The experiment involving Ferenc Krausz combined a subfemtosecond soft-X-ray pulse with a few-cycle visible electric field. Its delay-dependent krypton photoelectron spectra tested the field-controlled response. The reported pulse duration was $650\pm 150$ attoseconds. Changing delay compared the emitted electron distributions at different reference phases; the spectral-width trace supplied a further duration-sensitive observable.^[ref]

The distinction from RABBITT is the measurement regime and reconstruction, rather than merely calling one trace a clock. The attosecond streaking momentum map is a limiting model; a finite-duration wave packet requires a quantum transition calculation. Inferring an emission delay also requires separating the illuminating pulse's spectral phase from the target response. The measurement supports time-dependent photoionization in a specified model, with no unique classical electron trajectory inside an atomic orbital implied.

Choose $N$ equal-amplitude, monochromatic odd components with orders $q_j=q_0+2j$, $j=0,\dots ,N-1$, and odd $q_0$. Fix the complex-field conventionHere $I$ is a normalized light intensity after averaging the fast carrier oscillations. Setting every ${\varphi }_j=0$ and summing a geometric series givesAt the removable singularities take the continuous limit. Maxima repeat every $T/2=\pi /\omega$; the first zeros on either side of a maximum lie at $t=\pm T/(2N)$. The zero-to-zero width is $T/N$, which is not the full width at half maximum.

Choose orders $13,15,17,19,21$ and driving wavelength $\lambda =800\mathrm{nm}$. Using $T=\lambda /c$ with the speed of light $c$ gives $T=2.66851\mathrm{fs}$ and burst spacing $1.33426\mathrm{fs}$. Solving the exact half-maximum equation gives a full width at half maximum of approximately $240.6$ attoseconds. These are values of the specified five-component model, not measurements from the 2001 experiment.

A deliberately chosen spectral phase ${\varphi }_q=q\omega t_0$ translates the whole attosecond pulse train by $t_0$ without changing its shape. Adding $\kappa (q-17{)}^2/2$ makes neighboring phase increments unequal: the same component intensities then produce a different temporal profile. For the choice $t_0=0.20\mathrm{fs}$ and $\kappa =0.08$ radians per squared harmonic order, the adjacent-component delay estimates arenamely $0.0981$, $0.1660$, $0.2340$ and $0.3019\mathrm{fs}$ for $q=13,15,17,19$. This finite difference estimates the group delay at the pair's midpoint. It is not a single common arrival time.

The infinite monochromatic sum is periodic forever and has infinite total energy. A real attosecond pulse train has a finite envelope and finite component linewidths. This calculation isolates coherent addition; it cannot describe an isolated burst or establish its contrast against satellites. As a further check, averaging $I(t)$ over one half-period gives $N|a{|}^2$ for any constant component spectral phases, because the cross terms integrate to zero by Fourier orthogonality. Spectral phase redistributes the light intensity in time while leaving that average fixed.

The Michelson stellar interferometer, associated with Albert Abraham Michelson, uses separated collectors and beam-redirection optics to produce interference fringes. The collector spacing determines sensitivity to a stellar angular diameter, independently of the final beam spacing at the combiner. The first null for a uniform disk is $D=1.21967\lambda /\alpha$, where $\alpha$ is the full angular diameter. Different surface-brightness laws, including limb darkening, change the null.

The principal argument selects a single complex argument of a nonzero complex number, conventionally in $(-\pi ,\pi ]$. It is the imaginary part of the principal logarithm. The alternative interval $[-\pi ,\pi )$ changes the convention on the negative real axis. Neither convention makes the selected complex argument continuous across that axis.

One thousandth of an arcsecond: $1\mathrm{mas}=\pi /(648000000)\mathrm{rad}$.

For a piecewise differentiable curve given in polar coordinates by $r=r(\theta )$, differentiate $x=r\cos \theta$ and $y=r\sin \theta$. The cross terms cancel in $d{x}^2+d{y}^2$, leaving $d{s}^2=d{r}^2+r^{2}d{\theta }^2$. Therefore the arc length on an increasing parameter interval isThis formula remains valid through an isolated zero of the speed when the parametrization is piecewise smooth. A square root of a square must retain its absolute value.

