At fixed applied voltage, the mechanical potential of a capacitor and its source includes $-C{V}^2/2$. Increasing the liquid overlap of a partially filled coaxial capacitor therefore pulls the liquid upward. Balancing $V^2(dC/dh)/2$ against the excess hydrostatic pressure times the annular area gives the title formula for a wide external reservoir. Its differential sensitivity is $2\Delta h/V$; it vanishes at zero bias and measures the magnitude, rather than the sign, of voltage.

A pandigital prime is a prime number that is also a pandigital number for a specified radix, digit alphabet and repetition convention. For example, $2143$ uses each of $1,2,3,4$ exactly once and is prime; trial division by prime numbers at most its square root verifies the latter condition.

The digit sum supplies a useful obstruction. A decimal number using $1,\dots ,n$ exactly once has digit sum $n(n+1)/2$. If this sum is divisible by $3$ and the number exceeds $3$, the number is composite. In particular, a number using all ten decimal digits exactly once has digit sum $45$ and cannot be prime. The filter follows from the digit sum integer congruence, independently of the order in which the digits occur.

The 2-adic logarithm is the power series $\log (1+x)={\sum }_{n\ge 1}(-1{)}^{n+1}{x}^n/n$, convergent for $x\in 2{\mathbb{Z}}_2$. Thus it is defined on the odd 2-adic units $1+2{\mathbb{Z}}_2$.

It is inverse to the 2-adic exponential after restriction to $1+4{\mathbb{Z}}_2$, with image $4{\mathbb{Z}}_2$. On the larger domain, $\log (-1)=0$, so it is not injective there. Indeed each odd unit is uniquely a sign times an element of $1+4{\mathbb{Z}}_2$, and the logarithm forgets the sign. Distinguishing the convergence domain from the inverse-function domain avoids a frequent error.

The first overtone is the lowest oscillatory normal mode above the fundamental mode. It is an ordering by frequency, not a claim that its frequency is an exact integer multiple of the fundamental. If the fundamental is labeled $n=0$, the first overtone is $n=1$; another common convention labels them one and two.

For a string fixed at both ends, the frequencies are $f_n=nc/(2L)$ with $n\ge 1$, so the first overtone is the second harmonic at twice the fundamental. For an ideal pipe closed at one end, the next allowed mode is at three times the fundamental. Geometry and boundary conditions determine the relation.

In crown-based usage, the clade comprising the most recent common ancestor of living coelacanths, lungfish and tetrapods, and all its descendants. Lobed, internally supported paired appendages provide a traditional diagnostic character; inherited organization can persist despite conversion of fins into limbs.

A fossil bony vertebrate whose dermal odontodes preserve enamel despite teeth without an enamel coating, important for separating tissue origin from oral deployment.

A ray-finned vertebrate whose skin expression of enamel-matrix genes contributes to comparisons of enamel and ganoine evolution.

Measure the pressure difference across a solvent-permeable barrier that retains the solute, at known temperature. For dilute polymers, the zero-concentration intercept of reduced pressure determines the number-average molar mass. Leakage, aggregation and ionic contributions can change the inferred molecular biological population.

For a real orthonormal basis $u_1,\dots ,u_n$, each vector has expansion $x={\sum }_j{u}_j(u_j^{T}x)$. Since this holds for every $x$, the sum of the rank-one orthogonal projection matrices $u_j{u}_j^T$ is the identity matrix. Their pairwise products vanish for distinct basis vectors. For a complex inner product space, replace the transpose by the conjugate transpose.

Under orthogonal projection from one plane onto another, a circle of radius $R$ becomes an ellipse with semiaxes $R$ and $R|\cos \alpha |$, where $\alpha$ is the angle between the planes' unit normal vectors. To prove this, choose one in-plane unit vector parallel to the planes' intersection and the other perpendicular to it. The first vector is unchanged by the projection; the second has projected length $|\cos \alpha |$ and remains perpendicular to the first. Parametrizing the circle in that orthonormal basis therefore gives projected coordinates $(R\cos t,R\cos \alpha \sin t)$. Parallel planes have a circular image, and perpendicular planes give the degenerate case of a line segment. An arbitrary center simply projects to the center of the image.

For hydrogen approximated as a linear rigid rotor, the $\ell =1$ to $\ell =0$ energy decrease is ${\hslash }^2/I$, with moment of inertia $I\simeq m_p{r}^2/2$. A sample of mass $M$ initially containing an orthohydrogen fraction $x_o$ therefore releases $Q\simeq x_{o}M{\hslash }^2/(m_p^2{r}^2)$ on complete conversion to the lowest parahydrogen level. This estimate neglects the electron contribution to molecular mass and changes of the rotational constant with environment. It describes an internal heat load that external insulation cannot remove.

An optimal stopping problem in which observations update a posterior probability distribution over an unknown parameter. The decision state includes that posterior.

In a monochromatic two-path interferometer, an ideal phase plate of thickness $t$ and refractive index $n$ adds phase $2\pi (n-1)t/\lambda$ at normal incidence. The interference pattern is unchanged when $(n-1)t=m\lambda$ for integer $m$, provided transmitted amplitudes and alignment are unchanged. Calling the smallest index requires specifying the admissible range; $n>1$ excludes the optically null $n=1$ solution.

fringe tracking estimates changing optical delay or phase and adjusts delay compensation to keep an astronomical interferometer near a useful interference condition. The delay must first lie inside the finite-bandwidth coherence envelope; rapid tracking then controls atmospheric piston. This is optical-path stabilization, with sensitivity and reference-source limitations.

A nearly uniform wavefront phase over one collector is called its piston. Differential atmospheric piston changes the pairwise optical phase without requiring a change in the source. Delay tracking and closure phase address different consequences of this phase error; intensity correlations are unchanged by an ideal pure phase multiplication at either detector.

Place one pair of opposite tetrahedron vertices on the line $b+\lambda a$ and the other pair on $d+\mu c$, with fixed nonzero direction vectors $a,c$. Let $\Delta =d-b$ and let $\delta \lambda ,\delta \mu$ be the parameter differences within the two pairs. The scalar triple product of the three edge vectors from one vertex gives the displayed volume. Terms parallel to $a$ or $c$ vanish in the triple product, leaving only $\Delta \cdot (a\times c)$. Sliding either pair along its own line while preserving its separation leaves this volume unchanged. When the two lines are skew lines, it can also be written $V=d_{\perp }{\ell }_1{\ell }_2|\sin \alpha |/6$, where $d_{\perp }$ is their shortest distance, ${\ell }_1,{\ell }_2$ the two edge lengths and $\alpha$ their direction angle. Intersecting or parallel lines give zero volume because all four vertices are coplanar.

The loop gain is the product of the forward amplifier response and the return-network fraction, with the summing-junction minus sign kept explicit. Closed-loop poles are determined by $1+L=0$ in this convention. Large low-frequency magnitude controls gain error, while frequency-dependent phase affects stability.

The closed-loop gain is the input-to-output transfer after closing the feedback connection. In the elementary noninverting voltage model it tends to $1/\beta$ for large stable loop gain, rather than retaining the uncertain open-loop gain $A$. A frequency-dependent transfer must be checked for stability before that approximation is used.

The coefficient in $\mathbf{J}=-D\nabla \varphi$ for the scalar neutron flux convention. It has dimensions of length, unlike a particle-density diffusion coefficient with dimensions length squared per time. For a single fixed speed $v$, the corresponding density diffusion coefficient is $vD$.