For the defining sl2 Lie algebra module with basis $x,y$, put $A_k=x\otimes x^{3-k}{y}^k$ and $B_k=y\otimes x^{3-k}{y}^k$. The raising operator has $e{A}_k=k{A}_{k-1}$ and $e{B}_k=A_k+k{B}_{k-1}$; the lowering operator has $f{A}_k=B_k+(3-k)A_{k+1}$ and $f{B}_k=(3-k)B_{k+1}$. The highest-weight vectors $A_0$ and $B_0-A_1$ generate irreducible submodules of highest weights four and two. Bases are respectively $(A_0,3A_1+B_0,A_2+B_1,A_3+3B_2,B_3)$ and $(B_0-A_1,B_1-A_2,B_2-A_3)$. Their vectors are independent in every weight space, proving the displayed direct sum without only comparing dimensions.

A uniform cylindrical volume charge relaxes exponentially under Ohm's law and the continuity equation for electric charge. In a cylinder of radius $R$ initially carrying no surface charge, the radial conduction current deposits ${\rho }_s(t)=R{\rho }_0(1-e^{-\sigma t/{ϵ}_0})/2$ on its surface. Total charge per unit length remains $\pi R^2{\rho }_0$, so the exterior radial electric field is time independent even though the interior field decays. Neglecting the deposited surface charge violates charge conservation.

In the realization $\Phi (C_3)=\{\pm 2e_i,\pm e_i\pm e_j:i<j\}$, each coordinate two-plane cuts out a C2 root system. Each plane $\pm x_1\pm x_2\pm x_3=0$ cuts out an A2 root system; there are four such planes after identifying opposite normals. Thus the irreducible full rank-two root subsystems comprise three of type $C_2$ and four of type $A_2$. For an $A_2$ simple pair $p,q$, equal lengths and angle $120$ degrees give roots $\pm p,\pm q,\pm (p+q)$. For a mixed-length simple pair $s,l$ with $\Vert l{\Vert }^2=2\Vert s{\Vert }^2$ and $s\cdot l=-\Vert s{\Vert }^2$, the B2 root system has roots $\pm s,\pm l,\pm (s+l),\pm (2s+l)$. These lists also verify that the selected pairs are simple, rather than simply independent.

Under the hypotheses of Bishop volume comparison with a positive-curvature model, the Bonnet-Myers theorem gives diameter at most $D$. If total Riemannian volume equals $v_k(D)$, every comparison ball ratio is one. In a small normal ball the pointwise radial density bound therefore becomes equality. Writing $h={\partial }_r\log J$, the traced radial Riccati equation for distance spheres and Cauchy-Schwarz inequality force the radial operator to equal $(s_k^{\prime }/s_k)I$. The angular metric is consequently $s_k(r{)}^2{g}_{S^{n-1}}$, so the metric is locally round. The classification of complete positive constant-curvature manifolds gives a quotient $S_k^n/\Gamma$, whose volume is $v_k(D)/|\Gamma |$. Equality forces $\Gamma$ trivial.

For an $SU(2)$ bundle of charge $k$ on a closed oriented four-manifold, $\mathrm{ind}{D}_A=h_A^1-h_A^0-h_A^2=8k-3(1-b_1+b^{+})$. Equivalently it is $-2p_1(\mathrm{ad}E)[X]-3(\chi +\sigma )/2$, using $p_1(\mathrm{ad}E)=-4c_2(E)$. The Euler characteristic of the deformation complex has the opposite sign.

// Destination: analysis.bigb

On a simply connected negative-definite four-manifold, an ASD connection of full circle holonomy has $h_A^0=1$. If $h_A^2=0$, the index formula gives $h_A^1=8k-2$. The neutral diagonal deformation space vanishes because $b_1=0$, leaving a complex vector space of dimension $4k-1$ with stabilizer action $z\mapsto {\lambda }^{2}z$. Kuranishi reduction and the Coulomb slice give the local quotient ${\mathbb{C}}^{4k-1}/S^1$, equivalently the cone on ${\mathbb{CP}}^{4k-2}$.

