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The Gaussian representation of the delta function. introduced by Kirchhoff in his formulation of Huygens's principle [610], can also be obtained as another attempt of making (F.4) more mathematically correct. Instead of separating the integral in the last line of (F.4) into a positive and a negative k part in order to introduce the positive imaginary part ie to x for positive k and the negative imaginary part -ie to x for negative k, we can introduce the imaginary part iek to x such that when multiplied by ik, it always gives rise to a converging negative real term in the exponential regardless of the sign of k.



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Convert Image to PDF in C#, VB.NET. Step1: Use C#/VB.NET to create a PDF document. In this step, you need to create a new PDF file first, then, add a section in the newly built PDF, at last, add a page in the section that you just added. Step2: Load an image to PDF and set image location. Step3: Save the image to PDF ...

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Consider the probability of nding a polymer molecule containing x structural units. This is synonymous with the probability of nding a molecule with x 1 A groups reacted and one A group unreacted. The probability that an A group has reacted at time t is de ned as the extent of reaction p. The probability that x 1 A groups have reacted is the product of x 1 separate probabilities or px 1. Since the probability of an A group being unreacted is 1 p , the probability Nx of nding the molecule in question, with x structural units, is given by

n~()C)

N x px 1 1 p 2-86

(El3)

Since N x is synonymous with the mole or number fraction of molecules in the polymer mixture that are x-mers (i.e., that contain x structural units), then





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This is another common representation of the delta function in physics. To derive it, consider first the Fourier transform of 1/r:

Nx Npx 1 1 p 2-87

:F :;: (k)

where N is the total number of polymer molecules and Nx is the number that are x-mers. If the total number of structural units present initially is N0 , then N N0 1 p and Eq. 2-87 becomes

1 r dr 121< de/> 11< d( cos = 21T roo r dr d(

Nx N0 1 p 2 px 1 2-88

In the Navigation pane, click the library in which you want to save the file (here, Documents). In the file list, navigate to the folder in which you want to save the file. Type a name for the file in the File name field. Click Save.

Neglecting the weights of the end groups, the weight fraction wx of x-mers (i.e., the weight fraction of the molecules that contains x structural units) is given by wx xNx =N0 and Eq. 2-88 becomes

{))e ikr cos (}

wx x 1 p 2 px 1 2-89

eikrC:

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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

= 21T

the hydroxyl and nitrogen functions. The metal is separated by ltration of the polymer and can be recovered from the polymer by changing the pH. Other chelating polymers have been studied for removal of pertechnetate ion, TcO , from contaminated groundwater at nuclear 4 reactor facilities [Alexandratos and Natesan, 1999; Bonnesen et al., 2000]. Polymers containing crown ether or cryptand macrocyclic ligands bind and separate various cations [Jaycox and Smid, 1981; Mathias and Al-Jumah, 1980]. Polymers containing chiral groups are useful for resolving racemic mixtures into the individual enantiomers [Kiniwa et al., 1987; Mathur et al., 1980; Wulff et al., 1980]. For example, the copper(II) complex of XXXXIII (either the R- or S-enantiomer) resolves racemates of amino acids [Sugden et al., 1980]. The separation is based on the formation of a pair of diastereomeric complexes from the reaction of the polymer reagent with the two enantiomers. One of the enantiomers is complexed more strongly than the other and this achieves separation of the enantiomers. A polymer drug can be synthesized by the covalent bonding of a drug to a polymer or the synthesis and (co)polymerization of a monomer containing the drug moiety [Callant and Schacht, 1990; Donaruma and Vogl, 1978; Hsieh, 1988; Levenfeld et al., 1990; Ottenbrite

eikr -

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Sep 27, 2013 · Basically, I just want to convert an image to a PDF exactly as is (copying the page size from the image size and with no margin). The first step…












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