS

  • Flaherty, F.J.
    • 대한수학회보
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    • 제24권1호
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    • pp.31-37
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    • 1987
  • I want to focus on developments in the areas of general relativity and gauge theory. The topics to be considered are the singularity theorms of Hawking and Penrose, the positivity of mass, instantons on the four-dimensional sphere, and the string picture of quantum gravity. I should mention that I will not have time do discuss either classical mechanics or symplectic structures. This is especially unfortunate, because one of the roots of differential geometry is planted firmly in mechanics, Cf. [GS]. The French geometer Elie Cartan first formulated his invariant approach to geometry in a series of papers on affine connections and general relativity, Cf. [C]. Cartan was trying to recast the Newtonian theory of gravity in the same framework as Einstein's theory. From the historical perspective it is significant that Cartan found relativity a convenient framework for his ideas. As about the same time Hermann Weyl in troduced the idea of gauge theory into geometry for purposes much different than those for which it would ultimately prove successful, Cf. [W]. Weyl wanted to unify gravity with electromagnetism and though that a conformal structure would fulfill thel task but Einstein rebutted this approach.

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TOPOLOGICAL METHOD DOES NOT WORK FOR FRANKEL-MCDUFF CONJECTURE

  • Kim, Min Kyu
    • 충청수학회지
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    • 제20권1호
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    • pp.31-35
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    • 2007
  • In dealing with transformation group, topological approach is very natural. But, it is not sufficient to investigate geometric properties of transformation group and we need geometric method. Frankel-McDuff Conjecture is very interesting in the point that it shows struggling between topological method and geometric method. In this paper, the author suggest generalized Frankel-McDuff conjecture as a topological version of the conjecture and construct a counterexample for the generalized version, and from this we assert that topological method does not work for Frankel-McDuff Conjecture.

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