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Let $R$ be a Euclidean domain with Euclidean function $\delta$ and let $a, b \in R$. The Euclidean algorithm sets $r_0 = a$ and $r_1 = b$. While $r_k \neq 0$, it divides $r_{k-1}$ by $r_k$ and lets $r_{k+1}$ be the remainder. The values $\delta(r_k)$ decrease strictly, so the algorithm stops with some $r_{n+1} = 0$. The last nonzero remainder $r_n$ is a greatest common divisor of $a$ and $b$, if $b \neq 0$.

Let $D$ be an open subset of $\mathbb{R} \times \mathbb{R}^n$ and let $f : D \to \mathbb{R}^n$ be continuous. Assume that $f$ is locally Lipschitz in $x$. That is, each point of $D$ has a neighbourhood $V \subseteq D$ and a constant $K \ge 0$ such that $\|f(t, x) - f(t, y)\| \le K \|x - y\|$ whenever $(t, x), (t, y) \in V$. Let $I$ be an interval, and let $\gamma_1, \gamma_2 : I \to \mathbb{R}^n$ be two solutions of $x' = f(t, x)$ whose graphs lie in $D$. If $\gamma_1(t_0) = \gamma_2(t_0)$ for one $t_0 \in I$, then $\gamma_1(t) = \gamma_2(t)$ for all $t \in I$.

Conjunction is the logical operation "and". It joins two propositions into one proposition that is true only when both parts are true.

A first integral, or constant of motion, of a system of differential equations is a function of the state that stays constant along each solution. Energy, linear momentum and angular momentum are common examples in mechanics. See the definition.

Let $I$ be an interval and let $f \colon I \to \mathbb{R}$ be continuous, and differentiable at each interior point of $I$. If $f'(x) \ge 0$ for each interior point $x$ of $I$, then $f$ is increasing on $I$. That is, $f(x) \le f(y)$ for $x \le y$ in $I$.

Let $D \subseteq \mathbb{R}$ and let $f \colon D \to \mathbb{R}$ be uniformly continuous. Let $(x_n)$ be a Cauchy sequence in $D$. Then $(f(x_n))$ is a Cauchy sequence.

Let $V$ be an inner product space over $\R$ or $\C$, with the induced norm $\norm{\cdot}$. Then, for all $x, y \in V$, $$\norm{x + y}^2 + \norm{x - y}^2 = 2 \norm{x}^2 + 2 \norm{y}^2.$$

The flux of a vector field through an oriented surface measures how much of the field passes through the surface. It counts flow in the direction of the chosen normal. It is the surface integral of the normal component of the field. For a velocity field of a fluid, it is the volume of fluid that crosses the surface per unit time.

Let $G$ be a solvable group and let $\varphi \colon G \to K$ be a surjective group homomorphism. Then $K$ is solvable. In particular, $G/N$ is solvable for each normal subgroup $N$ of $G$.

Two elements of a commutative ring are associates when one is a unit times the other. The definition of associate elements states this precisely.

For all real numbers $x$ and $y$, $$\sin(x + y) = \sin x \cos y + \cos x \sin y,$$ $$\cos(x + y) = \cos x \cos y - \sin x \sin y.$$

There are infinitely many prime numbers. Equivalently, for each natural number $n$ there is a prime number $p$ with $p \ge n$.

A vector field on a subset of $\mathbb{R}^n$ assigns a vector of $\mathbb{R}^n$ to each point of the subset. It models quantities with a direction and a size at each point, such as the velocity of a fluid or a force field.

Negation is the logical operation that changes a proposition into its opposite. The negation of a true proposition is false, and the negation of a false proposition is true.

Let $c \in \mathbb{C}$, let $R \in (0, \infty]$, and let $D = \{z : |z - c| < R\}$ be the open disc with center $c$ and radius $R$. Let $f : D \to \mathbb{C}$ be holomorphic on $D$. Then there is a holomorphic function $F : D \to \mathbb{C}$ with $F'(z) = f(z)$ for each $z \in D$.

A set of points is convex if it contains the whole segment between each two of its points. A solid ball and a solid cube are convex; a crescent is not.

The additive group of integers $(\mathbb{Z}, +)$ is an infinite cyclic group. The element $1$ generates it, because each integer is a sum of copies of $1$ or of $-1$. Its only generators are $1$ and $-1$.

An open set is a member of a topology. In a metric space, a set is open when each of its points has a small ball around it that stays in the set.

Let $f = a_n x^n + a_{n-1} x^{n-1} + \dots + a_0 \in \mathbb{Z}[x]$ with $n \ge 1$, and let $p$ be a prime number with these three properties.

