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For a fixed affine spacetime point $x$, this linear relation cuts out a two-dimensional subspace $W_x=\{(ix\pi ,\pi )\}$ of twistor space. Projectivizing gives its twistor line. For a fixed projective twistor with $\pi \ne 0$, its solutions are $x=x_0+u^A{\pi }^{A^{\prime }}$, a totally null affine alpha-plane. The compact description replaces affine solutions by the two-planes containing the fixed twistor line in the vector space.

For $Z=(\omega ,\pi )$ and $W=(\eta ,\rho )$ the displayed Hermitian form has signature $(2,2)$. It is zero on every pair of vectors in $W_x=\{(ix\pi ,\pi )\}$ when $x$ is Hermitian, that is, Lorentzian real. Conversely a transverse totally isotropic two-plane is such a graph with $x$ Hermitian. This gives the Lorentzian reality structure on twistor space.

Choose an antilinear quaternionic structure $J$ on ${\mathbb{C}}^4$. It induces a fixed-point-free involution on projective twistor space. A real Euclidean spacetime point is a $J$-invariant complex two-plane, equivalently a quaternionic line in ${\mathbb{H}}^2$. These form ${\mathbb{HP}}^1\cong S^4$, the conformal compactification of Euclidean space. Every projective twistor lies on the unique invariant line $\mathbb{P}{\mathrm{span}}_{\mathbb{C}}(Z,JZ)$, giving a fibration ${\mathbb{CP}}^3\to S^4$ with fiber ${\mathbb{CP}}^1$.

The four-complex-dimensional Grassmannian of two-planes in twistor space. Its Plücker embedding is the Klein quadric. An affine chart consists of graphs $W_x=\{(ix\pi ,\pi )\}$. A Lorentzian reality structure on twistor space or an Euclidean reality structure on twistor space selects different four-real-dimensional compact spacetimes inside this same complex manifold.

The projective line associated to a two-plane $W_x$ representing a point of complexified compactified Minkowski space. Two distinct twistor lines intersect precisely when their spacetime points are null separated. On the Lorentzian real slice, the line lies entirely in projective null twistor space and parametrizes the null geodesics through the point.

An alpha-plane in the compact twistor correspondence is the set of two-planes containing a fixed line $\ell$ in twistor space. Its affine portion has tangent vectors $u^A{\pi }^{A^{\prime }}$ for fixed primed spinor $\pi$, so any two tangent vectors are orthogonal and null. Two distinct compact alpha-planes meet at the unique two-plane spanned by their defining lines; that point can lie at conformal infinity.

A holomorphic vector bundle on an appropriate twistor domain, trivial on its space-time incidence lines, determines a complex anti-self-dual gauge gauge connection. Split its patching matrix on each line and apply the tangent vector fields ${\pi }^{A^{\prime }}{\partial }_{A{A}^{\prime }}$. They annihilate the unsplit twistor patching matrix, so the two logarithmic derivatives of the factors agree and define a global degree-one function of $\pi$. It is consequently ${\pi }^{A^{\prime }}{A}_{A{A}^{\prime }}$. The inverse splitting bundle frame is parallel along each alpha-plane; commuting these parallel equations proves that the primed gauge curvature two-component spinor vanishes. Reality conditions recover unitary Euclidean gauge fields; Euclidean instanton bundles need triviality on the real twistor lines, not on every possible complex line.

Modulo the three quadric relations of the twisted cubic, replace $x_1^2$ by $x_0{x}_2$, $x_2^2$ by $x_1{x}_3$, and $x_1{x}_2$ by $x_0{x}_3$. Each replacement reduces the total exponent of $x_1,x_2$. The resulting monomials are $x_0^a{x}_3^b$, $x_0^a{x}_1{x}_3^b$, and $x_0^a{x}_2{x}_3^b$. In degree $m$ their images under $x_i\mapsto s^{3-i}{t}^i$ are all $3m+1$ distinct monomials of degree $3m$. They are linearly independent, so the three quadrics generate the full homogeneous ideal, not merely its radical. The quotient is the third Veronese subring of $k[s,t]$.

The sum of reciprocals of twin primes converges, whether or not there are infinitely many twin primes. The twin-prime upper bound from a quadratic sieve and reciprocal-sum convergence from a counting bound prove it. This contrasts with the Euler proof that the sum of reciprocals of primes diverges.

The conjecture asserts that infinitely many prime numbers $p$ have $p+2$ prime. The Hardy–Littlewood twin-prime asymptotic conjecture predicts their quantitative frequency and is stronger. The Selberg sieve supplies an upper bound of the predicted order but does not prove infinitude.

