Orbits of semigroups of truncated convolution operators

We prove that a semigroup generated by finitely many truncated convolution operators on $L_p[0, 1]$ with 1 ≤ p < ∞ is non-supercyclic. On the other hand, there is a truncated convolution operator, which possesses irregular vectors.

Existence of disjoint weakly mixing operators that fail to satisfy the Disjoint Hypercyclicity Criterion

Recently, Bès, Martin, and Sanders [11] provided examples of disjoint hypercyclic operators which fail to satisfy the Disjoint Hypercyclicity Criterion. However, their operators also fail to be disjoint weakly mixing. We show that every separable, infinite dimensional Banach space admits operators T1,T2,…,TN with N⩾2 which are disjoint weakly mixing, and still fail to satisfy the Disjoint Hypercyclicity Criterion, answering a question posed in [11]. Moreover, we provide examples of disjoint hypercyclic operators T1, T2 whose corresponding set of disjoint hypercyclic vectors is nowhere dense, answering another question posed in [11]. In fact, we explicitly describe their set of disjoint hypercyclic vectors. Those same disjoint hypercyclic operators fail to be disjoint topologically transitive. Lastly, we create examples of two families of d-hypercyclic operators which fail to have any d-hypercyclic vectors in common.

Formation of helical ion chains

We study the nonequilibrium dynamics of the linear to zigzag structural phase transition exhibited by an ion chain confined in a trap with periodic boundary conditions. The transition is driven by reducing the transverse confinement at a finite quench rate, which can be accurately controlled. This results in the formation of zigzag domains oriented along different transverse planes. The twists between different domains can be stabilized by the topology of the trap and under laser cooling the system has a chance to relax to a helical chain with nonzero winding number. Molecular dynamics simulations are used to obtain a large sample of possible trajectories for different quench rates. The scaling of the average winding number with different quench rates is compared to the prediction of the Kibble-Zurek theory, and a good quantitative agreement is found.

Optimization and Prediction of Mechanical and Thermal Properties of Graphene/LLDPE Nanocomposites by Using Artificial Neural Networks

The focus of this work is to develop the knowledge of prediction of the physical and chemical properties of processed linear low density polyethylene (LLDPE)/graphene nanoplatelets composites. Composites made from LLDPE reinforced with 1, 2, 4, 6, 8, and 10 wt% grade C graphene nanoplatelets (C-GNP) were processed in a twin screw extruder with three different screw speeds and feeder speeds (50, 100, and 150 rpm). These applied conditions are used to optimize the following properties: thermal conductivity, crystallization temperature, degradation temperature, and tensile strength while prediction of these properties was done through artificial neural network (ANN). The three first properties increased with increase in both screw speed and C-GNP content. The tensile strength reached a maximum value at 4 wt% C-GNP and a speed of 150 rpm as this represented the optimum condition for the stress transfer through the amorphous chains of the matrix to the C-GNP. ANN can be confidently used as a tool to predict the above material properties before investing in development programs and actual manufacturing, thus significantly saving money, time, and effort.