Let $A_i$ be real symmetric matrices with sum $I_n$. Their matrix ranks sum to at least $n$, because their ranges span ${\mathbb{R}}^n$. If the ranks sum to $n$, those ranges form a direct sum. For $y$ in the range of $A_i$, the identity $y={\sum }_j{A}_{j}y$ and uniqueness of its range decomposition imply $A_{j}y={\delta }_{ij}y$. Thus $A_i^2=A_i$ and $A_i{A}_j=0$ for $i\ne j$: the matrices are mutually orthogonal orthogonal projections. In a combined orthonormal basis, the forms $Z^T{A}_{i}Z$ for $Z\sim N_n(0,I_n)$ use disjoint sets of independent standard normal coordinates. They are independent chi-squared distributions with degrees of freedom equal to the ranks. Conversely, if all these forms have those chi-squared laws, their expectations and their sum give ${\sum }_i\mathrm{rank}{A}_i=n$.

For simple shear, $B=\left(\begin{array}{cc}1+{\gamma }^2& \gamma \\ \gamma & 1\end{array}\right)$. The off-diagonal entry of $\sigma B-B\sigma$ is $\gamma ({\sigma }_{11}-{\sigma }_{22})-{\gamma }^2{\sigma }_{12}$. Coaxiality of isotropic elastic stress therefore proves the identity for $\gamma \ne 0$. At $\gamma =0$, material isotropy makes the stress isotropic, proving the same identity.

For an objective material isotropy law, permutation of the principal stretches permutes the labelled principal stresses. This follows by conjugating the left stretch tensor with signed permutation rotation matrices. Each principal stress is symmetric in the other two stretches, but need not be symmetric in all three. The law $\sigma =V^2$ provides the example ${\sigma }_i={\lambda }_i^2$. In an invariant representation $\sigma ={\beta }_{0}I+{\beta }_{1}V+{\beta }_2{V}^2$, the scalar coefficients are symmetric functions of the unordered stretches.

For a CM-field, put $c$ for conjugation and $E_K={\mathcal{O}}_K^{\times }$. The map $u\mapsto u/c(u)$ takes values in the cyclic group ${\mu }_K$ of roots of unity: all conjugates have absolute value one, and the Kronecker theorem on algebraic integers in the unit disk applies. Its kernel is $E_{K^{+}}$, while its image on ${\mu }_K$ is ${\mu }_K^2$. It therefore embeds $E_K/(E_{K^{+}}{\mu }_K)$ into ${\mu }_K/{\mu }_K^2$, a group of order two. This proves the index bound without assuming that every unit is a real unit times a root of unity.

A clutching function is the equatorial transition map used in the clutching construction. For an oriented real vector bundle one may choose a fiber metric and orthonormal trivializations, so its clutching function takes values in the special orthogonal group.

Start with singleton clusters and repeatedly merge the two clusters with the smallest chosen intercluster dissimilarity. Single-linkage clustering uses the smallest cross-pair distance; complete-linkage clustering uses the largest. Merge heights and sensitivity to ties help assess a proposed number of groups.

For binary vectors, this similarity counts agreements on both presences and absences. Its complement is the normalized Hamming distance. Whether joint absences should count as similarity depends on the application; rare-presence data may call for a different coefficient.

A dissimilarity matrix records pairwise separation of observations or profiles. Usually it is symmetric, nonnegative and zero on the diagonal. A metric space additionally requires the triangle inequality and separation of distinct profiles. Agglomerative hierarchical clustering can be applied to many dissimilarity matrices that are not Euclidean distance matrices; the claimed geometry must be checked separately.

The index measures similarity of finite sets by shared presence relative to total presence. For binary profiles it ignores joint absences, unlike the simple matching coefficient. When both sets are empty, define the similarity to be one. The complementary Jaccard distance measures dissimilarity.

For a prescribed number of groups, minimize within-cluster squared Euclidean distances. At fixed assignments, differentiating the objective gives the mean as the optimal centroid. At fixed centroids, assigning each point to a nearest centroid minimizes its contribution. Alternating these steps cannot increase the objective but can converge to a local optimum, so multiple starts are useful. Empty clusters and ties need an explicit handling rule. The method favors roughly spherical groups and depends on variable scaling.

A cluster in cluster analysis is a nonempty group of observations treated as similar under the chosen representation and dissimilarity. A partition assigns each observation to exactly one cluster; agglomerative hierarchical clustering supplies a nested sequence of such partitions. An algorithmic cluster need not correspond to a distinct probabilistic population.

For a stable $n$-point closed-string scattering amplitude on a genus-$g$ surface, let $d_g$ count complex unmarked worldsheet moduli and $k_g$ count complex dimensions of the worldsheet conformal Killing group. Integrating the $n$ vertex positions and moduli, then fixing $k_g$ positions, leaves $d_g+n-k_g$ complex integrations. The punctured Riemann surface moduli dimension identifies this as $3g-3+n$. Faddeev-Popov ghost fields and antighost insertions implement the removed zero modes and the surviving modulus measure; they do not add independently integrated worldsheet coordinates.

