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In the prismatic triangular-channel gravity-current box model, take $h=h_0\sqrt{L/X}$ and $w(\varphi )=w_0(1-\varphi )$. Eliminating time gives $d\varphi /[\sqrt{\varphi }(1-\varphi )]=-2w_0{X}^{3/4}dX/(\mathrm{Fr}\sqrt{G}{h}_0^{3/2}{L}^{3/4})$. Integration yields the displayed invariant with $K=4w_0/(7\mathrm{Fr}\sqrt{G}{h}_0^{3/2}{L}^{3/4})$. The positive-concentration branch approaches a finite runout length of a gravity current as time tends to infinity.

In a complete discrete valuation ring of residue characteristic $p$, the power map $u\mapsto u^m$ is a bijection on $1+\mathfrak{m}$ when $p\nmid m$. For a principal unit $a$, the Hensel lemma lifts the simple residue root one of $T^m-a$ uniquely. This can absorb principal-unit factors in tame radical descriptions of local fields.

Pulling a principal connection back along a local section $s$ gives a Lie-algebra-valued one-form on the base. If the section changes to $sg$, the form has the displayed inhomogeneous transformation. Its gauge curvature $F=dA+A\wedge A$ transforms homogeneously as $F^{\prime }=g^{-1}Fg$. The collection of such local forms encodes a global principal connection even when no single section exists globally.

A gauge transformation is an automorphism of a principal bundle covering the identity on the base and commuting with its right group action. Pullback relates gauge-equivalent principal connections. Locally it acts by the displayed formula; the gauge curvature transforms by conjugation. Gauge functions on overlapping charts satisfy the bundle's transition compatibility. Infinitesimally $g=e^{\epsilon }$ gives $\delta A=D_A\epsilon$.

Suppose $A>0$ and every sufficiently high Nontrivial zero of the Riemann zeta function satisfies the displayed gap, with bounded heights also separated from one. The truncated explicit formula for the second Chebyshev function and the local zero count for the Riemann zeta function give errors $x{\log }^{2}x\exp (-c\log x/(\log T{)}^A)$ and $x{\log }^{2}x/T$. Balance them by $\log T\asymp (\log x{)}^{1/(A+1)}$. This explains how the width of a zero-free region of the Riemann zeta function determines the exponential scale in the Prime number theorem error.

Let $v_x,v_y$ and $u_x,u_y$ be the dust and gas velocity perturbations after subtracting the common background shear, suppressing primes. Put $t_s>0$ for the aerodynamic stopping time. Linear axisymmetric dust perturbations about a Keplerian shearing sheet obey $D_{ϵ}{v}_x-2\Omega v_y=ϵ\Omega u_x$ and $D_{ϵ}{v}_y+\Omega v_x/2=ϵ\Omega u_y$, with $ϵ\Omega =t_s^{-1}>0$. Eliminating $v_y$ givesThus a gas epicycle $u_x=u\cos (kx-\Omega t)$, $u_y=u\sin (kx-\Omega t)/2$ supplies both $2ϵ{\Omega }^{2}u\sin (kx-\Omega t)$ and ${ϵ}^2{\Omega }^{2}u\cos (kx-\Omega t)$. Keeping the latter is necessary for an exact linear equation beyond first order in drag. The homogeneous roots are $-ϵ\Omega \pm i\Omega$, separating damping from epicyclic frequency.

A null pregeodesic has a null tangent and can be parametrized as a null geodesic. In a two-dimensional Lorentzian metric, the acceleration of a regular null curve is orthogonal to its tangent and therefore proportional to it.

For a predictable process $H$ and deterministic $t>0$, $H_t$ is measurable for the left-limit sigma-algebra ${\mathcal{F}}_{t-}$. This follows by taking sections of the generating predictable rectangles. At time zero use ${\mathcal{F}}_{0-}={\mathcal{F}}_0$.

For coplanar circular dust, constant $A=\beta G{M}_{\star }/c$, fixed stellar radius $a_p$ and initial planetocentric radius $a_0>R_p$, integrating $\dot{a}=-3Aa/a_p^2$ givesCompare the inspiral time under Poynting–Robertson drag for a circular circumstellar orbit, $t_{\mathrm{cs}}=(a_p^2-R_{\star }^2)/(4A)$. Planetary arrival is not automatically faster; in the point-star limit it is faster only for $a_0/R_p<e^{3/4}$. A different constant orientation coefficient replaces three.

