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2. Absolute value (algebra) - Wikipedia

   en.wikipedia.org/wiki/Absolute_value_(algebra)

   The trivial value is the only possible absolute value on a finite field because any non-zero element can be raised to some power to yield 1. If an absolute value satisfies the stronger property |x + y| ≤ max(|x|, |y|) for all x and y, then |x| is called an ultrametric or non-Archimedean absolute value, and otherwise an Archimedean absolute value.

3. Triviality (mathematics) - Wikipedia

   en.wikipedia.org/wiki/Triviality_(mathematics)

   Triviality (mathematics) In mathematics, the adjective trivial is often used to refer to a claim or a case which can be readily obtained from context, or an object which possesses a simple structure (e.g., groups, topological spaces). [1][2] The noun triviality usually refers to a simple technical aspect of some proof or definition.

4. Absolute value - Wikipedia

   en.wikipedia.org/wiki/Absolute_value

   The absolute value of a number may be thought of as its distance from zero. In mathematics, the absolute value or modulus of a real number , denoted , is the non-negative value of without regard to its sign. Namely, if is a positive number, and if is negative (in which case negating makes positive), and . For example, the absolute value of 3 is ...

5. Archimedean property - Wikipedia

   en.wikipedia.org/wiki/Archimedean_property

   By Ostrowski's theorem, every non-trivial absolute value on the rational numbers is equivalent to either the usual absolute value or some -adic absolute value. The rational field is not complete with respect to non-trivial absolute values; with respect to the trivial absolute value, the rational field is a discrete topological space, so complete.

6. Quotient group - Wikipedia

   en.wikipedia.org/wiki/Quotient_group

   Quotient group. A quotient group or factor group is a mathematical group obtained by aggregating similar elements of a larger group using an equivalence relation that preserves some of the group structure (the rest of the structure is "factored out"). For example, the cyclic group of addition modulo n can be obtained from the group of integers ...

7. Homogeneous function - Wikipedia

   en.wikipedia.org/wiki/Homogeneous_function

   Homogeneous function. In mathematics, a homogeneous function is a function of several variables such that the following holds: If each of the function's arguments is multiplied by the same scalar, then the function's value is multiplied by some power of this scalar; the power is called the degree of homogeneity, or simply the degree.

8. Eigenvalues and eigenvectors - Wikipedia

   en.wikipedia.org/wiki/Eigenvalues_and_eigenvectors

   In linear algebra, an eigenvector (/ ˈaɪɡən -/ EYE-gən-) or characteristic vector is a vector that has its direction unchanged by a given linear transformation. More precisely, an eigenvector, , of a linear transformation, , is scaled by a constant factor, , when the linear transformation is applied to it: . It is often important to know ...

9. Norm (mathematics) - Wikipedia

   en.wikipedia.org/wiki/Norm_(mathematics)

   In mathematics, a norm is a function from a real or complex vector space to the non-negative real numbers that behaves in certain ways like the distance from the origin: it commutes with scaling, obeys a form of the triangle inequality, and is zero only at the origin. In particular, the Euclidean distance in a Euclidean space is defined by a ...