A nonconstant geodesic whose position and tangent velocity repeat after a positive time. On a flat torus it is a straight line modulo the lattice, closing when its displacement is a Euclidean lattice vector. A primitive closed geodesic traverses its image without being an iterate of a shorter periodic geodesic.

A filter is generic over its ground model exactly when it meets each ground-model maximal forcing antichain in one point. Extensions of a maximal antichain form a dense set, while any ground dense set contains an antichain maximal in the whole forcing. Directedness permits at most one antichain member in a filter.

For a proper subgroup $H$ of any group $G$, its complement is a generating set of a group. Choose $x\notin H$. For every $h\in H$, also $xh\notin H$, and $h=x^{-1}(xh)$ lies in the subgroup generated by the complement. That subgroup consequently contains both $H$ and its complement. No finiteness assumption is needed.

For prime $p$, every order-$p$ element of $G{L}_2({\mathbb{F}}_p)$ is conjugate to the displayed nonidentity unipotent matrix. Its cyclic subgroup has vector group orbits of sizes one or $p$; counting all $p^2$ vectors implies at least $p$ fixed vectors. Choose a nonzero fixed vector as first basis vector. The matrix becomes triangular with first diagonal entry one; its other diagonal entry satisfies $d^p=1$ and hence $d=1$ in the prime finite field. The off-diagonal entry is nonzero and can be rescaled to one. All matrices involved have invertible changes of basis.

For a character of modulus $p^k$, a primitive Dirichlet character attains the bound by finite Plancherel theorem. An imprimitive character with $k\ge 2$ has periodic values modulo $p^{k-1}$, so its frequency-one sum vanishes. The principal Dirichlet character modulo $p$ has unnormalized sum $-1$. The sign convention of the exponential does not affect its magnitude.

With gauge covariant derivative $D=d+A$, transforming the zero connection by a smooth group-valued map $g$ gives $A=-dg{g}^{-1}$. This potential has zero gauge curvature. A flat potential is locally of this form; global reconstruction can also depend on topology and holonomy. The sign changes when the definition of the covariant derivative changes.

Write $B_i=({\partial }_{i}g)g^{-1}$ and use $D=d+A$. The right Maurer-Cartan equation gives ${\partial }_x{B}_y-{\partial }_y{B}_x=[B_x,B_y]$. Thus $A_i=\alpha B_i$ has the displayed gauge curvature. Both $\alpha =0$ and $\alpha =-1$ are universally flat; the latter is a pure gauge potential. Commuting $B_x,B_y$ give additional flat cases. For the opposite convention $D=d-A$, the corresponding component formula instead has $\alpha (1-\alpha )$.

For ${\mathcal{F}}_{\mu \nu }={\partial }_{\mu }{\mathcal{A}}_{\nu }-{\partial }_{\nu }{\mathcal{A}}_{\mu }+ig[{\mathcal{A}}_{\mu },{\mathcal{A}}_{\nu }]$, the CP transformation of a non-Abelian gauge connection gives ${\mathcal{F}}_{\mu \nu }^{CP}(x)=-P_{\mu }{}^{\alpha }{P}_{\nu }{}^{\beta }{\mathcal{F}}_{\alpha \beta }^T(x_P)$. The commutator term transforms with the same sign as the derivative terms because $[X^T,Y^T]=-[X,Y{]}^T$. A simple component check is $F_{0i}\mapsto +F_{0i}^T$ and $F_{ij}\mapsto -F_{ij}^T$, evaluated at the reflected point.

If a complex scalar field uses $D_{\mu }\varphi =({\partial }_{\mu }+ie{A}_{\mu })\varphi$, its conjugate uses $(D_{\mu }\varphi {)}^{\ast }=({\partial }_{\mu }-ie{A}_{\mu }){\varphi }^{\ast }$. Under $\varphi \mapsto e^{-ie\alpha }\varphi$ and $A\mapsto A+d\alpha$, the two derivatives transform with opposite phases. Their product is gauge invariant. Applying the same plus-charge operator to both factors leaves an unwanted term proportional to ${\partial }_{\mu }\alpha$ and fails gauge invariance.

