The Milne universe is the empty $k=-1$, $\Lambda =0$ FLRW model with scale factor proportional to proper time. It is a hyperbolic-coordinate description of the interior of a future light cone in Minkowski spacetime.

Normalize the present scale factor to one. In a dust-only open cosmology with no cosmological constant, the Friedmann equation gives $-k{c}^2=H_0^2(1-{\Omega }_m)$. The present comoving and physical curvature radii are therefore $r_c=c/[H_0\sqrt{1-{\Omega }_m}]$. At another epoch the physical curvature radius is $a{r}_c$.

// Target: cosmic-microwave-background-anisotropy.bigb

For pressureless matter and a positive cosmological constant in a spatially flat universe, $H^2=H_0^2[{\Omega }_{m0}{a}^{-3}+1-{\Omega }_{m0}]$. Substituting $y=a^{3/2}$ reduces the expanding branch to $\dot{y}=(3H_0/2)\sqrt{{\Omega }_{m0}+(1-{\Omega }_{m0})y^2}$. With $a(0)=0$, integration gives $a=[{\Omega }_{m0}/(1-{\Omega }_{m0}){]}^{1/3}{\sinh }^{2/3}(3H_0\sqrt{1-{\Omega }_{m0}}t/2)$. Its matter-dominated early limit is $t^{2/3}$ and its vacuum-dominated late limit is exponential. Radiation must be restored to describe the actual earliest universe.

For a closed Friedmann equation normalized to $k{c}^2=1$, constant equation-of-state parameter $w$, and conformal time $\tau$, let $q=(1+3w)/2$. The cosmological continuity equation and Friedmann acceleration equation give ${\mathcal{H}}^{\prime }=-q(1+{\mathcal{H}}^2)$, where $\mathcal{H}=a^{\prime }/a$. For $q\ne 0$ this integrates to the stated scale factor on any interval where the sine is positive, with ${\alpha }^{2q}=8\pi G{\rho }_0/3$. For $w=-1/3$ the separate solution is $a=A{e}^{H_0\tau }$ with $H_0^2+1=8\pi G{\rho }_0/3$; the sine-power formula is inapplicable.

For pressureless matter and a positive cosmological constant, the Friedmann equation gives ${\dot{a}}^2=H_0^2[{\Omega }_{m,0}{a}_0^3/a+{\Omega }_{k,0}{a}_0^2+{\Omega }_{\Lambda ,0}{a}^2]$. Differentiating with respect to $a$ gives the displayed unique minimum of this expression for positive matter density. It is a minimum of the actual expanding solution's speed only if that branch reaches it; sufficient positive spatial curvature can cause recollapse first. At the minimum, ${\rho }_m=2{\rho }_{\Lambda }$ and the Friedmann acceleration equation changes sign.

For pressureless matter, put $C=8\pi G{\rho }_m{a}^3/3>0$ and use units $c=1$. The flat Friedmann equation is ${\dot{a}}^2=C/a+\Lambda a^2/3$. Substitution $u=a^{3/2}$ gives ${\dot{u}}^2=(9/4)(C+\Lambda u^2/3)$. With a Big Bang at $t=0$, positive cosmological constant gives $a=(3C/\Lambda {)}^{1/3}{\sinh }^{2/3}(\sqrt{3\Lambda }t/2)$. Negative cosmological constant replaces the hyperbolic sine by $\sin (\sqrt{3|\Lambda |}t/2)$ and produces recollapse at $2\pi /\sqrt{3|\Lambda |}$. Zero cosmological constant gives $a=(9C/4{)}^{1/3}{t}^{2/3}$.

For a perfect fluid in general relativity with constant equation-of-state parameter $w$, use conformal time $dt=ad\tau$ and the conformal Hubble parameter $\mathcal{H}=a^{\prime }/a=aH$. The Friedmann acceleration equation gives ${\mathcal{H}}^{\prime }=a\ddot{a}=-4\pi G{a}^2(1+3w)\rho /3$. The Friedmann equation eliminates $\rho$ and gives this Riccati equation. For positive energy density, $\alpha \ne 0$ and appropriately shifted conformal time, its usual branches are $\cot (\alpha \tau )$ for $k=1$, $1/(\alpha \tau )$ for $k=0$, and $\coth (\alpha \tau )$ for $k=-1$. Their scale factors are proportional to ${\sin }^{1/\alpha }(\alpha \tau )$, $(\alpha \tau {)}^{1/\alpha }$ and ${\sinh }^{1/\alpha }(\alpha \tau )$ respectively on intervals where the bases are positive. At $\alpha =0$ the conformal Hubble parameter is constant and the separate solution is $a=A{e}^{h\tau }$.