Consider an original two-path model with a locally flat ground baseline $d$ and a symmetric equivalent return height $h$. Replace the skywave by two straight segments traveling at the speed of light $c$, and replace the ground wave by a straight segment of length $d$. Their lengths and positive excess distance areThis is an equivalent geometric construction, not a metallic layer overhead. It neglects the curvature of Earth, refraction within the return region, ground-induced wave phase shifts and optical dispersion.

For received complex amplitudes $A_g$ and $A_s$, absorb their constant relative wave phase into ${\varphi }_0$ and choose their magnitudes nonnegative. An ideal detector reading proportional to squared received field hasAssume $\Delta L$ and ${\varphi }_0$ are independent of frequency across the sweep, with slowly changing amplitudes. Consecutive maxima differ by $2\pi$ in phase difference, soThe often-used approximation $\Delta f\simeq c/\Delta L$ therefore carries a frequency-independent-path assumption. Counting $N$ complete fringe periods between $f_1$ and $f_2$ gives $\Delta L\simeq Nc/|f_2-f_1|$; equivalently, $N=\Delta L|1/{\lambda }_2-1/{\lambda }_1|$ when the vacuum wavelength is $\lambda =c/f$. Count whole periods, not maxima plus minima as though each were a whole period.

Choose $d=150\mathrm{km}$ and $h=100\mathrm{km}$, with $c\simeq 3.00\times 10^8\mathrm{m}{\mathrm{s}}^{-1}$. Then $L_s=250\mathrm{km}$, $\Delta L=100\mathrm{km}$ and $\Delta f=3.00\mathrm{kHz}$. A $12.0\mathrm{kHz}$ sweep covers four full fringe periods. Conversely, the inferred equivalent height isThese chosen values illustrate the inversion; they are not a transcription of the 1924 measurements. With $A_g=1$ and $A_s=0.7$, the normalized reading ranges from $(1-0.7{)}^2=0.09$ to $(1+0.7{)}^2=2.89$: incomplete cancellation does not remove the distance information.

For a sweep with linearly changing frequency $f(t)=f_0+\dot{f}t$ and two constant delays ${\tau }_s>{\tau }_g$, the received instantaneous frequency difference has magnitude $|\dot{f}|({\tau }_s-{\tau }_g)$. In this model ${\tau }_s-{\tau }_g=\Delta L/c$, giving a beat frequency $|\dot{f}|\Delta L/c$. Equivalently, traversing each $\Delta f$ of the sweep crosses one fringe. This calculation requires a sufficiently slow sweep that propagation, tuning and amplitude changes can be treated consistently; it is not a beat between two independent transmitters.

More generally, measure the total phase difference $\Phi (f)$, including propagation and any reflection or apparatus contribution. Define the differential group delayFor a locally monotone wave phase with slowly varying derivative, the local spacing is $\Delta f\simeq 1/|\Delta {\tau }_g|$. If $\Phi =2\pi fD(f)/c+{\varphi }_r(f)$, this becomesHere $D(f)$ is a phase-equivalent optical path difference, not necessarily the geometric excess distance $\Delta L$. A drifting ionosphere, several simultaneous skywaves, changing wave polarization, a frequency-dependent reflection wave phase or significant loss can invalidate the simple constant-spacing inversion.

The dirty beam is the point spread function obtained by directly inverting the weighted sampled complex visibilities. For sampling positions $u_k$ and nonnegative weights $w_k$ with $W={\sum }_k{w}_k>0$, chooseInclude each $u_k$ with $-u_k$ and the same weight so that $B$ is real. Its unit peak convention means that an isolated unit point source contributes a shifted unit-height response. It is not a unit-integral convention. Sparse sampling can leave substantial subsidiary peaks, even when the longest interferometric baseline is large.

The One-Mile Telescope used three steerable 60-foot dishes: two fixed, one rail-mounted. The movable–fixed pairs supplied two baselines simultaneously. Twelve-hour tracks at successive rail positions sampled projected spacings. Phase-switching receivers measured in-phase and quadrature correlations; adjustable delay and phase rotation compensated tracking.^[ref]

The readout was correlation versus time. Reference-phase subtraction should leave constant point-source phase; position offsets produce predictable variation. A source near the field's declination supplies a collimation check. Structure, instrumental error and sampling artefacts need separating.^[ref]

Radio interferometry measures the spatial coherence of received radio waves. At two antennas, a receiver preserves the relative wave phase, compensates the geometric delay for a chosen phase centre, and averages a product of one signal with the complex conjugate of the other. In a complex-signal description this correlation is $⟨E_1{E}_2^{\ast }⟩$. Its real and imaginary parts carry two complementary components of the complex visibility; measuring only the received powers discards the relative wave phase needed for general imaging.

