For a finite-dimensional representation of a complex semisimple Lie algebra, select a maximal remaining dominant weight, subtract that multiplicity of the corresponding irreducible formal character, and repeat. The Weyl complete reducibility theorem guarantees nonnegative residual multiplicities and termination. Checking the full character or the exhausted total dimension confirms that no summands remain.

If $S\subseteq \kappa$ is stationary and $f(\alpha )\in A_{\alpha }$ for a kappa-filtration, then $f$ is constant on a stationary subset. Restrict to limit indices and use continuity to find a smaller stage containing each value. Fodor lemma fixes that stage on a stationary subset. Its size is less than $\kappa$, so club filter completeness makes one value fiber stationary.

A flat torus is ${\mathbb{R}}^d/\Lambda$ with its quotient Euclidean metric, where $\Lambda$ is a full-rank Euclidean lattice. Its volume is the lattice covolume. The dual lattice indexes its complete Fourier series of Laplacian eigenfunctions.

For a velocity Fourier mode with wavenumber $k$, define $a=\eta k^2$, $d=a^2+{\Omega }^2$ and $\mathbf{C}=i({\mathbf{B}}_0\cdot \mathbf{k})\hat{\mathbf{u}}$. A cosine-forced mode has periodic response coefficients $(\mathbf{p},\mathbf{q})=(a\mathbf{C},\Omega \mathbf{C})/d$, while a sine-forced mode has $(-\Omega \mathbf{C},a\mathbf{C})/d$. Homogeneous transients decay as $e^{-at}$. The periodic response is a particular solution, not arbitrary initial data.

For density proportional to ${\mathrm{sech}}^2(z/H)$, the magnetic vertical equation $F^{\prime \prime }+k^2{\mathrm{sech}}^2(z/H)F=0$ becomes the Legendre differential equation under $\xi =\tanh (z/H)$. Bounded solutions at both surfaces are Legendre polynomials, with $k_n=\sqrt{n(n+1)}/H$. The $n=0$ constant function has no magnetically coupled finite wavenumber. The smallest MRI-relevant eigenvalue is $k_1^2=2/H^2$, which makes sufficiently strong magnetic tension stabilize all allowed modes.

For morphisms of schemes $X\to S$, $Y\to S$, a pair of points $x,y$ lifts to a point of $X{\times }_{S}Y$ exactly when their base images agree. For sufficiency use the nonempty spectrum of a commutative ring of $\kappa (x){\otimes }_{\kappa (s)}\kappa (y)$, by tensor product of field extensions is nonzero. This establishes existence, not uniqueness, of a lift.

Opposite points on the orbit have opposite displacement and the same stresslet director up to sign. Their paired fields are $[G(\mathbf{r}-R\mathbf{p},\mathbf{e})+G(\mathbf{r}+R\mathbf{p},\mathbf{e})]/2$, so odd displacement powers cancel. Since a stresslet decays as $r^{-2}$, the first correction to its far-field orbit average of a tangent stresslet is $O(S{R}^2/(\mu r^4))$. The finite-radius mean need not be an exact point stresslet.

A measurable measure-preserving map intertwining the transformations of two probability measure-preserving systems. The equalities may be imposed on invariant full-measure domains. It represents the target's observable information inside the source. The conditional measures of a factor describe the information remaining on each fiber.

For canonical positive kinetic terms in global supersymmetry, $V={\sum }_k|W_k{|}^2$ has mixed Hessian matrix $V_{i\bar{j}}={\sum }_k{W}_{ki}\overline{W_{kj}}$ and holomorphic block $V_{ij}={\sum }_k\overline{W_k}{W}_{kij}$. The $2n$ real scalar squared-mass sum is $2\mathrm{tr}(m^{†}m)$ for the chiral-superfield fermion mass matrix $m=W_{ij}$. Each Weyl spinor contributes two states, giving the same fermion weighted sum. Thus their boson-minus-fermion supertrace vanishes even with F-term breaking. The holomorphic Hessian splits real scalar masses without changing their sum. Noncanonical Kähler metrics, supergravity, and radiative corrections require different formulas.

A column of viscosity $\mu$ in an annulus of viscosity $\lambda \mu$ and gap $h\ll a$ drives return flow of speed $O(a|w|/h)$ in a closed container. Annular shear scales as $\lambda \mu a|w|/h^2$, inducing relative core-velocity variation $O(\lambda a^2/h^2)$. Thus the extensional equations for a slender Newtonian column admit a leading plug-like axial velocity when $\lambda \ll h^2/a^2$. The small viscosity ratio alone is insufficient when the gap is very narrow.

