Polynomials satisfied by two linked matrices

Polynomials in two variables, evaluated at A and with A being a square complex matrix and being its transform belonging to the set {A⁼, A^†, A^∗}, in which A⁼, A^†, and A^∗ denote, respectively, any reflexive generalized inverse, the Moore-Penrose inverse, and the conjugate transpose of A, are considered. An essential role, in characterizing when such polynomials are satisfied by two matrices linked as above, is played by the condition that the column space of A is the column space of . The results given unify a number of prior, isolated results.     

Polymer‐Bound,Amphiphilic Hoveyda‐Grubbs‐Type Catalyst for Ring‐Closing Metathesis in Water

Summary: We report on the synthesis of a new amphiphilic, polymer‐bound variant of the Hoveyda‐Grubbs catalyst via the coupling reaction of a carboxylic acid‐functionalized poly(2‐oxazoline) block copolymer with 2‐isopropoxy‐5‐hydroxystyrene and subsequent reaction of the resulting macroligand with a second generation Grubbs catalyst. For the benchmark, the substrate diethyl diallylmalonate was studied in the ring‐closing metathesis (RCM) reaction and a turn‐over number (TON) of up to 390 in water was achieved. To the best of our knowledge, this is the highest value for any aqueous RCM reaction to date. For the first time, recycling of a ruthenium initiator in an aqueous RCM reaction has been successful to some extent. In addition, the micellar conditions accelerate the conversion of the hydrophobic diene and at the same time stabilize the active alkylidene species, although competing decomposition of the catalyst in water still impairs the catalyst performance. Residual ruthenium content was determined to be below 1 ppm in the product suggesting a very low leaching of the polymeric catalyst system.

Simplified chemical structure of the amphiphilic, polymer‐bound Grubbs‐Hoveyda catalyst.     

Synthesis and photoreactions of polyesters containing triazene and cinnamylidene malonyl units

The synthesis and photoreactions of polyesters containing the photolabile triazene group (Ph NN NR₂) as well as photocrosslinkable units based on substituted cinnamylidene malonyl moieties are reported. The polyesters were obtained by reacting a triazene containing difunctional alcohol with different substituted cinnamylidene malonic acid dichlorides. Irradiation at λ > 395 nm enables selective photocrosslinking of the cinnamylidene malonyl ester units via [2+2]π‐cycloaddition without affecting the triazene units. The photocrosslinking reaction of the triazene containing polyesters as well as triazene‐free reference polyesters were investigated by UV and IR spectroscopy. After selective photocrosslinking the irradiated polymers became insoluble and no longer exhibited a T_g below decomposition temperature. The crosslinked triazene containing polyesters show strong absorption at λ = 308 nm and are therefore interesting candidates for applications using XeCl excimer laser ablation. © 2000 John Wiley & Sons, Inc. J Polym Sci A: Polym Chem 38: 1903–1910, 2000     

On the (im)possibility of evaluating correct individual rate constants of chain propagation kp and chain termination kt by combining kp2/kt and kp/kt data for chain‐length dependent termination

On the basis of simulated data two ways of evaluating individual rate constants by combining k_p²/k_t and k_p /k_t (k_p , k_t = rate constants of chain propagation and termination, respectively) were checked considering the chain‐length dependence of k_t. The first way tried to make use of the fact that pseudostationary polymerization yields data for k_p²/k_t as well as for k_p /k_t referring to the very same experiment, in the second way k_p²/k_t (from steady state experiments) and k_p/k_t data referring to the same mean length of the terminating radical chains were compared. In the first case no meaningful data at all could be obtained because different averages of k_t are operative in the expressions for k_p /k_t and k_p²/k_t. In spite of the comparatively small difference between these two averages (≈15% only) this makes the method collapse. The second way, which can be regarded as an intelligent modification of the “classical” method of determining individual rate constants, at least succeeded in reproducing the correct order of magnitude of the individual rate constants. However, although stationary and pseudostationary experiments independently could be shown to return the same k_t for the same average chain‐length of terminating radicals within extremely narrow limits no reasonable chain‐length dependence of k_t could be derived in this way. The reason is an extreme sensitivity of the pair of equations for k_p/k_t and k_p²/k_t towards small errors and inconsistencies which renders the method unsuccessful even for the high quality simulation data and most probably makes it even collapse for real data. This casts a characteristic light on the unsatisfactory situation with respect to individual rate constants determined in the classical way, regardless of a chain‐length dependence of termination. As a consequence, all efforts of establishing the chain‐length dependence of k_t are recommended to avoid this way and should rather resort to methods based on inserting a directly determined k_p into the equations characteristic of k_p²/k_t or k_p/k_t, properly considering the chain‐length dependent character of k_t.     

