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Increasing primordial helium at the expense of hydrogen reduces fuel for long-lived main-sequence hydrogen burning and changes subsequent stellar evolution.

When protons outnumber neutrons, helium-4 production is neutron-limited and produces about one helium nucleus per two neutrons.

If $V$ is a simple left $R$-module and $D={\mathrm{End}}_R(V{)}^{\mathrm{op}}$, then every prescribed map on a finite $D$-linearly independent subset of $V$ is induced by an element of $R$.

The functions generated from the initial functions using composition alone are exactlyfor a constant $c\in \mathbb{N}$. Structural induction proves necessity, and iterating the successor function on zero or a projection proves sufficiency.

From $g:{\mathbb{N}}^k\to \mathbb{N}$ and $h:{\mathbb{N}}^{k+2}\to \mathbb{N}$, primitive recursion defines

Given a $m$-ary function $g$ and $k$-ary functions $h_1,\dots ,h_m$, composition forms

The initial functions are the zero function, the successor function, and all coordinate projections.

For a commutative unital ring $R$, an ideal $I$ is prime exactly when $R/I$ is an integral domain. The zero-product condition in the quotient translates directly to $ab\in I$.

An equation of state is a relation among thermodynamic state variables such as pressure, volume, temperature, energy, and particle number.

The Poynting theorem is the local energy balance of classical electromagnetism. For vacuum field laws, with current density $\mathbf{J}$, the electromagnetic energy per unit volume isandThus a decrease in local electromagnetic energy can supply outward energy flux or work on electric charge. The term $\mathbf{J}\cdot \mathbf{E}$ is power transferred to matter per unit volume. Its sign matters: positive values remove field energy, while negative values describe matter supplying the field. The magnetic part of the Lorentz force supplies no work term.

For a fixed volume $V$, the divergence theorem gives the directly measurable balanceFor example, $2.0\mathrm{W}$ entering a region with $0.50\mathrm{W}$ leaving and $1.20\mathrm{W}$ transferred to matter means stored electromagnetic energy increases at $0.30\mathrm{W}$. In source-free vacuum only the boundary flux changes the stored energy; a nonzero local Poynting vector can still have zero net closed-surface flux.

A brief proof takes the dot product of the Ampère-Maxwell equation with $\mathbf{E}/{\mu }_0$ and uses Faraday's law with $\mathbf{B}/{\mu }_0$. The identity $\nabla \cdot (\mathbf{E}\times \mathbf{B})=\mathbf{B}\cdot (\nabla \times \mathbf{E})-\mathbf{E}\cdot (\nabla \times \mathbf{B})$ turns the remaining terms into the divergence of the Poynting vector. ^[ref].

Shear thinning is the decrease of effective viscosity with increasing shear rate.

An irrelevant coupling has negative mass dimension and is suppressed toward long distances.

A relevant coupling has positive mass dimension and grows under scaling toward long distances.

A marginal coupling is dimensionless under classical power counting.

The marginal interaction coefficient is distinct from the operator it multiplies.

The superficial degree of divergence $D$ is the ultraviolet power predicted by rescaling all independent loop momenta together. For a scalar graph with $L$ loops and $I$ propagators behaving as $p^{-2}$ in $d$ dimensions, $D=dL-2I$ before cancellations and subdivergences are considered.

A causal effect compares potential outcomes under different interventions.

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 even potentialhas a cusp at the origin and approaches the finite value $V_{\infty }$. Motions below that value oscillate between symmetric turning points; motions above it 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 $V$.