The infinitesimal base change action on quiver representations is ${\xi }_x(u{)}_{\rho }=u_{t(\rho )}{x}_{\rho }-x_{\rho }{u}_{s(\rho )}$. Its image is the Zariski tangent space to the orbit, because the stabilizer is a smooth open subset of the endomorphism ring. The extension complex of quiver representations identifies the quotient of the ambient tangent space by this image with ${\mathrm{Ext}}_Q^1(X,X)$. Consequently rigid quiver representations have open orbits.

The stellar-to-halo mass ratio compares a galaxy's galactic stellar mass with its host's total halo mass. In a simple baryon-conversion model it is ${ϵ}_{\ast }=f_b{f}_{\ast }$. Observational comparisons must distinguish this total-mass ratio from the fraction of baryons converted, and must specify whether central, satellite or all stellar populations are included.

A monotonic luminosity assignment to a counted halo population converts its differential abundance per mass into a luminosity function by a change of variable. If $n_h\propto M_h^{-a}$ and $L\propto M_h^p$, then $\Phi \propto L^{-1-(a-1)/p}$. This simple mapping assumes one counted galaxy per halo, no scatter and a fixed selection. A mass-dependent baryon conversion efficiency of a halo, satellites and stellar-population differences modify it.

For $n,A>0$, the one-dimensional porous medium equation $h_t=A(h^n{h}_x{)}_x$ has a volume-preserving similarity solution $h=[c_n(R^2-x^2)/(At){]}_{+}^{1/n}$, where $c_n=n/[2(n+2)]$ and $R\propto t^{1/(n+2)}$. On a reflecting half-line with conserved area $M$, $R=[M(At/c_n{)}^{1/n}/I_n{]}^{n/(n+2)}$, where $I_n=\frac{1}{2}B(\frac{1}{2},1+1/n)$. For a whole symmetric release use half its total area as $M$. The Beta function fixes normalization, while finite-front zero volume flux fixes the edge condition.

The value of an account that continuously reinvests at the short rate $r$, with $B_0=1$. Its reciprocal is the discount factor $D_t=B_t^{-1}$.

A dense additive subgroup of a complete metrizable topological group cannot be proper if it is comeagre. Every translate is comeagre and meets the subgroup, so every translating element is a difference of two subgroup elements. Applied to a topologically complete normed space inside its completion, this proves that its original norm is complete.

As $\delta \uparrow 1$, both active players' effective prizes in every nonfinal subgame tend to the marginal valuation $v_{r+1}$. The two-player complete-information all-pay equilibrium then gives each a winning probability $1/2$. In the final stage, actual valuations determine the asymmetric winning probabilities. Taking this limit from discounted equilibria specifies the continuation selection instead of independently choosing an undiscounted game equilibrium.

A constant surviving buoyancy flux sets the amplitudes of a pure plume: $Q\propto F^{1/3}$ and $M\propto F^{2/3}$. Relative-flux perturbations have the neutral ratio $(\delta Q/Q,\delta M/M,\delta F/F)\propto (1,2,3)$. This is the zero-eigenvalue limit of growing mode of power-law plume similarity; nonzero endpoint stratification can still create cumulative logarithmic flux loss.

For pi-normalized top-hat fluxes, $b=Q/\sqrt{M}$. The pure similarity solution therefore has $b=(6\alpha /5)(z+z_0)$, so the radius grows linearly with slope $6\alpha /5$. The corresponding plume virtual origin is $-z_0$ and a finite source has $z_0=5b_s/(6\alpha )$. This law translates geometric plume-contact criteria into heights.

The six-element dihedral group generated by the displayed rotations already forces every invariant second-rank tensor to have the same form as an axially invariant second-rank tensor. The axial threefold rotation eliminates mixed entries and leaves only a planar scalar plus a planar antisymmetric part; the transverse half-turn eliminates the latter. No smaller finite subgroup suffices: cyclic rotation groups retain an axial antisymmetric tensor, while a four-element noncyclic rotation group retains arbitrary diagonal tensors along its three mutually perpendicular half-turn axes.

In strict axial gauge, the gauge-boson propagator obeys $n^{\mu }{D}_{\mu \nu }(k)=0$ and contains poles at $n\cdot k=0$. Its contraction with conserved external currents removes the gauge-choice terms. Consistent calculations require an axial-gauge pole prescription.

The inverse of $n\cdot \partial$ and the axial-gauge propagator are singular when $n\cdot k=0$. A pole prescription and residual-gauge boundary conditions specify these inverses. They must be compatible throughout a calculation; formal ghost decoupling alone does not specify them.

The defining averaging representation is equivalent to $\Vert Tz-Tw{\Vert }^2\le \Vert z-w{\Vert }^2-(1-\alpha )\Vert (I-T)z-(I-T)w{\Vert }^2/\alpha$. Expanding the norm of the convex combination proves both directions. Substituting a fixed point for $w$ gives Fejér monotonicity and summability of squared residuals.

In finite-dimensional Euclidean space, iterating an averaged operator with a nonempty fixed-point set converges in norm to a fixed point from every start. The averaged-operator inequality proves boundedness and a vanishing residual; compactness produces a fixed cluster point and Fejér monotonicity turns subsequential convergence into convergence of the full sequence. A fixed point must exist.

With nondegenerate white noise, a two-sided AR(1) equation has a unique weakly stationary solution exactly when $|\varphi |\ne 1$. The solution is causal for $|\varphi |<1$ and anticausal for $|\varphi |>1$. At $\varphi =\pm 1$, an n-term noise sum has variance $n{\sigma }^2$, while its difference-of-stationary-values representation has variance at most four times the stationary variance, a contradiction.

For a causal AR(1) plus uncorrelated observation white noise, cross covariances vanish by L2 convergence of the autoregressive noise expansion. Its covariance is ${\sigma }_z^2{\varphi }^{|h|}/(1-{\varphi }^2)+{\sigma }_w^2{1}_{\{h=0\}}$. The additional noise changes only lag zero, attenuating the normalized positive-lag correlations.

A unit root is a zero of an autoregressive polynomial on the unit circle. A zero at $1$ is removed by first differencing. A conjugate pair $e^{\pm i\omega }$ corresponds to the real factor $1-2\cos \omega B+B^2$. Nonzero white-noise excitation at an uncancelled unit root prevents a finite-variance stationary solution.

The group ring of an infinite cyclic group is the Laurent polynomial ring $R=\mathbb{Z}[t,t^{-1}]$, with augmentation $\epsilon (t)=1$. Multiply any element in the kernel by a power of $t$ to obtain a polynomial vanishing at one. The factor theorem makes it divisible by $t-1$. Thus the augmentation ideal is $(t-1)R$. Since $R$ is an integral domain, multiplication by $t-1$ is an injective module map, so this ideal is a free $R$-module of rank one, even though its additive group has infinite rank.

In finite dimensions, if the graph is locally closed at $(\bar{u},\bar{v})$, the Aubin property holds exactly when $D^{\ast }S(\bar{u}\mid \bar{v})(0)=\{0\}$. Thus no nonzero horizontal vector in the limiting normal cone may occur. The exact local Lipschitz modulus is the outer norm of this coderivative.

In hot hydrogen-rich thermochemical equilibrium, much carbon can be bound in carbon monoxide, consuming oxygen that is unavailable for water. At a C/O ratio near or above one, water can be strongly depleted. Cooler methane-dominated chemistry can release more oxygen for water at the same elemental inventory. Pressure, quenching, dissociation and atmospheric photochemistry can modify this temperature-dependent partition.