This tensor integrates the local pressure tensor, where ${\sigma }_{ij}=⟨(v_i-u_i)(v_j-u_j)⟩$ is the velocity covariance. Half its trace is the random kinetic energy. It enters the tensor virial theorem with coefficient one.

This force-moment tensor enters the tensor virial theorem. For an isolated self-gravitating population it is symmetric by pairwise interchange of the Newtonian interaction; for a tracer in an external field use the force-moment definition directly, with the appropriate symmetric combination in a time-dependent inertia identity.

The planar trace of the stellar pressure-energy tensor is the sum of its two Cartesian in-plane components. For an axisymmetric nonstreaming population it equals $\int \rho (⟨v_R^2⟩+⟨v_{\varphi }^2⟩)d^{3}x$, not merely the cylindrical radial second moment. A spherical isotropic population has ${\Pi }_{\perp }=2{\Pi }_{zz}$.

For a pressure-supported scale-free tracer $\rho \propto (R^2+z^2/q^2{)}^{-\gamma /2}$ in a spherical Newtonian gravitational potential, the tensor virial theorem constrains integrated directional velocity dispersions. Modest oblate flattening and a decreasing radial mass density require greater in-plane random support per direction. The displayed expansion assumes finite global virial integrals; the corresponding ratio of force moments also has a cutoff-independent angular interpretation.

With pressure amplitude $P=ZV$ and velocity $V$ directed into a boundary, the mean absorbed acoustic energy flux is $\frac{1}{2}\mathrm{Re}(P{V}^{\ast })=\frac{1}{2}\mathrm{Re}Z|V{|}^2$. A passive boundary cannot supply net energy, so $\mathrm{Re}Z\ge 0$ at a real frequency. Its reflection from a propagating acoustic plane wave has magnitude at most one. A pole continued to a complex incidence angle or frequency must be interpreted separately from real-frequency passive scattering.

A mean square in ANOVA divides a sum of squares in ANOVA by its positive number $d$ of statistical degrees of freedom. If the corresponding residual subspace has scalar error covariance $\lambda I$, its expectation is $\lambda$. Its scale depends on whether raw observations or group means were projected.

The Lebesgue decomposition theorem separates the absolutely continuous part from the singular part. BV fine structure then splits the latter into the jump part of a BV derivative and the Cantor part of a BV derivative. The density of the absolutely continuous part is the almost-everywhere approximate gradient. These three mutually singular components distinguish smooth variation, surface discontinuities and diffuse singular variation.

A continuous capacity-parameterized Loewner trace with finite, strictly ordered positive swallowing times visits every positive real point. An unvisited swallowed point has an interval disjoint from the compact trace up to its swallowing time; that interval is swallowed simultaneously, contradicting strict order. Reflection yields the negative-axis conclusion for chordal SLE at $\kappa \ge 8$.

By an orthonormal eigenbasis, the imaginary part of the Stieltjes matrix resolvent has eigenvalues $\eta /|\lambda -z{|}^2>0$. Thus its normalized trace has positive imaginary part, and its quadratic form at any vector has nonnegative imaginary part. This yields $|z+g|\ge \eta$ and $|z+q|\ge \eta$ for the normalized trace $g$ and a quadratic form $q$ used in Schur-complement estimates.

The eigenvalue interlacing of a Hermitian matrix and a principal minor bounds the difference of their resolvent traces by a constant times $|\mathrm{Im}z{|}^{-1}$. When both traces are normalized by the original dimension $N$, the bound acquires $N^{-1}$. One proof writes the difference using the interlacing counting functions, whose difference is at most one, and bounds the integral of $|t-z{|}^{-2}$ by $\pi /|\mathrm{Im}z|$. This gives the admissible absolute constant $c=\pi$.

For a zero-diagonal real symmetric matrix, put $q_i=x_i^T(X^{(i)}-zI{)}^{-1}{x}_i$ and $g_N=N^{-1}\mathrm{Tr}(X-zI{)}^{-1}$. The Schur complement formula for a diagonal resolvent entry gives $g_N=-N^{-1}{\sum }_i(z+q_i{)}^{-1}$. Subtracting the comparison value $-(z+g_N{)}^{-1}$ gives ${\epsilon }_N=N^{-1}{\sum }_i(q_i-g_N)/[(z+g_N)(z+q_i)]$. Upper-half-plane positivity and the principal minor resolvent trace bound control this defect by the average of $|q_i-g_N^{(i)}|$ plus a trace correction.

Summing the stationary law of a linear flow network over its through-flow count gives independent local geometric distributions with ratios $q_i={\rho }_i/(1-{\rho }_0)$. This gives the displayed means. The local counts are independent of one another in this marginal distribution; the through count is generally dependent on them. The independence is a stationary distributional fact, not an assertion that their dynamics evolve independently.

A sequence $D_{\alpha }\subseteq \alpha$, indexed by a stationary subset $S$ of a regular uncountable cardinal $\lambda$, guesses every $X\subseteq \lambda$ stationarily often: $\{\alpha \in S:D_{\alpha }=X\cap \alpha \}$ is stationary. This extends the usual stationary diamond principle at ${\omega }_1$.

The expected exit time from $(\ell ,r)$ for a one-dimensional diffusion is ${\int }_{\ell }^{r}K(x,v)m(v)dv$ when finite. Differentiating this expression on each side of $v=x$ gives the equation $\mathcal{L}u=-1$ with zero endpoint values. For the SLE two-boundary-point ratio diffusion, $s(v)-s(1)$ behaves as $(v-1{)}^{1-4/\kappa }$ and the speed density behaves as $(v-1{)}^{4/\kappa }$, so the product is integrable at $1$ for $\kappa >4$.

The real dispersion above is temporally stable for every real $k$, but its analytic continuation has stationary points at $k=\pm i$ with $\omega =i/2$. Writing $\omega =i\Omega$ gives $k^2=-1\pm \sqrt{1/2-\Omega }$; as $\Omega$ decreases to $1/2$, both branches at $+i$ approach from the upper half-plane and both at $-i$ from the lower. These are not spatial pinch points. This provides a concrete counterexample to treating growing algebraic double roots as sufficient for absolute wave-packet instability.

A sieve is J-closed when membership is local with respect to the Grothendieck topology $J$. Pullback preserves J-closedness. Their presheaf is the subobject classifier of the sheaf topos: a section of a sheaf is sent to the sieve of arrows on which it belongs to the chosen subsheaf.

For the shifted-exponential Poisson mixture, the displayed relation holds for $n\ge 2$, initialized by $p_0=e^{-\alpha }\beta /(\beta +1)$ and $p_1=(\alpha +1/(\beta +1))p_0$. It follows by differentiating the probability generating function and comparing coefficients. Nonnegativity follows from the independent Poisson-geometric convolution of independent random variables representation.

The BV lattice inequality $|D\min (v,w)|+|D\max (v,w)|\le |Dv|+|Dw|$ applied to two indicator functions gives the perimeter inequality. For smooth approximations, minima and maxima partition the two gradients; sequential lower semicontinuity passes the estimate to BV. It prevents a crossing of minimizers with strictly ordered set forcing.

The total variation seminorm of the indicator function within an open set counts only its interior interface. The perimeter of a zero extension may add a boundary contribution.

Here the sets of finite perimeter $E,F$ differ only compactly inside $U$. A minimizing set with bounded volume forcing satisfies this inequality. Interior regularity theorems give graph patches outside a singular set negligible for surface Hausdorff measure; bounded forcing allows the graph estimates used in the noncontact of ROF level boundaries.