If the rows of a Young tableau of shape $\lambda$ have no repeated intersection with the columns of one of shape $\mu$, then ${\sum }_{i\le k}{\lambda }_i\le {\sum }_{i\le k}{\mu }_i$ for every $k$. If also $\lambda \ge \mu$ in dictionary order on integer partitions, the shapes are equal. Saturation of the prefix bounds places one entry of each eligible row in each column, giving $u=rct$ with $r\in R(t)$ and $c\in C(t)$. A collision instead gives a transposition that makes row symmetrization followed by the relevant column antisymmetrization vanish.

For a local field with finite residue field, evaluate the Witt residue character on arithmetic Frobenius. This gives ${\mathrm{inv}}_K:\mathrm{Br}(K)\cong \mathbb{Q}/\mathbb{Z}$. For the unramified degree-$m$ cyclic algebra with arithmetic Frobenius generator and parameter $a$, the invariant is $v_K(a)/m$. Its reduced denominator is the index of the division representative.

A Wirtinger generator is the meridian of a knot associated with an arc of a knot diagram between successive undercrossings. The overpassing meridian conjugates the incoming meridian to the outgoing meridian, with exponent determined by the crossing sign.

If a circle-valued sequence is not an equidistributed sequence, some positive-shift difference is not equidistributed either. The contrapositive follows by applying the Van der Corput inequality for finite scalar sequences to every nonzero integer exponential sum. Thus cancellation for all fixed nonzero differences implies equidistribution of the original sequence.

For a finite-dimensional continuous complex representation $R$ of a compact group, normalize Haar measure to mass one and average any positive-definite Hermitian inner product:The resulting inner product is positive-definite and invariant. If $W$ is an invariant subspace, then $W^{\perp }$ is invariant as well, since $⟨R(g)v,w{⟩}_K=⟨v,R(g^{-1})w{⟩}_K$. Thus $V=W\oplus W^{\perp }$. Restriction to a compact real form, followed by integration of a Lie-algebra representation, proves the Weyl complete reducibility theorem.

A Lagrangian complement to $TX$ in $TM{|}_X$ is identified with $T^{\ast }X$ by $v\mapsto \omega (\cdot ,v){|}_{TX}$. This produces a vector bundle isomorphism $TX\oplus T^{\ast }X\to TM{|}_X$ that is the identity on $TX$ and carries the canonical pairing $\zeta (u)-\eta (v)$ to $\omega$. A suitably chosen tubular neighborhood map therefore matches the ambient two-forms along $X$, enabling the relative Moser theorem.

The homologous trial function $\xi =1$ in the weighted stellar pulsation Rayleigh quotient has numerator $-3{\int }_0^{R_s}{r}^{2}p(4-3\gamma )dr$. Positivity of that integral therefore proves radial instability. It is a sufficient test: a nonpositive integral does not guarantee stability against every radial shape. For constant $\gamma$, the homologous threshold is $4/3$.

Shifting the zero mode changes the algebra's central representative:The linear term follows from replacing $L_{m+n}$ by ${\mathcal{L}}_{m+n}+a{\delta }_{m+n,0}$. It changes the zero-mode convention, not the cubic coefficient or central charge. This distinction separates the intercept from the Virasoro central extension.

A smooth family $Y_t$ assigns a tangent vector at each space-time point. Its flow maps solve ${\partial }_t{\rho }_t=Y_t\circ {\rho }_t$, with ${\rho }_0$ the identity. For a varying differential form, differentiation gives ${\partial }_t({\rho }_t^{\ast }{\alpha }_t)={\rho }_t^{\ast }({\dot{\alpha }}_t+{\mathcal{L}}_{Y_t}{\alpha }_t)$.

Choose input eigenvectors $u_i$ and output eigenvectors $v_j$ for a unital quantum channel. Positivity makes the displayed entries nonnegative. Trace preservation makes every column sum one; unitality makes every row sum one. Thus $A$ is a doubly stochastic matrix and the output spectrum satisfies $s=Ar$. The mixing matrix need not be a matrix of squared overlaps of a single unitary, even though its row and column constraints are the same.

Let independent observations have uniform distributions on $(0,\theta )$ and let $\theta$ have a Pareto distribution with shape parameter $\alpha >0$ and lower bound $\beta >0$. For positive observations put $m=\max (\beta ,{\max }_j{y}_j)$. Integrating the likelihood function against the prior density gives the Bayesian model evidenceFor independent groups with shared fixed hyperparameters, the Bayesian model evidences multiply. With $I$ groups and $m_i=\max (\beta ,{\max }_j{y}_{ij})$, the exact hyperparameter log-likelihood function isThe term $-{\sum }_i{n}_i\log m_i$ may be omitted when optimizing $\alpha$ with $\beta$ fixed. It must remain when $\beta$ varies: each $m_i$ then changes whenever $\beta$ exceeds that group's sample maximum. Calling this term a constant in a joint fit would change the maximum-likelihood estimator.

