During binary mass-ratio reversal, an initially more massive donor star loses enough material that its companion becomes the more massive member. The stars retain their different prior nuclear evolution, resolving the Algol paradox. Conservative binary mass transfer shrinks the orbit before equal masses and widens it after reversal, because $a\propto (M_d{M}_a{)}^{-2}$ at fixed total mass and orbital angular momentum.

The Algol paradox is the apparently inconsistent evolutionary ordering in an Algol binary: the less massive star is more evolved, whereas higher-mass isolated main sequence stars normally exhaust core fuel sooner. Binary mass-ratio reversal resolves it by identifying the present donor as the initially more massive star.

The Alfvén frequency is the frequency of an Alfvén wave with wavevector $\mathbf{k}$ in a uniform magnetic field $\mathbf{B}$ and mass density $\rho$. For a vertical field and vertical wavenumber $k$, its square is ${\omega }_a^2=k^2{B}_z^2/({\mu }_0\rho )$. Its competition with orbital shear determines the ideal magnetorotational dispersion relation.

The Alfvén radius characterizes where a magnetized stellar wind crosses its Alfvén surface. For a roughly radial wind, $v_{\mathrm{wind}}\simeq B/\sqrt{{\mu }_0\rho }$ and ${\dot{M}}_w\simeq 4\pi R_A^2\rho v_{\mathrm{wind}}$. The wind's angular-momentum lever arm is of order $R_A$: its torque is $\dot{J}\sim -{\dot{M}}_w\Omega R_A^2$, up to geometry factors. The precise wind invariant uses the cylindrical radius of the surface crossing, rather than necessarily its spherical radius.

Named after George Biddell Airy, the standard Airy functions solve the Airy ordinary differential equation. Their Wronskian is $\mathrm{Ai}(Z){\mathrm{Bi}}^{\prime }(Z)-{\mathrm{Ai}}^{\prime }(Z)\mathrm{Bi}(Z)=1/\pi$, so they form a fundamental solution pair, including for complex $Z$.

For the Airy ordinary differential equation $y^{\prime \prime }=xy$, a power series obeys $a_2=0$ and $a_{m+3}=a_m/[(m+3)(m+2)]$. Setting its first two coefficients to $(1,0)$ or $(0,1)$ givesEmpty products are one. The ratio test proves convergence on the whole complex plane. Termwise differentiation verifies the equation, and the initial normalization gives Wronskian one. By the Abel identity, the Wronskian remains one everywhere, so these solutions span the whole solution space.

Connections ${\nabla }^1,{\nabla }^2$ on $T{M}_1,T{M}_2$ give a connection onby taking the direct sum of their pullback connections. In product coordinates its Christoffel symbols have the two factor blocks and zero mixed blocks. On arbitrary fields, the ordinary derivatives of their coefficients are still taken in both factors. On fields lifted separately from the factors it satisfies ${\nabla }_{Y_1+Y_2}(X_1+X_2)={\nabla }_{Y_1}^1{X}_1+{\nabla }_{Y_2}^2{X}_2$.

A SU(2) representation descends to the SO(3) group exactly when $-I$ acts trivially. In ${\mathrm{Sym}}^n({\mathbb{C}}^2)$ it acts by $(-1{)}^n$, so descent holds precisely for even $n$, or integer spin $j=n/2$. This is the inflation of a group representation criterion for the kernel $\{\pm I\}$. Half-integer-spin representations instead remain representations of the covering group.

Substitution of the adiabatic superbubble similarity solution into spherical mass conservation gives

If the similarity profiles account for all kinetic and internal energy inside an adiabatic strong shock, letThe injected energy $\dot{E}t$ equals $4\pi {\rho }_1{\dot{R}}^2{R}^{3}I$. Consequently conservation of energy fixesAn interior region not included in the profiles must have its energy added separately; a radiatively cooled shell changes this energy budget.

For point injection of constant power, the outward energy flux through a sphere tending to the source must tend to $\dot{E}$. In the adiabatic superbubble similarity solution, this requiresImposing finite regular central pressure and density with $h=O(\eta )$ would instead give zero flux and fail to supply the prescribed power. A resolved source region or an inner contact-boundary model can replace the point-source condition.

