An operator $({\bar{L}}_i\varphi e_{Rj})({\varphi }^{†}\varphi )/{\Lambda }^2$ contains three factors of $v+h$. Its contribution to the mass matrix is proportional to $v^3$, while its single-Higgs coupling is three times the corresponding mass-over-v contribution. With the normalization $-\sqrt{2}{\lambda }^{\prime }/{\Lambda }^2$, the mass-basis coupling is $Y_h=M_{\mathrm{diag}}/v+v^2{\hat{\lambda }}^{\prime }/{\Lambda }^2$. Nonzero off-diagonal coefficients permit charged-lepton flavour-violating Higgs decays.

For ${\psi }^C={\eta }_{C}C{\bar{\psi }}^T$ and ${\gamma }^{0T}={\gamma }^0=-C{\gamma }^0{C}^{-1}$, the Dirac adjoint transforms as ${\bar{\psi }}^C=-{\eta }_C^{\ast }{\psi }^T{C}^{†}$. Anticommuting the components of the fermion bilinear gives $(\bar{\psi }\psi {)}^C=\bar{\psi }(C^{†}C{)}^T\psi$. Invariance of the scalar bilinear therefore fixes $C^{†}C=I$.

If $C{\gamma }^{\mu T}{C}^{-1}=-{\gamma }^{\mu }$, transposition shows that $C^T{C}^{-1}$ commutes with every gamma matrix. Irreducibility of the complex Dirac representation and the Schur lemma make it scalar. Transposing $C^T=zC$ again yields $z^2=1$, so $C^T=\pm C$.

The unitary charge-conjugation operator acts on field operators and commutes with numerical chiral projectors, so $\hat{C}(P_L\psi ){\hat{C}}^{-1}=P_L{\psi }^c$. This component is left-chiral. The algebraic conjugate of an already projected spinor instead satisfies $(P_L\psi {)}^c=P_R{\psi }^c$, because its adjoint carries the opposite projector. Keeping the two operations separate avoids a false chirality conclusion.

For a self-adjoint element $h$, the exponentials $\exp (ith)$ are unitary. A continuous algebra character therefore satisfies $|\exp (it\varphi (h))|\le 1$ for every real $t$, forcing $\varphi (h)$ to be real. Decomposing an arbitrary element into real and imaginary self-adjoint parts proves the displayed identity. Thus every algebra character is a star-preserving scalar homomorphism; commutativity of the whole algebra is not needed for this fact.

If forcing has the $\kappa$-chain condition for a regular uncountable $\kappa$, every forced club subset of $\kappa$ contains a ground-model club below the forcing condition. Bound the possible witnesses above each ordinal using antichains, then take closure points of the bound function.

The Koszul complex on $f_1,\dots ,f_r$ has term $K_i={\bigwedge }^i{R}^r$ and differential $\partial (e_{j_1}\wedge \cdots \wedge e_{j_i})={\sum }_a(-1{)}^{a-1}{f}_{j_a}{e}_{j_1}\wedge \cdots \wedge \widehat{e_{j_a}}\wedge \cdots \wedge e_{j_i}$. Its degree-zero homology is $R/(f_1,\dots ,f_r)$. For a regular sequence it is exact in positive degrees, hence a Koszul resolution of that quotient. The variables of a polynomial ring form a regular sequence, so this resolves the coefficient field in length equal to the number of variables.

A double complex has modules $C_{p,q}$ and differentials lowering either index, each squaring to zero and anticommuting with the other. The total chain complex has term ${⨁}_{p+q=i}{C}_{p,q}$ and differential the sum of the two differentials. For the tensor product of two chain complexes, a sign $(-1{)}^p$ on the second differential ensures anticommutation. If the double complex is in the first quadrant and one direction has homology only in degree zero, its total homology is computed by the surviving degree-zero complex. Applying this twice to two free resolutions proves the balanced calculation of the Tor functor.

The chain complex with $R$ in a single degree $n$ and zero differential. A map from it into a complex chooses a cycle in degree $n$.

For $n\ge 1$, the chain complex with copies of $R$ in degrees $n,n-1$ and identity differential. It is acyclic; a map from it chooses an arbitrary degree-$n$ element and its boundary.

The tensor product of homologically graded chain complexes has differential $D(x_p\otimes y)=d{x}_p\otimes y+(-1{)}^p{x}_p\otimes dy$. The sign makes the two mixed terms in $D^2$ cancel. Total degree is the sum of the two degrees.

A graded Hom functor construction consists of maps shifting degree by $j$. One homological convention is $d_{M}f=f{d}_C+(-1{)}^{j-1}{d}_{C^{\prime }}f$. Its degree-zero cycles are chain maps, and its degree-zero boundaries are null-homotopic chain maps, so its zeroth homology is the module of chain homotopy classes. This convention differs by degree-dependent signs from the alternative convention $d^{\prime }f-(-1{)}^{j}fd$.

The reversed dual module of a chain complex becomes a chain complex with the unsigned precomposition differential above. A functional on $C_{-i}$ is sent to a functional on $C_{1-i}$, lowering the reversed degree by one.

Let $E_1,E_2,E_3$ be independent random variables with the standard normal distribution and set $Z_1=E_1$, $Z_2=Z_1+E_2$, $Z_3=Z_2-Z_1+E_3$. The complete acyclic graph with edges $1\to 2$, $1\to 3$, $2\to 3$ is causally minimal: deleting each edge imposes a false marginal or conditional independence. Yet $Z_1\perp Z_3$ because $Z_3=E_2+E_3$, so faithfulness of a directed acyclic graph fails. Nonzero structural coefficients alone do not ensure faithfulness.

For a prime divisor $p$ of the order of a finite group, rotate the tuples $(g_1,\dots ,g_p)$ whose product is the identity. Rotation preserves that constraint even for a nonabelian group, by conjugating the product. Every nonfixed orbit has $p$ elements, while the fixed tuples correspond to $g^p=1$. There are $|G{|}^{p-1}$ tuples, so the number of fixed tuples is divisible by $p$. Since the identity is one, another solution exists and has order exactly $p$. This supplies an elementary proof of Cauchy theorem for groups.

The set $D^{+}(\Sigma )$ of events for which every past-inextendible causal curve meets $\Sigma$. This causal characterization permits an initial sign condition on a null geodesic congruence to propagate throughout the regular region by integrating a focusing inequality from $\Sigma$.

A cataclysmic variable whose white dwarf magnetic field does not dominate the large-scale accretion flow. It normally supports an accretion disk fed through Roche-lobe overflow. The term does not rule out a magnetized donor star or magnetic braking of a binary star.

A cataclysmic variable with recurrent disk outbursts associated with the hydrogen-ionization disk instability. Enhanced accretion powers the bright phase; the event does not require thermonuclear runaway on the white dwarf.

A cataclysmic variable whose persistent luminous disk resembles a dwarf nova in outburst. A sufficiently high transfer rate can keep its accretion disk on the hot stable branch of the hydrogen-ionization disk instability, avoiding ordinary dwarf-nova cycles.

For an infinite cardinal number, $\gimel (\kappa )={\kappa }^{\mathrm{cf}(\kappa )}$. The singular cardinals hypothesis predicts its value when $\kappa$ is a singular cardinal and $2^{\mathrm{cf}(\kappa )}<\kappa$. For a regular cardinal, its value is $2^{\kappa }$.