With nonnegative conserved pressureless matter and radiation in cosmology, at least one of which has positive density, set $M=(8\pi G/3){\rho }_m{a}^3$, $Q=(8\pi G/3){\rho }_r{a}^4$, and $L=(8\pi G/3){\rho }_{\Lambda }$. The Friedmann equation becomes $(da/dt{)}^2+V(a)=-k$. For negative vacuum density the potential rises strictly from negative infinity to positive infinity, so an expanding branch has one maximum scale factor. For positive vacuum density it has one maximum: energy above that maximum gives monotonic expansion or contraction, energy below it gives disjoint recollapsing and bouncing branches, and equality admits an Einstein static universe together with asymptotic monotonic branches.

FRET optical crosstalk includes FRET donor emission leaking into the FRET acceptor detection band and direct excitation of the FRET acceptor by the nominal FRET donor excitation light. Measure donor-only, acceptor-only and unlabeled preparations with the same fluorescence microscope settings to separate these contributions and background.^[ref]

An explicit linear measurement model makes the controls' purpose clear. Let $D$ be background-subtracted donor-band counts under donor excitation, $A$ acceptor-band counts under acceptor excitation, and $F$ acceptor-band counts under donor excitation. Donor-only and acceptor-only preparations determine leakage coefficients $a$ and $b$. Thenis a corrected FRET sensitized emission signal under the assumed linear, fixed optical response. In a synthetic example $D=100$, $A=80$, $a=0.10$, $b=0.05$ and $F=34$ give $S=20$. Even a nontransferring sample would give $F=14$ from these two contaminants. Correcting that signal still does not convert $S$ into FRET efficiency without a calibration of excitation, yields and detection.

A biological comparison also needs a pairing-negative preparation and a known transferring positive preparation. Match label abundance and localization where possible; changing concentration can change encounter frequency as well as brightness. Check that attaching the labels preserves the protein function being investigated. Otherwise the measurement can faithfully report a perturbation introduced by the labels themselves.

The FRET donor lifetime is the mean time spent in the excited state. For the constant-rate model of FRET efficiency,After a sufficiently brief excitation pulse, the FRET donor's fluorescence follows exponential decay, $I(t)=I(0)e^{-t/\tau }$. With the illustrative values ${\tau }_D=4$ nanoseconds and ${\tau }_{DA}=2$ nanoseconds, the inferred FRET efficiency is $1/2$. Multiplying $I(0)$ changes brightness without changing this decay time.

The lifetime comparison requires the same nontransfer decay rates in the reference and paired samples. Changes in the FRET donor's environment or a different nonradiative loss can also shorten the FRET donor lifetime. Heterogeneous FRET efficiencies or slowly varying FRET dipole orientation factors need an appropriate decay model; substituting any fitted average lifetime into the single-pair formula can mislead. ^[ref]

The Förster distance is the separation giving FRET efficiency $E=1/2$ for the specified pair and conditions. It is a property of the optical pair and its environment, rather than a universal length. In the usual pair model,where ${\kappa }^2$ is the FRET dipole orientation factor, $Q_D$ the FRET donor quantum yield, $J$ the FRET spectral overlap integral, and $n$ the medium's refractive index. The omitted proportionality constant depends on the unit convention for $J$.^[ref]

If FRET efficiency is known and those assumptions hold, inversion givesThe derivative explains why errors become troublesome near $E=0$ or $E=1$. An illustrative pair with $R_0=4$ nanometres and another with $R_0=6$ nanometres both give $E=1/2$ at their different Förster distances: knowing FRET efficiency alone does not determine $r$.

A heterogeneous population adds another ambiguity. In a synthetic example with equal weights per FRET donor excitation, efficiencies $0.1$ and $0.9$ average to $0.5$, but their separations are $9^{1/6}{R}_0$ and $9^{-1/6}{R}_0$. Their average separation is approximately $1.068R_0$, whereas inverting the average FRET efficiency gives $R_0$. A measured emission ratio has additional detection weights; it need not equal even this excitation-weighted average. The transferable lesson is that an average readout does not uniquely specify a distribution of molecular separations.

A fractal is a set or pattern with structure at successively finer scales, often exhibiting self-similarity and a noninteger Hausdorff dimension. Fractal geometry studies such objects.

For $f\in L^1(G)$ the displayed integral defines a bounded continuous function on the Pontryagin dual group. It takes convolution to pointwise multiplication and left translation $T_x$ to multiplication by $\overline{\chi (x)}$. The transform depends on the scale of Haar measure on $G$; the compatible dual Haar measure scales reciprocally.