If $\Vert P{\Vert }_{L_0^2(\mu )}\le r<1$, a stationary reversible Markov chain satisfiesConsequently the variance of a sum of $N$ consecutive values is at most $N(1+r)/(1-r)$ times the stationary variance of one value. This bounds the Markov chain Monte Carlo asymptotic variance uniformly over square-integrable functions.

Interchanging two Abelian Higgs vortices replaces $q=z_1-z_2$ by $-q$. Thus $w=q^2$ is a single-valued relative coordinate on the unordered Abelian Higgs vortex moduli space. Coincidence is smooth in $w$; a geodesic crossing from positive real $w$ to negative real $w$ changes the line of separation by $\pi /2$, explaining right-angle scattering of Abelian Higgs vortices. The asymptotic cone in the $q$ coordinate has angular period $\pi$ and must not be treated as a singularity of the actual coincidence metric.

Fix a positive integer $e$. A partition of an integer is uniquely specified by its core of a partition for modulus $e$ and its ordered quotient of a partition for that modulus. More precisely, for every $e$-core $\gamma$ and ordered tuple $({\nu }_0,\dots ,{\nu }_{e-1})$ of partitions, there is a unique partition $\lambda$ such thatThe ordered runners use the usual abacus of a partition convention with a number of beads divisible by $e$. Adding $e$ beads preserves both the tuple and its runner labels. The core fixes the relative runner offsets, and each component partition fixes the displacement of the beads on that runner, giving existence and uniqueness.

For each real $c$, some maximizer of $e(G)-c{k}_3(G)$ among graphs on $n$ vertices is complete multipartite graph. Among maximizers maximize ${\sum }_{v}d(v{)}^2$. Cloning either of two nonadjacent vertices preserves the objective, since maximality forces their local contributions $d(v)-ce(G[N(v)])$ to agree. If their vertex neighbourhoods differ, the sum of the two changes in the squared-degree of a vertex objective is twice the size of their symmetric difference, a contradiction. Nonadjacency therefore partitions the vertices into classes with identical vertex neighbourhoods.

For $r=n+1/2$, $n\ge 0$, the convergent frequency zeta function, expressed through the Riemann zeta function, is ${\sum }_r{r}^{-s}=(2^s-1){\zeta }_R(s)$ when $\mathrm{Re}s>1$. Its analytic continuation at $s=-1$ gives $(1/2-1)(-1/12)=1/24$. A common exponential frequency cutoff gives ${\sum }_{r}r{e}^{-\epsilon r}={\epsilon }^{-2}+1/24+O({\epsilon }^2)$, so the same finite part follows by subtracting the leading divergence. In contrast, integer frequencies have regulated sum $-1/12$.

In a pointed category, the zero morphism $0_{A,B}:A\to B$ is the composite through its zero object. Composition on either side by any morphism remains zero. In a commutative-monoid enrichment on a pointed category, the additive zero must coincide with this zero morphism: morphisms to and from the zero object belong to singleton hom-sets.

For known size $r$ and ${\lambda }_i=e^{x_i^T\beta }$, the expectation-maximization algorithm gives structural-zero responsibility $t_i=0$ for positive counts and $t_i=\pi /[\pi +(1-\pi )(r/(r+{\lambda }_i){)}^r]$ for zero counts. This is Bayes' theorem. Maximizing the expected complete-data log-likelihood gives ${\pi }_{\mathrm{new}}=n^{-1}{\sum }_i{t}_i$ and a weighted negative binomial regression with weights $1-t_i$. The objective separates into a Bernoulli mixing term and a weighted count term, which proves the update.

ConsiderIts row sums vanish, so $(1,1,1{)}^T$ is an eigenvector for the zero eigenvalue. Its characteristic polynomial iswith sum of principal minors of order twoThe other eigenvalues are $(-S\pm \sqrt{S^2-4T})/2$. No symmetry is required for this formula; positive diagonal symmetrization of a matrix is a sufficient condition for real roots.

A semistandard Young tableau fills a Young diagram with positive integers, permitting repeated entries, so that entries weakly increase along rows and strictly increase down columns. Its content records the number of occurrences of each entry. The Kostka number $K_{\lambda \mu }$ counts such fillings of shape $\lambda$ and content $\mu$.