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The Airy disk is the central bright region, up to the first dark ring, in the scalar diffraction pattern of a uniformly illuminated, unobscured circular aperture. The whole pattern includes the weaker surrounding rings; its radius is distinct from its full width at half maximum.

For a circular optical pupil of diameter $D$ under normal, uniform monochromatic plane wave illumination, Fraunhofer diffraction givesHere $I_0$ is the on-axis light intensity, $\theta$ is the angle from the optical axis, $\lambda$ is the wavelength in the propagation medium, and $J_1$ is the order-one Bessel function of the first kind. The paraxial approximation replaces $\sin \theta$ by $\theta$, measured in radians. The first positive zero $j_{1,1}=3.831705970\dots$ givesThis is the angular radius of the Airy disk; the angular diameter is twice as large. The Airy disk derivation explains the origin of the Bessel function of the first kind and of the numerical coefficient.^[ref]

The image and both radial curves show the same calculated light intensity, divided by its peak. The image's logarithmic scale makes the weak rings visible; its colour is a display choice. The linear curve shows how small the ring peaks are compared with the centre. A dark ring marks a zero of wave amplitude, produced by cancellation across the optical pupil; it is not the edge of all transmitted light. The first Airy disk contains about $83.78\%$ of the total radiant flux, as calculated in Airy disk normalization and encircled energy.

For an original design example, take an ideal optical lens in air with focal length $f=100\mathrm{mm}$, circular optical pupil diameter $D=5.00\mathrm{mm}$ and wavelength $\lambda =550\mathrm{nm}$. In the focal plane, the radial coordinate $r\simeq f\theta$ givesThe first-zero diameter is $26.8327\mu \mathrm{m}$. Solving $P(u_{1/2})=1/2$ gives $u_{1/2}=1.616339948\dots$, so the diameter measured by full width at half maximum is onlyThus a quoted “spot diameter” is incomplete without its definition. Doubling $D$ halves both widths at fixed wavelength and focal length; increasing magnification alone changes the image scale without improving object-space discrimination.

The Airy disk supplies a calculable reference for a point spread function, allowing diffraction to be distinguished from defocus and optical aberrations. Its beauty is that one circular boundary determines an entire sequence of bright and dark rings through a Fourier transform. The ideal formula assumes an unobscured pupil, uniform illumination, ideal optical focus, a single wavelength, and a scalar description with small output angles. A blocked centre, nonuniform pupil transmission or a significant pupil wave phase error changes the response. High numerical aperture may require polarization of an electromagnetic wave and a vector field description. The circular-aperture diffraction laboratory experiment tests the far-field pattern directly; the Rayleigh criterion uses its first zero to define a two-source convention.

Antibody equilibrium occupancy counts occupied antibody binding sites, rather than retained assay label. For an original 1:1 model, let $A$ denote an unoccupied site, $E$ free antigen bearing one accessible epitope, and $AE$ their complex. At chemical equilibrium, writeSquare brackets mean concentrations. The dissociation constant $K_d$ here has concentration units. This is the same independent-site calculation as receptor binding occupancy, now applied to an antibody binding site. The last equality follows by substituting $[AE]=[A]c/K_d$ into the occupied fraction and cancelling $[A]$.

Assume identical independent sites, reversible 1:1 binding, dilute chemical solution behavior, fixed temperature and buffer solution, and enough time for chemical equilibrium. Exclude two-arm bridging, cross-reactivity and inaccessible sites from this model. Crucially, $c$ is free antigen concentration; substituting the amount supplied per volume requires negligible depletion by binding. Independent IgG arms could obey this model site by site, but it does not describe avidity from bridging a surface.

Stipulate $K_d=20\mathrm{nM}$, with $1\mathrm{nM}=10^{-9}\mathrm{mol}{\mathrm{L}}^{-1}$. Free antigen concentrations $2$, $20$ and $200\mathrm{nM}$ give $\theta =1/11$, $1/2$ and $10/11$. A second stipulated dissociation constant of $200\mathrm{nM}$ instead gives $\theta =1/11$ at $c=20\mathrm{nM}$. Both curves approach one at large $c$: a nearly saturated readout can conceal an antibody affinity difference. These are invented values, with no universal threshold for a useful antibody.

