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Jeans instability is the gravitational growth of density perturbations whose self-gravity overcomes pressure support. For physical sound speed $c_s$, the comoving threshold isModes with $k\ll k_J$ grow gravitationally, whereas modes with $k\gg k_J$ undergo pressure-supported acoustic oscillations.

For observations $X_1,\dots ,X_n$, the unbiased sample variance isFor independent identically distributed observations with finite variance ${\sigma }^2$, it converges in probability to ${\sigma }^2$.

For real periodic KdV data, translation by one period acts on a conjugate basis of scattering solutions byLax compatibility gives $\dot{T}=[\Lambda ,T]$ for the matrix of ${\partial }_t+A$ on the eigenspace. Hence $\mathrm{tr}T=2\mathrm{Re}a$ is conserved.

Ifthen every positive power has constant trace:The final equality is the cyclic property of the matrix trace.

A singularity is non-isolated when every punctured neighbourhood contains another singularity. An accumulation point of poles or essential singularities cannot itself be classified as a removable singularity, pole, or isolated essential singularity.

An isolated singularity is removable, a pole, or essential according as its Laurent principal part has zero, finitely many nonzero, or infinitely many nonzero terms.

The Zariski topology on affine space is Noetherian because polynomial rings are Noetherian. Every closed subset is a finite union of irreducible closed subsets: a minimal counterexample would split into two smaller closed subsets, each already having such a decomposition.

An irreducible component is a maximal irreducible closed subset. Every algebraic variety is a finite union of its irreducible components.

For $D_{2n}=⟨r,s:r^n=s^2=1,srs=r^{-1}⟩$, every irreducible complex representation has degree at most two. If $v$ is an eigenvector of $r$ with eigenvalue $\lambda$, then $sv$ has eigenvalue ${\lambda }^{-1}$, so irreducibility makes the representation equal to $\mathrm{span}\{v,sv\}$.

If conjugation sends an operator to its inverse, the conjugating element sends its $\lambda$-eigenspace to the ${\lambda }^{-1}$-eigenspace.

Under the same hypotheses, $g(T^p)$ is reducible exactly when every coefficient of $g$ is a $p$th power in $K$. If $g=\sum a_i^p{T}^i$, thenConversely, if $g(T^p)$ is reducible, all its irreducible factors are separable. Its zero derivative forces every factor multiplicity to be divisible by $p$, so it is a $p$th power in $K[T]$ and all its coefficients are $p$th powers.

The inverse sine is one member of the inverse trigonometric functions.

A boundary point with $\dot{V}=0$ can still enter the sublevel set; LaSalle analysis decides whether it belongs to an invariant zero-derivative trajectory.

The Lindeberg replacement method, inspired by the Lindeberg condition, compares functions of two independent random vectors by replacing their coordinates one at a time. Matching moments cancel the corresponding terms in a Taylor expansion.

For prescribed values $F(x_i)$ at ordered points $x_0<\cdots <x_n$, the connect-the-dots interval map is the unique piecewise linear function obtained by linear interpolation between consecutive data points.

For closed intervals $J$ and $K$, write $J⟶K$ under $F$ when $F(J)\supseteq K$. The intermediate value theorem implies that there is a closed subinterval $L\subseteq J$ with $F(L)=K$.

A point $x$ is periodic for an interval map $F$ when $F^n(x)=x$ for some positive integer $n$. Its least such $n$ is its period, and the finite setis its periodic orbit.

For an interval map, this is the usual periodic point condition. Its least positive return time is the period.

For a smooth closed embedding $i:X\hookrightarrow Y$, the normal bundle fits into $0\to TX\to i^{\ast }TY\to N_{X/Y}\to 0$.

For a pure $k$-dimensional projective scheme $X\subseteq {\mathbb{P}}^m$, its degree is $k!$ times the leading coefficient of its Hilbert polynomial, equivalently ${\int }_X{H}^k$.

This extends degree of an algebraic variety to retain scheme multiplicities.

Chern classes are characteristic classes of vector bundles satisfying functoriality and the Whitney product formula $c(E)=c(E^{\prime })c(E^{\prime \prime })$ for a short exact sequence.