A graduated tube for confining a gas, measuring its volume, and often examining changes after a chemical reaction. The reading requires temperature, pressure and vapor pressure corrections.

Two based edge loops are equivalent under insertion or deletion of stationary steps, cancellation or insertion of a backtracking spur, and replacement of two sides of a triangle by its third side. The edge-loop group consists of the resulting classes, with concatenation as product and reversal as inverse. Each elementary move can be performed inside a longer concatenation, so multiplication is well defined. Associativity follows from concatenation, the stationary loop is the identity, and a loop followed by its reversal cancels by spurs.

Realization induces a homomorphism to the fundamental group of the geometric realization, because each move is a homotopy in an edge or a triangle. The relative Simplicial approximation theorem applied to based circles makes this map surjective. The full edge-path theorem additionally proves injectivity by approximating homotopies; that stronger assertion is not needed for the group construction or the surjectivity argument.

For two mathematical functions $f,g:A\to B$ with the same domain and codomain, equality is equivalent to equality of their values at every argument. In the set-theoretic representation, their graphs of functions then contain exactly the same ordered pairs. This permits a function identity to be proved by fixing an arbitrary argument and proving the resulting value equality.

A dependent variable has its value specified by a function of one or more independent variables. In $y=x^2$, choosing the function input $x$ determines the function output $y$; equal values of $y$ can nevertheless arise from different inputs.

In an ordinary differential equation such as $dy/dx=2x$, $x$ is the independent variable and $y$ is the dependent variable. The unknown object being sought is a function $y(x)$, rather than one numerical value of $y$. Its derivative describes how the dependent variable changes with the independent variable.

The empty function has the empty set as both its domain of a function and its graph of a function: it assigns no values at all. For every specified codomain $Y$, there is exactly one function $\varnothing \to Y$, because there are no function inputs requiring a choice of function output. This includes $Y=\varnothing$. In contrast, a function from a nonempty set to the empty set cannot exist.

Consequently the set of functions from the empty set to any set has cardinality one. This is the reason for $0^0=1$ in cardinal exponentiation and for a Cartesian product indexed by the empty set having one element, the empty tuple.

A complex-valued function has values in the complex numbers. It has a unique decomposition $f=u+iv$, where $u$ and $v$ are its real-valued components. For example, $f(t)=e^{it}=\cos t+i\sin t$ combines two real-valued functions into one function of modulus one.

On a measure space, a measurable complex-valued function is Lebesgue integrable exactly when both components are integrable. The bounds $|u|,|v|\le |f|\le |u|+|v|$ prove this equivalence, and its Lebesgue integral isMultiplication by a measurable complex factor of modulus one leaves $|f|$ unchanged and therefore preserves absolute integrability.

An ice-crystal deposit formed directly from atmospheric water vapour on a sufficiently cold surface. It is distinguished from frozen dew, in which liquid water first condenses and subsequently freezes.^[ref]

Frozen dew is a surface ice deposit made by first condensing liquid water and then freezing those droplets. It differs in its sequence of phase changes from hoar frost, which grows directly from water vapor. A surface can first cool to its dew point while still above freezing and then cool further.

The distinction matters when interpreting visible deposits: the presence of ice alone does not establish direct vapour deposition. The relevant temperature is that of the surface, which may differ from a nearby sheltered air measurement because of thermal radiation and local heat exchange.^[ref]

The temperature at which the given water-vapour partial pressure equals saturation vapour pressure over ice. It is defined relative to ice, whereas the usual dew point uses equilibrium with liquid water.

In 1833 Michael Faraday compared electrical effects from different sources and then their quantitative delivery. His galvanometer test discharged a machine-charged jar battery through a wet thread and the detector, connecting frictionally produced charge to magnetic action.^[ref]

Let an endomorphism of the free group on $x,y$ fix $x$ and send $y$ to $yw$, where $w$ is a nonempty word containing only the positive letters $x,y$. Every iterate of $y$ is again a positive word containing $y$. Substitution increases its length by $|w|$ times the number of occurrences of $y$, hence by at least $|w|$. No cancellation can occur, so $|{\varphi }^n(y)|\ge 1+n|w|$.

The uniqueness of the reduced word in a free group now proves that no positive iterate of the endomorphism is the identity. This criterion does not require the endomorphism to be invertible; for example the replacement $y\mapsto y^2$ is included.

A drying front separates a positive liquid layer from a region where its thickness is zero. Its position is an unknown boundary determined with the wet-region solution. At a continuous zero-height front, a jump balance alone need not fix its speed; the vanishing-depth profile and removal law can supply that information. Source terms representing evaporation or infiltration must respect nonnegative water availability in the dry region.

A frame delimiter marks a data-link frame boundary. A transparency rule such as bit stuffing prevents data from accidentally imitating the reserved delimiter.

Bit stuffing inserts extra bits according to an agreed rule, for example after five consecutive ones, so that a reserved frame delimiter cannot occur inside encoded contents. The receiver reverses the insertion rule.

A crack is an internal separation surface, or a curve in a two-dimensional reduction, across which a displacement can have distinct traces. In the ideal traction-free model its two faces carry zero imposed traction. A crack tip or front terminates this separation and can produce a singular stress field. The stress intensity factor measures the leading singular amplitude, while the energy release rate measures the reduction in equilibrium potential energy per increment of crack extent.

A nonnegative fraction with positive denominator and numerator smaller than its denominator, so its value lies in $[0,1)$. The zero value has the reduced fraction representation $0/1$. The term is sometimes extended to negative fractions by comparing absolute values; the nonnegative convention is the one used here.

The diamond-shaped floor of the fourth ventricle, formed by the dorsal pons and upper medulla oblongata.

The fastigium is the apex of the tent-like roof of the fourth ventricle. It is a geometric landmark of the cavity, distinct from a neural nucleus or a fluid outlet. Its name helps orient descriptions of the superior and inferior roof.

A lateral recess of the fourth ventricle is a lateral extension of the ventricular cavity toward a lateral aperture of the fourth ventricle. Its direction differs from the main cranial-to-caudal ventricular axis, making three-dimensional relations important in interpreting a section.

The superior medullary velum is a thin neural sheet spanning between the superior cerebellar peduncles and contributing to the upper roof of the fourth ventricle. Its ventricular surface and cerebellar attachment provide landmarks for neighboring pathways.