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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 substitution into $\dot{X}=(1+X)(v-k+X^2)$ gives $\dot{X}=2X^2-2k-7Xk+9k^2+O(3)$, where $O(3)$ denotes total degree in $(X,k)$. Its leading saddle-node bifurcation balance is $\dot{X}=2(X^2-k)+O(X^3,Xk,k^2)$; the mixed and squared-parameter terms must not be hidden in a total-degree-three remainder.

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 $\mu =-a$.

Consider the cubic feedback systemAt the origin its eigenvalues are $0,-2,-1$. With $v=(x+y)/2$ and $w=(x-y)/2$, the center subspace is the $v$ axis and the two remaining directions are strictly stable. The equations becomeThe symmetry $(v,w,z)\mapsto (-v,-w,z)$ allows a center manifold graph with $w(v)$ odd and $z(v)$ even. Write $w=c_3{v}^3+O(v^5)$ and $z=d_2{v}^2+d_4{v}^4+O(v^6)$. The invariance equation for a graph requires $w^{\prime }(v)\dot{v}=\dot{w}$ and $z^{\prime }(v)\dot{v}=\dot{z}$, rather than setting the stable-coordinate derivatives to zero.

The quadratic $z$ equation gives $d_2=1$; the cubic $w$ equation gives $-2c_3+(a+1)/2=0$. At fourth order the $z$ invariance equation is $(a-1)v^4=-d_4{v}^4$, soSubstitution into the $v$ equation yieldsIn particular, the quintic coefficient is $[(3a+1)c_3-d_4]/2$, which records both the cubic stable graph and the quartic lag in $z$. For $a<1$, the leading odd term points toward zero on both sides; for $a>1$ it points outward. At $a=1$ the cubic vanishes but the quintic coefficient is one and still points outward. Since the transverse directions are strictly stable, the centre manifold theorem gives local asymptotic stability exactly when $a<1$.

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.

A Kepler orbit is a relative trajectory satisfyingFor nonzero specific angular momentum $h$, conservation of angular momentum confines motion to a plane, and the orbit is a conic section:Here $e$ is orbital eccentricity and $\nu$ the true anomaly. Negative specific orbital energy gives a bound ellipse; zero or positive energy gives a parabolic orbit or hyperbolic orbit. Other forces make the exact two-body description an approximation.^[ref]

A randomized controlled trial assigns experimental units to competing treatments by a specified random mechanism and compares their outcomes. Under complete randomization of $N$ units, exactly $n_1$ receive assignment one and $n_0=N-n_1$ assignment zero. With fixed potential outcomes, each unit has assignment probability $n_1/N$. ThereforeThis proves design-unbiasedness of the mean difference for the finite-population average treatment effect, assuming consistency in causal inference and no interference in causal inference.

Randomization controls the assignment distribution; it does not force every realized baseline covariate difference to be zero. Equal assignment probabilities yield unconditional exchangeability in the design, whereas covariate-dependent randomization can require conditional exchangeability and an analysis respecting that design.^[ref]

When assigned and received treatments differ, an intention-to-treat analysis estimates an assignment-policy effect. A treatment-receipt effect needs additional assumptions. Allocation concealment, blinded outcome assessment and handling of missing follow-up address other sources of distortion beyond random assignment itself.

Sensitivity analysis for unmeasured confounding asks how an estimated causal effect changes under explicit departures from the assumed conditional exchangeability. It varies parameters describing the omitted variable's assignment imbalance and outcome effect, rather than assuming that measured covariate balance proves the omission harmless.

For a transparent additive linear structural equation model, take a pretreatment variable $U$ andAssume intervening on $A$ leaves $U$ unchanged, so the constant treatment effect in this model is $\tau$. Subtracting the two conditional outcome means givesThus the adjusted effect for a hypothesized parameter pair is $\tau =D_{\mathrm{obs}}-\gamma {\Delta }_U$. An observed difference five would fall to three when $\gamma =2$ and ${\Delta }_U=1$; explaining it entirely would require $\gamma {\Delta }_U=5$.

The product is a bias term derived for this particular additive model. Other outcome scales and interactions need different formulas. Plausible ranges can be compared with measured pretreatment predictors or external evidence, but the observed treatment-outcome table alone cannot determine the omitted parameters. A sensitivity result quantifies a model-dependent robustness claim; it does not establish that unmeasured confounding is absent.

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.

In a binary-treatment potential outcomes model, conditional exchangeability given pretreatment covariates $X$ is $(Y(0),Y(1))\perp A\mid X$. Within each covariate stratum, treatment assignment then carries no information about the relevant potential outcomes. Together with consistency in causal inference and positivity, it permits outcome standardization to identify average causal contrasts. The assumption concerns unobserved counterfactual outcomes and cannot be established just by checking balance of the measured covariates. Including a large list of predictors is not by itself a guarantee that the assumption holds.

A confounder is a pretreatment variable relevant to removing confounding for a specified treatment-outcome question. A common cause $C$ with arrows $A\leftarrow C\to Y$ is a basic example: it creates a backdoor path between treatment $A$ and outcome $Y$.

Association with both variables is not a sufficient definition. In an appropriate causal directed acyclic graph, conditioning on a common effect can instead introduce collider bias. A confounder is a causal adjustment role, rather than any predictor with a small p-value.

One precise definition identifies a confounder as a member of some minimal sufficient adjustment set: there is a set containing $C$ that makes the relevant potential outcomes independent of treatment, and no proper subset of that set does so. The role can therefore depend on which other covariates are available and on the effect being estimated. A variable that merely reduces residual bias need not satisfy this definition.^[ref]

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 on a category consists of an endofunctor $T$ and natural transformations $\eta :1\Rightarrow T$, $\mu :T^2\Rightarrow T$ satisfying $\mu T\mu =\mu \mu T$ and $\mu T\eta =1_T=\mu \eta T$. These express associative flattening with a neutral insertion.

For the list monad on sets, $T(X)$ is the set of finite lists from $X$, $\eta$ makes a one-element list, and $\mu$ concatenates a list of lists. Both ways of flattening three nested levels agree. An algebra for a monad interprets these formal constructions in an object; list-monad algebras are monoids.

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.