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Isometric drawings
Published in Bob McFarlane, Beginning AutoCAD 2002, 2012
An isometric is a 2D representation of a 3D drawing and is useful as it can convey additional information about a component which is not always apparent with the traditional orthographic views. Although an isometric appears as a 3D drawing, the user should never forget that it is a ‘flat 2D’ drawing without any ‘depth’.
Tseng's methods for inclusion problems on Hadamard manifolds
Published in Optimization, 2022
Konrawut Khammahawong, Poom Kumam, Parin Chaipunya, Juan Martínez-Moreno
Let and . As [39], let be the Riemannian manifold, and the Riemannian metric defined by where is a diagonal metric defined by Tangent space at , denoted by , equals . Hence, the parallel transport is identity mapping. Additional, the mapping given by is isometry between the Euclidean space and the Riemannian manifold and the Riemannian distance is defined by Thus, is a Hadamard manifold. The exponential map on is defined by for and The inverse of the exponential map is defined by
Variable Selection for the Prediction of C[0,1]-Valued Autoregressive Processes using Reproducing Kernel Hilbert Spaces
Published in Technometrics, 2019
Beatriz Bueno-Larraz, Johannes Klepsch
These two spaces and can be connected using the following congruence (bijective transformation preserving the inner product), named Loève’s isometry (Berlinet and Thomas-Agnan 2004, Theorem 35 and Lukić and Beder 2001, Lemma 1.1) If the random variable U is an element of , that is, U = ∑pi = 1aiX(ti), its image by the isometry is given by Therefore, when applying the inverse of ΨX to a function in we obtain a finite combination of evaluations X(ti). That is, replacing the inner product in with Ψ− 1X we recover the Dirac’s delta behavior for the trajectories, since Ψ− 1X(c0(s, ·)) = X(s). Next we see how to use this isometry to perform variable selection in AR processes.
Multiple-image encryption scheme via compressive sensing and orthogonal encoding based on double random phase encoding
Published in Journal of Modern Optics, 2018
Dongming Huo, Xin Zhou, Luozhi Zhang, Yuanyuan Zhou, Huaidong Li, Shaoliang Yi
However, for each image encrypted method based on CS, a whole measurement matrix Φ with a large number of data is generated as a decryption key, similarly, image encryption technique based on DRPE usually takes two random phase masks and as keys, which increase the burden of the key's storage and transportation. Lei Yu et al. demonstrated that when the sampling distance is large enough, a chaotic matrix generated by the logistic map satisfies the restricted isometry property (RIP) with overwhelming probability, which indicates that the chaotic matrix has the similar performance to Bernoulli random matrix and Gaussian random matrix (32). As long as chaotic matrix's initial value is determined, the matrix will be determined. Similarly, we can use the same method to produce two random phase masks needed in DRPE. In other words, we use the initial value of the matrix to generate the whole matrix, and only the should be memorized.