• 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.

• 대한수학회지
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• 제42권1호
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• pp.65-83
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• 2005

In this paper, we prove that the two well-known natural normalizations of Hamiltonian functions on the symplectic manifold ($M,\;{\omega}$) canonically relate the action spectra of different normalized Hamiltonians on arbitrary symplectic manifolds ($M,\;{\omega}$). The natural classes of normalized Hamiltonians consist of those whose mean value is zero for the closed manifold, and those which are compactly supported in IntM for the open manifold. We also study the effect of the action spectrum under the ${\pi}_1$ of Hamiltonian diffeomorphism group. This forms a foundational basis for our study of spectral invariants of the Hamiltonian diffeomorphism in [8].

• 대한수학회지
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• 제46권2호
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• pp.363-447
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• 2009

The author previously defined the spectral invariants, denoted by $\rho(H;\;a)$, of a Hamiltonian function H as the mini-max value of the action functional ${\cal{A}}_H$ over the Novikov Floer cycles in the Floer homology class dual to the quantum cohomology class a. The spectrality axiom of the invariant $\rho(H;\;a)$ states that the mini-max value is a critical value of the action functional ${\cal{A}}_H$. The main purpose of the present paper is to prove this axiom for nondegenerate Hamiltonian functions in irrational symplectic manifolds (M, $\omega$). We also prove that the spectral invariant function ${\rho}_a$ : $H\;{\mapsto}\;\rho(H;\;a)$ can be pushed down to a continuous function defined on the universal (${\acute{e}}tale$) covering space $\widetilde{HAM}$(M, $\omega$) of the group Ham((M, $\omega$) of Hamiltonian diffeomorphisms on general (M, $\omega$). For a certain generic homotopy, which we call a Cerf homotopy ${\cal{H}}\;=\;\{H^s\}_{0{\leq}s{\leq}1}$ of Hamiltonians, the function ${\rho}_a\;{\circ}\;{\cal{H}}$ : $s\;{\mapsto}\;{\rho}(H^s;\;a)$ is piecewise smooth away from a countable subset of [0, 1] for each non-zero quantum cohomology class a. The proof of this nondegenerate spectrality relies on several new ingredients in the chain level Floer theory, which have their own independent interest: a structure theorem on the Cerf bifurcation diagram of the critical values of the action functionals associated to a generic one-parameter family of Hamiltonian functions, a general structure theorem and the handle sliding lemma of Novikov Floer cycles over such a family and a family version of new transversality statements involving the Floer chain map, and many others. We call this chain level Floer theory as a whole the Floer mini-max theory.

• 대한수학회보
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• 제54권5호
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• pp.1803-1825
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• 2017

Let X be a smooth scheme with an action of an algebraic group G. We establish an equivalence of two categories related to the corresponding moment map ${\mu}:T^{\ast}X{\rightarrow}g^{\ast}$ - the derived category of G-equivariant coherent sheaves on the derived fiber ${\mu}^{-1}(0)$ and the derived category of G-equivariant matrix factorizations on $T^{\ast}X{\times}g$ with potential given by ${\mu}$.

• 대한수학회지
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• 제60권2호
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• pp.375-393
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• 2023

Rabinowitz action functional is the Lagrange multiplier functional of the negative area functional to a constraint given by the mean value of a Hamiltonian. In this note we show that on a symplectization there is a one-to-one correspondence between gradient flow lines of Rabinowitz action functional and gradient flow lines of the restriction of the negative area functional to the constraint. In the appendix we explain the motivation behind this result. Namely that the restricted functional satisfies Chas-Sullivan additivity for concatenation of loops which the Rabinowitz action functional does in general not do.

• 대한수학회보
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• 제39권2호
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• pp.277-282
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• 2002

We Show that When ($M,\;\omega$) is a closed, simply connected, symplectic manifold for all $\gamma\;\in\;\pi_1(Ham(M),\;id)$ the following inequality holds: $\parallel\gamma\parallel\;{\geq}\;sup_{\={x}}\;|A(\={x})|,\;where\;\parallel\gamma\parallel$ is the coarse Hofer's norm, $\={x}$ run over all extensions to $D^2$ of an orbit $x(t)\;=\;{\varphi}_t(z)$ of a fixed point $z\;\in\;M,\;A(\={x})$ the symplectic action of $\={x}$, and the Hamiltonian diffeomorphisms {${\varphi}_t$} of M represent $\gamma$.

• 대한수학회보
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• 제38권4호
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• pp.741-751
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• 2001

In this paper when $(M, \omega)$ is a compact weakly exact symplectic manifold with nonempty boundary satisfying $c_1｜{\pi}_2(M)$ = 0, we construct an action spectrum bundle over the group of Hamil-tonian diffeomorphisms of the manifold M generated by the time-dependent Hamiltonian vector fields, whose fibre is nowhere dense and invariant under symplectic conjugation.

• Journal of Mechanical Science and Technology
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• 제17권12호
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• pp.1922-1927
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• 2003

Nonlinear normal mode (NNM) vibration, in a nonlinear dual mass Hamiltonian system, which has 6$\^$th/ order homogeneous polynomial as a nonlinear term, is studied in this paper. The existence, bifurcation, and the orbital stability of periodic motions are to be studied in the phase space. In order to find the analytic expression of the invariant curves in the Poincare Map, which is a mapping of a phase trajectory onto 2 dimensional surface in 4 dimensional phase space, Whittaker's Adelphic Integral, instead of the direct integration of the equations of motion or the Birkhoff-Gustavson (B-G) canonical transformation, is derived for small value of energy. It is revealed that the integral of motion by Adelphic Integral is essentially consistent with the one obtained from the B-G transformation method. The resulting expression of the invariant curves can be used for analyzing the behavior of NNM vibration in the Poincare Map.

• 디지털콘텐츠학회 논문지
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• 제18권6호
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• pp.1119-1125
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• 2017

장르 및 행위는 스토리의 분류뿐만 아니라 그 특성 데이터의 분포를 가시적으로 나타나는 데에도 효과적으로 사용될 수 있다. 본 논문에서는 스토리 특성 데이터의 분포를 장르-행위의 2차원 평면에서 가시화함에 있어, 인접한 장르 및 인접한 행위가 서로 유사성을 가지는 즉 공간적 특성을 가지는 장르-행위 좌표계를 제안한다. 제안된 장르-행위 좌표계를 이용하여 스토리 특성 데이터의 분포를 가시화 해본 결과 유사도가 높은 항목들이 연이여 좌표계의 항목을 이루고 또한 관련성 있는 특성 데이터들이 군집을 이루어 나타나는 등 공간적 의미를 가지도록 스토리 특성 데이터의 가시화가 가능함을 확인하였다.

• 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.