The stabilizer of a line in $O(n)$ is $O(1)\times O(n-1)$, giving the homogeneous space ${\mathbb{RP}}^{n-1}$. Its classifying-space Serre fibration has rationally acyclic fiber for odd $n$, so its cohomological Serre spectral sequence has only its degree-zero row and gives the displayed restriction isomorphism. The factor $BO(1)={\mathbb{RP}}^{\infty }$ is itself rationally acyclic. For even $n$, the fiber has a top rational class, with determinant-sign monodromy, so the one-row argument fails. The top universal Pontryagin class $p_{n/2}$ is nonzero on $BO(n)$ but vanishes on restriction to $BO(n-1)$, witnessing genuine failure of stabilization.

A classification tree partitions predictor space recursively and attaches a class probability vector to each terminal region. For a categorical response, a fitted leaf commonly uses its observed class proportions and predicts their largest entry. Binary numerical splits compare one coordinate with a threshold. Growing a tree greedily by reducing classification-tree deviance gives an interpretable rule but does not globally optimize the tree. Cost-complexity tree pruning and cross-validation control complexity and assess prediction error.

The misclassification rate is the expected zero-one loss of a classification in statistical learning rule. Its empirical version is the fraction of incorrectly labeled observations. Evaluation on the fitting sample gives a training error; evaluation on an independent test set estimates prediction error. For a majority-class leaf of a classification tree, the training error count is $n_t-{\max }_k{n}_{tk}$, unlike classification-tree deviance, which also uses the fitted class probabilities.

Let $A$ be a domain satisfying both Ore conditions, with quotient division ring $K$. A matrix over $A$ is regular exactly when it is invertible over $K$: a kernel vector of a singular matrix can have its denominators cleared to give a nonzero annihilator over $A$. An upper triangular matrix is regular exactly when all diagonal entries are nonzero. Each finite list of entries of a matrix over $K$ has a common left denominator $a\in A\setminus \{0\}$. Thus it is a fraction with denominator $a{I}_r$, proving the displayed classical localizations. The triangular localization retains its strictly upper triangular nilpotent ideal; it is not a semisimple matrix ring.

An adapted process is of class DL when, for every finite $T$, the family $\{X_{\tau }:\tau \le T\text{ a stopping time}\}$ is uniformly integrable. A local martingale is a true Martingale exactly when it has this property. This is a local-in-time condition, not Uniform integrability over the whole half-line.

// Target: probability-and-statistics.bigb

A one-period quadratic hedge minimizes $\mathbb{E}[(\xi -\varphi R-\pi \cdot S_1{)}^2]$ over deterministic holdings, where $R\ne 0$ is the riskless gross return, $\mu =\mathbb{E}{S}_1$, and the covariance matrix $V$ is invertible. For a square-integrable claim, the unique solution is ${\pi }^{\ast }=V^{-1}\mathrm{Cov}(S_1,\xi )$ and ${\varphi }^{\ast }=(\mathbb{E}\xi -{\pi }^{\ast }\cdot \mu )/R$. Centering separates the squared bias from the quadratic error. Completing the square gives the excess error as $(\pi -{\pi }^{\ast }{)}^{T}V(\pi -{\pi }^{\ast })$ after the bias is minimized. Unlike claim replication, the residual need not vanish.

For the circular topographic anticyclone in a positive uniform current $U$, the opposing disturbance speed is at most $Ca=fϵa/2$. Below the displayed threshold $u$ remains positive everywhere, so no streamline can close. Above it an interior stagnation point at $(0,-U/C)$ is a center, while the exterior point $(0,-C{a}^2/U)$ is a saddle. At equality the two meet at the rim. This is the onset criterion; once closed contours form, the potential vorticity of trapped fluid is not determined solely by its present upstream conditions.

Every supporting centre of an arc between fixed endpoints lies on their perpendicular bisector. Multiplicity at least $k$ therefore makes that line contain at least $k$ of the original points. For each such line containing $s$ points, its $s$ centres each support at most $t$ circles; on one circle at most two consecutive-point arcs have that symmetry axis. The rich line bound summed over dyadic occupancies gives the displayed bound.

Put $A_3=C_3+B_2\wedge dy$ and $F_4=G_4+H_3\wedge dy$, where $G_4=d{C}_3$. The circle component of $A_3\wedge F_4\wedge F_4$ is $[B_2\wedge G_4\wedge G_4+2C_3\wedge G_4\wedge H_3]\wedge dy$. Since $d(C_3\wedge B_2\wedge G_4)=B_2\wedge G_4\wedge G_4-C_3\wedge H_3\wedge G_4$, its integral equals three copies of $B_2\wedge G_4\wedge G_4$ up to a boundary term. The coefficient therefore becomes $-1/2$ after circle integration. Boundary terms and global potential patching must be retained when the spacetime has boundary or nontrivial flux topology.