A finite frequency band makes delay compensation essential. If the residual delay is $\Delta \tau$, the residual wave phase varies by $2\pi \Delta \nu \Delta \tau$ across a band of width $\Delta \nu$. Averaging these differently oriented complex contributions reduces the correlation. For an ideal rectangular band the multiplicative factor is $\sin (\pi \Delta \nu \Delta \tau )/(\pi \Delta \nu \Delta \tau )$. A phase correction at just the central frequency cannot remove this variation across the band. Long time averages can likewise smear a changing signal.

Nevill Francis Mott's 1977 lecture discusses silicon deposited in a glow discharge. Injected electrons provided a drift-mobility readout; electrical conductivity and temperature dependence tested localized-state transport and excitation toward a mobility edge. Varying temperature distinguishes transport regimes. The lecture gives no contact geometry or timing circuit here, so none is inferred. Its Figure 2 is schematic, not measured data.^[ref]

This evidence motivates a transport question beyond the closed-chain eigenvalue problem: how does a prepared charge disturbance move under the conditions of a real sample? The numerical figures above supply neither drift-mobility measurements nor a model of the deposition process. Finite-size concentration alone cannot reproduce an experimental electrical conductivity curve.

Finite temperature permits phonon-assisted hopping between localized eigenstates or thermal excitation toward a mobility edge; localized states therefore need not imply zero measured electrical conductivity at every nonzero temperature.^[ref]

Real dephasing destroys the relative wave phases needed for long-path interference. Observing coherent confinement requires the distance over which those wave phases survive to be long enough to explore the localized region, and the sample to be large enough to resolve it. Temperature alone is not an automatic switch: the actual inelastic processes set the relevant scales. Static elastic wave scattering preserves coherence within each realization. Loss of the ensemble-averaged coherent field through random wave phases is therefore not, by itself, physical dephasing of an individual sample.

Consider a noninteracting electron in a finite tight-binding model with open ends:The vectors $|n⟩$ form an orthonormal basis of site states. The hopping energy is $t$; it is not the time coordinate. Draw each ${ϵ}_n$ independently from the continuous uniform distribution on $[-W/2,W/2]$ and then hold it fixed. The disorder strength $W/t$ is dimensionless. Open ends mean no hopping bond connects sites $N$ and $1$; equivalently set ${\psi }_0={\psi }_{N+1}=0$. There are no electron interactions, phonons or wave absorption in this model.

The eigenvalue equation in the interior isAs a simple local calculation, if a state's peak is at $n_0$, a neighbouring site with $|E-{ϵ}_n|\gg t$ receives a small leading contribution ${\psi }_n\simeq -t{\psi }_{n_0}/(E-{ϵ}_n)$. Several successive large mismatches multiply small factors. This explains why large disorder can restrict spreading, but is not a proof of Anderson localization: near-resonant denominators and alternative interference paths prevent treating this expansion as uniformly valid.

For the first original calculation, take $N=80$, $t=1$, $W/t=6$ and the fixed random seed $20261015$. Diagonalizing the resulting real symmetric matrix with numpy.linalg.eigh and selecting the eigenvalue closest to zero gives $E/t\simeq -0.05962$, a probability peak at site $29$, and $P\simeq 6.0031$. The selected eigenvector has total probability one. This is one chosen realization, not a disorder average or a measured specimen.

The clean comparison needs care. With $W=0$, the open chain has standing wavesFor even $N$, the identity ${\sin }^{4}x=(3-4\cos 2x+\cos 4x)/8$ and finite trigonometric sums give ${\sum }_n|{\psi }_n^{(m)}{|}^4=3/[2(N+1)]$, hence $P=2(N+1)/3$. Thus the clean $80$-site comparison has $P=54$, although it extends across the chain. Extended does not mean uniform probability. For odd $N$, the special central mode $m=(N+1)/2$ has zeros on alternating sites and instead $P=(N+1)/2$; the even-size assumption avoids that exception.

