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For $Y^{2}Z=X(X-Z)(X-\lambda Z)$ over $\mathbb{C}$, order its three finite roots as $(r_0,r_1,r_2)$ and put $h=r_1-r_0$. The projective transformation $X^{\prime }=(X-r_{0}Z)/h$, $Y^{\prime }=Y/s$, $Z^{\prime }=Z$, with $s^2=h^3$, gives Legendre form of an elliptic curve with parameter $(r_2-r_0)/(r_1-r_0)$. The six root orderings give the six displayed values, all different from zero and one. These exhaust changes that act affinely on $X/Z$ and scale $Y/Z$, because such a change must send the finite root set onto the new finite root set. There are six distinct values except that $\lambda \in \{-1,2,1/2\}$ gives three and ${\lambda }^2-\lambda +1=0$ gives two.

Every nonzero holomorphic differential form on a compact Riemann surface of genus one is nowhere zero. Its canonical divisor is an effective divisor of degree $2g-2=0$, and hence has every coefficient zero. In particular, such a differential trivializes the canonical bundle. This assertion concerns the differential itself, not the individual factors in a representation $fdh$.

Metamorphism reaching conditions sufficient for diagnostic high-pressure phases such as coesite in an appropriate composition. Their growth must be distinguished from detrital inheritance or unrelated inclusions.

For a homogeneous polynomial $F$ of degree $d\ge 2$, the determinant of its three-by-three Hessian matrix is homogeneous of degree $3(d-2)$ if nonzero. Its zero locus is the Hessian curve; it can be reducible or degenerate, and a nonzero constant for $d=2$ has empty zero locus. Over $\mathbb{C}$, its intersections with the nonsingular locus of $V(F)$ are precisely the flexes, by the Hessian criterion for a flex, as long as the tangent is not a component.

For any qubit operator $A$,Expand $A$ in the Pauli operator basis. The identity term is fixed, while every nonidentity coefficient has two positive and two negative contributions and cancels. Thus forgetting a uniformly random Pauli key sends every density operator to the maximally mixed state. This is the dimension-two Heisenberg-Weyl twirling channel, since replacing $Y$ by $XZ$ changes only an irrelevant phase in the conjugation.

A mafic rock with a granulite-facies assemblage. Granulite alone specifies a metamorphic facies, not the mafic composition.

With insertion convention $i\Pi$, ghost propagator $i/p^2$, and oriented ghost-gluon Feynman rule $g{f}^{abc}{p}^{\mu }$ with $p$ pointing away along the ghost arrow, the ghost loop is $i{\Pi }_{\mathrm{gh}}=-g^2{C}_A{\delta }^{ab}\int d^{D}l(2\pi {)}^{-D}(l+k{)}^{\mu }{l}^{\nu }/[l^2(l+k{)}^2]$. In dimensional regularization with $D=4-ϵ$, a Feynman parameter and a shift give the displayed ultraviolet pole. Its contraction is $g^2{C}_A{\delta }^{ab}{k}^2{k}^{\nu }/(32{\pi }^2ϵ)$; it cancels the nontransverse part of the two-three-gluon-vertex contribution. A different insertion or ghost-vertex convention must be translated before comparing signs.

The six smallest complex irreducible representations of SU(3) have highest weights $(0,0),(1,0),(0,1),(2,0),(0,2),(1,1)$ and dimensions $1,3,3,6,6,8$. Their Young diagram row lengths are $(0,0),(1,0),(1,1),(2,0),(2,2),(2,1)$. They are realized by scalars, defining vectors, covectors (equivalently antisymmetric tensors of degree two), upper and lower symmetric tensors of degree two, and mixed tensors $T^i{}_j$ with zero trace, respectively. The Weyl dimension formula is $\dim (p,q)=(p+1)(q+1)(p+q+2)/2$. For $p+q\ge 3$ this is at least ten, proving that the list exhausts the six smallest representations.

In Feynman gauge, with insertion $-i\Sigma (p)$ and $d=4-2ϵ$, the one-loop QED result is ${\Sigma }_{\mathrm{UV}}=e^2(-/p+4m)/(16{\pi }^2ϵ)$. The coefficient of $/p$ is the electron kinetic counterterm ${\delta }_2=-e^2/(16{\pi }^2ϵ)$ when its insertion is $i/p{\delta }_2$. The scalar part fixes a separate mass counterterm in the chosen normalization convention.

An ideal planar conductor is an equipotential boundary in electrostatics. The electric field just outside is normal to it, while the electric field inside its metal vanishes. If grounded and infinite, the method of images implements the boundary by reflected opposite electric charges.

For a qubit, computational basis dephasing with complex overlap $c$ maps ${\rho }_{01}$ to $\overline{c}{\rho }_{01}$ and preserves the diagonal entries. Conjugation by the phase gate $P_{\phi }=\mathrm{diag}(1,e^{i\phi })$ maps the same entry to $e^{-i\phi }{\rho }_{01}$. Since these scalar factors commute, their quantum channels commute on every density matrix. More directly, an interaction $|j⟩|e⟩\mapsto |j⟩|e_j⟩$ and a diagonal phase gate give the same joint quantum state in either order on the specified initial environment state. This need not hold for an interaction that changes the quantum state populations.

Correlated dephasing caused by the same fluctuating $z$ rotation on all qubits. Averaging $U_{\varphi }\rho U_{\varphi }^{†}$ over the unknown angle suppresses quantum coherence between sectors with different total $Z$ eigenvalues. Within one eigenspace, however, every state acquires the same global phase, so its density matrix is preserved. Such an eigenspace is a decoherence-free subspace for this noise model. The assumption concerns the shared rotation; independent local phase noise generally destroys this protection.