A rational map is smooth as a map between compact round spheres, including at its poles in reciprocal coordinates. Its spherical differential norm therefore has a finite supremum $L$. Integrating along a shortest spherical geodesic gives $\rho (Rz,Rw)\le L\rho (z,w)$. Since $2\chi \le \rho \le \pi \chi$, we obtain $\chi (Rz,Rw)\le (\pi L/2)\chi (z,w)$. In finite coordinates the differential norm is $R^{\#}(z)=|R^{\prime }(z)|(1+|z{|}^2)/(1+|R(z){|}^2)$, extending continuously to the whole sphere. This uniform bound is useful for passing equicontinuity through compositions.

Use the unnormalized Choi matrix $J(\Phi )=(\Phi \otimes \mathrm{id})(|\tilde{\Psi }⟩⟨\tilde{\Psi }|)$, which has trace $d$ for a channel. A positive such matrix has a spectral decomposition with vectors $v_k=\sqrt{{\lambda }_k}{u}_k$. In paired bases define the Kraus operator $A_k$ by $(A_k{)}_{ij}=(v_k{)}_{ij}$, so $v_k=(A_k\otimes I){\sum }_j|j⟩|j⟩$. Partial contraction with a conjugate index vector gives $A_k|\varphi ⟩$, hence $\Phi (|\varphi ⟩⟨\varphi |)={\sum }_k{A}_k|\varphi ⟩⟨\varphi |A_k^{†}$. Rank-one projectors span the operator space, proving the Kraus representation on all inputs. Trace preservation is equivalent to ${\sum }_k{A}_k^{†}{A}_k=I$. This construction needs at most $\mathrm{rank}J$ nonzero Kraus operators.

Let $S=\bar{q}q$, $P_a=i\bar{q}{\gamma }_5{\tau }_{a}q$ and $Q_{5,a}=\int q^{†}{\gamma }_5{\tau }_{a}q/2d^{3}x$. The canonical anticommutation relations imply $[Q_{5,a},q]=-{\gamma }_5{\tau }_{a}q/2$ and $[Q_{5,a},\bar{q}]=-\bar{q}{\gamma }_5{\tau }_a/2$. The Leibniz rule for the commutator, together with $\{{\tau }_a,{\tau }_b\}=2{\delta }_{ab}$ and ${\gamma }_5^2=1$, gives the displayed transformations. These are equal-time operator relations; common nonzero quark masses remove conservation of the charges without changing their canonical action on the fields.

A nonzero vacuum value of the scalar quark density breaks the massless nonsinglet chiral symmetry to its diagonal vector subgroup. Indeed, the nonsinglet axial charge algebra of quark densities gives $⟨[Q_{5,a},P_b]⟩=-i{\delta }_{ab}⟨\bar{q}q⟩$, whereas invariance of the vacuum under every conserved axial charge would make these expectations vanish. The three broken generators produce three massless Goldstone bosons in the exact chiral limit. Small equal quark masses explicitly break that symmetry and turn the pions into pseudo-Goldstone bosons, with squared masses linear in the quark mass at leading order.

For $U\in SU(2)$ with $U\mapsto AU{B}^{-1}$, the chiral group acts transitively. A derivative-free invariant is therefore constant. At $U=I$, its diagonal stabilizer acts on tangent vectors by the irreducible three-dimensional rotation representation; the only invariant symmetric metric is a multiple of ${\delta }_{ab}$. Transporting this metric gives the displayed unique Lorentz-invariant two-derivative action up to normalization and total derivatives. Equivalently, $U^{†}{\partial }_{\mu }U$ is traceless, eliminating a separate double-trace invariant. Writing $U=\exp (i{\pi }_a{\tau }_a/F)$ and using $\mathrm{tr}({\tau }_a{\tau }_b)=2{\delta }_{ab}$ gives canonically normalized pion kinetic terms.

For condensate normalization $⟨{\bar{q}}_{Lj}{q}_{Ri}⟩=-v{\delta }_{ij}$, replacing the condensate matrix by $-vU$ maps the real quark mass matrix $M=\mathrm{diag}(m_u,m_d)$ to $v\mathrm{tr}[M(U+U^{†})]$. Since $(\mathbf{\pi }\cdot \mathbf{\tau }{)}^2=|\mathbf{\pi }{|}^{2}I$, expansion gives a constant minus $v(m_u+m_d)|\mathbf{\pi }{|}^2/F^2$. Comparison with $-m_{\pi }^2|\mathbf{\pi }{|}^2/2$ gives the displayed common leading mass. Strong isospin-breaking mass splitting enters beyond this leading action; electromagnetic effects are also excluded.

The relation uses $x=c_1(S)$ for the tautological line on the projectivization of $E$. It follows from the rank of the complementary bundle and Whitney multiplication. The alternating signs are ring subtraction; they do not assert that the First Chern class of a dual line is the additive negative in complex cobordism, where duality uses the formal-group inverse.

