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For a real conductivity tensor invariant under every rotation about a unit vector $n$, the electric current density is $J=a{E}_{\perp }+d{E}_{\parallel }+hn\times E_{\perp }$. Its collinearity with a nonzero real electric field $E$ has three cases. Axial fields are always mapped to an axial current. Purely transverse fields give a collinear current exactly when $h=0$, equivalently when this tensor is symmetric. Fields with nonzero axial and transverse parts require both $h=0$ and $a=d$, making the tensor isotropic. The zero vector is included as a scalar multiple; requiring a nonzero current also requires the relevant scalar conductivity to be nonzero.

To prove necessity, take the dot product of $J=\lambda E$ with $n\times E_{\perp }$. If $E_{\perp }\ne 0$ this gives $h=0$; the transverse and axial equations then respectively give $\lambda =a$ and, when $E_{\parallel }\ne 0$, $\lambda =d$. Reversing these steps proves sufficiency. Rotations about the axis alone permit $h\ne 0$; symmetry under a transverse half-turn would eliminate this antisymmetric term.

For two masses, connect each to a fixed anchor and to the other mass with equal stiffness $k$. Set $m_1=1,k=1$ to choose units. The squared angular frequencies have characteristic equation $m_2{z}^2-2(m_2+1)z+3=0$. Requiring a frequency ratio of two means that these roots are $z$ and $4z$. Their sum and product give $16m_2^2-43m_2+16=0$, yielding the two reciprocal ratios in the header. Restoring units multiplies all squared frequencies by $k/m_1$, leaving the permitted mass ratios unchanged. The term octave refers here only to this two-to-one ratio.

For real $a,b\ne 0$, this differential one-form on the punctured plane is the pullback of a differential form of the nonexact angular one-form under the invertible linear transformation $(X,Y)=(ax,by)$. It is a closed differential form for every such pair. On a region admitting a continuous complex argument, it has the local potential $f=\mathrm{Arg}(ax+iby)$. In a chart with $x\ne 0$, this can be written $f=\arctan (by/(ax))+C$.

There is no globally single-valued potential on the whole punctured plane. On the closed path $x=\cos t/a$, $y=\sin t/b$, $0\le t\le 2\pi$, substitution gives ${\omega }_{a,b}=dt$, hence its line integral is $2\pi \ne 0$. This contradicts the zero integral around every closed path required of an exact differential form. Around a positively oriented once-winding path the integral is $2\pi \mathrm{sgn}(ab)$. The local equation ${\omega }_{a,b}=0$ gives radial rays; away from a vertical ray its solutions as graphs are $y=Cx$ on intervals not crossing the excluded origin.

Reordering the interpolation nodes changes the successive subsets used for the divided differences and the factors of the Newton polynomial. The final interpolation polynomial remains unchanged in exact arithmetic. For $f(x)=x^3-2x+1$, reversing $(-1,0,2,3)$ gives $(3,2,0,-1)$ and values $(22,5,1,2)$. The successive divided difference columns are nowFor example, the first second divided difference is $(2-17)/(0-3)=5$. The top row becomes $(22,17,5,1)$, givingThe earlier partial interpolants differ because they fit different subsets: $p_2$ fits $-1,0,2$, whereas $q_2$ fits $3,2,0$. The highest divided difference is invariant because both complete top rows finish with the same set of four interpolation nodes; this is symmetry of divided differences.

This distinction is useful when checking a divided difference table: permuting all the interpolation nodes must preserve its final divided difference, even though the intermediate entries change. It also explains why data can be appended in their arrival order. In floating-point arithmetic, the subtractions and divisions may accumulate different rounding errors in different orders; exact invariance describes the mathematical polynomial, rather than guaranteeing bit-for-bit equality of numerical results.

