Appendix D using barcode encoding for vs .net control to generate, create qr codes image in vs .net applications. 2 of 5 Industrial FORMAL POWER SUMS D. . D A formal power sum F(D) o .net framework qr codes ver a eld F is a sum of the form F(D) =. Fn Dn , Fn F . Such objects appear frequ .net framework QR ently in this book: in the context of the nite-length analysis for the BEC, during the study of convolutional and turbo codes, or as generating functions of weight distributions. We collect the basic facts about formal power sums that we have used throughout the text.

. D. . B In the sequel we assume t Denso QR Bar Code for .NET hat the eld F is xed and we usually omit any reference to it. e two most important examples for us are F = F2 and F = R.

Given two formal power sums F(D) = n 0 Fn Dn and G(D) = n 0 Gn Dn , we de ne their addition by F(D) + G(D) = (Fn + Gn )Dn .. In a similar way we de ne their multiplication by F(D) G(D) =. n 0 i=0 Gi Fn i Dn ,. which is the rule familia r from polynomial multiplication. Note that this is well dened to compute the n-th coe cient of the product we only need to perform a nite number of operations. erefore, we do not encounter issues of convergence.

Also note that multiplication is commutative. Is is possible to de ne division Recall that over the reals we say that y is the multiplicative inverse of x, x 0, which we write as y = 1 x, if xy = 1. Dividing by x is then the same as multiplying by y.

We proceed along the same lines for formal power sums. Consider the formal power sum F(D). We want to nd the formal power sum G(D), if it exists, such that H(D) = F(D)G(D) = 1.

We say that G(D) is the multiplicative inverse of F(D) and dividing by F(D) corresponds to multiplying with G(D). Using the preceding multiplication rule, we get the following set of equations: 1 = H0 = F0 G0 ,. 0 = Hn = Gi Fn i , n 1. 0..

is set of equations has a solution, and this solution is unique, if and only if F0 In this case we get 1 , F0 1 Gn = F0 G0 =. Gi Fn i , n Since the evaluation of e ach coe cient Fn only involves a nite number of algebraic operations and only makes use of the values of Fi , 0 i < n, this gives rise to a well de ned formal power sum. In summary, a formal power sum F(D) has a multiplicative inverse if and only if F0 0. We write this multiplicative inverse as 1 F(D).

E D. (I F(D) = 1 D. Since F0 G0 = 1 D R).

Let F = R and consider the example 0, 1 (1 D) exists. We get. 1 1 1 = 1, G1 = G0 F1 = 1, G2 = (G0 F2 + G1 F1 ) = G1 = 1, F0 F0 F0 and, in general, Gn = e refore, G(D) =. 1 F0. Gi Fn i = 1 Gn 1 F1 = Gn 1 = 1. F0 Dn . e resemblance to the iden Visual Studio .NET QR-Code tity i=0 x i = 1 (1 x), which is familiar from analysis and valid for x < 1, is not a coincidence. In general, as a rule of thumb any identity which is valid for Taylor series and which can be meaningfully interpreted in the realm of formal power sums is still valid if considered as an identity of formal power sums.

Further basic properties of formal power sums are developed in Problem D. ..

D. . S Consider a formal power s Denso QR Bar Code for .NET um over R, F(D) =. Fn Dn ,. so that the corresponding Quick Response Code for .NET real sum n 0 Fn converges. Assume we have succeeded in nding a closed-form solution F(D).

en we have the compact representation of the sum n 0 Fn as F(D) D=1 . What can we say about the related power sum Fq (D) =. n 0n q N Fn Dn ,. and the related sum n 0 n q N Fn , where q is a natural number In words, we would like to nd the sum of all q-th terms. We claim that Fq (D) = is is true since.
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