For a prime $p$, take rank-$2p$ subsets of $[4p]$ with no distinct pair meeting in $p$ points. Split by membership of coordinate one. In the containing part, remove that coordinate; in the avoiding part, first take complements and then remove it. Both transformed families have rank $2p-1$ on $4p-1$ points. Their distinct intersections avoid residue $p-1$ modulo $p$. The uniform Frankl-Wilson theorem bounds each by $\left(\genfrac{}{}{0ex}{}{4p-1}{p-1}\right)$, yielding the displayed stronger form of the requested bound.

If member sizes modulo a prime number $p$ avoid $L$ while distinct intersections lie in $L$, evaluation at characteristic vectors of sets gives a diagonal matrix with nonzero diagonal. After multilinear reduction on the Boolean cube, these polynomials remain linearly independent and have degree at most $|L|$. Counting square-free monomials proves $|\mathcal{F}|\le {\sum }_{j=0}^{|L|}\left(\genfrac{}{}{0ex}{}{n}{j}\right)$. On a fixed uniform layer, the low-degree evaluation rank on a uniform layer sharpens this to $\left(\genfrac{}{}{0ex}{}{n}{|L|}\right)$ when $|L|$ does not exceed the layer rank.

For a prime number $p$, the vertices are $k$-subsets of $[N]$. An independent set avoids the only off-diagonal intersection congruent to $k$ modulo $p$, so the Frankl-Wilson theorem bounds its size by $\left(\genfrac{}{}{0ex}{}{N}{p-1}\right)$. A clique has constant intersection $p-1$ and has size at most $N$, by the same theorem modulo a prime larger than $k$. This explicit graph gives an asymmetric Ramsey number lower bound and an exponential chromatic number lower bound.

The degree-zero framed cobordism class is defined by the transfer of the subset cover of a framed vector bundle. With the canonical relative framing it equals $\left(\genfrac{}{}{0ex}{}{n}{k}\right)\cdot 1$ for a globally ordered framing of rank $n$. Put $l_0=1$ and $l_k=0$ outside the rank range. Disjoint-union additivity and fiber-product multiplicativity of transfer give $l_k(\eta \oplus \zeta )={\sum }_{i+j=k}{l}_i(\eta )l_j(\zeta )$.

The denominator is the nonzero lower entry $b$ in a fraction $a/b$. It specifies the division being performed; with $b>0$, the unit fraction $1/b$ is the size of one part. Thus $3/5$ and $3/10$ have the same numerator but different values.

An integer fraction can always be written with positive denominator. Dividing both entries by their greatest common divisor gives a reduced fraction, so $6/10$ and $3/5$ represent the same rational number despite different denominators.

A zero denominator is excluded from ordinary rational numbers. The symbol $1/0$ used at the boundary of the Stern-Brocot tree is a formal boundary symbol, not an additional rational number produced by division by zero.

The numerator is the upper entry $a$ in a fraction $a/b$. For an integer fraction, it records how many signed units of size $1/b$ are being taken when the denominator is positive: $3/5$ takes three fifths, while $-3/5$ reverses the sign.

A numerator belongs to a chosen representation, rather than being determined by the rational number alone. The equal fractions $3/5$ and $6/10$ have different numerators. The reduced fraction convention makes the numerator and positive denominator unique; zero then has numerator zero and denominator one.

For the dual Haar measure, the formula first follows for continuous compactly supported convolution squares from the Bochner theorem. More generally, if $f\in L^1(G)$ and $\hat{f}\in L^1(\hat{G})$, the right-hand side is a continuous representative of $f$; equality holds everywhere if $f$ is continuous, and otherwise at every point where the chosen representative is continuous. Convolution with normalized small-support positive convolution squares proves the extension: the spatial convolutions approach $f$ in $L^1$, while their transforms approach $\hat{f}$ in $L^1$ by equicontinuity of compact families of characters and an integrable-tail split.

A distinct experiment in An Automated, Highly Selective Reaction Monitoring Instrument Using Molecular Rotational Resonance Spectroscopy by Ann Adele Byars and others (Figure 1 and Figure 3) tracked exchange of hydrogen for deuterium in ethyl acetoacetate mixed with deuterated methanol. A pump circulated chemical solution through a valve; a calibrated loop injected aliquots into a temperature-controlled inlet, held at room temperature for this exchange experiment. Neon carried the vaporized sample to a cavity spectrometer. Separately recorded broadband spectra and fitted rotational Hamiltonians identified transitions for the four isotopic forms before targeted monitoring.^[ref]

The readout was each assigned line's signal over reaction time, normalized by the sum of the four signals. Transitions with the same quantum numbers reduced response-factor differences. A solvent wash addressed observed sample carryover. The signals distinguished the unsubstituted, two singly substituted, and doubly substituted forms. Only one gas-phase structural form was detected; the authors explicitly caution that their readout does not recover the structural chemical equilibrium in chemical solution.^[ref]

This example uses nonlinear molecules, so neither the single-constant linear rigid rotor nor the synthetic comb above is a fit to its spectrum. Its connection to the model is the sensitivity of moments of inertia to isotope placement. Line position helps assign a species; signal size additionally depends on sample transfer, initial quantum state populations and instrument response. Matching quantum numbers is a useful control, not a proof of identical response factors. These distinctions explain why resolving species and determining their concentrations are separate inference steps.

