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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.

For $d\ge 3$, 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.

Let $d\ge 3$, $p>1$ and $a=2/(p-1)<d-2$. Then the Generalized Lane-Emden equation has the positive singular solutionIndeed, the radial Laplacian satisfies $\Delta r^{-a}=a(a+2-d)r^{-a-2}$. Matching its exponent with $u^p$ gives $ap=a+2$; matching coefficients gives $C^{p-1}=a(d-2-a)>0$. The strict inequality on $a$ is exactly what makes that positive coefficient possible.

The solution is singular at the origin, so it is distinct from the regular Lane-Emden function normalized there. Under the Emden-Fowler transformation, it is the constant equilibrium $v=C$.

For a Gaussian field theory with quadratic kernel proportional to $t+k^2$, let $t>0$ be the distance from the critical point in rescaled units. The relevant heat capacity correction contains the integralwith fixed ultraviolet cutoff $\Lambda$. Radial integration followed by $k=\sqrt{t}x$ giveswhere $C_d$ is the area of the unit sphere in $d$ dimensions.

For $0<d<4$, the rescaled integral has a finite nonzero limit, so $I_d\propto t^{d/2-2}$. At $d=4$, its large-$x$ integrand behaves as $1/x$, givingFor $d>4$, the leading term instead tends to $I_d(0,\Lambda )=C_d{\Lambda }^{d-4}/(d-4)$. It is dominated by the short-distance cutoff. A bare power law cannot describe the full leading integral in every dimension. ^[ref]

The divergent correction below four dimensions warns that the neglected interactions can become important. It is not an exact interacting critical exponent; the Ginzburg criterion tests the validity of the approximation.

Consider $u_{tt}=c^2{u}_{xx}-c^2{A}^{2}u$ with $A,c>0$, initial displacement $f$ a Schwartz function, and zero initial velocity. With $a(k)={\int }_{\mathbb{R}}f(x)e^{-ikx}dx$,For a fixed $|V|<c$, set $s=\sqrt{c^2-V^2}$ and $k_0=AV/s$. The phases $Vk\pm \omega (k)$ have their unique stationary points at $\mp k_0$, with stationary values $\pm As$ and curvature magnitudes $d_0=s^3/(A{c}^2)$. The one-dimensional stationary-phase formula givesFor a real even initial profile, $a$ is real and even, and the bracket reduces to $2a(k_0)\cos (Ast+\pi /4)$. Thus the ray oscillation has angular frequency$As$ and generic decay$t^{-1/2}$. If the sampled amplitudes vanish, the displayed additive error still makes sense. The estimate is for a fixed subluminal ray; it is not uniform as $|V|\uparrow c$, where the selected wavenumbers escape to infinity.

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 of a linear operator $A$ for a scalar $\lambda$ is a nonzero vector $v$ with $(A-\lambda I{)}^{k}v=0$ for some positive integer $k$. An ordinary eigenvector is the case $k=1$.

For $A=\left(\begin{array}{cc}\lambda & 1\\ 0& \lambda \end{array}\right)$, the second coordinate vector $e_2$ is generalized: $(A-\lambda I)e_2=e_1$ and $(A-\lambda I{)}^2{e}_2=0$, although it is not an eigenvector. Such vectors form Jordan chains and complete an eigenvector set when the operator is not diagonalizable. Excluding the zero vector is part of the definition.

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.

The algebraic multiplicity of an eigenvalue is its multiplicity as a root of the characteristic polynomial. In Jordan form it is the sum of the sizes of all blocks carrying that eigenvalue.

If $F$ and $G$ have no explicit time dependence and satisfy $\{F,H\}=\{G,H\}=0$, the Jacobi identity gives $\{\{F,G\},H\}=0$. Their Poisson bracket is therefore also conserved.