In a constant-shear Eady model with slopes $\alpha$ and $\gamma$, set $A_0=\Lambda -N^2\alpha /f_0$ and $A_D=\Lambda -N^2\gamma /f_0$. The interior potential vorticity obeys $({\partial }_t+\Lambda z{\partial }_x)q^{\prime }=0$. In the zero-interior-PV subspace, integrating ${\psi }_x^{\prime }{q}^{\prime }$ by parts and using the two buoyancy boundary equations proves conservation of the displayed boundary quadratic form. Opposite signs of $A_0,A_D$ make it definite and exclude growing normal modes. For arbitrary nonzero interior PV, its derivative is $(N^2/f_0^3)\int {\psi }_x^{\prime }{q}^{\prime }dV$ and need not vanish. The restriction to zero interior PV is automatic for a growing smooth mode with complex phase speed.

When both rigid boundaries of a constant-shear Eady model have the same small slope $\alpha$, define $a=N^2\alpha /(f_0\Lambda )-1$, $M=N|k|D/|f_0|$ and $C=c/(\Lambda D)$. Zero-interior-PV modes have vertical structure $A\cosh (Mz/D)+B\sinh (Mz/D)$. The boundary conditions $(z/D-C){\hat{\psi }}_{z/D}+a\hat{\psi }=0$ at both endpoints give the displayed dispersion relation. For $a\ge 0$ its square root is real and there is no exponential baroclinic instability. The same-sign boundary-pseudomomentum criterion is necessary, not sufficient.

A dwarf spheroidal galaxy is a faint, diffuse, largely pressure-supported galaxy with little gas. A large dynamical mass-to-light ratio commonly indicates dark matter, subject to assumptions of equilibrium, member selection, and binary-star corrections. Its stellar light and total mass need not share one density profile.

Null dust is matter with a stress-energy tensor $T_{ab}=\epsilon k_a{k}_b$ for a future-directed null vector $k$. For every future-directed timelike vector $t$, the energy current is $-T^a{}_b{t}^b=\epsilon (-k\cdot t)k^a$. Therefore its dominant energy condition holds exactly when $\epsilon \ge 0$, allowing the zero current. Rescaling $k$ rescales $\epsilon$ inversely by the square and leaves the stress-energy tensor unchanged. Ingoing Vaidya metric matter has $k=-{\partial }_r$, $k_a=-(dv{)}_a$, and $\epsilon =\dot{m}/(4\pi r^2)$.

For a linear map $T:V\to W$, the dual map $T^{\ast }:W^{\ast }\to V^{\ast }$ sends a linear functional $\phi$ on the output space to the input measurement $T^{\ast }(\phi )=\phi \circ T$. It reverses direction because a measurement of the output can be evaluated on an input by first applying $T$:For $T(x,y)=(x+y,2y)$ and $\phi (u,v)=2u-v$, the input measurement is $T^{\ast }\phi (x,y)=2(x+y)-2y=2x$. The output measurement ignores the part of this transformation contributed by $y$.

If $T$ has matrix $A$ in specified bases, a row representing $\phi$ changes to $\ell A$. Storing the coefficients of a linear functional as a column in the corresponding dual basis gives $A^T{\ell }^T$, explaining why the transpose of a linear map is represented by the matrix transpose. This algebraic operation uses no inner product; it agrees with a matrix transpose even over the complex numbers, rather than automatically introducing complex conjugation. The adjoint operator is defined using an inner product and is a different construction. ^[ref].

In finite dimension, the kernel and image of the dual map are described by annihilators of vector subspaces:

Restricting a linear functional to each summand gives a family of functionals, with no finite-support restriction. Conversely an arbitrary such family acts on a vector by the finite sum of its nonzero coordinates. These constructions are inverse and prove the displayed algebraic dual space identity. A group action on the summands induces the product action by their algebraic contragredients.

The algebraic contragredient acts on the entire algebraic dual space. It need not be a smooth representation, even if the original representation is smooth. Taking the inverse in this formula is necessary to obtain a group action, and inverts a one-dimensional linear character.

An impermeable wall at $x=-\ell$ first receives the upstream-moving head at $t_1$. The leading downstream characteristic curve of the reflected adjustment obeys $\dot{x}=u+c=(12c_0+5x/t)/7$ in the unmodified rarefaction wave, giving the displayed curve. It stays behind the inviscid dry tip $x_f=6c_{0}t$ at every finite time. The finite reservoir drains and its upstream depth decreases, while only the region behind $x_r$ has yet received the wall signal. Bottom drag or other tip physics can change the front law.

Let two double circular cones have vertices $a_1,a_2$, common unit axis $n$, and semi-angle $0<\alpha <\pi /2$. Put $Q={\cos }^2\alpha I-n{n}^T$ and $b=a_1-a_2\ne 0$. Their equations are $(x-a_i{)}^{T}Q(x-a_i)=0$. Subtracting cancels the common quadratic term, so every intersection point satisfiesSince $Q$ has eigenvalues ${\cos }^2\alpha$ perpendicular to $n$ and $-{\sin }^2\alpha$ along $n$, $Qb\ne 0$. Thus the intersection is contained in a plane, possibly with an empty intersection. Its unit normal is $Qb/|Qb|$, with $|Qb{|}^2=|b{|}^2{\cos }^4\alpha +(n\cdot b{)}^2(1-2{\cos }^2\alpha )$. Degenerate endpoint semi-angles can make the normal vanish and must be treated separately.

In a finite zero-sum game, removing a strictly dominated strategy preserves the value of a zero-sum game: replace any probability on it by a dominating mixed strategy, including a pure one. This increases every payoff for the maximizing player, or decreases every payoff for the minimizing player. Repeat on the reduced game; new domination relations may appear after a deletion.

Hydroxyurea supplies the synthesis-blocking comparison in the experiment under EdU labeling.

Its inhibition of ribonucleotide reductase limits deoxynucleotide supply and can slow DNA replication. Reduced incorporation of a nucleotide label can therefore reflect limited precursor supply rather than disappearance of all polymerase proteins. ^[ref]

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.