The galaxy stellar mass function counts galaxies per stellar-mass interval and volume, often using comoving volume. Its stellar-mass moment gives the cosmic stellar mass density, while its number moment counts galaxies only if the low-mass population is integrable or truncated. It differs from the initial mass function of individual stars.

A thin ring of mass $m$ and radius $a$ has interior relative potential $Gm[1+R^2/(4a^2)+\cdots ]/a$. Its leading radial acceleration is outward, $GmR/(2a^3)$, unlike the zero interior force of a spherical shell. Exterior disc annuli therefore affect a galaxy rotation curve.

The spherical shell theorem yields $v_c^2=GM(<r)/r$ only in spherical geometry. A razor-thin axisymmetric disc instead requires $v_c^2=R{\partial }_R\Phi (R,0)$, including forces from exterior annuli. The interior gravitational force of an exterior thin ring gives a concrete counterexample with identical enclosed mass but different circular speeds.

Let a disk contain fraction $f_d$ of its host mass and let its edge specific angular momentum be $f_j{R}_h{v}_h$. With halo relation $v_h^2=G{M}_h/R_h$ and disk geometry coefficient $\kappa =v_d^2{R}_d/(G{M}_d)$, combining centrifugal balance and the edge angular momentum gives the displayed estimates. The coefficient records the radial mass profile and thickness: enclosed mass does not determine a disc rotation curve. Treating $\kappa =1$ as a monopole estimate must not be mistaken for an exact thin-disk theorem.

A starburst galaxy undergoes an episode of rapid star formation. An instantaneous-burst idealization places all newly formed stars at one stellar age, so their present evolutionary-state fractions follow from the initial mass function at that fixed age rather than from an average over formation times.

The differential Newtonian gravitational field of a galaxy across a smaller bound system. Its tidal tensor can slowly change the angular momentum of distant comets, changing their periapses and their access to planetary encounters.

This spherical gravitational potential, with $a,v_0>0$, has density $\rho =v_0^2(2a+r)/(4\pi Gr(a+r{)}^2)$ and circular speed $v_c=v_0\sqrt{r/(a+r)}$. It combines a central $1/r$ density cusp with an outer flat galaxy rotation curve, but its total mass is infinite. It differs from a logarithmic potential with a constant-density core.

The Oort constants describe local differential galactic rotation: $A=-R{\Omega }^{\prime }/2$ and $B=-\Omega -R{\Omega }^{\prime }/2$. Thus $\Omega =A-B$, the radial epicyclic frequency obeys ${\kappa }^2=-4\Omega B$, and the epicyclic ellipse has axis ratio $\sqrt{-B/(A-B)}$.

Violent relaxation is rapid redistribution of the orbital energies of collisionless matter in a time-dependent collective gravitational potential. It can occur during collapse or a major galaxy merger. It differs from collisional two-body relaxation and from phase mixing in a stationary potential; it does not imply perfect final thermal equilibrium.

A translation of a function shifts its function input without altering its values relative to that shift. On the real line, $(T_{h}f)(x)=f(x-h)$ moves the graph of a function rightward by $h$: if $f$ has its peak at zero, $T_{h}f$ has it at $h$. These shifts satisfy $T_h{T}_k=T_{h+k}$ and $T_{-h}=T_h^{-1}$.

Translation invariance of Lebesgue measure preserves each finite Lp norm. For the Fourier transform convention $\hat{f}(\xi )={\int }_{\mathbb{R}}f(x)e^{-ix\xi }dx$, a change of variables givesThe phase sign depends on the stated shift and Fourier transform conventions. In particular, a spatial shift changes phase while preserving the modulus of the transform.

For a function $f:X\to Y$ and $y\in Y$, its fiber of a function over $y$ is the set of all inputs producing that one output:Here $f^{-1}(y)$ is shorthand for a preimage of a singleton set, rather than a claim that an inverse function exists. A fiber can contain several inputs or be empty. It describes exactly which distinctions the function forgets.

For the finite function under image and preimage of a function, the fibers over $0,1,2,3$ are $\{a,b\}$, $\{c\}$, $\{d\}$ and the empty set, respectively. The nonempty fibers form a partition of a set: they are disjoint sets whose union is the domain of a function. Every input belongs to the fiber over its unique output, and two fibers with distinct output labels cannot share an input. A function is injective exactly when every fiber has at most one member; it is surjective exactly when every member of its specified codomain has a nonempty fiber.

More generally, $f^{-1}(U)$ is the union of the fibers over all $y\in U$. The relation $x\sim x^{\prime }$ defined by $f(x)=f(x^{\prime })$ is an equivalence relation, and its equivalence classes are precisely the nonempty fibers. The resulting quotient set is naturally in bijection with the image of a function. This links an input partition to the output values actually reached, without choosing one representative from each class.

