The zero-kick circular-parent threshold is $\beta =1/2$: below it grains are initially bound, at it their conservative release orbit is parabolic, and above it they are initially unbound. Poynting–Robertson drag can modify a marginal orbit, so the threshold is a statement about the instantaneous release orbit, not an exact capture criterion with dissipation included.

For spanning slab normals, apply the maximum of finitely many Rademacher linear forms bound to $u_i=(V{\Lambda }^{1/2}{)}^T{a}_i$, whose squared norms are $a_i^{T}X{a}_i\le 1$. Some sign vector has scaling denominator squared at most $2\log (2m)$. Rademacher rounding for a semidefinite relaxation then produces a feasible vector of squared norm at least the displayed fraction of the SDP optimum. This is an existence guarantee from the probabilistic method.

For independent Rademacher random variables and $\Vert u_i{\Vert }_2\le 1$, this bound holds for $\alpha >0$. The moment-generating function satisfies ${\prod }_j\cosh (t{u}_{ij})\le e^{t^2/2}$, with strict inequality for a nonzero vector and $t>0$. Applying the exponential Markov bound to both tails and then the union bound proves a strict failure bound; a zero vector has failure probability zero. At $\alpha =\sqrt{2\log (2m)}$, the success probability is strictly positive, ensuring a sign vector with all linear forms below that threshold.

The logit mean model $p_i=(1+e^{-x_i^T\beta }{)}^{-1}$ with working variance function $\varphi p_i(1-p_i)$ has the same coefficient estimates as ordinary binomial logistic regression, but estimates dispersion from residuals. For an actual individual binary random variable, $Y^2=Y$ forces variance $p(1-p)$, so a nonunit dispersion is a working specification rather than a new independent binary distribution.

For a Boussinesq approximation, constant buoyancy frequency $N$, and buoyancy heating rate $R$, the quasi-geostrophic potential-vorticity equation becomes $D_{g}q/Dt=(f_0/N^2)R_z$, where $q={\nabla }_h^2\psi +(f_0^2/N^2){\psi }_{zz}+\beta y$. Heating changes the vertical gradient of buoyancy and therefore produces a potential-vorticity tendency proportional to its height derivative.

Every quasi-coherent sheaf on an affine scheme has zero higher sheaf cohomology. Together with the equivalence between modules and quasi-coherent sheaves on an affine scheme, this makes affine varieties the basic acyclic pieces for computing cohomology. An affine open cover of a separated scheme has affine finite intersections, so the acyclic cover theorem identifies its Čech cohomology with sheaf cohomology.

The unassisted classical capacity is the supremum of classical bit rates achievable with asymptotically vanishing message error. It allows entanglement among different channel inputs, but no preshared sender-receiver entanglement. The Holevo-Schumacher-Westmoreland theorem gives its regularized entropy expression.

The Holevo capacity of a quantum channel is the maximum one-use output Holevo quantity,It is the unassisted classical communication limit for product state encodings with collective output measurements. Arbitrary entangled inputs require the regularized expression in the Holevo-Schumacher-Westmoreland theorem.

Along the dyadic partitions of an interval of $[0,1]$, the sums ${\sum }_i|{\Delta }_{i}M{|}^p$ of a continuous local martingale tend to zero in probability for $p>2$. If $0<p<2$ and these sums are eventually bounded almost surely, then $M$ is constant on $[0,1]$: compare them with the sums defining quadratic variation, using the maximum increment, which tends to zero by uniform continuity.

For homomorphisms $i_j:H\to G_j$, the pushout of groups is $(G_1\ast G_2)/⟨⟨i_1(h)i_2(h{)}^{-1}:h\in H⟩⟩$. It has the universal property for homomorphisms out of $G_1,G_2$ agreeing on $H$. When the maps $i_j$ are injective, it is the amalgamated free product, with the factors and common subgroup embedded.

For a rank-$r$ complex vector bundle and $u=c_1({\lambda }^{\ast })$, the cohomology of its projective bundle is free over the cohomology of the base on $1,u,\dots ,u^{r-1}$. Consequently pullback is injective, which yields the splitting principle for complex vector bundles by iteration.

For probit regression with $s_i=2Y_i-1$ and prior $N(0,{\sigma }^{2}I)$, the posterior score control variate isFor $h(\beta )=\Phi (x_{\ast }^T\beta )$, its covariance with the score is $-x_{\ast }\mathbb{E}[\varphi (x_{\ast }^T\beta )]$. It is nonzero whenever $x_{\ast }\ne 0$, ensuring strict improvement by the optimal control variate.