// Destination: geometry-and-topology.bigb

For $A_n$, the least non-linear ordinary degree is $3$ at $n=4,5$ and $n-1$ for $n\ge 6$. For $n\le 3$ no non-linear irreducible exists. The degree-$n-1$ representation is unique for $n\ge 2$, except $n=3$ where it does not exist and $n=6$ where two distinct degree-five irreducibles occur. The proof uses tableau branching to bound the degrees of nonstandard shapes and of the half-degree constituents of self-conjugate shapes, with the small cases checked by the hook-length formula.

Every nonconstant meromorphic function on the complex plane has a simple island over at least one of any five Jordan domains on the Riemann sphere with disjoint closures. The Ahlfors second fundamental theorem, area comparison over target domains, and a length-area exhaustion of the complex plane prove this: without a simple island every counted island has degree at least two, forcing $3S\le (5/2)S+O(L)$.

Choosing $N$ different complete one-particle states from an orthonormal basis of size $M$ gives $\left(\genfrac{}{}{0ex}{}{M}{N}\right)$ independent Slater determinants. This is the binomial coefficient because only the occupied subset matters: permutations change the sign of the Slater determinant, not the physical quantum state. Summing over particle numbers gives $2^M$ states in the finite fermionic Fock space.

For the diagram's two spatial orbitals $A,B$, take energies $0,\Delta$ with $\Delta >0$, each independent of spin. In the complete-state order $(A\uparrow ,A\downarrow ,B\uparrow ,B\downarrow )$, the six $N=2$ occupation number strings are $1100,1010,1001,0110,0101,0011$. Their total energies are $0,\Delta ,\Delta ,\Delta ,\Delta ,2\Delta$, giving energy multiplicities $1,4,1$. For $N=3$, the ground state energy is $\Delta$ with two independent ground states, according to which spin orbital in $B$ is occupied. Across $N=0,1,2,3,4$, the counts are $1,4,6,4,1$, summing to $16$. These are exact counts for this model, not measured atomic energies.

For a zero-mean magnetic shearing wave, the real wavevector satisfies $k_x=k_{x0}+2A{k}_{y}t$, with $k_y,k_z$ constant. Dotting the velocity-amplitude equation with its complex conjugate eliminates Coriolis force and pressure work, giving $d|\tilde{\mathbf{v}}{|}^2/dt=4A\mathrm{Re}({\tilde{v}}_x{\tilde{v}}_y^{\ast })-2\nu k^2|\tilde{\mathbf{v}}{|}^2$. The inequality $2|v_x{v}_y|\le |\mathbf{v}{|}^2$ proves the displayed bound; the magnetic bound has the same form with $\eta$ replacing $\nu$. NowFor $A{k}_y\ne 0$ and positive kinematic viscosity and magnetic diffusivity, the negative cubic exponent dominates the linear shear-work bound, so both amplitudes tend to zero. Transient growth is allowed. Positivity matters: an inviscid vertical velocity mode with $k_z=0$ can remain constant, and an ideal vertical magnetic mode with $k_z=0$ can also remain constant.

For a small Cartesian closed category, the Yoneda embedding into its presheaf category preserves exponential objects. At $C$, the exponential in a presheaf category has value $\mathrm{Nat}(YC\times YA,YB)$. Since $YC\times YA\cong Y(C\times A)$, the Yoneda lemma identifies this with $\mathcal{C}(C\times A,B)\cong \mathcal{C}(C,B^A)$. Every identification is natural in all three objects.

For $f:A\to B$, the Yoneda embedding sends $f$ to postcomposition $\mathcal{C}(-,A)\to \mathcal{C}(-,B)$. If this is epic in the presheaf category, its component at $B$ is surjective, so $1_B=fg$ for some $g:B\to A$. Conversely such a section makes the natural transformation split epic. Pointwise surjectivity follows directly by doubling the target presheaf along the image subpresheaf: the two inclusions agree on the image but differ on any element outside it.

