Cooling oceanic lithosphere becomes denser and subsides under isostasy. A simple calculation explains the square-root age law. Let depth $z$ be measured below a fixed cooling surface, take constant thermal diffusivity $\kappa$, and write the half-space cooling modelFor mantle reference mass density ${\rho }_m$ and small constant volumetric thermal expansion coefficient $\alpha$, the cold-column density excess relative to hot mantle is $\delta \rho ={\rho }_m\alpha (T_m-T)$. Its integral per unit horizontal area isThis uses ${\int }_0^{\infty }\mathrm{erfc}udu=1/\sqrt{\pi }$.

At a common compensation depth, additional water depth replaces the dense upper column by seawater of mass density ${\rho }_w$. Balancing column masses gives $({\rho }_m-{\rho }_w)[d(t)-d_r]=\Delta \Sigma$, henceThe relevant geophysical density contrast is therefore mantle versus seawater, and $t$ is seafloor age measured from the ridge reference depth $d_r$. The model neglects sediment loading, dynamic support and changes of material properties. A finite oceanic plate cooling model and basal heating limit indefinite square-root deepening at old ages.

For a positive integer $m$, take $f(x)=\sin (mx)$ on $0<x<\pi$ and form its half-range Fourier cosine series. The product-to-sum identity givesThus $a_n=4m/[\pi (m^2-n^2)]$ for opposite parity and zero for the same parity. The exceptional integral $a_m=(2/\pi ){\int }_0^{\pi }\sin (mx)\cos (mx)dx$ is zero. The constant term is $a_0/2$, not $a_0$. These coefficients represent the even periodic extension, not the original sine wave on negative arguments. Their quadratic decay gives absolute convergence.

For a parabolic profile $u(s)=U_{\max }(1-s^2/a^2)$ in a pipe of radius $a$, the mean speed is $U_{\max }/2$ and $P_{\mathrm{kin}}=\frac{1}{2}\rho \int u^{3}dA=\pi \rho a^2{U}_{\max }^3/8$. With pressure drop $\Delta p$ over length $\ell$ and kinematic viscosity $\nu$, $U_{\max }=\Delta p{a}^2/(4\rho \nu \ell )$, so $P_{\mathrm{kin}}=\pi \Delta p^3{a}^8/(512{\rho }^2{\nu }^3{\ell }^3)$. This is not the pressure-work dissipation $\Delta pQ$.

For $Q\propto a^4\Delta P/(\eta L)$ and independent small errors, ${\sigma }_Q^2/Q^2=16{\sigma }_a^2/a^2+{\sigma }_{\Delta P}^2/\Delta P^2+{\sigma }_{\eta }^2/{\eta }^2+{\sigma }_L^2/L^2$. Independent pressure errors add in quadrature before dividing by the pressure difference. Bounded worst-case errors instead add absolute derivative contributions.

A Hackenbush stalk is a Hackenbush position whose graph is a single path attached to the ground at one endpoint. Read its blue and red graph edges upward from the ground; Left may cut blue edges and Right red edges, with every edge above the cut falling away. Finite red-blue stalks are number games with dyadic rational values.

Here is the explicit value rule. Give blue sign $+1$ and red sign $-1$. If the initial monochromatic run has length $m$ and sign $\epsilon$, it contributes $\epsilon m$. If more edges remain, edge $j>m$, of sign ${\epsilon }_j$, contributes ${\epsilon }_j/2^{j-m}$. ThusFor example, a blue-red stalk has value $1-1/2=1/2$, blue-red-red has value $1/4$, and blue-blue-red has value $3/2$. The recursive number game rule verifies the first example directly: its Left option is $0$, its Right option is $1$, and the simplest surreal number between them is $1/2$. Repeating this comparison after successive cuts gives the dyadic rule.

H3K9 dimethylation is addition of two methyl groups to lysine 9 of histone H3, conventionally abbreviated H3K9me2. It is associated with repressive chromatin in several systems, including extensive domains in plants. Its distribution differs from the mono- and trimethylated states of the same residue. In Arabidopsis thaliana, joint changes in this mark and non-CG DNA methylation alter recombination near centromeres, linking chromatin state to inherited variation. ^[ref]

Poecilia reticulata is the scientific species name of the guppy. Its internal fertilization and production of live young distinguish its reproductive system from the externally spawning pattern of many familiar fish.

