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Restricted jaw opening, including restriction caused by muscle spasms.

A characteristic fixed facial expression produced by contraction of facial muscles.

A smooth muscle cell is a contractile cell without the regular visible striations of skeletal and cardiac muscle. It contributes to structures such as vessel walls and the digestive tract. Its activity can change both local force and the geometry of a hollow organ.

For nonnegative integers with binary representation $Y={\sum }_{i=0}^{w-1}{y}_i{2}^i$, multiply by $X$ by summing the shifted terms $y_i{2}^{i}X$. Each term is either zero or a left bit shift of $X$. An unsigned $w$-bit input pair needs at most $2w$ product bits, because $(2^w-1{)}^2<2^{2w}$. This expansion follows from distributivity and motivates a binary multiplier.

The source ${\Gamma }_i$ is the mass gained by phase $i$ per mixture volume and time, through processes such as evaporation or condensation. Internal exchange conserves total mass, so its phase sources sum to zero. In an immiscible material-interface model without phase change, every ${\Gamma }_i$ is zero. The interface may still move: absence of transfer means that mass does not cross it relative to its own motion.

Here $⟨\cdot ⟩$ means integration over a fixed pipe cross-section divided by its area, and ${\chi }_1$ is the indicator of one immiscible fluid. The volume fraction is $\alpha$; intrinsic density averages over the phase-occupied area rather than over the whole section. The velocity must be density-weighted for its product with mean phase density to represent mass flux exactly.

If there is no interphase mass transfer, integrating ${\partial }_t(\rho {\chi }_1)+\nabla \cdot (\rho v{\chi }_1)=0$ over a section with impermeable outer wall gives ${\partial }_t({\rho }_1\alpha )+{\partial }_x({\rho }_1\alpha u_1)=0$. The second phase uses $1-\alpha$ and its own intrinsic averages. For constant phase density the velocity reduces to the ordinary phase-area average. These definitions apply where the corresponding phase fraction is positive; zero-fraction intrinsic quantities need a separate convention.

Take incompressible phase mass equations for fractions $\alpha ,1-\alpha$ and momentum equations ${\rho }_i[(u_i{)}_t+u_i(u_i{)}_x]+p_x=0$ with a common pressure. At a uniform state with $0<\alpha <1$, elimination of the mass and momentum perturbation amplitudes givesIts roots are the displayed speeds, whose imaginary parts are $\pm s(u_1-u_2)/(1+s^2)$. A nonzero slip velocity therefore gives a Fourier branch growing as $e^{|k||\mathrm{Im}c|t}$. No finite-order Sobolev estimate can control all high wavenumbers, showing ill-posedness of this unregularized closure. At zero slip the roots coincide and this nonreal-root argument no longer settles the degenerate case. Phase-fraction endpoints also lie outside the root formula's derivation.

Unbiased random termination of a sampled contribution: retain it with probability $q>0$ and multiply surviving contributions by $1/q$. In path tracing this controls path length without the systematic truncation of a fixed bounce limit, provided the continuation probabilities and weights are handled consistently.

For a real number $x$, its reduction modulo one is its fractional partThe floor function is used, including at negative inputs: $-1.2-\lfloor -1.2\rfloor =-1.2-(-2)=0.8$. Truncating toward zero would instead give $-0.2$, outside the required interval.

Two real numbers have the same fractional part exactly when their difference is an integer. Thus reduction chooses representatives for the quotient group $\mathbb{R}/\mathbb{Z}$, the circle group. Applied to two coordinates, it selects representatives in $[0,1{)}^2$ for a torus, obtained from a closed unit square by identifying opposite edges.

A logic or fragment has the finite model property if every satisfiable sentence has a finite model. This does not require every model to be finite, or a uniform bound on all finite model sizes.

