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Every positive integer $m$ has a unique greedy representationThis is also called the binomial representation of $m$ at level $r$.

The identitycomes from an integer change-of-variable matrix of determinant $-1$. It therefore preserves the set of integers represented and reduces the form to $u^2+v^2$.

A positive definite integral form $[a,b,c]$ is reduced when $|b|\le a\le c$, with $b\ge 0$ if $|b|=a$ or $a=c$. Every proper equivalence class has a unique reduced representative under this boundary convention.

The discriminant of $[a,b,c]$ is $D=b^2-4ac$. It is invariant under proper equivalence, and a real form with $a>0$ is positive definite exactly when $D<0$.

An integral binary quadratic form $[a,b,c]$ is primitive when $\gcd (a,b,c)=1$. This property is invariant under proper equivalence because a unimodular change and its integral inverse express either coefficient triple as integer combinations of the other.

Integral forms $f$ and $g$ are properly equivalent whenfor some $\left(\begin{array}{cc}\alpha & \beta \\ \gamma & \delta \end{array}\right)\in {\mathrm{SL}}_2(\mathbb{Z})$. Proper equivalence preserves both represented integers and orientation.

If almost all neutrons enter helium-4 and $r=n_n/n_p$, its primordial mass fraction is $Y_p\simeq 2r/(1+r)$.

Light nuclei cannot accumulate until cooling makes deuterium sufficiently resistant to photodissociation, delaying nucleosynthesis after weak freeze-out.

Weak neutron--proton conversion freezes out when its interaction rate falls below the Hubble expansion rate.

With negligible lepton chemical potentials, weak equilibrium gives $n_n/n_p\simeq e^{-Q/(k_{B}T)}$, where $Q=(m_n-m_p)c^2$.

In a quadrant-invariant planar system, a stable interior equilibrium branch may emerge from one boundary equilibrium and terminate at another, transferring stability at each endpoint.

Fornonzero periodic orbits haveThey are born together at $\mu =-{\lambda }^2/4$; the inner unstable orbit then vanishes in a subcritical Hopf bifurcation at $\mu =0$, while the outer orbit is stable.

A stable and an unstable periodic orbit coalesce into one semistable periodic orbit and disappear at a saddle-node bifurcation of periodic orbits.

A Hopf bifurcation creates or destroys a small periodic orbit near an equilibrium point as a pair of complex conjugate eigenvalues crosses the imaginary axis. It explains how a resting state can give way to sustained oscillations: the linearization of a dynamical system supplies rotation and growth, while a nonlinear term can saturate that growth at a nonzero amplitude.

The simplest supercritical Hopf bifurcation is the exact Hopf normal formHere $z$ is a complex number, $\mu$ is the bifurcation parameter, and $\omega$ is the angular frequency. In polar coordinates $z=r{e}^{i\theta }$, the equations become $\dot{r}=\mu r-r^3$ and $\dot{\theta }=\omega$ for $r>0$. At $\mu <0$, each turn shrinks towards the equilibrium point. At $\mu >0$, small radii grow and large radii shrink towards $r=\sqrt{\mu }$, producing a limit cycle. For example, $\mu =0.04$ and $\omega =2$ give radius $0.2$, period $\pi$, and frequency $1/\pi$ cycles per unit time. The limiting angular frequency stays nonzero while the amplitude tends to zero: a tiny newborn oscillation can still repeat at a finite rate.

For a precise generic statement, consider a smooth function $f$ defining an autonomous differential equation $\dot{x}=f(x,\alpha )$ on ${\mathbb{R}}^n$, with a smooth branch of equilibrium points $x_{\ast }(\alpha )$ and critical parameter $\alpha =0$. Require:Then a locally unique branch of small periodic orbits, up to phase, emerges on the side $\beta a<0$. On the two-dimensional centre manifold, the branch attracts for $a<0$ and repels for $a>0$. Attraction in the full phase space additionally requires all remaining eigenvalues to have negative real part; any transverse unstable direction makes the periodic orbit unstable there.^[ref]

An eigenvalue-crossing bifurcation test alone cannot establish these nonlinear conclusions. For example, the purely linear equation $\dot{z}=(\mu +i\omega )z$ has the same crossing but no isolated periodic orbit: at $\mu =0$ every nonzero radius is constant, and at $\mu \ne 0$ none is. If the cubic coefficient vanishes, higher-order terms must be examined; a generalized Hopf bifurcation is one possible degeneracy. The usefulness of the generic theorem is that a large system can have its small oscillatory onset understood through one Hopf normal form, subject to these hypotheses and local limits.

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 $I_0$.

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