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A standard mollifier is a nonnegative test function $\rho$ with integral one. The rescaling ${\rho }_{\epsilon }(x)={\epsilon }^{-n}\rho (x/\epsilon )$ forms an approximation to the identity, and $u\ast {\rho }_{\epsilon }$ is smooth wherever the convolution stays inside the domain.

For a distribution $u$ and a multi-index $\alpha$, its distributional derivative is defined by

A multi-index is a tuple of nonnegative integers used to abbreviate products and partial derivatives: $|\alpha |={\sum }_j{\alpha }_j$, $x^{\alpha }={\prod }_j{x}_j^{{\alpha }_j}$, and $D^{\alpha }={\prod }_j{D}_j^{{\alpha }_j}$.

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.

Let $A,B,Z$ be pairwise disjoint sets of vertices in a directed acyclic graph. A graph path, traversed without requiring its edges to point in the direction of travel, is active given $Z$ precisely when every interior noncollider lies outside $Z$ and every collider either belongs to $Z$ or has a descendant in a directed graph in $Z$. The sets $A,B$ are D-separated by $Z$ when no active graph path connects them.^[ref]

In the three-vertex chain $X\to M\to Y$ and fork $X\leftarrow M\to Y$, conditioning on $M$ blocks the only graph path. In $X\to M\leftarrow Y$, the collider blocks the graph path before conditioning; conditioning on $M$ or its descendant in a directed graph activates it. Other vertices can still block a longer graph path.

Under the global Markov property of a directed acyclic graph, D-separation implies conditional independence. The reverse implication for a particular probability distribution requires faithfulness of a directed acyclic graph: additional conditional independences can arise through special parameter choices.

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 writes a real number as $[a_0;a_1,a_2,\dots ]=a_0+1/(a_1+1/(a_2+\cdots ))$, with integral $a_0$ and positive integral $a_i$ for $i\ge 1$. Repeatedly taking the integer part and inverting the remainder constructs the expansion.

For example, $17/7=[2;2,3]$. Rational inputs terminate; irrational inputs give infinite expansions. Truncations are continued fraction convergents, whose numerator and denominator satisfy short recurrences and provide particularly effective best rational approximations. The expansion of a quadratic irrational number is eventually periodic, linking these approximations with Pell equations.

A linear form in logarithms of algebraic numbers combines chosen complex logarithms of nonzero algebraic numbers:The coefficients ${\beta }_j$ may be algebraic numbers; the homogeneous integer-coefficient case has ${\beta }_0=0$ and ${\beta }_j=b_j\in \mathbb{Z}$. Each logarithm branch must be fixed, because changing it adds a multiple of $2\pi i$.

When all ${\alpha }_j$ are positive real and real logarithms are used, the homogeneous form is the logarithm of ${\prod }_j{\alpha }_j^{b_j}$. Thus a small nonzero linear form in logarithms of algebraic numbers expresses an unexpectedly close multiplicative relation. For instance,Baker's theorem proves qualitative nonvanishing and transcendence statements, while the general lower bound for a linear form in logarithms gives quantitative separation from zero. Comparing such a lower bound with a small upper bound supplied by a Diophantine equation can bound its exponents, as in the exponent bound for a binomial power equation.

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 $\alpha$.