Lattice Monte Carlo investigations on copolymer systems, 1. Diblock copolymers

Symmetric diblock copolymers in dilute solution were examined by means of Monte Carlo simulations on a cubic lattice with respect to chain- and block dimensions, shape, local structure and number of contacts. The solvent was either a common good one, a common θ-solvent or a selective one for the two blocks. In all cases, repulsive interactions are operative between the blocks. In addition, the underlying homopolymers (athermal and θ) were divided into two parts (and treated as a block copolymer) for comparison. Chain-length was varied from 40 to 1280 segments leading to the expected values for the critical exponent 2v ≈ 1.2 for good solvent quality and 2v ≈ 1.0 for θ-solvent. Copolymers in a selective solvent scale with an intermediate exponent, 2v ≈ 1.13. The deviation of the mean squared dimensions of the copolymers from the sum of those of two homopolymers of the same length and for the same solvent quality as the blocks is largest for block copolymers in a common θ-solvent (where it exceeds 20%), while the blocks themselves have mostly the same dimensions as their underlying homopolymers of equal length. The shape of the copolymers, expressed by the parameter δ (asphericity) becomes more rod-like with increasing chain-length if there are (compact) θ-blocks in the molecule which are subject to mutual repulsive interaction. In these cases, θ exceeds the value of the homopolymers in the limit of infinite chain-length. The number of contacts per segment approaches a limiting value with increasing chain-length which is ≈0.20 for athermal chains and athermal blocks. For θ-chains and θ-blocks, a limiting value is not yet reached within the range of chainlengths investigated. The number of contacts per segment between two different blocks quickly tends to zero with increasing chain-length.     

The incorporation of side reactions into pseudostationary polymerization, 3. The closed solution of a kinetic scheme for pulsed-laser polymerization comprising concomitant continuous initiation

A procedure is developed that allows the calculation of chain-length distributions of polymers prepared by periodic modulation of the initiation process considering concomitant continuous initiation. For the case of a (pseudostationary) laser-pulse initiated polymerization process a closed solution could be derived for the pseudostationary radical concentration and for the chain-length distribution of dead polymer terminated by disproportionation or stabilized by chain-transfer to monomer or solvent. The analysability of the characteristic peaks appearing in the chain-length distributions of laser-pulse initiated polymers (which is the key for determining the rate constant k_p) is only moderately influenced by continuous thermal radical formation if the extent of this side reaction is not pathologically large, i.e. as long as the amount of primary radicals created by the laser-pulse appreciably exceeds that produced in the dark reaction.     

Simulation of pseudostationary radical polymerization, 3. Periodic modulation of the termination process for termination by disproportionation

A procedure is developed that allows the calculation of chain-length distributions of polymers prepared by periodic modulation of the termination rate constant, considering termination by disproportionation. Applying some (hypothetical) field which suppresses — or at least diminishes — the termination rate constant periodically, results in step-function-like chain-length distributions. In principle, the same information may be obtained as in the case of periodical variation of the rate of initiation.     

Solvent stabilized transition metal cations as initiators for cyclopentadiene polymerization

Solvent stabilized transition metal cations are excellent initiators for the polymerization of cyclopentadiene. The product yield is – in first approximation – directly proportional to the reaction time. Partial substitution of the solvent molecules by ligands with different donor strength and charge allows the synthesis of initiators with tailor made activity. The presence of a metal-metal multiple bond seems not to be of significant importance for the catalytic activity of the initiator systems. The crucial point for the activity of the complexes described here is the presence of labile and therefore easily replacable ligands in equatorial coordination sites.     

Column space equalities for orthogonal projectors

In their paper [Y. Tian, G.P.H. Styan, Rank equalities for idempotent and involutory matrices. Linear Algebra Appl. 335 (2001) 101-117], Tian and Styan established several rank equalities involving a pair of idempotent matrices P and Q. Subsequently, these results are reinvestigated from the point of view of the following question: provided that idempotent P, Q are Hermitian, which relationships given in the aforementioned paper remain valid when ranks are replaced with column spaces? Simultaneously, some related results are established, which shed additional light on the links between subspaces attributed to various functions of a pair of orthogonal projectors.