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A causal effect compares potential outcomes under different specified interventions. For a binary treatment, a unit-level additive effect is $Y(1)-Y(0)$. Only one of those outcomes is observed for a given unit, so an observed treated-versus-untreated difference is not automatically the effect. For example, patients receiving treatment can differ in baseline severity from untreated patients. The average treatment effect averages the contrast over a target population; identifying it from data requires assumptions linking treatment assignment and the unobserved potential outcomes.

The escape velocity is the least initial speed that permits a freely coasting body to move arbitrarily far from an isolated attracting body. Despite its conventional name, it is a speed, not a unique velocity vector. For a fixed point mass $M$, or outside a body with spherical symmetry, set Newtonian gravitational potential energy zero at infinity. An exterior test particle of constant mass $m\ll M$ then hasHere $\mu =GM$ is the standard gravitational parameter, and $r$ is distance from the centre, not height above the surface. Set $r=R$ for surface escape velocity. Evaluate velocity in the body's rest inertial frame. Neglect drag, later rocket thrust, other bodies' gravity, and relativistic effects.^[ref]

Conservation of energy gives the threshold by requiring $E\ge 0$: at infinity the potential energy tends to zero and the kinetic energy cannot be negative. If $E=0$, the outward motion loses speed continuously toward zero. If $E>0$, the asymptotic speed is $v_{\infty }=\sqrt{2E/m}=\sqrt{v_0^2-v_{\mathrm{esc}}(r_0{)}^2}$. If $E<0$, infinity is inaccessible. For a radial outward launch, its outer turning point is $r_{\max }=-GMm/E$. For nonradial motion use angular momentum and the escape energy threshold to distinguish zero radial velocity from zero total speed.

Direction still determines the orbit and whether the body hits an obstacle. An initially inward velocity can cause collision even when $E\ge 0$. For an isolated spherical body, an exterior outward launch, or a tangential launch at or above escape velocity, has an outward escape branch without intersecting the body. A parabolic trajectory is the nonradial threshold path; a hyperbolic Kepler orbit has positive energy. The formula specifies their required local speed, not their orientation.^[ref]

The potential energyis even, has a cusp at its minimum and approaches the finite escape level $V_{\infty }$. Away from the origin, its force isFor a classical particle of mass $m$ and energy $0<E<V_{\infty }$, the turning points are $x=\pm x_T$, where $x_T=-\lambda \log (1-E/V_{\infty })$. The conservation of mechanical energy gives the oscillation periodThe second equality follows by substituting $y=\sqrt{V_{\infty }{e}^{-x/\lambda }-(V_{\infty }-E)}$.

At small energy, $T\sim 4\lambda \sqrt{2mE}/V_{\infty }$, rather than approaching a constant harmonic oscillator period: the cusp makes the leading restoring force approximately constant in magnitude on each side. As $E\to V_{\infty }^{-}$, the period diverges. For $E>V_{\infty }$, no finite turning point exists and the particle can escape.

Here inverse-square potential means the potential energy associated with an attractive inverse-square law for a central force: the potential energy itself varies as inverse distance. For $k>0$ and position $\mathbf{r}$ with $r=|\mathbf{r}|>0$,The gradient makes the distinction explicit: doubling $r$ halves the magnitude of potential energy but quarters the magnitude of force. The neighboring repulsive inverse-square potential instead has potential energy proportional to $r^{-2}$ and an inverse-cube force; the two naming conventions must be distinguished.

For Newton's law of universal gravitation, $k=GMm$, where $G$ is the gravitational constant. Treat the attracting mass $M$ as an isolated fixed point mass, or as a body with spherical symmetry and exterior test particle of constant mass $m\ll M$. The spherical shell theorem gives the same exterior Newtonian gravitational potential energy $U=-GMm/r$ without requiring uniform mass density. Here $r$ is measured from the centre, and the formula applies outside the body; it does not describe its interior. The Newtonian gravitational potential per unit mass is $\Phi =U/m=-GM/r$.^[ref]

The negative sign records binding relative to infinite separation, rather than a negative kinetic energy. Moving from $r_1$ to $r_2>r_1$ increases potential energy byBecause the force is a conservative force, this work depends on the endpoints, including for a curved path. Integrating $dV/dr=k/r^2$ with zero at infinity gives $V=-k/r$; adding a constant leaves the force unchanged but shifts the numerical escape energy threshold by that same constant. This is why the reference at infinity must accompany an escape velocity calculation.

For an exact synthetic comparison, stipulate the standard gravitational parameter $\mu =GM=8{\mathrm{m}}^3/{\mathrm{s}}^2$ and a $1\mathrm{kg}$ test particle. At $r=2\mathrm{m}$, $U=-4\mathrm{J}$ and the inward force has magnitude $2\mathrm{N}$; at $r=4\mathrm{m}$, $U=-2\mathrm{J}$ and the inward force has magnitude $0.5\mathrm{N}$. The increase is $2\mathrm{J}$ of potential energy. These stipulated parameters are an arithmetic example, not observations of a particular body. The escape-energy balance diagram uses the same parameters.