The discrete paraboloid is the graph of the sum-of-squares quadratic form in a vector space over a finite field. Its surface measure is normalized counting measure, of total mass one. A Fourier extension operator takes a function on this graph to its character sum in the ambient finite vector space. Ambient counting measure and normalized surface measure must be distinguished when defining a restriction constant.

For each integer $m\ge 1$, the set of continuous functions satisfying $|S_N(f,0)|\le m$ for every $N$ is closed in the uniform norm. If it contained a ball of radius $r$, subtracting two members of that ball would bound every partial-sum functional norm by $4m/r$. The Dirichlet kernel harmonic lower bound excludes this, so these sets are nowhere dense. By the Baire category theorem, their complement is a dense $G_{\delta }$ set. Every function in that complement has unbounded Fourier partial sums at zero. Translation gives the same conclusion at any prescribed point. This is a direct Uniform boundedness principle argument on the real Banach space of continuous functions on the circle.

The half-plane Fourier projection is $(I-i{H}_v)/2$, where $H_v$ is the directional Hilbert transform. A disk of radius $R$, modulated to put a boundary point at the origin, has symbol $1_{\{|\xi +Rv|\le R\}}$, which converges almost everywhere to this half-plane symbol. The Plancherel theorem makes this an $L^2$ multiplier limit.

For a function prescribed on $0<x<L$, its half-range Fourier cosine series is the Fourier series of the even extension, made $2L$-periodic. Its coefficients are $a_n=(2/L){\int }_0^{L}f(x)\cos (n\pi x/L)dx$, including $n=0$. At a jump the usual piecewise-smooth convergence theorem gives the average of the two one-sided limits of the periodic extension.

If $f$ is integrable and $2\pi$-periodic and $n$ is a positive integer, then ${\int }_0^{2\pi }f(nx)e^{imx}dx=0$ for every integer $m$ with $0<|m|<n$. Split the integral into $n$ equal intervals and substitute $y=nx-2\pi j$ to obtainThe root of unity sum vanishes by the finite geometric series formula. The zero mode is $\int f$, so a mean-zero dilated function is orthogonal to all trigonometric polynomials of degree at most $n-1$.

An incident particle of rest mass $m$ and energy $E$ striking a stationary identical target has total squared invariant energy $W^2=2m{c}^2(E+m{c}^2)$. Producing $N\ge 2$ final particles of the same rest mass requires $W\ge Nm{c}^2$, giving the threshold. Equality means no final kinetic energy in the centre-of-momentum frame. From the Lorentz factor, the corresponding incident speed is at least $cN\sqrt{N^2-4}/(N^2-2)$.

If the total incoming four-momentum is timelike and has centre-of-momentum energy $W$, production of particles with rest masses ${\mu }_i$ requires $W\ge {\sum }_i{\mu }_i{c}^2$. In the centre-of-momentum frame, total final energy is a sum of $\sqrt{{\mu }_i^2{c}^4+c^2|{\mathbf{p}}_i{|}^2}$, each at least its rest energy. For massive products equality occurs with all at rest in that frame. A single product requires equality of the entire invariant mass, not just an inequality; a massless product cannot have nonzero energy and be at rest.

A Fierz rearrangement rewrites a product of fermion bilinears with one spinor pairing as a sum of products with another pairing. For left-handed vector currents, it lets a charged-current four-fermion operator be compared directly with a neutral-current operator.

The four-dimensional membrane Fierz rearrangement identity $(C{\gamma }_{mn}{)}_{(\alpha \beta }(C{\gamma }^n{)}_{\gamma \delta )}=0$ makes the displayed supertranslation-invariant four-form closed. It is locally the exterior derivative of a three-form potential, which can change by an exact form under target supersymmetry. Pulling the potential back to a closed worldvolume gives an invariant Wess-Zumino brane coupling. The charge magnitude and normalization of this form are fixed together by cancellation of the area variation under kappa symmetry.

The scalar potential is a positive sum of squared commutators of the six adjoint scalar fields. It vanishes precisely when they commute. For a compact gauge group, simultaneous conjugation puts a commuting tuple in a Cartan subalgebra. Its arbitrary $6r$ real coordinates are classical flat directions of a scalar potential, with residual identification by the Weyl group. For $SU(N)$ the simultaneous eigenvalue vectors also obey the tracelessness constraint. Generic values break the gauge group to its maximal torus.

The FRET donor quantum yield is the emitted fluorescence photon fraction per absorbed excitation photon, measured without a FRET acceptor. In the constant-rate model used for FRET efficiency, $Q_D=k_{\mathrm{r}}/(k_{\mathrm{r}}+k_{\mathrm{nr}})$. Changing the FRET donor quantum yield changes the Förster distance, even if label positions remain fixed.

The FRET dipole orientation factor describes the geometry of the two transition electric dipoles relative to their separation. Let ${\hat{\mathbf{\mu }}}_D$, ${\hat{\mathbf{\mu }}}_A$ and $\hat{\mathbf{r}}$ be their unit vectors. ThenFor a separation along $z$, parallel electric dipoles along $z$ give ${\kappa }^2=4$, parallel ones along $x$ give $1$, and one along $x$ with the other along $y$ gives $0$. Thus rotating a pair can change FRET efficiency without changing its separation.

The familiar value ${\kappa }^2=2/3$ requires suitable rapid, independent orientation averaging during the FRET donor lifetime. Random orientations across a population of slowly rotating fluorescent proteins do not establish that dynamic condition. ^[ref]