A Western blot readout additionally depends on Western blot transfer, accessible epitopes, loss during washing and Western blot antibody detection. Western blot response saturation can flatten the final signal even when binding itself remains responsive. An ELISA also reports retained label through an enzyme readout, rather than automatically reporting $\theta$. Thus half of an assay's maximum signal need not occur at $c=K_d$. Varying free antigen through concentrations near the candidate dissociation constant, while checking depletion and the readout's response, is more informative than comparing a single saturated point.

Astronomical magnitude expresses brightness on a logarithmic scale: equal steps in astronomical magnitude represent equal ratios of received radiative flux. The scale runs backwards, so a brighter source has a smaller astronomical magnitude. A difference of five magnitudes means exactly a factor of $100$ in flux; one magnitude means a factor of $100^{1/5}\simeq 2.511886$, rather than exactly $2.5$. This compresses the enormous range of astronomical brightnesses into manageable numbers. ^[ref].

Apparent magnitude describes received brightness, while absolute magnitude compares sources at a shared reference distance. Both require a specified photometric passband and magnitude zero point. A magnitude difference turns a ratio of positive radiative fluxes into a difference of numbers; it does not measure an additive change in emitted luminosity.

For a positive random variable $X$, let $D\in \{1,\dots ,9\}$ be its first nonzero base-ten digit. The Benford law is the discrete probability distribution with probability mass functionHere $X$ need not be an integer: $0.0042$, $4.2$ and $4200$ all have $D=4$. Multiplication by an integer power of ten changes the decimal point's position and leaves the first nonzero digit unchanged. Zero has no first nonzero digit, so exclude it. For negative inputs one may explicitly use their absolute values, still excluding zero; that convention must be stated rather than applying a logarithm to a negative number.

The Benford law favors small digits: $\mathbb{P}(D=1)={\log }_{10}2\approx 0.301029996$, whereas $\mathbb{P}(D=9)={\log }_{10}(10/9)\approx 0.04575749$. Thus the model assigns about $30.10\%$ to digit one and $4.58\%$ to digit nine. The masses are positive and their normalization is a finite telescoping series:

Why can digit one be so common? In a decimal significand interval $[1,10)$, the digit-one region $[1,2)$ occupies a larger fraction of the logarithmic scale than the digit-nine region $[9,10)$. The precise assumption is that the fractional part of the base-ten logarithm is distributed uniformly on $[0,1)$; Benford law from logarithmic uniformity turns those unequal interval lengths into probabilities. This makes a surprising digit pattern into a simple geometry of multiplicative size. The logarithmic explanation is also presented in Peter Sheridan Dodds, Benford's Law, essential story.

The Benford law is useful as a proposed model for leading digits when the mechanism supports logarithmic uniformity or an appropriate limiting argument, such as Benford frequencies for powers of an integer. Merely spanning many powers of ten does not establish that assumption. A discrepancy from the Benford law can prompt investigation of the model and data collection; by itself it does not establish fraud, and agreement does not establish authenticity.

Simon Newcomb's 1881 paper, Note on the Frequency of Use of the Different Digits in Natural Numbers, appeared in the American Journal of Mathematics 4, pp. 39–40. Its opening connects unequal leading-digit frequencies with the faster wear of early pages in logarithm tables. The publication metadata and original opening excerpt are available in Newcomb (1881), original publication record and ^[ref]. Frank Benford's The Law of Anomalous Numbers, Proceedings of the American Philosophical Society 78 (1938), pp. 551–572, likewise discusses worn logarithm tables on p. 551 and tabulates observations on p. 553. See Benford (1938), transcription of the original paper with page numbers and publication metadata. These observations motivated study of the digit pattern; they do not make the Benford law universal.

