For a nonnegative integer claim count $N$ with $p=\mathbb{P}(N=0)<1$, the conditional law given $N>0$ has masses $\mathbb{P}(N=n)/(1-p)$ for $n\ge 1$. Its probability generating function is $(G_N(z)-p)/(1-p)$. Truncation at zero is meaningful only when the positive-count event has positive probability.

A scalar representation is the one-dimensional trivial representation: every group element acts as the identity. Scalar polarization labels are unchanged by the relevant rotation group.

This excited carbon-12 resonance lies close to the three-alpha threshold and enables the Triple-alpha process to make carbon efficiently. A transient beryllium-8 nucleus captures an alpha particle into the resonance; electromagnetic decay then stabilizes carbon-12. A small radiative branching probability still matters because the resonance enhances formation.

For $A\subseteq [n{]}^2$, use three disjoint labelled vertex classes $X=[n]$, $Y=[n]$, $Z=[2n]$. Each $(x,y)\in A$ inserts the three edges $xy,x(x+y),y(x+y)$, producing pairwise edge-disjoint triangles. Every triangle in a graph with labels $(x,y,z)$ corresponds to the three points $(x,y),(x,z-x),(z-y,y)$ of $A$. It is canonical when $z=x+y$; otherwise it gives a corner in an integer grid with nonzero displacement $z-x-y$.

For an even sextic Landau free energy perturbed by a conjugate field, a tricritical wing is a phase coexistence surface at nonzero field between ordered minima of different magnitudes. Two symmetry-related wings emerge from the three-phase line at zero field and terminate at ordinary critical edges. They meet the zero-field ordered coexistence sheet at the tricritical point.

In a metric space, applying the triangle inequality in both directions gives $|d(x,z)-d(y,z)|\le d(x,y)$. In a normed vector space, its corresponding form is $|\Vert x\Vert -\Vert y\Vert |\le \Vert x-y\Vert$. In particular, distance to a fixed point is a Lipschitz continuous real function.

A family of triangles in a graph is edge-disjoint if no edge belongs to two members. Any deletion of edges producing a triangle-free graph must remove at least one distinct edge for each member. The triangles in a graph may share vertices.

The assignment problem chooses a one-to-one pairing of jobs and agents minimizing their total cost. Its minimum-cost flow formulation puts unit supplies on job vertices and unit demands on agent vertices of a bipartite graph. Integral feasible flows are perfect matchings.

When source and target have continuous strictly increasing cumulative distributions, all transport maps are obtained by choosing a Lebesgue-measure-preserving map $S$ on the unit interval. The maps $F$ and $G$ turn the two measures into uniform measure. The choice $S=\mathrm{id}$ gives the monotone rearrangement.

For $0<\beta <4$, a nondegenerate transonic spherical flow in a power-law potential connected to infinity requires $\gamma <(4+\beta )/(4-\beta )$. For $\beta \ge 4$, every finite $\gamma >1$ satisfies the local crossing condition; there is no finite upper bound. The uniformly valid inequality is $2\beta -(4-\beta )(\gamma -1)>0$.

With reservoir density ${\rho }_{\infty }$ and adiabatic sound speed $c_{\infty }$, put $q=2/\beta -1/2$ and $h=1-q(\gamma -1)>0$. The selected mass accretion rate is $4\pi {\rho }_{\infty }(A\beta /2{)}^{2/\beta }{c}_{\infty }^{1-4/\beta }{h}^{q-1/(\gamma -1)}$. It follows by evaluating the conserved spherical flux at the sonic point.

A fixed sphere in a slowly varying incident Stokes flow requires an external force $\mathbf{F}=-6\pi \mu a{\mathbf{u}}_{\infty }$ and external couple $\mathbf{G}=-4\pi \mu a^3{\mathbf{\omega }}_{\infty }$ to leading order. These follow from Faxén translation law and Faxén rotation law; the incident Stokeslet supplies ${\mathbf{u}}_{\infty }$ and ${\mathbf{\omega }}_{\infty }$. Finite-radius Laplacian terms enter the translational force at the next nonzero order. The signs here refer to applied holding loads, not hydrodynamic loads.

For an infinite cardinal number $\kappa$, $H_{\kappa }$ consists of the sets $x$ for which $|\mathrm{tc}(\{x\})|<\kappa$, where $\mathrm{tc}$ is transitive closure. The entire transitive membership ancestry is bounded, not only the size of $x$. These collections are transitive sets. At $\kappa ={\omega }_1$ the elements are the hereditarily countable sets.

A positive transition intensity can be modeled as $q_{rs}(t\mid z)=\exp ({\theta }_0+{\theta }^{T}z(t))$. A unit increase in a fixed covariate multiplies the intensity by $e^{{\theta }_j}$. A changing covariate generally gives a time-dependent generator matrix.

A simple cooperative game is a monotone transferable utility game in which each coalition either loses, with value zero, or wins, with value one. In the usual nontrivial convention the empty coalition loses and the grand coalition wins. The Shapley value is then the probability that a player changes a losing coalition into a winning one when players enter in uniformly random order.

A weighted voting game assigns nonnegative weights $w_i$ to players and a threshold $t\ge 0$, and declares a coalition winning whenNonnegative weights make the rule monotone. The empty coalition loses, and the grand coalition wins when $t<{\sum }_i{w}_i$; under that nontrivial condition this is a simple cooperative game.

If all weights are integers, the strict threshold is equivalent to the weak quota ${\sum }_{i\in S}{w}_i\ge \lfloor t\rfloor +1$. It becomes quota $t+1$ when $t$ is itself an integer. This distinction matters for noninteger thresholds.

For example, three unit weights and $t=1$ give two-of-three majority. Nonnegative weights do not imply a convex cooperative game: $v(\{1,2\})+v(\{2,3\})=2$, while $v(\{1,2,3\})+v(\{2\})=1$, violating the supermodular inequality.

The Shapley value averages each player's marginal contribution over uniformly random player orderings. Exactly $|S|!(n-|S|-1)!$ orderings have $S$ immediately before player $i$, giving the formula. The values sum to $v(N)$ by telescoping each ordering. In a simple cooperative game this is the probability of being pivotal. For a convex cooperative game, Shapley value belongs to the core of a convex game guarantees a stable allocation as well.

The core consists of efficient payoff vectors for which each coalition receives at least its own attainable value. Thus no coalition can improve every member's payoff by leaving. It is a convex set defined by linear constraints, but can be empty. Every convex cooperative game has a nonempty core because each marginal contribution vector belongs to it.

A transferable utility game is convex when its coalition-value function is supermodular. Equivalently a player's marginal contribution cannot decrease when the preceding coalition grows. The implication follows by applying the displayed inequality to $A\cup \{i\}$ and $B$ with $A\subseteq B$, $i\notin B$; the reverse implication follows by adding successive marginal inequalities. This property concerns coalition values, not geometrical convexity of a strategy space.

A coalition is a subset of the players in a transferable utility game, considered as a group that can act together. The game assigns it value $v(S)$. The core of a cooperative game constrains the total payoff assigned to every coalition, while a marginal contribution measures the value gained by adding one player.