# Algebraic Combinatorics I: Association Schemes (Mathematics by Eiichi Bannai

By Eiichi Bannai

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Extra info for Algebraic Combinatorics I: Association Schemes (Mathematics lecture note series) (Bk. 1)

Sample text

This implies that the wave is split in a single rarefaction shock. Thus the number of waves decreases by 1. Therefore we conclude that a each interaction the number of wave fronts decreases at least by 1 and so the lemma is proved. 6 Wave-Front Tracking and Existence of Solutions ur ul PSfrag replacements ul 35 ur um Fig. 14. Interaction between two wave fronts. 5. The total variation of uν (t, ·) is not increasing with respect the time. Var. Var. u ¯. 38) Proof. It is clear that the total variation may vary only at interaction times.

1). 2. 13) is said entropy admissible if, for every C 1 function ϕ 0 with compact support in [0, T [×R and for every entropy–entropy flux pair (η, q), it holds T 0 R {η(u)ϕt + q(u)ϕx } dxdt 0. 14) We consider now an entropy admissible solution u and a sequence of entropy–entropy flux pairs (ην , qν ) such that ην → η and qν → q locally uniformly in u ∈ Rn . 15) for every ν ∈ N. 15), we obtain that T 0 R {η(u)ϕt + q(u)ϕx } dxdt 0. 16) This allows us to call a C 0 function η an entropy if there exists a sequence of entropies ην converging to η locally uniformly.

In this case the relation between C 1 entropy and entropy flux takes the form η (u)f (u) = q (u). 18) Therefore if we take a C 1 entropy η, every corresponding entropy flux q has the expression u q(u) = η (s)f (s)ds, u0 where u0 is an arbitrary element of R. 3. 17) satisfies the Kruzkov entropy admissibility condition if T 0 R {|u − k| ϕt + sgn (u − k) (f (u) − f (k)) ϕx } dxdt for every k ∈ R and every C 1 function ϕ [0, T [×R. 0 0 with compact support in We have the following theorem. 4. 17).