Consider $n\ge 2$ identical players, prizes $w_j>0$, unit effort costs, and exactly two entries per player. The symmetric equilibrium omission probabilities $q_j$ sum to one. The common per-contest payoff is $u=({\sum }_j{w}_j^{-1/(n-1)}{)}^{-(n-1)}$, and the unconditional rival distribution on $0\le b\le w_j-u$ is $G_j(b)=((b+u)/w_j{)}^{1/(n-1)}$. Conditional on entry, its distribution function is $(G_j(b)-q_j)/(1-q_j)$. Choosing the omitted contest according to $q$ and sampling the two active efforts independently gives a symmetric equilibrium with expected payoff $2u$. These formulas determine the marginal laws; they do not determine the dependence between a player's two efforts.

A simplicial cycle is a simplicial chain with zero boundary. Its homology class is unchanged on addition of a boundary; in the Mayer–Vietoris sequence, splitting a cycle between two subcomplexes constructs the connecting map.

For a linear program with unchanged cost and constraint matrix, an optimal simplex basis remains optimal while its basic solution remains feasible. Its reduced costs and dual certificate do not change. The new basic variables are $A_B^{-1}(b+ϵ)$, and the objective is the displayed affine function, with $y$ the basis's dual solution. This gives an exact piecewise-affine sensitivity region, determined by nonnegativity of the basic variables, rather than merely a formal first derivative.

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 $f:\mathcal{P}(V)\to \mathbb{R}$ is submodular whenfor every $A,B\subseteq V$. For finite $V$, this is equivalent to diminishing returns:The marginal increment of an element can only decrease as the surrounding set grows.^[ref]

For necessity, apply the defining inequality to $A\cup \{x\}$ and $B$. Conversely, enumerate $A\setminus B$ and compare adding its elements to $A\cap B$ with adding them to $B$. Summing the marginal inequalities gives $f(A)-f(A\cap B)\ge f(A\cup B)-f(B)$.

A coverage function is an example: $f(S)=|{\bigcup }_{v\in S}{C}_v|$ counts the covered objects in a finite universe. An added $C_x$ contributes only objects not already covered, so its gain decreases with $S$. Entropy submodularity is another fundamental example.

A set function is supermodular whenEquivalently, $-f$ is a submodular set function. On a finite ground set, this means increasing marginal increments: if $A\subseteq B$ and $x\notin B$, then ${\Delta }_{x}f(A)\le {\Delta }_{x}f(B)$.

For example, $f(S)=|S{|}^2$ has increment $2|S|+1$, which increases with the cardinality of $S$. In a convex cooperative game, supermodularity expresses increasing gains from joining larger coalitions; the corresponding marginal allocations lie in the core of a cooperative game.