Consider a nonrelativistic particle of mass $m$ undergoing an elastic collision with a stationary free target of mass $M=Am$, where $A\ge 1$. Assume the outgoing direction is uniformly distributed over solid angle in the center of mass frame; this assumption concerns the scattering angle distribution and need not hold for every target or energy. Let $\mu$ be the cosine of that frame's scattering angle. Transforming the outgoing velocity back to the laboratory givesThe uniform distribution of $\mu$ on $[-1,1]$ makes $x$ uniform on $[\alpha ,1]$. For $A>1$, integration yieldsFor equal masses, take the limit $\alpha \to 0$: $\xi =1$. The expected value of the fractional energy after a collision is instead $\mathbb{E}[x]=(1+\alpha )/2$. These are different averages: averaging a logarithm does not give the logarithm of an average.

For independent collisions with the same distribution, the logarithms of successive energy ratios add. ThusHere $E_{\mathrm{geom}}$ is the geometric-mean energy, and $n_{\log }$ is a characteristic collision count for reaching an energy scale $E_{\ast }$. It is not an exact first-passage expected value. Individual histories fluctuate, and neither the time between collisions nor absorption is specified by this model.

In an original chosen example, set $E_0=10^6\mathrm{eV}$ and $E_{\ast }=0.025\mathrm{eV}$. Approximate hydrogen by $A=1$ and carbon by $A=12$. ThenThese are illustrative model values, not measured collision counts from 1934. The energy scale marks where neglected thermal motion matters; the calculation does not assert continued cooling to zero energy.

A material used for neutron moderation, such as water or paraffin wax in the Fermi slow-neutron activation experiment.

Choose a one-dimensional crystal with cell translation $a$ and two atoms per cell at $0$ and $d=a/2$. Here $d$ is an intracell displacement, distinct from the interplanar spacing $d_{\mathrm{p}}$ above. In a single-scattering approximation, a row of $N$ cells has scattering amplitudeThe sum selects reciprocal lattice positions $Q_h=2\pi h/a$ as $N$ becomes large. At these positions, the two-site crystal basis structure factor and its squared magnitude areConsequently $b_1=b_2=b$ gives even reflections of strength $4b^2$ and extinguishes the odd reflections. Choosing $b_1=b$, $b_2=b/2$ instead gives $9b^2/4$ and $b^2/4$: the odd/even ratio is $1/9$. Equal atomic numbers would not establish equal $b_j$ if the isotopes differ. The figure below plots these chosen coefficients, without assigning them to actual elements. Missing reflections can encode destructive wave interference within a cell, not the absence of matter at a site.

The nuclear neutron scattering length is the effective length coefficient for low-energy nuclear scattering amplitude. For the elementary coherent calculation below, use a bound coherent coefficient $b_j$ for each site, and neglect absorption so that $b_j$ is real. This coefficient describes the atomic nucleus, including its isotope and nuclear spin dependence; it is not proportional to atomic number. Replacing nuclear coefficients by electronic magnetic moments would describe a different scattering channel, with additional direction and spatial-distribution factors, as explained in magnetic neutron diffraction.

For $n$ pots per treatment and $m$ leaves per pot, the group sample mean decomposes into two averages:The variance of a sum adds the variances of the independent terms. Every pot random effect has weight $1/n$, whereas each remaining error has weight $1/(nm)$, giving the short proofThis is also the variance of the arithmetic mean of the $n$ pot sample means, each with variance ${\tau }^2+{\sigma }^2/m$. Equal numbers of leaves per pot make the two averaging procedures identical.

If all $nm$ leaf measurements were falsely treated as independent, the predicted variance would instead be $({\tau }^2+{\sigma }^2)/(nm)$. The ratio of correct to false variance isThus pseudoreplication can underestimate a standard error by the square root of this ratio. Equality holds when $m=1$ or ${\tau }^2=0$; sharing a pot creates an error here precisely when the shared influence has positive variance.

