The norm induced by an inner product is the square root of the inner product of a vector with itself.

A finitely generated module is a module in which every element is a linear combination of a fixed finite set of elements.

A curve that almost solves the equation stays close to a true solution. The error has two sources: the difference of initial values and the defects $\varepsilon_i$. Both can grow at most exponentially in time.

An infinite set is a set that is not finite.

A set of numbers is bounded above when one number is greater than or equal to every element of the set. Such a number is an upper bound of the set. The same idea applies to a subset of any ordered set.

For a very large $x$, it is almost sure that $X \le x$, because $X$ is some real number. So the probability $F_X(x)$ gets as close to $1$ as we like.

If the slope is never negative, the graph never goes down. It may stay flat on some parts.

The number $\varepsilon$ is the allowed error, and $N$ is the point after which the error is always smaller than $\varepsilon$. For $a_n = 1/(n+1)$ and $L = 0$, the choice $N > 1/\varepsilon$ works.

The abelianisation of a group $G$ is the quotient $G/[G, G]$ by its commutator subgroup. It is the largest abelian quotient of $G$.

A binary operation is associative when the grouping of three elements does not change the result. Then the parentheses in a product of several elements can be omitted.

A ring isomorphism is a bijective ring homomorphism. Two rings are isomorphic when there is a ring isomorphism between them. The definition of a ring isomorphism gives the details.

The gcd of two elements is a combination of them with coefficients in the ring. For example, $\gcd(4, 6) = 2 = 2 \cdot 4 - 1 \cdot 6$. This is the main tool to solve linear equations $s a + t b = c$ in a Euclidean domain.

To compare two sequences that grow without bound, it is enough to compare their steps. If each step of $a_n$ is about $L$ times the step of $b_n$, then $a_n$ is about $L$ times $b_n$ in the long run.

The triple integral of a function of three variables over a solid region is a limit of Riemann sums. These sums use partitions of the region into small boxes. It extends the double integral to three dimensions, and it computes volumes, masses and centres of mass of solids.

The domain of a function is the set of its inputs.

A solution of a linear recurrence is a sequence that satisfies the recurrence for all indices. The solutions form a module, and they are determined by their first terms. The node solution of a linear recurrence gives the precise definition.

Let $f$ be a differentiable real-valued function on an open subset of $\mathbb{R}^n$. Its gradient at a point $a$ is the vector $\nabla f(a)$ of its partial derivatives at $a$. It points in the direction in which $f$ increases fastest, and its length is that fastest rate of increase. The derivative of $f$ at $a$ is the dot product with the gradient: $Df(a)(v) = \nabla f(a) \cdot v$.

Think of a line of dominoes that never ends. If the first domino falls, and each falling domino knocks over the next one, then every domino falls. The base case is the first domino, and the induction step is the rule that each domino knocks over the next one.

Maximum likelihood chooses the parameter that makes the observed data most probable. If a coin shows $7$ heads in $10$ tosses, the maximum likelihood estimate of the probability of heads is $7/10$.

A minimum spanning tree of a weighted connected graph is a spanning tree of least total weight. It is a classical problem of combinatorial optimization, solved by greedy algorithms. The node minimum spanning tree gives the precise definition.