Carousels,Zindler curves and the floating body problem

A carousel is a dynamical system that describes the movement of an equilateral linkage in which the midpoint of each rod travels parallel to it. They are closely related to the floating body problem. We prove, using the work of Auerbach, that any figure that floats in equilibrium in every position is drawn by a carousel. Of special interest are such figures with rational perimetral density of the floating chords, which are then drawn by carousels. In particular, we prove that for some perimetral densities the only such figure is the circle, as the problem suggests.     

Flat transversals to flats and convex sets of a fixed dimension

Helly and Hadwiger type theorems for transversal m-flats to families of flats and, respectively, convex sets of dimension n are proved in the case of general position. The proofs rely on Helly type theorems for “linear partitions” and “convex partitions,” so that a general theory of Helly numbers is also developed.     

Paths of Trains with Two-Wheeled Cars

In this paper we study the following simple and mind-puzzling problem: Can a model train car, whichruns along an intricate track, complete a full cycle around it? In our paper a track will be represented by a simple closed curve, and the cars of our model train by segments whose endpoints lie on the curve.     

A variation of Menger's theorem for long paths

In this paper we prove a Mengerian theorem for long paths, namely, that if in order to cut every uv-path of length at least n (n ≥ 2), in a diagraph D, we need to remove at least h points, then there exist {$\text{h}\text{(3n ? 5)}$} interior disjoint uv-paths in D of length at least n. Some variations and applications of this theorem are given as well.     

A colorful theorem on transversal lines to plane convex sets

We prove a colorful version of Hadwiger’s transversal line theorem: if a family of colored and numbered convex sets in the plane has the property that any three differently colored members have a transversal line that meet the sets consistently with the numbering, then there exists a color such that all the convex sets of that color have a transversal line. All authors are partially supported by CONACYT research grant 5040017.     

Photophysical properties of safranine O in protic solvents

Spectroscopic and photophysical properties of safranine O (Sf) were investigated in binary water/solvent mixtures. It was found that these properties are strongly solvent-dependent. A blue shift is observed for both the ground-state absorption and the triplet-triplet main absorption band when the solvent polarity augments. At the same time a red shift of the fluorescence emission band takes place. These facts are interpreted in terms of higher dipole moment of the dye molecule in the S(1) state as compared with the S(0) state, while a decrease in the dipole moment of the triplet state T(n) with respect to the triplet state T(1) occurs. The Stokes' shift and the fluorescence lifetime shows a linear correlation with the E(T)(30) parameter, while a non-linear behavior is observed when a correlation with models of a continuous dielectric solvent is attempted. These results suggest the operation of strong specific interactions of Sf with solvent molecules, most likely hydrogen bonding. From fluorescence lifetime and quantum yield determinations, as well as intersystem-crossing quantum yields, the solvent dependence of the photophysical kinetic parameters were obtained. The radiative fluorescence rate constant can be adequately reproduced by calculations based on the UV-Vis absorption and emission spectra, as given by the Strickler-Berg equation.     

Synthesis and functionalization of new polyhalogenated BODIPY dyes. Study of their photophysical properties and singlet oxygen generation

A theoretical and experimental study on the iodination of BODIPY dyes with different degrees of substitution has been developed. Polyhalogenated BODIPYs synthesized in this work are the first examples of this type of dyes with more than two halogen atoms in the BODIPY core and they can be selectively functionalized. Surprisingly, the position in which halogen is attached has a marked effect in the photophysical properties and modulates the fluorescence capacity of the resulting BODIPY. These iodinated BODIPYs are efficient singlet oxygen generators.