Homogeneous turbulence has velocity statistics unchanged by translating all observation points. Mean spatial divergences vanish when the moments exist and averaging commutes with derivatives. Homogeneity is distinct from isotropy: it does not require equal statistics in different directions.

An energy cascade transfers kinetic energy among scales through nonlinear motion. A forward cascade carries energy toward smaller spatial scales, where kinematic viscosity can dissipate it. Conservation and a mean scale-space flux do not uniquely prescribe the geometry or locality of the transfer mechanism.

Internal intermittency is the fluctuating concentration of gradients, transfer and viscous dissipation inside turbulent fluid. It is distinct from an interface between turbulent and non-turbulent regions intermittently passing an observer. Its scale dependence is reflected in non-Gaussian increments and coarse-grained viscous dissipation moments.

Velocity-fluctuation covariances transport mean momentum. For positive wall shear, $-\overline{u^{\prime }{w}^{\prime }}$ is the downward momentum-transport magnitude and contributes to the total shear stress. Some stress conventions include the minus sign in the definition of the tensor itself; the momentum equation must use the same convention.

Kinetic energy per unit mass in velocity fluctuations. Mean shear, buoyancy and transport supply or redistribute it, while molecular viscosity dissipates it. Its local budget motivates the local turbulent kinetic energy balance when storage and transport are negligible.

A mixing length is the characteristic transverse displacement over which a turbulent parcel retains its streamwise momentum before mixing with its surroundings. Taylor expansion of a mean velocity profile gives a fluctuation of magnitude $l|U_y|$. Combining comparable transverse fluctuations with down-gradient Reynolds stress transport gives the closure $\overline{u^{\prime }{v}^{\prime }}=-l^2|U_y|U_y$. This is a phenomenological approximation, not a consequence of the Navier-Stokes equation. In a free jet, a prescribed local mixing length must also be compatible with centreline regularity and vanishing outer momentum flux.

For a slender plane jet with conserved specific momentum flux $M_0=\int U^{2}dy$ and a mixing length proportional to transverse position, dimensional balance gives width proportional to $x$ and centreline speed proportional to $\sqrt{M_0/x}$. The streamfunction $\psi =\sqrt{M_{0}x}f(\eta )$, $\eta =y/x$, yields $U=\sqrt{M_0/x}{f}^{\prime }$ and $V=\sqrt{M_0/x}(\eta f^{\prime }-f/2)$. With $l=C_1|y|$, the upper-half Reynolds-averaged momentum equation integrates to $f{f}^{\prime }=2C_1^2{\eta }^2(f^{\prime \prime }{)}^2$ for a finite centreline velocity and zero centreline stress. The momentum normalization is $2{\int }_0^{{\eta }_w}(f^{\prime }{)}^{2}d\eta =1$ for a compact jet, or the same integral to infinity for a decaying jet. These formal equations do not justify an independently imposed edge condition.

Turbulent mixing redistributes a transported scalar, such as temperature or particle concentration, through irregular fluid motions and eventual molecular diffusion. A top-hat plume model assumes this mixing keeps the scalar approximately uniform within a selected cross section; that uniformity is an additional closure, not a consequence of thin geometry alone.

A two-equation closure transporting turbulent kinetic energy $k$ and its turbulent kinetic energy dissipation rate $ϵ$. Its eddy viscosity is $C_{\mu }{k}^2/ϵ$. In a high-Reynolds-number wall-normal reduction, turbulent diffusion has coefficients ${\nu }_T/{\sigma }_k$ and ${\nu }_T/{\sigma }_{ϵ}$; the energy equation has shear production $P$ and destruction $ϵ$, while the dissipation equation has modeled production $C_{ϵ1}Pϵ/k$ and destruction $C_{ϵ2}{ϵ}^2/k$. The dissipation production/destruction terms are closure terms, not a second conservation of energy.

A turbulence closure approximates the unclosed higher-order moments in evolution equations for lower-order turbulent statistics. For example, an equation for the velocity correlation tensor involves triple correlations, whose evolution involves fourth-order correlations and pressure. A closure must be checked against conservation laws, realizability and boundary conditions; matching an energy spectrum alone does not establish all these properties.

Magnetohydrodynamic turbulence is irregular nonlinear motion of an electrically conducting fluid coupled to a magnetic field. Alfvén waves and magnetosonic waves give propagation times in addition to the nonlinear mixing time. A strong guide field distinguishes directions parallel and perpendicular to it, so the resulting cascades need not be isotropic.