A compact interval in the real numbers is either the empty set or a closed bounded interval $[a,b]$ with $a\le b$. Its compactness follows from the Heine-Borel theorem.

For an irreducible closed migration process with finite population $N$, positive departure rates and routing matrix $P$, choose positive traffic weights satisfying $\theta =\theta P$. On ${\sum }_j{n}_j=N$, its stationary distribution has the displayed form. For a predecessor $m=n+e_j-e_k$, its weighted transition into $n$ is $\pi (n)({\theta }_j/{\theta }_k)P_{jk}{\varphi }_k(n_k)$. Summing over $j$ gives $\pi (n){\varphi }_k(n_k)$ by the traffic equation, and summing over $k$ proves global balance for a continuous-time Markov chain. The distribution need not satisfy detailed balance.

// Target: probability-and-statistics.bigb

Represent $N$ telephone lines by a closed migration process with free-line, operator and automated-service colonies. Accepted arrivals have rate $\nu$ when a free line exists, $c$ operators complete at rate $\lambda \min (k,c)$, and $i$ automated calls complete at rate $\mu i$. Define $D_c(k)={\prod }_{h=1}^k\min (h,c)$ andThe product-form stationary distribution of a closed migration process and Poisson arrivals see time averages give the displayed blocking probability. This extends a single-operator model without treating the independent operators as a server with rate $c\lambda$ when fewer than $c$ calls are present.

// Target: probability-and-statistics.bigb

A primitive closed geodesic is a closed geodesic with its smallest positive period, rather than a repeated traversal of a shorter periodic geodesic. A degree-$m$ lift of a primitive base geodesic is still primitive upstairs when its monodromy permutation has a cycle of exact length $m$: no smaller number of traversals returns to the starting sheet.

On the punctured plane, this one-form is closed and locally the differential of the polar angle. Its line integral around a closed curve is $2\pi$ times the winding number. A once-winding circle therefore has nonzero integral, proving that no global single-valued angle potential exists and the form is not an exact differential form.

For a counterclockwise circle of radius $R>0$, substitute $x=R\cos t$, $y=R\sin t$. The pullback of a differential form isThe radius cancels. In coordinates, $P=-y/(x^2+y^2)$ and $Q=x/(x^2+y^2)$ satisfy ${\partial }_{x}Q={\partial }_{y}P$ away from the origin, so $d\omega =0$ there. The nonzero integral is a global obstruction to being an exact differential form, not a nonzero local exterior derivative.

Compare this with the smooth one-form $\eta =(-ydx+xdy)/2$ on the whole plane. Here $d\eta =dx\wedge dy$, and its integral around a radius-$R$ circle is $\pi R^2$, the enclosed area. The Generalized Stokes theorem applies to the full disc for $\eta$. For $\omega$, that disc crosses the excluded origin, where the one-form is undefined. On an annulus the theorem applies, but its induced inner boundary is clockwise: the two integrals $2\pi$ and $-2\pi$ cancel. This explains exactly why closedness and nonzero circulation can coexist.

On $[0,\infty )$ the utility functions $u_n(x)=x/n$ converge uniformly on every bounded set to $u(x)=0$. Their preference relations remain the usual order, while the limiting utility function makes all pairs indifferent. The weak comparison $0⪰1$ in the limit cannot be approximated by comparisons from the varying graphs. Uniform convergence does guarantee the upper graph-limit condition, because convergent pairs remain in a fixed bounded set.

With gross star formation rate $\psi =k{M}_g$, constant returned mass fraction $F$, and the instantaneous recycling approximation, the net locked mass rate is $(1-F)\psi$. Mass conservation gives ${\dot{M}}_g=-(1-F)k{M}_g$. For constant stellar yield $Y$ per unit locked mass and initially pristine gas, metal conservation gives $M_g\dot{Z}=Y(1-F)\psi$, so gas-phase metallicity grows as $Z=Y(1-F)kt$. Gross star formation and permanent locking have different normalizations.

For a pristine closed-box model of galactic chemical evolution, $M_{g}dZ=Yd{M}_s$ and $d{M}_s=-d{M}_g$ imply $M_g=M_0{e}^{-Z/Y}$ independently of the star formation rate. The resulting metallicity distribution function is exponential on $0\le Z\le Z_f$. Its normalized density is $e^{-Z/Y}/[Y(1-e^{-Z_f/Y})]$. A fixed initial mass function and survival selection are needed to turn locked stellar mass into a stellar number distribution. Logarithmic abundance intervals have the additional factor $(\ln 10)Z$.