Stipulate pKa $5$ and $K_w=10^{-14}$; these are model inputs, not measurements of an identified substance. At pH $5$ the pair contribution is $0.575646\dots C$. For analytical concentrations $C=0.010$ and $0.100\mathrm{mol}{\mathrm{L}}^{-1}$, respectively, it isThe free hydrogen ion and hydroxide contribution is the same at this pH, ${\beta }_{\mathrm{water}}=(\ln 10)(10^{-5}+10^{-9})\mathrm{mol}{\mathrm{L}}^{-1}=0.0000230282\dots \mathrm{mol}{\mathrm{L}}^{-1}$ per pH. The full buffer capacities are therefore approximately $0.00577949$ and $0.0575877\mathrm{mol}{\mathrm{L}}^{-1}$ per pH. Identical pH fixes the free hydrogen ion thermodynamic activity; it does not fix the inventory of weak acid and conjugate base available for neutralization.

At one unit of pH on either side of pKa, $r=10$ or $0.1$, giving ${\beta }_{\mathrm{pair}}/{\beta }_{\mathrm{pair},\max }=40/121\approx 0.331$. The partners still form a buffer solution, but its local resistance is only about one third of the maximum for that analytical concentration. At fixed pH, the pair contribution scales exactly with $C$; the full buffer capacity contains the additional $C$-independent water contribution. Consequently, extreme dilution does not justify dropping self-ionization of water.

Let $P$ be the Bernoulli distribution with success probability $0.2$ and $Q$ the Bernoulli distribution with success probability $0.8$. For $0\le s\le 1$,The two summands have product $0.16$. Their arithmetic mean is at least their geometric mean, with equality at $s=1/2$. ThusThe same argument gives $C=-\ln (2\sqrt{p(1-p)})$ for complementary parameters $0<p<1$. At $p=1/2$ the laws agree and $C=0$.

With equal prior probabilities, count the successes $K$ among $n$ observations. The likelihood ratio is $Q^{\otimes n}/P^{\otimes n}=4^{2K-n}$, so choose $Q$ when $K>n/2$ and $P$ when $K<n/2$. If $n$ is even, either decision or a fair tie randomization gives the same average error at $K=n/2$. Under $P$, $K$ has the binomial distribution with parameters $n,0.2$. Symmetry therefore gives the exact optimal average errorThe last term applies only for even $n$; it represents half the tie probability. Direct finite sums give $p_1^{\ast }=0.2$, $p_5^{\ast }=0.05792$ and $p_{20}^{\ast }\simeq 0.00157912$. Twenty observations reduce the optimal average error to about 0.158%. The finite-sample Chernoff error bound at $n=20$ is $\frac{1}{2}(0.8{)}^{20}\simeq 0.00576461$, a guarantee rather than the exact error.

Reducing the exact error from $p_{20}^{\ast }$ to $p_{50}^{\ast }\simeq 1.29363\times 10^{-6}$ illustrates the value of repeated evidence. The example assumes a single fixed coin law throughout the experiment. If repeated sensor readings share an unmodelled common error, increasing their number need not supply this much independent information.

The New Jersey Institute of Technology laboratory uses a silicon 1N4148 diode. For its forward current-voltage characteristic, the supply operates in current-source mode; a benchtop voltmeter independently reads the diode electric potential difference. Small electric current steps resolve the steep branch. For reverse measurements, a sensitive ammeter sits in series; the handout suggests a $10$–$100\mathrm{k\Omega }$ series resistor. Reversing the applied polarity supplies the comparison. Meter placement must avoid counting the voltmeter's own electric current, and the zero reading may have an offset. The same diode is then measured with an Agilent U2722A source measure unit for graphical comparison.^[ref]

The prescribed analysis uses a logarithmic scale for forward electric current to infer $I_S$ and $n$ from the straight region. Large-electric current departures can indicate series electrical resistance. The handout cautions that the accessible range may not determine it precisely; meter loading also matters. This is a measurement procedure, without a published set of device readings.^[ref]

The reason for the logarithmic scale follows directly from the Shockley diode equation. If $I_D\gg I_S$ and series electrical resistance is negligible,where $I_0>0$ is a fixed reference electric current making the logarithm dimensionless. The slope is $1/(n{V}_T)$, so extracting $n$ needs the junction temperature. Fitting only an appropriate interval matters: a straight-looking subset cannot establish the model at all electric currents.

