Dynamic data structures for fat objects and their applications

We present several efficient dynamic data structures for point-enclosure queries, involving convex fat objects in or . Our planar structures are actually fitted for a more general class of objects – (β,δ)-covered objects – which are not necessarily convex, see definition below. These structures are more efficient than alternative known structures, because they exploit the fatness of the objects. We then apply these structures to obtain efficient solutions to two problems: (i) finding a perfect containment matching between a set of points and a set of convex fat objects, and (ii) finding a piercing set for a collection of convex fat objects, whose size is optimal up to some constant factor.     

Non-linear ground state representations and sharp Hardy inequalities

We determine the sharp constant in the Hardy inequality for fractional Sobolev spaces. To do so, we develop a non-linear and non-local version of the ground state representation, which even yields a remainder term. From the sharp Hardy inequality we deduce the sharp constant in a Sobolev embedding which is optimal in the Lorentz scale. In the appendix, we characterize the cases of equality in the rearrangement inequality in fractional Sobolev spaces.     

Propensity rules for orientation by atom impact: II. Many-state description of direct transitions

This paper contains a theoretical analysis of orientation and alignment created in direct, collision-induced transitions among atomic states with arbitrary angular momentum. Using the natural coordinate frame, general propensity rules are derived in the velocity region of maximum transition probability and their range of validity is investigated. The predictions are tested and illustrated by nine-state calculations for Li(n=2, 3) transitions in Li-He collisions.     

Propensity rules for orientation by atom impact: I. three-state description of directS ↔P transitions

P state orientation and alignment created in direct, collision-inducedS ?P transitions of a quasi-one-electron atom are analyzed in the natural coordinate frame. In the velocity region of maximum transition probability, propensity rules for the orientation are derived, and their range of validity in impact parameter and velocity is discussed. The predictions are tested and illustrated by three-state calculations for Na(3s ? 3p) transitions in Na — He collisions.     

Beyond linear least squares regression

Regression is a collection of statistical methods that are used to study relationships among predictor and response variables. In addition to the most popular linear model, solved by least squares, several other techniques have found an application in analytical chemistry. Biased methods such as stepwise regression, ridge regression, principal components regression, and partial least squares regression are especially useful in cases of poorly or underdetermined systems with collinearity. When structural and/or distributional assumptions associated with linear least squares are violated, nonlinear regression, robust regression or generalized least squares estimators may offer potential remedies.     

Regulating complex dynamics in firms and economic systems

The decisions of economic firms are often guided by simple routines. We demonstrate that such a routine may generate suboptimal chaotic dynamics, where the suboptimality is due to the fact that decision outcomes are in regions of low performance. We discuss a simple, but effective, method to regulate such dynamics in order to improve the performance. The method works as follows: In a first step, from time series information one has to identify critical starting areas which may lead to a crash or an outcome with low performance. In a second step, one has to perturb the system when it enters these critical areas. Due to sensitive dependence on initial conditions, already minor interventions suffice to prevent harmful events and to obtain better results.