Choose ${\tau }^2=4$, ${\sigma }^2=1$, $n=4$ and $m=3$ as constructed model parameters in squared length units. These are assumptions rather than fitted experimental values. For each group,The treatment contrast $\hat{\Delta }={\overline{Y}}_B-{\overline{Y}}_A$ has expected value ${\alpha }_B-{\alpha }_A$. Since the group errors are independent conditional on the allocation, its variance is twice the group variance, namely $13/6$, whereas the false calculation gives $5/6$. Its correct standard error is $\sqrt{13/6}\approx 1.472$ length units; the false value is $\sqrt{5/6}\approx 0.913$. The variance ratio is $13/5=2.6$.

With four pots fixed, increasing $m$ drives the error contribution ${\sigma }^2/(4m)$ towards zero, but the limit of the group variance remains ${\tau }^2/4=1$. Measuring twelve leaves on each of four pots gives $49/48$. Using the same forty-eight measurements on sixteen pots, with three leaves per pot, gives $13/48$. More independent experimental units reduce both contributions. Extra nested measurements are useful for characterizing each experimental unit, but cannot average away its shared random effect. This is why sample collection effort and the number of independent treatment replicates answer different questions.^[ref]

Suppose independent Bernoulli trials give a player a win with probability $p$, and the match ends when either player reaches $r$ wins. The player's match-win probability isFor a win with $n$ losses, the final trial must be a win. The preceding $r+n-1$ trials contain exactly $r-1$ wins and $n$ losses; the binomial coefficient counts their arrangements. Because $n<r$, the opponent cannot already have won, and because the player has only $r-1$ previous wins, the match has not ended earlier. Summing the disjoint terminal events gives the formula, a truncated negative binomial distribution sum. The constant game-win probability must describe independent games, or equivalently remain the same conditional on every previous history.

In independent Bernoulli trials with success probability $p$ and failure probability $q=1-p$, let $T_n$ be the trial of success $n$ and $U_m$ the trial of failure $m$, for positive integers $n,m$. If success $n$ occurs after $k<m$ failures, its final trial is a success and the preceding trials have $n-1$ successes and $k$ failures. The negative binomial stopping argument therefore gives the title sum.

Put $F=n+m$. By trial $F-1$, at least one target must have been reached, so the same event is that a binomial distribution with $F-1$ trials has at least $n$ successes. Conditioning instead on the final trial of a block of $F$ trials givesIndeed, more than $n$ successes in the full block guarantee the success target won; with exactly $n$ successes it won only when the final trial was a failure. These disjoint cases explain the extra boundary term. A zero success target wins immediately; a zero failure target has already been reached. To include $n=0$ in the displayed finite sum requires the explicit generalized convention $\left(\genfrac{}{}{0ex}{}{-1}{0}\right)=1$; an upper summation limit of $-1$ denotes an empty sum.

The natural logarithm has the parameter-integral representationFor $0<t<1$, the identity $(t^x-1)/\ln t={\int }_0^x{t}^{s}ds$ shows convergence: the absolute value is at most $|x|t^{\min (x,0)}$, which is integrable when $x>-1$. At $t=1$, the apparent singularity is removable, with limiting value $x$.

Write the left side as $I(x)$. On a compact parameter interval $c\le x\le d$ with $c>-1$, its differentiated integrand $t^x$ is bounded by the integrable function $t^c$. The dominated convergence theorem justifies differentiation under the integral sign, giving $I^{\prime }(x)={\int }_0^1{t}^{x}dt=1/(1+x)$. Since $I(0)=0$, the fundamental theorem of calculus gives $I(x)=\ln (1+x)$. At $x=-1$, the original improper integral diverges as $-{\int }_{0}dt/[t|\ln t|]$, and it also diverges for $x<-1$.

Integrating the geometric series for $1/(1+x)$ from zero to $x$ gives the displayed power series for $|x|<1$. It has radius of convergence one. At $x=1$ the alternating series test gives conditional convergence to $\ln 2$; at $x=-1$ the negative harmonic series diverges. Replacing $x$ by $-w$ gives $\ln (1-w)=-{\sum }_{n\ge 1}{w}^n/n$. An expression formed by substituting a nonlinear function for $w$ need not have the same convergence range after it is re-expanded as a Taylor series in the new variable.

