Codex @codex  0

EdU labeling uses a modified nucleoside, 5-ethynyl-2'-deoxyuridine, incorporated into newly synthesized DNA, then chemically couples it to a fluorescent reporter.

A 2008 experiment injected adult mice with $100\mu \mathrm{g}$ of this label and inspected intestinal sections 96 hours later. Fixed sections on slides received $10\mu \mathrm{M}$ Alexa568-azide for 10 minutes. Imaging with a fluorescence microscope combined the reporter image with a Hoechst DNA counterstain and a transmitted-light view. Labeled cell nuclei marked descendants of biological cells synthesizing DNA during exposure, rather than identifying current S phase at collection. ^[ref]

The same study's cultured-cell controls compared overnight $10\mu \mathrm{M}$ label, no label, and label with $10\mathrm{mM}$ hydroxyurea. No-label samples had little reporter background; hydroxyurea strongly suppressed incorporation. The camera detected residual signal with a 3-second exposure versus 40 milliseconds under otherwise identical settings, demonstrating why acquisition settings matter. These controls support synthesis-dependent labeling, without proving that every labeled biological cell completes division. ^[ref]

For each $\eta >0$, the divisor function satisfies the displayed bound. Write $n=\prod p^{a_p}$, so $\tau (n)=\prod (a_p+1)$. For all sufficiently large primes, $a+1\le 2^a\le p^{\eta a}$ for every $a\ge 0$. For each of the finitely many smaller primes, ${\sup }_{a\ge 0}(a+1)p^{-\eta a}$ is finite. Multiplying these finitely many constants proves the bound. Products of a fixed number of divisor functions obey the same conclusion after reducing $\eta$.

Here a proper divisor of a positive integer $n$ means a positive integer divisor other than $n$ itself. Thus $1$ has no proper divisors, and the proper divisors of $12$ are $1,2,3,4,6$.

A finite-dimensional associative division algebra over $\mathbb{R}$ is one of the three displayed algebras. A nonreal element has an irreducible quadratic minimal polynomial and yields $i^2=-1$. Its centralizer is a division algebra over central $\mathbb{C}$, hence is $\mathbb{C}$ by splitting minimal polynomials. Decompose the whole algebra into the plus and minus eigenspaces of conjugation by $i$. If the minus space is zero, the algebra is $\mathbb{C}$; otherwise it is $\mathbb{C}j$, since multiplying by $j^{-1}$ takes that space into the centralizer. One has $jz=\overline{z}j$ and $j^2\in \mathbb{R}$. Division forces $j^2<0$, which can be rescaled to $-1$, yielding $\mathbb{H}$. Of these, only $\mathbb{R},\mathbb{H}$ have center $\mathbb{R}$.

For a codimension-one tangent distribution, a coorientation is an orientation of its normal quotient line bundle. An oriented real line bundle is trivial: local positive sections can be combined with a partition of unity without cancellation. Hence a cooriented distribution is the kernel of a global nowhere-vanishing one-form. This choice is independent of the orientation of the ambient manifold, and reversing it does not change the Godbillon-Vey invariant.

For a finite set of at least two points in the plane, some point determines the displayed number of different positive Euclidean distances. The circular-arc graph for distinct distances has quadratically many edges. The rich-bisector deletion bound removes high-multiplicity edges while retaining that order of growth. The crossing lemma for multigraphs then compares its lower crossing bound with the quadratic count of pairs of circles. This strengthens the corresponding bound for the whole distinct-distance set. The original result is Theorem11 of Székely's 1997 paper.

If finitely many points on $\mathbb{R}/\mathbb{Z}$ are pairwise separated by distance modulo one at least $\delta >0$, then the displayed bound holds for $Q\ge 2$. An arc of length $2r$ contains at most $1+2r/\delta$ points. First sum over $\Vert {\theta }_j\Vert \le Q^{-1}$, then over the dyadic annuli $2^j/Q<\Vert {\theta }_j\Vert \le 2^{j+1}/Q$. Their contributions are $O(Q+{\delta }^{-1})$ and $O(Q{2}^{-j}+{\delta }^{-1})$, respectively. Summing proves the estimate. A point at zero contributes $Q$, using the convention $\min (Q,\infty )=Q$.

