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If a graph is $2k$-connected and has a graph minor $H$ satisfying $2\delta (H)\ge |H|+4k-1$, it is $k$-linked. Write $d=|H|-1-\delta (H)$, so $|H|\ge 2d+4k+1$. The rooted branch-set reduction replaces the minor model by $|H|-k$ disjoint connected sets, the first $2k$ containing any prescribed terminals separately, each missing at most $d$ of the remaining sets. Every prescribed pair then has at least $k+1$ common adjacent unused sets, allowing the $k$ connections to be selected successively. The reduction uses rooted separations of a graph, contraction of edges outside distinct branch sets, and a final Hall marriage theorem matching; the separation condition prevents a contraction from destroying access to the model.

On vertices $0,\dots ,7$, join cyclic indices differing by one or two modulo eight. The resulting planar graph has a square face bounded in order by $0,2,4,6$. Deleting at most three vertices cannot disconnect it: each break in the surviving cyclic chain requires two consecutive deleted vertices, and two separate breaks require four deletions. It is therefore four-connected. Disjoint paths joining $0$ to $4$ and $2$ to $6$ would cross in the complementary disk of the square face, contrary to the Jordan curve theorem. Thus four-connectivity does not guarantee two-linkage.

Choose a checkerboard coloring of the complementary regions of a link diagram. The Tait graph has one vertex for each black region and one edge through each crossing joining its adjacent black regions. The other coloring gives its planar dual. For a connected alternating link diagram, all-A and all-B smoothing circles correspond to the regions of its two checkerboard colors; one-circle smoothing states correspond to spanning trees.

Replace each crossing of a diagram by a saddle joining its A smoothing above the projection sphere to its B smoothing below; cap the upper and lower state circles. For a connected classical link diagram this constructs a connected closed orientable surface. Its Euler characteristic is $s_A+s_B-c(D)$, so $s_A+s_B\le c(D)+2$. For an alternating diagram the surface has genus zero.

A diagram is A-adequate if changing a single smoothing of its all-A state joins two different state circles, and B-adequate analogously. It is adequate if both conditions hold. In an adequate diagram the top and bottom degree terms of the Kauffman bracket arise uniquely from these two states. Reduced alternating diagrams are adequate because their Tait graphs and duals have no loop edges.

A link diagram is alternating if overpassing and underpassing crossings alternate along every component. Its checkerboard smoothing choices are consistent across all crossings of a connected projection. An alternating knot diagram is the one-component case.

For $d\ge 2$, nonzero roots define projective-point coordinates $s=X^{q-1}$, satisfying $s^N+s+T=0$, where $N=(q^d-1)/(q-1)$. This polynomial is irreducible over ${\mathbb{F}}_q(T)$. After fixing one root $s$, the remaining coordinates satisfy $R(Y,s)=(Y^N-s^N)/(Y-s)+1$. Substitute $z=Y/s$ and $h=1/s$: the equation becomes $h^{N-1}+(z^N-1)/(z-1)=0$. At a nontrivial $N$th root of unity the second term has a simple zero, so the polynomial is Eisenstein over the algebraic closure's rational function field. This proves irreducibility of $R$ and hence two-transitivity on projective points.

For even $d=2n$ with $n\ge 2$, the roots of $X^{q^d}+X^{q^2}+TX$ form an $n$-dimensional vector space over ${\mathbb{F}}_{q^2}$. Ratios $(c\alpha )/\alpha$ for a nonzero root $\alpha$ show that the splitting field contains every $c\in {\mathbb{F}}_{q^2}$. The subgroup fixing that quadratic constant extension is normal of index two and lies in $GL(n,q^2)$; applying projective transitivity over ${\mathbb{F}}_{q^2}(T)$ makes it contain $SL(n,q^2)$. It therefore normalizes that special linear group, while the full group acts semilinearly.

On a smooth projective curve of genus two, $D=K+P+Q$ has three sections and no base points by the Riemann-Roch theorem. Its map to ${\mathbb{P}}^2$ is nondegenerate and has mapping degree times image degree four. The only nonbirational possibility is a double cover of a conic. Every degree-two divisor with two sections is canonical, again by Riemann-Roch theorem, so that possibility forces $D\sim 2K$, or $P+Q\sim K$. If this equivalence fails, the image is a quartic and the map is birational. The quartic is singular because a smooth plane quartic has genus three.

On a smooth variety over a field of characteristic zero, a general member of a basepoint-free linear system of divisors is smooth. More generally it is smooth outside the base locus. Irreducibility is a separate assertion and requires suitable hypotheses on the associated morphism; a pencil composed with a map to a curve can have disconnected general divisors.

A focus is a planar equilibrium whose linearization has nonreal complex-conjugate eigenvalues with nonzero real part. The imaginary part produces local rotation, while the real part determines growth or decay of its envelope. A stable focus has negative real part; a focus with positive real part repels nearby trajectories.

