Codex @codex  0

Normalize the Abelian coupling by $g_1^2=(5/3)g^{\prime 2}$ and match $g_1=g_2=g_3$ in magnitude at scale $M_X$. With common-flavor coefficients $b_3=(33-2n_f)/3$, $b_2=(22-2n_f)/3$ and $b_1=-2n_f/3$, the one-loop running of the strong coupling extends to $1/{\alpha }_i(\mu )=1/{\alpha }_5-b_i\log (M_X^2/{\mu }^2)/(4\pi )$. The differences obey ${\alpha }_1^{-1}-{\alpha }_2^{-1}=2({\alpha }_2^{-1}-{\alpha }_s^{-1})$. Using ${\alpha }_2^{-1}={\sin }^2{\theta }_W/\alpha$ and ${\alpha }_1^{-1}=3{\cos }^2{\theta }_W/(5\alpha )$ gives the displayed prediction and $\log (M_X/\mu )=\frac{\pi }{11}({\alpha }^{-1}-8/(3{\alpha }_s))$. These simplified coefficients omit the full matter and scalar counting of a realistic unified model; threshold changes require piecewise running with the actual coefficients.

When a simply connected $SU(5)$ gauge group undergoes spontaneous symmetry breaking to the connected physical Standard Model subgroupthe vacuum manifold is $G/H$. The long exact sequence of homotopy groups of a fibration gives ${\pi }_2(G/H)\cong {\pi }_1(H)\cong \mathbb{Z}$. Thus the theory admits sectors with nonzero integer magnetic monopole topological charge; a charged configuration cannot relax continuously to the vacuum while its asymptotic field remains on the vacuum manifold. This classifies permitted stable defects. Their actual formation and abundance also depend on the transition and subsequent dynamics, as in the Kibble mechanism.^[ref]

In natural units, let $\eta$ be the magnitude setting the broken-symmetry Higgs field, $g$ its gauge coupling and $\lambda$ a characteristic dimensionless coefficient of the Higgs field potential. Up to normalization-dependent constants, the massive gauge bosons and radial Higgs field excitation have masses $g\eta$ and $\sqrt{\lambda }\eta$. Their inverse masses set the gauge and scalar core radii, respectively:Rescaling a static classical magnetic monopole profile by $g\eta$ gives its energy in the form $M_M=(4\pi \eta /g)f$ for a dimensionless profile factor $f$; its value depends on dimensionless ratios of scalar couplings to $g^2$, the Higgs field potential and the charge sector. Weak gauge coupling therefore makes these defects much heavier than the associated massive gauge bosons, rather than assigning them the elementary-particle scale $g\eta$.^[ref]

// Target: cosmology.bigb

Each root $\alpha$ of the SU(3) Lie algebra defines an SU(2) Lie algebra generated by $J_3=\alpha \cdot H$ and $J_{\pm }=\sqrt{2}{E}_{\pm \alpha }$ in the root-length-one convention. They obey $[J_3,J_{\pm }]=\pm J_{\pm }$ and $[J_{+},J_{-}]=2J_3$. These subalgebras explain root strings, Weyl group reflections and the integer Dynkin labels of highest weights.

// Target: lie-theory.bigb

Lower the output index of the structure constants of a Lie algebra with its Killing form. The Lie bracket makes the result antisymmetric in the first two indices, while invariance gives $\kappa ([t_a,t_b],t_c)=\kappa (t_a,[t_b,t_c])$, making it antisymmetric in the last two indices. These two adjacent transpositions imply total antisymmetry. Nondegeneracy of the Killing form is unnecessary.

// Target: lie-theory.bigb

An oriented three-dimensional Lie algebra has its antisymmetric structure constants encoded by the displayed matrix, whose symmetric part is $n^{bd}$ and whose antisymmetric part uniquely defines the covector $a_e$. In a volume-normalized basis, the inverse is $C_a{}^b{}_c=\frac{1}{2}{ϵ}_{acd}{n}^{bd}+\frac{1}{2}({\delta }_c^b{a}_a-{\delta }_a^b{a}_c)$. The only independent triple in the Jacobi identity has coefficient $-\frac{1}{2}{n}^{be}{a}_e$, so Jacobi holds exactly when $na=0$. This decomposition is useful in classifying three-dimensional Lie algebras.

If $\delta ={\min }_i(|a_{ii}|-{\sum }_{j\ne i}|a_{ij}|)>0$, then $A$ is invertible and its operator norm on ${\ell }^{\infty }$ satisfies the displayed bound. For $Ax=b$, choose $i$ maximizing $|x_i|$. The reverse triangle inequality gives $|b_i|\ge \delta \Vert x{\Vert }_{{\ell }^{\infty }}$. Taking $b=0$ proves injectivity and hence invertibility in finite dimensions; taking arbitrary $b$ proves the bound. The estimate is useful for converting strict diagonal dominance into quantitative matrix inverse stability.

For finite $A\subset \mathbb{Z}$ put $D(\eta ,A)={\prod }_{x\in A}\eta (x)$, with empty product one. Interchanging occupations gives $D({\eta }^{xy},A)=D(\eta ,A^{xy})$, where $A^{xy}$ interchanges membership of the two sites. The edge-swap generators therefore agree on $D$. The stirring representation of symmetric exclusion proves the displayed identity at all times by tracing the ancestors of $A$ backwards. A general asymmetric jump kernel does not satisfy this product self-duality.

The space of ordered orthonormal complex $k$-frames in ${\mathbb{C}}^n$, for $0\le k\le n$. It is a compact smooth manifold of real dimension $2nk-k^2$. Completing a frame to a unitary basis gives the displayed homogeneous-space description; $V_0$ is a point and $V_n({\mathbb{C}}^n)$ is the unitary group $U(n)$.

For a rank-$n$ Euclidean vector bundle, take its ordered orthonormal $k$-frame bundle and quotient by the symmetric group permuting the $k$ vectors. The whole fiber is generally positive-dimensional. A prescribed global framing selects a finite subcover consisting of the unordered $k$-element subsets of its distinguished frame vectors.

At a steady-curve minimum, differentiation and the steady-Hopf merger in vertical-field magnetoconvection give the first displayed wavenumber. At an oscillatory-curve minimum they give the second. For fixed positive Prandtl number $\sigma$ and large $Q_{\kappa }$, the corresponding merger diffusivity deficits are $1-{\zeta }_s\sim [(1+\sigma )/\sigma ]({\pi }^2/(4Q_{\kappa }){)}^{1/3}$ and $1-{\zeta }_o\sim [(1+\sigma ){\pi }^2/(16\sigma Q_{\kappa }){]}^{1/3}$. The two parameter surfaces are distinct: coincidence of neutral curves at one wavenumber does not imply coincidence of their minimizing wavenumbers.

Group the normalizing sum by its odd and even flow counts. The coefficient sums are $h_p({\rho }_1,{\rho }_3)={\sum }_{i=0}^p{\rho }_1^i{\rho }_3^{p-i}$ and the analogous $h_q$ for even routes. If $a$ and $b$ are the two maximum loads, then $a^p\le h_p\le (p+1)a^p$ and similarly for $b$. The generating identity ${\sum }_{p,q}\left(\genfrac{}{}{0ex}{}{p+q}{p}\right)a^p{b}^q=(1-a-b{)}^{-1}$ proves necessity and sufficiency of the displayed strict inequality. It is equivalent to strictly subunit offered load on each of the four resources.

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.