A halo is associated with the first upcrossing of the linear spherical-collapse threshold by the smoothed density contrast as its variance increases. With a sharp-k smoothing filter, the Brownian reflection principle makes the crossing probability twice the one-scale Gaussian upper tail. This handles trajectories that crossed earlier but finish below the barrier and leads to the Press-Schechter halo mass function.

An elementary predictable process may use finitely many ordered stopping times ${\tau }_j$, with bounded ${\mathcal{F}}_{{\tau }_j}$-measurable coefficients $h_j$ on $({\tau }_j,{\tau }_{j+1}]$. Its elementary stochastic integral is ${\sum }_j{h}_j(X_{t\wedge {\tau }_{j+1}}-X_{t\wedge {\tau }_j})$. The stopped indicators are left-continuous and adapted, so the process is predictable; common refinement shows the sum is independent of its representation. Deterministic endpoints recover the usual simple predictable process. These integrands give the precise good-integrator test in the Bichteler-Dellacherie theorem.

For spherical planetesimals with mass density $\rho$ and total-number distribution $n(D)=K{D}^{-\alpha }$ between $D_{\min }$ and $D_{\max }$,Thus $K=6M_{\mathrm{tot}}(4-\alpha )/[\pi \rho (D_{\max }^{4-\alpha }-D_{\min }^{4-\alpha })]$ for $\alpha \ne 4$. At $\alpha =4$, use $K=6M_{\mathrm{tot}}/[\pi \rho \log (D_{\max }/D_{\min })]$. Divide $n(D)$ by belt volume to obtain the number-density distribution used in a planetesimal collision rate.

The log-posterior is the logarithm of a positive posterior density. Its gradient is the posterior score control variate when boundary terms vanish. An additive constant independent of the parameter does not affect optimization or differentiation.

For real positive semidefinite matrices $A,B$, their product need not be symmetric, but its matrix trace is nonnegative:The equality uses the cyclic property of the matrix trace, and the last matrix is a positive semidefinite matrix. Consequently the Loewner order inequality $A⪯C$ implies $\mathrm{tr}(AB)\le \mathrm{tr}(CB)$ whenever $B⪰0$.

If $P$ is a real positive semidefinite matrix, then $x^{T}Px=0$ if and only if $Px=0$. Indeed $x^{T}Px=\Vert \sqrt{P}x{\Vert }^2$ by the principal square root of a positive semidefinite matrix, so a zero quadratic form puts $x$ in the kernel of $\sqrt{P}$ and hence of $P$.

The interior Helmholtz equation for a polytrope of index one also admits a positive solution vanishing on the faces of a cube:Its central value is ${\rho }_c$ and its average is $\bar{\rho }=8{\rho }_c/{\pi }^3$. Together with $\Phi =C-2K\rho$, it solves the interior hydrostatic equilibrium and Poisson equation for Newtonian gravity. A local interior solution need not match the external field of its own mass.

The left Schouten-Nijenhuis bracket on polynomial polyvector fields is the degree-minus-one Gerstenhaber bracket extending the commutator of derivations and $[D,f]=D(f)$ by graded antisymmetry and the left graded Leibniz rule. For $\Pi ={\partial }_X\wedge {\partial }_Y$, this convention gives $[h\Pi ,f]=h(f_Y{\partial }_X-f_X{\partial }_Y)$ and $[F{\partial }_X+G{\partial }_Y,h\Pi ]=(D(h)-h(F_X+G_Y))\Pi$. A right insertion convention reverses the degree-two-with-function formula.

In a Poisson regression with $\log \mu ={\beta }_0+{\beta }_{T}T+{\beta }_{B}b+{\beta }_{TB}Tb$, the treatment logarithm of the mean ratio at baseline $b$ is ${\beta }_T+b{\beta }_{TB}$. The main treatment coefficient tests the contrast at $b=0$. An overall no-treatment test requires both ${\beta }_T=0$ and ${\beta }_{TB}=0$, with their joint covariance matrix for a Wald test.

The Schwarz integral formula reconstructs a holomorphic function from boundary values of its real part. For a real boundary function $f$ on the complex upper half-plane satisfying ${\int }_{\mathbb{R}}|f(s)|/(1+s^2)ds<\infty$, the regularized form isIts real part is the Poisson integral of $f$. The subtraction makes the integral converge and changes only the imaginary normalization. Integration by parts, when the boundary terms vanish, gives $W^{\prime }(w)=(\pi i{)}^{-1}\int f^{\prime }(s)/(s-w)ds$; jumps of $f$ are included as Dirac delta function terms in its distributional derivative.

A Poisson distribution conditioned on a positive count has probability mass function $\mathbb{P}(Y=y)={\theta }^y/[y!(e^{\theta }-1)]$ for $y\ge 1$ and $\theta >0$.