The distance modulus compares received brightness with intrinsic brightness through $\mu =m-M$, the difference between apparent magnitude and absolute magnitude in the same photometric passband and magnitude zero point. With no interstellar extinction, fixed emitted spectrum and luminosity, isotropy of emission, and the local inverse-square law, it determines Euclidean distance:The ratio inside the logarithm is dimensionless. To see the factor $5$, use the inverse-square law in the magnitude difference formula:For example, $m=5$ and $M=0$ give $\mu =5$, hence $d=10\mathrm{pc}\times 10=100\mathrm{pc}$. A tenfold increase in distance adds five magnitudes; an increase of one magnitude multiplies distance by $10^{1/5}\simeq 1.584893$. The distance modulus is a logarithmic scale for distance, not a distance in length units. ^[ref].

At $10$ parsecs, $\mu =0$ and $m=M$. Within $10$ parsecs, the distance modulus is negative; for example, $\mu =-5$ gives the positive distance $d=1\mathrm{pc}$. A negative distance modulus does not imply negative distance. At cosmological distances, the inference needs luminosity distance and a K correction; the simple local formula must not silently absorb these effects.

Interstellar extinction increases apparent magnitude. The observed distance modulus then includes both geometric dimming and attenuation; extinction bias in photometric distance explains how to separate them.

The harmonic series is the positive seriesIts partial sums are the harmonic numbers $H_N$. Although the individual terms tend to zero, $H_N$ grows without bound, so the harmonic series is not a convergent series. This is a basic warning against using the size of one term as a bound on all the terms still to come.

The growth is slow enough to be visually deceptive. The partial sums at $N=10,100,1000,10000$ are approximately $2.92896825,5.18737752,7.48547086,9.78760604$. Adding ten times as many terms increases the accumulated total by roughly $\log 10\approx 2.30258509$, rather than multiplying the total by ten. The telescoping series $\sum 1/[n(n+1)]$ has similarly vanishing terms but approaches the finite sum one.

A short block argument proves the divergence without trusting a finite plot. For each integer $k\ge 0$, the block from $2^k+1$ through $2^{k+1}$ contains $2^k$ terms, each at least $1/2^{k+1}$. ThereforeEvery doubling supplies at least another half, and arbitrarily many doublings are possible. A collection of tiny terms can still have a substantial combined contribution. See Jiří Lebl, Basic Analysis I, Example 2.5.11 for this grouping proof.

Integral comparison sharpens the growth estimate. Here $\log$ denotes the natural logarithm. Since $1/x$ is a strictly decreasing function for positive $x$, comparison of rectangles with the integral givesThus the divergence is logarithmic. The finer difference $H_N-\log N$ tends to the Euler--Mascheroni constant, approximately $0.57721566$; the harmonic number expansion supplies higher-order corrections. This turns the counterexample into a usable quantitative estimate of accumulated reciprocal contributions.

Euler's De progressionibus harmonicis observationes, written in 1734 and published in 1740, studies these sums beyond the yes-or-no question of convergence. Sections 2–3 compare long blocks to show continued growth, and later sections compare the sums with logarithms and calculate the constant offset. The ^[ref] identifies the dates and original publication; ^[ref] makes the calculations accessible. This is not a claim that Euler first discovered divergence: the paper's interest includes measuring how slowly a divergent sum grows.

The Henderson-Hasselbalch equation estimates buffer solution pH using a weak acid's acid dissociation constant and the conjugate base to weak acid concentration ratio.

