Krdytkyu

2 1 $begingroup$ I studied that a measure valued solution of a conservation law $u_t+f(u)_x=0$ is a measurable map $eta : y rightarrow eta_y in Prob(mathbb{R^n})$ which satisfies $partial_t(eta_y, lambda) +partial_x (eta_y, f(lambda)) =0$ in the sense of distribution on $mathbb{R^2_+}$ . In particular, when the conservation law admits $L^{infty}$ solution then $eta_y=delta_{u(y)}$ . Now I am trying to read "Delta-shock Wave Type Solution of Hyperbolic Systems of Conservation Laws" by Danilov and Shelkovich [1]. In this article what do they mean by $delta$ -shocks? in which sense these $delta$ -shocks are different from the shocks of the conservation laws? According to the definition which I stated in the begining any shock solution $u in L^{infty}$ can be written as a dirac measure $eta_y

2 1 $begingroup$ Consider the function $f$ on natural numbers defined by the following recursion: $f(1)=1$ $f(3)=3$ $f(2n)=f(n)$ $f(4n+1)=2f(2n+1)-f(n)$ $f(4n+3)=3f(2n+1)-2f(n)$ Numerical evidence shows that for odd $k$ we have $f(f(k))=k$ , but I have no clue on how to prove it. Any ideas? recurrence-relations recursion share | cite | improve this question asked Dec 19 '18 at 18:37 A. Bellmunt A. Bellmunt 895 5 15 $endgroup$ add a comment  |