We develop a consistent model of a quantum damped harmonic oscillator with arbitrary time-dependent frequency and damping coefficients within the framework of the Heisenberg–Langevin equations with two noncommuting delta-correlated noise operators justifying the choice of the “minimal noise” set of damping coefficients and correlation functions.

Let A be an alphabet and let F a set of words with letters in A. We show that the sum all no consecutive subwords F, as formal power series noncommuting variables, is reciprocal coefficients 0, 1 or -1. also explain how this result related to Curtis Greene on lattices M\"obius function 1,

Abstract We introduce a family of norms on the $n \times n$ complex matrices. These arise from probabilistic framework, and their construction validation involve probability theory, partition combinatorics, trace polynomials in noncommuting variables. As consequence, we obtain generalization Hunter’s positivity theorem for complete homogeneous symmetric polynomials.

Abstract We discuss quantum mechanical detection models in the weak limit context of conservation laws physical quantities. In particular, we analyze what kind system–detector interaction can preserve global or related symmetry, and how final measurement on detector affects measured observable systems its presumed conservation. It turns out that order noncommuting measurements results differenc...

let $r$ be a commutative ring with identity. let $g(r)$ denote the maximal graph associated to $r$, i.e., $g(r)$ is a graph with vertices as the elements of $r$, where two distinct vertices $a$ and $b$ are adjacent if and only if there is a maximal ideal of $r$ containing both. let $gamma(r)$ denote the restriction of $g(r)$ to non-unit elements of $r$. in this paper we study the various graphi...