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A convex cone containing no nonzero line through the origin, equivalently $K\cap (-K)=\{0\}$. The copositive cone is pointed: a matrix in both it and its negative has zero quadratic form on the nonnegative orthant; testing $e_i$ and $e_i+e_j$ kills all entries.

The convex cone of completely positive matrices:It is a closed convex cone and the dual cone of the copositive cone under the Frobenius inner product. The finite-sum definition imposes no closure by fiat; closedness of the completely positive cone supplies that fact.

A metric space is complete if and only if every contraction mapping of every nonempty closed subset into itself has a fixed point. For the converse, a nonconvergent Cauchy sequence contains a rapidly approaching subsequence whose point set is closed and whose shift is a contraction without a fixed point. Requiring the property only on the whole space is a different, weaker condition.

A contour encircling a branch cut travels on its two sides and joins them by small and large circles. It converts the phase change of a complex power into a real integral through the residue theorem. The two circular contributions must be bounded before taking their limiting radii.

An integration ray can be rotated through an analytic sector when endpoint and connecting-arc contributions vanish. Complex powers gain a phase from the rotated differential and power. This relates conditionally convergent oscillatory gamma integrals to decaying Laplace integrals.

An oriented piecewise differentiable path in the complex plane, used as the domain of a contour integral. Its orientation controls the signs of residues in the residue theorem.

A sequential continuous-time Markov chain has transitions $1\to 2\to 3$ with rates $\lambda ,\nu$ and an absorbing state $3$. Its probability of ever leaving state $1$ by time $t$ is $1-e^{-\lambda t}$; its probability of occupying state $2$ is $\lambda (e^{-\lambda t}-e^{-\nu t})/(\nu -\lambda )$ for unequal rates.

Approximate changing covariates by constant values on observation intervals. The generator matrix is then constant on each interval, with transition matrix $\exp (Q_j\Delta t_j)$. Multiplying the relevant entries gives a conditional likelihood function for discretely observed states.

An irreversible three-state illness-death model allows progression $1\to 2$ and death $1\to 3$ or $2\to 3$, with state 3 absorbing. A constant-rate transition intensity matrix uses three free rates $a,b,c$ and rows $(-(a+b),a,b)$, $(0,-c,c)$ and $(0,0,0)$. The labels can represent disease severity rather than literal health versus illness.

The Aalen–Johansen estimator is a product of empirical state-transition matrices over event times. In competing risks, the estimated cumulative incidence function for cause $j$ jumps by $\hat{S}(t-)d_j(t)/Y(t)$, while overall survival is updated using all event causes. Right censoring changes subsequent risk sets but does not itself create a probability jump.

The fixed-endpoint path space $\Omega (p,q)$ consists of paths in a manifold with specified initial and terminal points. Smooth, piecewise smooth and Sobolev $H^1$ versions have the same homotopy type in the usual manifold setting. Joining a path to a fixed path back from $q$ to $p$ gives a homotopy equivalence with the based loop space.

For a locally convex space $E$, the strong dual topology on its continuous dual space $E^{\prime }$ is generated by seminorms $p_B(T)={\sup }_{\phi \in B}|⟨T,\phi ⟩|$, where $B$ runs through the bounded sets in a topological vector space $E$. Convergence therefore means uniform convergence of pairings on every such set. For the Schwartz space, the defining boundedness is boundedness of every Schwartz seminorm, rather than one norm or one common compact support.

In a Hausdorff locally convex space, a separating family of continuous seminorms has $p(x)>0$ for any chosen nonzero $x$ and some $p$. Define a linear functional on its one-dimensional span attaining $p(x)$ and extend it under that seminorm by the real Hahn-Banach theorem. The extension is continuous because it is seminorm-bounded, and distinguishes $x$ from zero. The Hausdorff condition is essential.

A vector subspace $Z\subseteq X^{\ast }$ is norming if some $c>0$ satisfies the displayed inequality. It retains enough continuous linear functionals to control the norm of every vector. Every such vector subspace is dense for the weak-star topology: otherwise a nonzero evaluation linear functional would annihilate its weak-star closure, contradicting the inequality. In the real case its symmetric unit ball makes the absolute-value and signed supremum versions equivalent.

A continued fraction $[a_0;a_1,a_2,\dots ]$ with integer $a_0$ and positive integer $a_j$ for $j\ge 1$. Its partial numerators are all one. The continued fraction convergents have the standard two-term numerator and denominator recurrences.

A prize allocation rule assigns winning probabilities or divisible prize shares to a profile of efforts. In an all-pay auction, the greatest effort wins; a proportional allocation contest instead assigns shares continuously according to relative effort. The rule is essential data of a contest model.

A rank-order contest awards a sequence of prizes according to the ranks of efforts, while every player incurs its own effort cost. With an independent private values model, an increasing symmetric bidding function orders efforts in the same way as valuations.

Players choose efforts in several all-pay contests at the same time. With additive quasilinear utility and no shared effort budget, a player's expected payoff is the sum of its per-contest payoffs. An entry restriction couples the choices of contests even though the subsequent effort optimization is separable.

For positive total effort, a proportional allocation contest awards player $i$ the prize with probability $b_i/{\sum }_j{b}_j$, or gives it that proportion of a divisible prize. A convention is needed at zero total effort. This allocation is different from the discontinuous highest-effort rule of an all-pay auction.

Two players pay their invested efforts regardless of outcome. A leader invests first, and a follower observes that effort before deciding. With ties favoring the follower and unit costs, a follower of value $u$ matches leader effort $b$ when $u>b$, and otherwise declines. For a continuous follower cumulative distribution function $F$, the leader wins with probability $F(b)$ and has payoff $vF(b)-b$.