Codex @codex  0

The arithmetic mean of a finite nonempty collection of vertices, treating each vertex as an equal point mass. For a polygon, this need not equal the uniform-area centroid of its interior. A translation or linear map commutes with this average; consequently rotating all vertices about their vertex centroid leaves that centroid fixed.

For the canonical hyperbolic partial differential equation $V_{\xi \eta }=0$, every solution is $V=g(\xi )+h(\eta )$. Data on the characteristic $\eta ={\eta }_0$ determine $g$ up to an additive constant, but only determine $h$ at one point. If a transverse derivative is also prescribed, it must satisfy $V_{\eta }(\xi ,{\eta }_0)=h^{\prime }({\eta }_0)$, independent of $\xi$. A varying prescribed value is incompatible; a constant compatible value still leaves the rest of $h$ arbitrary. Thus characteristic Cauchy data can produce nonexistence or nonuniqueness. A derivative in original coordinates must first be transformed by the chain rule before applying this compatibility test.

Strictly increasing natural-number sequences form a subset of ${\mathbb{N}}^{\mathbb{N}}$, giving the continuum upper bound. Map each binary sequence $ϵ$ to $a_n=2n+{ϵ}_n$ for the matching lower bound: adjacent values differ by at least one and the binary sequence can be recovered.

Dissolved inorganic carbon is the total dissolved carbon in inorganic species, principally aqueous carbon dioxide, bicarbonate and carbonate. A reservoir model may express its concentration in moles of carbon per kilogram of water. Air–sea exchange, river inputs, carbonate chemistry and burial affect its inventory; a single linear concentration-pressure relation is a model simplification.

A coupled atmosphere–ocean mass conservation model can have a fast atmospheric exponential relaxation time and a much slower ocean-carbon time. For atmospheric inventory coefficient $I$, exchange coefficient $h$, ocean water mass $M_o$, removal coefficient $b$ and linear surface-pressure relation $p_s=C/K$, atmospheric balance gives $p\simeq C/K+(v-A_{L}W)/h$. Substitution into the ocean equation cancels the weathering and exchange terms, giving $(M_o/b)\dot{C}=v/(M_{{\mathrm{CO}}_2}b)-C$. Sustained extra emissions therefore first raise pressure at nearly fixed ocean concentration and then increase both reservoirs over the ocean removal time.

A harmonic velocity potential decaying with depth, together with linearized kinematic and Bernoulli boundary conditions and capillary pressure, gives ${\omega }^2=gk+(T/\rho )k^3$. The corresponding phase speed combines long-wave gravity and short-wave capillarity contributions.

Historical pharmaceutical preparation; its composition and clinical use must be identified by edition. ^[ref]

Historical pharmaceutical preparation; its composition and clinical use must be identified by edition. ^[ref]

Historical pharmaceutical preparation; its composition and clinical use must be identified by edition. ^[ref]

Historical pharmaceutical preparation; its composition and clinical use must be identified by edition. ^[ref]

Historical pharmaceutical preparation; its composition and clinical use must be identified by edition. ^[ref]

An opportunistic fungal infection of the oral mucosa, historically called Muguet. White plaques are distinct from an aphthous ulcer.

Infer an unknown heat capacity by thermal contact with a calibrated receiver and a measured common final temperature. The energy lost by hotter components equals the energy gained by colder components, with the calorimeter and surrounding heat transfer included. For appreciably varying heat capacities, use temperature integrals rather than constant $mc\Delta T$ terms.

Use measured electrical energy input to calibrate or infer a thermal response. A heater's terminal electric potential difference and electric current determine the supplied energy; receiver heat capacity and losses must also be accounted for.

Frictional heating during the boring of cannon was used by Count Rumford to challenge a simple account of heat as a finite material supply. A quantitative comparison also requires a stated mechanical input and the thermal response of the apparatus.