Assume an ideal-gas grey atmosphere with $P(0)=0$, constant surface gravity of a star $g$, $T^4=T_e^4(1+3\tau /2)/2$ and $\kappa ={\kappa }_0\rho T^{4\beta +1}$ for $\beta >1$. Using $\rho =\mu P/(\mathcal{R}T)$ in $dP/d\tau =g/\kappa$ and integrating givesDifferentiation yields $d\log T/d\log P=[(1+3\tau /2{)}^{\beta -1}-1]/[2(\beta -1)]$. Applying the Schwarzschild criterion with monatomic adiabatic gradient $2/5$ gives ${\tau }_b=\frac{2}{3}[((4\beta +1)/5{)}^{1/(\beta -1)}-1]$. Since ${\tau }_b$ depends only on $\beta$, matching to $P=K{T}^{5/2}$ fixes $K\propto g^{1/2}{T}_e^{-2\beta -5/2}$.

For a plane-parallel atmosphere in radiative equilibrium, constant surface gravity of a star $g$ and opacity $\kappa ={\kappa }_0{P}^{\alpha -1}{T}^{4-4\beta }$, match the stellar structure interior where the radiative flux satisfies $F=\sigma T^4$. The grey atmosphere law then puts the surface at optical depth $\tau =2/3$ with $T^4=2T_0^4$. Integrating hydrostatic equilibrium in optical depth from $P(0)=0$ gives $P^{\alpha }=2\alpha g(T^{4\beta }-T_0^{4\beta })/(3\beta {\kappa }_0{T}_0^4)$. Multiplication by ${\kappa }_0{T}^{4-4\beta }/g$ proves the displayed boundary condition for $\alpha >0$, $\beta \ne 0$. At $\beta =0$ the integral is logarithmic and the continuous limit is $P\kappa /g=4\alpha \log 2/3$. The relation assumes negligible radiation pressure and describes the Eddington closure approximation, rather than exact angular transfer.

Inside a rooted $d$-ary subtree of a degree-$(d+1)$ regular tree, pair each cell root with one distinguished child. Keep reinfection inside each pair, suppress infection back into its parent cell, and allow each child cell to be activated only once. Offspring counts of different cells are independent and identically distributed, so cell genealogies form a Galton-Watson process. A cell root has $d-1$ outgoing child-cell edges and its paired vertex has $d$. Starting at the root, the probabilities to send an arrow along one specified outgoing edge from these two vertices before internal extinction are respectivelyThey follow by first-step equations on the three nonempty pair states. The offspring mean is $M=(d-1)a+db$, andContinuity makes $M>1$ at some strictly smaller positive rate. The branching-process extinction criterion then gives positive survival probability for the suppressed process, hence for the original contact process.

For positive rates in congestion control, put $x_r={\kappa }_r{z}_r^2/4$ and $V(z)=U(x(z))$, where $U$ is the primal congestion potential. The equation ${\dot{x}}_r={\kappa }_r{x}_r{\partial }_{r}U$ becomes ${\dot{z}}_r={\partial }_{r}V$. Positive logarithmic utility weights make ${\sum }_r{w}_r\log ({\kappa }_r{z}_r^2/4)$ strictly concave on the positive orthant. Each integrated nonnegative nondecreasing resource price is a convex nondecreasing function of its load; composing it with the convex quadratic load in $z$ preserves convexity. Hence $V$ is a strictly concave function, even for prices that are merely continuous. For two solutions, $\frac{1}{2}\frac{d}{dt}\Vert z-\tilde{z}{\Vert }^2=(z-\tilde{z})\cdot [\nabla V(z)-\nabla V(\tilde{z})]\le 0$. This proves uniqueness of the flow without assuming local Lipschitz continuity of the price functions.

For a cumulative distribution function $G$, let $Q(u)=\inf \{x:G(x)\ge u\}$, $0<u<1$. Right continuity gives $Q(u)\le x$ exactly when $u\le G(x)$, including at atoms. Thus applying $Q$ to independent uniform variables $U_i$ constructs an independent sample with distribution function $G$. Its empirical distribution function is $G_n(x)=F_n(G(x))$, where $F_n$ is the uniform-sample empirical distribution function. Consequently ${\sup }_x|G_n(x)-G(x)|\le {\sup }_{t\in [0,1]}|F_n(t)-t|$. For the uniform sample, monotonicity and a grid of mesh $1/m$ bound the last supremum by the largest error at grid points plus $1/m$. The strong law of large numbers on the countable union of all such grids proves the almost-sure limit. This construction proves the theorem for distributions with atoms as well as continuous distributions.

