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The Dirac delta is the distribution defined by $⟨\delta ,\phi ⟩=\phi (0)$.

A distributional identity is an equality that holds after both sides act on every smooth compactly supported test function.

A test function is a smooth function with compact support. Weak and distributional equations are identities obtained by integrating against every test function.

Taylor dispersion is enhanced longitudinal spreading produced when transverse molecular diffusion repeatedly moves solute between streamlines of different velocity.

The Péclet number compares advection over length $L$ at speed $U$ with molecular diffusion of diffusivity $D$.

Charge conjugation exchanges particles with antiparticles. For a matrix-valued gauge potential it acts as $A_{\mu }\mapsto -A_{\mu }^T$.

A period-doubling bifurcation occurs when a fixed-point multiplier crosses $-1$ and a nearby period-two orbit is created. Under the generic nondegeneracy conditions, the orbit amplitude is proportional to the square root of the parameter displacement.

For a period-$k$ orbit $x_0,\dots ,x_{k-1}$ of a differentiable map, the multiplier isThe orbit is locally asymptotically stable when the modulus of this product is less than one.

The Jury criterion tests whether every root of a real discrete-time characteristic polynomial lies strictly inside the unit circle. For $z^2+az+b$, this is equivalent to $|b|<1$, $1+a+b>0$, and $1-a+b>0$.

An interval map is a continuous function $F:I\to I$ from a real interval to itself. Its iterates form a one-dimensional discrete dynamical system.

D-separation blocks paths through conditioned noncolliders and through unconditioned colliders without conditioned descendants.

A topological ordering is a topological order: a list containing every vertex once and placing the source of every directed edge before its destination. In the parent-and-child language of a Directed acyclic graph, every parent precedes every child, and hence every ancestor precedes every descendant along a directed path. This is an ordering of the vertices, whereas topological sorting is the task of finding it. See topological order for the precise position condition and Kahn's algorithm for a construction.

If an irrational algebraic number has degree $d$, then $|\alpha -p/q|>C(\alpha )q^{-d}$. Approximations of arbitrarily larger polynomial order therefore prove that a Liouville number is transcendental.

A simple continued fraction recursively expands a real number by taking integer parts and reciprocals. Its convergents are exceptionally good rational approximations.

Baker's theorem gives transcendence and quantitative nonvanishing results for such logarithmic forms.

A homogeneous linear form in logarithms of nonzero algebraic numbers has the form

Let $L_1,\dots ,L_n$ be linearly independent linear forms in $n$ variables with algebraic coefficients. For every $\epsilon >0$, the nonzero $\mathbf{x}\in {\mathbb{Z}}^n$ satisfyinglie in finitely many proper linear subspaces of ${\mathbb{Q}}^n$. There are variants involving a finite set of places of a number field.

A reduced fraction $p/q$ is a best rational approximation of the first kind to a real number $\alpha$ when no rational number with denominator at most $q$ is closer to $\alpha$.

For a smooth function $f(x,y)$,where the smooth remainder is

The Grassmannian $\mathrm{Gr}(k,V)$ parametrizes the $k$-dimensional linear subspaces of a finite-dimensional vector space $V$. Its Plücker embedding makes it a projective algebraic variety.

The real symplectic group consists of matrices $A$ satisfying $A^{T}JA=J$. It is an embedded matrix Lie group of dimension $2n^2+n$ in size $2n$.