For $Ly=(x^2{y}^{\prime }{)}^{\prime }-2y$ on $x>0$, choose solutions bounded at zero and infinity. The homogeneous Cauchy-Euler equation has basis $x,x^{-2}$. Joining the left-regular and right-regular solutions continuously and imposing $s^2[G_x{]}_{s^{-}}^{s^{+}}=1$ gives the displayed Green's function with ordinary integration measure $ds$. For bounded forcing,The integrals converge and give the unique bounded solution: no nonzero homogeneous combination is bounded at both ends. If $0\le f\le M$, negativity of the kernel gives $-M/2\le y\le 0$. Differentiation cancels the two moving-endpoint terms, leavingwhich implies $|y^{\prime }(x)|\le M/(3x)$. For constant forcing $f=M$, the bounded solution is $-M/2$; the endpoint conditions are boundedness, rather than zero endpoint values.

For $Ly=y^{\prime \prime }+y^{\prime }/x-y/x^2$ on $x>0$, impose $y(0)=0$ by a continuous limit and $y(x)\to 0$ at infinity. The adapted homogeneous solutions are $x$ at zero and $1/x$ at infinity, with Wronskian $-2/x$. The displayed Green function is continuous and its first derivative has jump one at $x=s$, so $L_{x}G(x,s)=\delta (x-s)$ with ordinary source measure $ds$. For continuous $f$ with ${\int }_0^{\infty }(1+s^2)|f(s)|ds<\infty$,This solves $Ly=f$ with the stated endpoint limits. No nonzero combination of $x,1/x$ meets both homogeneous endpoint conditions, proving uniqueness. Multiplying $L$ by $x$ gives the self-adjoint expression $(x{y}^{\prime }{)}^{\prime }-y/x$, explaining the weighted reciprocity $xG(x,s)=sG(s,x)$.

For a smooth bounded connected domain $V$ with boundary $S$, integrating the Green function equation and applying the divergence theorem forces ${\int }_S{\partial }_{n}GdS=1$. Thus zero Neumann boundary data are incompatible with an unmodified unit point source. The displayed constant boundary derivative has the required total flux. For $\Delta \Phi =\sigma$ and ${\partial }_n\Phi =f$, the compatibility condition is ${\int }_V\sigma dV={\int }_{S}fdS$. Green's second identity then yieldsThe last term is an undetermined additive constant. Adding a constant to $G$ does not change the other two terms together, because of the compatibility condition. This construction keeps the unit Dirac delta source; using a mean-subtracted source with zero normal derivative is a different normalization.

A static but spatially inhomogeneous linear operator has time translation symmetry without spatial translation symmetry. At fixed source position ${\mathbf{r}}^{\prime }$, one may still Fourier transform its Green function in $\mathbf{x}=\mathbf{r}-{\mathbf{r}}^{\prime }$. The coefficient retains ${\mathbf{r}}^{\prime }$ as a parameter. A multiplication by $V(\mathbf{r})$ becomes a convolution with $V_{{\mathbf{r}}^{\prime }}(\mathbf{q})=e^{i\mathbf{q}\cdot {\mathbf{r}}^{\prime }}\tilde{V}(\mathbf{q})$, not a spatially diagonal multiplier unless $V$ is constant. Suppressing the source-position subscript is only a notational convention.

An advanced Green function has support only before its source time. It is selected by a future boundary condition, rather than an initial causal response. For $L={\partial }_t^2+2\lambda {\partial }_t+{\omega }_0^2$ with $\lambda <0$ and ${\Omega }^2={\omega }_0^2-{\lambda }^2>0$, the real-axis inverse Fourier transform of $(-{\omega }^2-2i\lambda \omega +{\omega }_0^2{)}^{-1}$ is $-H(-t)e^{-\lambda t}\sin (\Omega t)/\Omega$. Its first derivative jumps by one at zero. A growing retarded solution exists with a different inverse contour; this illustrates causality versus stability of a Green function.

A static background permits a Fourier transform of time difference, but a spatially varying coefficient generally prevents dependence on position difference alone. A momentum-space Green function then carries an incoming and an outgoing momentum, or equivalently one momentum together with the source position. Multiplication by a background field becomes a convolution, with a source-position phase when the Fourier basis is written as $e^{i\mathbf{k}\cdot (\mathbf{r}-{\mathbf{r}}^{\prime })}$. A diagonal single-momentum representation is justified only with additional spatial translation symmetry or an explicitly fixed source origin.

For the inverse transform $G(\tau )=\int d\omega e^{-i\omega \tau }g(\omega )/(2\pi )$, positive $\tau$ requires closing a decaying contour below the real axis, clockwise; negative $\tau$ requires closing above, counterclockwise. Poles strictly below give a retarded Green function, while poles strictly above give an advanced Green function. Real-axis poles need a prescription. The sign convention in the Fourier exponential is essential.

gravity reduced to a reference elevation while retaining the gravitational contribution of the intervening topographic mass. The local elevation correction is approximately $2gh/R$ for a spherical reference, with sign chosen to undo the measured decrease with height.

A gravity anomaly after removing an approximate topographic mass contribution, often an infinite slab $2\pi G\rho h$, in addition to free-air reduction. Terrain and spherical-curvature corrections refine the slab model.

A spectral transfer estimate $Z(k)=⟨g_k{h}_k^{\ast }⟩/⟨|h_k{|}^2⟩$. Its wavelength dependence constrains compensation, density structure and support, but one transfer curve does not uniquely determine them.

A static emitter at radius $R>2GM/c^2$ and a static receiver at infinity compare photon wavelengths with $1+z=(1-2GM/(R{c}^2){)}^{-1/2}$. Conservation of the time-translation Killing energy and the static clock rate $d\tau =\sqrt{1-2GM/(r{c}^2)}dt$ prove the relation. Motion of the emitter adds a Doppler factor; a rotating stellar exterior may require a different metric.

In geometric optics without absorption, phase-space conservation preserves $I_{\nu }/{\nu }^3$ along a ray. Relative images of the same source at a common redshift therefore preserve surface brightness while changing apparent angular area and integrated flux. Source structure and propagation effects can introduce observed wavelength dependence.

The merger separates X-ray-emitting collisional gas from galaxies and lensing mass concentrations. Comparing those observables provides evidence for a dominant less-collisional unseen gravitating component, without identifying its particle species.

The gravitational lensing approximation in which deflection is concentrated in a plane whose thickness is small relative to the relevant observer-lens-source distances. The reduced angular deflection includes the distance ratio. A projected surface mass density then determines the lensing potential and the image-to-source mapping. This is distinct from a refracting thin optical lens.

In a spherical collapse model a freely falling surface can cross its event horizon after finite proper time, while outgoing signals received far away have increasing delay and redshift. Schwarzschild-coordinate freezing does not mean collapse physically stops or leaves a permanently bright surface at the horizon.

A collapse front propagates out through an initially singular isothermal envelope at $r_f=a_{T}t$. The only steady accretion scale is $\dot{M}\propto a_T^3/G\propto T^{3/2}$ for fixed mean particle mass. The front sweeps original mass at $2a_T^3/G$, but the exact central accretion coefficient requires the dynamical similarity solution and cannot be identified with front sweeping alone.

For an alphabet of size $q$, the order-$n$ directed graph has words of length $n-1$ as vertices and words of length $n$ as edges. An edge goes from its prefix of length $n-1$ to its suffix of that length. An Eulerian cycle therefore lists every length-$n$ word once as a cyclic window, producing a De Bruijn sequence.

A discrete process in which a vertex holding at least its degree in chips may send one chip along each incident edge. On the infinite line, a firing removes two chips from a site and gives one to each neighbor. When stabilization exists, commuting local firings explain why their order does not change the final stable configuration.

A path decomposition of a finite graph is a sequence of vertex sets, or bags, $B_1,\dots ,B_t$, such that every vertex occurs, every edge has both endpoints in a common bag, and the bags containing any fixed vertex form a consecutive interval. Its width is ${\max }_i|B_i|-1$; pathwidth is the minimum width.

A path with at least one edge has pathwidth one using bags $\{v_i,v_{i+1}\}$. A complete graph on $n$ vertices has pathwidth $n-1$. The consecutive-occurrence condition means a vertex can be forgotten permanently after its last bag, which supports dynamic programming that stores information only about the current bag. Pathwidth is at least treewidth because a path is a special tree.

An interval graph is a graph whose vertices can be represented by intervals on a line, with two vertices adjacent exactly when their intervals intersect. For example, intervals $[0,2],[1,3],[4,5]$ represent a single edge and an isolated vertex.

Interpreting an interval as the duration of a task makes an edge mean that the tasks overlap. A proper vertex coloring then assigns resources so simultaneous tasks use different resources. For interval graphs, the minimum number of colors equals the largest number of mutually overlapping intervals, their clique number. Sorting endpoints provides an efficient way to recognize and exploit this structure.