The second original calculation compares $N=40$ and $80$ over $W/t=0,0.5,1,2,3,4,6,8$. For each size, draw $24$ independent site patterns using seed $20261016$, with the size included in the random-seed construction. Scale each fixed pattern across disorder strengths. At every point, average $P$ over the four energy eigenstates closest to $E=0$. The solid line is the median across these sample averages; shading spans their $10$th–$90$th percentiles. The selected states can change as $W/t$ varies: lines connect one sample's selections rather than tracking a single mode.

At $W/t=0$ the two clean values are $27\frac{1}{3}$ and $54$. At $W/t=8$, the computed median sample averages are approximately $3.17$ and $3.01$. In the latter regime, doubling $N$ hardly changes the typical occupied site count, so $P/N$ falls. At weak disorder the larger chain permits more spatial extent. Neither a single curve nor the overlap of finite-size bands proves the infinite-system statement under mobility edge; a long localization length can hide confinement in either size.

A mobility edge separates localized from extended energy eigenstates on the energy axis. Three-dimensional disordered models can have both. In the infinite one-dimensional tight-binding model with independent continuous random site energies, all states are localized for any nonzero disorder, with probability one. This statement excludes specially correlated disorder.^[ref]

A mobility edge is not a band gap: states can exist on both sides of $E_c$, but differ in spatial extent. For electrons, the location of the Fermi energy relative to a mobility edge matters for low-temperature electrical conductivity. A three-dimensional sample need not have an extended interval at every disorder strength. In particular, the finite one-dimensional calculations below neither locate a three-dimensional mobility edge nor establish a nonzero universal critical disorder.

A localized eigenstate is an energy eigenstate whose spatial weight remains concentrated near some position as the system grows. An exponentially localized eigenstate has a decaying envelope characterized by a localization length. A small participation ratio measures concentration in a chosen site basis; it does not by itself establish an exponential tail or a transport law.

An energy eigenstate changes only by an overall wave phase under time evolution, whether localized or extended. Its stationary probability distribution alone therefore does not distinguish transport mechanisms. The important distinction concerns spatial extent and how superpositions of energy eigenstates carry or retain a disturbance. A wave packet assembled from localized eigenstates with nearby centres can remain confined, whereas extended modes can support propagation.

An energy interval with no extended bulk energy eigenstates available for transport. It can contain localized eigenstates and have a nonzero density of states; a mobility gap need not be a spectral gap.

The accounting method charges operations enough to pay for their work and save credits for later work. A nonnegative balance from an initially empty account makes every prefix's total charge an upper bound on its actual cost. Choosing the balance to be $\Phi$ gives the potential method; the ledger for dynamic array doubling illustrates this equivalence.^[ref]

// Destination: computer-science.bigb

Aggregate analysis bounds the total cost of a whole sequence directly, then divides by its length to obtain an amortized bound. Summing the copied capacities as a geometric series in dynamic array doubling is an example.

The potential method assigns a number $\Phi (D)$ to each state $D$ of a data structure. Let $D_i$ be the state after operation $i$, ${\Phi }_i=\Phi (D_i)$, and $c_i$ its actual cost. Its amortized cost is the actual cost plus the change in potential. Summing and cancelling adjacent potentials gives a telescoping sum:If ${\Phi }_0=0$ and every reachable ${\Phi }_i\ge 0$, the sum of amortized costs bounds the actual total for every prefix. A nonzero initial potential contributes the explicit additive term ${\Phi }_0$. Potential is proof bookkeeping; the algorithm need not store it.^[ref]

For dynamic array doubling, chooseThe potential method now starts at $\Phi (0,1)=0$. After every nonempty reachable state, $C<2n$, so the maximum equals $2n-C$. There are just three cases for append:

For every $n\ge 1$, the total charge is $2+3(n-1)=3n-1$, and the telescoping sum provesThis agrees exactly with $T(n)=n+C_n-1$. The unmodified choice $2n-C$ has initial value $-1$, so calling it nonnegative at the empty state would be incorrect. It also works if that initial constant is retained in the identity; alternatively, $2n-C+1$ starts at zero and is nonnegative on these reachable states. The maximum above keeps a literal nonnegative credit balance from the beginning. The proof covers append alone; allowing reserve or arbitrary deletion changes the reachable states and requires a new argument.