Fibers are useful when an output is known but the input is not uniquely determined. A solution set $g^{-1}(\{0\})$ is one fiber, and factorization through a surjection explains when another function depends only on the retained output. Quantum hidden subgroup problem algorithms use functions whose fibers are cosets, giving the same idea a structured form in group theory.

A function collision is a pair of distinct inputs $x\ne x^{\prime }$ with $f(x)=f(x^{\prime })$. An injective function has no collisions. For a two-to-one function every nonempty fiber has two elements and hence one unordered collision pair. Finding such a pair through an oracle is the task of quantum collision finding.

In the finite example under image and preimage of a function, the pair $\{a,b\}$ is a function collision: both inputs give zero. The function retains their shared output but loses the distinction between them. If a finite set has more inputs than the number of available outputs, the pigeonhole principle forces a function collision. Equal source and target cardinalities alone do not prevent one.

For a finite function, let $m_y$ be the cardinality of its fiber of a function over $y$. The number of unordered function collisions is ${\sum }_y\left(\genfrac{}{}{0ex}{}{m_y}{2}\right)$, because each pair must lie in one common fiber. The example's nonempty fiber sizes are $2,1,1$, giving exactly one collision pair.

A set function assigns a value to each set in a specified family of sets. In finite optimization the usual form is $f:\mathcal{P}(V)\to \mathbb{R}$, a real-valued function on the power set of a ground set $V$.

For a finite set $V$, the rule $f(S)=|S|$ is a set function counting selected elements; $f(S)={\sum }_{v\in S}w(v)$ instead totals their weights. A measure is also a set function, but its domain of a function is a specified collection of measurable sets, and its values can include infinity. The intended family and codomain are therefore part of the definition.

A vector-valued function is a function $F:X\to V$ taking values in a vector space. For example, $F(t)=(\cos t,\sin t)$ describes the unit circle in ${\mathbb{R}}^2$: one real number controls a two-component vector.

In a real or complex finite-dimensional vector space equipped with a norm, choosing a basis writes $F={\sum }_j{f}_j{e}_j$ in scalar-valued functions. Continuity and differentiability can then be checked in these coordinates, and $F^{\prime }={\sum }_j{f}_j^{\prime }{e}_j$ when the derivatives exist. Finitely many coordinates matter: coordinatewise convergence in an infinite-dimensional vector space need not imply norm convergence.

An accessory parameter is a coefficient not determined by the singular points and their indicial exponents. A second-order Fuchsian differential equation with exactly three regular singular points has no such independent parameter.

For $m\ge 2$ integrable weak velocity derivatives on a unit-period spatial circle, the Fourier transform of a derivative gives $|{\hat{f}}_0(k,kt)|\le \Vert {\partial }_v^m{f}_0{\Vert }_1/(2\pi |kt|{)}^m$ for $k\ne 0$. Summing these modes uses ${\sum }_{k\ne 0}|k{|}^{-m}=2\zeta (m)$. The resulting absolute Fourier summability gives the displayed uniform bound for the continuous representative of the density. For $m=1$ the same summation diverges, so this argument needs additional input.

A subset $X$ is a free basis of a group $G$ when the natural map from the free group on $X$ to $G$ is an isomorphism. Equivalently, $X$ generates $G$ and no nonempty freely reduced word in $X\cup X^{-1}$ equals the identity. A generating set is not automatically a free basis.

In an Earth-fixed rotating reference frame, the horizontal displacement of a freely falling body receives contributions from Coriolis acceleration and centrifugal acceleration. A short-fall approximation substitutes the unperturbed downward velocity into the Coriolis term and the Earth's surface position into the centrifugal term. If gravitational acceleration excludes centrifugal acceleration, both contributions must be included explicitly.

A free-energy functional assigns a free energy to an entire spatial order parameter configuration rather than only to one uniform value. In Landau-Ginzburg theory, a local derivative expansion gives $\mathcal{A}[m]=\int d^{D}x\{\kappa (\nabla m{)}^2/2+V(m)\}$, with appropriate boundary terms and regular backgrounds. The Landau approximation minimizes this functional; a fluctuating statistical field theory instead integrates its Boltzmann weight over configurations. These procedures need not give identical critical exponents.

A nonzero element of a free abelian group is primitive when it is part of an integral basis, equivalently when some homomorphism to $\mathbb{Z}$ takes it to one. Its coordinates have greatest common divisor one. This lattice meaning differs from a primitive homology class in the coalgebra sense.