For $0<x<M$, a three-dimensional Bessel process stopped at $M$ has the law of Brownian motion conditioned to hit $M$ before zero and stopped at $M$. On $(\epsilon ,M)$, the Girsanov theorem and the Itô formula give terminal density $X_{\tau }/x$. Its value is $M/x$ at the upper exit and $\epsilon /x$ at the lower exit. Letting $\epsilon \downarrow 0$ in bounded stopped-path functionals gives density $(M/x)1_{\{T_M<T_0\}}$ relative to the corresponding stopped Wiener measure. The exit probability is $x/M$ by the optional stopping theorem, so the density is normalized. On the unstopped full path space the same density defines the conditioned Brownian law; pushing it forward by the stopping map gives the stopped Bessel law. The already-stopped law is not absolutely continuous relative to unstopped Wiener measure, since eventually constant paths have Wiener probability zero.

For canonically normalized fields with action $-\int [\frac{1}{4}{F}^2+\frac{1}{2}(\partial \cdot A{)}^2+\partial \bar{c}\cdot Dc]$, use signature $(-,+,+,+)$ and incoming Fourier modes $e^{ipx}$. The ghost-gluon vertex is $g{p}_{\mu }{f}_{abc}$ and the ghost propagator is $-i/(p^2-i0)$. The open ghost-line one-particle-irreducible two-point vertex at one loop is $i{g}^{2}C{I}_d(p)$, whereA counterterm ${\mathcal{L}}_{\mathrm{ct}}=-\delta Z_c\partial \bar{c}\cdot \partial c$ inserts $-i\delta Z_c{p}^2$ and cancels this pole. Equivalently, writing $d=4-2ϵ$ gives $\delta Z_c=g^{2}C/(32{\pi }^2ϵ)$. There is no closed ghost-loop sign for this open-line graph. The Feynman i-epsilon prescription, Fourier convention and antighost sign must be kept consistent throughout.

The differential operator $L=\frac{1}{2}{x}^2{\partial }_{xx}+\frac{1}{2}x{\partial }_x-\frac{1}{2}$ has solution operator $P_{t}g(x)=e^{-t/2}\mathbb{E}[g(x{e}^{B_t})]$. For $x\ne 0$ its integral kernel with respect to Lebesgue measure is $p(t,x,y)=e^{-t/2}(|y|\sqrt{2\pi t}{)}^{-1}\exp [-{\log }^2(|y/x|)/(2t)]$ when $xy>0$, and zero otherwise. For $x=0$ the kernel is the measure $e^{-t/2}{\delta }_0$, not a density. The Itô formula applied to $e^{-s/2}u(t-s,X_s)$ proves this Feynman-Kac formula.

For geometric-attempt exponential backoff, assume each attempted transmission fails independently with a fixed probability $q<1$. The probability of reaching stage $k$ is $q^k$, and its mean waiting time is $2^k/p_0$. The Tonelli theorem gives $\mathbb{E}T=p_0^{-1}{\sum }_{k\ge 0}(2q{)}^k$, finite exactly when $q<1/2$. The packet succeeds almost surely for every $q<1$, so almost-sure success does not guarantee finite mean delay. A fixed independent failure probability is a simplifying assumption, not a theorem about collisions in an interacting network.

// Target: probability-and-statistics.bigb

For sampling times $t_i$ and $\bar{t}=n^{-1}{\sum }_i{t}_i$, the geometric mean $A=({\prod }_i{S}_{t_i}{)}^{1/n}$ has $\log A$ normally distributed under the stock-numeraire measure in the Black-Scholes model. Its mean is $\log S_0+(\rho +{\sigma }^2/2)\bar{t}$, and its variance is $v={\sigma }^2{n}^{-2}{\sum }_{i,j}\min (t_i,t_j)$. For descending times the sum is ${\sum }_i(2i-1)t_i$. The drift shift comes from exponential tilting, and must be included when the delivered payment contains a factor $S_T$.

For $I_t={\int }_0^t\log S_{u}du$ and $G_T=\exp (T^{-1}{I}_T)$ in the Black-Scholes model, the conditional law of $\log G_T$ under the risk-neutral measure has a normal distribution with mean $m_t$ and variance $q_t$. The future Brownian integral equals ${\int }_t^T(T-u)d{W}_u$, so the Itô isometry gives the variance. For strike $K>0$ and $t<T$, the call price is $e^{-\rho (T-t)}[e^{m_t+q_t/2}\Phi (d_1)-K\Phi (d_2)]$, where $d_2=(m_t-\log K)/\sqrt{q_t}$ and $d_1=d_2+\sqrt{q_t}$. This follows by completing the square in the truncated normal exponential moment; the known past integral must be included in the mean.

