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For rational $a$, a multiple of its denominator gives exact integer recurrence. For irrational $a$, Weyl differencing and the Weyl criterion give an equidistributed sequence $a{m}^2$ modulo one. Thus every parameter belongs to one of the open sets $\{a:\Vert a{m}^2\Vert <ϵ\}$. Compactness of the circle group gives a finite subcover, hence a bound $N$ uniform in $a$; no explicit rate follows from this compactness step.

For $|a|\ll \mu$, the equal-density ion-neutral cubic has a rapidly damped mode ${\omega }_d=-2i\mu +i{a}^2/(4\mu )+O(a^4/{\mu }^3)$ and two propagating modes ${\omega }_{\pm }=\pm a/\sqrt{2}-i{a}^2/(8\mu )+O(a^3/{\mu }^2)$. The reduced phase speed uses the total density ${\rho }_i+{\rho }_n=2{\rho }_i$. The exact displacement ratio is ${\xi }_n/{\xi }_i=\mu /(\mu -i\omega )$. It is nearly $-1$ for the rapidly damped counterflow and nearly $1\pm ia/(\sqrt{2}\mu )$ for the propagating modes. Thus the latter have small ion-neutral slippage of relative size $|a|/\mu$, responsible for ambipolar damping.

For a characteristic-p local field with residue field ${\mathbb{F}}_q$, the simple roots of $X^q-X$ lift uniquely by Hensel lemma. Their lifts form a copy of ${\mathbb{F}}_q$, since qth powers preserve addition and multiplication. Choose a uniformizer $\pi$. Repeatedly subtracting the lifted residue and dividing by $\pi$ produces a unique convergent expansion ${\sum }_{i\ge 0}{a}_i{\pi }^i$ for each integral element. Allowing finitely many negative powers gives a valued-field isomorphism $K\cong {\mathbb{F}}_q((T))$, $T\mapsto \pi$.

First prove the Engel lemma by induction on the dimension of a Lie algebra of nilpotent endomorphisms. If $x^m=0$, then $\mathrm{ad}x=L_x-R_x$ on the endomorphism algebra has $(\mathrm{ad}x{)}^{2m-1}=0$, since left and right multiplication commute. Thus a proper Lie subalgebra $M$ acts nilpotently on $L/M$, and induction gives a nonzero class normalized by $M$. This proves the Engel normalizer lemma.

For a maximal proper $M$, its normalizer is all of $L$, so $M$ is an ideal. Maximality then forces $\dim L/M=1$, since every line in a Lie algebra is a subalgebra. Induction gives a nonzero common annihilator $W$ of $M$ in the representation space. Ideality makes $W$ stable under $L$. A generator $x$ of $L/M$ is nilpotent on $W$ and has nonzero kernel there, providing a vector killed by all of $L$. In the possibly nonfaithful quotient actions, apply induction to the image, whose dimension is at most $\dim M$.

Repeatedly apply this common-kernel statement to quotient representation spaces to obtain a complete flag with $x{V}_j\subseteq V_{j-1}$. Hence the operators are simultaneously strictly upper triangular. Applying this to the Adjoint representation shows that a finite-dimensional Lie algebra whose every adjoint map is nilpotent has terminating lower central series. Conversely, termination of that series immediately makes every adjoint map nilpotent. This proves Engel theorem.

// Target: semisimple-lie-algebra.bigb

For dipole pressure $\delta p\sim Cx/(x^2+y^2)$, choose a rectangular control volume whose transverse size tends to infinity faster than its streamwise size. Each end face contributes $\pi C$ to $\int \delta p{n}_{x}ds$, so fixed-throughflow boundary work gives $\Delta \mathcal{D}=-2\pi UC$, the displayed result. This control-surface choice removes the integrated far-field velocity correction. A circular cutoff of the pressure-only functional gives $-\pi UC$ instead and does not implement the same throughflow normalization.

After eliminating a vertex set $E$, a generic edge between remaining vertices $i,j$ exists exactly when the original matrix graph contains a path from $i$ to $j$ whose internal vertices all belong to $E$. Inductively, the new edges at pivot $k$ concatenate two such paths through $k$; conversely split any allowed path at $k$, or retain it when it avoids $k$. This characterizes fill by graph reachability.

An electrical conductor supports mobile charge carriers and hence an electric current under a field. In a band description, a partially filled band or available overlapping bands supply nearby accessible states. The contrast with a band insulator depends on band filling as well as spectral gaps; semiconductors provide carriers through thermal excitation or doping.

A uniaxial electrical conductivity has one distinguished direction $n$ and equal response in its orthogonal plane. Its eigenvalues are $\alpha$ in $n^{\perp }$ and $\alpha +\gamma$ along $n$, so it is invertible precisely when $\alpha (\alpha +\gamma )\ne 0$. For an electric field perpendicular to $n$, the electric current density is $J=\alpha E$. Positive dissipation requires both eigenvalues to be nonnegative, a physical constraint separate from algebraic invertibility. For $D_{ij}={ϵ}_{ijk}{n}_k$, one has $D^2=n{n}^{\mathsf{T}}-I$, and $\sigma D^2=-\sigma$ is equivalent to $\gamma =-\alpha$; the conductivity then vanishes along $n$.

Electrical resistivity is the inverse of scalar electrical conductivity in an isotropic Ohmic medium. In its rest frame, $\mathbf{E}={\rho }_{\mathrm{el}}\mathbf{j}$, so the electric field does work per unit volume $\mathbf{j}\cdot \mathbf{E}={\rho }_{\mathrm{el}}|\mathbf{j}{|}^2$. In Gaussian units, finite conductivity gives magnetic diffusivity $\eta =c^2/(4\pi \sigma )$; neglecting this diffusion is a distinct approximation from neglecting electron inertia in Hall magnetohydrodynamics.

