A wavelet is a localized analyzing function whose shifted and scaled copies examine a signal at different positions and resolutions. Its usefulness is to separate where a change occurs from how large a region that change spans. A brief feature can contribute to a few nearby wavelet coefficients instead of being described entirely through global oscillations.

The Haar wavelet is the simplest example: compare the first half of an interval with the second. A constant value contributes nothing to this contrast, while a difference between the halves survives. Moving the function changes which part of the signal is examined; shrinking it compares smaller neighbouring regions. In an orthonormal wavelet construction, these comparisons fit together without duplicating squared-norm contributions.

A scaling function retains the coarse information that such contrasts omit. The discrete wavelet transform computes both coarse values and details, and the wavelet series recombines them. This gives signal processing a representation that can expose local structure and support image compression. The transform itself is reversible in the complete orthonormal representation; discarding coefficients introduces approximation error. ^[ref]

For integers $m$ and $n$,This orthogonality extracts matching coefficients when two Fourier series are integrated over one period.

The Poisson summation formula equates sample sums of a function and its Fourier transform. Exchanging broad and narrow sums can save work. Use the conventionFor every Schwartz function $f\in \mathcal{S}(\mathbb{R})$,Both series have absolute convergence. A sufficient hypothesis is that $f$ is smooth and ${\sup }_x(1+|x|{)}^M|f^{(r)}(x)|<\infty$ for all nonnegative integers $M,r$; its Fourier transform is then also a Schwartz function.^[ref] This assumption controls both decay and regularity; mere membership of an L1 space does not control values at individual sampling points, as periodization of an integrable function illustrates.

Here $\xi$ counts cycles per unit length. In the angular convention used elsewhere in Fourier analysis, $\tilde{f}(\omega )=\int f(x)e^{-i\omega x}dx$, so $\hat{f}(\xi )=\tilde{f}(2\pi \xi )$ and the right-hand side becomes ${\sum }_k\tilde{f}(2\pi k)$. Keeping the Fourier transform convention explicit prevents stray $2\pi$ factors. Changing the sample spacing leads to the Poisson summation formula for a Euclidean lattice.

With signature $(-,+,+,+)$ and units $c=1$, parametrizing a timelike worldline by proper time makes its four-velocity satisfy $U^a{U}_a=-1$.

A supermultiplet is an irreducible representation of the Super-Poincaré algebra. Conserved supercharges pair its bosonic and fermionic states at equal four-momentum.

The super-Poincaré algebra extends the Poincare algebra by odd spinor generators. In four-dimensional $\mathcal{N}=1$ supersymmetry, $\{Q_{\alpha },{\bar{Q}}_{\dot{\beta }}\}=2{\sigma }_{\alpha \dot{\beta }}^{\mu }{P}_{\mu }$ and the equal-chirality anticommutators vanish.

A field is perfect when every algebraic extension is separable. In positive characteristic $p$, this is equivalent to surjectivity of the Frobenius endomorphism $x\mapsto x^p$; every finite field is perfect.

BCS theory describes a superconducting instability caused by attractive interactions between opposite-momentum fermions near a Fermi surface.

John Bardeen, Leon Neil Cooper and John Robert Schrieffer developed BCS theory in 1957, using attraction mediated by phonons.^[ref]

Weak-coupling Cooper pairs overlap. The Cooper instability reorganizes an unpaired Fermi surface; it is not Bose-Einstein condensation of compact, pre-existing molecules. The original paper explicitly distinguishes these pictures.^[ref]

Two complementary clues made the microscopic problem concrete. The superconducting isotope effect connected the transition to phonons.^[ref]

A 1957 transmission experiment used thin lead and tin films as samples in far-infrared and millimeter-microwave measurements. The measured transmission was converted to complex electrical conductivity, normalized to the normal-state value. Comparing film thickness, resistance, annealing and the two metals tested the response's reproducibility. Weak low-photon-energy absorption followed by an onset supported a superconducting energy gap; its extraction required a thin-film electrodynamic model.^[ref]

The later Giaever superconducting tunnelling experiment provided another way to resolve the superconducting energy gap through quantum tunnelling.^[ref] The derivations below explain what the model predicts, including the distinction between a single quasiparticle's minimum energy and the threshold for making two of them.

Fermi-surface renormalization-group scaling sends frequency and normal displacement from the Fermi surface to zero while leaving tangential momentum fixed. Generic interactions are irrelevant, while forward-scattering and opposite-momentum Cooper channels can remain marginal.

A Fermi surface is the codimension-one surface in momentum space separating occupied and unoccupied zero-temperature states of a fermionic system.

A symmetric chain in the Boolean lattice $\mathcal{P}([n])$ contains one set of each size from $r$ through $n-r$. A symmetric chain decomposition partitions the Boolean lattice into such chains.

An exoplanet interior is inferred from mass, radius, age, irradiation, and atmospheric composition. Distinct mixtures of metal, rock, water, and hydrogen-helium can share the same bulk mass and radius.

Given sites in a Euclidean space, the Voronoi cell of a site is the set of points at least as close to it as to every other site. The Voronoi diagram is the subdivision formed by all of these cells.

Modified gravity changes the Einstein-Hilbert action or its field equations in order to produce gravitational dynamics different from general relativity.

For $h_{ab}=\delta g_{ab}$, variation of the inverse metric and metric volume form givesThe Levi-Civita connection varies by

The Einstein-Hilbert action is the integral of the Ricci scalar against the metric volume form. Varying it with respect to the metric gives the vacuum Einstein field equations, up to a boundary term.

A four-fermion interaction is a local operator containing four fermion fields. In four spacetime dimensions it has mass dimension six, so its coefficient has mass dimension minus two; writing it as the square of a current multiplied by a coefficient inside the square gives that coefficient mass dimension minus one.

An estimator is doubly robust when it remains consistent if either of two nuisance models is correctly specified, commonly an outcome-regression model or a propensity-score model.

For a compact convex set $K\subset {\mathbb{R}}^n$, the Fourier transform gives a bijection between distributions supported in $K$ and entire functions satisfyingfor some $C,N$. Here $H_K$ is the support function. In the closed-ball case $K={\overline{B}}_R$, the exponential factor is $e^{R|\mathrm{Im}\zeta |}$. Convexity is essential to infer support in $K$ from this bound: the support function cannot distinguish a nonconvex set from its convex hull.

The singular support of a distribution is the complement of the largest open set on which it is represented by a smooth function.