The riemann matrix of (2 + 1)‐dimensional symmetric‐hyperbolic systems

Numerical Measure of a Complex Matrix

“…see [26,16,17] and Section 5 below. In the generic case where λ 1 (θ) < λ 2 (θ) < · · · < λ n (θ) for all θ ∈ [0, π], the bitangent set C ′ A is empty and Σ A = C A .…”

“…In Section 4, we derive a representation formula for the numerical density using the inversion of the Radon transformation. Section 5 collects a few results on the geometry of the critical set Σ A , which are mainly borrowed from [19,26,17]. These informations are used in Section 6 to derive an explicit formula for the derivatives of order n − 2 of the numerical density, which allows us to obtain generic regularity results and to express the fundamental solution of the hyperbolic system (4) in terms of derivatives of the numerical density.…”

“…The algebraic and geometric properties of the map Φ A have been extensively studied, see [19,26,4,9,16,17]. In particular, the set of all critical values of Φ A , which we denote by Σ A ⊂ C, has received a lot of attention, because this is an interesting object which contains a lot of information on the matrix A.…”

“…For any z ∈ C \ Σ A , let N (z) be the number of straight lines containing z which are tangent to the curve C A , see (39) below for a precise definition where possible multiplicities are taken into account. It is easy to verify that N (z) is constant in each connected component of C \ Σ A , and that N (z) ≤ n [26]. The distinguished regions where f A is polynomial of degree n − 3 (if n ≥ 3) or f A ≡ 0 (if n = 2) are exactly those connected components of C \ Σ A on which N (z) takes its maximal value n. This remarkable property of the numerical density, which is one of our main results, will be established in Section 6.1.…”

“…The properties of the fundamental solution of symmetric hyperbolic systems have been studied by many authors, see e.g. [22,26,4,2,3]. In the particular case of system (4), J. Bazer and D. Yen have shown that, if one identifies C with R 2 , the singular support of the distribution E * is contained in the critical set Σ A , and the (stable) lacunas of system (4) are exactly the regions described above where the numerical density f A is polynomial of degree n − 3.…”