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A compactly supported distribution of order of a distribution at most $M$ has a smooth function as its Fourier transform, bounded by $C⟨\xi {⟩}^M$. It therefore belongs to every Sobolev space $H^s$ with $s<-M-n/2$.

The dual-number ring over a field $k$ consists of elements $a+b\epsilon$ with ${\epsilon }^2=0$. The element $\epsilon$ is a nonzero nilpotent element; its ideal is the ring's only prime ideal. Hence its affine scheme has the same one-point topology as $\mathrm{Spec}k$ while its structure sheaf differs.

A regular sequence in a commutative ring $A$ is a finite list such that $(f_1,\dots ,f_r)$ is proper and $f_j$ is a non-zero-divisor on $A/(f_1,\dots ,f_{j-1})$ for each $j$. In a graded ring with homogeneous $f_j$, injectivity can be checked on homogeneous elements because components of distinct degrees cannot cancel. It gives short exact sequences $0\to R_{j-1}(-\mathrm{deg}{f}_j)\to R_{j-1}\to R_j\to 0$ for the successive graded quotients.

For a multiplicative monoid automorphism $\varphi :\mathbb{N}\to \mathbb{N}$ fixing $0$, define $a\oplus b={\varphi }^{-1}(\varphi (a)+\varphi (b))$. This transports a semiring structure to the fixed underlying multiplication, hence a commutative-monoid enrichment on its one-object category. Swapping prime $2$ with an odd prime $p$ gives $1\oplus 1=p$. Infinitely many such primes therefore give infinitely many distinct additions, although the transported enriched categories are isomorphic.

For catastrophic disruption threshold $Q_D^{\ast }=Q_b{D}^b$, encounter speed $v_{\mathrm{rel}}=v_p{D}^p$, fixed belt volume, and small impactors, the catastrophic projectile diameter scales as $D_{\mathrm{cc}}\propto D^{1+(b-2p)/3}$. If $\alpha >1$, small destructive impactors dominate the catastrophic planetesimal collision rate, givingMass per logarithmic bin scales as $D^{4-\alpha }$. Constant mass flux in a collisional cascade then givesSize-independent speed and strength recover the Dohnanyi collisional cascade slope $7/2$. A size-dependent belt thickness changes this derivation. The broader coupled size-and-velocity problem is studied by Pan and Schlichting.

In a steady collisional cascade supplied at large sizes and depleted only at small sizes, mass conservation makes the downward mass flux across every intermediate size threshold equal. With self-similar fragmentation and comparable transfer between logarithmic bins, the mass removed per bin per unit time is consequently independent of size. Nonlocal or size-dependent fragmentation, additional sources, and proximity to the cutoffs can invalidate the simplified bin-by-bin relation.

An unshielded collider in a Directed acyclic graph is a triple $a\to b\leftarrow c$ with $a$ and $c$ nonadjacent. The absence of an edge between its outer vertices makes this orientation detectable from conditional independence information under faithfulness of a directed acyclic graph.

A pushout of $B\leftarrow A\to C$ is a colimit with morphisms $B\to P$ and $C\to P$ making the square commute and universal among such commuting pairs. In the Category of sets, form the disjoint union of $B,C$ and identify the two images of each element of $A$.

For a finite diagram in a category $D:\mathcal{J}\to \mathcal{C}$, set $X={\coprod }_{j}D(j)$ and $Y={\coprod }_{a:i\to j}D(i)$. The two maps $s,t:Y\rightrightarrows X$ on the $a$-summand are the injection of $D(i)$ and the injection of $D(j)$ after $D(a)$. Their coequalizer is the colimit of $D$. The pair $[s,1_X],[t,1_X]:Y⨿X\rightrightarrows X$ is a reflexive pair with the same coequalizer. Thus finite coproduct in a category constructions and coequalizers of reflexive pairs suffice, including the empty coproduct for the empty diagram. Finite products cannot replace coproducts here: the poset with elements $0,a,b,u_0,u_1,\dots$, order $0<a,b<u_{n+1}<u_n$, and incomparable $a,b$ has finite meets and top $u_0$, hence finite categorical products. Every reflexive parallel pair is an equal pair and has its identity as coequalizer, but $a,b$ have no least upper bound and therefore no coproduct.

Let $i:{\mathbb{RP}}^d\hookrightarrow {\mathbb{CP}}^d$ be the standard inclusion. If $h=c_1(\mathcal{O}(1))\mathrm{mod}2$ and $a=w_1({\gamma }_d)$, thenThe restricted complex tautological bundle is the complexification of a real vector bundle ${\gamma }_d{\otimes }_{\mathbb{R}}\mathbb{C}$. Its underlying real bundle is ${\gamma }_d\oplus {\gamma }_d$. The Whitney product formula for Stiefel–Whitney classes gives total class $(1+a{)}^2=1+a^2$, while the Stiefel–Whitney class of the underlying real bundle of a complex line identifies its second class with the reduced First Chern class. Dualizing changes the integral sign but has no effect modulo two. Thus the full induced map on cohomology sends $h^j$ to $a^{2j}$.

A subset $E$ of a set $X$ is cofinite in $X$ when its complement $X\setminus E$ is a finite set. For infinite $X$, the cofinite subsets form the cofinite filter. A nonprincipal ultrafilter on $\mathbb{N}$ contains every cofinite subset: it contains no singleton, hence no finite set, and decides between each set and its complement.

