It is well known that a continuous piecewise monotone interval map with positive topological entropy is semiconjugate to a map of a constant slope and the same entropy, and if it is additionally transitive then this semiconjugacy is actually a conjugacy. We generalize this result to piecewise continuous piecewise monotone interval maps, and as a consequence, get it also for piecewise monotone g...

Nonlinear dynamic systems with positive solutions and the parametric problem of entropy maximization (entropy operator) are considered. The continuity, differentiability and boundedness of the entropy operator and the boundedness of the solutions of the dynamic system are derived using the global implicit function theorem. The technique of positive dynamic systems is applied to modelling of the...

We consider unbounded fanin depth-2 circuits with arbitrary boolean functions as gates. The entropy of an operator f : {0, 1}n → {0, 1}m is defined as the logarithm of the maximum number of vectors distinguishable by at least one special subfunction of f . We prove that every depth-2 circuit for f requires at least entropy(f) wires. This generalizes and substantially simplifies the argument use...

The quantum derivatives of e, A and logA, which play a basic role in quantum statistical physics, are derived and their convergence is proven for an unbounded positive operator A in a Hilbert space. Using the quantum analysis based on these quantum derivatives, a basic equation for the entropy operator in nonequilibrium systems is derived, and Zubarev’s theory is extended to infinite order with...

We derive new characterizations of the matrix Φ-entropy functionals introduced in Chen & Tropp (Chen, Tropp 2014 Electron. J. Prob.19, 1-30. (doi:10.1214/ejp.v19-2964)). These characterizations help us to better understand the properties of matrix Φ-entropies, and are a powerful tool for establishing matrix concentration inequalities for random matrices. Then, we propose an operator-valued gene...

The present work focuses on entropy solutions for the fractional Cauchy problem of nonsymmetric systems. We impose sufficient conditions on the parameters to obtain bounded solutions of L∞. The solutions attained are unique and exclusive. Performance is established by utilizing the maximum principle for certain generalized time and space-fractional diffusion equations. The fractional differenti...

In this paper, we propose a theoretical scheme for the generation of non-classical states of the cavity field in a system of a single trapped atom via controlling the Lamb-Dicke parameter. By exploiting the super-operator method, we obtain an analytical expression for the density operator of the system by which we examine the dynamical behaviors of the atomic population inversion, the phase-spa...