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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 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.

For a functor between Eilenberg-Moore categories lying over a right adjoint on the base categories and compatible with the monad structures, an adjoint lifting theorem constructs a left adjoint when the required reflexive coequalizers of algebra presentations exist. Applied through power-object monadicity, this converts a left adjoint of a logical functor into a right adjoint of that logical functor. The coequalizer hypotheses are part of the theorem; commutation with the forgetful functors alone is insufficient.

For a monad $(T,\eta ,\mu )$, the free algebra functor sends $X$ to $(TX,{\mu }_X)$ and is a left adjoint to the forgetful functor from the Eilenberg-Moore category. The transpose bijection sends an monad algebra morphism $h:TX\to A$ to $h{\eta }_X$; its inverse sends $k:X\to A$ to $aT(k)$, where $a:TA\to A$ is the algebra structure. The monad and algebra laws make these inverse and natural. The adjunction unit is $\eta$ and the adjunction counit at $(A,a)$ is $a$, so the induced monad has exactly the original endofunctor, unit and multiplication.

Killed planar Brownian motion conditioned to exit at a smooth boundary point $a$ is the Doob h-transform by its positive Poisson kernel, normalized at the starting point. Its generator is $\frac{1}{2}\Delta +\nabla \log h\cdot \nabla$. Conditioning on shrinking boundary arcs yields the same localized laws and specifies a canonical version of an otherwise probability-zero conditioning event.

For positive $n,m$, $\mathrm{Cl}({\mathbb{P}}^n\times {\mathbb{P}}^m)={\mathbb{Z}}^2$, generated freely by the pullbacks of coordinate hyperplanes. Restrict a divisor to the standard affine chart, where unique factorization makes it principal, and subtract this principal Weil divisor to leave only the two boundary hyperplanes. Any principal relation between them would be given by a rational function whose divisor vanishes on the chart; unique factorization then makes that function a constant, proving independence. The same argument gives $\mathrm{Cl}({\mathbb{P}}^r)=\mathbb{Z}$.

An abelian group $A$ is $p$-divisible when multiplication by $p$ is surjective, equivalently $A/pA=0$. This does not mean it has no $p$-torsion: $\mathbb{Q}/\mathbb{Z}$ is divisible and has elements of every finite order. The Kummer cohomology divisibility criterion relates this property to Galois cohomology.

If a degree-$n$ characteristic polynomial has distinct roots of a polynomial ${\lambda }_j$ of multiplicities $m_j$, all homogeneous distribution solutions are the displayed combinations. There are ${\sum }_j{m}_j=n$ independent coefficients. Repeated characteristic roots of a constant-coefficient differential equation account for the polynomial factors multiplying exponentials; conjugate pairs give real sine and cosine combinations when real solutions are desired.

For a positive-order constant-coefficient ordinary differential equation, the kernel of multiplication by a coordinate gives $xP(D)u=0$ exactly when $P(D)u=C{\delta }_0$. Hence every solution is $u=v+CE$, where $P(D)v=0$ and $E$ is any fundamental solution of a linear differential operator. Taking a retarded fundamental solution of a constant-coefficient ordinary differential operator shows that the extra freedom is a single jump in the derivative of order one below the operator order. The coordinate multiplies the already differentiated distribution: this is a different operator from $P(D)(xu)$.

For a smooth function $u$, the distributional derivative of the product of $u$ with the Heaviside function is $D(Hu)=H{u}^{\prime }+u(0){\delta }_0$. Induction gives the displayed formula. It records all boundary sources, including derivatives of the Dirac delta distribution, and determines which initial derivatives must vanish in a retarded fundamental solution.

The Lighthill acoustic analogy for a moving body has a volume acoustic quadrupole, a surface acoustic monopole from acoustic thickness noise, and a surface acoustic dipole from acoustic loading noise. For an impermeable moving surface, with outward normal $n$, normal surface speed $v_n$, and fluid and body normal velocities equal, the surface mass coefficient is ${\rho }_0{v}_n$ and the surface loading is the force exerted on the fluid. Schematically, the density-source equation isThe density perturbation is understood with the chosen interior extension. Permeable-surface versions have additional mass and momentum flux terms. Source approximations must distinguish local thickness radiation from cancellation of its compact net-volume contribution.

The response to a localized perturbation must be followed in space as well as time. Growth at a fixed position is absolute wave-packet instability; growth transported away while the original position decays is convective wave-packet instability. The Briggs-Bers criterion evaluates the relevant complex spatial branches of a dispersion relation, while a real-wavenumber temporal mode test alone does not generally distinguish those behaviors.

For a primitive Dirichlet character whose conductor of a Dirichlet character is $q>1$ and whose character parity is $a$, Poisson summation gives ${\theta }_{\chi }(x)={\epsilon }_{\chi }{x}^{-a-1/2}{\theta }_{\overline{\chi }}(1/x)$, where ${\epsilon }_{\chi }=\tau (\chi )/(i^a\sqrt{q})$. Its rapid decay at both ends yields the entire Mellin representation of the completed Dirichlet L-function. The case of conductor of a Dirichlet character equal to one uses the ordinary Jacobi theta function and a subtracted constant term.

For a primitive nonprincipal Dirichlet character, the theta Mellin integral makes this an entire function satisfying $\Lambda (s,\chi )={\epsilon }_{\chi }\Lambda (1-s,\overline{\chi })$. The gamma poles cancel the trivial zeros of a Dirichlet L-function. For the principal Dirichlet character whose conductor of a Dirichlet character is one the completion is meromorphic, with poles at zero and one.