Cochran's statistical theorem for outer inverses of matrices and matrix quadratic forms

We extend the matrix version of Cochran's statistical theorem to outer inverses of a matrix. As applications, we investigate the Wishartness and independence of matrix quadratic forms for Kronecker product covariance structures.     

Effects of diamines and their fluorinated groups on the color lightness and preparation of organosoluble aromatic polyimides from 2,2‐bis[4‐(4‐amino‐2‐trifluoromethylphenoxy)phenyl]‐hexafluoropropane

To investigate the position and amount of the CF₃ group affecting the coloration of polyimides (PIs), we prepared 2,2‐bis[4‐(4‐amino‐2‐trifluoromethylphenoxy)phenyl]hexafluoropropane ( 2 ) with four CF₃ groups with 2‐chloro‐5‐nitrobenzotrifluoride and 2,2‐bis(4‐hydroxyphenol)hexafluoropropane. A series of soluble and light‐colored fluorinated PIs ( 5 ) were synthesized from 2 and various aromatic dianhydrides ( 3a – 3f ). 5a – 5f had inherent viscosities ranging from 0.80 to 1.19 dL/g and were soluble in amide polar solvents and even in less polar solvents. The glass‐transition temperatures of 5 were 221–265 °C, and the 10% weight‐loss temperatures were above 493 °C. Their films had cutoff wavelengths between 343 and 390 nm, b* values (a yellowness index) ranging from 5 to 41, dielectric constants of 2.68–3.01 (1 MHz), and moisture absorptions of 0.03–0.29 wt %. In a comparison of the PI series 6 – 8 based on 2,2‐bis[4‐(4‐aminophenoxy)phenyl]hexafluoropropane, 2,2‐bis[4‐(4‐amino‐2‐trifluoromethylphenoxy)phenyl]propane, and 2,2‐bis[4‐(4‐aminophenoxy)phenyl]propane, we found that the CF₃ group close to the imide group was more effective in lowering the color; this means that CF₃ of 5 , 7 , and 8f was more effective than that of 6c . The color intensity of the four PI series was lowered in the following order: 5 > 7 > 6 > 8 . The PI 5f , synthesized from diamine 2 and 4,4′‐hexafluoroisopropylidenediphthalic anhydride, had six CF₃ groups in a repeated segment, so it exhibited the lightest color among the four series. © 2003 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 41: 922–938, 2003     

1H and 13C NMR spectroscopic study of oxidation of D,L‐cystine and 3,3′‐dithiobis(propionic acid) with hydrogen peroxide in aqueous solution

The oxidation of symmetrical disulfides [D ,L ‐cystine ( 1 ) and 3,3′‐dithiobis(propionic acid) ( 2 )] with hydrogen peroxide in D₂O–NaOH solution (pH 10–11) was studied by NMR spectroscopy. Assignments of the proton and carbon NMR signals of starting materials ( 1 and 2 ) and products of oxidation are based on conventional 1D NMR methods (DEPT, selective spin decoupling). Formation of C—S bond cleavage products or, in case of 2 , partially oxidized intermediates was not detected. The accelerating effect of Cu²⁺ cations, but not Fe³⁺ cations, on the oxidation rate of 1 in basic medium was demonstrated. Copyright © 2002 John Wiley & Sons, Ltd.     

Novel,biodegradable, functional poly(ester‐carbonate)s by copolymerization of trans‐4‐hydroxy‐L‐proline with cyclic carbonate bearing a pendent carboxylic group

Water‐soluble poly(ester‐carbonate) having pendent amino and carboxylic groups on the main‐chain carbon is reported for the first time. This article describes the melt ring‐opening/condensation reaction of trans‐4‐hydroxy‐N‐benzyloxycarbonyl‐L ‐proline (N‐CBz‐Hpr) with 5‐methyl‐5‐benzyloxycarbonyl‐1,3‐dioxan‐2‐one (MBC) at a wide range of molar fractions. The influence of reaction conditions such as catalyst concentration, polymerization time, and temperature on the number average molecular weight (M_n) and molecular weight distribution (M_w/M_n) of the copolymers was investigated. The polymerizations were carried out in bulk at 110 °C with 3 wt % stannous octoate as a catalyst for 16 h. The poly(ester‐carbonate)s obtained were characterized by Fourier transform infrared spectroscopy, ¹H NMR, differential scanning calorimetry, and gel permeation chromatography. The copolymers synthesized exhibited moderate molecular weights (M_n = 6000–14,700 g mol^?1) with reasonable molecular weight distributions (M_w/M_n = 1.11–2.23). The values of the glass‐transition temperature (T_g) of the copolymers depended on the molar fractions of cyclic carbonate. When the MBC content decreased from 76 to 12 mol %, the T_g increased from 16 to 48 °C. The relationship between the poly(N‐CBz‐Hpr‐co‐MBC) T_g and the compositions was in approximation with the Fox equation. In vitro degradation of these poly(N‐CBz‐Hpr‐co‐MBC)s was evaluated from weight‐loss measurements and the change of M_n and M_w/M_n. Debenzylation of 3 by catalytic hydrogenation led to the corresponding linear poly(ester‐carbonate), 4 , with pendent amino and carboxylic groups. © 2004 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 42: 2303–2312, 2004     

Two‐stage copolycondensation of preformed oligomers with bisphenols by tosyl chloride/dimethylformamide/pyridine from distributions of resulting oligomers determined by a combination of gel permeation chromatography and NMR

