BROWSE
ALPHABETICALLY
LEVEL:
Elementary
Advanced
Both
INCLUDE TOPICS:
Basic Math
Algebra
Analysis
Biography
Calculus
Comp Sci
Discrete
Economics
Foundations
Geometry
Graph Thry
History
Number Thry
Phys Sci
Statistics
Topology
Trigonometry
|
|
accumulation point – associative property
accumulation point
Given a set X, an accumulation point of X is a point p (which may or may not be in X) such that every neighborhood of p contains points of X different from p. See also: limit point, perfect set.
acute
An angle is called acute if it is less than a right angle, that is, if its measure is less than 90° (p/2 radians). A triangle is called acute if all of its angles are acute.
Cf. obtuse.
add
To perform the operation of addition on numbers or quantities to obtain a total value. See addition.
addition
A binary operation on numbers or quantities to obtain a total value, called the sum. For natural numbers this operation is defined recursively by the Peano axioms (i) a + 0 = a, and (ii) a + Sb = S(a + b), where Sa is the successor of a. For sets, addition is defined by the cardinality of the disjoint union. Addition on other kinds of numbers or structures is typically defined as an extension of these operations.
adjacent
Two vertices of a graph are said to be adjacent when they have an edge that is incident on both of them. Also occasionally used of edges that are incident on at least one vertex in common.
a.e.
See: almost everywhere.
algebra
The term algebra has broadened enormously in modern mathematics from its original meaning as the abstract study of the laws of arithmetic, in which letters or other symbols are used in place of specific numbers in equations or other arithmetic statements. The term algebra should now be understood to denote any set of objects together with a collection of finitary operations defined on it. Thus, we may have an algebra of sets, an algebra of numbers, an algebra of functions, and so on. See the following entries for many particular uses of the words "algebra" and "algebaic."
algebraic function
A function which operates on its variable(s) by addition, subtraction, multiplication, division, raising to a power, or extraction of roots. Compare: transcendental function.
algebraic number
A real or complex number which is the root of a polynomial with rational coefficients. Numbers that cannot be so expressed are called transcendental numbers. The algebraic numbers are countable.
algebra of functions
Given a topological space X, an algebra of functions on X is a subset A of the space of all continuous functions on X such that for all functions f and g in X we have:- fg is in A,
- f + g is in A, and
- for all constants c, cf is in A
whenever f and g are in A.
algebra of sets
Given a set X, an algebra on X is a collection of subsets of X which is closed under finite unions and complements. Such an algebra always includes the empty set and X itself. An algebra may also be defined as a ring of sets which includes X. An algebra of sets which is closed under countable unions is called a s-algebra.
almost everywhere
Given a measure m on a measure space X, a condition (e.g., continuity of functions, etc.) is said to hold almost everwhere if it holds on X – N, i.e., on the set difference of X and N, where N is a null set. This is often abbreviated by "a.e."
alternating series test
ARTICLE
A test for the convergence of a series. See the article for a complete description.
angle
A figure formed by two line segments that extend from a common point. Also refers to the measure of the angle, in degrees or radians, indicating the amount by which one of the line segments must be rotated about the common point to make it coincide with the other segment. The angle between two planes is the angle between two intersecting lines, one lying in each plane and perpendicular to the line of intersection of the planes. Angles are frequently denoted by lower-case Greek letters.
Cf. acute, obtuse, right angle.
antitone function
A function that is order preserving and decreasing.
anxiety, math
A fear or emotional dislike of mathematics, characterized by difficulty learning or mastering mathematical techniques or concepts, poor performance on exams despite careful preparation, etc.
Related MiniText: Coping With Math Anxiety
arc
Graph Theory: Another name for a directed edge.
arithmetic
The mathematical theory of addition, subtraction, multiplication, and division of integers. Formally, arithmetic is usually axiomatized by the so-called Peano axioms, and the theory is then often referred to as PA (for “Peano arithmetic”).
Ascoli-Arzelŕ Theorem
If {X} is a compact Hausdorff space and F is an equicontinuous, pointwise bounded subset of the space C(X) of continuous functions on X, then F is totally bounded in the uniform metric and the closure of F in C(X) is compact. Also, if X is a s-compact, locally compact Hausdorff space and if (fn} is an equicontinuous, pointwise bounded sequence in C(X), then there exists a subsequence and an f in C(X) such that the subsequence converges to f uniformly on compact sets.
associative
An operation “ · ” on a set A is associative if for all a, b, and c in A, (a · b) · c = a · (b · c).
Cf. commutative, distributive.
associative property
A property of numbers which states that the operations of addition and multiplication are associative.
Cf. commutative, distributive.
|
|
 
|