For a normal domain $A$ with fraction field $K$, the field trace of an integral element in a finite separable field extension $L/K$ belongs to $A$. Its conjugates are integral, so their sum is integral; it lies in $K$ and hence in $A$ by normality. This places the integral closure inside a trace-dual lattice.

A fiber bundle whose fibers are affine spaces and whose transition functions are affine. Differences of points in one fiber lie in a corresponding model vector bundle, but there need not be a preferred zero in each fiber.

A closed chain of Dirac propagators is a fermion loop. Its closed fermion loop sign distinguishes it from a bosonic loop. Tracing spinor indices produces gamma matrix traces, and the high-momentum powers of the propagators determine possible ultraviolet divergences.

A box diagram has a one-loop cycle of four propagator edges and four vertices connecting external legs. Its internal particle types depend on the theory. In a charged weak box contribution to kaon mixing, two charged weak-boson propagators connect two up-type quark lines, with flavor-changing charged-current vertices.

In a Dirac field mode expansion, the antiparticle contribution to $\bar{\psi }\Gamma \psi$ first appears with operator order $d{d}^{†}$. Normal ordering changes it to $-d^{†}d$ using the fermionic anticommutation relations. Thus its scattering coefficient is $-{\bar{v}}_{\mathrm{in}}\Gamma v_{\mathrm{out}}$. Pair-annihilation and creation terms have the operator orders $db$ and $b^{†}{d}^{†}$. Keeping these orders and a fixed external-state convention determines interference signs, especially between exchange and annihilation graphs in a Yukawa interaction.

A member of a Farey sequence. Two distinct level-$Q$ fractions have ordinary distance at least $Q^{-2}$. On the quotient circle identify zero and one before forming a set; the shorter wrapped distance of two remaining distinct fractions obeys the same bound.

For compact Hausdorff $K$, the extreme points of the real unit ball in the space of continuous functions on a compact space are precisely its continuous sign-valued functions. A point with $|f|<1$ permits a nonzero continuous bump perturbation supported where there is uniform slack, while pointwise equality at an endpoint of $[-1,1]$ forces both functions in a midpoint decomposition to agree. For the Cantor set, clopen indicators give the perturbations directly.

A quadratic exponential sum has a polynomial phase of degree two. Its multiplicative derivative at lag $h$ has the linear phase $2\alpha hn$ up to a constant factor. This degree reduction lets the Van der Corput inequality for finite scalar sequences reduce its size to estimates for finite geometric series.

For known good probability $p={\sin }^2\theta$, choose $j=\lceil \pi /(4\theta )-1/2\rceil$. A known reversible preparation of one state with good probability $p_{\ast }$ permits $j$ ordinary amplitude amplification iterations to reach the good subspace exactly. With ancillary marking access, prepare a flag with one-probability $q=p_{\ast }/p$ and define joint success as good data and flag one. Reflection about the prepared product state is known, while the joint phase test requires a controlled marking oracle or a computed Boolean output. This changes the success projector and enlarges the state space; it does not evade the obstruction to uniform probability lowering by a unitary. Query costs of preparation, its inverse and marking must all be counted.

For $f,g:\kappa \to \kappa$, write $f{<}^{\ast }g$ when there is $\gamma <\kappa$ such that $f(\delta )<g(\delta )$ whenever $\gamma <\delta <\kappa$. For an infinite regular cardinal, suprema of fewer than $\kappa$ ordinals below $\kappa$ remain below it, permitting the long chain under eventual domination construction.

An Euler circuit, also called an Euler tour, is an Eulerian trail whose final vertex equals its initial vertex. Its length is $|E|$. It can revisit other vertices, unlike a Hamiltonian cycle, which visits each vertex once before returning to the start. For a finite undirected multigraph with at least one edge, an Euler circuit exists exactly when the edge support of a graph is a connected graph and every vertex degree is even.

For example, let $p,q$ be distinct parallel edges joining $A,B$, let $\ell$ be a self-loop at $A$, and add an isolated vertex $Z$. Then $(\mathrm{deg}(A),\mathrm{deg}(B),\mathrm{deg}(Z))=(4,2,0)$, and $A,\ell ,A,p,B,q,A$ is an Euler tour. The self-loop supplies one traversal and two incidences; $Z$ need not be visited. Replacing $p,q$ by a single edge changes both vertex degrees to odd numbers and removes the possibility of an Euler circuit. Thus collapsing parallel edges can change the answer.

