Ball-Morph: Definition, Implementation, and Comparative Evaluation
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Abstract:
We define b-compatibility for planar curves and propose three ball morphing techniques between pairs of b-compatible curves. Ball-morphs use the automatic ball-map correspondence, proposed by Chazal et al. [1], from which we derive different vertex trajectories (linear, circular, and parabolic). All three morphs are symmetric, meeting both curves with the same angle, which is a right angle for the circular and parabolic. We provide simple constructions for these ball-morphs and compare them to each other and other simple morphs (linear-interpolation, closest-projection, curvature-interpolation, Laplace-blending, and heat-propagation) using six cost measures (travel-distance, distortion, stretch, local acceleration, average squared mean curvature, and maximum squared mean curvature). The results depend heavily on the input curves. Nevertheless, we found that the linear ball-morph has consistently the shortest travel-distance and the circular ball-morph has the least amount of distortion.Keywords:
Ball (mathematics)
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In this note we study the behaviour of holomorphic functions in the unit ball BJV in on one-dimensional complex subspaces of C N .The behaviour of functions is described in terms of L 2 -integrability with weights on the sets L n B/v, where L runs over different families E of one-dimensional complex subspaces of C N .
Ball (mathematics)
Bergman space
Bergman kernel
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A local curvature multi-vertex model was developed. This model is a straightforward two-dimensional topological network model based on physical principles that consider the local curvatures of grain boundaries and the grain boundary tensions at triple junctions. Virtual vertices are set on the grain boundaries in order to calculate the driving forces of grain boundary and triple junction migration. Therefore, the accuracy of the developed model is higher than that of the conventional curvature model and the vertex model. In the proposed model, the generation and annihilation of virtual vertices maintained a proper configuration of virtual vertices, and high accuracy is expected with a suitable set of simulation parameters. The proposed model was verified by the grain growth simulation using adequately determined parameters for the artificially generated specimens with 5040 grains.
Vertex model
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One of the most widely known topological descriptors is the Wiener index or (Wiener number) named after American chemist Harold Wiener in 1947.Wiener number of a connected graph G is defined as the sum of the distances between distinct pairs of vertices of G..It correlates between physico-chemical and structural properties.The hyper wiener index denoted by WW of a graph G was introduced by Randic and his definition is applicable to trees only.Klein, Lukovits and Gutman introduced the formula for both trees and cycle containing structures.The Hyper Wiener index is defined as WW (G) = (∑d 2 (u ,v)+∑d(u,v))/2, where d (u,v) denotes the distance between the vertices u and v in the graph G and the summations run over all distinct pairs of vertices of G.Recently an edge version of Hyper Wiener Index was introduced by Ali Iranmanesh.In this paper, we have determined Hyper Wiener numbers of some Cluster graphs and also for some bipartite cluster graphs
Wiener index
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Intensity
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In this paper, He's Variational Iteration Method (HVIM) is used to study the nonlinear timeinvariant and time-varying singular systems.The results obtained using He's Variational Iteration Method and the methods taken from the literature [18] were compared with the exact solutions of the nonlinear timeinvariant and time-varying singular systems.It is found that the solution obtained using the He's Variational Iteration Method is closer to the exact solutions of the nonlinear time-invariant and time-varying singular systems.Error Calculations for discrete and exact solutions are presented in a table form to highlight the efficiency of this method.
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Based on the properties of curvature connection and theory of differential geometry, a sufficient condition of curvature connection between two adjacent surfaces is obtained, and then a new method of curvature connection of surface patches around a common vertex is put forward by using these conditions and consistency. As a result, the obtained surface patches have the lowest degrees, whose degrees are five, and the relevant system of equations can be solved easily.
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In some recent papers the method of partial differences introduced by the author in [4] was very helpful in the construction of cyclic cycle systems. Here we use and describe in all details this method for the purpose of constructing, more generally, cycle decompositions with a sharply vertex transitive automorphism group not necessarily cyclic.
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A local curvature multi-vertex model was developed. This model is a straightforward two-dimensional topological network model based on physical principles that consider the local curvatures of grain boundaries and the grain boundary tensions at triple junctions. Virtual vertices are set on the grain boundaries in order to calculate the driving forces of grain boundary and triple junction migration. Therefore, the accuracy of the developed model is higher than that of the conventional curvature model and the vertex model. In the proposed model, the generation and annihilation of virtual vertices maintained a proper configuration of virtual vertices, and high accuracy is expected with a suitable set of simulation parameters. The proposed model was verified by the grain growth simulation using adequately determined parameters for the artificially generated specimens with 5040 grains.
Vertex model
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