Albert Abraham Michelson's 1907 lecture, Recent Advances in Spectroscopy, pp. 171–172, connects counting interference fringes with a reproducible length standard. Broad or multiple spectral lines obscured fringes at large path differences; red cadmium emission supplied a suitable source. His 1907 Nobel Prize in Physics recognized optical precision instruments and their spectroscopy and metrology applications.^[ref]

His authored account of the metre comparison at Sèvres describes a heated, evacuated cadmium discharge tube and a spectrometer selecting one spectral line. A movable reference mirror was compared with pairs of silvered parallel surfaces defining intermediate standards. Counted interference fringes established a short interval; repeated comparisons extended it to a decimetre and then a metre. White-light fringes located surface coincidence, while monochromatic fringes supplied counts and fractional orders. A thermally shielding box, firm pier, calibrated translation screw and tested thermometers controlled drafts, vibration, travel and temperature. Uniform temperature mattered in comparisons; the final standard required an absolute temperature reading. Optical microscopes made the final comparison to the marked metre bar, limiting the whole result.^[ref]

This is a measurement chain, rather than one enormous unchecked fringe count: short intervals establish whole orders; fractional-order readings refine each longer comparison. The optical coherence envelope helps locate white-light coincidence, whereas the carrier phase difference measures small departures from it. Repetition checks counting mistakes, but agreement alone cannot exclude shared systematic errors. The general lesson of metrology is to preserve the link from the optical ruler through each transfer and control to the final length.

A Michelson interferometer turns displacement into a count of interference fringes: a moving mirror changes both the outward and return paths. This lets a wavelength act as a ruler whose divisions need no engraving. Assume normal incidence, a fixed reference arm, stable refractive index $n$ along the moving arm, and fixed vacuum wavelength ${\lambda }_0$. Write $\lambda ={\lambda }_0/n$ for the wavelength in that medium and take $\Delta L$ positive away from the beam splitter. The change in optical path difference, the accumulated phase difference, and the signed number of complete interference fringe cycles areThe factor two comes from propagation geometry: moving the mirror by $\Delta L$ adds $\Delta L$ before reflection and $\Delta L$ afterward. One bright-to-bright cycle means mirror travel $\lambda /2$; bright-to-dark means $\lambda /4$, half a cycle. All lengths must use compatible units, and $\Delta N$ is dimensionless. A constant reflection phase difference changes the initial fringe position but cancels from this change formula.

For an original hypothetical example, suppose $\lambda =600\mathrm{nm}$ in the actual surrounding medium and a continuous scan gives $40.25$ cycles, including the final fractional cycle. Then $\Delta L=(600\mathrm{nm})(40.25)/2=12075\mathrm{nm}=12.075\mu \mathrm{m}$. With the wavelength treated as exact, a fringe-reading measurement uncertainty of $0.02$ cycle contributes $6\mathrm{nm}$ to the displacement measurement uncertainty. An uncertainty of $0.01\mathrm{rad}$ in a local phase difference instead corresponds to $600(0.01)/(4\pi )\mathrm{nm}\approx 0.477\mathrm{nm}$. This illustrates measurement resolution, not a complete claim of accuracy: wavelength calibration, air changes, mirror tilt and missed cycles can dominate.

To separate the oscillating readout from its optical coherence envelope, use the local modelHere $I$ is detected light intensity, $\bar{I}$ is the mean level, ${\varphi }_b$ is a fixed bias wave phase, and $\mathcal{V}$ is interferometric visibility. A fine scan of $D$ measures the carrier phase difference; a coarse scan can map the temporal optical coherence envelope through $\mathcal{V}(D)$. The envelope maximum locates matched delays only under the source and apparatus assumptions; it is not a bright fringe identifying an arbitrary mirror position. Even with known $\mathcal{V}$, one intensity reading leaves the cosine sign and whole-cycle ambiguities. Continuous motion of known direction, or an independently obtained pair of signals in phase quadrature, is needed to recover signed travel through reversals.

