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A bifurcation diagram plots equilibrium values against a parameter and distinguishes stable, unstable, and nonhyperbolic branch segments.

The supercritical pitchfork normal form is $\dot{x}=\mu x-x^3$: the trivial branch is stable for $\mu <0$ and two stable nonzero branches emerge for $\mu >0$.

A saddle-node bifurcation occurs when a stable and an unstable equilibrium coalesce at a nonhyperbolic equilibrium and disappear as a parameter crosses a critical value.

A simple eigenvalue crossing the imaginary axis signals loss of hyperbolicity and a possible local bifurcation, while the remaining eigenvalues stay away from it.

The modified Bessel equation of order zero isIts solution regular at zero and normalized by $I_0(0)=1$ is the modified Bessel function $I_0$.

Quantum dense coding uses one shared Bell pair and one transmitted qubit to communicate two classical bits. Alice applies one of four Pauli operators to her qubit, producing four orthogonal Bell states that Bob can distinguish jointly. The transmitted qubit alone is maximally mixed for every message and reveals no information to an interceptor.

Local operations and classical communication allow separated parties to measure or transform their own subsystems and exchange ordinary classical messages.

For a bipartite density operator ${\rho }_{AB}$, the reduced state of subsystem $A$ is the partial trace ${\rho }_A={\mathrm{Tr}}_B{\rho }_{AB}$. Every measurement performed only on $A$ has outcome probabilities determined entirely by ${\rho }_A$.

For independent $X_i\sim N(\theta ,{\sigma }^2)$ and prior $\theta \sim N(\mu ,{\nu }^2)$, the posterior has variance and meanAs $n\to \infty$, $m_n-\overline{X}=O_p(n^{-1})$ and $v_n^{1/2}\sim \sigma /\sqrt{n}$, so fixed-level posterior credible intervals agree asymptotically with the corresponding known-variance confidence intervals.

An equilibrium point, or fixed point, of $\dot{z}=f(z)$ is a state $z_{\ast }$ with $f(z_{\ast })=0$.

An autocatalytic biochemical switch uses bistability and a transient input to move a system from one stable concentration to another. A saddle-node bifurcation supplies a threshold above which the low-concentration equilibrium disappears.

Cross-fitting trains nuisance estimators outside each observation's fold and evaluates the target score on the held-out fold.

A contour integral with a large exponential parameter is deformed through saddle points along curves on which the real part of the phase decreases most rapidly and the imaginary part is locally constant.

If a smooth real phase $\varphi$ has a unique nondegenerate interior maximum at $t_0$, thenFor a maximum at an endpoint $b$ with ${\varphi }^{\prime }(b)>0$, the leading factor is $a(b)e^{x\varphi (b)}/(x{\varphi }^{\prime }(b))$.

An asymptotic sequence orders the scales against which successive errors are compared. On a specified domain approaching $x_0$, suppose each ${\varphi }_n$ is nonzero sufficiently close to the limit andEquivalently, ${\varphi }_{n+1}=o({\varphi }_n)$ in little o notation. The notationmeans that for every $N\ge 0$,Each $N$ stays fixed while the parameter tends to its limit.^[ref]

As $z\to +\infty$, $1,z^{-1},z^{-2},\dots$ is an asymptotic sequence, because each adjacent ratio is $1/z$. Fractional scales $1,z^{-1/2},z^{-1},\dots$ work too, with ratio $z^{-1/2}$. By contrast, $1,1+z^{-1},z^{-1},\dots$ fails at its first pair: the second scale is not smaller than the first. Ordering by written position alone does not create a hierarchy of sizes.

Successive division of the remainder by ${\varphi }_N$ uniquely recovers every coefficient:If two proposed expansions first differ at order $N$, subtracting them and dividing by ${\varphi }_N$ would give a nonzero constant tending to zero, a contradiction. This is uniqueness of the coefficients in a fixed scale, not uniqueness of the represented function.

For the factorial integral example, the first three limiting responses are $\lim F(z)=1$, $\lim z(F(z)-1)=-1$, and $\lim z^2(F(z)-1+z^{-1})=2$. They extract exactly the coefficients $1,-1,2!$. Choosing an asymptotic sequence therefore turns “the next small correction” into a precise and reusable operation.

Magnetohydrodynamics treats an electrically conducting fluid coupled to a magnetic field through the Lorentz force and electromagnetic induction.

A polytropic atmosphere obeys $p=K{\rho }^{1+1/m}$ for polytropic index $m$. Under uniform vertical gravity and a free surface, its density and pressure are powers of depth below that surface.

An affine stellar model represents a fluid body by a time-dependent ellipsoid whose internal velocity is linear in position. Its axis lengths become finitely many dynamical degrees of freedom.

Disk self-gravity can overcome pressure and rotational support, producing growing density disturbances, spiral structure, or fragmentation.

An accretion disk transports mass inward and angular momentum outward through stresses, while dissipating orbital energy as heat.