A benzene-fused five-membered lactone containing a gem-dimethyl-substituted saturated ring carbon. It can result from double methyl addition to one carbonyl of dimethyl phthalate followed by ring closure.

For counts $N(p)dp$ per electron momentum interval, the allowed spectrum is proportional to $p^{2}F(p)(Q-T{)}^{2}dp$ when recoil and neutrino mass are neglected. Thus the displayed ordinate is linear in kinetic energy under a constant-shape assumption. Counts per kinetic-energy interval instead have the factor $p{E}_e$ in natural units; the Jacobian $dT/dp=p/E_e$ accounts for the difference. Mixing those two binning conventions changes the inferred ordinate.

If a beta spectrum has nonconstant shape factor $C(T)$, divide by it as well as by the Coulomb and phase-space factors before taking a square root. A leading unique first-forbidden shape contains $p_e^2+p_{\nu }^2$ in natural units, with Coulomb and nuclear corrections. Dividing only by a Coulomb Fermi function does not remove that shape. A straight uncorrected line must therefore be treated as an approximation, not an exact consequence of forbidden-transition selection rules.

A periodic array of local moments coupled to conduction electrons. Competition between screening, coherent hybridization and intersite magnetic interactions can produce a heavy fermion material, magnetic order or a correlated insulating state.

The maximum equilibrium supersaturation along a solution-droplet growth curve. Above this threshold, an aerosol particle can grow into an activated cloud droplet under the conditions of the parcel. In Köhler theory, larger dry diameter or greater hygroscopicity generally lowers the threshold; aerosol number alone does not specify it.

The bounded knapsack problem adds per-type upper limits $0\le x_i\le b_i$ to the integer knapsack problem. It interpolates between 0-1 knapsack problem ($b_i=1$) and the unbounded knapsack problem.

Processing types one at a time gives $F_i(c)={\max }_{0\le k\le b_i,k{w}_i\le c}(k{v}_i+F_{i-1}(c-k{w}_i))$. The previous-type index prevents the recurrence from exceeding the allowed count. For large $b_i$, splitting counts into power-of-two bundles and a final remainder reduces the bounded type to logarithmically many 0-1 knapsack problem items while preserving every allowable count.

In the unbounded knapsack problem, each item type may be used any nonnegative integer number of times. With positive integer weights $w_i$, values $v_i$ and capacity $W$, maximize ${\sum }_i{v}_i{x}_i$ subject to ${\sum }_i{w}_i{x}_i\le W$ and $x_i\in {\mathbb{Z}}_{\ge 0}$.

A dynamic programming recurrence for capacity-at-most value is $F(c)=\max (0,{\max }_{w_i\le c}(v_i+F(c-w_i)))$. Evaluate capacities in increasing order so repeated uses are available. With weights three and five, values four and seven, and capacity ten, two weight-five items give value fourteen. A zero-weight positive-value item would make the objective unbounded, explaining the positive-weight hypothesis.

For a complex Klein-Gordon field, $j^{\mu }=i({\varphi }^{\ast }{\partial }^{\mu }\varphi -\varphi {\partial }^{\mu }{\varphi }^{\ast })$ is conserved. A mode $\varphi =N{e}^{-iEt+i\mathbf{p}\cdot \mathbf{x}}$ has $j^0=2E|N{|}^2$, so unrestricted positive- and negative-frequency solutions do not have a positive probability norm under this current. In the quantized complex field, the associated conserved quantity is charge: particles and antiparticles contribute with opposite signs while both have positive excitation energy and positive Fock space norm.

For unidirectional flow through area $A$ with mean speed $\overline{u}$, the outgoing kinetic-energy flux is $\frac{1}{2}\rho A\alpha {\overline{u}}^3$, where $\alpha =(A{\overline{u}}^3{)}^{-1}{\int }_A{u}^{3}dA$. Cubing the mean speed generally differs from averaging the cube. A uniform profile has $\alpha =1$; circular laminar Hagen-Poiseuille flow has $\alpha =2$.

Viscous transfer of tangential momentum smooths gradients of velocity. For unidirectional incompressible flow without a pressure gradient, the Navier-Stokes equation reduces to the displayed diffusion equation, whose diffusion coefficient is the kinematic viscosity $\nu =\eta /\rho$.

Competition among a restricted set of neighbours or relatives. Benefits to related recipients can be offset when those recipients compete for the same limited breeding opportunities.

An intensity-dependent refractive index produced by the optical field itself. In a simple instantaneous isotropic model, $n=n_0+n_{2}I$. A transversely nonuniform intensity produces self-focusing for positive $n_2$ and self-defocusing for negative $n_2$. Temporal intensity variation produces self-phase modulation. These are consequences of a third-order nonlinear optical susceptibility, distinct from a static applied-field electro-optic Kerr effect.