A holding-dependent cost proportional to the absolute position size. In a one-period gain objective it changes the sample payoff to $\theta X-\epsilon |\theta |$, giving gain parameter $X-\epsilon$ for positive holdings and $X+\epsilon$ for negative holdings. The objective is concave but can have a kink at zero.

For a positive claim and $r>0$, the tail integral formula for moments gives the displayed expression. In particular the first moment is ${\int }_0^M\overline{F}(x)dx$ and the raw second moment is $2{\int }_0^{M}x\overline{F}(x)dx$. If the claim law has a density, capping adds an atom of a measure at $M$ with mass $\mathbb{P}(X\ge M)$.

For positive claim sizes with finite second moment in a compound Poisson distribution aggregate of parameter $\lambda$, the two excess of loss reinsurance payouts are $\min (X,M)$ and $(X-M{)}_{+}$. The derivative of their total aggregate variance is $2\lambda (M\overline{F}(M)-\mathbb{E}[(X-M{)}_{+}])$. Thus the displayed condition characterizes stationarity. When the tail probability is positive it says that the retention equals the mean residual life. Global minimality requires an additional sign or comparison argument.

One exact parity query determines whether the first two bits agree. If they agree, a second query reads one of them; otherwise it reads the third bit. Those values respectively determine the three-bit majority function. The matching lower bound follows from its degree-three multilinear polynomial.

Suppose a function has least period $r\mid N$ and distinct values on distinct period cosets. Two independent Fourier samples have the form $c_j=s_{j}N/r$ with uniform $s_j\in {\mathbb{Z}}_r$. The displayed candidate is $r/\gcd (r,s_1,s_2)$, so exact recovery occurs when no prime divisor of $r$ divides both residues. The Chinese remainder theorem gives probability ${\prod }_{\ell \mid r}(1-{\ell }^{-2})\ge 1/\zeta (2)=6/{\pi }^2$, using the Euler product. Unlike a single-sample coprimality probability, this bound is uniformly greater than one half. A candidate-period test makes successful recovery heralded when such a test is available.

For every infinite regular cardinal $\kappa$, there is a strictly eventually increasing sequence of ${\kappa }^{+}$ functions $\kappa \to \kappa$. At stage $0<\zeta <{\kappa }^{+}$ enumerate predecessors by $e:\kappa \to \zeta$ and put $f_{\zeta }(\delta )={\sup }_{\eta <\delta }(f_{e(\eta )}(\delta )+1)$. Regularity bounds each value below $\kappa$, while any predecessor's index is eventually included in the supremum.

For $s\ge 0$ and $\epsilon >0$, integrate over $0\le K\le s$ to obtain the displayed identity. Tonelli theorem then expresses the moment of a nonnegative random variable as the same integral of expected call payoffs, even when the moment is infinite. This is a static representation across strikes rather than a dynamic replicating strategy in a restricted finite-asset market.

For $\sigma >1$, logarithmic expansion of the Euler product reduces positivity to $3+4\cos \theta +\cos 2\theta =2(1+\cos \theta {)}^2$. At primes dividing the modulus the zeta contribution is positive by itself. If the last factor is bounded at the line-one point, a zero of the middle factor would outweigh the zeta pole and force this product to zero. This proves the relevant nonvanishing of Dirichlet L-functions on the line one.

A U-statistic averages a symmetric function of $m$ distinct sample observations over all unordered $m$-tuples. For independent and identically distributed observations it is an unbiased estimator of $\mathbb{E}H(X_1,\dots ,X_m)$. Omitting repeated indices can remove diagonal contributions in quadratic plug-in estimators.

An edge density exceeding the Turan theorem threshold for $K_{r+1}$ by a fixed positive amount forces a balanced $K_{r+1}(t)$ subgraph with logarithmic part size. Combine clique supersaturation by sampling with the dense clique family blow-up lemma. Dense binomial random graphs show that the logarithmic order is optimal for a uniform density guarantee.

For $X_j$ the number of tree components of order $j$ in a binomial random graph, choose their vertices, choose one of $j^{j-2}$ labelled trees by the Cayley formula, require its $j-1$ edges, and exclude both the remaining internal edges and all crossing edges. For $j=1$ interpret $j^{j-2}=1$, recovering the isolated vertex count. If $p=\lambda /n$ for fixed positive $\lambda$ and $j=O(\log n)$, the Stirling formula gives $\mathbb{E}{X}_j\sim n(\lambda e^{1-\lambda }{)}^j/(\lambda \sqrt{2\pi }{j}^{5/2})$. Distinct overlapping vertex sets cannot both be graph components; for disjoint sets their joint occurrence gains the factor $(1-p{)}^{-j^2}$ relative to the product, because their between-set edges must be absent only once.