A force is conservative when its work depends only on the endpoints. On a simply connected domain this is equivalent to $F=-\nabla V$ for a potential energy $V$.

For a positive semidefinite matrix, a trace bound is a bound on the sum of its nonnegative eigenvalues.

In a strictly stationary spacetime, the timelike Killing flow gives a preferred positive-frequency splitting. The corresponding vacuum is annihilated by every annihilation operator associated with a positive-frequency mode.

LetThenThus the one-soliton Schrodinger operator is paired with the free Schrodinger operator. Its normalizable state annihilated by $A$ is proportional to $\mathrm{sech}(\chi x)$.

In a neutrally stratified polytropic atmosphere, vertically trapped polynomial solutions of degree $n\ge 1$ describe acoustic pressure modes. The degree-zero member is the surface-gravity mode.

A surface gravito-inertial wave combines free surface gravitational restoration with the Coriolis force. For a uniform incompressible fluid occupying $z<0$, take gravity downward, rotation $\mathbf{\Omega }=\Omega {\mathbf{e}}_z$ normal to the surface and $f=2\Omega$. A mode proportional to $e^{i(kx-\omega t)+qz}$ must decay downward, so $q>0$.

The linear rotating momentum equations and incompressibility giveThe kinematic boundary condition is $w=-i\omega \eta$ at the surface. The dynamic condition is $p^{\prime }=\rho g\eta$, so vertical momentum gives ${\omega }^2=gq$. Eliminating $q$ yieldsFor $f=2\Omega$, this isFor nonzero horizontal wavenumber, this branch has ${\omega }^2>f^2$ and an evanescent vertical structure. With no rotation it reduces to the deep-water gravity wave relation ${\omega }^2=g|k|$.

The orientation and incompressible deep-fluid assumptions matter. A tilted rotation axis, finite depth or compressible polytropic atmosphere has a different eigenvalue problem; the displayed formula is the stated incompressible model.

A polytropic atmosphere is neutrally stratified when $1+1/m=\gamma$, so its equilibrium pressure-density relation matches the adiabatic relation and its buoyancy frequency vanishes.

A polynomial $f\in R[t]$ is a zero divisor exactly when some nonzero scalar $a\in R$ annihilates it. One proof chooses a nonzero polynomial $g$ of minimal degree with $fg=0$ and successively kills its leading coefficient.

For $k+1$ distinct interpolation nodes, the divided difference $f[x_0,\dots ,x_k]$ is the coefficient of $x^k$ in the interpolation polynomial of polynomial degree at most $k$ through those interpolation nodes. It is the leading coefficient when that polynomial degree is exactly $k$, and zero when it is smaller. Explicitly,starting with $f[x_i]=f(x_i)$. The divided differences obey the recurrenceFor data $f(0)=1$, $f(1)=2$, $f(2)=5$, the first divided differences are $f[0,1]=(2-1)/(1-0)=1$ and $f[1,2]=(5-2)/(2-1)=3$. The second divided difference is $f[0,1,2]=(3-1)/(2-0)=1$, the leading coefficient of $x^2+1$. The first divided difference is the slope of a secant line; higher divided differences supply successive coefficients for the Newton polynomial.

For a monic separable polynomial over a field of characteristic other than two, its Galois group is contained in the alternating group exactly when its discriminant is a square in the base field. The Galois action multiplies the Vandermonde product by the sign of the induced root permutation.

The depressed cubic $x^3+ax+b$ has discriminant $-4a^3-27b^2$.

The Vandermonde product ${\prod }_{i<j}(r_i-r_j)$ is alternating, and its square is the discriminant of the monic polynomial with roots $r_i$.

For a polynomial over a unique factorization domain, primitive means that its polynomial content, in the gcd convention, is a unit. Thus its coefficients have no common nonunit divisor. For example, $2X+3$ is primitive in $\mathbb{Z}[X]$, whereas $2X+4$ has content two. Gauss lemma for polynomials makes products of primitive polynomials primitive.

This convention differs from requiring the coefficient ideal to be the unit ideal over an arbitrary commutative ring. The polynomial $xT+y$ over the unique factorization domain $k[x,y]$ is primitive in the gcd sense, although its coefficient ideal $(x,y)$ is proper. The distinction matters when transferring integer content arguments to more general coefficient rings.

Gauge fixing worldsheet diffeomorphism and Weyl symmetry introduces an anticommuting vector ghost $c^{\alpha }$ and a symmetric traceless antighost $b_{\alpha \beta }$. In complex coordinates they form holomorphic and antiholomorphic $bc$ systems.

Conformal gauge uses worldsheet diffeomorphisms and Weyl transformations to set $g_{\alpha \beta }=e^{2\varphi }{\eta }_{\alpha \beta }$ locally.

A Weyl transformation locally rescales a metric tensor, $g\mapsto e^{2\omega }g$, while retaining the same points and coordinates. Applied to a worldsheet metric, it leaves the embedding $X^{\mu }$ unchanged, and the classical Polyakov action is invariant under these Weyl transformations. Applied to a spacetime metric, a Weyl rescaling can relate different frame conventions, such as a string metric and an Einstein metric; the accompanying field normalizations must be transformed as well. It is not itself a coordinate transformation.