The uncollided component decays exponentially when its removal probability per unit path length is constant. Scattered contributions require a separate transport treatment or a specified radiation buildup factor.

A coefficient describing the locally absorbed fraction of an incident photon energy flow per path length, under a stated averaging convention. It is distinct from the total linear attenuation coefficient, which counts removal from an uncollided beam. Dividing by mass density gives an energy-absorption coefficient per mass.

A conservative material-screening upper bound obtained by requiring wall penetration before fast fracture even at a stated yield-limited stress. Substituting a thickness-dependent pressure stress instead yields a different fixed-pressure inequality; the two comparisons must not be conflated.

A Late Devonian fossil lamprey from South Africa preserving diagnostic anatomy in an exceptional soft-tissue window.

Set $z=E-V_0+i0$ and continue both roots from the upper half-plane. The right-hand wavenumber is positive above the positive-energy threshold, positive imaginary inside the gap, and negative below the negative-energy threshold. Its group velocity in the last region remains rightward because both momentum and kinetic energy are negative. Choosing a positive real root there instead specifies an incoming wavepacket from the right.

For the outgoing negative-energy branch $q=-Q$, matching the field and its spatial derivative gives these signed charge-current ratios. Their sum is one although reflection exceeds one. The ideal sharp step has a singular stationary matching denominator at $E=V_0/2$ in the supercritical range. It is not a finite single-particle probability prediction; wavepackets, source preparation and field quantization matter for its physical interpretation.

For $E\epsilon +\eta \dot{\epsilon }={\sigma }_0\Theta (t)$ with initially zero strain, $\epsilon =({\sigma }_0/E)(1-e^{-t/\tau })$ and $\tau =\eta /E$. Strain is continuous at load application, approaches a bounded value and recovers exponentially after unloading.

Under harmonic stress, strain amplitude is ${\sigma }_0/\sqrt{E^2+{\omega }^2{\eta }^2}$ and its lag is $\arctan (\omega \eta /E)$. Low frequencies are nearly elastic; high-frequency stress-controlled deformation is small and nearly a quarter cycle behind stress.

The edges of an itinerant spin-flip particle-hole continuum follow from both excitation energy and occupation constraints. Consider a zero-temperature three-dimensional parabolic-band model with majority states occupied for $|\mathbf{k}|\le k_F$ and an entirely empty minority band. For exchange splitting $2M$ and ${ϵ}_k={\hslash }^2{k}^2/(2m)$, a majority-to-minority flip carrying wavevector $\mathbf{q}$ hasFor $q>0$, the scalar product is bounded by $\pm k_{F}q$, giving continuum edgesThe extremes occur at opposite points of the occupied Fermi surface along $\mathbf{q}$. At $q=0$ the interval collapses to the single exchange energy $2M$.

The empty-minority assumption requires sufficient splitting; otherwise Pauli exclusion principle blocks some final states and the edges must be recomputed over the allowed region. These are independent particle-hole thresholds, not a calculation of the collective ferromagnetic magnon dispersion. A collective mode below the continuum can therefore have a different low-momentum energy scale.

Every three-dimensional fourth-rank Cartesian tensor invariant under all proper rotations has the displayed form, without assuming index-interchange symmetry. Half-turns about the coordinate axes force every nonzero component to contain each coordinate label an even number of times. The surviving patterns have four equal labels or two equal pairs. Axis permutations identify the latter with $c_{1122}=\alpha$, $c_{1212}=\beta$ and $c_{1221}=\gamma$, while all four-equal components share a value $A$.

Evaluate the component with four copies of $(e_1+e_2)/\sqrt{2}$. Multilinearity gives $(A+\alpha +\beta +\gamma )/2$, which must equal $A$ by rotational invariance. Hence $A=\alpha +\beta +\gamma$, proving the formula. Conversely each pairing of Kronecker deltas is invariant under any orthogonal transformation. For symmetric strain only $\alpha$ and $\beta +\gamma$ matter, giving the two Lamé parameters of isotropic elasticity on tensor components.

For a uniformly distributed unit vector in three dimensions, $⟨u_i{u}_j{u}_k{u}_l⟩=({\delta }_{ij}{\delta }_{kl}+{\delta }_{ik}{\delta }_{jl}+{\delta }_{il}{\delta }_{jk})/15$. Contracting indices fixes the coefficient. This identity is useful for incoherent scattering averages and random-axis material models.

The pool adjacent violators algorithm solves weighted isotonic regression: minimize ${\sum }_i{w}_i(y_i-{\theta }_i{)}^2$ with $w_i>0$ and ${\theta }_1\le \cdots \le {\theta }_n$. Start with singleton blocks. Whenever consecutive fitted block means decrease, merge the two blocks and replace them by their weighted mean; continue until no violation remains. For equal weights and observations $(3,1,2)$, merging the first two gives the solution $(2,2,2)$. A stack implementation revisits each block only when merging it, giving linear time. Pooling is required by the order-constrained optimum: two reversed adjacent blocks cannot remain separate fitted levels.

Heating over land often favours a deeper cyclonic thermal low, while cooling favours a shallower anticyclonic thermal high. Diabatic heating and surface friction violate the assumptions of simple parcel PV conservation; a vertical profile is needed for a quantitative statement.

${\sigma }_{\theta }=-(1/g)\partial p/\partial \theta$ measures mass per horizontal area per unit potential-temperature interval under hydrostatic balance. It is positive in a statically stable atmosphere.

An ideal operational amplifier with a grounded noninverting input, input resistor $R_i$, and parallel feedback resistor $R_f$ and capacitor $C$ implements a low-pass filter. The virtual short and Kirchhoff's current law give $V_{\mathrm{out}}/V_{\mathrm{in}}=-Z_f/R_i$, where $Z_f=R_f/(1+i\omega R_{f}C)$. The DC gain magnitude is $R_f/R_i$ and the cutoff frequency is $1/(2\pi R_{f}C)$. The capacitance reduces high-frequency feedback electrical impedance, causing the gain magnitude to fall.

An ideal inverting operational amplifier with series $R_1,C_1$ input and parallel $R_2,C_2$ feedback has the displayed transfer function, where $K=R_2/R_1$ and ${\tau }_i=R_i{C}_i$. Its peak is at ${\omega }_0=1/\sqrt{{\tau }_1{\tau }_2}$ with voltage gain $K{\tau }_1/({\tau }_1+{\tau }_2)$. A wide input-high-pass/feedback-low-pass separation produces a broad plateau, not a narrow resonance. Validity requires stable negative feedback, sufficient loop gain and linear output operation.

Terms of a sequence satisfying the third-order linear recurrence $T_{n+3}=T_n+T_{n+1}+T_{n+2}$. Initial values vary by convention; some counting problems specify $T_1=T_2=T_3=1$. To study divisibility by $m$, retain three consecutive remainders modulo $m$, a deterministic state with at most $m^3$ possibilities.

The nondecreasing sequence of positive integers beginning with $G(1)=1$ in which the integer $n$ occurs exactly $G(n)$ times. Its first terms are $1,2,2,3,3,4,4,4,5,5,5,\dots$. The prefix sum ${\sum }_{j=1}^{k}G(j)$ gives the last position occupied by $k$.

Starting with two distinct positive integers $a<b$, repeatedly append the smallest integer larger than the previous term that is expressible in exactly one way as the sum of two distinct earlier terms. Requiring distinct summands and counting unordered pairs are both essential parts of the definition.