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At first order, geodesic deviation gives ${\ddot{\xi }}_i={\ddot{h}}_{ij}{X}^j/2$ for a small detector in a plane gravitational wave in linearized gravity. For physical initial displacement zero and initial velocity $V_i$, integration gives ${\xi }_i=[h_{ij}(t)-h_{ij}(0)]X^j/2+[V_i-{\dot{h}}_{ij}(0)X^j/2]t$. A bounded periodic response selects $V_i={\dot{h}}_{ij}(0)X^j/2$. Prescribed positions alone do not select that response when the wave has no wave-free past.

For a uniform rod of length $L$ and twist per unit length $\tau$, the end-rotation angle is $\phi =L\tau$. The chain rule gives $K_{\tau }=L{K}_{\phi }$, where $K_{\phi }$ is the torsional stiffness. If a source defines a normalized torque $M=N/(2\pi )$, its derivative is $dM/d\tau =K_{\tau }/(2\pi )$.

Let $V$ be a finite-dimensional symplectic vector space over an algebraically closed field of characteristic zero. For $a\in \mathfrak{sp}(V)$, write its Jordan–Chevalley decomposition as $a=S+N$. Applying the symplectic adjoint to this decomposition and using $a^{\ast }=-a$, uniqueness gives $S^{\ast }=-S$ and $N^{\ast }=-N$. Thus both parts belong to the symplectic Lie algebra. The operators $\mathrm{ad}S$ and $\mathrm{ad}N$ commute. The former is diagonalizable on $\mathrm{End}(V)$ in a matrix unit basis for an eigenbasis of $S$, and the latter is a nilpotent linear map, as follows fromThe restrictions to the invariant symplectic Lie algebra retain these properties. Hence these restrictions form the Jordan–Chevalley decomposition of $\mathrm{ad}a$.

If $\mathrm{ad}a$ is diagonalizable, $\mathrm{ad}N=0$. For every $v\in V$, the rank-one map $T_v(w)=\omega (v,w)v$ lies in $\mathfrak{sp}(V)$, since the two terms in $\omega (T_{v}w,u)+\omega (w,T_{v}u)$ cancel by alternation. For $v\ne 0$ its image is $kv$, so commuting with $T_v$ forces $N$ to preserve $kv$. A linear map preserving every line is scalar: compare its action on two linearly independent vectors and on their sum. Since $N$ is a nilpotent linear map, that scalar is zero. Thus $a=S$ is diagonalizable. Conversely, if $a$ is diagonalizable on $V$, its operator commutator action is diagonalizable on $\mathrm{End}(V)$ and therefore on its invariant symplectic Lie algebra. This proves both directions, including $V=0$.

Let $\mathfrak{a}$ be a toral subalgebra and an abelian Lie algebra in the symplectic Lie algebra, over an algebraically closed field of characteristic zero. Its defining representation has a weight decomposition. The symplectic Lie algebra identity gives $(\lambda (a)+\mu (a))\omega (v,w)=0$ for $v\in V_{\lambda }$, $w\in V_{\mu }$, so $\omega (V_{\lambda },V_{\mu })=0$ unless $\mu =-\lambda$. Since $\omega$ is a nondegenerate bilinear form, its restrictions give a perfect pairing between $V_{\lambda }$ and $V_{-\lambda }$, and a nondegenerate bilinear form on $V_0$.

An element of the centralizer of a Lie subalgebra preserves each weight space. On $V_0$ it is any element of $\mathfrak{sp}(V_0)$. On each nonzero pair, its action $A$ on $V_{\lambda }$ is arbitrary, and the perfect pairing uniquely forces the action on $V_{-\lambda }$ to be the negative dual map $-A^{\ast }$. These independent choices preserve Lie brackets, proving the displayed direct sum of Lie algebras; the sum runs over one representative per nonzero pair.

This centralizer of a Lie subalgebra is an abelian Lie algebra precisely when $V_0=0$ and all the nonzero weight spaces are one-dimensional. Indeed, ${\mathfrak{gl}}_m$ is nonabelian for $m\ge 2$ by $[E_{11},E_{12}]=E_{12}$, while every nonzero symplectic vector space contains a two-dimensional symplectic subspace on which $[\mathrm{diag}(1,-1),E_{12}]=2E_{12}$. In the stated case the surviving summands are copies of ${\mathfrak{gl}}_1$ and commute. Repeated weights of a representation, including a zero one, are therefore exactly what obstructs this commutativity.

A vector area element combines the scalar surface area element with a chosen unit normal vector. Reversing the order of the two parameter directions reverses its sign. Flux through an oriented surface is the integral of the dot product of a vector field with this vector element. For a graph $z=f(x,y)$, the upward choice is $(-f_x,-f_y,1)dxdy$.

If the dimensional linear drag acceleration is $-r\mathbf{u}$, then $r$ has units of inverse time and Stommel boundary layer balance $\beta v=-r{v}_x$ gives ${\delta }_S=r/\beta$. For returned volume flux $Q$ in depth $H$, the boundary speed is $Q/(H{\delta }_S)$ and relative vorticity is of order $Q/(H{\delta }_S^2)$. A gyre return varying along a distance $B$ has a generic nonlinear-to-planetary vorticity advection ratio $V_b/(\beta B{\delta }_S)$. Large relative vorticity alone does not force nonlinear advection: a perfectly parallel $v=v(x)$, $u=0$ has zero material relative-vorticity advection.

