Young's inequality bounds a product by a sum of powers. It splits the product in a way that depends on the pair of exponents $p$ and $q$.

The group of units of a ring is the set of its units, with the multiplication of the ring. The definition of the group of units gives the details.

The area over $[a, b]$ is the area over $[a, c]$ plus the area over $[c, b]$. We can cut the interval at any point and add the two pieces.

A complex derivative must be the same in each direction. If you compare the derivative along the real axis with the derivative along the imaginary axis, you get the two equations. So the equations are a test with partial derivatives for a complex derivative. For example, $\bar z = x - i y$ has $u_x = 1$ and $v_y = -1$, so it fails the test.

A relation is antisymmetric when two different elements are never related in both directions.

Two elements $a$ and $b$ of a group $G$ are conjugate when $b = gag^{-1}$ for some $g \in G$. Conjugate elements play the same role in the group, up to a change of names given by $g$.

The graph of $e^x$ bends upward everywhere. Every chord of the graph lies above the graph.

The product of the two neighbours of a Fibonacci number differs from its square by exactly $1$. The sign alternates: $1 \cdot 3 - 2^2 = -1$, then $2 \cdot 5 - 3^2 = 1$, then $3 \cdot 8 - 5^2 = -1$.

The $n$th roots of unity sit on the unit circle at equal angles of $2\pi/n$, starting at $1$. For $n = 6$, they are the six corners of a regular hexagon.

The principal branch of a multivalued function is one branch of it that a standard convention chooses, see the definition. Its values are the principal values. An example is $\operatorname{Log} z = \ln |z| + i \operatorname{Arg} z$ with $\operatorname{Arg} z \in (-\pi, \pi]$.

A stochastic process has the Markov property when its future, given the present, is independent of its past. The node Markov property gives the precise definition.

To reach a point $c$ of $C$, first find a point of $B$ that $g$ sends to $c$. Then find a point of $A$ that $f$ sends to that point of $B$.

On a closed bounded interval, continuity at each point upgrades to uniform continuity. There is no room for the function to become steeper and steeper.

An Archimedean ordered field is an ordered field that has no infinitely large and no infinitely small positive elements. In such a field, the natural numbers are not bounded above. The fields $\mathbb{Q}$ and $\mathbb{R}$ are Archimedean.

A perfect number is a positive integer that is equal to the sum of its positive divisors other than itself. The definition of a perfect number gives the precise form.

A nilpotent endomorphism is a linear map from a space to itself that becomes zero after finitely many compositions with itself.

The moment-generating function of a real random variable $X$ is the function $t \mapsto E[e^{tX}]$. When it is finite near $0$, its derivatives at $0$ are the moments of $X$. The node moment-generating function gives the precise definition.

The determinant turns a product of matrices into a product of numbers. That is exactly what a homomorphism does. The matrices that it sends to $1$ form an important subgroup, the special linear group.

An invertible matrix is a square matrix that has a two-sided inverse for matrix multiplication.

Vieta's formulas say that the sum of the roots, the sum of their products in pairs, and so on, are given by the coefficients. You can know these sums without finding the roots.