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 $vF(b)-b$.

A finite sequence of all-pay contests in which each stage awards one prize and its winner leaves. Losing players remain eligible for later stages. A player can receive at most one prize, while its effort costs are incurred whenever it participates. Subgame perfect equilibrium accounts for both the immediate prize and the value of remaining eligible.

An optimization constraint restricts the feasible points in constrained optimization, for example by an equality $g(x)=0$ or an inequality $h(x)\le 0$. Extrema are considered on the resulting feasible set rather than on the whole domain of the objective.

Let $m\ge t\ge 3$ distinct sets have every $t$-fold intersection of size $\lambda$. Either all contain a common set of size $\lambda$, or $m\le k+t-2$, where $k$ is the minimum size of a $(t-2)$-fold intersection. Fix such a minimum intersection and restrict the remaining sets to it. Equal traces give the common set; distinct traces satisfy the constant-intersection family bound. The qualification $m\ge t$ excludes vacuous counterexamples.

For independent exponentially distributed failure times subject to independent right censoring, let $d$ be observed failures and $E={\sum }_i{x}_i$ total follow-up exposure. The parameter-dependent likelihood is ${\theta }^d{e}^{-\theta E}$, giving maximum-likelihood estimator $\hat{\theta }=d/E$. The fitted individual cumulative hazards sum to $d$, but this estimating identity alone does not establish that a constant hazard model fits.

For a time-stepping formula written with an unscaled step residual $R_h$, order at least $p$ means that substituting a smooth exact solution gives $R_h=O(h^{p+1})$. Under the relevant stability of a numerical method assumptions and starting errors $O(h^p)$, this produces global error $O(h^p)$. Residuals divided by the step size instead use the equivalent convention $O(h^p)$.

Conic optimization using the copositive cone. The constraint $Q-\lambda I\in {\mathrm{COP}}_n$ supplies an exact reformulation of a Rayleigh quotient minimum on the nonnegative orthant. Exact conic formulation does not itself provide an efficient membership algorithm.

Conic optimization using the completely positive cone. For a real symmetric matrix $Q$, the trace-normalized program $\min \{⟨Q,X{⟩}_F:X\in {\mathrm{CP}}_n,\mathrm{tr}X=1\}$ minimizes a weighted average of nonnegative-unit-vector Rayleigh quotients. The weights are the squared norms of the factors in $X={\sum }_j{x}_j{x}_j^T$. Hence a minimizing rank-one factor attains the same value as the original orthant minimum.

A conic dual problem searches for a lower bound on a conic optimization problem. A feasible point exhibits an attainable cost; a feasible dual variable explains why every allowed decision costs at least a specified amount. If the two costs agree, the pair certifies a global minimum without relying on how the points were found.

Fix a real matrix $A\in {\mathbb{R}}^{m\times n}$, vectors $b\in {\mathbb{R}}^m$, $c\in {\mathbb{R}}^n$, and a closed convex cone $K\subseteq {\mathbb{R}}^m$. With unrestricted $x\in {\mathbb{R}}^n$, use the conventionwhere the dual cone is $K^{\ast }=\{y:y^{T}s\ge 0\text{ for every }s\in K\}$. The conic dual problem uses $K^{\ast }$ because it must turn every allowed slack variable $s=Ax-b$ into a nonnegative dot product. Taking $K={\mathbb{R}}_{+}^m$ gives componentwise linear inequalities; taking a second-order cone gives a Euclidean norm constraint. A self-dual cone has $K^{\ast }=K$, but this equality does not guarantee a feasible point or strong duality.

The Lagrangian supplies a short derivation. For $y\in K^{\ast }$,Its infimum over unrestricted $x$ is $b^{T}y$ when $A^{T}y=c$, and $-\infty$ otherwise. Maximizing the resulting Lagrange dual function gives the stated conic dual problem. The minus sign matches the convention $Ax-b\in K$.

An interior-point method solves constrained mathematical optimization problems by maintaining variables in the interiors of their feasible cones or inequality domains. Barrier derivatives are defined there, and damped Newton methods keep subsequent iterates in their domains. A central path gives a family of barrier-regularized optima whose duality gap approaches zero; a conic phase-I problem can supply a strict feasible starting point.

For the extended canonical variables with $\{t,E\}=-1$ and $\{x,p\}=1$, the charge $K=t^{2}E-txp+x^2/2$ generates a special conformal transformation. Its time derivative is $2te(E-H)$, so it is conserved on the constraint surface. With a time-dependent parameter $\beta$, the multiplier varies as $\delta e=-2te\beta$ and the action variation is $\dot{\beta }K-d(\beta x^2/2)/d\lambda$.

For four-dimensional Minkowski spacetime, set $p=\arctan ((t-r)/L)$ and $q=\arctan ((t+r)/L)$, with $L>0$, then $T=p+q$, $R=q-p$. Multiplication by the square of the conformal factor $2\cos p\cos q/L$ yields the displayed metric on $R\ge 0$, $|T|+R<\pi$. Its radial Penrose diagram is a triangle: the timelike edge $R=0$ is the ordinary centre, while the two sloping edges are future null infinity and past null infinity. A signed Cartesian spatial coordinate in two dimensions gives a full diamond instead.

For a sample of size $n$ from a continuous probability distribution, this interval covers the median $F^{-1}(1/2)$ with probability $1-2{\sum }_{r=0}^{j-1}\left(\genfrac{}{}{0ex}{}{n}{r}\right)2^{-n}$, for $1\le j<n/2$. Count observations below the median: their number has the binomial distribution with parameters $n,1/2$. No strict increase of the distribution function is needed.

If a positive rate $q$ has a confidence interval $[L,U]$, monotonicity transforms it into $[1/U,1/L]$ for the reciprocal mean $1/q$. This preserves the coverage event exactly and reverses the endpoints. It applies directly to mean exponential holding times.

For nontrivial events, strict attraction under conditional probability is equivalent to a positive covariance of their indicator variables. This condition is symmetric in the two events. Complementing exactly one of the events reverses the strict inequality. This two-event condition should be distinguished from association of an entire family of random variables.

A family of probability measures $K_y$ on a measurable space for which $y\mapsto K_y(A)$ is measurable for every measurable set $A$. This combines measurable dependence on a parameter with a probability measure at each parameter. The conditional measures of a factor form such a kernel, with the additional fiber-support property from disintegration.

For a probability measure on a standard Borel space and a measurable map $\pi$ to another standard Borel space, there is an almost-everywhere unique measurable probability kernel $y\mapsto {\mu }_y$ supported on ${\pi }^{-1}\{y\}$. It integrates to $\mu$, and $\int fd{\mu }_{\pi (x)}$ realizes conditional expectation given $\pi$. These measures are the conditional measures of a factor.

For classical finite random variables, $H(X,Z\mid Y)=H(X\mid Y)+H(Z\mid X,Y)=H(Z\mid Y)+H(X\mid Z,Y)$. Each identity follows by inserting the corresponding joint entropy into $H(X,Z,Y)-H(Y)$. The two orders are particularly useful when one variable is a deterministic function of the others, as in Fano's inequality via an error indicator.