For streamfunction $\psi =\Psi (x)\cos (my)$ with impermeable walls at $x=0,L$, let $A=-F_0/(r{m}^2)$ and positive exponents satisfy $\lambda -\gamma =\beta /r$, $\lambda \gamma =m^2$. The exact profile is $\Psi =A[1-C_W{e}^{-\lambda x}-C_E{e}^{\gamma (x-L)}]$, where $C_W=(1-e^{-\gamma L})/(1-e^{-(\lambda +\gamma )L})$ and $C_E=(1-e^{-\lambda L})/(1-e^{-(\lambda +\gamma )L})$. Substitution proves the ordinary differential equation and both boundary conditions. The usual narrow western boundary current limit requires $\lambda L\gg 1$ and $\gamma L\ll 1$, with $\lambda \simeq \beta /r$ and $\gamma \simeq r{m}^2/\beta$; the opposite first inequality does not describe weak drag.

For equal-luminosity stars in a spherical isotropic equilibrium, both $I(R)/L$ and $I(R){\sigma }_{\mathrm{los}}^2(R)/L$ are Abel projections of $\nu$ and $\nu {\sigma }_r^2$. Separate inversion recovers the number density and radial stress; $M(r)=-r^2[d(\nu {\sigma }_r^2)/dr]/(G\nu )$. The stellar luminosity cancels in this mass ratio. Unknown anisotropy, noisy derivatives and limited outer coverage require additional modelling.

For a solenoidal vector field $J$ with compact support and suitable integrability, apply the divergence theorem to $x_i{x}_{j}J$. It gives zero boundary term and hence the displayed antisymmetry. Equivalently, for any constant vector $C$, the vector triple product identity yields $\int (C\cdot r)JdV=\frac{1}{2}C\times \int (J\times r)dV$. A piecewise smooth field must have no normal jump at its support boundary; otherwise its weak divergence contains a surface contribution and the argument fails.

A regime in which an initially descending slab becomes nearly horizontal beneath the overriding plate for a substantial inland distance before descending again. It can narrow or eliminate a hot mantle wedge, shift deformation inland and coincide with a volcanic gap. Incoming buoyant crust, hydration, upper-plate motion and mantle forces can all contribute; association with an oceanic ridge does not isolate one cause.

With ${\beta }_0=-ϵg_0$, the chain rule gives $\beta (g)={\beta }_{0}dg/d{g}_0$. Inverting the displayed coupling expansion yields $g_0=g-a_1{g}^2/ϵ+(2a_1^2-b_1)g^3/{ϵ}^2$. Substitution gives $\beta =-ϵg-a_1{g}^2+2(a_1^2-b_1)g^3/ϵ+O(g^4)$, so finiteness requires the displayed double-pole condition. An omitted simple-pole term $c_1{g}_0^3/ϵ$ supplies a finite $-2c_1{g}^3$ term; higher-pole consistency does not by itself determine the finite two-loop beta function.

The coefficient of a simple one-particle pole in a propagator is its pole residue. If $G(p)=i/[p^2-m_0^2-\Sigma (p^2)]$ and $m$ is the pole mass, then $Z_{\mathrm{pole}}=[1-{\Sigma }^{\prime }(m^2){]}^{-1}$. An on-shell renormalization scheme may impose unit residue by a finite or divergent wave-function renormalization; mere ultraviolet finiteness does not force the residue to be one.

The two-point, one-particle-irreducible correction for a scalar field. With an insertion denoted $-i\Sigma (p^2)$, the full scalar propagator is $i/[p^2-m^2-\Sigma (p^2)+i0]$. A divergent coefficient of $p^2$ requires wave-function renormalization; a momentum-independent divergent term requires a mass counterterm.

The proper two-point correction to the gluon propagator in QCD. At one loop it receives quark, gluon, Faddeev-Popov ghost field and four-gluon tadpole diagram contributions. The Slavnov-Taylor identity makes the total tensor transverse, $k_{\mu }{\Pi }^{\mu \nu ,ab}=0$, although the individual gluon and ghost loop contributions need not be transverse. The sign of the symbol must be tied to whether the amputated insertion is defined as $i\Pi$ or $-i\Pi$.

With query and key dimension $d_k$, form the score matrix $Q{K}^T/\sqrt{d_k}$, apply the softmax function row by row, and multiply by the values $V$. If query and key coordinates have independent unit variance, their dot product has variance $d_k$; division by $\sqrt{d_k}$ keeps that score scale of order one. Optional additive masks exclude forbidden keys before the softmax function.

Several independently learned query, key, and value projections apply scaled dot-product attention in parallel. Concatenating their outputs and applying a learned output projection allows different heads to mix sequence information using different learned representations.

Self-attention whose output at sequence position $t$ may depend only on positions at or before $t$. In a score matrix, set forbidden future-position scores to negative infinity before applying the softmax function; the corresponding weights become zero. This permits parallel computation over a known training sequence without allowing future tokens to influence earlier predictions.

A sedimentary rock composed predominantly of sand-sized grains. Cementation, porosity, pore fluid and effective stress strongly affect its seismic velocity.

A zircon grain eroded from its source and deposited as a sedimentary particle. Its crystallization age can precede deposition by a long interval.

A deposit formed in a lake. Fine lacustrine deposits can archive past water chemistry and preserve organic matter.