In a closed-base cylindrical container, mass conservation makes the annular volume flux opposite to the core flux. With $h=R-a\ll a$ and annular viscosity $\lambda \mu$, Couette-Poiseuille flow in a thin gap gives $q_g=wh/2-h^3{P}_z/(12\lambda \mu )\simeq -aw/2$, where $P$ removes the annular hydrostatic pressure. Hence $P_z=6\lambda \mu aw/h^3[1+O(h/a)]$. The resulting column equation is $(a^2{w}_z{)}_z/a^2=\Delta \rho g/(3\mu )+2\lambda aw/h^3$. Curvature and the Couette flux change only the neglected relative order $h/a$.

A short exact sequence in an abelian category is a sequence of the form $0\to A\stackrel{i}{\to }B\stackrel{p}{\to }C\to 0$. Equivalently, $i$ is the kernel in a category of the epimorphism $p$. It splits if $p$ has a section; in that case the additive category splitting argument identifies $B$ with the biproduct $A\oplus C$. Categories of modules give familiar examples.

A complex rank-$r$ vector bundle has a canonical real orientation, and its Euler class equals its top Chern class. For a complex line this is $e(L_{\mathbb{R}})=c_1(L)$. The splitting principle for complex vector bundles gives an injective cohomology pullback on which the bundle splits into lines. The Whitney product formula for Euler classes and Whitney sum formula for Chern classes then identify both sides with the product of those first Chern classes, proving the general equality.

Set $y_j=-\log (1-B_j)$ and let $e_C(a)=E(a,C)$. For positive integer capacities,The function $U$ is the carried load of an Erlang loss resource evaluated at its inverse blocking parametrization. It is continuous and strictly increasing from zero to $C$. Hence $F$ is a coercive function and a strictly convex function on the nonnegative orthant. Its first-order conditions are the fixed-routing Erlang fixed point equations, including zero prices at unused links.

A conditional probability of an invariant probability measure given its invariant sigma-algebra. In a standard Borel probability system, almost every such probability is invariant and ergodic. Their integral recovers the original measure. The countable-test proof of ergodicity of conditional components explains why invariance alone is not the whole conclusion.

For rotation by $p/q$ with coprime integers and Haar probability on the circle, components are the equal measures on the $q$-point orbits. The quotient map is $x\mapsto qx(\mathrm{mod}1)$. Integration over the quotient recovers Haar measure, and the rotation permutes each orbit transitively, making its probability measure ergodic.

A two-cell-thick strip of triangles with degree-six vertices in a dual arrangement of a planar point set produces three indexed primal point families with collinearities $a_i,b_j,c_k$ whenever $i+j+k=0$. A plane cubic can be fitted to nine initial points because its homogeneous coefficient space has dimension ten. Overlapping nine-point configurations then force each next point onto the same cubic by eight-point cubic completion for two triples of lines. Safe neighbourhoods guarantee the distinctness needed in these local completion steps. A possible seed is $a_{-1},a_0,a_1,a_2,b_{-3},b_{-2},b_{-1},c_1,c_2$. Adjacent completion blocks force $c_3,b_{-4},c_4,a_{-2}$, followed by the alternating continuation along the two long transverse families.

For the effective-temperature profile of a zero-torque disk, differentiation of $y^{-3}(1-y^{-1/2})$ with $y=r/r_{\mathrm{in}}$ gives $y_{\max }=49/36$. Thus $T_{\max }=(6/7{)}^{3/2}{7}^{-1/4}{T}_{\mathrm{in}}\simeq 0.488T_{\mathrm{in}}$. This distinguishes a genuine maximum effective temperature from the normalization used in a power-law approximation.

With fixed total mass and orbital angular momentum and $R_L\propto a(M_d/M{)}^{1/3}$, the Roche-lobe radius response exponent is $2q-5/3$. A donor with stellar radius response exponent ${\zeta }_{\ast }$ is stable to infinitesimal mass loss when ${\zeta }_{\ast }>2q-5/3$, giving $q<({\zeta }_{\ast }+5/3)/2$. The threshold depends on the donor response.

Let $t_k$ count vertices where $k$ lines meet and $f_j$ count faces with $j$ sides in a nonpencil projective line arrangement. Using Euler characteristic one, $v=\sum t_k$, $e=\sum k{t}_k$, and $2e=\sum j{f}_j$ gives $t_2=3+{\sum }_{k\ge 4}(k-3)t_k+{\sum }_{j\ge 4}(j-3)f_j$. Thus few double vertices control both high-multiplicity vertices and nontriangular faces. Discarding the face term gives the usual arrangement form of the ordinary-line inequality.