Expand a smooth wavefunction and potential in the normalized short-time Schrödinger kernel. Its odd moment vanishes and its second moment is $i\Delta t$, so one step changes the wavefunction by $i\Delta t{\psi }^{\prime \prime }/2-i\Delta tV\psi +O(\Delta t^2)$. Division by the time step gives the Time-dependent Schrödinger equation.

For a closed string, the currents $J_{\pm }=P\pm T{X}^{\prime }$ have vanishing mixed Poisson brackets. The densities ${\mathcal{H}}_{\pm }=J_{\pm }^2/(4T)$ therefore generate commuting constraint algebras. With opposite Fourier orientations their modes satisfy $\{L_m,L_n\}=-i(m-n)L_{m+n}$ and the identical relation for ${\tilde{L}}_n$. This is a direct sum of Lie algebras consisting of two Witt algebras.

At an initially resting instant with negligible pressure, sum the horizontal components of the magnetized-fluid tensor virial theorem. The horizontal magnetic trace is $-B_z^2/{\mu }_0$ and the gravitational force-stress trace is $({\partial }_z\Phi {)}^2/(4\pi G)$. Their negatives give the displayed radial second-moment balance. Vacuum field stresses contribute too; being initially at rest does not imply zero acceleration.

In a strictly planar nematic liquid crystal, a charge-one nematic disclination is a winding-two map into the circle ${\mathbb{RP}}^1$. Its nonzero fundamental group class prevents complete unwinding while the surrounding loop stays fixed, but permits splitting into two charge-$1/2$ defects. For separation $r_0\ll d\ll L$, the two near fields contribute $\pi K{\lambda }_0^2(1/4+1/4)\log (d/r_0)$ while their common charge-one far field contributes $\pi K{\lambda }_0^2\log (L/d)$. Compared with the unsplit defect, the change isIt is negative at sufficiently large scale separation. Thus integer-defect fission is energetically favorable despite conservation of total topological charge. This leading-log argument proves failure of global energetic stability, not absence of kinetic barriers or pinning. If escape into the third dimension is allowed, an integer winding can also become topologically trivial in ${\mathbb{RP}}^2$.

After time $t$, map remaining domains by $g_t-U_t$. The new mapping-out function at elapsed time $s$ is ${\hat{g}}_s(z)=g_{t+s}(g_t^{-1}(z+U_t))-U_t$. Direct differentiation of the Chordal Loewner equation proves that its driver is $U_{t+s}-U_t$, and its half-plane capacity is $2s$. Defining transformed hulls as complements of transformed domains avoids ambiguity at the old hull boundary.

The domains of a capacity-parametrized Loewner chain determine their hydrodynamically normalized mapping-out functions of compact H-hulls. At a surviving point, the Chordal Loewner equation gives $U_t=g_t(z)-2/{\partial }_t{g}_t(z)$. For a continuous driver and $t>0$, the left derivative uses only the hull past. High points $in$, $n\in \mathbb{N}$, survive up to any fixed finite time for sufficiently large $n$, making this a countable local reconstruction of the driver and its past filtration.

For a small secondary-to-primary mass ratio, the unstable exponent at the outer collinear equilibrium is $n_{\mathrm{bin}}\sqrt{1+2\sqrt{7}}$. Its inverse is approximately $0.0634$ binary periods. Reaching a finite target distance takes this time multiplied by the logarithm of the ratio of final to initial unstable amplitudes.

For purely meridional wavevectors, $k=0$, $l\ne 0$, the growth rate of a uniform meridional upper-layer flow $V$ over a resting lower layer isGrowth requires $V\ne 0$ and $0<l^2<2F$. The perturbation has no northward velocity, so planetary vorticity advection does not explicitly stabilize it. At fixed normalized wind stress curl, $V=W/\beta$ means stronger $|\beta |$ nevertheless reduces the shear and growth rate.

For the local flow ${\varphi }_t$ of a smooth vector field $X$, pull a tensor field $T$ back to the fixed base point before differentiating. Pullback uses $d{\varphi }_t^{-1}$ on vector factors and the dual of $d{\varphi }_t$ on covector factors. The chain rule gives ${\mathcal{L}}_{X}f=Xf$. In coordinates, $d{\varphi }_t^{-1}Y({\varphi }_t(x))=Y+t(dYX-dXY)+O(t^2)$, so ${\mathcal{L}}_{X}Y=[X,Y]$. Differentiation of pullback preserves tensor contractions and obeys the Leibniz rule for tensor products; hence this is the tensor derivation defining the Lie derivative of a tensor field.

