A concentration gradient is the spatial gradient $\nabla c$ of a scalar concentration field. It points toward the greatest local increase of $c$; its components are spatial partial derivatives, so its units are those of concentration divided by length.

For a one-dimensional profile $c(x)=c_0+ax$, the concentration gradient is the constant $a\hat{\mathbf{x}}$. Fick's first law gives a diffusive flux proportional to $-\nabla c$ when that constitutive model applies: the minus sign describes transport toward lower concentration. A fluid velocity can simultaneously advect the constituent in another direction, so a concentration gradient alone does not determine its total flux.

Protein concentration specifies protein amount per sample volume, commonly as mass concentration or molar concentration. A molar value must state the counted entity, such as chains or oligomeric particles, and quantification must account for contaminants and optical properties.

An equilibrium concentration is a constant concentration satisfying the fixed point equations of a specified dynamical model. Balance between production and loss can maintain such a value without implying thermodynamic chemical equilibrium.

The recorded origins and transformations of data, including references, versions, preprocessing and identifiers.

A model representing a passage by its varying cross-sectional area and approximately uniform section-averaged flow variables. It neglects detailed transverse velocity and thermodynamic structure. In a smooth lossless nozzle, conservation laws then connect area to Mach number; this model does not by itself determine the curved sonic surface of an arbitrary multidimensional passage.

Flow whose mass flux is limited by a sonic condition in a controlling section, for fixed upstream thermodynamic conditions. In a smooth isentropic converging-diverging nozzle, the throat reaches Mach number one. Lowering downstream pressure further cannot indefinitely increase that fixed-upstream throat mass flux; shocks and downstream structure can still change.

The temperature reached by bringing a flow to rest adiabatically while converting its kinetic energy to enthalpy. For a calorically perfect gas, $T_{\mathrm{stag}}=T+u^2/(2c_p)=T[1+(\gamma -1)M^2/2]$. It is conserved across an adiabatic normal shock even though entropy increases and stagnation pressure decreases.

A compressible flow that passes between subsonic and supersonic regions. A smooth steady crossing of a sonic point requires the corresponding regularity condition; a discontinuous transition can instead involve a shock wave.

Carbon-fibre-reinforced polymer is a composite material in which carbon fibres reinforce a polymer composite matrix. The fibres carry axial tension efficiently, while the composite matrix maintains their arrangement and transfers load through interfacial shear stress. A component's response therefore depends on fibre direction, fibre volume fraction and bonding as well as the constituent properties.

For continuous aligned reinforcement, the longitudinal rule of mixtures predicts the initial axial Young's modulus. A stack with fibres in several directions trades some axial performance for reinforcement in those directions. Such an arrangement can reduce in-plane anisotropy, but does not automatically make the through-thickness response identical. Comparing tensile testing along the fibres, across them and through the assembled layers reveals different possible failure mechanisms, including fibre fracture and loss of fibre-matrix load transfer.

A long-fibre composite contains reinforcing fibres whose lengths are large enough that end regions do not dominate their load-carrying response. In a well-bonded continuous aligned specimen pulled along the fibres, the fibres and composite matrix undergo approximately the same axial strain. Their different Young's moduli then produce different axial stresses, and adding their forces gives the longitudinal rule of mixtures.

Load enters a fibre through shear stress at its interface with the composite matrix. Near a free fibre end the axial stress must rise from zero, so a short fibre can spend much of its length below its useful load. The critical fibre length expresses the associated scale in a simple strength-transfer model. Whether a fibre is mechanically long therefore depends on its diameter, the interface and the required stress, rather than on an absolute length in millimetres. Changing fibre direction still changes the response: long reinforcement does not remove anisotropy.

A glass fibre is a thin strand of glass used as a reinforcement. When embedded in a composite matrix, many glass fibres can carry axial tension while the surrounding material maintains their positions and transfers load between them. The reinforcement is effective only if the interface transmits the required shear stress; an unbonded fibre can slide without carrying the expected load.