Two triangles in a graph sharing one vertex $A$ give another instructive Euler circuit: $A,B,C,A,D,E,A$. The shared vertex has vertex degree four, the others have vertex degree two, and $A$ is revisited between the two closed trails. This is precisely the splice used by Hierholzer's algorithm.

An instanton is a localized finite-action classical solution in Euclidean spacetime, used in a semiclassical description of quantum tunnelling and other nonperturbative effects. In one-dimensional quantum mechanics it can connect distinct degenerate potential minima over infinite Euclidean time. A quartic double-well instanton is an elementary example; a Yang-Mills instanton is a gauge-field example with different geometric structure.

For the skew matrix $S=\left(\begin{array}{cccc}0& 1& 1& 1\\ -1& 0& -1& 1\\ -1& 1& 0& -1\\ -1& -1& 1& 0\end{array}\right)$, both Euclidean lattices reduce to the tetracode modulo $3$. A reflection depending on the reduction coset matches their vectors coordinatewise in absolute value. Their weighted length spectra agree for all positive diagonal axis weights. Rationally independent weights force any possible linear isometry to be a coordinate sign change; the code then forces an overall sign, which does not identify the two lattices.

For independent clusters with mean vector $m_i(\theta )$ and positive definite working covariance matrix $V_i$, a generalized estimating equation solvesCorrect mean specification and regularity can give a consistent estimator of identifiable mean coefficients despite a misspecified working covariance. A sandwich covariance matrix accounts for actual cluster variation. Mean equations alone cannot identify two parameters that always enter through the same combination.

For a nonzero elliptic function with period lattice $\Lambda$, its zeros of a function $a_j$ and poles $b_j$, repeated with their multiplicities, have equal total number and satisfy ${\sum }_j{a}_j-{\sum }_j{b}_j\in \Lambda$. Choose a fundamental parallelogram avoiding all zeros and poles on its boundary. The argument principle applied to $f^{\prime }/f$ gives equality of the numbers. Applying the residue theorem to $z{f}^{\prime }/f$ gives their difference of sums. Pairing opposite edges, the extra factors are the two lattice generators multiplied by integrals of $f^{\prime }/f$ along an edge. These integrals are integer multiples of $2\pi i$, because the endpoints have the same nonzero function value. Thus the difference of sums is a lattice element. This links zeros of a pulled-back line to the chord-and-tangent group law.

A power object $PA$ represents parameterized subobjects of $A$: $\mathcal{E}(X,PA)\cong \mathrm{Sub}(X\times A)$. In an elementary topos, it is ${\Omega }^A$, with the membership subobject supplied by evaluation at the subobject classifier.

A logical functor between elementary toposes preserves finite limits, exponential objects and the subobject classifier, with their canonical comparison maps. It therefore preserves power objects and the double-power-object monad. If it has a left adjoint, power-object monadicity and the adjoint lifting theorem for monad algebra functors give it a right adjoint as well.

A Boolean topos is an elementary topos in which every subobject has a complement, equivalently its internal truth-value Heyting algebra satisfies excluded middle. Its internal first-order logic is classical. Being Boolean alone does not imply that it has enough set-valued points.

A noninitial object is irreducible, or supercompact, when every family of subobjects with union the whole object contains the whole object as one member. This concerns unions of subobjects rather than disjoint coproduct decompositions. Irreducible representable sheaves for the regular coverage follow from the one-arrow nature of that coverage.

For the elastic filament operator $K=A{\partial }_x^4$, two integration by parts operations give boundary form $A[\overline{u}{v}^{\prime \prime \prime }-\overline{u^{\prime }}{v}^{\prime \prime }+\overline{u^{\prime \prime }}{v}^{\prime }-\overline{u^{\prime \prime \prime }}v]$. Four elementary homogeneous choices at each endpoint are $h=h^{\prime }=0$ (clamped), $h=h^{\prime \prime }=0$ (hinged), $h^{\prime \prime }=h^{\prime \prime \prime }=0$ (free), and $h^{\prime }=h^{\prime \prime \prime }=0$ (slope-constrained and transverse-force-free). Each defines a self-adjoint operator on a finite interval when imposed at both ends. More general real Robin boundary conditions also give self-adjointness; the four choices are not an exhaustive classification of all boundary subspaces.