A linear transformation on a plane can be represented by a corresponding matrix. If we take a point (x,y) then its transformation can be written as follows: The simplest (not degenerate) ...

The OpenSCAD manual section for multmatrix gives some details of how this works. We'll elaborate a bit more below. An affine transformation matrix for three dimensional data is a 4x4 matrix. The top ...

The modern era of digital communication facilities bring digital watermarking techniques which is required to copyright and authenticate owner's information which further tracks control of ...

Motion equation for the density matrix; Fundamental properties of a free particle density matrix; Scaling of density matrix: a free particle; Density matrix for a particle in a one-dimensional ...

Jakub Hyla received the Ph.D. degree in automation, electronics, electrical engineering, and space technologies from the Silesian University of Technology, Gliwice, Poland, in 2024. His doctoral ...

The reversible formation of hydrogen bonds is a ubiquitous mechanism for controlling molecular assembly in biological systems. However, achieving predictable reversibility in artificial ...

Mechanical controllability of recently discovered topological defects (e.g., skyrmions) in ferroelectric materials is of interest for the development of ultralow-power mechano-electronics that are ...