A linear recurrence ${\sum }_{j=0}^r{p}_j(n)a_{n+j}=0$ whose coefficients $p_j(n)$ are polynomials in the index. A sequence satisfying such a relation is P-recursive.

For a rectangular membrane mode with peak antinode displacement A, the integrals of each squared sine factor are half the corresponding side length. Integrating local kinetic energy density $\rho ({\partial }_t\psi {)}^2/2$ gives $K=\rho ab{\omega }^2{A}^2{\sin }^2(\omega t+\varphi )/8$. The total energy is $\rho ab{\omega }^2{A}^2/8$, and potential energy has the complementary cosine-squared factor. Its time-averaged kinetic energy is $\rho ab{\omega }^2{A}^2/16$. At equal peak displacement, frequency and density, such mean energies are proportional to area.

For a thick rectangular barrier with equal exterior potentials, constant mass and $0<E<U$, the exact matched transmission probability has the displayed limit when $\kappa a\gg 1$. The exponential is fixed by the evanescent action; the interface prefactor remains energy-dependent. A leading-exponential estimate can therefore differ by an order-one multiplier from the matched result.

Programmed ribosomal frameshifting changes the reading frame during biological translation at a specific recoding signal. A slippery sequence and a downstream RNA structure can bias the event, allowing one transcript to produce different polypeptides in controlled proportions.

A change of reading frame during biological translation. It need not change the underlying DNA sequence, unlike a frameshift mutation. It can be programmed by a recoding signal or occur as a biological translation error.

A sequence interpreted in a particular codon frame without an internal in-frame stop, with the boundary convention stated. Prediction of an open reading frame alone does not demonstrate biological translation or a stable functional product.

For $A\rightleftharpoons 2B$, ${\alpha }^2/(1-\alpha )=K_N/(4M)$. The left side increases monotonically on $0<\alpha <1$, so increasing the conserved formula-unit density at fixed temperature reduces dissociation in the ideal model.

Let $s=r(1-r)$ and $p=r(1-r^2)$. Under reflection $r\mapsto 1-r$, $s$ is symmetric and $p=(3/2)s+a$, where $a=r(1-r)(r-1/2)$ is antisymmetric. Their cross terms vanish in both the $L^2$ inner product and the Dirichlet energy form: each integrand is odd about the midpoint. Their separate Rayleigh quotients are $R[s]=10$ and $R[a]=42$. Thus every quotient on their two-dimensional span is a positive weighted average of ten and forty-two, so the minimum remains ten. Adding a direction with the wrong symmetry need not improve the ground-state estimate.

The symmetric quartic $s^2=r^2(1-r{)}^2$ does add a useful direction. On the span of $s,s^2$, the generalized eigenvalue problem has stiffness matrix $K=\left(\begin{array}{cc}1/3& 1/15\\ 1/15& 2/105\end{array}\right)$ and mass matrix $M=\left(\begin{array}{cc}1/30& 1/140\\ 1/140& 1/630\end{array}\right)$. Its least eigenvalue is $56-4\sqrt{133}\simeq 9.86974962$, close to the exact Dirichlet ground value ${\pi }^2$. This is an explicit useful enlargement of a polynomial Rayleigh-Ritz trial space for a one-dimensional box.

A rank function of a partially ordered set assigns an integer $\rho (x)$ to each element so that $\rho (y)=\rho (x)+1$ whenever $y$ covers $x$. In a finite ranked partially ordered set with minimal elements normalized to zero, $\rho (x)$ is the length of every saturated chain from a minimal element to $x$.

For subsets ordered by inclusion, cardinality is a rank function of a partially ordered set. For a chain $a<b<c$, the normalized ranks are $0,1,2$. An order with two saturated chains of unequal lengths between the same endpoints cannot have a rank function of a partially ordered set, because the required increments would assign conflicting values. This notion is distinct from the rank of a set in cumulative set theory.

