When D: ξ→η is a linear differential operator, “direct problem” to find the generating compatibility conditions (CC) in form of an operator D1: η→ξ such that Dξ=η implies D1η=0. D involutive, procedure provides successive first order involutive operators D1, ..., Dn, when ground manifold has dimension n, result found by M. Janet as early 1920, footnote. However, link between this “Janet sequenc...

In this work‎, ‎an iterative method based on a matrix form of LSQR algorithm is constructed for solving the linear operator equation $mathcal{A}(X)=B$‎ ‎and the minimum Frobenius norm residual problem $||mathcal{A}(X)-B||_F$‎ ‎where $Xin mathcal{S}:={Xin textsf{R}^{ntimes n}~|~X=mathcal{G}(X)}$‎, ‎$mathcal{F}$ is the linear operator from $textsf{R}^{ntimes n}$ onto $textsf{R}^{rtimes s}$‎, ‎$ma...

in this work‎, ‎an iterative method based on a matrix form of lsqr algorithm is constructed for solving the linear operator equation $mathcal{a}(x)=b$‎ ‎and the minimum frobenius norm residual problem $||mathcal{a}(x)-b||_f$‎ ‎where $xin mathcal{s}:={xin textsf{r}^{ntimes n}~|~x=mathcal{g}(x)}$‎, ‎$mathcal{f}$ is the linear operator from $textsf{r}^{ntimes n}$ onto $textsf{r}^{rtimes s}$‎, ‎$ma...

Both the coherent states and also the squeezed states of the harmonic oscillator have long been understood from the three classical points of view: the 1) displacement operator, 2) annihilation(or ladder-) operator, and minimum-uncertainty methods. For general systems, there is the same understanding except for ladder-operator and displacement-operator squeezed states. After reviewing the known...

The Quantum Alternating Operator Ansatz (QAOA+) is an extension of the Approximate Optimization Algorithm (QAOA), where search space smaller in solving constrained combinatorial optimization problems. However, QAOA+ requires a trivial feasible solution as initial state, so it cannot be applied directly for problems that are difficult to find solution. For simplicity, we call them Non-Trivial-Fe...

In this paper, chaos based a new arithmetic crossover operator on the genetic algorithm has been proposed. The most frequent issue for the optimization algorithms is stuck on problem's defined local minimum points and it needs excessive amount of time to escape from them; therefore, these algorithms may never find global minimum points. To avoid and escape from local minimums, a chaotic arithme...