A flower bud is an undeveloped flower in which the floral organs are forming or remain enclosed before anthesis. It differs from a vegetative bud, which will extend a shoot with leaves. A flower bud is already reproductive even while its external coverings hide the future stamens and carpels.^[ref]

Dissecting a series of flower buds of increasing size can reveal the order in which floral structures appear. Opening the largest bud is not equivalent to observing the original switch to flowering: that developmental decision occurred earlier. This makes flower bud size a useful staging observation, provided the relation between size and internal development is checked for the particular plant.

Robust linear constraints for bounded covariance models require a linear inequality to hold for every error state admitted by the selected bounded model. A covariance matrix fixes second moments; the robust guarantee additionally depends on the support of a probability distribution or other specified uncertainty set. This distinction matters when a physical limit must never be crossed within the model.

Let $E=LS$ as in finite-support realization of a covariance matrix, where all coordinates of $S$ are independent fair signs. Then $\mathrm{Cov}(E)=L{L}^{\mathsf{T}}$. For nominal input $\mu$, require $a^{\mathsf{T}}(\mu +E)\ge b$ for every sign state. Writing $c=L^{\mathsf{T}}a$, direct minimization givesEach sign can independently oppose its coefficient, including when some coefficients vanish. Thus the exact necessary and sufficient condition is $a^{\mathsf{T}}\mu -\Vert L^{\mathsf{T}}a{\Vert }_1\ge b$, using the L1 norm. The same extremum holds over the full box $s_j\in [-1,1]$ because a linear function attains an extreme at a vertex. This is a support function calculation in robust optimization; the general worst-case formulation and its dependence on the uncertainty set are illustrated in the primary optimization text.^[ref]

For an original two-input example in volts, let $E_1=(1\mathrm{V})Z$ and $E_2=(1\mathrm{V})(0.6Z+0.8W)$, with independent fair signs $Z,W$. The covariance matrix is $\left(\begin{array}{cc}1& 0.6\\ 0.6& 1\end{array}\right){\mathrm{V}}^2$. To keep the actual first input at least as large as the second, let the nominal difference be $d={\mu }_1-{\mu }_2$. ThenA nominal difference of $1.0\mathrm{V}$ fails in the state $Z=-1,W=1$, despite being positive nominally. A difference of $1.3\mathrm{V}$ leaves a guaranteed $0.1\mathrm{V}$ margin under this model. Using only the separate bounds $|E_1|\le 1\mathrm{V}$, $|E_2|\le 1.4\mathrm{V}$ would demand $d\ge 2.4\mathrm{V}$, because that larger box admits joint states the sign model forbids.

The standard deviation of $E_1-E_2$ is instead $\sqrt{0.8}\mathrm{V}$. It is not the hard bound $1.2\mathrm{V}$. A different bounded law with the same covariance matrix can change the hard bound, as nonuniqueness of bounded uncertainty laws with a fixed covariance explains. For several linear inequalities, apply the support calculation to each row. This verifies an assumed model's feasible states; it does not establish that the chosen support contains every error of an actual instrument.

Nonuniqueness of bounded uncertainty laws with a fixed covariance means that specifying finite bounds, expected values and a covariance matrix can still leave several different joint probability distributions. They agree on the variance of every linear combination of the errors, but can disagree on threshold probabilities and nonlinear outputs. A particular construction in bounded errors with prescribed correlation therefore requires a distributional assumption beyond its second moments.

Here is an original example with a common allowed box $[-\sqrt{2},\sqrt{2}{]}^2$. In model A, $X$ and $Y$ are independent fair signs: the four points $(1,1),(1,-1),(-1,1),(-1,-1)$ each have probability $1/4$. In model B, the four points $(\sqrt{2},0),(-\sqrt{2},0),(0,\sqrt{2}),(0,-\sqrt{2})$ each have probability $1/4$. Both are bounded, have zero expected value, and satisfyFor model B, each squared coordinate is two at two of the four points and zero otherwise, proving its unit variance directly. Its zero covariance does not make the coordinates independent random variables: when $X$ is nonzero, $Y$ must vanish. The common box is an allowed deterministic bound; its boundary need not be attained by every model.

The linear output $U=X+Y$ has expected value zero and variance two under either model. Yet under model A it takes $-2,0,2$ with probabilities $1/4,1/2,1/4$, while model B gives only $-\sqrt{2},+\sqrt{2}$ with equal probabilities. Consequently,Even a linear output's threshold risk is not fixed by its variance. For the nonlinear output $F=X^2{Y}^2$, model A gives $F=1$ always and model B gives $F=0$ always. Thus the complete output law, and even a nonlinear output's expected value, can differ.