For a probability density function $f$ on $\mathbb{R}$ with Lipschitz bound $L>0$, the triangular lower envelope $f(x+u)\ge (f(x)-L|u|{)}_{+}$ gives $f(x{)}^2\le L$. If $f$ is strictly positive everywhere, the inequality is strict. Equality forces the entire probability density function to coincide with the triangular envelope, which vanishes outside a finite interval.

A probability density function $f$ is log-concave if, for every $0<t<1$,Equivalently, the extended-valued function $\log f$, with $\log 0=-\infty$, is a concave function. In particular, the positive-density region must be a convex set. Stating concavity only on an arbitrarily disconnected support misses this requirement.

For example, the Gaussian distribution has log density equal to a constant minus a positive quadratic, hence is log-concave. A uniform distribution on a convex set of finite positive volume is also log-concave. But density $1/2$ on $[-2,-1]\cup [1,2]$ is not: its values at $-3/2$ and $3/2$ are positive while its value at their midpoint is zero.

At an interior point $x_0$ where $\log f$ is differentiable, a supporting tangent gives $\log f(x)\le \log f(x_0)+\nabla \log f(x_0)\cdot (x-x_0)$. Exponentiating produces an upper envelope used in rejection sampling. The envelope must also have finite integral on the sampling region to serve as a normalized proposal probability density function.

For a strictly positive probability density function with Lipschitz bound $L$, $p_x(r)={\int }_{x-r}^{x+r}f(y)dy$ satisfies $|p_x(r)-2rf(x)|\le L{r}^2$. If $0<s<1$ and $s\le f(x{)}^2/L$, monotonicity gives $p_x^{-1}(s)\le s/f(x)$ and hence $|2f(x)p_x^{-1}(s)-s|\le L{s}^2/f(x{)}^2$. The Lipschitz density height bound prevents extending this local argument to every $s\in (0,1)$ by assuming $f(x)\ge \sqrt{L}$ for a strictly positive density.

The space-time Probabilists' Hermite polynomial $H_n(x,t)=t^{n/2}{\mathrm{He}}_n(x/\sqrt{t})$ extends polynomially to $t=0$ and solves ${\partial }_t{H}_n+\frac{1}{2}{\partial }_x^2{H}_n=0$. Therefore $H_n(B_t,t)$ is a martingale for a standard Brownian motion, by the Itô formula and Gaussian moments.

Suppose $K=1+a+a^2[C-b\ln (q^2/{\mu }^2)]$, with $da/d\ln {\mu }^2=-b{a}^2$. Differentiation gives $dK/d\ln {\mu }^2=-2b{a}^3[C-b\ln (q^2/{\mu }^2)]$. For nonzero $a,b$, the principle of minimal sensitivity therefore gives $\boxed{{\mu }_{\ast }^2=q^2{e}^{-C/b}}$. The derivative vanishes identically through order $a^2$: this prescription retains the residual $a^3$ term generated by one-loop running of the truncated expression. Higher-order running and coefficients can shift the stationary scale. For $b=0$ or $a=0$, no unique scale is selected.

Let $A,B$ be finite-dimensional commutative algebras, let $I\subset A$ and $J\subset B$ be ideals, and let $f:I\to J$ be a surjective algebra homomorphism. The ideals need not have identities. Then $f$ gives a bijection between the primitive idempotents of $A$ lying in $I$ with nonzero image and the primitive idempotents of $J$.

Decompose $A$ into its commutative Artinian local rings $A_i$, with identities $e_i$. Each $I\cap A_i$ is either all of $A_i$, in which case $e_i\in I$, or a proper ideal and therefore a nilpotent ideal. The sum of the latter ideals is nilpotent, so its image contributes no idempotents. Each nonzero $f(e_i)$ has local corner algebra $f(e_i)J=f(A_i)$, a quotient of $A_i$, so is primitive. These idempotents are orthogonal, and the remaining part of $J$ is nilpotent; hence they are exactly its primitive idempotents. This statement does not require the entire kernel of $f$ to be nilpotent.

For the pressure-density Hugoniot relation for a perfect gas, the change in specific entropy is $[s]/c_v=\log P-\gamma \log D(P)$, with $c_v$ the specific heat capacity at constant volume. Its derivative isIt is positive for a compressive normal shock wave with $P>1$. For a weak shock with $P=1+\delta$, integrating the leading term givesThus the entropy production is cubic in the small pressure jump, even though the mass density and temperature changes already appear at first order.

A sharp-k filter includes every Fourier mode below a cutoff with unit weight and excludes all modes above it. For a Gaussian random field, increasing the cutoff adds independent Fourier shells, so the smoothed density contrast is a Brownian motion when parameterized by its variance. The real-space window is not a localized spherical top-hat, so its halo mass assignment needs a specified convention.