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Resistive-force theory approximates the local viscous force on a slender filament by separate drag coefficients multiplying its velocity components parallel and perpendicular to the tangent.

Cytoplasmic streaming is directed cytoplasmic flow within a cell, often driven by molecular motors acting along the boundary and used to enhance intracellular transport.

Microcirculation is blood flow through arterioles, capillaries, and venules, where vessel dimensions are comparable with cellular length scales.

For a nonnegative integer $n$ and commuting quantities $x,y$,In expanding the $n$ factors, obtain $x^k{y}^{n-k}$ by selecting the $k$ positions that contribute $x$. There are $\left(\genfrac{}{}{0ex}{}{n}{k}\right)$ such choices. For example,The coefficients are row five of Pascal's triangle, in reverse order under the displayed $x^k{y}^{n-k}$ convention; the row is symmetric. Setting $x=y=1$ gives $1+5+10+10+5+1=32$, the number of subsets of a five-element set. Each member has two inclusion choices, agreeing with the rule of product and the cardinality of the power set.

For nonnegative integers with $n_1+\cdots +n_r=n$, the multinomial coefficientcounts arrangements of a multiset containing $n_j$ copies of symbol $j$.

The central binomial coefficient is the largest coefficient in the expansion of $(1+1{)}^{2n}$:

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.