Diagonalize the self-adjoint Jacobi curvature operator at the initial point and transport its orthonormal eigenbasis parallel along the geodesic. The Jacobi field equation reduces to $j_i^{\prime \prime }+{\lambda }_i{j}_i=0$. For zero initial position and initial derivative $b_i$, the solution is $b_i\sin (\sqrt{{\lambda }_i}t)/\sqrt{{\lambda }_i}$ when ${\lambda }_i>0$, $b_{i}t$ when ${\lambda }_i=0$, and $b_i\sinh (\sqrt{-{\lambda }_i}t)/\sqrt{-{\lambda }_i}$ when ${\lambda }_i<0$. Only positive eigenvalues contribute conjugate points, at $t=k\pi /\sqrt{{\lambda }_i}$. At a common zero the multiplicity is the sum of the contributing eigenspace dimensions. The displayed convention uses the curvature operator positive on a positively curved plane; reversing the tensor convention also reverses its tensor formula.

A rank-one subsheaf of a vector bundle on a smooth algebraic curve is saturated when the quotient is torsion-free. Intersecting a one-dimensional subspace of the generic fibre with each stalk constructs a saturated subsheaf. The local ring of a smooth algebraic curve is a discrete valuation ring, so the subsheaf and its torsion-free quotient are locally free. Saturating a nonzero map from a line bundle enlarges that line bundle by an effective divisor, recording the map's zeros.

A free sheaf of finite rank on a scheme is a sheaf of modules globally isomorphic to ${\mathcal{O}}_X^{\oplus r}$. A locally free sheaf only requires such an isomorphism on an open cover; its local frames need not glue to one global frame.

For $1\to N\to G\to Q\to 1$ with $N,Q$ locally finite, take a finitely generated subgroup $L\le G$. Its image in $Q$ is finite, so $L\cap N$ has finite index in $L$ and is finitely generated by Schreier's lemma. Local finiteness of $N$ makes that intersection finite. Thus $L$ itself is finite, proving local finiteness of $G$.

// Target: algebra.bigb

For a time update with one-step defect ${\delta }_{h,k}$ after exact solution values are inserted, the normalized defect divides by the time step $k$. If ${\delta }_{h,k}=O(k^{p+1})$, the normalized local truncation error is $O(k^p)$. A spatial mesh contributes its own powers: for a diffusion update, ${\mathcal{T}}_{h,k}=O(k+h^2)$ means ${\delta }_{h,k}=O(k^2+k{h}^2)$. Under $k=O(h^2)$ these are respectively second and fourth powers of $h$. Neither normalization changes the actual order of a numerical method; it changes the power used to express its residual.

For a sinusoidal mean jet on an $f$ plane, its QG potential-vorticity gradient is $({\pi }^2/L^2+k_d^2)U$. Ground-stationary modes satisfy the displayed circle and can solve the full stationary shear equation even though their scale does not meet a local short-wave assumption. The local stationary group vector is $2Uk(k,l)/(k^2+l^2+k_d^2)$, reducing to a zonal scalar formula only for $l=0$.

At $\chi ={\chi }^w{\delta }^{-1}$, a determinant-character twist reduces to $\chi (\mathrm{diag}(a,d))=|a|$. In the open Bruhat coordinate, a section $g_0(x)=1$ for $|x|\le 1$ and $g_0(x)=|x{|}^{-1}$ otherwise has $\pi (\mathrm{diag}(\varpi ,1))g_0-q^{-1}{g}_0=(1-q^{-1})1_{\varpi \mathfrak{o}}$. Its additive Haar integral is nonzero, whereas every unipotent translation difference has zero such integral. The difference therefore survives in the Jacquet module, proving that this two-dimensional self-extension is nonsplit and has just one quotient functional of the common linear character.

The local Steinberg representation is the quotient of locally constant functions on the projective line by constant functions, with the action induced from GL2. In the unnormalized convention, $V(1)$ fits into $0\to 1\to V(1)\to \mathrm{St}\to 0$. Its scalar endomorphism algebra obstructs a splitting. This local infinite-dimensional representation differs from the finite-field Steinberg representation.

For the standard representation of the symmetric group $S_3$, the unramified local factor of an Artin L-function, with $t=p^{-s}$, is respectively the displayed factor for identity, transposition, and three-cycle Frobenius cycle types. These follow by removing the trivial summand from the three-point permutation representation. If the inertia group contains a three-cycle, its invariant subspace in the standard representation is zero, so the ramified local factor is one.

In dimension two, $z\mapsto z^k$ for positive $k$ has local degree $k$ at zero, and $z\mapsto {\overline{z}}^{|k|}$ has degree $k$ for negative $k$. The proper map $z\mapsto |z{|}^2\in \mathbb{C}$ has isolated zero fiber and local degree zero, since its image misses every nearby nonreal target. These proper maps extend to the two-sphere. Taking their product with the identity of ${\mathbb{R}}^{n-2}$ realizes the same local degrees in every dimension $n\ge 2$. In dimension one an isolated local degree is restricted to $0,1,-1$.

Its fiber at $x$ is $H_n(M,M\setminus \{x\};\mathbb{Z})\cong \mathbb{Z}$. Transport changes its generator by the sign of local orientation along the path. For a connected nonorientable manifold, its degree-zero cohomology is zero, because no nonzero integer is invariant under sign reversal, whereas its degree-zero homology is $\mathbb{Z}/2$, the coinvariants under that reversal.

