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Given morphisms $f:A\to B$ and $g:B\to C$, their composite is $gf:A\to C$. Composition is associative and has the identity morphisms as units.

A generating set is a set-indexed family $(P_i)$ such that unequal parallel morphisms $f,g:A\to B$ are distinguished by some $h:P_i\to A$, with $fh\ne gh$. It is dual to a cogenerating set.

A full subcategory is coreflective when its inclusion $I$ has a right adjoint $q$. The counit $IqX\to X$ is universal for maps to $X$ from objects of the subcategory. The comonad $Iq$ is idempotent. For a finite-limit-closed coreflective subcategory of a topos, its coalgebra construction can prove that the subcategory is itself a topos.

A category is preadditive if every hom-set is an abelian group and composition is additive in each variable. Zero morphisms and differences of parallel morphisms are therefore defined. A zero object or finite biproducts need not exist; these are extra requirements in an additive category.

A finite tuple is an ordered list of finitely many entries; coordinates can repeat, and their positions matter. A tuple of length $n$ from a set $M$ is an element of the Cartesian product $M^n$. The empty tuple has length zero. Finite tuples support Scott formulas in a back-and-forth method construction, where matching tuples determine a partial isomorphism of structures.

An infinite cardinal number is a limit cardinal if it is not a successor cardinal; equivalently it is an aleph number with zero or a limit ordinal as index. Uncountable limit cardinals are either singular cardinals or weakly inaccessible cardinals. This concerns succession among cardinals, rather than merely being a limit ordinal.

A cap set in ${\mathbb{F}}_3^n$ has subset density $O(1/n)$. A Fourier analysis on a finite abelian group proof uses a hyperplane density increment for cap sets repeatedly: absence of nondiagonal zero-sum triples forces a large finite abelian Fourier coefficient, and hence a denser slice. This is a weaker bound than the Ellenberg–Gijswijt cap-set bound, but illustrates the density increment method.

In the de Sitter approximation, a canonically rescaled scalar has superhorizon powers $(-\tau {)}^{1/2\pm \nu }$. The physical field has powers $(-\tau {)}^{3/2\pm \nu }$. For $H^2<m^2<2H^2$, both physical branches decay, while the slower rescaled branch grows. The generic late-time dimensionless power spectrum is proportional to $(k/(aH){)}^{3-2\nu }$.

The Maxwell Lagrangian density contains no time derivative of $A_0$, so its canonical momentum vanishes. The condition is a first-class constraint. Its time preservation yields the Gauss law constraint in gauge theory; $A_0$ acts as the corresponding multiplier in the Hamiltonian.

The free transverse photon field is a Hermitian sum of two transverse-polarization plane-wave modes with creation and annihilation operators. Their standard bosonic commutator gives the transverse equal-time commutator. The normal-ordered Hamiltonian is a sum of positive oscillator energies.

A contraction semigroup is a strongly continuous semigroup of bounded linear operators satisfying $\Vert S(t)\Vert \le 1$ for every $t\ge 0$. The Hille-Yosida theorem characterizes its infinitesimal generator of a semigroup by dense domain, closedness, and the resolvent estimates $\Vert (\lambda -A{)}^{-n}\Vert \le {\lambda }^{-n}$ for real $\lambda >0$ and integers $n\ge 1$.

A transport plan is strongly c-monotone if Borel Kantorovich potentials satisfy the displayed feasible inequality everywhere and equality almost everywhere for the plan. Extended values $-\infty$ away from the full marginal-measure sets may be permitted. This is a potential certificate, not a strict version of every cyclic inequality. A finite continuous cost allows the transport potential path construction.

An optimal transport plan of finite cost for a finite continuous cost function is c-cyclically monotone. A strict violation at finitely many support points persists in small product neighborhoods; subtracting small normalized pieces and inserting cyclically re-paired product measures preserves both marginals and strictly improves the finite cost. The finite-value hypothesis matters: if every competitor has infinite cost, a nonmonotone plan may still be an extended-value minimizer.

The expected process variance in the Bühlmann model averages within-risk conditional variance over the population of risks. It measures variation that repeated observations can average out and contributes $v/n$ to the variance of the sample mean.

The variance of hypothetical means in the Bühlmann model is the between-risk variance of the conditional claim expected value. It creates the shared covariance $a$ between different years of the same risk and determines how much individual experience should influence the premium.

The credibility factor in the Bühlmann model is the weight $Z=na/(na+v)$ placed on the sample mean after $n$ observations. The remaining weight is placed on the population expected value. More experience or larger variance of hypothetical means increases $Z$; larger expected process variance decreases it.

The Bühlmann credibility premium is the best affine mean squared error estimate of a risk’s conditional claim expected value. In the Bühlmann model it is $Z\overline{X}+(1-Z)m$, where $Z=na/(na+v)$. The linear least-squares projection equations use $\mathrm{Var}(X_j)=a+v$ and $\mathrm{Cov}(X_i,X_j)=a$ for distinct years.

Conditional on a common latent variable $\Theta$, annual averages $X_j$ are independent, have common conditional expectation $\mu (\Theta )$, and have conditional variance ${\sigma }^2(\Theta )/m_j$ for known positive exposures $m_j$. Its structural parameters are $m_0=\mathbb{E}\mu (\Theta )$, expected process variance $v=\mathbb{E}{\sigma }^2(\Theta )$ and variance of hypothetical means $a=\mathrm{Var}(\mu (\Theta ))$. It generalizes the equal-exposure Bühlmann model.

If $u$ is a bounded harmonic function on a domain $D$, continuous on its closure, and the Brownian exit time $T_D$ is almost surely finite from $z$, then $u(z)={\mathbb{E}}_z[u(B_{T_D})]$. The Itô formula and compact localization make $u(B_{t\wedge T_D})$ a bounded martingale. bounded convergence theorem at the exit time proves the representation even if the domain is unbounded. Removing boundedness of $u$ can invalidate this conclusion through loss of uniform integrability.

For a bounded domain whose boundary points are all regular boundary points, continuous Dirichlet boundary data $f$ have a unique solutionHere ${\tau }_D$ is the Brownian exit time. The Strong Markov property gives the mean value property for harmonic functions; a barrier for the Dirichlet problem gives continuity at the boundary. This is an existence theorem as well as the Brownian representation of the Dirichlet problem for an already known solution.