For offered loads on consecutive routes $a,b,c,d$ of the reversible proportionally fair allocation on a four-cycle, the normalizing sum of the stationary distribution isThis formula applies in the stability region for the reversible four-cycle flow network. Grouping populations on opposite routes gives ${\sum }_{s,t}\left(\genfrac{}{}{0ex}{}{s+t}{s}\right)h_s(a,c)h_t(b,d)$, where $h_s(a,c)={\sum }_{i=0}^s{a}^i{c}^{s-i}$. For distinct opposite loads, substituting $h_s=(a^{s+1}-c^{s+1})/(a-c)$ and summing the four geometric generating series proves the formula; continuity covers coincident loads. If all four loads equal $z<1/2$, it reduces to $(1-2z+2z^2)/(1-2z{)}^3$.

Here $n=|\lambda |-|\mu |$, with zero entries for negative factorial arguments. Extract the squarefree monomial $x_1\cdots x_n$ from the skew Jacobi–Trudi identity. A product $h_{r_1}\cdots h_{r_{\ell }}$ contributes $n!/\prod r_i!$ because its distinct labels are assigned to the factors in that many ways. Summing determinant signs proves the formula, since a squarefree semistandard Young tableau is a standard skew Young tableau.

For source-free Keplerian viscous diffusion equation, the variables $y=\sqrt{r}$ and $g=r^{3/2}\Sigma$ give $g_t=(3{\nu }_0/4)g_{yy}$ when $\bar{\nu }={\nu }_{0}r$. Rescaling y gives the displayed heat equation. Initial and radial boundary conditions must also be transformed; the reduction does not specify them.

For the torsion-free spin connection with $[{\nabla }_{\nu },{\nabla }_{\rho }]=R_{\nu \rho ab}{\gamma }^{ab}/4$, Clifford expansion and the first Bianchi identity give ${\gamma }^{\mu \nu \rho }{R}_{\nu \rho ab}{\gamma }^{ab}=4R^{\mu }{}_{\lambda }{\gamma }^{\lambda }-2R{\gamma }^{\mu }$. Thus the antisymmetric double derivative equals $G^{\mu }{}_{\lambda }{\gamma }^{\lambda }/2$. This is the shared algebra behind quadratic-order minimal four-dimensional supergravity invariance and the Nester two-form divergence.

For $\{{\gamma }_a,{\gamma }_b\}=2s{g}_{ab}$, the compatible spinor covariant derivative has curvature $R_{abcd}{\gamma }^{cd}/(4s)$. The torsion-free connection preserves Clifford multiplication, so the mixed derivative terms cancel for $D_a={\nabla }_a+c{\gamma }_a$. Its integrability is therefore $(R_{abcd}{\gamma }^{cd}+8s{c}^2{\gamma }_{ab})ϵ=0$. Changing antisymmetrization weight cannot change the relative coefficient. The common normalization uses $s=1$; $s=2$ gives the coefficient $16c^2$.

A steady galactic distribution function depending only on relative energy $E=\psi -v^2/2$ is isotropic and has zero mean velocity. If its one-dimensional velocity dispersion is constant, the Spherical Jeans equation reduces to ${\sigma }_0^2{\rho }^{\prime }=\rho {\psi }^{\prime }$, yielding the displayed density. This is a tracer relation in a specified gravitational potential, not a requirement that the tracer generate that potential. With no escape-energy cutoff its local velocity distribution is Maxwell-Boltzmann velocity distribution.

For a spherical nonrotating tracer with luminosity density $\lambda$, radial velocity dispersion ${\sigma }_r$, enclosed luminosity $L(r)$, and constant mass-to-light ratio ${\rm Y}$, let $q=\lambda {\sigma }_r^2$ and $\gamma =G{\rm Y}$. DefineThe Spherical Jeans equation and the projected second stellar velocity moment give $p=2A+R{A}^{\prime }$. To differentiate without a singular lower-end contribution, write $A(R)={\int }_0^{\infty }q(\sqrt{R^2+z^2})dz$. If $R^{2}A$ vanishes at both endpoints, ${\int }_0^{\infty }Rp(R)dR=0$. A finite outer aperture instead retains its boundary stress term.

For an isotropic self-gravitating collisionless stellar system with $k<3$ and no central point mass, $M(r)=4\pi A{r}^{3-k}/(3-k)$. Integrating the Spherical Jeans equation to an outer radius $b$ gives the displayed pressure, with $P_b=\rho (b){\sigma }_r^2(b)$. For $k<1$ the inner pressure is set by outer radii and ${\sigma }_r^2\sim C{r}^k$, with $C$ depending on the boundary. An unbounded pure power law has divergent pressure in this case. For $1<k<3$, inner radii dominate and ${\sigma }_r^2\sim 2\pi GA{r}^{2-k}/[(3-k)(k-1)]$. At $k=1$ a logarithm appears. For $k\ge 3$ the enclosed central mass diverges unless the density law is modified at small radii.

In a constant-speed spherical shell with outer radius $R$ and inner radius $a$, a symmetric single reflection from the inner boundary has the displayed travel time. The boundary lies on the angular bisector of source and receiver. The branch exists only for $0\le \Delta \le 2\arccos (a/R)$, since the incident line must remain in the shell. At the endpoint it joins the mantle-only chord branch with the same slope $a/c$. Continuing the formula beyond that endpoint would make the line intersect the inner boundary before its supposed reflection point.

