On combinatorial and projective geometry

Cross ratios constitute an important tool in classical projective geometry. Using the theory of Tutte groups as discussed in [6] it will be shown in this note that the concept of cross ratios extends naturally to combinatorial geometries or matroids. From a thorough study of these cross ratios which, among other observations, includes a new matroid theoretic version and proof of the Pappos theorem, it will be deduced that for any projective space M= _n (K) of dimension n2 of M over some skewfield K the inner Tutte group is isomorphic to the commutator factor group K ^*/[K ^*, K ^*] of K ^*K{0}. This shows not only that in case M= _n (K) our matroidal cross ratios are nothing but the classical ones. It can also be used to correlate orientations of the matroid M= _n (K) with the orderings of K. And it implies that Dieudonné's (non-commutative) determinants which, by Dieudonné's definition, take their values in K ^*/[K ^*, K ^*] as well, can be viewed as a special case of a determinant construction which works for just every combinatorial geometry.Research supported by the DFG (Deutsche Forschungsgemeinschaft).     

The Real Tree

In this note, it is shown that there exists a natural metric on the set[formula]of bounded subsets of containing their infimum which endowsTwith the structure of an -tree so that, for everyt∈T, the “number” of connected components ofT\{t} coincides with the cardinality #() of the set of subsets of . In addition, the set of ends ofTis explicitly determined, and various further features ofTare discussed, too.     

A photochemically driven molecular-level abacus

A molecular-level abacus-like system driven by light inputs has been designed in the form of a [2]rotaxane, comprising the pi-electron-donating macrocyclic polyether bis-p-phenylene-34-crown-10 (BPP34C10) and a dumbbell-shaped component that contains 1) a Ru(II) polypyridine complex as one of its stoppers in the form of a photoactive unit, 2) a p-terphenyl-type ring system as a rigid spacer, 3) a 4,4'-bipyridinium unit and a 3,3'-dimethyl-4,4'-bipyridinium unit as pi-electron-accepting stations, and 4) a tetraarylmethane group as the second stopper. The synthesis of the [2]rotaxane was accomplished in four successive stages. First of all, the dumbbell-shaped component of the [2]rotaxane was constructed by using conventional synthetic methodology to make 1) the so-called "west-side" comprised of the Ru(II) polypyridine complex linked by a bismethylene spacer to the p-terphenyl-type ring system terminated by a benzylic bromomethyl function and 2) the so-called "east-side" comprised of the tetraarylmethane group, attached by a polyether linkage to the bipyridinium unit, itself joined in turn by a trismethylene spacer to an incipient 3,3'-dimethyl-4,4'-bipyridinium unit. Next, 3) the "west-side" and "east-side" were fused together by means of an alkylation to give the dumbbell-shaped compound, which was 4) finally subjected to a thermodynamically driven slippage reaction, with BPP34C10 as the ring, to afford the [2]rotaxane. The structure of this interlocked molecular compound was characterized by mass spectrometry and NMR spectroscopy, which also established, along with cyclic voltammetry, the co-conformational behavior of the molecular shuttle. The stable translational isomer is the one in which the BPP34C10 component encircles the 4,4'-bipyridinium unit, in keeping with the fact that this station is a better pi-electron acceptor than the other station. This observation raises the question- can the BPP34C10 macrocycle be made to shuttle between the two stations by a sequence of photoinduced electron transfer processes? In order to find an answer to this question, the electrochemical, photophysical, and photochemical (under continuous and pulsed excitation) properties of the [2]rotaxane, its dumbbell-shaped component, and some model compounds containing electro- and photoactive units have been investigated. In an attempt to obtain the photoinduced abacus-like movement of the BPP34C10 macrocycle between the two stations, two strategies have been employed-one was based fully on processes that involved only the rotaxane components (intramolecular mechanism), while the other one required the help of external reactants (sacrificial mechanism). Both mechanisms imply a sequence of four steps (destabilization of the stable translational isomer, macrocyclic ring displacement, electronic reset, and nuclear reset) that have to compete with energy-wasteful steps. The results have demonstrated that photochemically driven switching can be performed successfully by the sacrificial mechanism, whereas, in the case of the intramolecular mechanism, it would appear that the electronic reset of the system is faster than the ring displacement.     

