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The Discrete Sine Transform
Published in Humberto Ochoa-Domínguez, K. R. Rao, Discrete Cosine Transform, 2019
Humberto Ochoa-Domínguez, K. R. Rao
Similar to the DCTDCT matrix and the SCTSCT matrix, the DSTDST matrix and SSTSST mstrix matrices follow certain symmetries whose center depends on the order (length) of the row vector. The symmetry indicates that the fast forward and inverse algorithms are identical. Also, one can notice that the matrices [SN−1I], [SNIV] and [S~NI] are self-inverse. In other words, they are involutory matrices. The inverse matricesInverse matrices of [SNII] and [SNIII] are their transpose matrices, which mean that the fast algorithms for the inverse matrices can be obtained by reversing the arrows of the flowgraph of the forward transform.
Stress constrained thermo-elastic topology optimization based on stabilizing control schemes
Published in Journal of Thermal Stresses, 2020
Qingxuan Meng, Bin Xu, Chao Wang, Lei Zhao
For chaos control of nonlinear dynamics, Schmelcher and Diakonos [41] presented an effective linear transformation which can revise the eigenvalues of Jacobian matrix of dynamical system and realize the oscillation control of original system. This method is called as the stability transformation method given by where is the iteration variable vector, represents an implicit iteration function and denotes the involutory matrix. Generally, when ( means the identity matrix), Eq. (31) becomes
Synergy between manipulative and digital artefacts: a teaching experiment on axial symmetry at primary school
Published in International Journal of Mathematical Education in Science and Technology, 2018
Eleonora Faggiano, Antonella Montone, Maria Alessandra Mariotti
The geometric concept addressed is axial symmetry, in the sense of the isometric transformation of the plane in itself, with a line of fixed points (the axis). From this definition, it can be deduced that: axial symmetry preserves the distances and the amplitudes of the angles, and consequently transforms segments into congruent segments and straight lines into straight lines; it is an involutory function [13].