A finite-metal-fraction variant of the closed-box model of galactic chemical evolution uses effective stellar yield $Y(1-Z_g)$, rather than constant yield. With pristine initial gas and $Y>0$, metal accounting gives $d{Z}_g/d\log (1/\mu )=Y(1-Z_g)$ and hence $Z_g=1-{\mu }^Y$. The mass-weighted mean stellar metallicity is $[Y-(Y+1)\mu +{\mu }^{Y+1}]/[(Y+1)(1-\mu )]$. At final gas fraction ${\mu }_f$, the metallicity distribution function is $(1-z{)}^{1/Y-1}/[Y(1-{\mu }_f)]$ on $0<z<1-{\mu }_f^Y$. For trace abundances, $Z_g\simeq -Y\log \mu$ and the usual closed-box metallicity distribution is recovered to leading order.

Realize $V_n={\mathrm{Sym}}^n{\mathbb{C}}^2$. Contract $r$ pairs of indices, one from each symmetric tensor, with the alternating invariant tensor $ϵ$, and symmetrize the remaining indices. This is a nonzero intertwining operator into $V_{m+n-2r}$. In the bihomogeneous polynomial realization, $(z_1{w}_2-z_2{w}_1{)}^r{z}_1^{m-r}{w}_1^{n-r}$ is a highest-weight vector of weight $m+n-2r$. Its lowering string has length $m+n-2r+1$. These inequivalent irreducible summands exhaust the tensor product, since their dimensions sum to $(m+1)(n+1)$.

Let $\mathbf{FinPos}$ contain finite partially ordered sets and all order-preserving functions, including the empty object. The covariant functor category $[\mathbf{FinPos},\mathbf{Set}]$ classifies weak partial orders. Finite positive presentations reduce to finite preorders with mutually comparable cycles identified. An internal model $M$ gives a flat functor $P\mapsto \mathrm{Hom}(P,M)$ on ${\mathbf{FinPos}}^{\mathrm{op}}$; conversely the corresponding filtered presentation by finite models recovers its carrier and order relation.

Let $\mathbf{FinSPos}$ contain finite strict partial orders and strict order-preserving functions. Its covariant set-valued functor category classifies the strict theory. A finite presentation with a strict cycle is contradictory; every other finite diagram gives a finite strict partial order. Equalizing finitely many element identifications either gives such an order or forces a forbidden cycle, so the internal model-to-flat-functor construction remains valid. The reflexification of a strict partial order induces a surjective geometric morphism to the weak-order classifier, but not a geometric embedding because strict maps exclude collapse of comparable elements.

With empty objects allowed, finite totally ordered sets and all weak order-preserving functions give a presheaf classifier for weak total orders. Finite tuples in a model can locally be merged into a sorted tuple using finitely many totality disjunctions. Equalizing two maps between finite chains amounts to collapsing the convex intervals forced equal. These two constructions give the flatness conditions without assuming decidable equality.

Finite strict total orders and increasing injections, including the empty order, give a presheaf classifier for strict total orders. Trichotomy locally merges two finite sorted tuples, identifying equal entries. A strict map from a chain is injective, so parallel maps admitting the same realization are already equal. Reflexification gives a geometric morphism to the classifying topos of linear orders; pulling back weak totality gives exactly strict trichotomy.

In the sheaf topos for the Zariski coverage on finitely presented rings, the universal local ring is represented by the polynomial ring $\mathbb{Z}[t]$. At the stage of a finitely presented ring $A$, its section set is $A$, and restriction along $A\to B$ is that ring map. It is internally local because the localizations at $x$ and $1-x$ cover each stage. Its square-zero infinitesimals are represented by $\mathbb{Z}[\epsilon ]/({\epsilon }^2)$. Stage rings need not themselves be local or reduced; their variation supplies the internal local and infinitesimal semantics.

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.