A two‐stage co‐oligomerization of the oligomers initially formed from an equimolar mixture of isophthalic acid (IPA) and terephthalic acid (TPA) and 2,2‐bis(4‐hydroxyphenyl)propane (BPA, 50 mol %) with bisphenols (BPs, 20 mol %) was carried out using a tosyl chloride/dimethylformamide/pyridine condensing agent. The distributions of the resulting oligomers (n_x‐mers), which were quenched with methanol, were determined by a combination of gel permeation chromatography (GPC) and NMR. These distributions (presented by molar percentage) were conveniently calculated with the equation n_x (mol %) = n_x (% mol by GPC) × n₀ (mol % by NMR)/n₀ (% mol by GPC), where n_x (% mol) = n_x (wt % by GPC)/its molecular weight. The results showed the distributions of the preformed IPA/TPA‐BPA oligomers to be in fairly good accord with those obtained directly from GPC and to be supported by the NMR results. The calculation was applied to the co‐oligomers prepared up to a reaction of 0.7, at which there was an increase in the number of higher oligomers indivisible by GPC and the distributions could no longer be determined by molar percentage. The calculated distributions are discussed in relation to the results of copolycondensation. The sequence distributions in the resulting co‐oligomers, which were also examined by NMR, are compared with those in the copolymers. © 2003 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 42: 44–51, 2004     

Multicyclic polyethers derived from 1,1,1‐tris(4‐hydroxyphenyl)ethane and 2,6‐dihalopyridines

Poly(pyridine ether)s were prepared in two ways: the polycondensation of silylated 1,1,1‐tris(4‐hydroxyphenyl)ethane (THPE) with 2,6‐difluoropyridine (method A) and the polycondensation of free THPE with 2,6‐dichloropyridine (method B). With method A, the THPE/difluoropyridine feed ratio was varied from 1.0:1.0 to 1.0:1.6. Cycles, bicycles, and multicycles were the main reaction products, and crosslinking was never observed. When ideal stoichiometry was used exclusively, multicycles free of functional groups were obtained. These multicycles were detectable in matrix‐assisted laser desorption/ionization time‐of‐flight (MALDI‐TOF) mass spectra up to B₃₈C₇₆ with a mass of approximately 32,000 Da. With method B, the reaction conditions were varied at a fixed feed ratio to achieve an optimum for the preparation of multicyclic polyethers, but because of the lower reactivity of 2,6‐dichloropyridine, a quantitative conversion was not achieved. The reaction products were characterized with MALDI‐TOF mass spectrometry, viscosity measurements, and size exclusion chromatography. © 2004 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 42: 5725–5735, 2004     

Structure and properties of the mesophase of syndiotactic polystyrene. VIII. Solvent sorption behavior of syndiotactic polystyrene/p‐chlorotoluene mesophase membranes

A crystalline δ form of a syndiotactic polystyrene (sPS) membrane was prepared from a solution of sPS (1 wt %) and p‐chlorotoluene (p‐CT) by a solution‐casting method. The mesophase (δ empty form) of sPS was obtained by the extraction of the guest solvent from the δ form of sPS by a stepwise solvent‐extraction method. The sPS/p‐CT mesophase membrane [p‐CT (A‐M)] was used for the sorption of 1 mol % p‐CT for different times and for the sorption of different concentrations of p‐CT, chlorobenzene (CB), p‐xylene (p‐X), toluene, and chloroform for 48 h. The presence of solvents in the sPS membrane was confirmed by IR analysis. A thermal study revealed that the sorption amount of 1 mol % p‐CT increased with increasing immersion time, and the sorption amounts of different solvents increased with increasing solvent concentration. Differential scanning calorimetry results showed that the desorption peak temperature increased as the amount of the solvent increased in the clathrated sPS membrane. Wide‐angle X‐ray diffraction results showed that 2θ at 8.25° was slightly shifted toward 8°, and there was no change in the peak position at 10° for p‐CT (A‐M), which was immersed in different solvents (1 mol %); however, the intensity of 2θ at 10° was not similar for all the samples. Among the solvents used for the sorption studies at 1 mol %, p‐CT (A‐M) could sorb more p‐CT and CB than p‐X, toluene, and chloroform. The solvent sorption isotherm was the Langmuir sorption mechanism. © 2004 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 42: 3439–3446, 2004     

Pressure and temperature dependence of the gas‐transport properties of dense poly[2,6‐toluene‐2,2‐bis(3,4‐dicarboxylphenyl)hexafluoropropane diimide] membranes

The gas‐transport properties of poly[2,6‐toluene‐2,2‐bis(3,4‐dicarboxylphenyl)hexafluoropropane diimide] (6FDA‐2,6‐DAT) have been investigated. The sorption behavior of dense 6FDA‐2,6‐DAT membranes is well described by the dual‐mode sorption model and has certain relationships with the critical temperatures of the penetrants. The solubility coefficient decreases with an increase in either the pressure or temperature. The temperature dependence of the diffusivity coefficient increases with an increase in the penetrant size, as the order of the activation energy for the diffusion jump is CH₄ > N₂ > O₂ > CO₂. Also, the average diffusion coefficient increases with increasing pressure for all the gases tested. As a combined contribution from sorption and diffusion, permeability decreases with increases in the pressure and the kinetic diameter of the penetrant molecules. Even up to 32.7 atm, no plasticization phenomenon can be observed on flat dense 6FDA‐2,6‐DAT membranes from their permeability–pressure curves. However, just as for other gases, the absolute value of the heat of sorption of CO₂ decreases with increasing pressure at a low‐pressure range, but the trend changes when the feed pressure is greater than 10 atm. This implies that CO₂‐induced plasticization may occur and reduce the positive enthalpy required to create a site into which a penetrant can be sorbed. Therefore, a better diagnosis of the inherent threshold pressure for the plasticization of a glassy polymer membrane may involve examining the absolute value of the heat of sorption as a function of pressure and identifying the turning point at which the gradient of the absolute value of the heat of sorption against pressure turns from a negative value to a positive one. © 2003 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 42: 354–364, 2004