A Victorian Age Proof of the Four Color Theorem

In this paper we have investigated some old issues concerning four color map problem. We have given a general method for constructing counter-examples to Kempe's proof of the four color theorem and then show that all counterexamples can be rule out by re-constructing special 2-colored two paths decomposition in the form of a double-spiral chain of the maximal planar graph. In the second part of the paper we have given an algorithmic proof of the four color theorem which is based only on the coloring faces (regions) of a cubic planar maps. Our algorithmic proof has been given in three steps. The first two steps are the maximal mono-chromatic and then maximal dichromatic coloring of the faces in such a way that the resulting uncolored (white) regions of the incomplete two-colored map induce no odd-cycles so that in the (final) third step four coloring of the map has been obtained almost trivially.Comment: 27 pages, 18 figures, revised versio

Evaluation of the nasal mucociliary transport rate by rhinoscintigraphy before and after surgery in patients with deviated nasal septum

In this study, we have investigated the effect of nasal septal deviation (NSD) on nasal mucociliary activity and how does a septoplasty operation affecs the nasal mucociliary transport rate in the first and third months during the post-operative period. Twenty-two patients who were diagnosed with NSD and 22 healthy controls were studied using rhinoscintigraphy with Tc-99m-macroaggregated albumin (Tc-99m-MAA). On each case, the nasal mucociliary transport rate (NMTR) was measured pre-operatively only on five cases, on the first and third months of post-operative period. The NMTRs of patients with a deviated septum were significantly lower than the NMTRs of the healthy controls on both the convex and concave sides. Significant improvement was observed in the first post-operative month. On the concave and convex sides, the average postop third month post-operative NMTR value was higher than the first month post-operative NMTR values. It was concluded that the septoplasty operation improves reduced NMTRs after surgery. The effect of nasal surgery on nasal mucociliary activity may be more accurately evaluated in the third month than the first month of post-operative period

Internal and External Validity in Experimental Economics: A Review on the Methodology

Internal and external validity in experimental economics: A review on the methodology. The aim of this study is to consider the methodology of the experimental economics. Economic research relies increasingly on experimental methods as a relatively new approach to collecting data in economics. Therefore, in this paper, we analyze and discuss the methodological foundations of experimental economics. The study starts with the elaboration on internal validity, followed by a general overview on external validity. In this context, the influence of experimental economics on the use of scientific methods in economic methodology and the dynamics of the trade-off between internal validity-external validity are discussed. The main result of the study is that it is more convenient to adopt a methodology compatible with aims and nature of each study instead of a standardized methodological approach, when the methodological dimensions of an experimental design are evaluated.Wo