Trees,tight extensions of metric spaces,and the cohomological dimension of certain groups: A note on combinatorial properties of metric spaces

The concept of tight extensions of a metric space is introduced, the existence of an essentially unique maximal tight extension T_x—the “tight span,” being an abstract analogon of the convex hull—is established for any given metric space X and its properties are studied. Applications with respect to (1) the existence of embeddings of a metric space into trees, (2) optimal graphs realizing a metric space, and (3) the cohomological dimension of groups with specific length functions are discussed.     

Study of satellite structure found in the photoelectron spectra of gaseous alkenes

The satellite structure in the photoelectron spectrum of the C(1s) shell has been studied for the following compounds: ethylene, propylene, 1-butene, trans-2-butene, cis-2-butene, butadiene and, for comparison, propane. The structure can be divided into two parts: a relatively sharp peak due to π → π* excitation and a broad complex band involving excitation of σ orbitals. Calculation of electron shakeup probabilities has been carried out using CNDO wave functions and compared with experiment. Reasonably good agreement was obtained, and the general problem of characterizing carbon—carbon double bond behavior by satellite structure due to electron shakeup is discussed.     

On some combinatorial properties of algebraic matroids

It was proved implicitly by Ingleton and Main and explicitly by Lindström that if three lines in the algebraic matroid consisting of all elements of an algebraically closed field are not coplanar, but any two of them are, then they pass through one point. This theorem is extended to a more general result about the intersection of subspaces in full algebraic matroids. This result is used to show that the minimax theorem for matroid matching, proved for linear matroids by Lovász, remains valid for algebraic matroids.     

Effects of different composition ratio on the dielectric relaxation and dynamic mechanical properties of poly(ω‐dodecalactam‐co‐ε‐caprolactam‐co‐propylene oxide) copolymers

The molecular dynamics of new poly (ω‐dodecalactam‐co‐ε‐caprolactam‐co‐propylene oxide) copolymers (DL/CL/PAC) has been investigated by using dynamic mechanical thermal analysis (DMTA) and dielectric relaxation spectroscopy (DRS) measurements. The copolymers were synthesized via anionic polymerization of relevant lactams activated with carbamoyl derivatives of telechelic hydroxyl terminated polypropylene oxide with isophorone diisocyanate (PAC). The calorimetric, X‐ray diffraction, and DMTA measurements were performed to recognize the influence of the composition ratio and the type of PAC on the physical, thermal, and mechanical properties of the synthesized copolymers. The DRS was used to study the frequency dependence of the dielectric permittivity of some isotherms from ?110 to 145 °C. Copolymerization of ε‐caprolactam with about 10 wt % ω‐dodecalactam results in a copolymer that has lower water absorption, a melting point close to that of polyamide 6 and has a high enough degree of crystallinity in respect to high storage modulus. Five dielectric relaxations have been observed in the dielectric spectra, three at lower temperature and two at higher temperature. The copolymers have two glass transition temperatures for polyamide segments and polyether blocks, indicating microphase separation in the copolymers. Other studies directed toward molecular dynamics of polyamide DL/CL/PAC copolymers have not been reported. © 2010 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys, 2010     

Two finiteness theorems for periodic tilings of d-dimensional euclidean space

Consider the d -dimensional euclidean space E ^d . Two main results are presented: First, for any N∈ N, the number of types of periodic equivariant tilings that have precisely N orbits of (2,4,6, . . . ) -flags with respect to the symmetry group Γ , is finite. Second, for any N∈ N, the number of types of convex, periodic equivariant tilings that have precisely N orbits of tiles with respect to the symmetry group Γ , is finite. The former result (and some generalizations) is proved combinatorially, using Delaney symbols, whereas the proof of the latter result is based on both geometric arguments and Delaney symbols. <lsiheader> <onlinepub>7 August, 1998 <editor>Editors-in-Chief: &lsilt;a href=../edboard.html#chiefs&lsigt;Jacob E. Goodman, Richard Pollack&lsilt;/a&lsigt; <pdfname>20n2p143.pdf <pdfexist>yes <htmlexist>no <htmlfexist>no <texexist>no <sectionname> </lsiheader> Received September 5, 1996, and in revised form January 6, 1997.