Fix $p$ on a Riemannian surface. For sufficiently small $r>0$, the geodesic circles of radius $r$ centered at $p$ and at every sufficiently nearby distinct $q$ meet transversally at exactly two points. In a strongly convex normal neighbourhood, write $q={\exp }_p(\epsilon v)$ and $x(\theta )={\exp }_p(re(\theta ))$. First variation givesThe quotient of the bracketed function by $\epsilon$ extends smoothly and has two simple zeros at zero. The implicit function theorem, uniformly over unit $v$, preserves these two zeros and excludes others. The nonzero angular derivatives prove transversality. The distinct-centre hypothesis is essential: coincident circles are not transverse.

Blow up each distinct singular point of an integral projective curve in the projective plane once. The arithmetic genus drop under a point blowup and the genus-degree formula give arithmetic genus $(d-1)(d-2)/2-{\sum }_P{m}_P(m_P-1)/2$ for the strict transform. Its arithmetic genus is nonnegative, proving the bound in every characteristic. Infinitely near singularities need not have been resolved.

At unit speed on a Riemannian surface, the right-hand side is $⟨E,iv⟩iv$. This removes the velocity-parallel component of the forcing, so differentiating $|\dot{\gamma }{|}^2$ gives zero. The dynamics need not preserve the ordinary Liouville volume of a surface geodesic flow.

Write a tangent variation of a surface thermostat as $aF+yH+zV$ in the canonical frame. The commutators $[F,H]=-\lambda F+(K-H\lambda +{\lambda }^2)V$ and $[F,V]=-H-(V\lambda )V$ give $\dot{a}=\lambda y$ and the stated transverse equation. Near the vertical projective direction, $r=y/z$ obeys $\dot{r}=1-(V\lambda )r+(K-H\lambda +{\lambda }^2)r^2$.

For steady injection into a Keplerian disk between two zero-torque boundaries, conservation of mass gives $\dot{M}={\dot{M}}_{\mathrm{in}}+{\dot{M}}_{\mathrm{out}}$. Conservation of angular momentum gives $\dot{M}{h}_0={\dot{M}}_{\mathrm{in}}{h}_{\mathrm{in}}+{\dot{M}}_{\mathrm{out}}{h}_{\mathrm{out}}$, where $h(R)=\sqrt{GMR}$. Eliminating the outward mass flux gives the displayed inward fraction. This fraction does not depend on the viscosity law: the two boundary fluxes must carry away the injected mass and angular momentum. Equivalently, continuity of the viscous torque in an accretion disk joins $\nu \Sigma \sqrt{R}={\dot{M}}_{\mathrm{in}}(\sqrt{R}-\sqrt{R_{\mathrm{in}}})/(3\pi )$ inside the source to ${\dot{M}}_{\mathrm{out}}(\sqrt{R_{\mathrm{out}}}-\sqrt{R})/(3\pi )$ outside.

For total streamfunction $\Psi =-Uy+A{e}^{-(x^2+y^2)/a^2}$, the maximum opposing zonal disturbance speed is $\sqrt{2/e}|A|/a$. Below the displayed threshold the zonal velocity never changes sign and streamlines cannot close. Above it two stagnation points appear on the symmetry axis: one has definite Hessian and is a center, the other has indefinite Hessian and is a saddle. Closed contours surround the center. Equality is a degenerate onset, not a finite-area closed-cell regime.

When $a_{ij}=\exp ({\beta }_0+{\beta }^{\mathsf{T}}{x}_{ij})$, replacing $U_i$ by $c{U}_i$ and the fixed intercept by ${\beta }_0-\log c$ leaves every conditional count mean and the integrated data law unchanged. Thus without fixing the random multiplier's mean, the gamma random-intercept Poisson model identifies the marginal intercept ${\beta }_0+\log \tau$ and relative heterogeneity $\theta /{\tau }^2$, rather than these three parameters separately. Normalizing $\tau =1$ removes this scale ambiguity.

Each pion has G parity minus one, so an $n$-pion state has $G=(-1{)}^n$. In the isospin-symmetric strong interaction, a parent with positive G parity can decay only to an even pion multiplicity, and a negative parent only to an odd multiplicity. The rule does not forbid small isospin-breaking or electromagnetic amplitudes.

A probability measure-preserving system has the SZ property at level $k$ if the displayed lower limit is positive for every measurable set $A$ of positive measure. This convention counts $k$ nonzero shifts, hence progressions of length $k+1$. The Furstenberg correspondence principle turns this property at all levels into the Szemerédi theorem. A compact measure-preserving system has the property because syndetic near returns of an almost periodic observable applied to an indicator give uniformly positive multiple intersections at bounded-gap times.