Near-resonance approximation of rotating Navier–Stokes equations

Evident from dispersion relation (2.10), the Coriolis term does not generate (linear) oscillatory dynamics in horizontal modes. Such a large set of non-oscillatory modes is a common feature that separates many fluid models (even in the inviscid case) from classically known dispersive PDEs.

For set S, the associated indicator function is defined as ${\boldsymbol 1}_S(x) = 1$ if $x\in S$ and ${\boldsymbol 1}_S(x) = 0$ if $x\notin S$.

The zero-mean assumption is imposed WLOG and this property propagates in time—see the start of section 2.

Such study can appear in literature concerning the van der Corput lemma.

A simple curve is a curve that does not cross itself.

In fact, it suffices to know that ${\mathcal T}$ has zero Lebesgue measure in $\mathbb{R}$ by Sard's theorem.

Rewrite the numerator of (5.14) as: $\frac12\cos\left( \sigma_k \theta_k +\sigma_n \theta_n \right) +\frac12\cos\left( \sigma_k \theta_k -\sigma_n \theta_n \right) -\frac12\cos\left( 2\theta_m + \sigma_k \theta_k +\sigma_n \theta_n \right) -\frac12\cos\left( \sigma_k \theta_k +\sigma_n \theta_n \right) = \sin\left( \theta_m + \sigma_k \theta_k \right)\sin\left( \theta_m +\sigma_n \theta_n \right)$, and the denominator as $\sin(\theta_m+\sigma_k\theta_k)\sin(\theta_m-\sigma_k\theta_n)$.

Together with identity, they are the only self-inverse, parity-preserving members of the 4-permutation group, and form a subgroup. Then, necessarily, they are commutative and the three non-identity elements are cyclic under the group operation. This subgroup is isomorphic to the Klein 4-group and to $\mathbb Z_2 \times \mathbb Z_2$.

if the largest and smallest elements of $x, y, z, w$ form set $\{x,w\}$ or set $\{y,z\}$, then $(x-y)(z-w)\gt0$.

e.g. $\min\{a, b\} = \frac12(a+b)-\frac12|a-b|$ and $\varsigma_3 = c_n^2+c_k^2+c_m^2-\min\{c_n^2,c_k^2,c_m^2\}+\min\{-c_n^2,-c_k^2,-c_m^2\}$.

Consider summability and $\mathbb{R}^+ = \cup_{j\in \mathbb{Z}}[2^{j},2^{j+1})$.