Fix a holomorphic function on a neighbourhood of a closed disk, with $f(0)\ne 0$. On an interval of positive radii whose circles contain no zeros of a function, differentiating Jensen's formula with respect to $x=\log R$ givesEach interior zero of a function contributes slope one for each unit of its multiplicity of a zero. Between consecutive zero moduli, $L_f(e^x)$ is an affine function; passing the modulus of a zero of multiplicity of a zero $m$ increases the adjacent slopes by $m$. The finite sum $\log |f(0)|+{\sum }_j\max (0,x-\log |a_j|)$ defines a continuous convex function of $x$ whose values agree with the circle log-mean of a holomorphic function at every boundary-zero-free radius. This describes the adjacent line segments without asserting a boundary-singularity extension of the integral.

For a polynomial $p(z)=c{\prod }_j(z-a_j)$ with $c\ne 0$, no polynomial root at zero, and $R>0$ different from every $|a_j|$, Jensen's formula gives the displayed expression. The polynomial roots are repeated according to their multiplicity of a zero: an exterior root contributes $\log |a_j|$, whereas an interior root contributes $\log R$.

For an explicit example, take the polynomialIts two simple zeros are $1/4$ and $-1/2$. The circle log-mean of a holomorphic function is thereforeAt $R=0.2$ the open disc contains no zeros of a function, at $R=0.4$ it contains $1/4$ only, and at $R=0.75$ it contains both. These three circles avoid the polynomial roots; the radii $1/4$ and $1/2$ are excluded from the displayed version.

The corresponding values of the circle log-mean of a holomorphic function and the circle geometric mean of a holomorphic function areFor example, at $R=0.4$ the contribution of the sole interior zero of a function is $\log (0.4/0.25)=\log (8/5)$; adding it to $\log (3/8)$ gives $\log (3/5)$. The second polynomial root still affects $f(0)$, but contributes no additional interior-zero term at this radius.

The continuous angular geometric mean of the positive boundary complex modulus isHere the empty product is one. This differs from the angular arithmetic mean $A_f(R)=(2\pi {)}^{-1}{\int }_0^{2\pi }|f(R{e}^{i\theta })|d\theta$. In general $G_f(R)\le A_f(R)$ by the concavity of the logarithm; equality requires constant boundary complex modulus. For a holomorphic function with no zeros of a function in the closed disk, Jensen's formula gives $G_f(R)=|f(0)|$, even when the boundary complex modulus varies with angle. Thus replacing the mean of a logarithm by the logarithm of an arithmetic mean loses the exact zero identity.

For every finite simple graph, Vizing's theorem givesHere $\Delta (G)$ is the maximum degree of a graph, taken as zero when there are no vertices. A bipartite graph has the stronger equality $\chi \prime (G)=\Delta (G)$. The six-job complete bipartite graph above attains this lower bound, while the triangle in a graph attains the extra-colour alternative. The simple-graph restriction in the upper bound matters.^[ref]

The bipartite graph equality still holds for finite loopless multigraphs, with parallel edges counted separately in each vertex degree. For instance, $r$ parallel jobs between two resources need exactly $r$ slots: every pair conflicts, and $\Delta =\chi \prime =r$. For a general finite loopless multigraph, however, the simple graph upper bound can fail. Put two parallel edges on each side of a triangle in a graph. All six edges pairwise share a vertex, so each needs its own colour; every vertex has degree of a vertex four. Consequently $\chi \prime =6>\Delta +1=5$. A multiplicity-aware bound is $\chi \prime (G)\le \Delta (G)+\mu (G)$, where $\mu (G)$ is the largest number of parallel edges between a pair of vertices. The doubled triangle in a graph has $\mu =2$ and attains that bound. These statements exclude self-loops; the two-distinct-resource scheduling model excludes them too.^[ref]

For historical orientation, Graph Theory by Reinhard Diestel (Colouring preview, §5.3, p. 132) dates the bipartite equality to 1916 and Vizing's theorem to 1964. These are textbook attributions; the original papers have not been inspected here. The results state existence of optimal or nearly optimal proper edge colourings, rather than saying that arbitrary greedy choices will find them. The list-chromatic index asks the further question of success for every sufficiently large family of allowed-colour lists.

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