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Neyman allocation assigns sample sizes in proportion to outcome standard deviations to minimize variance.

A continuous variational symmetry gives a current satisfying ${\partial }_{\mu }{j}^{\mu }=0$ on shell. Integrating $j^0$ over space gives a conserved charge when boundary flux vanishes.

A Lagrangian density is integrated over spacetime to give the action $S=\int d^{d}x\mathcal{L}$ in Lagrangian.

For $R(r)\propto r^l{e}^{-r/[a(l+1)]}$, the radial probability density is proportional to $r^{2l+2}{e}^{-2r/[a(l+1)]}$ and

For curvature $F=dA+A\wedge A$, a standard normalization of the second Chern form is

A chemotactic pattern-forming instability occurs when cells produce an attractant and drift up its gradient strongly enough that positive aggregation feedback overcomes diffusion and turnover at a finite wavelength.

Adding linear drag to cyclotron motion produces an exponentially contracting spiral.

If a centre manifold is the graph $v=h(u)$ for $\dot{u}=f(u,v)$ and $\dot{v}=g(u,v)$, its coefficients satisfy $Dh(u)f(u,h(u))=g(u,h(u))$.

Appending $\dot{\mu }=0$ turns a system parameter into a centre variable, allowing one invariant graph to describe nearby parameter values.

The reduced mass is the effective mass for the two-particle relative coordinate of two bodies with constant positive masses $m_1,m_2$:Both bodies can respond to their mutual force, so their relative acceleration is greater than if one were fixed. Consequently $0<\mu <\min (m_1,m_2)$. This explains why using either individual mass, or their sum, generally gives the wrong internal oscillation rate. The same reduced mass describes a two-body problem, a free two-mass spring, and the approximate molecular vibration of a diatomic molecule.^[ref]

For equal masses $m_1=m_2=m$, $\mu =m/2$. Two $1\mathrm{kg}$ bodies therefore have $\mu =0.5\mathrm{kg}$: in their stationary centre of mass frame a relative speed $2m/s$ means individual speeds $1m/s$ and total kinetic energy $1\mathrm{J}$. With $m_1=1\mathrm{kg}$ and $m_2=100\mathrm{kg}$, instead $\mu =100/101\mathrm{kg}\approx 0.990099\mathrm{kg}$. At the same relative speed, the individual speeds are $200/101$ and $2/101m/s$, and the total kinetic energy is $200/101\mathrm{J}$. These are calculations for stipulated masses and velocities.

For fixed $m_1$ and $q=m_2/m_1>0$, $\mu /m_1=q/(1+q)$. The heavy-partner limit $q\to \infty$ gives $\mu \to m_1$ from below; the relative error incurred by replacing $\mu$ with $m_1$ is $(m_1-\mu )/\mu =1/q$. At $q=100$ this is $1\%$. Treating the heavy body as fixed is thus a controlled approximation to internal motion, rather than an exact consequence of its being heavier.

The center of mass moves as if the total external force acted on the total mass; internal forces cancel in its equation.

Forthe invariant center manifold near $(x,y,k)=(0,0,0)$ is $y=0$, with reduced equation $\dot{x}=x(-k-3x+x^2)$ and hence a transcritical bifurcation. Near $(x,y,k)=(1,2,0)$, put $X=x-1$ and $v=y-2-X$. The extended center manifold beginsand its reduced equation is $\dot{X}=2(X^2-k)+O(3)$, a saddle-node bifurcation.

Forset $X=x-a^2$ and $Y=y-a^2$, and append $\dot{a}=0$. Near $(X,Y,a)=(0,0,0)$, the extended centre manifold and its reduced dynamics areThe reduced equation is a subcritical pitchfork with reversed parameter $\mu =-a$.

For the system in the 2024 Part II Paper 2 question at $r=1$, with
$v=(x+y)/2$ and $w=(x-y)/2$, the center manifold isIts reduced equation is

An orbital resonance occurs when orbital frequencies satisfy an approximately integer relation, allowing perturbations to accumulate coherently.

The restricted three-body problem follows a negligibly massive particle in the gravitational field of two massive bodies whose motion is unaffected by the particle.

An inverse-square central force produces conic-section trajectories with the force centre at a focus.

Random treatment assignment makes treatment independent of baseline potential outcomes in expectation.

This sensitivity analysis quantifies how strong an omitted association must be to alter a causal conclusion.

A negative control outcome cannot plausibly be caused by treatment, so an estimated effect can reveal bias.

Unlike treatment comparison alone, this scientific control checks for an association that should vanish under the intended causal model.