For a symmetric localized turbulent plane jet similarity profile, integrated momentum balance gives $f{f}^{\prime }=2C_1^2{\eta }^2(f^{\prime \prime }{)}^2$ on $\eta >0$. At a finite free edge ${\eta }_w>0$, $f^{\prime }({\eta }_w)=0$ and finite $f({\eta }_w)$ force $f^{\prime \prime }({\eta }_w)=0$. Thus the mixing length model $l=C_1|y|$ cannot simultaneously impose zero velocity and a nonzero maximum velocity gradient at the edge. Its eddy viscosity vanishes there. Moreover, a nonzero finite centreline speed $f^{\prime }(0)=a$ forces $|f^{\prime \prime }|\sim a/(\sqrt{2}{C}_1\sqrt{\eta })$ near the centre, displaying a singular-shear defect of that local closure. The molecular-viscosity neglect cannot be uniform in this core.

With boundary fluxes suppressed, let $B$ transfer kinetic energy to potential energy, $M$ transfer available potential energy irreversibly to background potential energy, and $W$ transfer internal energy to potential energy. Then $\dot{P}=B+W$, ${\dot{P}}_b=M+W$, $\dot{A}=B-M$, and $\dot{K}=F-B-ϵ$. The internal reservoir has $\dot{I}=ϵ-W$, conserving total energy apart from forcing $F$. The distinction between reversible buoyancy exchange and irreversible mixing is essential outside a quasi-steady negligible-$A$ regime.

For $A=U_0^2(1/L_f-1/L_{\rho })>0$, define $K_{\infty }=A{L}_v$ and $y=\sqrt{K/K_{\infty }}$. The positive-energy solution is $y(t)=[y_0+\tanh (\sqrt{K_{\infty }}t/(2L_v))]/[1+y_0\tanh (\sqrt{K_{\infty }}t/(2L_v))]$. Energies above and below $K_{\infty }$ approach it from their respective sides. The zero-energy equation is not Lipschitz and admits both permanent rest and delayed startup, so a unique positive steady limit needs a positive seed or an additional startup prescription. With negligible available and internal energy exchange, the long-time instantaneous mixing efficiency fraction is $L_f/L_{\rho }$, independent of $L_v$.

Taylor turbulent dispersion describes the absolute displacement of a marked fluid element in stationary homogeneous turbulence. Here $R_L$ is its Lagrangian velocity autocorrelation. Continuous correlation at zero gives ballistic spreading at short times, and an integrable correlation gives diffusive spreading at long times. This is distinct from shear-induced Taylor dispersion and from Richardson pair dispersion.

Richardson pair dispersion concerns relative separation of two marked fluid elements, rather than their centre-of-mass motion. In an inertial range, scale-local relative diffusivity is of order ${ϵ}^{1/3}{r}^{4/3}$. A self-similar closure gives $d⟨r^2⟩/dt=A{ϵ}^{1/3}⟨r^2{⟩}^{2/3}$ and hence a $t^3$ regime after loss of memory of initial separation. Stationarity and inertial-range separation alone do not imply this law at arbitrarily short times.

In stationary locally equilibrated turbulence, mean-shear production and buoyancy production balance viscous dissipation after transport terms are neglected. Stable buoyancy flux is negative and consumes part of the shear input. This approximation gives $ϵ=u_{\ast }^3/(\kappa z)$ in a neutral logarithmic layer and $ϵ\sim B_0$ in buoyancy-dominated free convection.

The work of the Reynolds stress against the mean velocity gradient transfers kinetic energy from the mean flow into fluctuations. In a plane shear flow $U(y)$ this is $P=-\overline{u^{\prime }{v}^{\prime }}{U}_y$. The eddy viscosity closure gives $P={\nu }_T{U}_y^2\ge 0$. Buoyancy production and transport are separate terms in the turbulent kinetic energy budget.

In a neutral turbulent constant-stress region, the stress velocity and wall distance give mean shear $U_z=u_{\ast }/(\kappa z)$. Its logarithmic integral uses the roughness length. The law applies outside the inner sublayer and below the outer boundary-layer scale, not all the way to a smooth wall.

A local similarity description for approximately stationary homogeneous surface-layer turbulence with nearly constant momentum and buoyancy fluxes. Besides height and friction velocity, the Obukhov length enters as a stability parameter. The hypothesis fixes the dimensional shear prefactor and function argument, while empirical or dynamical closures determine the functions themselves.

The near-wall region where molecular momentum diffusion dominates turbulent transport. Using the shear velocity $u_{\ast }=\sqrt{{\tau }_d/\rho }$ gives the viscous wall length $\nu /u_{\ast }$. The precise extent depends on the chosen crossover criterion and wall condition. In this region constant stress and no slip give $U\simeq u_{\ast }^{2}z/\nu$; the logarithmic law of the wall applies outside it.