Extreme norms on ℝ n

LetN ¹,N be thel ¹ andl norms on ⁿ . We denote by the set of all normsN on ⁿ such thatN NN ¹. The aim of the paper is to present a characterization of the extreme points of .This paper has been written while the author was a research fellow of the Alexander von Humboldt-Stiftung at the Mathematisches Institut der Eberhardt-Karls-Universität, Tübingen.     

Chemistry of the 2,3-oxaphosphabicyclo[2.2.2]octene ring system: Extrusion of metaphosphates

The 2,3-oxaphosphabicyclo[2.2.2]octene 3-oxide (or sulfide) ring system is of considerable value because it easily fragments on being heated or irradiated (254 nm) to provide three-coordinate phosphoryl species. The system is synthesized by O-insertion with peracids into a C P bond of 7-phosphanorbornene derivatives with a variety of P-substituents. With rare exception, the insertion has been found to proceed with retention of the configuration at phosphorus, as established by X-ray and NMR techniques. The thermal fragmentation that produces the metaphosphate derivatives EtO PO₂, EtO P(S)O, and Et₂N PO₂ follows first-order kinetics, and is independent of the concentration of a trapping agent for these species. Solvent effects and activation parameters join in defining a retrocycloaddition mechanism that ejects the free metaphosphate. The species Ph PO₂ can also be easily generated either thermally or photochemically. Metaphosphates have been found to attack ethereal oxygen in epoxides and oxetanes, and may undergo anchimeric participation with a properly placed methoxy group on the substituent used in the 2,3-oxaphosphabicyclo[2.2.2]octene precursor.     

Intermolecular chain transfer to polymer with chain scission: general treatment and determination of kp/ktr in L,L-lactide polymerization

A general kinetic treatment of the system with intermolecular chain transfer followed by fast reinitiation is given. It leads to the broadening of the molecular weight distribution (MWD), the number of growing chains being invariable. Thus, this system can be considered as a special case of living polymerization. A general method has been elaborated allowing the determination of the ratio of the rate constant of propagation (k_p) to the rate constant of the bimolecular transfer (k⁽²⁾_tr) from the dependence of the MWD on monomer conversion. Numerical values of k_p/k⁽²⁾_tr equal to ≈ 10² and 25 were thus determined for the polymerization of L , L -lactide (L , L -dilactide) initiated with aluminium tris(isopropoxide) trimer ({Al(OⁱPr)₃}₃) and tributyltin ethoxide (ⁿBu₃SnOEt), respectively.     

Towards periodic copolymers with flexible and mesogenic units in the main chain. Copolymerization of tetrahydrofuran with 7-oxabicyclo[2.2.1]heptane

An attempt was made to prepare periodic polymers having promesogenic units of various lengths. Due to a certain distribution within these units, copolymers of this structure should rather be called pseudoperiodic. In order to understand better the behaviour of such systems, copolymerization of tetrahydrofuran (THF — soft units) with 7-oxabicyclo[2.2.1]-heptane (BC — hard units) was studied. The kinetically and thermodynamically controllable copolymerizations were elaborated, leading to products of different distribution of the THF and BC units. The apparent reactivity ratios are dependent on the copolymerization conditions due to the reversibility of the propagations on the THF type of active species. They vary: r̄₁ from 0.09 to 2.72, r̄₂ from 0.31 to 1.57 (1 stands for BC, 2 for THF). Pseudoperiodic copolymers of the structure $\rlap{-} [\rlap{–} ({\rm BC}\rlap{-} )_x {\rm THF}\rlap{-} ]_n $, where x̄ = 1 to 4, were prepared carrying out the copolymerization above the ceiling temperature of THF.     

Thermodynamics of homogeneous equilibrium copolymerization from the point of view of the Flory-Huggins theory

A general treatment of the thermodynamics of binary copolymerization from the point of view of the Flory-Huggins theory is given. The relations allow one to determine, on the basis of experimental data, the thermodynamic parameters of copolymerization and the parameters of intercomponent interaction (χ). On the other hand, when these parameters are known, it is possible to predict, for any initial set of conditions, the equilibrium concentrations of comonomers and the equilibrium microstructure of a copolymer. The presented simulations indicate that positive values of interaction parameters increase the volume fractions of copolymer units in equilibrium with comonomers, in comparison to an analogous copolymerization system without intercomponent interactions.     

Increase of the average reactivity of active species due to a shift to the more reactive species in ionic propagation accompanied by termination

The presented simulations demonstrate that in polymerizations proceeding on two kinds of species, differing in reactivity and being in equilibrium, the expected decrease of the rate of polymerization due to termination may happen to be compensated by the relative increase of concentration of the more reactive species. This takes place, for instance, in the polymerization proceeding simultaneously on ions and ion pairs if ions are more reactive. Because of termination the total concentration of ionic species during the course of polymerization decreases while the proportion of ions increases due to increasing dilution. The maximum compensation is observed when simultaneously k_(ions)/k_{(ion pairs)} → and K_d/[I]₀ → 0, where k are the propagation rate constants, K_d is the equilibrium constant of dissociation and [I]₀ is the starting concentration of initiator. Then, the degree of compensation (the ratio of the rate with compensation to the rate without termination) is becoming equal to ([P*]/[P*]₀)^1/2, where [P*] is the actual, total concentration of the growing species and [P*]₀ is the initial total concentration (before any termination has taken place).