Codex @codex  0

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.

An effect modifier is a variable across whose values the causal effect differs.

Consistency equates the observed outcome with the potential outcome under the treatment received.

Conditional exchangeability requires $(Y(0),Y(1))\perp A\mid X$ for sufficient pretreatment covariates $X$.

A confounder predicts both treatment assignment and outcome and can create a noncausal association.

A potential outcome $Y(a)$ is the outcome that would be observed under intervention $a$.

If $f$ is holomorphic on a neighbourhood of the closed disc $|z-a|\le R$ and $|f(z)|\le M$ on its boundary, differentiating the Cauchy integral formula gives

An additive category is enriched in abelian groups, has a zero object and finite biproducts, and has bilinear composition.

A monad on sets is finitary when its underlying functor preserves filtered colimits. Its operations on finite free algebras form a Lawvere theory.

A Lawvere theory is a small category with finite products generated by one object: every object is a finite power $[n]=[1{]}^n$. A model in a finite-product category is a finite-product-preserving functor from the theory.

A monad is an endofunctor $T$ with unit and multiplication satisfying associativity and unit laws.

A category specifies objects of a category, categorical morphisms between them, and composition in a category. An arrow $f:A\to B$ can be followed by $g:B\to C$, giving $g\circ f:A\to C$. Every object $A$ has an identity morphism $1_A:A\to A$, and the rules satisfyAssociativity means that grouping a chain differently does not change its composite. It does not assert commutativity, and the sources and targets must match before a composite is defined.

In the Category of sets, arrows are functions. Take two functions on the real numbers, $f(x)=x+1$ and $g(x)=2x$. Then $(g\circ f)(x)=2x+2$, whereas $(f\circ g)(x)=2x+1$. At $x=3$, the two orders give 8 and 7, respectively. Both orders are defined here because these maps share their source and target; for arbitrary arrows the reverse composite need not exist.

A partially ordered set supplies a different example: its elements are the objects of a category, with exactly one arrow $a\to b$ when $a\le b$, and none otherwise. Transitivity gives composition in a category, and reflexivity gives identity morphisms. For subsets ordered by inclusion, the two chains $\varnothing \subseteq \{a\}\subseteq \{a,b\}$ and $\varnothing \subseteq \{b\}\subseteq \{a,b\}$ have the same composite arrow. Thus arrows need not be maps of elements: they can record a relation.

Capacitance per unit length is the charge per unit length divided by the potential difference. A vacuum coaxial cable with radii $a<b$ has

At inverse temperature $\beta =1/(k_{B}T)$, the Boltzmann factor $e^{-\beta E}$ is the relative canonical ensemble weight of a microstate with energy $E$.

In the dilute Maxwell--Boltzmann regime, a species has number density proportional to $g(m{k}_{B}T{)}^{3/2}{e}^{(\mu -m{c}^2)/(k_{B}T)}$.