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An autoregressive polynomial with a root at one gives a unit-root model. The basic example $X_t=X_{t-1}+{\epsilon }_t$ is a random walk, whose variance grows with time when the increments have positive variance. Differencing removes this root and recovers its white noise increments.

A weakly stationary process satisfying an autoregressive model can depend on future driving white noise. For $X_t=\varphi X_{t-1}+{\epsilon }_t$ with $|\varphi |>1$, the convergent in the sense of mean-square convergence solution is $X_t=-{\sum }_{j\ge 1}{\varphi }^{-j}{\epsilon }_{t+j}$. Its autocovariance is ${\sigma }^2{\varphi }^{-|h|}/({\varphi }^2-1)$. This is different from a causal time series.

A causal time series satisfying $\Phi (B)X_t={\epsilon }_t$ with nondegenerate white noise has a square-summable present-and-past filter. The identity $\Phi (z)\Psi (z)=1$ excludes roots inside the unit disk; a pole of $1/\Phi$ on its boundary also prevents square summability. Thus all roots of $\Phi$ lie strictly outside the unit circle.

The autoregressive polynomial is $\Phi (z)=1-{\sum }_{j=1}^p{\varphi }_j{z}^j$. Substituting the backshift operator gives the autoregressive filter. Zeros outside the unit disk give the causality root criterion for an autoregressive model; zeros on the unit circle obstruct a nondegenerate stationary innovation-driven solution.

With two fixed initial states and iid N(0,1) errors, the parameter likelihood is the product of conditional transition densities. For $X_0=X_1=0$, it is proportional to $\exp [-\frac{1}{2}{\sum }_{t=0}^{n-2}(x_{t+2}-a{x}_{t+1}-b{x}_t{)}^2]$. The initial observation is a point mass and the first nonzero observation has a parameter-independent likelihood term.

Fixing a Krull-Schmidt decomposition embeds $L={\prod }_a{\mathrm{GL}}_{m_a}(k)$ by its actions on multiplicity spaces. This splits reduction modulo the Jacobson radical. Hence ${\mathrm{Aut}}_Q(X)=(1+J({\mathrm{End}}_Q(X)))⋊L$, with a closed normal unipotent first factor. The explicit section works in every characteristic.

A group multiplier automaton accepts exactly the padded convolutions of words $u\otimes v$ with $u,v\in L$ and $\overline{u}x=\overline{v}$. The case $x=\epsilon$ recognizes equality in the group, rather than equality of literal words over an alphabet.

A private-value auction gives each bidder its own value for receiving an item; the value is determined by its own type rather than by another bidder's information. An independent private values model also assumes independence between types.

For a risk-neutral single-parameter bidder with incentive-compatible type reports, interim utility satisfies $u^{\prime }(\theta )=G(\theta )$ wherever the winning probability is continuous. Consequently expected payment is determined by allocation probabilities and the utility of the lowest type. The common normalization $u(0)=0$ must be justified, not obtained from allocation alone.

Each bidder submits one private bid; the highest bidder wins and pays its own bid, while losers pay nothing. A multi-unit version can award one item to each of the highest bidders, each paying its own bid. In a monotone symmetric equilibrium, a bidder can imitate another type's bid when checking incentive constraints.

In a standard all-pay auction, every player pays its bid or effort cost, and the highest bid receives the prize. With a value $v_i$, unit effort cost, and winning probability $x_i(b)$, player $i$ has quasilinear utility $v_i{x}_i(b)-b_i$. Equal highest bids require an explicit tie rule.

With well strength $\lambda$ and spacing $a$, negative-energy Bloch states obey $\tanh y\le 2y/(\lambda a)\le \coth y$, where $y=\kappa a/2$. There is one negative band: it reaches zero when $\lambda a\le 2$ and is separated from zero when $\lambda a>2$.

Complexifying the directional transport equation yields spectral solutions analytic inside and outside the unit circle. An integrating factor built from known attenuation removes its multiplication term. The limiting Green kernels encode weighted line measurements and transverse Hilbert transforms. Their jump gives a Riemann-Hilbert problem; its solution recovers the source from an asymptotic coefficient. A general primary treatment of this spectral construction is www.stat.uchicago.edu/~guillaumebal/PAPERS/AttRadTh.pdf .

For a thin hydrostatic isothermal atmosphere, $\Delta D\simeq 2R_p{N}_{H}H/R_{\star }^2$ and $H=k_{B}T/(\mu m_{u}g)$. Hence $T\simeq \mu m_{u}g\Delta D{R}_{\star }^2/(2k_B{R}_p{N}_H)$. A measured exoplanet transmission spectrum amplitude constrains $N_{H}H$, so gravity, mean molecular mass and the number $N_H$ of sampled atmospheric scale heights are additional inputs. In a clear fixed-composition model, $N_H\simeq \log ({\kappa }_{\mathrm{band}}/{\kappa }_{\mathrm{cont}})$; clouds or abundance gradients can change it.

For $P_0>0$, $\alpha \ne 0$, $T\ge T_0$ and physical $T>0$, the atmospheric pressure-temperature profile $P=P_0\exp (\alpha \sqrt{T-T_0})$ is equivalent toThe branch restriction is essential: squaring does not create a second physical branch. Its temperature gradient is $dT/d\log P=2\sqrt{T-T_0}/\alpha$. Thus, in hydrostatic equilibrium, $\alpha <0$ gives an atmospheric thermal inversion, while $\alpha >0$ gives outward cooling. At $T=T_0$ the inverse branch has zero derivative; $\alpha =0$ gives constant pressure and cannot describe a nontrivial hydrostatic vertical interval.

The lower part of a planetary atmosphere where temperature generally decreases outward and convection can carry intrinsic or surface heat. Its upper boundary is a tropopause. Stellar irradiation, composition gradients and clouds affect the extent of this region; pressure alone does not universally identify it.

The exosphere is the dilute outer atmospheric region in which a particle can travel a substantial distance without a collision. Its lower boundary, the exobase, is near equality of the mean free path and the local atmospheric scale height. Escaping and gravitationally bound trajectories can coexist.

Jeans escape is collisionless atmospheric escape by particles in the high-speed tail of a thermal distribution at the exobase. The Jeans escape parameter $\lambda =GMm/(k_{B}T{r}_e)$ controls the exponential suppression, and the Jeans escape flux is proportional to $(1+\lambda )e^{-\lambda }$. There is no exact positive-temperature threshold below which the tail vanishes. For $\lambda$ of order a few or smaller, a purely hydrostatic tail model can fail and hydrodynamic atmospheric escape becomes important.

Nonthermal atmospheric escape supplies particle energy through processes other than the local thermal distribution. Examples include dissociation producing fast atoms, ion pickup by a stellar-wind electric field, and sputtering caused by impacting energetic particles. Radiation pressure can also accelerate material in an extended exosphere.

For two uniform angular motions with $P_b<P_c$, an inner exoplanet transit at zero and next outer exoplanet transit at $\tau$, the next conjunction solves $(n_b-n_c)t+n_c\tau =2\pi$. Hence its longitude modulo $2\pi$ isFor an eccentric outer body this is only an approximation. Let $f_L,M_L$ be its line-of-sight anomalies and ${\Delta }_e(t)=f(t)-M(t)-(f_L-M_L)$. Exact true alignment instead requires $(n_b-n_c)t+n_c\tau -{\Delta }_e(t)=2\pi$. The correction follows directly from uniform mean anomaly and nonuniform true anomaly.