A model with $Y\mid \Lambda \sim \mathrm{Poisson}(\Lambda )$ for a nonnegative random intensity. Conditional expectation and total variance give $\mathbb{E}Y=\mathbb{E}\Lambda$ and $\mathrm{Var}Y=\mathbb{E}\Lambda +\mathrm{Var}\Lambda$, explaining overdispersion. A Poisson-gamma mixture gives the negative binomial distribution.

For a horizontal forced plume, the plume virtual origin lies downstream and below the physical source on the far-field vertical asymptote. The origin shift in centreline arc length is distinct from its vertical height: if $z=s-D+o(1)$ and $\hat{s}=s+s_0$, the virtual height is $z_v=-D-s_0$. Both geometric shifts scale with the jet length at fixed entrainment coefficient.

The collision-ejection lifetime of a comet combines two ways to remove it from a specified orbital population: impact and gravitational ejection. If these can be approximated as constant competing Poisson process rates ${\lambda }_c$ and ${\lambda }_e$, the survival probability and mean lifetime areEquivalently, $1/\tau =1/{\tau }_c+1/{\tau }_e$. The probability that collision is the first removal is ${\lambda }_c/({\lambda }_c+{\lambda }_e)$.

Coplanar comet collision time and planetary ejection of a comet estimate the constituent processes under different encounter assumptions. Their reciprocal times should only be added when they refer to a consistent population and approximately stationary rates. Repeated planetary scattering can change the orbit and correlate encounters, invalidating a single exponential lifetime.

Suppose the apparent capillary pressure is $p_1-p_2=\gamma (1+\beta x)({\kappa }_0+\nabla \cdot \mathbf{n})$, with normal towards the displaced fluid and positive apparent tension at the flat front $x=X$. Linearizing the Darcy law interface problem gives the local growth rateThe pressure increase experienced by a forward protrusion is stabilizing when ${\kappa }_0\beta >0$. It suppresses every positive wavenumber when $U\le k\gamma {\kappa }_0\beta /({\mu }_2-{\mu }_1)$. For a fixed spatial gradient, $X=Ut/\varphi$ makes the rate time dependent; constant exponential growth is a frozen-position approximation.

The excursion from $0$ to infinity is the Doob h-transform of killed planar Brownian motion with $h(z)=\mathrm{Im}z$, using the entrance law at zero. Equivalently, its real part is a standard Brownian motion and its independent imaginary part is a Bessel process of dimension three started at zero. This planar, infinite-duration excursion differs from the one-dimensional excursion on a finite time interval.

An eutectic system has a composition and temperature at which a liquid is in equilibrium with two distinct solid phases. Cooling liquid of that composition through the eutectic temperature produces both solid phases together.

For $\kappa >8$, the Loewner trace visits every point of the complex upper half-plane almost surely. Choose $\rho =\kappa -8$ in the SLE interior-point martingale to obtain $M_t={{\rm Y}}_t^{(\kappa -8)/8}{S}_t^{-(\kappa -8)/\kappa }$. The maximal inequality for a nonnegative supermartingale bounds its running supremum. If a fixed interior ball were avoided, its center would have ${{\rm Y}}_t\ge c_1>0$ before swallowing, forcing $S_t\ge c_2>0$. But the Chordal Loewner equation gives $d{Y}_t^2/dt=-4S_t^2$, forcing a finite lifetime with $Y_t\to 0$, and $d\log {{\rm Y}}_t/d\log Y_t=2S_t^2$ then forces ${{\rm Y}}_t\to 0$, a contradiction. Countably many rational balls give a dense trace; continuity and Transience of chordal SLE make its image relatively closed, so it is all of the complex upper half-plane. This argument does not cover the critical parameter eight.

A zero-mean Gaussian process need not have zero average in each realization. For the periodic covariance function, the constant Fourier component iswhere $I_0$ is a modified Bessel function. Replacing $f(t)$ by $f(t)-P^{-1}{\int }_0^{P}f(u)du$ gives a Gaussian process with kernel $k_P(t,t^{\prime })-c_0$. This is a positive-semidefinite kernel because it is a covariance after a linear transformation, and each sample has zero period average. The real integral representation of the modified Bessel function I0 yields the displayed constant.

The percolation critical probability of independent bond percolation on the infinite square lattice is exactlyIn this model, $\theta (p)=0$ for $p\le 1/2$ and $\theta (p)>0$ for $p>1/2$. The particular graph, independent edge decisions, and distinction between bond percolation and site percolation matter: this value is not a universal threshold for arbitrary random networks. The equality of open and closed edge probabilities at $1/2$ suggests a relation to dual bond percolation, but that symmetry alone is not a proof of long-distance connectivity.

Harris established the lower bound, including absence of an infinite percolation cluster at $p=1/2$, in 1960. Kesten completed the equality in 1980. His original paper states the threshold result as Theorem 1 and the positive/zero percolation probability alternatives as Theorem 2: ^[ref]. These are claims about the infinite square lattice, rather than deductions from a simulated finite sample.