For a smooth function $f$ and a vector field $X$, the Lie derivative is ${\mathcal{L}}_{X}f=X(f)=df(X)$. It is the derivative of $f$ along the local flow of $X$.

For vector fields $X,Y$, the Lie derivative is the Lie bracket of vector fields ${\mathcal{L}}_{X}Y=[X,Y]$. Pulling $Y$ back by the local flow of $X$ and differentiating gives $dY(X)-dX(Y)$. No affine connection is needed.

The Lie derivative of a tensor field satisfies $[{\mathcal{L}}_X,{\mathcal{L}}_Y]={\mathcal{L}}_{[X,Y]}$. On smooth functions this is the definition of the Lie bracket of vector fields; on vector fields it follows from the Jacobi identity. The Leibniz rule and contraction compatibility extend the equality to tensor fields. The cyclic double-commutator identity also follows directly from associativity of operator composition.

Wedge multiplication by a closed $(1,1)$ form induces maps on Dolbeault cohomology. If ${\omega }^{\prime }-\omega =\bar{\partial }\theta$, then $({\omega }^{\prime }{)}^k-{\omega }^k=\bar{\partial }(\theta \wedge {\sum }_{j=0}^{k-1}({\omega }^{\prime }{)}^j\wedge {\omega }^{k-1-j})$. Wedging with a closed representative shows that the maps agree. For a change of Kähler metric potential, take $\theta =-i\partial f$.

Represent total machine configuration transitions and tests on Church numerals and Church pairs, then use a fixed-point combinator to iterate until the halting test selects the output. Normal-order beta reduction follows the computation. A nonhalting machine gives an infinite required reduction, so the normal-order normalization theorem rules out a numeral normal form.

Let $E:y^2=f(x)$ with $f$ monic and separable, and let $T=(\theta ,0)$. Over $K$ containing $\theta$, the map ${\alpha }_T(P)=[x(P)-\theta ]$ for $P\ne O,T$, completed by ${\alpha }_T(O)=1$ and ${\alpha }_T(T)=[f^{\prime }(\theta )]$, is a group homomorphism to the square-class group of a field. A nonvertical chord has the product of its three $x-\theta$ values a square; a chord through $T$ instead has the product of the other two values equal to $f^{\prime }(\theta )$. These identities include tangencies.

The radical of a bilinear form of the Killing form is the ideal of a Lie algebra $\{x:B(x,y)=0\text{ for all }y\}$. It is zero for a complex semisimple Lie algebra. Every element of the center of a Lie algebra lies in this radical.

For an ideal of a Lie algebra $I$ and nondegenerate Killing form, invariance makes $I^{\perp }$ an ideal. An element of $I\cap I^{\perp }$ commutes with $I$, so the intersection is an abelian ideal. Abelian ideals lie in the radical of the Killing form, forcing the intersection to vanish. Thus $\mathfrak{g}=I\oplus I^{\perp }$ as commuting ideals. Iterating minimal nonzero ideals gives the direct-sum formulation of a semisimple Lie algebra.

A geometric Kähler potential is locally a real smooth function $\phi$ with $\omega =i\partial \bar{\partial }\phi$ for the Kähler form. Its complex Hessian must be positive definite to define a Kähler metric. Conventions sometimes absorb a factor of $1/2$ into $\phi$. Potentials differing by the real part of a holomorphic function determine the same form. This topic concerns ordinary complex geometry; the existing undisambiguated Kähler potential entry concerns superfields.

For a Kähler metric $g$ with complex structure $J$, the associated real two-form is $\omega (u,v)=g(Ju,v)$. It has type $(1,1)$, is positive in the sense $\omega (v,Jv)=g(v,v)>0$ for $v\ne 0$, and is closed. Conversely a real closed positive $(1,1)$ form determines a Kähler metric. A local geometric Kähler potential writes it as $i\partial \bar{\partial }\phi$.

At any point of a Kähler manifold, a complex-linear coordinate change normalizes the Hermitian metric matrix. Closedness makes its holomorphic first derivatives symmetric in the two unbarred indices. A quadratic holomorphic coordinate change removes those derivatives, and Hermitian symmetry removes the antiholomorphic derivatives. Conversely, having this first-order normalization at every point makes the fundamental Hermitian form closed.