// Target: geometry-and-topology.bigb

For a smooth degree-$d$ projective hypersurface $X\subset {\mathbb{P}}^n$, put $h=c_1({\mathcal{O}}_X(1))$. The cotangent Euler sequence in homogeneous coordinates gives $c({\Omega }_{{\mathbb{P}}^n}^1{|}_X)=(1-h{)}^{n+1}$. The Conormal exact sequence for Kähler differentials and Whitney sum formula for Chern classes therefore yield $c({\Omega }_X^1)=(1-h{)}^{n+1}/(1-dh)$. In codimension $i\le n-1$, its coefficient is $c_i({\Omega }_X^1)={\sum }_{j=0}^i(-1{)}^j(\genfrac{}{}{0ex}{}{n+1}{j})d^{i-j}{h}^i$. Higher classes vanish by rank and dimension; the formal quotient is truncated in the Chow ring.

For relativistically normalized kaon states and $Q=(\bar{s}{\gamma }_{\mu }(1-{\gamma }^5)d)(\bar{s}{\gamma }^{\mu }(1-{\gamma }^5)d)$, define $⟨{\bar{K}}^0|Q|K^0⟩=(8/3)f_K^2{m}_K^2{B}_K$. The dimensionless kaon bag parameter measures strong interaction corrections to the vacuum-insertion normalization. The coefficient and matrix element must be defined in the same renormalization scheme and at the same scale. Dividing by $2m_K$ gives the corresponding two-state mixing-Hamiltonian normalization.

Charge carrier mobility is the magnitude of steady drift velocity per unit weak electric field in a scalar transport model. In the Drude model, ${\mathbf{v}}_d=\mathrm{sgn}(q)\mu \mathbf{E}$ and $\sigma =n|q|\mu$. The positive charge carrier mobility separates ease of motion from the sign of electric charge; its SI unit is ${\mathrm{m}}^2/(\mathrm{V}\mathrm{s})$.

The drift velocity is a charge carrier population's mean velocity relative to its material's rest frame. It describes the net transport superposed on microscopic motion. For a single species, current density is $\mathbf{J}=nq{\mathbf{v}}_d$; for electrons, $q=-e$ reverses the direction. It is distinct from the propagation speed of an electrical disturbance. ^[ref].

A nontrivial group with no proper nontrivial characteristic subgroup. A minimal nontrivial normal subgroup of a finite group has this property: every characteristic subgroup of it is normal in the ambient group.

With the convention $\mathbb{E}{e}^{iu{X}_t}=e^{t\psi (u)}$, this complex process is a martingale for each real $u$. Its absolute value is the deterministic finite number $e^{-t\mathrm{Re}\psi (u)}$. Conditioning on the past at time $s$ and using independent increments multiplies $e^{iu{X}_s-t\psi (u)}$ by $e^{(t-s)\psi (u)}$, giving $M_s^u$. Under the opposite exponent convention the sign of the compensating term reverses.

For a unital commutative complex Banach algebra $A$, the Gelfand transform sends $a\in A$ to the continuous function $\hat{a}(\phi )=\phi (a)$ on its character space of an algebra. It is a contractive unital algebra homomorphism into $C({\Phi }_A)$.

For a commutative complex unital Banach algebra, maximal ideals correspond bijectively to characters of an algebra by taking kernels. Give the maximal ideal space the Gelfand topology through this correspondence. It is a compact Hausdorff space, since the characters form a weak-star closed subset of the dual unit ball, which is compact by the Banach-Alaoglu theorem.

An algebraic torus character is a morphism of algebraic groups $T\to {\mathbb{G}}_m$. For a split torus with character lattice $M$, these are precisely the Laurent monomials ${\chi }^m$, with ${\chi }^{m+m^{\prime }}={\chi }^m{\chi }^{m^{\prime }}$.

// Target: algebra.bigb

Orthogonal coordinates have pairwise orthogonal coordinate tangent directions. Their metric is diagonal, with scale factors of orthogonal coordinates $h_i=|\partial \mathbf{x}/\partial q_i|$. Spherical coordinates and cylindrical coordinates are examples.

In the open first quadrant this is one-to-one, with $x=\sqrt{(u+v)/2}$, $y=\sqrt{(u-v)/2}$ and Jacobian determinant $-8xy$. Its image satisfies $u>v$ and $u+v>0$. In particular $1\le v\le u\le U$ is a triangular image for a circular-hyperbolic region. A factor $xy$ in the original integrand cancels the inverse-Jacobian factor $1/(8xy)$; moreover $x^4-y^4=uv$ and $4x^2{y}^2=u^2-v^2$. Boundary degeneracy on $y=0$ is confined to a measure-zero set and must not be confused with an interior failure of invertibility.

On the positive quadrant, the inverse coordinate change is $x=\sqrt{uv}$, $y=\sqrt{v/u}$. Its Jacobian determinant is $1/(2u)>0$. Regions bounded by positive constant-ratio lines and constant-product hyperbolas become rectangles in these coordinates. An integrand $x/y=u$ then cancels the Jacobian factor, making weighted area integrals particularly simple. The positivity restriction selects a unique square-root branch and fixes orientation.

On $x>0$, $0<y<1$, $0<z<x$, this change of variables is a bijection onto $0<\gamma <\beta <\alpha$. Its inverse is $x=\sqrt{\alpha \beta }$, $y=\sqrt{\beta /\alpha }$, $z=\gamma \sqrt{\alpha /\beta }$, and its forward Jacobian determinant is $2x$. Thus $xdxdydz=\frac{1}{2}d\alpha d\beta d\gamma$. Passing to increments $(\alpha -\beta ,\beta -\gamma ,\gamma )$ then maps the ordered cone to the positive octant, turning an exponential with linear exponent into a product of three one-dimensional integrals.