For $s=|z_1{|}^2+|z_2{|}^2>0$ and $a>0$, a geometric Kähler potential with derivative ${\phi }^{\prime }(s)=\sqrt{s^2+a^2}/(2s)$ defines a Kähler metric on ${\mathbb{C}}^2\setminus \{0\}$. Its tangential and radial eigenvalues are $\sqrt{s^2+a^2}/(2s)$ and $s/(2\sqrt{s^2+a^2})$. Their product is $1/4$, giving the same Riemannian volume form as ${\omega }_0=(i/2)\sum d{z}_j\wedge d{\bar{z}}_j$, although the metric is different. This construction makes preservation of volume compatible with anisotropic stretching.

For the Chern connection of a Hermitian holomorphic vector bundle on a Kähler manifold, the two Dolbeault Laplacians differ by $[iF\wedge ,\Lambda ]$, where $F$ is its curvature form of a connection and $\Lambda$ is the adjoint Lefschetz operator. The curvature term controls vanishing estimates.

For a closed real $(1,1)$ form, the Poincaré lemma gives a real one-form $\eta$ with $d\eta =\omega$. Its $(0,1)$ part is $\bar{\partial }$-closed, so the Dolbeault-Poincaré lemma gives ${\eta }^{0,1}=\bar{\partial }\psi$. Reality gives ${\eta }^{1,0}=\partial \bar{\psi }$, hence $\omega =i\partial \bar{\partial }(2\mathrm{Im}\psi )$. Positivity is a separate requirement for this form to define a metric.

An indicial exponent is a root of the indicial equation at a regular singular point. It gives the leading local power in a Frobenius method; resonant roots can require logarithmic terms.

An indicial root is a root of the indicial equation for a regular singular point. For a leading term $x^r$ as $x\downarrow 0$, its magnitude is $x^{\mathrm{Re}r}$, so positive real part gives decay and negative real part gives divergence. At zero real part, logarithmic factors and oscillation must also be checked.

For $x^2{y}^{\prime \prime }+x{y}^{\prime }=0$, the repeated root is zero. Its independent solutions $1$ and $\log x$ have the same indicial root but different boundedness. For roots $r=\pm i\mu$, nonzero real solutions oscillate in $\log x$ with bounded magnitude and usually no limit at zero. Thus the sign of a real root is useful away from the borderline, but is not a complete boundedness rule for all Frobenius solutions.

For a closed point $P$ with residue field $k(P)$, evaluation gives a morphism from the structure sheaf onto the skyscraper sheaf $k_P$. Its kernel is the ideal sheaf of $P$. On a Noetherian scheme it is a coherent ideal sheaf.

At a noncorner cell of a Young diagram, this random walk chooses uniformly one of the other cells in its hook of a Young diagram. It stops at a corner. Every step moves right or down, so it terminates. Its corner probabilities provide a probabilistic proof of the hook-length formula.

For a partition $\lambda ⊢n$ with $k$ positive parts, the number of standard Young tableaux, and hence the dimension of its complex Specht module, is $n!\det (1/({\lambda }_i-i+j)!)$. Negative factorial arguments give zero entries. Setting $L_i={\lambda }_i+k-i$ turns the determinant into ${\prod }_{i<j}(L_i-L_j)/{\prod }_i{L}_i!$ by the Vandermonde determinant, matching the hook-length formula.

On a complex manifold, ${\Omega }_X^p$ is the sheaf of holomorphic degree-$p$ forms. It is the kernel of the Dolbeault operator on ${\mathcal{A}}^{p,0}$. The Dolbeault theorem computes its sheaf cohomology using smooth forms.

If two simple roots have angle $5\pi /6$, normalize them to unit vectors $u,v$ with $(u,v)=-\sqrt{3}/2$. For any other unit simple root $w$, set $r=(u,w)\le 0$, $s=(v,w)\le 0$. Its three-root Gram matrix has determinant $1/4-r^2-s^2-\sqrt{3}rs$. Any nonzero $r$ or $s$ has absolute value at least $1/2$, contradicting positive definiteness. Thus all other simple roots are orthogonal to $u,v$. An irreducible root system containing this triple bond must be the rank-two G2 root system.

Using the extensional screening length of a slumping column, an initially uniform column of length $\Lambda$ obeys $W_{ZZ}=1+W$, $W(0)=0$ and $W_Z(\Lambda )=0$. Thus $W=\cosh (\Lambda -Z)/\cosh \Lambda -1$ and its initial area-fraction growth is ${\varphi }_T={\varphi }_0\sinh (\Lambda -Z)/\cosh \Lambda$. For $\Lambda \gg 1$, the velocity is $e^{-Z}-1$ outside exponentially small end corrections; for $\Lambda \ll 1$, it is $Z^2/2-\Lambda Z$. The first limit balances excess weight with annular hydraulic drag, the second with extensional viscous stress.

For a finite connected unweighted loopless graph with $n$ graph vertices,The edge-inclusion formula for a uniform spanning tree gives the left-hand side as $\mathbb{E}|T|$, and every spanning tree has $n-1$ edges. On the same finite connected graph with positive edge conductances $c_e>0$, the corresponding identity is ${\sum }_e{c}_e{R}_{\mathrm{eff}}(e)=n-1$.

For $n\ge 1$, the finite Dynkin diagram $A_n$ is a chain of $n$ vertices joined by single bonds. It represents the root system of ${\mathfrak{sl}}_{n+1}$.

For $n\ge 2$, the finite Dynkin diagram $C_n$ is a chain with a terminal double bond. Its arrow points from the terminal long root toward the adjacent short root. It represents the Cn root system; $C_2$ is isomorphic to $B_2$ after relabelling.

For $n\ge 4$, the finite Dynkin diagram $D_n$ is a simply laced tree with one trivalent vertex and three arms containing $1,1,n-3$ edges. All its roots have the same length.

The finite Dynkin diagrams $E_6,E_7,E_8$ are simply laced trees with one trivalent vertex. Their three arm lengths, counting edges, are respectively $(1,2,2)$, $(1,2,3)$ and $(1,2,4)$.

The finite Dynkin diagram $F_4$ is a four-vertex chain whose central bond is double. The two consecutive long roots lie on one side, the two consecutive short roots on the other, and the arrow points toward the short pair. The remaining two bonds are single.

The finite Dynkin diagram $G_2$ has two vertices joined by a triple bond, with the arrow toward the short root. The squared root-length ratio is three. It represents the G2 root system.

The underlying graph of a finite Dynkin diagram is a forest. Normalize every simple root to length one. Each bonded pair has inner product at most $-1/2$, and every other distinct pair has nonpositive inner product. A cycle with $m$ vertices would make the squared norm of their sum at most $m-m=0$, contradicting their linear independence in an inner product space. Multiple bonds denote root-length information, not graph-theoretic parallel-edge cycles.

The Dolbeault-Poincaré lemma makes this a locally exact complex of sheaves, with degree-zero kernel the holomorphic differential forms. The sheaves of smooth forms are fine sheaves by a partition of unity. The acyclic resolution theorem therefore computes the cohomology of ${\Omega }^p$ by the global Dolbeault operator complex.