Complex conductivity relates the complex amplitudes of current density and electric field at an angular frequency $\omega$ in sinusoidal response. Its modulus gives the amplitude ratio, while its argument encodes their relative phase. Both the exponential time convention and whether amplitudes are peak or root mean square values must be stated when interpreting phase or average power.

Select the three downgoing or depth-decaying elastic modes for a solid below a plane free surface. For each mode use its reduced traction column $b=(Tq+R)a$. Zero total boundary traction gives $BA=-b_0$ for an incident mode of reduced traction $b_0$. Invertibility gives the displayed amplitudes. A singular matrix instead permits an outgoing homogeneous surface mode, so a radiation-limit or solvability analysis is necessary.

Given a random-cluster configuration, its components receive independent uniform colors under the Edwards-Sokal coupling. Two vertices have equal Potts spins with probability $1$ if connected and $1/q$ otherwise. Averaging proves the identity. With a wired component fixed to color $1$, the same argument gives normalized spontaneous magnetization $(q{\pi }^1({\sigma }_0=1)-1)/(q-1)={\varphi }^{\mathrm{w}}(0\leftrightarrow \infty )$ in infinite volume. Thus its onset is the wired random-cluster critical probability, transformed to ${\beta }_c=-\log (1-p_c)$ in this convention.

The Cornish-Fisher expansion inverts an Edgeworth expansion to approximate quantiles. For a standardized iid sum, let ${\gamma }_1$ be the skewness of one summand and $z_{\alpha }$ the standard normal quantile. The first distribution function correction is $-{\gamma }_1(z^2-1)\varphi (z)/(6\sqrt{n})$. Substituting $y_{\alpha }=z_{\alpha }+c/\sqrt{n}$ and expanding $\Phi$ gives $c={\gamma }_1(z_{\alpha }^2-1)/6$. This is a fixed-quantile asymptotic, with $\alpha$ fixed in $(0,1)$.

For a standardized iid sum with a regular Edgeworth expansion, the first distribution correction is proportional to $\varphi (t)(t^2-1)$ and is even. Subtracting the two endpoint distribution functions therefore cancels the $n^{-1/2}$ contribution. Fixed symmetric pivot cutoffs can consequently have $O(n^{-1})$ coverage error even though each one-sided approximation has error $O(n^{-1/2})$. This cancellation does not automatically hold for asymmetric or estimated random cutoffs.

Use the rational principal-parts resolution on an algebraic curve to represent a class by finitely many rational differential principal parts. Sum their algebraic residues of a rational differential. The sum of residues of a global rational differential is zero: trace to the projective line, where partial fractions prove the assertion. Hence the sum descends to cohomology. Pairing an invertible sheaf with its differential-valued dual gives the perfect Serre duality pairing. Since the global regular functions on an integral projective curve over an algebraically closed field are $k$, evaluation at the constant function $1$ identifies this trace canonically with an isomorphism to $k$.

The elapsed-time $h>0$ transition distribution of a driftless square-root diffusion starting at $x$ is the compound Poisson distributionwhere all variables on the right are independent and the empty sum is zero. Here the exponential distribution parameter is its rate. The Laplace transform of this sum iswhich is the Laplace transform of the driftless square-root diffusion; uniqueness of the Laplace transform proves the distributional identity. In particular, for $0\le v<2/h$, the exponential moment isFor $x>0$ this exponential moment is infinite when $v\ge 2/h$, since the event $N=1$ has positive probability and one exponential jump already has an infinite moment there. At $x=0$ the process remains zero and every such moment is one.

In a circular binary star let the donor star lose mass ${\dot{M}}_1<0$, with constant retained fraction $f$ and ${\dot{M}}_2=-f{\dot{M}}_1$. A donor star stellar wind escaping with the donor star's orbital specific angular momentum has $\dot{J}/J=(1-f){\dot{M}}_1{M}_2/[M_1(M_1+M_2)]$. Differentiating $J=G^{2/3}(P_{\mathrm{orb}}/2\pi {)}^{1/3}{M}_1{M}_2(M_1+M_2{)}^{-1/3}$ and using this loss law givesFor constant $f$ this integrates to $P_{\mathrm{orb}}{M}_1^{3f}{M}_2^3(M_1+M_2{)}^2=\mathrm{constant}$. If $f$ varies, the differential relation remains valid but the first factor must be replaced by $\exp (3\int fd\log M_1)$; treating a time-dependent exponent as a fixed power introduces an extra $\dot{f}\log M_1$ term.

Iteration of the pricing Euler equation givesThe displayed fundamental price follows when the future dividend series is integrable and ${\mathbb{E}}_t[{\zeta }_N{S}_N]\to 0$. Without this dividend-price transversality condition, a deflated price martingale component can survive. For individual holdings, the analogous terminal term is $\mathbb{E}[{\zeta }_N{S}_N{\theta }_{N+1}]$. Telescoping the discrete dividend budget equation gives the initial wealth cost of consumption minus this terminal term; its vanishing is needed for budget equality.

Nematic splay describes local spreading or convergence of the nematic director field. Its leading elastic cost is quadratic in the divergence and invariant under the head-tail identification $\mathbf{n}\sim -\mathbf{n}$.

Nematic twist rotates the nematic director around an axis perpendicular to it. The helix $\mathbf{n}=(\cos qz,\sin qz,0)$ has zero nematic splay and nematic bend, and twist density $K_2{q}^2/2$.