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For a monad $(T,\eta ,\mu )$, the free functor $J:\mathcal{C}\to {\mathcal{C}}_T$ and functor $U:{\mathcal{C}}_T\to \mathcal{C}$ with $UA=TA$ and $U(f)={\mu }_{B}T(f)$ form an adjunction inducing that monad. Every other adjunction inducing the same monad receives the unique Kleisli comparison functor commuting with its left and right adjoints and preserving the adjunction structure. This is initiality among adjunctions inducing the specified monad, with morphisms required to respect that structure.

For a scattered field satisfying ${\psi }_s=e^{ikr}{f}_{\infty }(\hat{\mathbf{r}})/r+O(r^{-2})$, the Kirchhoff–Helmholtz representation givesThe sign assumes the normal points out of the obstacle.

For smooth inviscid barotropic fluid motion without gravity or magnetic forces, let $w$ be the specific enthalpy with $\nabla w={\rho }^{-1}\nabla p$, and set $q=\mathbf{u}\cdot \mathbf{\omega }$. ThenThis follows by dotting the barotropic vorticity transport equation with $\mathbf{u}$ and using the divergence of a cross product. A volume integral is constant when the resulting boundary flux vanishes.

The Kelvin-Helmholtz cooling time estimates how long a self-gravitating body can radiate a reservoir of energy of order its gravitational binding energy. It uses intrinsic luminosity $L_{\mathrm{int}}$, excluding reradiated stellar light. At fixed radius and structure, this reservoir scales as $M^2$; the approximation changes when degeneracy or other internal physics changes the accessible energy.

Kantorovich potentials are the functions in the dual of the Kantorovich optimal transport problem. A feasible pair satisfies $u(x)+v(y)\le c(x,y)$ and provides a lower bound $\int ud\mu +\int vd\nu$ for every transport plan. An optimal pair attaining the dual value provides an optimality certificate.

If $D$ is a nontrivial nef divisor on a K3 surface with $D^2=0$, its complete linear system of a divisor has no fixed part. Write $D=F+M$. Nefness of $D$ and $M$ gives $D\cdot F=D\cdot M=M^2=M\cdot F=F^2=0$. But a nonzero fixed part satisfies $h^0(F)=1$, while Riemann–Roch theorem for algebraic surfaces and Serre duality would give $h^0(F)\ge 2$ if $F^2=0$. Hence $F=0$. Two movable members with no common component have intersection zero and are disjoint, so the system is basepoint-free and has Iitaka dimension one.

The K-theory transfer of a finite covering distributes over a pulled-back tensor product of vector bundles:For an $n$-sheeted covering map, $p_{!}(1)$ is the permutation vector bundle of rank $n$. Its difference from $n$ is a nilpotent element by nilpotence of rank-zero K-theory classes, so it is a unit after inverting $n$. Thus pullback on Topological K-theory becomes injective after this localization of a ring.

A family of random variables is exchangeable if its joint probability distribution is unchanged by finite permutations of its coordinates. Independent and identically distributed random variables are exchangeable, but exchangeable random variables need not be independent random variables.

In dimension $n\ge 2$, along a nonconstant geodesic, such a Jacobi field is determined injectively by $D_{t}J(0)$, which lies in the $(n-1)$-dimensional normal space. On the round unit sphere the geodesic from a point to its antipode, parametrized with speed $\pi$ on $[0,1]$, has fields $J(t)=\sin (\pi t)E(t)$ for all parallel normal fields $E$, attaining the bound. For a constant geodesic the endpoint-vanishing Jacobi field space is zero.

A frozen-age approximation holds an age covariate constant at the start of each observation interval, allowing homogeneous transition probabilities to enter a panel-observed multi-state likelihood. It approximates a model with continuously changing age. Reciprocal rate calibration of mean waiting times is exact for a fixed-age exponential distribution, but not for a continuously ageing Gompertz distribution.

For a finite group $G$ and prime number $p$, the notationdenotes the irreducible characters whose irreducible character degree is coprime to $p$.

Let $F$ be a linear map on a vector space over a field of characteristic different from two, with $F^2=I$. ThenThe projections satisfy $P_{+}{P}_{-}=0$ and $P_{+}+P_{-}=I$, giving existence and uniqueness of the decomposition. In characteristic two, the two signs coincide and the conclusion need not hold.

Let $\zeta =z/L\in [-1,1]$, $h=L_s/L\in (0,1)$, and normalize velocity by its maximum magnitude. The profile is $U=\zeta /h$ for $|\zeta |<h$ and $U=\mathrm{sgn}\zeta$ outside. Each region has exponential vertical-velocity modes; the two walls and vorticity-jump matching for an inviscid shear flow determine their coupling. With $\alpha =kL$, $q=\alpha h$, $X=\tanh q$, $Y=\tanh [\alpha (1-h)]$, $H=q(1+XY)$ and $J=q(X+Y)$, elimination gives $c^2=(H-Y)(J-XY)/(HJ)$, with $c$ normalized by the velocity scale.

Consumption in a self-financing portfolio model is wealth withdrawn for use outside the portfolio. With no initial capital, a one-period position of nonpositive initial cost and nonnegative terminal payoff permits nonnegative consumption at both dates.

A pure-investment arbitrage has zero initial cost, no intervening consumption, and a nonnegative terminal payoff that is strictly positive with positive probability.

A left inverse of a function $f:X\to Y$ is a function $g:Y\to X$ satisfying $g\circ f={\mathrm{id}}_X$. It forces $f$ to be injective, since equal images can be mapped back to equal original elements.

A right inverse of a function $f:X\to Y$ is a function $g:Y\to X$ satisfying $f\circ g={\mathrm{id}}_Y$. It forces $f$ to be a surjective function. A surjective linear map onto a finite-dimensional vector space has a linear right inverse: lift a basis of the target and extend by linearity.

A first-order formula $A$ is stable when intuitionistic first-order logic proves $\neg \neg A\to A$. Double negations and logical falsity are stable. Logical conjunction of stable first-order formulas is stable; a logical implication is stable when its consequent is stable; and universal quantification preserves stability. For logical implication, from $\neg \neg (A\to B)$ and $A$, assuming $\neg B$ contradicts the first hypothesis, giving $\neg \neg B$ and then $B$. For universality, obtain $\neg \neg A(t)$ from $\neg \neg \forall xA(x)$ for arbitrary $t$, then use stability and generalize.

For an atomic formula $P$ (including logical equality), put $P^N=\neg \neg P$, and put ${\perp }^N=\perp$. Preserve logical conjunction, logical implication and universal quantification recursively, while settingLogical negation is logical implication to logical falsity, so $(\neg A{)}^N=\neg A^N$. The displayed logical disjunction and existence clauses are intuitionistically equivalent to the usual negative forms $\neg (\neg A^N\wedge \neg B^N)$ and $\neg \forall x\neg A^N$. Every translated first-order formula is stable by stability of a formula under double negation. In classical logic the translation is equivalent to the original first-order formula by mathematical induction and double-negation elimination.

A boundary wavelet replaces a translated interior wavelet whose support of a function meets a domain endpoint. Compatible finite boundary refinement matrices preserve orthogonality and localization across scales. If the coarse boundary spaces reproduce polynomials of degree less than $q$, their detail-space orthogonal complements have $q$ vanishing moments.