Show that there is only one linear fractional transformation that maps three given distinct points z1, z2 and z3 in the extended z plane onto three specified distinct points w1, w2 and w3 in the extended w plane.
A linear fractional transformation has the form [ w=\phi(z)=\frac{az+b}{cz+d}, ] where (a,b,c,d) are complex constants not all zero and (ad-bc\neq 0). Such a transformation is completely determined up to a nonzero common factor multiplying (a,b,c,d). Consider three distinct points (z_1,z_2,z_3) in the extended (z)-plane and three distinct points (w_1,w_2,w_3) in the extended (w)-plane. A fundamental property of linear fractional transformations is the invariance of the cross-ratio. For any _____ ___ _______ _______ __________ ______ ______ _____ ______ __________.
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Find the radius of convergence of the following series.
i)
ii)
Let C be a cyclic code of length eight over with generator polynomial
. Find the generator matrix of C, the generator polynomial of C and the parity check matrix of C.
Let and
be two binary codes with generator matrices
respectively.
i) Find the minimum distance of both the codes.
Table 1: Table for F16.
| 0000 | 0 | 1000 | α³ | 1011 | α⁷ | 1110 | α¹¹ |
|---|---|---|---|---|---|---|---|
| 0001 | 1 | 0011 | α⁴ | 0101 | α⁸ | 1111 | α¹² |
| 0010 | α | 0110 | α⁵ | 1010 | α⁹ | 1101 | α¹³ |
| 0100 | α² | 1100 | α⁶ | 0111 | α¹⁰ | 1001 | α¹⁴ |
Which of the following binary codes are linear?
i)
ii)
Justify your answer.
Evaluate where c is the circle
.
Evaluate , where c is the eight like figure shown in Fig. 1.
Find the image of the semi-infinite strip x > 0, 0 < y < 1 when . Sketch the strip and its image.