For a spherically symmetric speed with strictly increasing $\eta (r)=r/v(r)$, a turning ray of parameter $p$ has $\Delta (p)=2p{\int }_p^{{\eta }_a}(d\log r/d\eta )({\eta }^2-p^2{)}^{-1/2}d\eta$. Integrate this against $(p^2-{\mu }^2{)}^{-1/2}$ and interchange the triangular integration region. The inner kernel integral is $\pi /2$, giving the displayed radius and then $v(r(\mu ))=r(\mu )/\mu$. Complete turning-ray branch data are required.

A meromorphic function on a punctured disk has finite pulled-back spherical area precisely when it extends to the Riemann sphere at the puncture. A removable singularity or pole gives a smooth spherical extension and finite area. At an essential singularity, the meromorphic version of the Great Picard theorem gives infinitely many preimages of all but at most two spherical targets. The area formula then makes the area infinite. The logarithmically weighted annular area divided by $\log (1/r)$ tends to the total unweighted area divided by $4\pi$, by averaging a nondecreasing area function.

For vectors of norm at least one, the number of signed sums in an open unit ball is at most the central binomial coefficient. In one dimension this follows from the antichain bound. In Euclidean spaces a separated block decomposition for vector subset sums supplies the same bound without assuming all vectors have positive projection in one common direction. Signed sums are counted with multiplicity, by their sign choices.

With the defining two-dimensional matrix trace and inner product $⟨X,Y⟩=-\mathrm{tr}(XY)$, the displayed Pauli matrix basis is orthonormal. The Pauli matrix multiplication law gives the indicated structure constants of a Lie algebra. With the alternative form $-2\mathrm{tr}(XY)$, the orthonormal basis is $-i{\sigma }_a/2$ and the structure constant is one. The bracket and inner-product normalizations must be kept consistent in projected gauge-field formulas.

For antisymmetric Maurer–Cartan forms with $d{\lambda }_{ij}={\lambda }_{ik}\wedge {\lambda }_{kj}$, let ${\alpha }^{\pm }=({\lambda }_{12}\pm {\lambda }_{43},{\lambda }_{23}\pm {\lambda }_{41},{\lambda }_{31}\pm {\lambda }_{42})$. Direct expansion gives $d{\alpha }_1^{\pm }=-{\alpha }_2^{\pm }\wedge {\alpha }_3^{\pm }$ and cyclic analogues, with no mixed terms. The dual triples form two commuting ideals, each with the SO(3) Lie algebra brackets. This is a real compact splitting, distinct from the splitting of the complexified algebra into two copies of ${\mathfrak{sl}}_2(\mathbb{C})$.

For the complexified Special orthogonal Lie algebra ${\mathfrak{so}}_{2n}$, $n\ge 2$, choose positive roots of a root system $e_i-e_j$ and $e_i+e_j$ for $i<j$. Its Weyl vector is $\rho =(n-1,n-2,\dots ,0)$. Dominant coordinates satisfy ${\lambda }_1\ge \cdots \ge {\lambda }_{n-1}\ge |{\lambda }_n|$. The Weyl dimension formula becomes ${\prod }_{i<j}(({\lambda }_i+{\rho }_i{)}^2-({\lambda }_j+{\rho }_j{)}^2)/({\rho }_i^2-{\rho }_j^2)$. The case $D_2$ is two independent A1 root systems, while $D_3$ is isomorphic to the A3 root system.

For a complex vector space with a nondegenerate symmetric bilinear form $Q$, the map $u\wedge v\mapsto [w\mapsto Q(v,w)u-Q(u,w)v]$ identifies the exterior square of the defining representation with the Adjoint representation of a Lie algebra of $\mathfrak{so}(V)$. Equivariance follows from invariance of $Q$; an orthogonal basis verifies bijectivity. It is irreducible when $\mathfrak{so}(V)$ is simple. In dimension four it splits into the adjoint modules of the two commuting sl2 Lie algebras.

For a real inner product space $V$, the displayed map identifies ${\Lambda }^{2}V$ with its Special orthogonal Lie algebra. It sends a wedge of distinct orthonormal basis vectors to the corresponding elementary skew-symmetric matrix, so it is an isomorphism. For an orthogonal transformation $g$, invariance of the inner product gives $\Phi (gu\wedge gv)=g\Phi (u\wedge v)g^{-1}$. Thus the exterior square of the defining representation is the Adjoint representation of the orthogonal group. Complexification identifies their weight spaces, allowing the root system to be read from the exterior-square weights and the zero-weight multiplicity.

Over a field of characteristic zero, an sl2 triple consists of elements of a Lie algebra satisfying the displayed brackets. It defines a homomorphism from ${\mathfrak{sl}}_2$ by sending its standard generators to $h,e,f$. A nonzero homomorphism is injective because ${\mathfrak{sl}}_2$ is a simple Lie algebra. Root vectors and their coroot give such triples in a semisimple Lie algebra, and their representations supply finite weight strings.

Construct countable levels of bounded increasing rational sequences, maintaining the bounded rational extension property. At successor stages append every allowed rational. At a countable limit, handle each earlier-node/rational-cap pair by a cofinal sequence of extensions with values approaching a rational strictly below the cap. Its union has that rational supremum. Countably many requirements give a countable new level. Strictly increasing labels exclude uncountable chains, and label fibers are tree antichains, so the result is a special Aronszajn tree.

For these constants, every real-wavenumber growth rate is $-k^2-1/10<0$. The real-contour impulse integral decays with envelope $e^{-t/10}{t}^{-1/2}$ at fixed position. Yet the stationary candidate $k=-2i/3$ has $\mathrm{Im}\omega =4/27-1/10>0$. It cannot be a contributing spatial pinch point; otherwise it would contradict the real-contour growth bound. This explicit example shows why solving the algebraic saddle equations is insufficient for absolute wave-packet instability.