For $N$ observations, let $m_X$ and $m_Y$ count missing values of two variables, and let $n_{XY}$ count observations on which both are available. The number missing at least one variable is $N-n_{XY}$. Applying the inclusion-exclusion principle to the two missing-value sets gives the displayed number missing both. The disjoint counts are consequently $m_X-m_{XY}$ missing only $X$, $m_Y-m_{XY}$ missing only $Y$, $m_{XY}$ missing both, and $n_{XY}$ complete cases. Their nonnegativity is a necessary consistency check on the supplied totals.

This identifies an aggregate missingness pattern without reconstructing individual records. It does not identify a missing-data mechanism, establish missing completely at random, or justify treating a substantive response such as an explicit unknown category as a missing observation. Counts from different cohorts cannot be inserted into the identity without first establishing a common denominator.

A microscope slide is a support, usually a flat glass plate, that holds a specimen for an optical microscope. A thin specimen and a cover glass can provide a reproducible optical path through the sample; the appropriate preparation depends on whether the observation uses transmitted or reflected light.

Preparation affects the inference. A thick or uneven specimen can superpose structures at different depths, while compression or drying can change the object being observed. Magnification alone therefore does not establish that an apparent boundary corresponds to a structure in the original specimen.

An anemometer measures air motion. In a cup anemometer, moving air exerts different drag on the two sides of the cups, turning a shaft; a calibration converts its rotation rate into wind speed. A separate directional measurement is needed to recover a horizontal velocity vector. A sonic anemometer instead compares sound travel times along different paths and can operate without rotating parts.^[ref]

The instrument measures the flow at its own location, so obstacles, mounting height and gust averaging affect the result. The reported wind is therefore not simply an intrinsic property of an entire town or atmospheric layer. Comparing observations requires a consistent exposure and a stated sampling procedure.

A weather station is a set of instruments measuring atmospheric conditions at a specified site. A thermometer records temperature, a barometer records pressure, an anemometer records wind, and a rain gauge measures collected rainfall. Moisture observations may be reported as relative humidity or dew point.

Instrument exposure is part of the measurement. Direct sunlight can heat a temperature sensor above the surrounding air; nearby buildings alter wind; the height and averaging interval affect comparability. A long record therefore needs instrument and site metadata as well as numbers. Consistent observations support both immediate meteorology and the statistics used in climatology.

An atmospheric optical feature produced by interactions with ice crystals. A common $22^{\circ }$ halo comes from a $60^{\circ }$ prism path in hexagonal crystals; minimum deviation concentrates rays near its inner angular boundary. This differs from the liquid-water path of a rainbow.^[ref]

A rotating atmospheric column over water, with visible mist and sometimes surface spray. Fair-weather waterspouts and thunderstorm-associated tornadic waterspouts have distinct formation settings.^[ref]

Frost is ice deposited or formed on a surface under sufficiently cold conditions. Hoar frost grows directly from water vapor, whereas frozen dew begins as liquid droplets that subsequently freeze. The frost point is the saturation condition relative to ice; it is distinct from the dew point defined relative to liquid water.

At night a surface can lose thermal radiation faster than surrounding air replaces the energy, so the surface temperature can fall below the air temperature measured at standard height. Frost can therefore occur even when that air observation remains above freezing.^[ref] Clouds and stronger mixing alter this energy exchange and hence the likelihood of a surface deposit.

Ice precipitation grown within thunderstorms, chiefly by collection and freezing of supercooled liquid droplets. A strong updraft can retain stones in a growth region; changing growth conditions can create clear and opaque layers without requiring one complete up-and-down trip per layer.^[ref]

A region of upward air motion. In a thunderstorm, an updraft can support and transport precipitation particles and supply the liquid-water environment in which hail grows.

Wind is motion of atmospheric gas relative to the underlying surface. A pressure gradient accelerates air toward lower pressure, while the Coriolis force deflects moving air in a rotating frame and friction alters flow near the surface. Consequently, large-scale wind need not point directly from high to low pressure: approximate geostrophic balance can produce flow nearly along pressure contours.

An anemometer measures the local motion. A stated wind observation needs its height, averaging interval and directional convention, because gusts and surface obstacles produce variability. Vertical air motion is also physically important: an updraft transports heat and moisture and can support growing hail.