For $q\ge 1$, finite random-cluster measures with all boundary graph vertices identified have a weak infinite-volume limit of a sequence. On a regular lattice it is invariant under translations and lattice reflections and has positive association of the random-cluster model. Its conditional single-edge open probability lies between $p/(p+q(1-p))$ and $p$. Connection events can be approximated by increasing finite-volume path events, so their positive-association inequalities pass to the limit of a sequence.

For a root system with $N$ positive roots, put $A_{\nu }(H)={\sum }_{w\in W}\det (w)e^{(w\nu ,H)}$. Its degree-$m$ Taylor coefficient is alternating separately in $\nu$ and $H$, and hence divisible in each set of variables by the product of positive-root linear forms. Thus it vanishes for $m<N$ and its degree-$N$ term is $C{\prod }_{\alpha >0}(\nu ,\alpha ){\prod }_{\alpha >0}(H,\alpha )$. The Weyl denominator formula at $\nu =\rho$ determines the nonzero constant $C$. Taking the ratio $A_{\lambda +\rho }(tH)/A_{\rho }(tH)$ for regular $H$ and letting $t\to 0$ gives the Weyl dimension formula $\dim L(\lambda )={\prod }_{\alpha >0}(\lambda +\rho ,\alpha )/(\rho ,\alpha )$. Factors converting roots to coroots cancel.

The Weyl denominator is the alternating expression in the Weyl denominator formula. Its product form uses the Weyl vector $\rho$, half the positive-root sum. On a compact maximal torus, the factor $e^{\rho }$ has unit modulus; if it is only defined on a covering torus, the squared modulus still descends. Thus $|\Delta (t){|}^2={\prod }_{\alpha >0}|1-\alpha (t){|}^2$ is a globally defined density in the Weyl integration formula. For the unitary group an integer central shift gives the Vandermonde determinant $\det (z_i^{n-j})$, whose modulus is the same.

For a representation on which a commuting torus or Cartan action is semisimple, the weight decomposition is the direct sum of simultaneous weight spaces. A finite-dimensional representation of a compact torus can be made unitary by Haar measure averaging; its commuting unitary matrices are simultaneously diagonalizable. Each diagonal character is an integral weight of a representation. The decomposition makes tensor and exterior-power weights additive. For nonsemisimple actions, simultaneous generalized eigenspaces give a generalized weight decomposition instead.

A global ordered framing selects the displayed $\left(\genfrac{}{}{0ex}{}{n}{k}\right)$-sheeted cover inside the unordered Stiefel bundle. It is a trivial cover because the frame labels are global. Selecting subsets of a direct-sum framing decomposes the cover as the disjoint union of fiber products of the two summands' subset covers.

In a local parallel orthonormal frame a holomorphic section has holomorphic coefficients $f_j$. Its squared norm satisfies ${\partial }_z{\partial }_{\bar{z}}{\sum }_j|f_j{|}^2={\sum }_j|f_j^{\prime }{|}^2\ge 0$. On a compact connected complex curve, integration of the Laplacian makes the right side zero. Thus every coefficient is locally constant and the section is parallel. A global parallel section is determined by a vector fixed by all monodromy operators, and every such vector gives a global section.

The group of unitary bundle gauge transformations under composition. It acts on connections and their solution spaces. The stabilizer of a connection is the centralizer of its holonomy acting by parallel automorphisms, so a quotient can have singularities when this stabilizer increases.

// Destination: geometry-and-topology.bigb

Set the lower half-space load to $q_b={\mu }_b{p}_{\beta ,b}$ using the causal vertical wavenumber branch, which excludes a wave arriving from depth infinity. Apply the uniform-layer SH impedance update upwards through every layer, keeping particle velocity and traction continuous. For upper half-space impedance $q_a$, incident velocity $V_0$ and reflected velocity $R{V}_0$, the top values are $V=V_0(1+R)$ and $T=-q_a{V}_0(1-R)$. Their ratio gives the displayed reflection coefficient, incorporating every underlying interface and multiple reflection.