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3. Is there a Z4 version? 31 pairs of matroids, and attached to such pairs a three-variable analogue of the Tutte polynomial. This polynomial gives some features of C as specialisations. The following section is entirely based on her work. Let M1 and M2 be two matroids on the same set E. We say that (M1 , M2 ) is a matroid pair, or that M1 is a quotient of M2 , if there is a matroid N on a set E ∪ X such that M1 = N/X and M2 = N\X. ) It can be shown that we may choose N and X so that |X| = ρM2 (E) − ρM1 (E), and X is independent and E is spanning in N.

4. Find an example of two matroids M1 and M2 on a set E such that both M1 and M2 are representable over Z2 and (M1 , M2 ) is a matroid pair but not a matroid chain over Z2 . 5. Let (M1 , M2 ) be a matroid pair on E, and let ρi be the rank function of Mi for i = 1, 2. Prove that 0 ≤ ρ2 A − ρ1 A ≤ ρ2 E − ρ1 E for any set A ⊆ E. 6. Calculate the weight enumerator of the code associated with a representation of U3,n over GF(q). Find examples with n = q + 1. CHAPTER 5 Permutation groups In the second half of the notes, we introduce the last strand, permutation groups, and braid it together with codes and matroids.

We compute α1 g. If this is not in the G-orbit of α1 , then g ∈ / G, and we are done. Otherwise, there is a unique x1 ∈ X1 such that α1 g = α1 x1 . Now gx1−1 fixes α1 , and we apply the test recursively to decide whether gx1−1 ∈ G1 ; for we have g ∈ G if and only if gx1−1 ∈ G1 in this case. If the test succeeds, then we will eventually find that gx1−1 · · · xr−1 = 1, that is, g = xr · · · x1 , with xi ∈ Xi for i = 1, . . , r. This expression is unique, so |G| = |Xr | · · · |X1 |, and we have found the order of G.

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A Determination of the Solar Motion and the Stream Motion Based on Radial Velocities and Absolute Ma by Strömberg G.


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