Its activity form is an exact rearrangement of the chosen chemical equilibrium relation. Write the acid dissociation step as $\mathrm{HA}\rightleftharpoons {\mathrm{H}}^{+}+{\mathrm{A}}^{-}$, with hydrated hydrogen ions understood in water and the solvent convention incorporated into the acid dissociation constant. For consistent standard states,Here each $a_i$ is a dimensionless thermodynamic activity at chemical equilibrium. Exactness refers to this specified acid dissociation constant relation and the chosen single-ion thermodynamic activity convention, not to identifying the ratio with quantities initially mixed. If water is instead written explicitly as a reactant, its thermodynamic activity must be treated consistently. See Hippler and Metcalfe, Using activities to correct the Henderson-Hasselbalch equation (2020), equations 3–4.

For a toy weak acid with stipulated pKa $5$, use the ideal dilute concentration approximation and put $r=[{\mathrm{A}}^{-}]/[\mathrm{HA}]$. The Henderson-Hasselbalch equation becomes $\mathrm{pH}\approx 5+{\log }_{10}r$:More conjugate base relative to weak acid means higher pH. Each tenfold increase in the ratio contributes exactly one to the calculated pH. The stipulated pKa is not a tabulated value for an identified substance.

The distinction between thermodynamic activity and concentration is explicit when $a_i={\gamma }_i[i]/c^{\circ }$, where ${\gamma }_i$ is an activity coefficient and $c^{\circ }$ is the reference concentration:Neglecting the final term gives the usual approximate Henderson-Hasselbalch equation. It is justified when the ratio of activity coefficients is sufficiently close to one, as in a suitable dilute ideal approximation. Ionic strength, including contributions from other dissolved ions, affects activity coefficients; a nominally dilute weak acid alone does not establish that all corrections are small. The ratio must use chemical equilibrium concentrations. Replacing these by prepared amounts or post-neutralization inventories is a further approximation, suitable only when subsequent redistribution is small. See Hippler and Metcalfe, concentration, activity, and analytical-concentration approximations.

The short derivation is to solve the acid dissociation constant relation for $a_{{\mathrm{H}}^{+}}$, take a negative base-ten logarithm, and use the definitions of pH and pKa. Its useful beauty is that a chemical equilibrium becomes a simple rule about a ratio: select a weak acid with pKa near the desired pH, then adjust the partner ratio. Scaling both partner concentrations equally leaves the ideal pH prediction unchanged, while changing buffer capacity. In practical buffer solutions, temperature, ionic strength, and the available inventories still matter.

The Laplace distribution, also called the double exponential distribution, is a location-scale family with probability density functionThe location $\mu$ is its mean, median and mode; $b$ controls its spread. In the centred rate convention, the same law has $\mu =0$ and $\theta =1/b>0$, giving $f(x)=\theta e^{-\theta |x|}/2$. Keeping track of whether a parameter is a scale or its reciprocal prevents reversing the effect of increasing it. ^[ref]

The two halves of the probability density function are reflected exponential decay curves, each with total probability $1/2$. Their meeting point has a sharp corner: the left and right derivatives are $1/(2b^2)$ and $-1/(2b^2)$. This finite peak is not a positive probability mass at $\mu$. Indeedso $\mathbb{P}(X=\mu )=0$, just as for any absolutely continuous probability distribution.

The figure compares a Laplace distribution with $\mu =0$, $b=1$ against a normal distribution with mean zero and standard deviation $\sigma =\sqrt{2}$: both have variance 2. The right panel uses a logarithmic scale for the two-sided tail probability $\mathbb{P}(|X|>r)$. Its straight Laplace distribution curve comes from $e^{-r}$; the normal distribution tail falls faster. Thus values far from the center can be more plausible under a Laplace distribution even after matching the variance. These are theoretical curves, not measured frequencies. The tails still have exponential decay; this is not a heavy-tailed distribution in the convention requiring every positive exponential moment to diverge.

Integrating either half of the probability density function gives the cumulative distribution functionIn particular $F(\mu )=1/2$. The basic moments areThus the scale is the expected value of the absolute distance from the center, and is not the standard deviation. ^[ref] A short way to obtain the moments is to note that $R=|X-\mu |$ has an exponential distribution with rate $1/b$: $\mathbb{P}(R>r)=e^{-r/b}$ for $r\ge 0$. Integrating gives $\mathbb{E}R=b$ and $\mathbb{E}{R}^2=2b^2$; the symmetric distribution gives the centered first moment zero.