For spherical fragments of common mass density $\rho$, with $n(D)=K{D}^{-\alpha }$ between specified diameter cutoffs, mass normalization of a power-law size distribution and integration of the geometric cross-section giveIndeed $m=(\pi \rho K/6)\int D^{3-\alpha }dD$ and ${\sigma }_{\mathrm{tot}}=(\pi K/4)\int D^{2-\alpha }dD$; eliminating $K$ proves the formula. If both cutoff ratios raised to $\alpha -3$ and $4-\alpha$ are small, respectively, this reduces to $3m(4-\alpha )D_{\min }^{3-\alpha }{D}_{\max }^{\alpha -4}/[2\rho (\alpha -3)]$. Approaching either endpoint requires the corresponding logarithmic integral.

For equal-density spherical bodies with differential number distribution $n(D)\propto D^{-\alpha }$, a fixed logarithmic bin has width $dD=Dd\log D$. Multiplying the body mass proportional to $D^3$ by $n(D)dD$ gives $m_{\log }\propto D^{4-\alpha }$. Distinguishing differential-number and logarithmic-mass slopes is essential in collisional cascade plots.

For a nonnegative $H^1$ weak solution of a homogeneous divergence-form elliptic equation on $B_1\subset {\mathbb{R}}^{\ell }$, $\ell \ge 2$, with symmetric measurable $\lambda I\le A\le \Lambda I$, set $R=\Lambda /\lambda$ and $\alpha =1+1/\ell$. There is $D_{\ell }\ge 1$ independent of $q\ge 2$ such thatTest the equation with a cutoff times $u^{q-1}$, first regularized or truncated, to get $\int {\zeta }^2|\nabla u^{q/2}{|}^2\le [q/(q-1){]}^2{R}^2\int u^q|\nabla \zeta {|}^2$. Apply a fixed-domain Sobolev embedding theorem to $\zeta u^{q/2}$ and take the $2/q$ power. Keeping the constant inside that power is crucial for Moser iteration.

In unforced adiabatic inviscid motion, potential vorticity is conserved following a parcel:The linearization about a basic state is $({\partial }_t+\bar{\mathbf{u}}\cdot \nabla )q^{\prime }+{\mathbf{u}}^{\prime }\cdot \nabla \bar{q}=0$. This separates advection of a disturbance from its generation by displacement across a basic potential-vorticity gradient.

A posterior predictive check compares a summary of the observed data with the same summary in replicated data drawn from a model's posterior predictive distribution. The replication must match the target: replicating an existing group's observations conditional on its effect differs from predicting a new group with a newly drawn effect. Such checks expose mismatches in dispersion, tails or other features; using the observations to fit and check the model means they are not automatically calibrated frequentist tests.

A real symmetric matrix admitting a finite sum of nonnegative rank-one matrices $x_j{x}_j^T$. It is both a positive semidefinite matrix and a nonnegative matrix, but these two properties alone need not imply membership in the completely positive cone in arbitrary dimension. This matrix notion is distinct from a completely positive map.

For positive block sizes, the real matrix $\left(\begin{array}{cc}aJ& bJ\\ bJ& aJ\end{array}\right)$ is a positive semidefinite matrix exactly when $a\ge |b|$. For test-vector block sums $s,t$, its quadratic form is $(a+b)(s+t{)}^2/2+(a-b)(s-t{)}^2/2$. This proves sufficiency; testing $s=\pm t$ proves necessity. Zero sums in both blocks describe part of its kernel.

Every density operator $\sigma$ with positive partial transpose has squared quantum fidelity at most $1/d$ with a fixed maximally entangled state on $d\otimes d$. The partial transpose of a maximally entangled projector and the trace-adjoint identity for partial transpose express that squared overlap as $\mathrm{Tr}(S{\sigma }^{T_B})/d$. Since ${\sigma }^{T_B}$ is positive of trace one and the swap operator has operator norm one, trace duality gives the bound. A matching product state attains equality.

For a positive Harris recurrent Markov chain with invariant probability $\pi$ and $\int |h|d\pi <\infty$, its time average converges almost surely to the invariant expectation:This is the consistency result for Markov chain Monte Carlo averages. Independent-observation variance formulas do not follow from this theorem.

Apply the homology cross product and then the map induced by loop concatenation. This degree-additive bilinear product supplies the Pontryagin ring. The Bott–Samelson theorem computes it for looped suspensions with free homology.

For $n\ge 1$, the Bott–Samelson theorem gives a single homology generator of degree $2n$ with no word relations. Its Pontryagin product satisfies $x^i\ast x^j=x^{i+j}$. Unlike this polynomial homology product, the cohomology cup product has divided-power coefficients.