The biconjugate of $f$ is the convex conjugate of its convex conjugate:It is the supremum of all affine minorants. For a proper function with an affine minorant, the Fenchel-Moreau theorem identifies it with the closed convex envelope; a proper lower semicontinuous convex function equals its biconjugate.

The set of all finite conic combinations of elements of $S$, including zero. It is the smallest convex cone containing $S$. It need not be closed even when $S$ is closed: the closed set $S=\{(t,1):t\ge 1\}$ generates vectors converging to $(1,0)$ via $(t,1)/t$, but no nonzero generated vector has second coordinate zero.

A convex cone that is a closed set in its ambient topology. In a finite-dimensional normed vector space, this means it contains every limit of its convergent sequences. For example, the positive semidefinite cone is closed because each test $X\mapsto x^{T}Xx$ is continuous.

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.

For points $p,q$ in a topological space $X$, the fixed-endpoint path space iswith the subspace compact-open topology from the continuous-map space. For a metric space, this agrees with uniform convergence of paths on $[0,1]$. If $p,q$ lie in different path components, this path space is empty.

If $p,q$ lie in the same path component, choose one continuous path $\alpha$ from $p$ to $q$. Appending the reversed $\alpha$ sends a path from $p$ to $q$ to a based loop at $p$. Conversely, append $\alpha$ to a based loop. The backtracking pieces contract through paths with fixed endpoints, so these maps are homotopy inverses. Hence $\Omega (p,q)$ has the homotopy type of the based loop space ${\Omega }_{p}X$. In the usual smooth manifold setting, smooth, piecewise smooth and Sobolev $H^1$ path models give the same homotopy type.

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.

The continuous dual space of a Hausdorff locally convex space separates points: for every nonzero $x$ there is a continuous linear functional $f$ with $f(x)\ne 0$.

A separating family of continuous seminorms supplies $p(x)>0$. In the real case, define $g(tx)=tp(x)$ on the span of $x$ and extend it under $p$ by the Hahn-Banach theorem. The bound $|g(y)|\le p(y)$ gives continuity. In a complex vector space, apply the real argument to the underlying real vector space, then use $f(y)=g(y)-ig(iy)$ to obtain a complex-linear functional with nonzero real part at $x$. The Hausdorff condition is essential: a nontrivial vector invisible to every continuous seminorm is also invisible to every continuous linear functional.

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.