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A rotating detonation combustor exhibits corotating 
$N$-wave modes with 
$N$ detonation waves propagating in the same direction. These modes and their responses to ignition conditions and disturbances were studied using a surrogate model. Through numerical continuation, a mode curve (MC) is obtained, depicting the relationship between the wave speed of the one-wave mode and a defined baseline of the combustor circumference (
$L_{{base}}$) under fixed equation parameters, limited by deflagration and flow choking. The modes’ existence is confirmed by the equivalence between a one-wave mode within a combustor with circumference 
$L_{{base}}$/
$N$ on the MC and an 
$N$-wave mode in an 
$L_{{base}}$ combustor. The stability, measured by the real part of the eigenvalue from linear stability analysis (LSA), revealed the dynamic properties. When multiple stable modes exist under the same parameters, ignition conditions with a spatial period of 
$L_{{base}}$/
$N$ are more likely to form 
$N$-wave modes. An unstable evolution in formed modes, occurs in the dynamics from stable to unstable modes through saddle-node bifurcation and Hopf bifurcation induced by parameter perturbations and from unstable to stable modes induced by state disturbances. Eigenmodes from LSA reveal mechanisms of the unstable evolution, including the effect of secondary deflagration in the unstable one-wave mode and competitive interaction between detonation waves in the unstable multiwave mode, crucial for the combustor to mode transition.
For each 
$n\geq 1$, let 
$FT_n$ be the free tree monoid of rank n and 
$E_n$ the full extensive transformation monoid over the finite chain 
$\{1, 2, \ldots , n\}$. It is shown that the monoids 
$FT_n$ and 
$E_{n+1}$ satisfy the same identities. Therefore, 
$FT_n$ is finitely based if and only if 
$n\leq 3$.
${\mathbb R}^3$ and 
${\mathbb R}^4$, and certain cases in higher dimensions
Let 
$\mu _{M,D}$ be the self-similar measure generated by 
$M=RN^q$ and the product-form digit set 
$D=\{0,1,\ldots ,N-1\}\oplus N^{p_1}\{0,1,\ldots ,N-1\}\oplus \cdots \oplus N^{p_s}\{0,1,\ldots ,N-1\}$, where 
$R\geq 2$, 
$N\geq 2$, q, 
$p_i(1\leq i\leq s)$ are integers with 
$\gcd (R,N)=1$ and 
$1\leq p_1<p_2<\cdots <p_s<q$. In this paper, we first show that 
$\mu _{M,D}$ is a spectral measure with a model spectrum 
$\Lambda $. Then, we completely settle two types of spectral eigenvalue problems for 
$\mu _{M,D}$. In the first case, for a real t, we give a necessary and sufficient condition under which 
$t\Lambda $ is also a spectrum of 
$\mu _{M,D}$. In the second case, we characterize all possible real numbers t such that 
$\Lambda '\subset \mathbb {R}$ and 
$t\Lambda '$ are both spectra of 
$\mu _{M,D}$.