A linear scale places a value $x$ at a position $u=ax+c$, where $a>0$ and $c$ are constants. Equal differences in $x$ occupy equal distances because $u_2-u_1=a(x_2-x_1)$. This contrasts with a logarithmic scale, which assigns equal distances to equal ratios.

For positive endpoints $a,b$, halfway along a logarithmic scale meansHere $\beta >1$ is the base. Equivalently, using the natural logarithm, $2\ln (m/x_0)=\ln (a/x_0)+\ln (b/x_0)$. The product of numbers identity for logarithms gives $m^2=ab$, so the midpoint is the geometric mean $\sqrt{ab}$. The two endpoints lie at equal multiplicative distances because $m/a=b/m$.

For $a=10$ and $b=100$, this midpoint is $\sqrt{1000}\approx 31.6228$, giving a factor of $\sqrt{10}$ on each side. The arithmetic mean $55$ is halfway on a linear scale, but its two ratios are $55/10=5.5$ and $100/55\approx 1.81818$. Thus reading halfway between the marks $10$ and $100$ as $55$ on a logarithmic scale gives the wrong value.

A logarithmic scale places a positive quantity $x$ according to a logarithm, so equal ratios occupy equal distances. For example, $1,2,4,8$ are equally spaced under the natural logarithm because every successive difference is $\ln 2$. An ordinary linear scale instead gives equal distances to equal differences: $1,2,3,4$.

For a base $b>1$ and a positive reference $x_0$, define the axis positionThe reference has the same units as $x$, making $x/x_0$ a dimensionless quantity. The distance between two positions isMultiplying both values by the same positive factor preserves that distance. Replacing $x_0$ by $c{x}_0$ translates every position by $-{\log }_{b}c$; changing the base rescales positions through ${\log }_{b}z=\ln z/\ln b$. Neither change alters which ratios receive equal gaps.

In base ten, the values $1,10,100,1000$ occupy positions $0,1,2,3$ when $x_0=1$. The differences between neighboring values grow from nine to ninety to nine hundred, but every step spans one order of magnitude. The values $2,20,200$ occupy positions $0.30103,1.30103,2.30103$: the same spacing shifted by ${\log }_{10}2$. This lets one plot small and large positive quantities together while keeping equal proportional changes equally visible.

A logarithmic scale is a rule for position, not an instruction to relabel the tick marks with their logarithms. A mark labelled $100$ still means the original value one hundred. Ordinary real logarithms exclude zero and negative inputs. Omitting such values or clipping them to a plotting boundary does not give them genuine logarithmic positions; the plotting choice needs to be clear.^[ref]

A log-log plot uses logarithmic scales on both axes. For a positive power law $y=C{x}^p$ with $x,C>0$, positive references $x_0,y_0$ giveIts transformed slope is the exponent $p$ when both axes use the same logarithm base and the slope is measured in logarithmic coordinate units. For example, $y=2x^2$ takes the values $2,8,32$ at $x=1,2,4$. Every doubling of $x$ quadruples $y$, giving slope two on a log-log plot.

This differs from a semi-log plot: a power law has constant proportional response to a proportional change of input, whereas exponential growth has constant proportional response to an additive change of input. Both can appear straight with appropriate axes, so one must read the scale labels before interpreting a slope.

The natural logarithm $\ln x$ is the logarithm with base Euler's number, and the inverse function of the real exponential function. A particularly geometric definition isFor $x>1$, it is the area under the hyperbola $y=1/t$ from $1$ to $x$. For $0<x<1$, the reversed integration limits give a negative value. The fundamental theorem of calculus gives $(\ln x{)}^{\prime }=1/x$, so the natural logarithm is increasing, while $(\ln x{)}^{\prime \prime }=-1/x^2$ gives concavity.

