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An ideal-MHD simple wave traces a one-parameter state-space curve whose tangent is a right eigenvector of the characteristic matrix. Its parameter obeys $q_t+v(q)q_x=0$ and its characteristic speed obeys the Inviscid Burgers equation. Nonlinear Alfvén waves are a linearly degenerate example.

A characteristic field is linearly degenerate if the directional derivative of its speed along its right eigenvector vanishes. Its simple waves have constant propagation speed along the state curve, so they do not steepen by amplitude-dependent transport.

A normal subgroup of $A_6$ has order dividing 360 and consists of the identity plus whole conjugacy classes of the alternating group on six letters. If it omits the odd-sized class of size 45, its order is an odd divisor and no nontrivial class sum fits. If it includes that class, the possible class sums not exceeding 180 are $46,86,118,126,136,158,176$, none dividing 360. Every proper subgroup has order at most 180; therefore $A_6$ is a simple group.

A nonempty collection of subsets of a fixed universe closed under complements and finite unions, hence also finite intersections. It generates a sigma-algebra by allowing countable set operations. A countable generating algebra provides one simultaneous family of test indicators for measure-theoretic arguments.

A short exact sequence with specified kernel module $A$ and quotient module $C$. Its equivalence class is determined by the degree-one Ext functor; equivalence must preserve both end modules.

For a member of a shifted set family that is an intersecting family, find the first prefix with more selected than unselected positions. Reflect membership in that prefix. The result has one fewer element and lies in the lower shadow; the first prefix with more unselected positions gives the inverse. This injection proves the intersecting shadow lemma.

A dilute dust component can be approximated as a pressureless fluid with velocity $\mathbf{v}$ and surface density of a disk ${\Sigma }_d$. In a gas velocity $\mathbf{u}$, linear drag force gives acceleration $-(\mathbf{v}-\mathbf{u})/t_s$ for positive aerodynamic stopping time $t_s$. Its equations include Coriolis acceleration, the shearing-sheet tidal potential, and ${\partial }_t{\Sigma }_d+\nabla \cdot ({\Sigma }_d\mathbf{v})=0$. Neglecting gas backreaction is justified only when the prescribed-gas approximation is appropriate. Strong orbit crossing or appreciable dust velocity dispersion can invalidate the single-valued pressureless-fluid description.

In an unstratified Keplerian shearing sheet with sound speed $c$, the four linear velocity-density equations yield ${\omega }^4-({\Omega }^2+c^2{k}^2){\omega }^2+{\Omega }^2{c}^2{k}_z^2=0$. The lower-frequency branch approaches the inertial wave relation ${\omega }^2={\Omega }^2{k}_z^2/k^2$ for $ck\gg \Omega$; the upper branch is predominantly acoustic.

The steady planar velocity $\mathbf{u}=\alpha y{\mathbf{e}}_x-\beta x{\mathbf{e}}_y$ supports a bounded polytropic patch when $\alpha =({\beta }^2-3{\Omega }^2)/\beta$ and $\sqrt{3}\Omega <\beta <2\Omega$. Its polytropic enthalpy is $Q=Q_0-(2\Omega -\beta )(\beta x^2+\alpha y^2)/2$ with $Q_0>0$. Its fluid free boundary is the material ellipse $\beta x^2+\alpha y^2=2Q_0/(2\Omega -\beta )$. No additional self-gravity potential is part of this model.

An infinite gravitating particle lattice with spacing $h$ in the azimuthal direction is a shearing-sheet equilibrium when all particles sit at $x_n=0,y_n=nh$. Opposite-neighbor accelerations cancel absolutely. The linear gravitational response to pair displacement differences has transverse and longitudinal coefficients $1$ and $-2$, divided by $h^3|j-n{|}^3$.

On a ringed space $(X,{\mathcal{O}}_X)$, an injective module sheaf is an injective object in the category of sheaves of modules over ${\mathcal{O}}_X$. Its defining extension property concerns module-sheaf morphisms; it is distinct from injectivity solely in the category of sheaves of abelian groups.

The dual is the sheaf of modules whose sections on $U$ are module-sheaf morphisms $\mathcal{F}{|}_U\to {\mathcal{O}}_U$. For a locally free sheaf of finite rank, the dual is locally free of the same rank and the natural double-dual map is an isomorphism. General coherent sheaves need not have this property: a skyscraper at a point of positive-dimensional projective space has zero ordinary sheaf dual.

On a complex manifold, a sheaf of modules over the sheaf of holomorphic functions is coherent analytic if it is locally finitely generated and the kernel of every morphism ${\mathcal{O}}^r\to \mathcal{F}$ is locally finitely generated. The structure sheaf and locally free finite-rank sheaves are examples. Cartan theorem B gives vanishing of their higher sheaf cohomology on a Stein manifold.

The graph is $\mathrm{gph}S=\{(u,v):v\in S(u)\}$. Local graph normals encode the Mordukhovich coderivative and hence the Aubin property. Graph closedness is a local hypothesis in the sensitivity criterion.

Near $(\bar{u},\bar{v})$ in a graph, this property requires neighborhoods $U,V$ and a finite $\kappa$ such that $S(u)\cap V\subset S(u^{\prime })+\kappa \Vert u-u^{\prime }\Vert \mathbb{B}$ for all $u,u^{\prime }\in U$. Nearby solutions can be matched under nearby perturbations with linear displacement control. Single-valued maps reduce to local Lipschitz continuity.

A set function is submodular if $f(A)+f(B)\ge f(A\cup B)+f(A\cap B)$ for every pair of subsets. Entropy submodularity is a fundamental example.

A real-valued set function is supermodular if it satisfies the displayed inequality. Its negative is a submodular set function. Equivalently, the gain from adding an element cannot decrease as the set grows. This is the defining property of a convex cooperative game, and makes coalition marginal allocations lie in the core of a cooperative game.

For a Serre fibration, this first-quadrant spectral sequence has differential of bidegree $(r,1-r)$ and a fiber-cohomology local coefficient system on the base. A simply connected base makes that system constant. Its edge map is restriction to the fiber; a surviving fiber class is the restriction of a total-space class. For commutative ring coefficients it is multiplicative and its differentials are graded derivations.

For a series ${\sum }_n{a}_n$, let $r={\mathrm{limsup}}_{n\to \infty }|a_n{|}^{1/n}$. If $r<1$, choose $r<\rho <1$; eventually $|a_n|\le {\rho }^n$, so geometric comparison gives absolute convergence. If $r>1$, the terms fail to tend to zero along a subsequence and the series diverges. At $r=1$, the test is inconclusive.

With follower distribution $F$, a leader of value $v$ chooses $b\in \arg {\max }_{0\le b\le 1}\{vF(b)-b\}$. If $F$ is a concave function, the objective is concave and an interior optimum satisfies $v{F}^{\prime }(b)=1$. Endpoint derivatives handle zero or maximal effort. Zero effort guarantees nonnegative payoff, so every optimum satisfies $b\le vF(b)\le v$.