The law of propagation of uncertainty computes first-order output variance from sensitivity coefficients and input covariance; it does not identify all probabilities. NIST correspondingly requires distributional justification to assign a confidence level to an uncertainty interval.^[ref] The finite counterexample above is an original exact calculation. For guaranteed linear limits, robust linear constraints for bounded covariance models must use the chosen support of a probability distribution, rather than treating a covariance matrix as a support description.

Boundary-layer transition is the development of turbulent flow from an initially laminar flow within a boundary layer. A laminar boundary layer can contain small, organized disturbances before the fluctuations become turbulent. Thus detecting an oscillation is evidence of a disturbance, not by itself proof that transition is complete.

A motivating experiment by G. B. Schubauer and H. K. Skramstad studied a flat plate while reducing the surrounding stream's turbulence intensity with damping screens. At unusually low stream disturbances, they observed oscillations in the laminar boundary layer. Controlling the external disturbances made internal disturbance development experimentally distinguishable.^[ref]

The local Reynolds numbercompares inertia with viscosity using external speed $U$, distance $x$ from the leading edge and kinematic viscosity $\nu$. For stipulated $U=10\mathrm{m}{\mathrm{s}}^{-1}$, $x=0.50\mathrm{m}$ and $\nu =1.5\times 10^{-5}{\mathrm{m}}^2{\mathrm{s}}^{-1}$, it is approximately $3.33\times 10^5$. This calculation specifies a flow scale; it is not a universal transition criterion. Surface condition, surrounding disturbances and the pressure gradient also matter.

A simple model of boundary-layer disturbance amplification makes this sensitivity explicit. Suppose a small disturbance amplitude obeysIf a stipulated breakdown amplitude is 100 times the initial amplitude, the integrated growth needed is $\ln 100\approx 4.61$. Increasing the initial amplitude tenfold reduces that requirement to $\ln 10\approx 2.30$. This is an illustrative linear growth model, not a calibrated predictor of turbulent breakdown. It explains why specifying Reynolds number without specifying the disturbance environment can leave the observed transition location undetermined.

A bosonic string is a relativistic one-dimensional object whose embedding coordinates are bosonic fields on a two-dimensional worldsheet. Its classical motion can be described by the Nambu–Goto action, proportional to the worldsheet area, or by the equivalent Polyakov action with an auxiliary worldsheet metric.

An open string has endpoints; a closed string forms a loop and has independent left- and right-moving oscillations. Quantization produces a tower of particle states. In the ordinary flat critical bosonic string theory, anomaly cancellation selects 26 spacetime dimensions, and the closed-string spectrum includes a massless graviton. The tachyonic ground state indicates instability of the chosen vacuum.^[ref]

A borehole televiewer produces an oriented image of a borehole wall, often by scanning with acoustic pulses. Recorded echo amplitude and travel time reveal changes in wall reflectivity and radius as a function of depth and azimuth.

An enlarged cross-section can indicate borehole breakout, while fractures produce other characteristic traces. Inferring the surrounding stress requires a mechanical interpretation of those features; the image alone does not directly measure every stress component. Borehole conditions and tool orientation affect the result.

A proper border of a string is a string prefix that is also a string suffix and is shorter than the complete string. With zero-based lengths, its length $b$ satisfies $0\le b<m$ for a length-$m$ string, if the empty string is allowed. Specify that convention when counting borders.

For ababa, the nonempty proper borders are a and aba; the complete ababa is a border of a string but not a proper one. A proper border of length $b$ allows two copies of the string to overlap with displacement $m-b$. The border aba therefore explains the shift of two between overlapping occurrences of ababa in abababa.

The prefix function stores the longest proper-border length of each prefix. For ababa, following these lengths gives $3\to 1\to 0$, listing successively shorter candidate overlaps. The Knuth-Morris-Pratt algorithm uses this chain after a mismatch, retaining a suffix already known to match rather than comparing the whole pattern again.

Diborane, ${\mathrm{B}}_2{\mathrm{H}}_6$, is an electron-deficient borane with four terminal B–H connections and two bridging hydrogen atoms. The bridges are described by three-centre two-electron bonds, with one bonding electron pair shared over each B–H–B unit.

The valence-electron count explains why the multicentre model is useful: $2(3)+6(1)=12$ valence electrons. Four ordinary terminal bonds use eight, leaving four for the two bridges. Assigning a separate ordinary two-electron bond to every boron–hydrogen contact would demand more electrons than the neutral molecule has. Diborane therefore provides a concrete counterexample to treating every drawn connection as an independent localized electron-pair bond.