The change of variables $t=au$ shows ${\int }_a^{ab}dt/t={\int }_1^{b}du/u=\ln b$ for $a,b>0$. Thus the intervals from $1$ to $2$ and from $2$ to $4$ each contribute area $\ln 2$, explaining $\ln 4=2\ln 2$. Equal ratios give equal increments on a logarithmic scale, even though the ordinary differences grow. This makes the natural logarithm useful for comparing multiplicative changes and solving for time in exponential growth or exponential decay. The integral construction and inverse function properties are discussed in OpenStax, the natural logarithm as an integral.

Numerical integration approximates a definite integral using finitely many sampled function values. For an integrand $f$ on a finite interval $[a,b]$, $a<b$, a quadrature rule has the formThe quadrature nodes $x_j$ tell us where to evaluate $f$; the quadrature weights $w_j$ turn those values into an approximation to the accumulated signed area. The exact definite integral depends on the integrand throughout the interval, whereas the quadrature rule sees only these samples. Integrating a chosen interpolant exactly still generally approximates the original integrand's integral.^[ref]

The fundamental theorem of calculus evaluates $I(f)$ as $F(b)-F(a)$ when an antiderivative $F$ is available. Numerical integration is useful when no convenient antiderivative is known, or each evaluation of the integrand requires an expensive computation. The task then becomes choosing informative quadrature nodes and controlling the quadrature error with as few evaluations as possible. Fixed data also permit a quadrature rule, but refinement cannot supply new information between observations.^[ref]

For an exact comparison, take the polynomial $f(x)=x^2$ on $[0,1]$. Its antiderivative $x^3/3$ gives $I(f)=1/3$. On a partition of an interval into $n\ge 1$ equal panels, the composite midpoint quadrature rule uses $n$ midpoint samples, each with quadrature weight $1/n$; the composite trapezoidal rule uses $n+1$ endpoint samples, with outer quadrature weights $1/(2n)$ and interior quadrature weights $1/n$. The existing derivations under these two quadrature rules giveIn particular, $M_1=1/4$ and $T_1=1/2$ already bracket $1/3$; two panels give $M_2=5/16$ and $T_2=3/8$. The convex function $x^2$ explains the opposite signs. A midpoint rectangle can cross the curve within its panel, so its total signed area, rather than pointwise ordering, determines the underestimate.

Define the signed approximation error as $E_n=Q_n(f)-I(f)$. In this example the absolute errors are $1/(12n^2)$ and $1/(6n^2)$: doubling $n$ reduces each by exactly four. A plot with both axes on a logarithmic scale therefore has slope $-2$, expressing order two with respect to panel refinement. This is an exact calculation for this polynomial, including $n=1$, rather than an observed rate extrapolated from a few computed answers.

For a general integrand, the composite midpoint quadrature rule and composite trapezoidal rule give second-order quadrature error bounds when $f$ has two continuous derivatives on $[a,b]$ and a known bound on $|f^{\prime \prime }|$. These hypotheses describe behavior between samples. A formal error formula containing an unknown derivative is not a numerical accuracy certificate or a justified sample count. Finite samples do not certify quadrature accuracy makes the missing information explicit. Errors in the evaluations and roundoff error must also be considered; noise sensitivity of a quadrature rule treats the former.

More elaborate choices exploit other information: Gaussian quadrature obtains high polynomial exactness from carefully chosen quadrature nodes, and Simpson's rule uses a quadratic interpolant. Adaptive numerical integration compares approximations and concentrates samples where they disagree, often saving evaluations. Such a difference is an error estimate, not a certified bound without further assumptions: both approximations can miss the same feature.^[ref]

There are three regimes. At the lowest temperatures, most donor impurities retain their electrons; thermal ionization gives semiconductor carrier freeze-out. The result from part (b) isThis produces the approximately straight descending low-temperature tail on the logarithmic scale plot against $1/T$. The activation energy in the exponential is $E_i/2$, because the chemical potential adjusts with donor ionization.

At intermediate temperatures, the donor impurities are essentially fully ionized while intrinsic excitation remains small. Charge neutrality gives $n\simeq N_D$, producing the extrinsic plateau, roughly from $20$ to $40\mathrm{K}$ in the figure.