A differential form is coclosed when its codifferential vanishes. On a closed Riemannian manifold, a form is a harmonic differential form exactly when it is both a closed differential form and coclosed: the identity $⟨\Delta \alpha ,\alpha ⟩=\Vert d\alpha {\Vert }^2+\Vert \delta \alpha {\Vert }^2$ proves the converse as well as necessity.

An unbounded class in set theory of ordinals is closed if every nonempty set-sized increasing sequence of its members has its supremum in the class. This is the proper-class analogue of a club set. The reflection theorem for definable hierarchies gives such a class for every finite collection of first-order formulas.

Let $f:X\to Y$ be a continuous map that is a closed map and whose fibres are compact. Then $f^{-1}(K)$ is compact for every compact set $K\subseteq Y$, without a Hausdorff hypothesis. Given an open cover, cover each fibre by a finite union $V_y$ of its members. The set $W_y=Y\setminus f(X\setminus V_y)$ is open, contains $y$, and has $f^{-1}(W_y)\subseteq V_y$. A finite subcover of $K$ by the $W_y$ uses only finitely many original cover members and covers $f^{-1}(K)$. Empty fibres cause no exception.

For connected oriented closed $d$-dimensional manifolds with $d\ge 2$, their connected sum removes an open $d$-dimensional ball from each and identifies the resulting boundary spheres by an orientation-reversing diffeomorphism. The remaining orientations fit together. Collapsing the separating sphere gives a pinch map to the wedge sum of the two closed manifolds, and projection to either summand has degree of a continuous mapping one. In intermediate positive degrees the cohomology is the direct sum of the summand groups. Products of classes from different summands vanish; products landing in top degree use the single common orientation class. These statements follow from excision, the Mayer–Vietoris sequence and the degree-one projections.

A compact two-dimensional manifold without boundary. A connected oriented surface is classified by its genus; for genus $g$ its Euler characteristic is $2-2g$.

The notation $C_b^{1,2}([0,\infty )\times \mathbb{R})$ ordinarily denotes functions with continuous time derivative and two continuous space derivatives, with the function and indicated derivatives bounded globally. A different condition, often intended in finite-horizon partial differential equations, is membership in $C_b^{1,2}([0,T]\times \mathbb{R})$ for each finite $T$, allowing bounds to grow with $T$. These conventions must be distinguished in a long-time growth problem: the second permits divergence with time; the first does not. There is no universal notation for the finite-horizon condition, so it should be stated explicitly.

The unforced planar equation of motion in a rotating frame near a circular Kepler orbit has components $\ddot{x}-2n\dot{y}-3n^{2}x=0$ and $\ddot{y}+2n\dot{x}=0$. They describe epicyclic motion and Keplerian shear; mutual gravity adds close-encounter forcing.

An achromatic lens combines optical lens elements with different optical dispersion so that two specified wavelengths have the same focal length in the paraxial approximation. This brings their parallel incoming light rays to a common optical focus, helping an optical camera form a sharper image under mixed-colour illumination. A common implementation is an achromatic doublet: a positive crown glass element paired with a negative flint glass element. ^[ref].

The achromatic doublet does not require either material to have zero optical dispersion. Instead, the positive element gains positive optical power toward blue, while the negative element becomes more negative. Suitable Abbe numbers and signed optical powers let these changes cancel even though their reference optical powers do not cancel. For example, the achromatic doublet power balance calculation below combines $+10\mathrm{D}$ with $-5\mathrm{D}$ to obtain $+5\mathrm{D}$, or focal length $0.2\mathrm{m}$, with zero primary F–C optical power difference. The smaller Abbe number of the negative element makes its weaker reference optical power sufficient to oppose the positive element's colour change.

An achromatic lens is corrected at two chosen wavelengths, not necessarily at every wavelength between them. The remaining focal length variation is the secondary spectrum. Correcting axial chromatic aberration also does not automatically remove lateral chromatic aberration throughout an imaging system, spherical aberration, or diffraction. Actual designs therefore need ray tracing with the intended wavelengths, element thicknesses, separations and apertures. ^[ref]; ^[ref].

For a specified chemical reaction and standard state convention, the dimensionless equilibrium constant is the equilibrium value of the reaction quotient $Q={\prod }_i{a}_i^{{\nu }_i}$, where $a_i$ denotes thermodynamic activity and ${\nu }_i$ a signed stoichiometric vector entry. Thermodynamic activities incorporate the reference state and departures from ideal behavior. See IUPAC Gold Book, equilibrium constant.

In the ideal concentration model for A converting to B, use $a_{\mathrm{A}}=[\mathrm{A}]/c^{\circ }$ and $a_{\mathrm{B}}=[\mathrm{B}]/c^{\circ }$ with the same reference $c^{\circ }=1\mathrm{M}$. Their ratio gives $Q=[\mathrm{B}]/[\mathrm{A}]$ and $K=0.75/0.25=3$. Equality $K=k_f/k_r$ applies to this elementary first-order pair; it is not a universal ratio of two measured rate constants for an arbitrary multistep chemical reaction.

At a specified constant temperature, with consistent standard states, the standard molar reaction Gibbs free energy isHere ${\Delta }_{r}G$ is the local slope per mole of extent of reaction at fixed temperature and pressure, rather than the finite total Gibbs free energy change between two mixtures. At equilibrium ${\Delta }_{r}G=0$, while ${\Delta }_r{G}^{\circ }$ generally remains nonzero. For $K=3$ at $298\mathrm{K}$, it is about $-2.72kJ/mol$. For positive activities, $Q<K$ favors forward conversion and $Q>K$ favors reverse conversion; neither sign supplies a reaction rate. See MIT 5.111, Lecture 18, reaction free energy and equilibrium.