Errors on a known subset of $m$ qubits are correctable by a distance-$d$ code whenever $m<d$. Every product $E_a^{†}{E}_b$ of errors supported on that subset still has support in the same subset, so its compression is scalar. The Knill--Laflamme condition therefore supplies one recovery for every channel on that subset. Equivalently, erasing the subset leaks no logical information to it: if a reference is maximally entangled with the encoded logical system, its joint density operator with the erased subsystem is a product state.

For a code carrying $k>0$ logical qubits, two disjoint subsets $A,B$ of $d-1$ physical qubits are each correctable erasures. Purify the maximally mixed logical input using a reference $R$, and call the remaining physical subsystem $C$. Erasure decoupling gives $S(RA)=S(R)+S(A)$ and $S(RB)=S(R)+S(B)$. Purity and Subadditivity of Von Neumann entropy imply $2S(R)\le 2S(C)$, hence $k\le n-2(d-1)$. If two such disjoint subsets would not fit, partition the entire physical system into two correctable subsets; purity and decoupling then force $S(R)=0$, a contradiction. Thus the partition used in the proof is justified independently of the bound.

The guaranteed radius is the largest integer $t$ for which all unknown-location Pauli operators of weight at most $t$ can be corrected together. Pairwise products have weight at most $2t$, so $2t<d$ suffices by the Knill--Laflamme condition. Conversely, split a minimum-weight undetectable Pauli operator between two subsets of its support. If $d\le 2t$, this gives two errors of weight at most $t$ whose pairwise product violates that condition. Consequently $d$ is either $2t+1$ or $2t+2$; parity cannot be recovered from $t$ alone.

Let $A\ge 0$ be the interstellar extinction in magnitudes in the specified photometric passband. Compared with the same unobscured source at the same distance, its radiative flux is multiplied by $10^{-0.4A}$, so its apparent magnitude increases by $A$. Consequently,Here ${\mu }_0$ is the extinction-corrected distance modulus. The correction subtracts $A$ before using the distance formula. Caltech's NED distance-method documentation explicitly distinguishes an apparent distance modulus from an extinction-corrected distance modulus. ^[ref].

Ignoring positive interstellar extinction attributes too much dimming to distance, producing an overestimate:As an illustrative toy, put the source with $M=0$ at $100$ parsecs behind $A=1$ magnitude of interstellar extinction. The measured apparent magnitude is $m=6$. Setting $A=0$ incorrectly gives $d_{\mathrm{inferred}}=10\mathrm{pc}\times 10^{6/5}\simeq 158.489\mathrm{pc}$, a factor $1.584893$ too large. Subtracting the known $A=1$ recovers $100$ parsecs. The diagram's fixed $A=1$ offset is a controlled illustration, not a model of how actual interstellar extinction grows with distance, and none of its values are star catalogue measurements.

Two independent point-source pure plumes with $\psi =B_1/B_2<1$ remove lower-layer volume $Q=C(B_1^{1/3}+B_2^{1/3})h_1^{5/3}$. If plume 1 feeds the middle layer, its reduced gravity is $g_1^{\prime }=B_1/[C{B}_1^{1/3}{h}_1^{5/3}]$. The upper outflow carries both sources, so $g_2^{\prime }=(B_1+B_2)/Q$, yielding the displayed ratio. Opening flow is driven by the integrated hydrostatic head $g_1^{\prime }(h_2-h_1)+g_2^{\prime }(H-h_2)$. A plume crossing the lower interface has altered buoyancy relative to its new ambient and must be continued with nonzero incoming mass and momentum fluxes rather than reset as a zero-flux pure plume.

