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The derived series is ${\mathfrak{g}}^{(0)}=\mathfrak{g}$ and ${\mathfrak{g}}^{(n+1)}=[{\mathfrak{g}}^{(n)},{\mathfrak{g}}^{(n)}]$.

A vector subspace $I\subseteq \mathfrak{g}$ is an ideal when $[\mathfrak{g},I]\subseteq I$. It is therefore the kernel of a Lie-algebra quotient map.

A Lie algebra is abelian when every Lie bracket vanishes.

The Lie bracket is the product in a Lie algebra: a bilinear map with an Alternating Lie bracket and the Jacobi identity. Bilinearity means, for $a,b\in k$,Unlike matrix multiplication, a Lie bracket need not satisfy associativity. In the sl2 matrix commutator example, $[H,[E,F]]=0$ but $[[H,E],F]=2H$. The replacement compatibility law is the Jacobi identity, which makes bracketing with a fixed element a derivation of a Lie algebra.

For Matrix Lie groups, the Lie bracket is the matrix commutator $[X,Y]=XY-YX$ on their tangent space. For the cross-product Lie algebra, it is $[x,y]=x\times y$. Both products measure a failure of infinitesimal motions to commute. In a real Lie group, a zero Lie bracket means that the two corresponding one-parameter subgroups commute for all real parameters.

If a Lévy process has mean zero and $\mathrm{Var}(X_1)={\sigma }^2<\infty$, then $\mathrm{Var}(X_t)=t{\sigma }^2$ and $X_t^2-t{\sigma }^2$ is a martingale.

The intersection $L^1\cap L^2$ is Banach for $\Vert f{\Vert }_1+\Vert f{\Vert }_2$, but it is generally incomplete when equipped with the $L^1$ norm alone. Truncations of an $L^1$ function outside $L^2$ provide an $L^1$-Cauchy sequence with no limit in the intersection.

For $1\le p\le \infty$, the Lebesgue space $L^p$ is complete in its usual norm.

For a small-noise diffusion $dX=a(X)dt+\sqrt{ϵ}\sigma (X)dW$, the Freidlin--Wentzell action of a smooth path is

A rate function is a lower-semicontinuous map $I$ into $[0,\infty ]$. It is good when every sublevel set $\{x:I(x)\le c\}$ is compact.

Axisymmetric harmonic functions separate into radial powers times Legendre polynomials.

A Dirichlet Laplacian eigenfunction vanishes on the boundary and satisfies $-\Delta u=\lambda u$.

The normalized order-zero function has the representation

For the Laguerre equationthe ansatz $y={\int }_{\gamma }{e}^{zt}f(t)dt$ givesThe contour must make the endpoint contributionvanish.

For $x>0$, substituting $y(x)={\int }_{\gamma }{e}^{xt}f(t)dt$ into $x{y}^{(3)}+2y=0$ and integrating by parts givesThe amplitude equation $(t^{3}f{)}^{\prime }=2f$ has solution $f(t)=C{t}^{-3}{e}^{-1/t^2}$. On $\gamma =(-\infty ,0)$ both endpoint terms vanish, and hence

In plane polar coordinates, Laplace's equation isSeparation of variables gives radial powers $r^n,r^{-n}$ for positive angular modes and $1,\log r$ for the zero mode.

In three dimensions,Consequently the decaying solution of $-{\nabla }^{2}u=f$ in free space is

This is the free-space Green function for the negative Laplace operator.

At the energy-critical exponent $p=(d+2)/(d-2)$, the normalized Aubin-Talenti bubblesolves $\Delta W+W^p=0$ and optimizes the critical Sobolev embedding theorem.

When $a=2/(p-1)$ and $a<d-2$, the Lane-Emden equation has the singular scale-invariant solution

For a quadratic kernel $K(t,k)\sim t+k^2$, the singular heat-capacity correction is proportional to

For an objective $f(x)$ with equality constraints $g_i(x)=0$, the Lagrangian function is