A bracket $[\ell ,u]$ contains every measurable function $h$ satisfying $\ell \le h\le u$ pointwise. For a Probability measure $P$, its $L^1(P)$ width is $P(u-\ell )$, with integrable endpoints. Finite bracketing at every positive width is a sufficient complexity condition for a uniform law of large numbers. Pointwise inequalities, or inequalities outside a single common null set, allow empirical inequalities to hold simultaneously throughout the class.

A measurable-function class is pointwise separable if it has a countable subclass such that every member is the pointwise limit of a sequence from that subclass. With an integrable common envelope, dominated convergence theorem makes both empirical averages and population expectations converge along each such sequence. Thus the supremum of the absolute empirical errors equals the supremum over the countable subclass and is measurable. A finite integrable bracketing cover provides an integrable envelope by taking the maximum absolute value of all its endpoints.

For a connected Riemannian manifold, two-point homogeneity is equivalent to the isometry group being transitive on the unit tangent bundle. One direction follows by taking short equal-length radial geodesic segments: the Gauss lemma identifies their distance, and injectivity of the exponential map identifies the initial directions after the endpoints are matched. Conversely, unit tangent transitivity implies point homogeneity, hence completeness. The Hopf-Rinow theorem supplies minimizing geodesics for arbitrary equal-distance pairs. Matching their initial unit tangent vectors and using uniqueness of the geodesic equation matches their other endpoints.

At fixed position a two-integral distribution depends on $v_R,v_z$ only through $v_R^2+v_z^2$. Swapping these integration variables makes their second moments equal. Their odd moments and mixed second moments vanish by reflection symmetry. For an even function of $L_z$ the azimuthal velocity is also centered, so all mixed entries of the velocity ellipsoid vanish.

The singular axisymmetric logarithmic gravitational potential admits this stationary galactic distribution function withThe zero of specific orbital energy is the printed logarithmic convention. Products of Gaussian integrals normalize its two spatial density terms. For $1/2\le q^2\le 1$ the two terms are separately nonnegative; mildly prolate models require a bound on their sum.

A sufficiently regular four-partite three-uniform hypergraph, supported on sufficiently regular pair cells of positive density and with positive relative triple densities, has a positive fourth-power number of three-uniform tetrahedra. On complete pair supports, telescoping the four edge functions shows that an error of at most $\xi$ in each three-dimensional box norm gives a counting error at most $4\xi$. The relative version additionally requires control of the pair-support densities.

For a plane gravitational wave in linearized gravity with $k_{\mu }=(-\omega ,0,0,\omega )$, Lorenz gauge in linearized gravity gives $H_{3\nu }=-H_{0\nu }$. Under $\delta H_{\mu \nu }=i(k_{\mu }{X}_{\nu }+k_{\nu }{X}_{\mu }-{\eta }_{\mu \nu }{k}^{\rho }{X}_{\rho })$, choose $X_i=H_{0i}/(i\omega )$ for $i=1,2$, $X_3-X_0=i{H}_{00}/\omega$, and $X_0+X_3=(H_{11}+H_{22})/(2i\omega )$. This residual gauge symmetry of linearized gravity removes all time and longitudinal entries and the transverse trace. The remaining amplitudes $H_{+},H_{\times }$ are the two gravitational wave polarizations. The formula uses the plus-sign convention for $\delta h=2{\partial }_{(\mu }{\xi }_{\nu )}$; reversing the definition of $\xi$ reverses the gauge change.

Reflection across a cut splits Laplacian eigenfunctions into odd and even parts. The odd restriction has a Dirichlet boundary condition on the cut; the even restriction has a Neumann boundary condition there. The displayed matrix intertwines the two-tile swap with $\mathrm{diag}(-1,1)$. A connected uniform-Dirichlet rectangle of width $2a$ is therefore isospectral to the disjoint union of width-$a$ rectangles with all-Dirichlet and three-Dirichlet/one-Neumann conditions. This illustrates why changing boundary conditions can conceal connectedness in the spectrum.

A narrow dipolar flux-tube area $A(z)\propto z^3$ gives $A\rho v=\dot{M}$ for inward speed $v>0$. The isothermal magnetic Bernoulli integral in $\Phi =-GM/z$ then gives $(v-c_s^2/v)v\prime =3c_s^2/z-GM/z^2$. A smooth transonic branch crosses at $z_s=GM/(3c_s^2)$ with $v_s=c_s$, ${\rho }_s=e^{5/2}{\rho }_{\infty }$ and $\dot{M}={\rho }_s{c}_{s}A(z_s)$. The crossing must lie in the domain; subsonic branches need not attain it.