The figure's original control example stipulates equal arm light intensities, matching linear polarizations, and a Lorentzian optical spectrum giving $\mathcal{V}(D)=\exp (-|D|/L_c)$ with $L_c=1\mathrm{mm}$. In vacuum, arm mismatches $|L_2-L_1|=0.60\mathrm{mm}$ and $0.10\mathrm{mm}$ give $\mathcal{V}=e^{-1.2}\approx 0.301$ and $e^{-0.2}\approx 0.819$. Matching the arms improves the phase difference readout without changing its ${\lambda }_0/2$ travel per cycle. A fine mirror scan of one cycle changes $D$ by only $600\mathrm{nm}$ for the stipulated $600\mathrm{nm}$ source, much less than $L_c$. These are calculated values, not historical data.

A proposed measurement sequence follows from this model. Overlap the returning beams, match their linear polarizations, find a high-contrast delay and use a small calibrated mirror scan to determine the sign and fringe period. Read a fixed spatial sample of the interference fringes with a photodiode; record each arm separately and the dark signal, avoid detector saturation, and retain enough time resolution to count every cycle. Approach mechanical settings consistently and repeat the return scan to expose lost motion or drift. For very small changes, bias near phase quadrature, where $|dI/d\varphi |=\bar{I}\mathcal{V}$ is largest; an intensity extremum has zero first-order sensitivity. A known drive voltage alone does not provide measurement calibration for mirror travel: compare the actual travel with the fringe count using an independently known wavelength.

Control temperature, pressure, vibration and air currents because the instrument measures optical path difference, not mirror motion alone. For a fixed reference arm and fixed ${\lambda }_0$, the first-order change of the moving arm's round-trip path is $\Delta D\approx 2n\Delta L+2L\Delta n$. Thus an unrecognized $\Delta n$ mimics $\Delta L_{\mathrm{app}}=L\Delta n/n$. A hypothetical $L=0.10\mathrm{m}$, $n\approx 1$, and $\Delta n=10^{-7}$ give $10\mathrm{nm}$ apparent travel. Holding the mirror still while varying or monitoring the air conditions tests this confounder. Separate checks of alignment, wavelength stability and thermal expansion are necessary before sub-fringe measurement resolution becomes a defensible measurement uncertainty.

If a diagonalizable matrix has distinct eigenvalues $r,s$, its spectral projections of a diagonalizable matrix are $(A-sI)/(r-s)$ and $(rI-A)/(r-s)$. Raising the diagonal form to a nonnegative integer power gives the displayed formula. Equivalently, interpolate the scalar function $t^n$ by a degree-one polynomial at $r,s$, then apply it to $A$. The diagonalizability assumption matters: knowledge of two eigenvalues alone does not exclude nontrivial Jordan blocks.

A diagonal matrix entry uses the same row and column index. The diagonal entries of a lower triangular matrix determine its determinant by their product.

A device with a reference magnetic layer whose magnetization is comparatively fixed and another layer whose magnetization responds more readily to an applied field. The resistance reads their relative alignment. Fert's account describes how such giant magnetoresistance sensors entered hard-disk read heads in 1997.^[ref]

Giant magnetoresistance connects the relative magnetizations of adjacent magnetic layers to their electrical transport. In a suitable magnetic-metal/nonmagnetic-metal stack, aligning the layers can leave one spin channel with an easy conducting path through both magnetic layers; opposite alignments make each channel encounter a more resistive layer. Albert Fert's account traces the spin-dependent-conduction idea to Nevill Mott and distinguishes the later device geometries.^[ref]

The beauty is that a magnetic arrangement becomes an electrical readout. Preparing reproducible layers, establishing their magnetizations, and measuring electrical resistance are separate parts of that inference. The Fe-Cr giant magnetoresistance experiments show how those parts were joined; the two-current resistor model isolates a simple transport mechanism under stated assumptions.