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A rough path augments a path with compatible higher-order iterated-integral data. The extra data records effects, such as signed area, that are not continuously determined by the first-level path in the uniform topology. A rough differential equation uses this enhancement to recover a continuous solution map for irregular drivers.

The Faraday constant is the electric charge associated with one mole of elementary charges, $F=N_{\mathrm{A}}e\approx 96485.33212\mathrm{C}\mathrm{mo}{\mathrm{l}}^{-1}$. The product of the Avogadro constant and elementary charge is exact under the SI definitions; the displayed decimal is rounded. For ions with signed charge number $z$, the molar electrical contribution to an electrochemical potential is $zF\psi$, where $\psi$ is electric potential. ^[ref].

The Avogadro constant converts amount of substance into a dimensionless entity count: $N=n{N}_{\mathrm{A}}$. Its value is exact under the SI definition of the mole. ^[ref]

Online text: telescope-optics.net/Seidel_aberrations.htm

Cited passages: Amateur Telescope Optics, §3.5.1, opening paragraph: historical classification and equation 5.

Online text: www.telescope-optics.net/coma.htm

Cited passages: Amateur Telescope Optics, §4.2, Figure 41: geometrical spots versus diffraction images; Amateur Telescope Optics, §4.2, classical coma formula after equation 15 and equation 13: pupil-cubic and field-linear dependence; Amateur Telescope Optics, §4.2, opening paragraphs and Figure 39: field coma versus misalignment coma.

For fixed-length messages in ${\mathbb{F}}_q^L$, choose independent uniform field elements $a,b$. Each output is uniform because $b$ masks it. For distinct messages and prescribed output difference $d$, the equation ${\sum }_{j=1}^L(m_j^{\prime }-m_j)a^j=d$ is a nonzero polynomial equation of degree at most $L$, so the root bound for a polynomial gives at most $L$ choices of $a$. The joint output probability is at most $L/q^2$. Thus the family is $(L/q)$-almost strongly universal when $L<q$. Starting the message polynomial at power one matters: using a freely varying message coefficient at power zero would permit a known constant tag shift after observing one tag.

The allele frequency is the proportion of gene copies at a specified genetic locus carrying an allele. Two unrelated copies can have the same observed allele without being identical by descent; under independent sampling their matching probability is the sum of squared allele frequencies. Pedigree autozygosity instead traces copies to the same ancestral copy.

Online text: raw.githubusercontent.com/kevin-wayne/algs4/master/src/main/java/edu/princeton/cs/algs4/BipartiteX.java

Cited passages: BipartiteX.java, class documentation: breadth-first search, linear construction time and vertex-linear extra space; bfs method: odd-cycle reconstruction; BipartiteX.java, constructor component loop and bfs method: discovery colours, parent pointers and same-colour conflict.

A statement true whenever execution reaches a chosen point in a loop, usually immediately before its condition is tested. Establish it initially, preserve it through each iteration, and combine it with the exit condition to establish algorithm correctness. Termination needs a separate argument.^[ref]

Conformance of an implementation's behaviour to its specified functional requirements. The exact obligations come from the formal specification: correct delivery of an IPC payload and correct computation of a file read are different requirements. Neither requirement by itself supplies a numerical response-time bound.

Establishing specified properties of a system using formal proofs about a precise model. Formal verification connects an implementation to a formal specification; correctness of the model and its assumptions still matters when applying the result to a physical system.

The tensor product of fields $L{\otimes }_{K}M$ is a $K$-algebra, not necessarily a field. If $L$ and $M$ sit in one overfield, the map $l\otimes m\mapsto lm$ has an image that generates their compositum as a field. When the extensions are algebraic, this image is already a field. Injectivity of this multiplication map is equivalent to linear disjointness.

The eight Gell-Mann matrices are traceless Hermitian $3\times 3$ matrices forming a conventional basis for the Lie algebra $\mathfrak{su}(3)$ after multiplication by $i$.

The Pauli matrices ${\sigma }_x,{\sigma }_y,{\sigma }_z$ are traceless Hermitian $2\times 2$ matrices that, together with the identity, form a basis of the complex $2\times 2$ matrices.

For integers $a,b$ not both zero, there are integers $x,y$ such that $ax+by=\gcd (a,b)$. The extended Euclidean algorithm constructs them. For example, $4\cdot 252-5\cdot 198=18$ and $7\cdot 5-2\cdot 17=1$; the second identity also gives the modular inverse of $5$ modulo $17$.

The identity converts a greatest common divisor into a solvability certificate. The linear Diophantine equation $ax+by=c$ is solvable in integers precisely when $\gcd (a,b)$ divides $c$. Every common divisor divides any integer linear combination; conversely, multiplying a Bézout identity by $c/\gcd (a,b)$ constructs a solution when that quotient is integral. The coefficients generally are not unique, as the description of all solutions of a linear Diophantine equation makes explicit.