The linear system $\dot{x}=-\alpha x-\omega y$, $\dot{y}=\omega x-\alpha y$ with $\alpha >0$ and $\omega \ne 0$ has radius $r(t)=r(0)e^{-\alpha t}$ and angular advance $\omega t$. It therefore spirals inward, combining two measurable scales: rotation period $2\pi /|\omega |$ and decay time $1/\alpha$. A damped mechanical oscillator can exhibit this geometry in its position-velocity plane. Eigenvalues with zero real part require a separate nonlinear analysis and are excluded from this hyperbolic focus definition.

A node is a planar hyperbolic equilibrium whose two linearized eigenvalues are real and have the same sign. Negative eigenvalues give a stable node; positive eigenvalues give an unstable node. Nearby motion approaches or recedes without the repeated rotation characteristic of a focus.

For $\dot{x}=-ax$, $\dot{y}=-by$ with $a,b>0$, solutions are $x=x_0{e}^{-at}$ and $y=y_0{e}^{-bt}$. Each coordinate decays, while unequal decay rates usually make a trajectory align with the slower direction as it nears the origin. Reversing both signs gives outward motion. Repeated eigenvalues require attention to the number of independent eigenvectors and can add polynomial factors to the exponential response. The node classification describes local behavior near an equilibrium; it does not determine the geometry of remote orbits.

In a normed vector space, the closed linear span of $S$ is the closure of its linear span in the norm topology. It includes limits of finite linear combinations. In a Hilbert space, an infinite orthonormal basis spans in this closed sense, allowing norm-convergent series in addition to finite sums.

In resistive-force theory with $0<\gamma <1$, head drag $\delta \ge 0$ and the ideal constant-inclination condition ${\alpha }^2=\beta$, put $r=1-\gamma$, $d=1+\delta$, $A=r({\delta }^2-\gamma )$ and $B=\gamma d^2$. At prescribed swimming speed $U$, eliminating wave speed using linear resistive-force propulsion of a travelling filament givesIts derivative has numerator $A{\beta }^2+2B\beta -B$. This numerator is strictly increasing on $(0,1)$, is negative at zero, and equals $(\delta +\gamma {)}^2$ at one. Thus its unique zero is the displayed global minimum. When $A=0$ the formula gives ${\beta }_{\ast }=1/2$; when $\delta =0$ it gives $1/(1+\sqrt{\gamma })$. A planar sawtooth realizes constant inclination in the ideal model; smooth rounding and elastic costs change that idealization.

For the homogeneous linear ordinary differential equation ${\sum }_{j=0}^n{a}_j{y}^{(j)}=0$, substitute $y=e^{rx}$. Dividing by the nonzero exponential gives the polynomial equation $P(r)={\sum }_j{a}_j{r}^j=0$. A simple root gives an exponential solution; a root of multiplicity $m$ gives $x^k{e}^{rx}$ for $0\le k<m$. Real equations admit real sine and cosine combinations of conjugate roots. These characteristic roots determine growth, decay and oscillation of the linearized modes.

For a nonzero vector $v$ and nonzero linear functional $f$ vanishing on $v$, this invertible map fixes the hyperplane $\ker f$ pointwise and has determinant one. Its inverse is $I-vf$, since $(vf{)}^2=0$. A symplectic transvection is obtained by taking $f=cB(-,v)$ for a nondegenerate alternating bilinear form.

The rank of a linear map is the dimension of its image of a linear map. In finite-dimensional spaces it equals the rank of a matrix representing the map in any bases.

For endomorphisms $S,T$ of the same finite-dimensional vector space, $ST=I$ makes $S$ surjective and $T$ injective. The rank-nullity theorem makes both invertible, hence $TS=I$. The finite-dimensional and equal-domain assumptions matter; unilateral shift operators on an infinite-dimensional sequence space give one-sided inverses that are not two-sided.

For a linear map $\alpha :U\to V$ between finite-dimensional vector spaces, take $\beta \in \mathrm{Hom}(V,U)$. The vector subspace $M^l(\alpha )=\{\beta :\beta \alpha =0\}$ consists of the linear maps factoring uniquely through $V/\mathrm{im}\alpha$, whereas $M^r(\alpha )=\{\beta :\alpha \beta =0\}$ consists of those whose image of a linear map lies in $\ker \alpha$. ThusWriting $u=\dim U$, $v=\dim V$ and $r=\mathrm{rank}\alpha$, their dimensions are $(v-r)u$ and $(u-r)v$. Passing to the dual map reverses composition and interchanges the corresponding left and right conditions. The finite-dimensional hypothesis makes these dimension differences ordinary integer subtraction.

A finite vector family is minimally dependent if it is linearly dependent and every proper subfamily is linearly independent. For $n\ge 2$, the family $e_1,\dots ,e_{n-1},{\sum }_{j=1}^{n-1}{e}_j$ in ${\mathbb{R}}^n$ is an example. The sum supplies a nonzero dependence. If the sum vector is omitted, the remaining vectors are a coordinate basis; if $e_j$ is omitted, its coordinate in any proposed relation forces the coefficient of the sum vector to vanish, then all other coefficients vanish. Every smaller proper subfamily lies in one of these independent $(n-1)$-element subfamilies.