Wave turbulence describes the statistical transfer of energy between interacting waves. The wave period, nonlinear interaction time, phase correlations and resonances determine the transfer rate. A weak wave cascade treats the change in one interaction as small; critical balance instead makes the nonlinear interaction rate comparable to the linear wave frequency.

A fixed observation point alternates between turbulent and non-turbulent fluid because a fluctuating interface passes it. This is distinct from internal intermittency, which concerns variations within fluid that remains turbulent.

Reynolds averaging separates a resolved mean from zero-mean fluctuations. The idealized averaging operation is linear and idempotent, commutes with the derivatives under consideration and treats resolved mean factors as constant inside its averaging scale. Products then satisfy $\overline{fg}=\overline{f}\overline{g}+\overline{f^{\prime }{g}^{\prime }}$. Actual finite spatial filters need scale separation or corrections when these rules are only approximate.

Glyceraldehyde 3-phosphate is a three-carbon sugar phosphate with an aldehyde group. It is an intermediate of glycolysis and of carbon assimilation in photosynthesis.

In glycolysis, oxidation of glyceraldehyde 3-phosphate is coupled to reduction of NAD+ and incorporation of inorganic phosphate. This connects oxidation of a carbon substrate to the later formation of ATP, rather than releasing all of the available energy directly as heat.

The tangent half-angle substitution $t=\tan (x/2)$ converts rational trigonometric expressions into rational functions: $\sin x=2t/(1+t^2)$, $\cos x=(1-t^2)/(1+t^2)$ and $dx=2dt/(1+t^2)$.

The inverse trigonometric functions invert trigonometric functions on branches where they are one-to-one. Standard real ranges are $[-\pi /2,\pi /2]$ for $\arcsin$, $[0,\pi ]$ for $\arccos$, and $(-\pi /2,\pi /2)$ for $\arctan$. Thus $\arcsin (1/2)=\pi /6$ but $\arcsin (\sin (5\pi /6))=\pi /6$: the inverse returns the selected branch. The notation ${\sin }^{-1}x$ usually means $\arcsin x$, not $1/\sin x$. Implicit differentiation gives $(\arcsin x{)}^{\prime }=(1-x^2{)}^{-1/2}$ for $|x|<1$.

Every infinite finitely branching rooted tree has an infinite branch. Starting at the root, repeatedly choose a child above which infinitely many vertices remain.

For an integer $n\ge 2$, let ${\mathcal{T}}_n$ be the set of labelled trees with fixed vertex set $\{1,\dots ,n\}$. Each tree is a finite simple graph, with no distinguished root of a tree, and two trees are different exactly when their sets of edges differ. ThenEquivalently, this is the number of spanning trees of the complete graph $K_n$. The labelled combinatorial structure includes the fixed labels: drawings, positions, and the order in which edges are listed do not create extra trees. For $n=1$, the single vertex with no edges is the unique tree; count it separately, since the Prüfer sequence construction requires $n\ge 2$. For $n=2$, the unique tree has edge $\{1,2\}$ and corresponds to the empty Prüfer sequence. ^[ref]

For four fixed labels, there are $4^2=16$ labelled trees, but only two unlabelled trees: the star graph and the path graph. There are four star graphs, one for each choice of centre. There are $4!/2=12$ path graphs: arrange the labels along a graph path, then identify each ordering with its reversal, which describes the same set of edges. Thus $4+12=16$. For instance, star graphs centred at $1$ and at $2$ have different sets of edges even though they have the same shape.

The striking point is that connectedness and the absence of a graph cycle are global restrictions on edges, yet the answer is a power with no correction factor. The Prüfer sequence supplies a bijection that turns those restrictions into $n-2$ freely chosen labels. The rule of product then gives $n^{n-2}$ immediately. The same representation makes vertex degrees easy to count and supports uniform labelled tree generation for simulations and algorithm experiments.

Arthur Cayley's 1889 paper "A theorem on trees", published in the Quarterly Journal of Pure and Applied Mathematics, volume 23, pages 376–378, treats the counting problem and its refinement by vertex degrees. It was reprinted as paper 895 in volume XIII of his Collected Mathematical Papers, pages 26–28. This publication is the historical context for the name Cayley's formula; the name alone is not a claim of priority. ^{[ref] [ref]}

An oriented tree is an orientation of every edge of an undirected tree. Its arrows need not all point towards, or away from, any one root. It is acyclic as a directed graph, but different orientations can have different embedding behaviour in tournaments.