For nonzero $f$ with $f,xf,f^{\prime }\in L^2(\mathbb{R})$, integration by parts gives $\Vert f{\Vert }_2^2=-2\mathrm{Re}⟨f^{\prime },xf⟩$. The Cauchy-Schwarz inequality and Plancherel theorem yield the moment product bound $1/4$ with the angular-frequency Fourier convention. Equality occurs exactly for $f(x)=C{e}^{-a{x}^2/2}$ with $a>0$ real and $C\ne 0$. Translation, modulation or a quadratic phase increase one of these uncentred moments; the equality family is narrower than that for centred variances.

A window function is a function that vanishes outside a chosen finite interval; multiplying an input by it selects and weights that part of the history before a Fourier transform. The rectangular window $w_T$ is the indicator function of $[0,T]$. Applied to the weighted causal extension by zero of an input, it produces the finite-history integral ${\int }_0^T{e}^{-st}f(t)dt$ used to approximate a Laplace transform.

For the period-one continuation of $\log |x-1/2|$, symmetry and integration by parts give $c_m=-(-1{)}^m\mathrm{Si}(\pi m)/(\pi m)$ for $m\ne 0$. Thus the sine integral asymptotic gives $c_m=-(-1{)}^m/(2m)+1/({\pi }^2{m}^2)+O(m^{-4})$ as $m\to +\infty$. Translation of the logarithmic singularity causes the alternating phase; the endpoint derivative mismatch causes the next term.

A smooth compactly supported window with ${\sum }_{j\in \mathbb{Z}}\chi (x+j)=1$ permits multiplying every periodic distribution unambiguously. Its Fourier transform sampled at integer harmonics gives the periodic coefficients. The sharp window $H(x)-H(x-1)$ works for locally integrable functions but can be undefined at singular period endpoints, such as delta derivatives.

The Du Bois-Reymond theorem says that continuity of a periodic function does not guarantee convergence of its ordinary Fourier series at every point. At any prescribed circle point $x_0$, there is even a real continuous $f$ with unbounded Fourier partial sums $S_{N}f(x_0)$.

For $x_0=0$, view $T_N:f\mapsto S_{N}f(0)$ as bounded linear functionals on the real Banach space $C(\mathbb{T})$ with its supremum norm. The Dirichlet kernel harmonic lower bound givesIf every continuous input had bounded values under all $T_N$, the Uniform boundedness principle would bound these operator norms, a contradiction. Translation moves the exceptional point to any specified $x_0$. The generic unbounded Fourier sums at a fixed point strengthens this existence proof to a category statement.^[ref]

This does not contradict mean-square convergence: a continuous function is square-integrable, and its series still converges in $L^2$. Nor does it obstruct Fejér's theorem, which averages partial sums and gives uniform convergence for every continuous periodic input. The Fejér kernel is positive and its sums are a uniform-norm contraction, but the sums themselves need not be nonnegative: for $f=-1$, every Fejér sum equals $-1$.

For $0<\epsilon \le 1/2$, this continuous function on $\mathbb{R}/\mathbb{Z}$ has nonnegative Fourier coefficientsIt is ${\epsilon }^{-1}$ times the periodic convolution of two indicators of $[-\epsilon /2,\epsilon /2]$. Its Fourier series is absolutely convergent, ${\sum }_r{\hat{b}}_{\epsilon }(r)=b_{\epsilon }(0)=1$, and ${\sum }_{|r|>R}{\hat{b}}_{\epsilon }(r)\ll 1/(\epsilon R)$. Consequently, if all $N$ points $t_n$ avoid the interval $\Vert t\Vert <\epsilon$, some frequency $1\le r\ll {\epsilon }^{-2}$ satisfies $|{\sum }_{n}e(r{t}_n)|\gg N\epsilon$. This detects missing small neighborhoods by a large exponential sum.

If symmetric partial sums ${\sum }_{r=-n}^n{a}_r{e}^{irt}$ converge almost everywhere on the circle, then $a_n,a_{-n}\to 0$. Their successive differences $T_n=a_n{e}^{int}+a_{-n}{e}^{-int}$ tend to zero almost everywhere. If $M_n=|a_n|+|a_{-n}|$ failed to tend to zero, choose a subsequence with $M_n\ge \epsilon$. The normalized $T_n/M_n$ are bounded by one and tend to zero almost everywhere, so dominated convergence makes their squared integrals tend to zero. Fourier orthogonality instead makes those integrals $(|a_n{|}^2+|a_{-n}{|}^2)/M_n^2\ge 1/2$, a contradiction.