For finite nimbers, multiplication is recursively defined from smaller arguments bywhere $\oplus$ is Nim-sum and mex selects the smallest missing nonnegative integer. Thus zero times anything is zero and one times $b$ is $b$. For $a=b=2$, the displayed set is $\{0,2,3\}$, so $2\otimes 2=1$.

Together with Nim-sum, this is field arithmetic on finite nimbers, rather than ordinary integer multiplication. In particular $2\otimes 2=1$ cannot be computed by a usual integer product or bitwise exclusive or.^[ref]

The distance combination $D_{\Delta t}=(1+z_l)D_l{D}_s/D_{ls}$ converts a dimensionless thin-lens delay difference into an observed time delay. Inference from a variable lensed source also requires a mass model and control of external convergence.

Conservation of orbital energy and angular momentum in a two-body encounter gives $\sigma =\pi R_c^2(1+v_{\mathrm{esc}}^2/v_{\infty }^2)$. At fixed density and encounter speed in the strong-focusing limit, $R_c\propto M_c^{1/3}$ and $\sigma \propto M_c^{4/3}$. This yields runaway fractional growth until stirring or supply changes the assumptions.

A thin rod of length $2R$ and linear mass density $\lambda$ slides along a diameter of an infinite uniform source cylinder of radius $R$. Neglect the narrow hole's perturbation of the source field. If the rod centre is displaced by $x$, $F(x)=\lambda {\int }_{x-R}^{x+R}{g}_y(y)dy$. The endpoint derivative gives $F^{\prime }(0)=\lambda [g_y(R)-g_y(-R)]=-4\pi G\rho \lambda R$. Dividing the restoring coefficient by rod mass $2R\lambda$ gives the displayed small-oscillation frequency.

A pendulum of length $L$ in vertical gravity $g$ is weakly attracted horizontally by an added long circular cylinder of line density $\lambda$. Outside it, $g_h=2G\lambda /r$ at axis distance $r$. Small-angle equilibrium gives horizontal deflection $d\simeq L{g}_h/g=2GL\lambda /(rg)$. Removing the cylinder's mass reverses the change. It is a before/after differential estimate; finite length, a nonhorizontal radial direction and mechanical motion require corrections.

A mass above a sufficiently wide estuary experiences additional downward gravity when extra water is present. Approximating the tide as a plane slab of density $\rho$ and added thickness $h$ gives $\Delta g=2\pi G\rho h$. A linear spring with initial extension $x_0=mg/k$ changes by $\Delta x=(m/k)\Delta g=x_0\Delta g/g$. This isolates direct water attraction; it omits finite shoreline geometry, buoyancy, structural motion and terrestrial tides.

A formal grating order must have a real outgoing angle. For normal incidence, $|m|\lambda \le d$ is necessary. Oblique incidence can access orders outside that normal-incidence range; reflection Littrow geometry has $m\lambda =2d\sin \theta$. The ideal resolving power of a diffraction grating applies only to an order that exists and has sufficient signal.

A potassium graphite intercalation compound, with KC-eight a common stage-one composition. Charge transfer from potassium to the graphitic pi system can enhance electrical conductivity.

A graphene vacancy defect with one carbon removed. A common reconstruction has a pentagon and nine-membered ring with residual under-coordination. The reconstructed geometry need not retain the symmetry of the unrelaxed missing-atom site.

A graphene vacancy defect with two carbon atoms removed. A common adjacent-pair reconstruction has a 5–8–5 ring arrangement; other reconstructed arrangements exist and can be selected by energetic processing.

Maintain a frontier of partial states in a priority queue and expand the state with the best current priority. Correctness of a particular stopping rule depends on what the priority bounds. If every descendant has smaller priority than its ancestor, the frontier maximum also bounds the priority of every unseen descendant.

A quark-lepton unification framework with conventional minimal gauge group $SU(4{)}_C\times SU(2{)}_L\times SU(2{)}_R$. leptons supply a fourth color charge, and a right-handed neutrino completes each family. fermion masses additionally depend on symmetry breaking and Yukawa couplings.

The empirical scaling ${\sigma }_y={\sigma }_0+k_y{d}^{-1/2}$ of yield stress with average grain size in its applicable regime. It is not a universal law for arbitrarily small grains or high-temperature deformation.