An eigenvalue equation representing symmetric (dual) quantum equation is introduced. The particle is described by two scalar wavefunctions, and two vector wavefunctions. The eigenfunction is found to satisfy the quantum Telegraph equation keeping the form of the particle fixed but decaying its amplitude. An analogy with Maxwellian equations is presented. Massive electromagnetic field will satisfy a quantum Telegraph equation instead of a pure wave equation. This equation resembles the motion of the electromagnetic field in a conducting medium. With a particular setting of the scalar and vector wavefunctions, the dual quantum equations are found to yield the quantized Maxwell's equations. The total energy of the particle is related to the phase velocity ($v_p$) of the wave representing it by $E=p\,|v_p|$, where $p$ is the matter wave momentum. A particular solution which describes the process under which the particle undergoes a creation and annihilation is derived. The force acting on the moving particle is expressed in a dual Lorentz-like form. If the particle moves like a fluid, a dissipative (drag) quantum force will arise.

In this paper, we propose the set-membership quaternion normalized least-mean-square (SM-QNLMS) algorithm. For this purpose, first, we review the quaternion least-mean-square (QLMS) algorithm, then go into the quaternion normalized least-mean-square (QNLMS) algorithm. By having the QNLMS algorithm, we propose the SM-QNLMS algorithm in order to reduce the update rate of the QNLMS algorithm and avoid updating the system parameters when there is not enough innovation in upcoming data. Moreover, the SM-QNLMS algorithm, thanks to the time-varying step-size, has higher convergence rate as compared to the QNLMS algorithm. Finally, the proposed algorithm is utilized in wind profile prediction and quaternionic adaptive beamforming. The simulation results demonstrate that the SM-QNLMS algorithm outperforms the QNLMS algorithm and it has higher convergence speed and lower update rate.