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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

For zero initial velocity and Fourier amplitude $a(k)$, observation along $x=Vt$ with $0\le V<c$ selects the two stationary wavenumbers $\pm k_0$, whereWriting $s=\sqrt{c^2-V^2}$, the leading oscillation has angular frequency $As$ and amplitude proportional to $t^{-1/2}$.

In a flat fundamental square, straight horizontal, vertical and rational-slope lines project to closed geodesics. The central horizontal glide axis is one-sided, a vertical line is two-sided, and the line $(2t,t)$ closes with one transverse self-intersection.

Forthe transformations with even $c$ form an index-two translation subgroup. Its quotient is a torus, giving the orientation double cover $T^2\to K$.

If spatial reflection leaves a Stokes configuration and forcing equivalent to flow reversal while preserving one candidate velocity component, uniqueness forces that component to vanish.

The Penrose process extracts rotational energy by allowing a fragment with negative stationary Killing energy to enter the black hole while another fragment escapes with more energy than the original body carried.

Kernel principal component analysis diagonalizes a centered kernel matrix to perform principal component analysis in an implicit feature space.

For three transverse families of $R^{1/2}\times R^{1/2}\times R$ tubes in ${\mathbb{R}}^3$,

A generalized eigenvector lies in the kernel of a positive power of A minus lambda I.

Over an algebraically closed field, the characteristic polynomial gives the total size of the Jordan blocks for each eigenvalue, while the minimal polynomial gives the largest block size. For total dimension at most three, these two numbers determine the partition into block sizes: a multiplicity-three eigenvalue has partition $3$, $2+1$, or $1+1+1$ according as its largest block has size $3$, $2$, or $1$. Hence two matrices of size at most three with the same characteristic and minimal polynomials are similar.

If $\beta$ has eigenvalues $\lambda ,\mu \ne 0$ and is diagonalizable on a two-dimensional complex vector space, thenIf $\beta$ has one size-two Jordan block, then

A Jordan block has one eigenvalue on its diagonal and ones on its superdiagonal.

The geometric multiplicity of an eigenvalue is the dimension of its eigenspace. In Jordan form it equals the number of blocks carrying that eigenvalue.