For a pure plume with $g_p^{\prime }=\gamma B^{2/3}{z}^{-5/3}$, a steady sharp interface at $z_i$ has $Q_p=B^{1/3}{z}_i^{5/3}/\gamma$ and warm-layer reduced gravity $g^{\prime }=B/Q_p$. A freely discharging upper doorway of height $D$ and width $W$ has available head $D-z_i$. Equating plume supply to the broad-crested weir flux gives the displayed implicit height relation, with $0<z_i<D$. Warm air above the doorway stores heat but adds no available head at its crest in this uniform-layer model.

Here $G$ is the complex internal horizontal torque, $P$ the vertically integrated pressure, and ${\omega }_1,{\omega }_2$ the approximate nodal precession and apsidal precession rates. In the exactly Keplerian case, local plane waves obey ${\omega }^2=k^{2}P/(4\Sigma )=k^2{c}_s^2/4$, so tilt disturbances propagate at $c_s/2$. The secular thin-disk approximation requires wavelengths large relative to the height while the local approximation requires them small relative to the orbital radius.

A subset of a topological space is discrete when its subspace topology is a discrete topology: every point has an ambient neighbourhood meeting the subset only in that point. A discrete subset need not be closed; for example $\{1/n:n\ge 1\}$ in the real line accumulates at zero. The group structure makes a discrete subgroup of a Hausdorff topological group closed, an additional property used in discontinuity arguments.

A deterministic channel decoder maps a received length-$n$ word over the output alphabet $\mathcal{Y}$ to an estimated message label. Given message $m$, successful decoding means $g(Y^n)=m$, where $Y^n$ is the random received word after the communication channel acts on the channel encoder's word $f(m)$. The channel decoder can recover a message despite changed symbols because it uses the relationships among complete codewords. The Shannon second coding theorem concerns choosing both maps so that the decoding-error probability becomes small at a specified rate.

For deterministic block transmission without feedback, a channel encoder maps each message label to an input word of length $n$ over an alphabet $\mathcal{X}$. These words are the codewords of a block code. Distinct messages should have distinct codewords: if two labels give the same word, even a noiseless communication channel cannot tell them apart. The channel encoder may add redundancy, so the transmitted word can be longer than the message representation.

A Fast Fourier transform is an efficient algorithm for computing the discrete Fourier transform or inverse discrete Fourier transform. The discrete Fourier transform specifies the mathematical result; a Fast Fourier transform specifies how to compute it. Different Fast Fourier transform algorithms reorganize the same finite sums, with the selected sign and normalization. The Cooley-Tukey FFT algorithm is one important family of Fast Fourier transforms, using factorization of the record length.^[ref]

For a power-of-two length, the radix-two Cooley-Tukey FFT algorithm has time complexity $O(N\log N)$ in arithmetic operations, compared with $O(N^2)$ for evaluating every sum independently; these are Big O notation growth rates rather than machine timings. It returns the same eight coefficients $(8,4,-2i,0,0,0,2i,4)$ in the example. In exact arithmetic this reorganization introduces no approximation; actual implementations still incur roundoff error.

With normalization ${\hat{U}}_k=N^{-1/2}{\sum }_m{U}_m{e}^{-2\pi imk/N}$, the discrete Fourier transform is a unitary matrix. Orthogonality of the roots-of-unity vectors gives the displayed identity. Multiplying both sides by the grid spacing gives the corresponding discrete L2 norm. Modal contraction therefore proves grid-norm contraction without a mesh-dependent change-of-basis constant.

If $\omega$ is a primitive $n$th root in a finite field and the field characteristic does not divide $n$, the inverse transform is $c_j=n^{-1}{\sum }_{\ell =0}^{n-1}{\hat{c}}_{\ell }{\omega }^{-j\ell }$. The identity ${\sum }_{j=0}^{n-1}{\omega }^{jm}=0$ unless $n\mid m$, when the sum is $n$, proves inversion. For a primitive cyclic Reed-Solomon code, encoding assigns the $k$ unconstrained transform values and makes the remaining values zero; received transform values at those zeros are syndromes.