Consider a dynamic array holding fixed-size entries. Its size $n$ is the number of occupied slots and its capacity $C$ is the number of allocated slots. Start with $n=0$, $C=1$. The only operation is append: if $n=C$, allocate a new array of capacity $2C$, copy the $C$ existing entries, and release the old array; then write the new entry and increase $n$ by one. There is no reserve operation, deletion or shrinking in this model.

Count one unit for writing the new entry and one unit per existing entry copied during resizing. Thus the $i$th append has actual cost $c_i=1$ without resizing and $c_i=C+1$ when the old capacity $C$ is full. Copying an entry, including its read and write, is one counted action. Allocation, deallocation, index arithmetic operations and the initial one-slot allocation are outside this exact count. If these add only a constant per ordinary append and a linear cost per resize, the Big O notation conclusions still hold, with different constants; costly entry copies require their own cost model.

The first ten costs are $1,2,3,1,5,1,1,1,9,1$. For example, append $9$ copies eight entries and writes one, so $c_9=9$. At that point the capacity becomes $16$, but the total cost of all nine appends is only $24$.

For $n\ge 1$, the final capacity is $C_n=2^{\lceil {\log }_{2}n\rceil }$. Every earlier capacity $1,2,4,\dots ,C_n/2$ was copied once. This geometric series sums to $C_n-1$, including the empty sum when $n=1$. Therefore aggregate analysis gives the exact totalConsequently $T(n)<3n$, so append has amortized $O(1)$ cost in Big O notation for every prefix from this empty initial state. The values being appended do not affect the count. In contrast, an append to a full dynamic array with $n$ entries costs $n+1$: its worst-case cost is $O(n)$, and actually grows linearly along full-capacity states. A bound on total work does not promise a constant response time for each call.

The same policy controls computer memory: after each completed append, $n\le C_n<2n$ for $n\ge 1$, so fewer than $n$ slots are unused. This is a bound on the retained backing array, with fixed-size slots. During a resize from full capacity $k$ to $2k$, a copying implementation can temporarily hold both arrays, totalling $3k$ allocated slots before releasing the old one. Thus the retained-capacity bound does not bound this temporary peak by twice the live size. Allocator metadata and separately allocated objects are additional computer memory costs.

The geometric series is the source of the efficiency. If capacity instead increases by one each time, the same count becomes $n+1+2+\cdots +(n-1)=n(n+1)/2$, a quadratic total. Dynamic array doubling buys a linear total with bounded spare capacity, even when the final length is unknown.

Why add the daughter four-momenta? Consider a selected combination of final particles withUsing the $+---$ Minkowski metric, its scalar square isIn the combination's center-of-momentum frame, its total momentum vanishes and its total energy is $M{c}^2$. Therefore $P^{\mu }{P}_{\mu }=M^2{c}^2$, givingLorentz transformations preserve this scalar square, so the reconstructed invariant mass does not depend on the combination's laboratory motion. It includes relative kinetic energy and need not equal the sum of the daughter rest masses. A random pair has an invariant mass too; that number does not itself establish a common parent.

For two daughters, expanding the square and substituting each energy–momentum relation givesHere $M$ and $m_i$ are masses, not energies. Reporting $M$ in $\mathrm{GeV}/c^2$ and momenta in $\mathrm{GeV}/c$ is equivalent to calculating numerically with $M{c}^2$ and $c{\mathbf{p}}_i$ in $\mathrm{GeV}$. Setting $c=1$ in natural units suppresses these conversion factors; it does not change the measured quantity.

Alvarez's lecture illustrates invariant mass searches with $K\pi$ combinations and contrasts resonance lifetimes near $10^{-23}\mathrm{s}$ with the much longer lived states the chambers originally targeted. These resonances decayed before producing a resolvable parent track.^[ref]

Operationally, choose a reaction hypothesis, require an acceptable bubble-chamber kinematic fitting result, and form specified daughter combinations. Accumulate their invariant masses in a histogram across events. A particle resonance is inferred from the ensemble's dynamics; its short-lived parent is not directly photographed. Daughter tracks can be visible even when the production and decay positions of the parent cannot be separated. Wrong combinations create background, and several combinations from one event are statistically correlated.