A lazy random walk remains at its current position with positive probability and otherwise makes an ordinary random walk step. A common convention replaces transition matrix $P$ by $P_{\mathrm{lazy}}=(I+P)/2$. This preserves every stationary distribution because $\pi P=\pi$ implies $\pi P_{\mathrm{lazy}}=\pi$. It also removes periodic alternation in an irreducible discrete-time chain: the self-loop permits return in one step. A lazy symmetric walk on the integers has increments $-1,0,1$ with probabilities $1/4,1/2,1/4$, so its step variance is $1/2$.

For $N$ equal antennas, weak-source independent receiver noise, natural weighting and total on-source time $t$, the point-source rms in one polarization of waves is the displayed value. There are $N(N-1)/2$ independent cross-correlation baselines, each with real-component error $\mathrm{SEFD}/\sqrt{2\Delta \nu t}$. Combining $n_{\mathrm{pol}}$ independent polarization of waves estimates reduces the rms by $\sqrt{n_{\mathrm{pol}}}$. Correlation losses divide the sensitivity by an efficiency factor; weighting, flagging and an extended source change the effective sensitivity.^[ref]

Independent visibility samples are combined with weights proportional to inverse noise variance. This maximizes the point-source sensitivity for correctly calibrated samples. Dense regions of the sampled Fourier plane receive more total weight; tapering the long baselines can further emphasize extended structure at the cost of angular resolution. A source-template filter instead also weights by predicted source response.

A residual delay $\delta \tau$ in a rectangular band of width $\Delta \nu$ attenuates the correlated amplitude by $\mathrm{sinc}(\pi \Delta \nu \delta \tau )$. Coarse delay quantization must keep this factor near one; the relevant path scale is $c/\Delta \nu$. Separately, an untracked carrier phase has error $2\pi \delta \ell /\lambda$. Fine phase rotation can remove that central-frequency phase while leaving the band-dependent decorrelation, so delay-step specifications must state both bandwidth and compensation architecture.

very-long-baseline interferometry correlates accurately time-referenced radio signals recorded at widely separated collectors, giving angular resolution of order wavelength divided by projected baseline. Geometric delays, frequency standards, atmospheric propagation and source structure enter the calibration. Compact maser positions and motions are one application.

For $x(t)=A\cos {\omega }_{1}t+B\cos {\omega }_{2}t$ with irrational frequency ratio and nonzero amplitudes, $|x|\le |A|+|B|$. The phase trajectory is dense on the two-torus, so this bound is a supremum. The required exact phase alignment may never occur at finite time. Thus an exam's maximum-amplitude estimate can be a limiting bound rather than an attained displacement; finite observation intervals have smaller maxima.

Periodically poled lithium niobate is lithium niobate whose ferroelectric domain orientation is reversed periodically along an optical interaction path. Reversal changes the sign of the relevant second-order nonlinear optical susceptibility, supplying the spatial modulation needed for quasi-phase-matching.

For a uniform collinear conversion mismatch $\Delta k$, a first-order poling period $\Lambda$ can compensate it through $\Delta k-2\pi /\Lambda \simeq 0$, with signs chosen consistently. The generated contributions then reinforce over many domains instead of repeatedly canceling. The poling period, temperature and input wavelengths determine the useful conversion band; domain defects and finite length affect efficiency. ^[ref]

Return amplitudes sum even numbers of well-to-well events, while transfer amplitudes sum odd numbers. A dilute event gas gives cosh and sinh factors and a doublet splitting $2\hslash A{e}^{-S_0/\hslash }$.

Two indistinguishable bosons entering separate inputs of an ideal balanced beam splitter have zero probability of leaving one per output. The two alternatives for that occupancy interfere destructively. Distinguishability in temporal, spatial or internal modes weakens the cancellation.

For linear spin waves, $\delta S=(2N{)}^{-1}{\sum }_k[(1-{\gamma }_k^2{)}^{-1/2}-1]$. The chain has ${\gamma }_k=\cos k$ and a logarithmic infrared divergence. This signals failure of the ordered expansion, not a physical negative staggered moment.