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Linear Algebra for Quantum Mechanics
Published in Caio Lima Firme, Quantum Mechanics, 2022
The complex numbers are an extension of the real numbers where there is an element which represents the square root of −1, called the imaginary unit, i. A complex number is represented by Z composed of a real part, a, the imaginary part, bi. Z=a+bi,i=−1
Inner Product Spaces
Published in James R. Kirkwood, Bessie H. Kirkwood, Linear Algebra, 2020
James R. Kirkwood, Bessie H. Kirkwood
The modulus of a complex number z, denoted |z|, is the distance of the number z from the origin. For z = a + bi, we have |z|2=a2+b2=zz¯.
The complex plane and conformal mappings
Published in Martin Vermeer, Antti Rasila, Map of the World, 2019
A complex number z = x + iy can be identified with the point (x, y) in the plane, which is also called the complex plane (see Figure 3.2). The argument or phase angleargz of a complex number z = x + iy, z ≠ 0 corresponds to the angle, at the origin, between the point corresponding to the number z and the positive half of the x-axis, which is also called the real axis. The argument is defined via standard trigonometry by tan(argz)=defyx,x≠0.
Direct Method for Generating Floor Response Spectra considering Soil–Structure Interaction
Published in Journal of Earthquake Engineering, 2022
Wei Jiang, Yang Zhou, Wei-Chau Xie, Mahesh D. Pandey
where and represent the power spectral density functions of the excitations of FLIRS and that of FIRS, respectively. In Equation (4.4), denotes a matrix in which each element is equal to the squared modulus of the corresponding element in . For a complex number , its modulus is defined as . The physical meaning of the off-diagonal terms represents contribution from the free-field motion FIRS in direction j to the equivalent fixed base excitation FLIRS (which is the foundation motion in a coupled soil-structure system) in direction i. This assumption is valid when the foundation can be considered as rigid and the centre of gravity of the structure (including the foundation mat) is not significantly offset from the vertical centerline of the structure foundation. Most of safety related structures in nuclear power plant fall into this category, and the foundation are mainly of circle or rectangle shapes. For other civil structures with irregular foundation shapes, the accuracy of this assumption should be further evaluated.
Comparative analysis of quaternion modulation system with OFDM systems
Published in International Journal of Electronics Letters, 2021
Anam Zahra, Qasim Umar Khan, Shahzad Amin Sheikh
A complex number is defined by, where are real numbers and is an imaginary number such that. Complex numbers are two-dimensional vectors space over the real numbers. In addition to, quaternions are constructed by adding two new imaginary units and with one real part. A quaternion is an extension of the complex number system (Catoni, Bordoni, Cannata, & Zampetti, 1997). In 1843, Irish mathematician William Rowan Hamilton was the first person who described quaternions and practically applied them in three-dimensional mechanics (Farouki, Al-Kandari, & Sakkalis, 2002). The mathematical notation of quaternions represents three-dimensional rotations of objects. A quaternion can be written as a sum of one real part and three imaginary parts.
Amplitude, period and orientation of the moiré patterns in barrier 3D displays
Published in Journal of Information Display, 2018
Vladimir Saveljev, Sung-Kyu Kim
The moiré wavevector is the ray OD from the origin to the spectral component of the rotated plate whose center of rotation is point C. The equation of the trajectory with radius k1 is shown as: The complex number z can be rewritten in the polar form , where modulus a and argument ϕ are the wavenumber and orientation, respectively. Then the wavenumber of the moiré pattern is as follows (see also Equation (12) in Ref. [17]): where is the ratio of the periods of the layers and αMAX is the moiré angle. The period of the moiré patterns appears in the literature (see, e.g. Equation (9) in Ref. [18]); it is the inverse wavevector Equation (2).