The list-chromatic index is the least integer $q$ such that every assignment of at least $q$ allowed colors to each edge admits a list edge coloring. Uniform identical lists show it is at least the chromatic index. The Galvin theorem for bipartite list edge coloring gives equality for bipartite graphs.

For a Lagrangian submanifold $L$ in an exact symplectic manifold with primitive $\lambda$, its period subgroup is $\{{\int }_c\lambda :c\in H_1(L;\mathbb{Z})\}\subset \mathbb{R}$. It is rational when this subgroup is discrete. If it is nonzero, it is $\gamma \mathbb{Z}$ for a unique $\gamma >0$. Some conventions reserve the word rational for this nonzero case. The zero subgroup corresponds to an exact differential form $\lambda {|}_L$; its least positive period is interpreted as $+\infty$ if that endpoint convention is admitted.

For a smooth nonconstant holomorphic disk $h$ in ${\mathbb{C}}^n$ with boundary on a Lagrangian submanifold $L$, its symplectic area is ${\int }_{D^2}|{\partial }_{s}h{|}^{2}dsdt>0$. The Cauchy-Riemann equations give ${\partial }_{t}h=J{\partial }_{s}h$, so this is the pullback of the standard symplectic form. The Stokes theorem identifies the area with the period of a primitive along the boundary loop. Thus the Liouville class of a Lagrangian submanifold evaluates nontrivially on this loop and cannot vanish.

The logit link maps a mean probability to its log odds: $g(p)=\log [p/(1-p)]$, $0<p<1$. Its inverse is $p=(1+e^{-\eta }{)}^{-1}$, used in logistic regression.

The identity link function equates the response mean with the linear predictor. In a generalized linear model it gives ${\mu }_i=x_i^T\beta$. It is the canonical link function for the normal distribution. For response families whose means must remain positive or between zero and one, using the identity link requires corresponding restrictions on the linear predictor.

For $y_{tt}+2k{y}_t=c^2{y}_{xx}$ with fixed endpoints, initial displacement zero and initial velocity a Dirac delta at $a\in (0,l)$, the Fourier sine series coefficient of that velocity is $2\sin (n\pi a/l)/l$. Each coefficient evolves by $T_n^{\prime \prime }+2k{T}_n^{\prime }+(n\pi c/l{)}^2{T}_n=0$. Here ${\omega }_n^2=(n\pi c/l{)}^2-k^2$. At critical damping, replace $\sin ({\omega }_{n}t)/{\omega }_n$ by $t$; for an overdamped mode replace it by $\sinh (\sqrt{k^2-(n\pi c/l{)}^2}t)/\sqrt{k^2-(n\pi c/l{)}^2}$. The initial conditions hold in the sense of distributions.

For nonresonant incompressible normal modes of a uniform-density shear flow with aligned uniform magnetic field, let $S=\sigma +k_{x}U(z)$, $k^2=k_x^2+k_y^2$ and $f=w/S$. The normal displacement is $f/i$. The linearized horizontal equations and incompressibility give $i{k}^2{p}_1=\rho (S{w}^{\prime }-S^{\prime }w)=\rho S^2{f}^{\prime }$. Substituting the ideal magnetohydrodynamic induction equation into normal momentum gives the displayed conservative equation. Across a zero-thickness shear layer, continuity of $f$ and of $(\rho S^2-k_x^2{B}^2)f^{\prime }$ supplies the interface conditions. They express common normal displacement and continuous perturbed magnetohydrodynamic total pressure, avoiding undefined products of distributional shear gradients.

Pair $2n$ linearly ordered endpoints uniformly. Each pair is a chord; its larger endpoint ends a vertex block, with all endpoints after the previous right endpoint merged into that vertex. Orient the chord from its right-endpoint block to its left-endpoint block. Removing the last endpoint and its mate leaves a uniform smaller pairing. Reinserting the mate into one of $2n-1$ equally likely gaps chooses an existing block proportionally to its endpoint count, or the final gap creates a self-loop. This proves equivalence with the linearized chord diagram model.

An equivalent continuous representation draws $n$ independent pairs of independent uniform points on $[0,1]$, orders the pairs by their maxima $R_1<\cdots <R_n$, and calls their minima $L_i$. The $R_i^2$ are uniform order statistics; conditional on all the $R_i$, the $L_i$ are independent uniform points on $[0,R_i]$.

In a uniform linearized chord diagram, $\mathrm{Pr}(\sqrt{n}{R}_1\ge x)=(1-x^2/n{)}^n\to e^{-x^2}$. Conditional on the right endpoints, $d_1(n)=2+{\sum }_{i=2}^n{1}_{\{L_i\le R_1\}}$, with independent success probabilities $R_1/R_i$. A Chernoff bound gives $R_i=(1+o(1))\sqrt{i/n}$ uniformly for $i\ge n^{1/10}$. The early indicators are $o(\sqrt{n})$, and the remaining conditional mean is $(2+o(1))n{R}_1$. Their conditional variance is $O_{\mathbb{P}}(\sqrt{n})$. Hence $d_1(n)/\sqrt{n}-2\sqrt{n}{R}_1\to 0$ in probability, proving the displayed tail limit. A self-loop's contribution is included in the constant two.