Experimentation and the evaluation of algorithms have a long history in algebra. In this paper we follow single test example over more than 250 years. 1685, John Wallis published A treatise algebra, both historical practical, containing solution Colonel Titus’ problem that was proposed to him around 1650. The Titus consists three algebraic quadratic equations unknowns, which transformed into fi...

We propose a novel stochastic method to generate paths conditioned to start in an initial state and end in a given final state during a certain time t(f). These paths are weighted with a probability given by the overdamped Langevin dynamics. We show that these paths can be exactly generated by a non-local stochastic differential equation. In the limit of short times, we show that this complicat...

The optical solitons in Bragg gratings fibers are studied for NLSE having cubic-quartic dispersive reflectivity with parabolic non-local combo law of refractive index. extended auxiliary equation method and the addendum to Kudryashov’s introduced. existence criteria such indicated.

This paper is concerned with risk-sensitive performance analysis for linear quantum stochastic systems interacting with external bosonic fields. We consider a cost functional in the form of the exponential moment of the integral of a quadratic polynomial of the system variables over a bounded time interval. An integro-differential equation is obtained for the time evolution of this quadratic-ex...

Let n ≥ 3, v1(x), . . . , vn−1(x) ∈ Z[x] and u ∈ {−1, 1}, then x(x− v1(a)y) · · · (x− vn−1(a)y) + uy = ±1 is called a parametrized familiy of Thue equations, if a ∈ Z and the solutions x, y are searched in Z; cf. Thomas (1993). There are several results concerning parametrized families of cubic and quartic families of Thue equations, see Mignotte et al. (1996) and the references therein. Thomas...