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Resonances, Euler’s Figures, and Particle-Waves
Published in Shamil U. Galiev, Evolution of Extreme Waves and Resonances, 2020
Since all references to imaginary number have been eliminated from this equation, it can be applied to fields that are real-valued as well as those that have complex values. We will consider cases when ∇2φ=∑n=1Nφxnxn, n = 1, 2, 3, …, N, where xn are axes of a rectangular coordinate system.
Conclusions
Published in Charlie Cullen, Learn Audio Electronics with Arduino, 2020
Audio electronics can become quite complex due to the use of AC signals, and it is important to understand that this book has not provided a detailed discussion of several important areas. Although the theoretical complexity of AC signals has been reduced where possible to allow time-varying signals to be used, there are significant elements (notably phase) that require further study if you wish to extend your knowledge of audio electronics. For phase calculations, complex numbers can be daunting for many learners (the clue is in the title!) but they are a very elegant solution to the problem of representing a dimension of variance (in this case magnitude over time) within a set of data. When AC circuit theory was first proposed, the use of imaginary numbers was a very clever way of linking the magnitude and phase of a signal together. In this system, the symbol j represents the value of −1 (in mathematics this is the symbol i, but as i is used for current in electronics the next letterj is used), which helps to define the portions of a sine wave where the amplitude is negative. Although initially more difficult mathematically, complex numbers are nevertheless essential to progressing your knowledge of audio electronics beyond the basic usage discussed in this book.
Heat Conduction
Published in Greg F. Naterer, Advanced Heat Transfer, 2018
Conformal mappings are based on functional analysis of complex variables. A complex number can be expressed in the form z = x + iy, where x and y are real numbers, and i is the “imaginary number” (i2= –1). The real part of the complex number is denoted by Re(z) = x and the imaginary part is denoted by Im(z) = y.
An upper bound on the minimum rank of a symmetric Toeplitz matrix completion problem
Published in Optimization, 2023
Yi Xu, Xihong Yan, Jiahao Guo, Cheng Ma
Notation. Let be an n-dimensional Euclidean space and be the set of real matrices. denotes the rank of a matrix T. A matrix T is symmetric positive definite (resp. positive semidefinite) and is denoted by (resp. ). represents the jth component of vector x. i represents the unit imaginary number. is the number of elements of set A. The symbol represents the transpose. For a measure , its supporting set is defined as , where is the definition domain of .
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.
Prospective mathematics teachers learning complex numbers using technology
Published in International Journal of Mathematical Education in Science and Technology, 2022
Jorge Gaona, Silvia Soledad López, Elizabeth Montoya-Delgadillo
To represent the square root of a negative number other than , the form bi is used, where b is any real number and i is the imaginary unit (called a pure imaginary number).