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Calculus on Manifolds
Published in Mattias Blennow, Mathematical Methods for Physics and Engineering, 2018
Thanks to the metric tensor, we have now recovered the relation between tangent vectors and dual vectors that was previously lost by not being able to refer back to an underlying Cartesian coordinate system. As a type (0, 2) tensor, the metric is a linear map from tangent vectors to dual vectors, while the inverse metric as a type (2, 0) tensor is a linear map from dual vectors to tangent vectors. In a similar fashion, when there exists a metric tensor, we can lower and raise the indices of tensors of arbitrary type and although a tensor such as the curvature may be naturally considered as a type (1, 3) tensor, it then generally suffices to refer to it as a rank four tensor.
A Lagrange–Newton algorithm for tensor sparse principal component analysis
Published in Optimization, 2023
Shuai Li, Ziyan Luo, Yang Chen
The tensor Principal Component Analysis (PCA), as a higher-order generalization of the traditional PCA, is an important and prevalent approach for modern data analysis, with wide applications in computer vision, diffusion magnetic resonance imaging, signal processing, spectral hypergraph theory and higher-order statistics [1–5]. To solve principal components (PCs, i.e. linear combinations of original variables) of tensors, Qi et al. [6] have proposed a class of Z-eigenvalue methods for four tensor cases with different orders and dimensions, and showed the effectiveness and prospect of methods through numerical experiments. Under the assumption of rank-one tensor data, Jiang et al. [7] have equivalently reformulated the tensor PCA problem as the matrix optimization with a rank-one constraint, and proposed two solution methods for solving the new matricization model. Huang et al. [8] have studied the convergence and statistical inference aspects of the power iteration algorithm for solving the tensor PCA model.
Development of a constitutive model for fibre reinforced cemented Toyoura sand
Published in European Journal of Environmental and Civil Engineering, 2022
Muhammad Safdar, Tim Newson, Faheem Shah
Recently, there is growing interest in hypoplastic models (Fuentes, Triantafyllidis, & Lizcano, 2012; Grabe & Heins, 2016; Herle & Kolymbas, 2004; Hleibieh, Wegener, & Herle, 2014; Huang et al., 2006; Jiang & Liu, 2016; Kolymbas & Herle, 1998; Li et al., 2016; Lin & Wu, 2016; Mašín, 2012, 2013; Peng et al., 2015). Hypoplastic constitutive equations are based on nonlinear tensor functions and are characterized by simple formulation and few parameters. The early hypoplastic models contain four tensor polynomial terms with four material parameters as coefficients, usually with two linear terms and two nonlinear terms in strain rate. A major advantage of the basic model is the fact that it requires only four parameters, which can be easily identified with a single triaxial compression test. The stress tensor is considered as the only state variable in such basic models. As a consequence, the basic models cannot account for the complex history dependence of soil. Moreover, the constitutive model needs to be re-calibrated for the same material but with different initial densities. The hypoplastic constitutive model with critical state presents a major achievement by introducing the void ratio as an additional state variable into the basic model (Mašín, 2013; Niemunis, 2003). For a given soil, the hypoplastic model with critical state requires a single set of parameters for the entire range of densities. Since the critical model is built on the basic model, the performance of the critical state model depends on the quality of the basic model.
p †q : a tool for prototyping many-body methods for quantum chemistry
Published in Molecular Physics, 2021
Nicholas C. Rubin, A. Eugene DePrince
The code generated above produces a four-tensor which can be reshaped into the matrix . The reduced Hamiltonian matrix elements and 2-RDM are stored in OpenFermion order–. Combining this result with the 1-RDM based metric () defines the generalised eigenvalue problem that is the extended RPA equations. Further simplifications can be made to resolve the delta functions but are left here, represented by identity matrices kd[:, :], for clarity. Though further modifications can be applied, such as delta function simplification or taking advantage of antisymmetry of , the resulting code is fully functioning computational kernel ready for deployment in an appropriate solver.