Spectral embedding and spectral clustering are common methods for non-linear dimensionality reduction and clustering of complex high dimensional datasets. In this paper we provide a diffusion based probabilistic analy...
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(纸本)9783540737490
Spectral embedding and spectral clustering are common methods for non-linear dimensionality reduction and clustering of complex high dimensional datasets. In this paper we provide a diffusion based probabilistic analysis of algorithms that use the normalized graph Laplacian. Given the pairwise adjacency matrix of all points in a dataset, we define a random walk on the graph of points and a diffusion distance between any two points. We show that the diffusion distance is equal to the Euclidean distance in the embedded space with all eigenvectors of the normalized graph Laplacian. This identity shows that characteristic relaxation times and processes of the random walk on the graph are the key concept that governs the properties of these spectral clustering and spectral embedding algorithms. Specifically, for spectral clustering to succeed, a necessary condition is that the mean exit times from each cluster need to be significantly larger than the largest (slowest) of all relaxation times inside all of the individual clusters. For complex, multiscale data, this condition may not hold and multiscale methods need to be developed to handle such situations.
The stochastic generalized Ginzburg-Landau equation with additive noise can be solved pathwise and the unique solution generates a random system. Then we prove the random system possesses a global random attractor in ...
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The stochastic generalized Ginzburg-Landau equation with additive noise can be solved pathwise and the unique solution generates a random system. Then we prove the random system possesses a global random attractor in H 0 1 .
Compositional lipid domains (“lipid rafts”) in plasma membranes are believed to be important components of many cellular processes. The mechanisms by which cells regulate the sizes and lifetimes of these spatially e...
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Compositional lipid domains (“lipid rafts”) in plasma membranes are believed to be important components of many cellular processes. The mechanisms by which cells regulate the sizes and lifetimes of these spatially extended domains are poorly understood at the moment. Here we show that the competition between phase separation in an immiscible lipid system and active cellular lipid transport processes naturally leads to the formation of such domains. Furthermore, we demonstrate that local interactions with immobile membrane proteins can spatially localize the rafts and lead to further clustering.
We study the stability of a two-component Bose-Einstein condensate (BEC) in the parameter regime in which its classical counterpart has regular motion. The stability is characterized by the fidelity for both the same ...
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We study the stability of a two-component Bose-Einstein condensate (BEC) in the parameter regime in which its classical counterpart has regular motion. The stability is characterized by the fidelity for both the same and different initial states. We study as initial states the Fock states with definite numbers of atoms in each component of the BEC. It is found that for some initial times the two Fock states with all the atoms in the same component of the BEC are more stable than Fock states with atoms distributed in the two components. An experimental scheme is discussed, in which the fidelity can be measured in a direct way.
We investigate the state-specified capture process of antiprotons by helium. Freezing one of the two electrons, we reduce this four-body rearrangement problem into a three-body problem. The capture cross sections are ...
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We investigate the state-specified capture process of antiprotons by helium. Freezing one of the two electrons, we reduce this four-body rearrangement problem into a three-body problem. The capture cross sections are calculated by solving the Chew-Goldberger-type integral equation. Differing from the capture of antiprotons by hydrogen atoms, the bumpy structures are revealed in the total angular momentum dependent capture cross sections. Further analysis shows that the bumps arise from the partial channel closing due to the removal of the energy degeneracy in the antiprotonic helium.
We show that edge stresses introduce intrinsic ripples in freestanding graphene sheets even in the absence of any thermal effects. Compressive edge stresses along zigzag and armchair edges of the sheet cause out-of-pl...
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We show that edge stresses introduce intrinsic ripples in freestanding graphene sheets even in the absence of any thermal effects. Compressive edge stresses along zigzag and armchair edges of the sheet cause out-of-plane warping to attain several degenerate mode shapes. Based on elastic plate theory, we identify scaling laws for the amplitude and penetration depth of edge ripples as a function of wavelength. We also demonstrate that edge stresses can lead to twisting and scrolling of nanoribbons as seen in experiments. Our results underscore the importance of accounting for edge stresses in thermal theories and electronic structure calculations for freestanding graphene sheets.
Stationary rotating strings can be viewed as geodesic motions in appropriate metrics in two-dimensional space. We obtain all solutions describing stationary rotating strings in flat spacetime as an application. These ...
Stationary rotating strings can be viewed as geodesic motions in appropriate metrics in two-dimensional space. We obtain all solutions describing stationary rotating strings in flat spacetime as an application. These rotating strings have infinite length with various wiggly shapes. Averaged value of the string energy, the angular momentum, and the linear momentum along the string are discussed.
We show that under tension a classical many-body system with only isotropic pair interactions in a crystalline state can, counterintuitively, have a negative Poisson’s ratio, or auxetic behavior. We derive the condit...
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We show that under tension a classical many-body system with only isotropic pair interactions in a crystalline state can, counterintuitively, have a negative Poisson’s ratio, or auxetic behavior. We derive the conditions under which the triangular lattice in two dimensions and lattices with cubic symmetry in three dimensions exhibit a negative Poisson’s ratio. In the former case, the simple Lennard-Jones potential can give rise to auxetic behavior. In the latter case, a negative Poisson’s ratio can be exhibited even when the material is constrained to be elastically isotropic.
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