Normalization and Fermi–Coulomb and Coulomb hole sum rules for approximate wave functions

For approximate wave functions, we prove the theorem that there is a one‐to‐one correspondence between the constraints of normalization and of the Fermi–Coulomb and Coulomb hole charge sum rules at each electron position. This correspondence is surprising in light of the fact that normalization depends on the probability of finding an electron at some position. In contrast, the Fermi–Coulomb hole sum rule depends on the probability of two electrons staying apart because of correlations due to the Pauli exclusion principle and Coulomb repulsion, while the Coulomb hole sum rule depends on Coulomb repulsion. We demonstrate the theorem for the ground state of the He atom by the use of two different approximate wave functions that are functionals rather than functions. The first of these wave function functionals is constructed to satisfy the constraint of normalization, and the second that of the Coulomb hole sum rule for each electron position. Each is then shown to satisfy the other corresponding sum rule. The significance of the theorem for the construction of approximate “exchange‐correlation” and “correlation” energy functionals of density functional theory is also discussed. © 2006 Wiley Periodicals, Inc. Int J Quantum Chem, 2007     

Stability control of the electrooptic effect with new maleimide copolymers containing photoreactive tricyanopyrrolidene‐based chromophores

New photocrosslinkable maleimide copolymers have been synthesized by the attachment of a tricyanopyrrolidene‐based chromophore. The 2‐(3‐cyano‐4‐(2‐{4‐[hexyl‐(6‐hydroxy‐hexyl)‐amino]‐phenyl}‐vinyl)‐5‐oxo‐1‐{4‐[4‐(3‐oxo‐3‐phenyl‐propenyl)‐ phenoxy]‐butyl}‐1,5‐dihydro‐pyrrol‐2‐ylidene)‐malononitrile chromophore exhibits nonlinear optical activity and contains a chalcone moiety that is sensitive to UV light (λ = 330–360 nm) for crosslink formation. The maleimide monomers have also been functionalized with chalcone moieties. The resultant copolymers exhibit great processability, and one of them shows a maximum electrooptic coefficient of 90 pm/V at 1300 nm. We could control the thermal stability of the electrooptic coefficient with the newly synthesized photoreactive copolymers successfully. © 2006 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 45: 531–542, 2007     

Degradative chain transfer in vinyl acetate polymerizations using toluene as solvent

In this work, we propose that retardation in vinyl acetate polymerization rate in the presence of toluene is due to degradative chain transfer. The transfer constant to toluene (C_trs) determined using the Mayo method is equal to 3.8 × 10^?3, which is remarkably similar to the value calculated from the rate data, assuming degradative chain transfer (2.7 × 10^?3). Simulations, including chain‐length‐dependent termination, were carried out to compare our degradative chain transfer model with experimental results. The conversion–time profiles showed excellent agreement between experiment and simulation. Good agreement was found for the M_n data as a function of conversion. The experimental and simulation data strongly support the postulate that degradative chain transfer is the dominant kinetic mechanism. © 2007 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 45: 3620–3625, 2007     

Glass transitions and mixed phases in block SBS

Differential scanning calorimetry (DSC) does not allow for easy determination of the glass‐transition temperature (T_g) of the polystyrene (PS) block in styrene–butadiene–styrene (SBS) block copolymers. Modulated DSC (MDSC), which deconvolutes the standard DSC signal into reversing and nonreversing signals, was used to determine the (T_g) of both the polybutadiene (PB) and PS blocks in SBS. The T_g of the PB block was sharp, at ?92 °C, but that for the PS blocks was extremely broad, from ?60 to 125 °C with a maximum at 68 °C because of blending with PB. PS blocks were found only to exist in a mixed PS–PB phase. This concurred with the results from dynamic mechanical analysis. Annealing did not allow for a segregation of the PS blocks into a pure phase, but allowed for the segregation of the mixed phase into two mixed phases, one that was PB‐rich and the other that was PS‐rich. It is concluded that three phases coexist in SBS: PB, PB‐rich, and PS‐rich phases. © 2004 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 43: 276–279, 2005     

Chemical Constituents of Ligusticum sinensis Oliv.

The new phenylpropanoid diglycoside ligusinenoside A ( 1 ), and the two new 8,4′‐oxyneolignan(‘8‐O‐4′‐neolignan’) diglycosides ligusinenosides B ( 2 ) and C ( 3 ), together with nine known compounds, were isolated from the rhizomes of Ligusticum sinensis Oliv. The structures of 1 – 3 were elucidated on the basis of spectroscopic analyses.     

Banach spaces of distributions having two module structures

After a discussion of a space of test functions and the corresponding space of distributions, a family of Banach spaces (B, ∥ ∥_B) in standard situation is described. These are spaces of distributions having a pointwise module structure and also a module structure with respect to convolution. The main results concern relations between the different spaces associated to B established by means of well-known methods from the theory of Banach modules, among them B₀ and $\text{B}\text{?}$, the closure of the test functions in B and the weak relative completion of B, respectively. The latter is shown to be always a dual Banach space. The main diagram, given in Theorem 4.7, gives full information concerning inclusions between these spaces, showing also a complete symmetry. A great number of corresponding formulas is established. How they can be applied is indicated by selected examples, in particular by certain Segal algebras and the A_p-algebras of Herz. Various further applications are to be given elsewhere.     

Jetmetriken und hyperbolische Geometrie

Ohne ZusammenfassungHerrn E. Heinz zum 65. Geburtstag gewidmet