Start with the empty graph and reveal the edges of the complete graph in a uniformly random order. After $m$ steps its edge set is a uniform $m$-element subset. Independent uniform edge labels couple this process with all binomial random graphs: $G(n,p)=G_{n,M(p)}$, where $M(p)$ has a binomial distribution with parameters $\left(\genfrac{}{}{0ex}{}{n}{2}\right),p$. For a monotone graph property, concentration of $M(p)$ transfers a threshold estimate between edge probability and edge count.

During breadth-first search of $G(n,p)$ let $U_t$ count unseen vertices, $A_t$ active vertices, and $t$ explored vertices. When $A_{t-1}=0$ start a new unseen root, recorded by $b_t=1$; otherwise $b_t=0$. The newly discovered count is conditionally $\mathrm{Bin}(U_{t-1}-b_t,p)$. Subtract its conditional mean to obtain a martingale difference $D_t$. With $q=1-p$, $U_0=n$ and $A_0=0$, iteration givesThe last sum is a martingale. Positive $A_t$ over an interval means that the graph component under exploration does not finish there. Independently adding phantom children couples each component exploration below a binomial branching process with offspring law $\mathrm{Bin}(n,p)$.

An equivalence relation on the natural numbers is co-computably enumerable when its complement as a set of pairs is computably enumerable. Inequivalence can then be positively recognized, even when equivalence cannot be decided. This is sufficient for a semidecidable least-representative transversal.

The intersection of any family of equivalence relations on one set is an equivalence relation. Reflexivity and symmetry hold in every component. A pair of consecutive related points belongs to every component, so each component's transitivity supplies the required final pair. Its equivalence classes refine the component partitions. The empty intersection is the universal relation.

A union of a nonempty family of equivalence relations on the same set is reflexive and symmetric, but can fail transitivity. On a three-point set, partitions into blocks $\{1,2\},\{3\}$ and $\{1\},\{2,3\}$ make the union relate $1$ to $2$ and $2$ to $3$ without relating $1$ to $3$. An equivalence relation containing both requires transitive closure as well.

A nonzero entire function can be described by its discrete zero locations and multiplicities, a product of Weierstrass elementary factors and a zero-free exponential factor. Conversely, consistent prescribed multiplicities at distinct points escaping every compact subset can be realized by choosing factor orders large enough for infinite product convergence from logarithmic tails. Repeated locations with incompatible exact multiplicities are not admissible data, and a finite accumulation of distinct zeros is excluded by the identity theorem.

Finite places of a number field above $p$ correspond to absolute Galois group orbits of embeddings of a number field into a p-adic algebraic closure. The absolute value is invariant under the group. Conversely, embeddings inducing the same place extend to embeddings of the same local completion and are conjugate.

A subset of the space of infinite subsets of the natural numbers has this property when it differs from an Ellentuck topology open set by a meagre set in the same topology. By the complete-Ramsey characterization, these are precisely the completely Ramsey sets. Using meagreness from a different topology changes the definition.

Let $T$ and $S$ be monads on categories $\mathcal{C}$ and $\mathcal{D}$, with forgetful functors $U_T,U_S$. Suppose $K:{\mathcal{D}}^S\to {\mathcal{C}}^T$ and $U:\mathcal{D}\to \mathcal{C}$ satisfy $U_{T}K=U{U}_S$. If $F⊣U$ and ${\mathcal{D}}^S$ has coequalizers of reflexive pairs, then $K$ has a left adjoint $L$. Thus the square must specify an actual lifting to Eilenberg-Moore categories; the required coequalizers are taken in the category of algebras.^[ref]

Writing $F_T,F_S$ for the free algebra functors, the construction starts withbecause the adjunctions give natural bijectionsAn algebra for a monad $(A,a)$ has the reflexive free-algebra presentation of a monad algebra $F_T(TA)\rightrightarrows F_{T}A\to (A,a)$. Transporting its two arrows through these bijections gives a reflexive pair between $F_{S}F(TA)$ and $F_{S}FA$; its coequalizer defines $L(A,a)$. The universal property supplies $L⊣K$.

This explains the theorem’s use in constructing adjoints of logical functors through power-object monadicity: the algebra presentation makes the missing adjoint explicit.