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A bounded linear operator on a Hilbert space is Hilbert-Schmidt when ${\sum }_n\Vert T{e}_n{\Vert }^2<\infty$ for one, equivalently every, orthonormal basis. Every Hilbert-Schmidt operator is a compact operator.

Peculiar velocity is motion relative to the homogeneous Hubble flow. In comoving coordinates,where $\mathbf{u}$ is physical velocity and $\mathbf{v}$ is peculiar velocity.

The abelianization of a group $G$ is the quotient groupwhere $G^{\prime }$ is the commutator subgroup. Its elements are cosets, and its operation is $(g{G}^{\prime })(h{G}^{\prime })=gh{G}^{\prime }$. The result is an abelian group: the two products $gh$ and $hg$ differ by a group commutator, which disappears in the quotient group.

For the symmetric group $S_3$, the sign of a permutation realizes $S_3^{\mathrm{ab}}\cong C_2$, as shown in commutator subgroup of the symmetric group on three letters. For the dihedral group of a square, abelianization of the square symmetry group gives $C_2\times C_2$: two independent yes-or-no pieces of information survive. An abelian group already has $G^{\prime }=\{e\}$, so its abelianization is isomorphic to itself. At the other extreme, a perfect group has trivial abelianization.

Calling this the largest abelian quotient means that every group homomorphism into any abelian group factors uniquely through $q$. It does not require $G$ to be finite, or compare the sizes of unrelated abelian groups. The universal property of abelianization makes this precise. It lets us solve all such mapping problems in one simpler group: any invariant that respects multiplication and takes values in an abelian group sees only $G^{\mathrm{ab}}$.

For every finite alphabet $X$ and every positive integer $k$, there is $n$ such that every $k$-coloring of $X^n$ contains a monochromatic combinatorial line.

The impulse of a force over a time interval is its vector integral:Its International System of Units unit is $\mathrm{N}\mathrm{s}=\mathrm{kg}m/s$. The impulse-momentum theorem equates the net impulse to the change in momentum. For one fixed-mass point mass, integrate its net force; for a closed collection obeying Newton's third law, integrate the sum of external forces. In one dimension, the signed area under the force versus time graph gives the impulse; negative force contributes negative area. ^[ref]

Three stipulated rightward net force pulses illustrate the same $J=6\mathrm{N}\mathrm{s}$: constant $12\mathrm{N}$ for $0.5\mathrm{s}$, a triangle rising from zero to $24\mathrm{N}$ at $0.25\mathrm{s}$ and falling to zero at $0.5\mathrm{s}$, and constant $3\mathrm{N}$ for $2\mathrm{s}$. Their areas areApplied separately to the same fixed mass $m=2\mathrm{kg}$, each gives $\Delta v=J/m=3m/s$. Their peak forces differ: $12$, $24$, and $3\mathrm{N}$. Thus impulse fixes the final momentum change, while the force history determines how abruptly it occurs.

The mean net force over the chosen interval is $\overline{\mathbf{F}}=\mathbf{J}/(t_f-t_i)$. Extending the interval for a fixed impulse reduces this mean force. A lower peak force additionally depends on the pulse shape: the two short pulses above have the same mean $12\mathrm{N}$ but different peaks. This explains why controlling the duration and shape of a stopping impulse matters in cushioning a collision.

In a frictionless oblique collision, impulse acts along the common normal and tangential velocity components remain unchanged.

An elastic collision preserves total kinetic energy between the incoming and outgoing states. For isolated colliding bodies, momentum conservation also holds. The kinetic energy condition supplies information beyond momentum conservation alone. ^[ref]

Consider a stipulated one-dimensional example with rightward positive: $m_1=2\mathrm{kg}$, $m_2=1\mathrm{kg}$, incoming velocities $u_1=3m/s$ and $u_2=0$. The total momentum is $P=6\mathrm{kg}m/s$ and total kinetic energy is $K=9\mathrm{J}$. In SI numerical values, outgoing velocities $v_1,v_2$ satisfySubstituting $v_2=6-2v_1$ gives $3(v_1-1)(v_1-3)=0$. The root $v_1=3$, $v_2=0$ repeats the incoming state; the outgoing separating state is $v_1=1m/s$, $v_2=4m/s$. Its momenta add to $2+4=6\mathrm{kg}m/s$ and its kinetic energies add to $1+8=9\mathrm{J}$. The incoming and outgoing relative velocities both have magnitude $3m/s$, consistent with coefficient of restitution $e=1$.

The inelastic collision example below uses exactly these incoming data. Both outcomes preserve total momentum, while only the elastic collision preserves total kinetic energy. These are ideal numerical examples, not experimental measurements.

For chain complexes of free modules over a principal ideal domain $R$, the Künneth theorem gives a natural split short exact sequencealthough the splitting is not natural. Over a field the Tor term vanishes, and the cohomological cross product gives $H^{\ast }(X\times Y;k)\cong H^{\ast }(X;k){\otimes }_k{H}^{\ast }(Y;k)$ as graded rings under standard finiteness hypotheses.

The cup product is a natural bilinear operation $H^p(X;R)\times H^q(X;R)\to H^{p+q}(X;R)$. It is graded-commutative: $a⌣b=(-1{)}^{pq}b⌣a$.

The cap product pairs a homology class and a cohomology class to lower degree:

Compactly supported cohomology is the cohomology of cochains vanishing outside a compact subset. Equivalently,

This is the cohomological form of the universal coefficient theorem, with Ext and Hom terms rather than the homological tensor and Tor terms.

For an abelian coefficient group $G$, cohomology fits into a split short exact sequence

A limit point of a subset $A$ of a topological space is a point $x$ whose every neighbourhood meets $A\setminus \{x\}$. Thus $1$ is a limit point of $[0,1)$ in the real line, even though $1\notin [0,1)$. In contrast, an isolated point of $A$ belongs to its closure but is not a limit point of $A$. This distinction separates limit points from the more inclusive notion of an adherent point.

For $\beta =4$ and $a(0)=0$, the expanding branch isIt grows as $2\sqrt{H_{0}t}$ near zero and approaches $\sqrt{2}$ exponentially at late times.

For $\beta >4$, the turning polynomial has two positive roots and its smaller root lies below $2$. A universe expanding from $a=0$ reaches that root with $H=0$ and $\dot{H}<0$, then contracts to a finite-time Big Crunch.

If $X,Y$ are metric spaces, $Y$ is compact, and the graph of $f:X\to Y$ is closed, then $f$ is continuous. Any failure of sequential continuity gives a subsequence whose images stay away from the proposed limit; compactness produces a convergent image subsequence, and closedness of the graph forces its limit to be the correct value.

Integrating Maxwell–Boltzmann statistics over momentum for nonrelativistic particles of mass $m$, spin-state count $g_s$, and energy $m{c}^2+p^2/(2m)$ gives

A class $M$ is transitive when $x\in y\in M$ implies $x\in M$. Consequently, every member of a member of $M$ is available as an element of the structure $(M,\in )$.

A proper class is a class in set theory that is not a set in the ambient universe.

For the clamped-free beam under uniform load and endpoint force, the additional minimized internal energy isIts derivative is the endpoint displacement caused by the endpoint force: