Schulz function

Fig. 11.8 The molecular-weight distribution calculated using the Schulz function...

Obviously, the theory outhned above can be applied to two- and three-dimensional systems. In the case of a two-dimensional system the Fourier transforms of the two-particle function coefficients are carried out by using an algorithm, developed by Lado [85], that preserves orthogonality. A monolayer of adsorbed colloidal particles, having a continuous distribution of diameters, has been investigated by Lado. Specific calculations have been carried out for the system with the Schulz distribution [86]... 

Fig. 6. Dependence of the degree of polymerization corresponding to the maximum of the GPC elution curve, GFC m x °f the monofunctional (black) and bifunctional (open) growing species as a function of the conversion X (V. Warzelhan, H. Hocker, G. V. Schulz, Ref.31>)...

To generate the necessary distribution functions, the ratio of is used to approximate the true molecular weight distribution by a Schulz-Zimm distribution. It is also assumed that the reactive functional groups are distributed randomly on the polymer chain. The Schulz-Zimm parameters used to calculate distribution functions and probability generating functions (see below) are defined as follows ... 

Equations 22 and 23 can be solved numerically using the method described in Ref. 5. For oligomers, the probability generating functions are calculated by the appropriate sums. For random copolymers analytical expressions for and t can be written for a polymer or crosslinker using the appropriate Schulz-Zimm parameters (5) ... 

These parameters are used calculate the site and mass distribution functions assuming a Schulz-Zimm molecular weight distribution. The Schulz-Zimm parameters are calculated in lines 930-950. The weight fraction of diluent (as a fraction of the amount of polymer) is then sought. If there is no diluent enter 0. If there is a diluent, the functionality and molecular weight of the diluent is requested (line 1040). The necessary expectation values are computed (lines 1060-1150). 

Reaction mechanisms and molar mass distributions The molar mass distribution of a synthetic polymer strongly depends on the polymerization mechanism, and sole knowledge of some average molar mass may be of little help if the distribution function, or at least its second moment, is not known. To illustrate this, we will discuss two prominent distribution functions, as examples the Poisson distribution and the Schulz-Flory distribution, and refer the reader to the literature [7] for a more detailed discussion. 

Schulze TG, Schumacher J, Muller DJ, Krauss H, Alfter D, Maroldt A, et al. Lack of association between a functional polymorphism of the cytochrome P450 1A2 (CYP1A2) gene and tardive dyskinesia in schizophrenia. Am J Med Genet 2001 105(6) 498-501. 

Elliott, C., Muller, J., Miklis, M., Bhat, R. A., Schulze-Lefert, P. and Panstruga, R. (2005). Conserved extracellular cysteine residues and cytoplasmic loop-loop interplay are required for functionality of the heptahe-lical MLO protein. Biochem J. 385, 243-54. 

Muller, R.H., Ruhl, D., Runge, S., Schulze-Forster, K., and Mehnert, W., Cytotoxicity of solid lipid nanoparticles as a function of the lipid matrix and the surfactant, Pharmaceutical Research, 1997, 14, 458-462. 

Patil AO, Schulz DN, Novak BM (eds) (1998) Functional polymers modern synthetic methods and novel structures. ACS Symposium Series, vol 704. American Chemical Society, Washington, DC... 

In 1939, Schulz [92-94] first reported that 12 (X=CN in 21) served as an initiator for the radical polymerization of MM A and St. Thereafter, Hey and Misra [95] also reported the polymerization of St with 12 or its p-methoxy substituted derivatives. Borsig et al. [96,97] reported in 1967 the polymerization of MMA and St with 3,3,4,4-tetraphenylcyclohexane (21b) and 1,1,2,2-tetraphenylcyclopentane (21c) and that the reaction orders of the polymerization rates with respect to the concentrations of 21b and 21c were 0.25 and 0.20, respectively, and concluded that the primary radical termination predominantly occurred. It was noted that in these polymerizations the average molecular weight of the polymer increased as a function of the polymerization time, although the clear reason was not described in these papers. It was also reported by the same authors that the resulting polymer could further induce block copolymerization [98]. 

The formulation of the engineered nonlinear short-term model presented is a variant of an MINLP model described in the dissertation by Schulz [5], In this subsection, all necessary indices, parameters and variables are introduced, and the constraints and the objective function are derived. In the following section, the nonlinear formulation is linearized yielding a MILP model. In order to keep track of the variables used in the MINLP and in the MILP formulation, they are displayed in Figure 7.3 along with some key parameters. 

Schulz, A., Grosse, R., Schultz, G., Gudermann, T., and Schoneberg, T. (2000) structural implication for receptor oligomerization from functional reconstitution studies of mutant V2 vasopressin receptors. J. Biol. Chem. 275, 2381-2389. 

The normal distribution function, also referred to as the Flory-Schulz distribution, relates the fraction of an x-mer (a polymer molecule consisting of x repeat units) in the entire assembly of molecules to its formation probability. It can be defined either as a number distribution function or as a weight distribution function. The number of moles of an x-mer (Nx) is given by the normal number distribution as follows ... 

Amination. The synthesis of polymers with primary amine end-group functionality has been a challenge because the primary amine group can undergo rapid chain transfer and termination reactions with car-banionic chain ends (14). Schulz and Halasa (15) used a phenyllith-ium initiator with a bis(trimethylsilyl)-protected amine group to prepare amine-terminated polydienes. Nakahama and coworkers (16,17)... 

Schaefgen and Flory [79] were the first to observe this effect. They prepared star-branched polyamides by co-condensation of A-B types of monomers with central units which carried/-functional A groups. By this technique star molecules were obtained in which the arms are not monodisperse in length. They rather obeyed the Schulz-Flory most probable length distribution with polydis-persity index However, the coupling of f arms onto a star center leads... 

Levings MK, Bachetta R, Schulz U, Roncarolo MG The role of IL-10 and TGF-P in the differentiation and effector function of T regulatory cells. Int Arch Allergy Appl Immunol 2002 129 263-276. 

Schoneberg, T., Schulz, A., and Gudermann, T. (2002) The structural basis of G-protein-coupled receptor function and dysfunction in human diseases. Rev. Physiol. Biochem. Pharmacol. 144, 143-227. 

Equations 2-86 and 2-89 give the number- and weight-distribution functions, respectively, for step polymerizations at the extent of polymerization p. These distributions are usually referred to as the most probable or Flory or Flory-Schulz distributions. Plots of the two distribution functions for several values of p are shown in Figs. 2-9 and 2-10. It is seen that on a... 

Muller DJ, Schulze TG, Macciardi F, et al (2002) Moclobemide response in depressed patients association study with a functional polymorphism in the monoamine oxidase A promoter. Pharmacopsychiatry 35 157-158... 

Recently Schulz et aland Fischer et al have had some difficulty in applying the CDW-EIS theory successfully for fully differential cross sections in fast ion-atom collisions at large perturbations. These ionization cross sections are expected to be sensitive to the quality of the target wave function and therefore accurate wave functions are needed to calculate these cross sections. Thus one purpose of this paper is to address this problem theoretically by re-examining the CDW-EIS model and the assumptions on which it is based. We will explore this by employing different potentials to represent the interaction between the ionized electron, projectile ion and residual target ion. For other recent work carried out on fully differential cross sections see and references therein. This discussion is presented in section 4. 

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