Permutations,periodicity, and chaos

Let f: I → I be a continuous function on a closed interval I. If there exists x?I which has period 3 with respect to f, then Li and Yorke [1] proved that f is chaotic in the sense that there are not only points x?I of arbitrarily large period, but also uncountably many points x?I which are not even asymptotically periodic with respect to f. By using only elementary combinatorial facts about permutations, it is shown that if there is a point x?I of period p with respect to f, where p is divisible by 3, 5, or 7, then f is chaotic. The proof is followed by a study of some related combinatorial problems in symmetric groups.     

On optimal embeddings of metrics in graphs

This paper extends previous results of the authors. In particular, non-treerealizable metrics are investigated and it is shown that every finite metric has an optimal realization by a graph.     

Reactions on solid supports part I: Novel preparation of α-formyl pyrroles from α-methylpyrroles by oxidation with thallium (III) nitrate on clay

Treatment of α-methylpyrroles with thallium (III) nitrate/Montmorillonite clay affords the corresponding α-formylpyrroles in excellent yields.     

Time-resolved studies of backscattered light from a plasma generated by a 0.52 μm laser

A frequency doubled Nd glass laser was used to irradiate plane solid targets. Temporally and spectrally resolved measurements of the backscattered light were obtained together with calorimeter measurements of the backreflected energy. The results show the presence of stimulated Brillouin scattering and an estimate is made of the coronal electron temperature.     

Does QCD have infrared renormalons?

We give a naïve perturbative proof of the existence of infrared renormalons for large distance asymptotically free theories. We argue that the extension of the result for small distance asymptotically free theories is not obvious. Indeed we do not find infrared renormalons for QCD.     

The Assessment of Reservation Prices of an Intermediate R&D Result and Patent Reward Under Oligopolistic Competitive Behaviour

One of the shortcomings of R&D evaluation is a lack of emphasis on analytical assessment of the value of an on-going R&D project. This paper addresses the problem concerning the distributive aspect of access to superior knowledge. Decision tree analysis and probability models appear to be appropriate tools for assessing the values of an intermediate result and patent reward of a firm's R&D decision in an environment of perfect information and oligopolistic competition. The assessed values are used as minimum prices acceptable to the firm when the knowledge is disseminated to the public. This paper will attempt to resolve this problem through the determination of the appropriate values of the reservation price of the first-stage invention in terms of the final reward and of a patent reward for which the inventor is willing to apply.