For $\mathcal{L}=-\mathcal{N}{F}_{\mu \nu }^a{F}^{a\mu \nu }/4$ with $\mathcal{N}>0$, rescale ${\mathcal{A}}^a=\sqrt{\mathcal{N}}{A}^a$ and $g_c=g/\sqrt{\mathcal{N}}$. The gauge field strength then has canonical component normalization. Masses inferred from matter couplings must use $g_c$, not an unnormalized coefficient. A matrix-trace convention $\mathrm{Tr}(t^a{t}^b)=\mathcal{N}{\delta }^{ab}$ fixes this factor.

For an odd prime $p$, embed the elementary abelian group $(C_p{)}^3$ and the Heisenberg group over a prime field regularly in $S_{p^3}$. Every nonidentity element has order $p$ and hence regular permutation cycle type $p^{p^2}$. The two subgroups have equal intersection counts with every symmetric-group conjugacy class, so they are Gassmann equivalent, but they are not isomorphic because one is abelian and the other is not. Using a compact simply connected isometric cover with ambient deck group $S_{p^3}$ gives quotients with these nonisomorphic fundamental groups; Sunada theorem makes them isospectral while their topology differs.

For signature $+---$, use ${\gamma }^0=\left(\begin{array}{cc}0& I_2\\ I_2& 0\end{array}\right)$ and ${\gamma }^i=\left(\begin{array}{cc}0& {\sigma }^i\\ -{\sigma }^i& 0\end{array}\right)$, where ${\sigma }^i$ are Pauli matrices. Their Clifford algebra follows directly from the Pauli anticommutator. The chirality matrix is ${\gamma }^5=\mathrm{diag}(-I_2,I_2)$, separating the two chiral components of a Dirac spinor.

For a field $F$ whose characteristic is different from $p$, the cyclotomic character is defined by $g(\zeta )={\zeta }^{\kappa (g)}$ on compatible $p$-power roots of unity. On the full cyclotomic tower of $\mathbb{Q}$, it identifies the Galois group with ${\mathbb{Z}}_p^{\times }$. Its finite-order part is the Teichmüller character and its pro-$p$ part acts through $1+p{\mathbb{Z}}_p$, or $1+4{\mathbb{Z}}_2$.

A Galois character is a continuous one-dimensional Galois representation: the absolute Galois group acts by multiplication by a scalar in a coefficient field $F$. A quadratic character over a coefficient field of characteristic different from two has values in $\{1,-1\}$; after identifying this group with $\mathbb{Z}/2\mathbb{Z}$, its kernel fixes a quadratic extension, or the base field for the trivial character. Its restriction to an inertia group is trivial exactly when that extension is unramified at the corresponding place.

The module ${\mu }_m$ consists of the $m$th roots of unity in a separable closure. If the field characteristic does not divide $m$, it has $m$ elements and fits into the Kummer sequence $1\to {\mu }_m\to K_s^{\times }\stackrel{m}{\to }{K}_s^{\times }\to 1$. Its Galois action is the cyclotomic action; a generator gives a noncanonical identification with $\mathbb{Z}/m\mathbb{Z}$.

For an extension $L/K$ with Galois group $G$, a semilinear action on an $L$-space is an additive group action satisfying $\sigma (av)=\sigma (a)\sigma (v)$. Scalar coefficients are transformed along with vectors. Invariants form a $K$-space, and Galois descent of vector spaces reconstructs the original space from it.

A splitting isomorphism of a central simple algebra with $M_n(L)$ transports its Galois action to $T_{\sigma }=c_{\sigma }{\sigma }_0$, giving a nonabelian first cohomology class. Conversely a cocycle gives a semilinear algebra action whose invariants descend to a central simple algebra. Equivalent cocycles give isomorphic fixed algebras. This gives a pointed-set bijection ${\mathrm{CSA}}_n(L/K)\cong H^1(\mathrm{Gal}(L/K),{\mathrm{PGL}}_n(L))$.

A small-study effect is a systematic relationship between study precision or size and estimated effect. It may result from selective publication, design differences, effect modification or other heterogeneity. A regression of effect on its standard error tests one such relationship; it does not by itself determine which mechanism generated it.

For a specified diagram shape, a functor reflects categorical limits when a cone over a diagram is limiting whenever its image is limiting. This condition tests an existing categorical cone. It does not assert that every limiting categorical cone in the codomain has a lift.

A functor is conservative when it reflects isomorphisms: if $F(f)$ is invertible then $f$ is invertible. It need not be full or faithful functor. The crude monadicity theorem uses this reflection property to lift invertibility of the comparison-adjunction counit from the base category.