Event detection and analytic calculations in both Wide Area Measurement Systems (WAMS) and in distribution systems including Phasor Measurement Unit (PMU) based microgrid control systems often use the independent components of the phasors (e.g. frequency or Voltage) rather than the vector representation of a phasor. This paper introduces the use of the discrete time complex Fourier Transform ba...

We take non-Hermitian Aubry-Andr\'{e}-Harper models and quasiperiodic Kitaev chains as examples to demonstrate the equivalence superposition of real imaginary potentials (QPs) on inducing localization single-particle states. prove this by analytically computing Lyapunov exponents (or inverse lengths) for systems with purely QPs. Moreover, when superposed same frequency, QPs are coherent localiz...

Abstract This paper first reviews how anti-symmetric matrices in two dimensions yield imaginary eigenvalues and complex eigenvectors. It is shown how this carries on to rotations by means of the Cayley transformation. Then the necessary tools from real geometric algebra are introduced and a real geometric interpretation is given to the eigenvalues and eigenvectors. The latter are seen to be two...

the analysis of city as an approach in reading has drawn a  lot of attention from the end of nineteenth century till present time and the concept of city, real or imaginary, utopia or  promise land has been one of  the most significant frameworks for narrative techniques. the present research attempts to introduce george wells' theories concerning utopia, and apply them to analyze suhayl i...

In this paper we are using Fast Fourier Transform to generate the feature vector which considers the mean real and mean imaginary parts of complex numbers of polar coordinates in frequency domain. The method proposed here considers 12 mean values of 6 upper half sectors real and imaginary parts of each R, G and B components of an image. The algorithm proposed uses 36 mean values of real and ima...

The advantages of solving potential problems using an overdetermined boundary integral element method are examined. Representing a 2-dimensional potential solution by an analytic complex function forms two algebraic systems from the real and imaginary parts of the discretized form of the Cauchy theorem. Depending on which boundary condition is prescribed, the real or the imaginary algebraic sys...

Expressions, as well as necessary and sufficient conditions are given for the existence of the real and pure imaginary solutions to the consistent quaternion matrix equation AXB+CY D = E. Formulas are established for the extreme ranks of real matrices Xi, Yi, i = 1, · · · , 4, in a solution pair X = X1 +X2i+X3j+X4k and Y = Y1+Y2i+Y3j+Y4k to this equation. Moreover, necessary and sufficient cond...