At higher temperatures, excitation from the valence band to the conduction band creates electron-hole pairs. The intrinsic carrier concentration grows as $n_i\propto T^{3/2}{e}^{-E_g/(2k_{B}T)}$ for constant parabolic band effective masses and constant band gap. With ionized donor impurities, charge neutrality and the semiconductor mass-action law giveThus $n\simeq n_i+N_D/2$ when $n_i\gg N_D$. This explains the renewed rapid rise and the eventual intrinsic branch; that branch is governed by the band gap, rather than the donor binding energy.

For a stipulated population of amplifiable copies, let $N_n$ be the amount after cycle $n$ and define PCR efficiency by the fractional new yieldHere $E_n=1$ means one additional copy per existing copy, hence ideal doubling. If every cycle has the same PCR efficiency $E$, the exponential growth model gives $N_n=N_0(1+E{)}^n$. Starting with $N_0=100$ copies, twenty ideal cycles give $104857600$ copies; the stipulated value $E=0.8$ gives about $12748236$. An efficiency of 80 percent therefore means multiplication by 1.8 each cycle, not retention of 80 percent of the final ideal yield.

In the ideal strand-counting example, all duplex-equivalent material doubles, but exact-length amplicons lag initially. Starting from one longer duplex and retaining its two original strands, cycle $n\ge 1$ contains $2(2^n-n-1)$ exact-length strands out of $2^{n+1}$ total strands. Thus their fraction is $1-(n+1)/2^n$: zero after cycle 1, one quarter after cycle 2 and one half after cycle 3. This counts strands rather than assuming every strand immediately finds an exact-length partner. The generic yield model above ignores this early distinction.

A power law is a relation $f(x)=C{x}^p$, with fixed constants $C$ and $p$. For positive $x$, any real number $p$ is allowed. Its useful feature is a simple scaling rule:The same proportional change of input produces the same proportional change of output, independently of the starting value. The exponent controls that response; the coefficient fixes the magnitude.

For example, the area of a circle is $A(r)=\pi r^2$. Doubling the radius multiplies its area by four, and tripling it multiplies the area by nine. The volume of a sphere is $V(r)=4\pi r^3/3$: the same changes multiply its volume by eight and twenty-seven. The distinction explains why similarly shaped objects do not preserve their ratio of surface area to volume when their size changes.

When the output is positive, a logarithm turns the power law into a straight-line relation. Using positive reference values $x_0$ and $f_0=f(x_0)$,These ratios are dimensionless quantities, so the formula also applies to physical measurements with units. Plotting both ratios on a logarithmic scale gives a line of slope $p$. Changing a unit changes the numerical coefficient, while the exponent stays the same. Positive $p$ describes increasing values; negative $p$ describes decreasing values; $p=0$ gives a constant. The zero coefficient is a degenerate case with no useful logarithmic slope.

Exact scaling is stronger than regular variation. The function $x^2\log x$ has a doubling ratio tending to four, but its logarithmic correction keeps growing. The diagram contrasts it with the exact power law $x^2$.

A power law for a probability density and a power law for a tail probability have different exponents. For a Pareto distribution with $\alpha >0$ and lower endpoint one,The tail probability has exponent $-\alpha$; differentiating it and changing sign gives the probability density exponent $-\alpha -1$. Confusing them changes the interpretation of a fitted exponent. The normalizing coefficient and lower endpoint also matter: a decreasing power law is not automatically a valid probability density over the entire positive real line.

The appearance of a line over a finite logarithmic range is evidence to investigate, rather than a proof of an exact power law or its mechanism. In their study of power-law distributions, Clauset, Shalizi and Newman combine maximum likelihood estimation, goodness-of-fit tests and comparisons with alternative models; their abstract explains why ordinary least-squares fitting can give misleading results for tail data. See Aaron Clauset, Cosma Rohilla Shalizi and M. E. J. Newman, Power-law distributions in empirical data, SIAM Review 51 (2009), 661–703. The scaling rule is mathematically exact when it defines a model; whether a dataset supports that model is a question of statistical inference.