Continuous aligned glass fibres reinforce one direction strongly, whereas short fibres with varied orientations distribute reinforcement over more directions.^[ref] Even if the glass itself is approximately isotropic, a composite material containing aligned glass fibres is generally anisotropic. Comparing specimens with equal fibre volume fraction but different orientations separates the effects of constituent quantity and arrangement.

A carbon fibre is a slender reinforcement made predominantly of carbon, used to carry tension along its length. Its axial Young's modulus, tensile strength and mass density are especially useful when comparing how much elastic load a lightweight component can carry. A fibre's measured tensile strength also depends on defects and the length tested, so one strength value does not characterize every specimen.

Embedding carbon fibres in a polymer produces carbon-fibre-reinforced polymer. The composite matrix supports the fibre arrangement and permits fibre-matrix load transfer; the axial performance of an aligned assembly does not imply equal transverse performance. Tensile testing of single fibres measures the reinforcement response, while testing composite strips at different orientations measures how that response survives assembly into a composite material.

Take a complex logarithm whose radial branch cut has angle $\beta$. On a positively oriented circle, use the one-sided branch values on either side of its crossing with the cut. Parametrize from $\theta =\beta$ to $\beta +2\pi$; adding an integer multiple of $2\pi i$ to the branch does not change the integral because the integral of $dz$ is zero. Along the open arc, a primitive is $z\mathrm{Log}z-z$. Its endpoint difference is $2\pi i$ times the cut-crossing point $R{e}^{i\beta }$, proving the displayed formula. The circle crosses the cut, so this is a piecewise contour integral of boundary values, not the integral of a single holomorphic logarithm on a neighborhood of a closed circle. The Cauchy integral theorem therefore does not make it zero.

Synteny describes the occurrence of genomic loci on the same chromosome; comparative usage often additionally concerns conserved neighbourhoods or order between genomes. A block containing corresponding genes can support assignment of an uncertain region even when individual sequences have diverged. Rearrangements such as a chromosomal inversion can preserve membership of a block while reversing its internal orientation. The intended sense of conserved location versus conserved order should be stated. ^[ref]

A nonempty finite word that is strictly smaller in lexicographic order than each of its nontrivial cyclic rotations. For example, $001$ is a Lyndon word, whereas $010$ is not because its rotation $001$ is smaller.

For slope $0\le \alpha \le 1$ and intercept $\rho$, the lower mechanical word has letters $w_n=\lfloor (n+1)\alpha +\rho \rfloor -\lfloor n\alpha +\rho \rfloor$. These letters are zero or one. Replacing floors by ceilings defines the upper mechanical word. An irrational slope gives a Sturmian word.

For a word $w=w_1\cdots w_n$ over a totally ordered alphabet, an inversion of a word is a pair of positions $i<j$ with $w_i>w_j$. Equal symbols contribute no inversion of a word. In baba, with $a<b$, the pairs are $(1,2),(1,4),(3,4)$, so the word inversion number is three. Counting these pairs records distance from sorted order even when letters repeat; replacing the letters by distinct ordered labels gives the inversion of a permutation case.

Color is a perceptual attribute arising from the optical spectrum reaching an observer and the observer's visual processing and viewing conditions. Cone cells have overlapping spectral sensitivities, so their responses do not provide a separate measurement of every wavelength. Different optical spectra can therefore produce the same color match, an instance of metamerism.

For a transmitting sample, the incident optical spectrum is modified by wavelength-dependent transmittance. Removing part of the blue band can make the remaining light appear yellow, but the absorption maximum alone does not specify the resulting shade. Changing the illuminant or background can change color appearance without changing the sample.

Colour vision based on three classes of photoreceptor input. Functional discrimination must be tested rather than inferred solely from gene count.

Dichromacy is color vision based on two functionally independent classes of photoreceptor input. Their different spectral sensitivities permit some optical spectra to be distinguished, but distinct spectra can still produce identical pairs of receptor responses and hence be metamers under the stated conditions.

A color matching experiment asks an observer to adjust primary lights to match a comparison field while controlling luminance cues. The available matches constrain the independent spectral information used by that observer. Two receptor classes do not mean that every appearance can be identified with one of only two named colors.