The cardinality of a finite set is the number of elements of the set. It is the basic quantity that all counting results compute. The node cardinality of a finite set gives the precise definition.

Let $U \subseteq \mathbb{R}^n$ be open, let $f \colon U \to \mathbb{R}^m$ be differentiable on $U$, and let $x, y \in U$ be points such that the segment $[x, y] = \{x + t (y - x) : 0 \le t \le 1\}$ lies in $U$. Suppose that $M \ge 0$ and $\|Df(z)\| \le M$ for each $z \in [x, y]$, where $\|Df(z)\| = \sup_{\|h\| \le 1} \|Df(z) h\|$ is the operator norm. Then $$\|f(y) - f(x)\| \le M \, \|y - x\|.$$

The sign of a permutation is $+1$ or $-1$, according to the parity of the number of transpositions in a decomposition of the permutation. The sign map is a homomorphism from $S_n$ to $\{1, -1\}$.

The power of a point with respect to a circle is the square of its distance to the centre minus the square of the radius. It is positive outside the circle, zero on the circle and negative inside.

Let $(X, d)$ be a metric space. Then the set of all open balls $B(x, r)$, with $x \in X$ and real $r > 0$, is a basis of the metric topology of $X$. Also, the balls $B(x, 1/n)$ with $x \in X$ and integer $n \ge 1$ form a basis.

The prime factorization of a positive integer writes it as a product of prime numbers. The problem of integer factorization is to find this product for a given integer. The definition of a prime factorization gives the precise form.

Let $F$ be a field, let $A$ be an $m \times n$ matrix and $B$ an $n \times p$ matrix over $F$. Then $$\rank(AB) \le \min(\rank A, \rank B),$$ where $\rank$ is the rank of a matrix.

A group action is transitive when it has exactly one orbit. Then each point can be moved onto each other point by some element of the group.

Let $T(t)$ be the temperature of a body at time $t$. Let $T_a$ be the constant temperature of the surroundings, and let $k > 0$ be a constant. Newton's law of cooling is the equation $$T'(t) = -k \left(T(t) - T_a\right).$$ With the initial condition $T(0) = T_0$, its unique solution is $$T(t) = T_a + (T_0 - T_a)\, e^{-k t}.$$

Let $c \in \mathbb{C}$, let $r > 0$, and let $f$ be holomorphic on the punctured disc $0 < |z - c| < r$. Assume that $c$ is an essential singularity of $f$. Then for each $\delta$ with $0 < \delta < r$, the image $f(\{z : 0 < |z - c| < \delta\})$ is dense in $\mathbb{C}$. Thus for each $w \in \mathbb{C}$ there is a sequence $z_n \to c$ with $z_n \neq c$ and $f(z_n) \to w$.

Let $R$ be a ring and let $I$ be an ideal of $R$ with $I \neq R$. Then there is a maximal ideal $M$ of $R$ with $I \subseteq M$. In particular, every nonzero ring has a maximal ideal.

An isometry is a map between metric spaces that keeps all distances.

A nullcline of a system of autonomous equations $x_j' = f_j(x)$ is a set on which one component $f_j$ of the vector field vanishes. The equilibrium points of the system are the common points of all the nullclines. See the definition.

Let $X$ be a real random variable with finite mean $\mu$ and finite variance $\sigma^2$. Then for each $a > 0$, $$P(|X - \mu| \ge a) \le \frac{\sigma^2}{a^2}.$$ Equivalently, if $\sigma > 0$, then $P(|X - \mu| \ge k\sigma) \le 1/k^2$ for each $k > 0$.

Let $X$ and $Y$ be metric spaces and let $f \colon X \to Y$ be an isometry. Then $f$ is injective.

A monotone sequence is a real sequence that never decreases or never increases. A bounded monotone sequence always converges. The node monotone sequence gives the precise definition.

Let $X_1, \dots, X_n$ be independent real random variables, and let $S = X_1 + \dots + X_n$. Then for each $t \in \mathbb{R}$, the characteristic functions satisfy $$\varphi_S(t) = \varphi_{X_1}(t) \, \varphi_{X_2}(t) \cdots \varphi_{X_n}(t).$$

For every real $x$, the series $\sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{(2n+1)!}$ converges absolutely, and $$\sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots.$$ This series is the Maclaurin series of the sine function.

To evaluate a polynomial at an element, you replace the variable by the element and compute the result in the ring. The definition of evaluation gives the formula.

The discrete uniform distribution on a finite set gives the same probability to each element of the set. The node discrete uniform distribution gives the precise definition.

For each $z \in \mathbb{C}$, $$e^{z + 2 \pi i} = e^{z},$$ where $e^z$ is the complex exponential of $z$.