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Refraction
Published in Marlos A. G. Viana, Vasudevan Lakshminarayanan, Symmetry in Optics and Vision Studies, 2019
Marlos A. G. Viana, Vasudevan Lakshminarayanan
The condition |a|2−|b|2=1 characterizes the group SU1,1, homeomorphic to the open solid torus D×S1. See, for example, [18, pg. 57]. The immediate data-analytic consequence is that the torus renders itself as a summary space for the homogeneous (projective) data (4.12). For refractive matrices as points in the dihedral group algebra, the non-zero scalar coefficients are, from (4.1), x0=μ+1,h0=C+, and h1=C×, so that a=μ and b=C++iC×, whereas the projective condition (4.13) becomes, for μ≠−1, 0=0μ+1=y1y0=C×C+=sin[2α]cos[2α]=tan[2α],
Existence of SRB measures for hyperbolic maps with weak regularity
Published in Dynamical Systems, 2023
Fix a expanding map T of the circle that does not have an absolutely continuous invariant measure (see [21]), and has a physical measure (see [9]). Let be the open solid torus, and define by The map f is a hyperbolic on and any invariant measure of f is supported in Λ. If f has an SRB measure μ, then this measure projects to an absolutely continuous invariant measure for T, so f has no SRB measure. The product of the physical measure of T and the Bernoulli measure on the cantor set is an invariant physical measure of f.
Automatic method to calculate the storage volume of refrigerator assembly based on binary voxel images
Published in Journal of the Chinese Institute of Engineers, 2021
Jianhui Fu, Yoongho Jung, Jin Wang, Guodong Lu, Wei Wang, Xinwei Zhang
The second category focuses on filling the holes and gaps of a model to construct a closed boundary surface. Many scholars locate geometric holes or gaps by searching loops comprising linear edges contained in only one polygon and incrementally triangulating the loop to generate a smooth and continuous surface (Guo, Xiao, and Wang 2018; Pérez et al. 2016; Attene, Campen, and Kobbelt 2013; Altantsetseg et al. 2017; Wang et al. 2018). This method is inapplicable to a refrigerator assembly because the holes and gaps are topological. Topological holes or gaps occur when a closed path on an object cannot be transformed into a single point by a sequence of elementary local deformations (Kong 1989), such as a solid torus having one topological hole but no geometric hole. Examples of topological holes and gaps include tunnels and handles. Bischoff, Pavic, and Kobbelt (2005) developed a morphological closing operation to fill topological holes on the outer hull of an object. However, this operation cannot handle cases of holes in the internal structure. Aktouf, Bertrand, and Perroton (2002) proposed a tunnel-closing algorithm that fills topological holes by repeatedly checking the morphological features of background voxels based on the digital topology. However, this algorithm cannot deal with the solid tentative to trap a space, such as a thin solid sphere with a small tunnel (Janaszewski, Postolski, and Babout 2011).
The 0:1 resonance bifurcation associated with the supercritical Hamiltonian pitchfork bifurcation
Published in Dynamical Systems, 2023
Now, we look back to the full system (11) in domain . On the 3-dimensional manifold in the -space, there exists two stable (elliptic) periodic solutions of the form which are related to the centres of (12) (here and are determined by the given initial conditions). In fact, by using the -action generated by (), all closed orbits (e.g. ) in the planar system (12) correspond to 2-tori in the 4-dimensional system . Specifically, the periodic solutions (e.g. ) with period (see (23) in Section 3) of (12) for , create quasi-periodic solutions in the full system (11). They live in a 2-torus in which is bounded component of Cartesian product with . For different values of , a family of 2-tori is created with respect to the energy value h and the parameter λ. Similarly, the closed orbits and of (12) for give rise to two other families of 2-tori. On the 3-dimensional manifold in -space, there exists an unstable (hyperbolic) periodic solution of the form which is related to the saddle O of (12). The periodic solution (17) has 2-dimensional unstable and stable manifolds, respectively, which intersect transversally at (17), and are also connected to each other to form two 2-dimensional (symmetric homoclinic) manifolds in homoclinic to the periodic solution (17). This 2-dimensional symmetric homoclinic manifold is surrounded by a solid 2-torus (i.e. (16)). And this solid torus is foliated by parallel invariant 2-tori (i.e. (15)), shrinking down to the 2-dimensional homoclinic manifold. To sum up, the unfolding () undergoes the supercritical HPB of a periodic orbit at the half straight line .