A principal fractional ideal of an integral domain $A$ with fraction field $K$ is $aA$ for a nonzero $a\in K$. Its generators differ by a unit in a ring in $A$. For a number field, taking $A={\mathcal{O}}_K$ and quotienting nonzero fractional ideals by principal fractional ideals gives the ideal class group; requiring congruences and signs of the generators instead gives a ray class group.

Relative to a nontrivial additive character, the self-dual additive Haar measure makes the twice-applied Fourier transform over a local field equal to reflection. If the annihilator of the valuation ring is ${\pi }^c{\mathcal{O}}_F$ and its residue field has $q$ elements, the normalization is $\mathrm{vol}({\mathcal{O}}_F)=q^{c/2}$. Volumes of a compact open subgroup and its character annihilator then multiply to one.

For a compact open additive subgroup $H$ of a non-Archimedean local field, translation and additive character orthogonality give the displayed transform of a coset indicator. With the self-dual Haar measure, transforming again gives $1_{a+H}(-x)$. Every Schwartz-Bruhat function is a finite linear combination of these indicators.

With kernel $e^{-2\pi i⟨u,x⟩}$ and $\mathrm{Im}\tau >0$, the Fourier transform of $e^{\pi i\tau |x{|}^2}$ is $(-i\tau {)}^{-n/2}{e}^{-\pi i|u{|}^2/\tau }$. Choose the logarithm of $-i\tau$ on the right half-plane. The ordinary real Gaussian Fourier transform proves the identity at $\tau =it$; holomorphic continuation proves the complex formula.

For sufficiently regular integrable functions on a half-line, Fourier sine transform inversion recovers the odd extension at continuity points. A zero endpoint value is required when demanding continuity at the endpoint.

The functions $\sqrt{2/\pi }\sin (nx)$, $n\ge 1$, form a complete orthonormal basis of $L^2(0,\pi )$. An odd extension to $(-\pi ,\pi )$ reduces completeness to that of the Fourier series. Equality of all sine coefficients therefore implies equality almost everywhere; if both functions are continuous, it implies equality everywhere.

If a periodic integrable function has all Fourier coefficients zero, it vanishes almost everywhere. Its Fejér sums vanish, whereas the Fejér kernel is an approximate identity and converges to that function in the integrable norm. Applied to a periodized squared Fourier transform, this detects orthogonality of integer translates.

A real closed field is a formally real field with no proper formally real algebraic extension. Equivalently every positive element in its unique compatible order is a square and every odd-degree polynomial has a root. Its positive cone is definable in the field language as the nonzero squares. The Theory of real closed fields is model-complete.

A field can be ordered compatibly with addition and multiplication if and only if it is a formally real field. This ordering criterion concerns real algebra, and is different from Artin–Schreier theory of cyclic extensions in positive characteristic.

A two-component spanning forest of a finite connected graph is an acyclic spanning subgraph with exactly two connected components of a graph. It has $|V|-2$ edges. For distinct terminals $a,z$, a separating two-component forest places them in different components. Deleting an edge of the $a$-to-$z$ graph path of a spanning tree produces such a forest; adjoining any original edge across its cut reverses this operation. This is the counting correspondence behind the mean spanning-tree path current.

In standard notation for forcing, an order is separative if $p\nleqq q$ implies that some $r\le p$ is incompatible with $q$. Separativity does not imply an atomless forcing order: the one-point order is separative and has a forcing atom. Its only generic filter is a ground-model set. Atomlessness is the hypothesis in generic filter for an atomless order is new.

A forcing order that is countable inside the ground model. It has the countable chain condition and preserves ${\omega }_1$. Internal countability is distinct from external countability of the entire model.

A forcing is cofinality-preserving over a transitive model $M$ if every generic extension has the same cofinality for every ground ordinal. The values are compared as ordinals, which forcing preserves.

A forcing collapses a ground-model cardinal $\kappa$ if it forces a bijection from a smaller ordinal onto the ground ordinal $\kappa$. The ordinal itself is preserved; its being an initial ordinal is lost.

A subset of a forcing order whose distinct members have no common stronger extension. A maximal forcing antichain has a compatible member for every condition. For forcing by nodes of a set-theoretic tree, ordered by extension, this coincides with a tree antichain.

The drift vector is the expected infinitesimal displacement per unit time. For jumps $r$ with rates $W(x,r)$, it is $A_i(x)={\sum }_r{r}_{i}W(x,r)$. It is the first jump moment entering the Fokker-Planck equation.

For jumps $r$ from state $x$ at nonnegative rates $W(x,r)$, the Fokker-Planck diffusion matrix in the infinitesimal-covariance convention isAssume the second jump moments are finite. Over a short interval $dt$, the conditional covariance of the state increment is $B(x)dt+o(dt)$. Subtracting the product of the mean increments changes only order-$d{t}^2$ terms. The matrix is symmetric and positive semidefinite because $v^{\mathsf{T}}Bv={\sum }_{r}W(x,r)(v\cdot r{)}^2\ge 0$ for every real vector $v$.

With this definition, its term in the Fokker-Planck equation is $\frac{1}{2}{\sum }_{i,j}{\partial }_i{\partial }_j(B_{ij}p)$. It is therefore $B/2$, not $B$, that corresponds to a scalar diffusion coefficient multiplying ${\partial }_x^{2}p$ in one dimension. Unit births and deaths at rates $b,d$ give drift $b-d$ and increment variance $B=b+d$: opposite mean displacements cancel in the drift but both contribute positively to the variance. The chemical diffusion matrix applies the same outer-product construction to reaction channels.

The probability-density equation for the Ornstein-Uhlenbeck process with drift $-v$ and SDE noise amplitude $\sqrt{2}$, corresponding to diffusion coefficient one in the displayed Fokker-Planck equation. The unit-covariance Gaussian density is stationary because its Fokker-Planck probability current vanishes. This normalization makes mean velocity decay at rate one and energy excess decay at rate two.

A probability density left unchanged by the Fokker-Planck equation ${\partial }_{t}f=L^{\ast }f$. With the boundary conditions of the process imposed, its Fokker-Planck probability current has zero divergence; the current itself need not vanish. For the Ornstein-Uhlenbeck Fokker-Planck equation, the normalized standard Gaussian density has zero current and is stationary.

The eigenvalue of a mode number operator counts excitations in that mode. Bosonic occupation numbers range over all nonnegative integers; fermionic occupation numbers are zero or one.