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abacus – algebra of functions
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Chinese abacus
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abacus
An ancient calculator consisting of a frame, rods, and beads. A horizontal beam separates the frame into an upper deck and a lower deck, and a place notation is used to represent numbers and calculations by sliding the beads into various positions on the two decks. The abacus illustrated is the Chinese or "2/5" abacus (having 2 beads on each rod on the upper deck and 5 beads on each rod on the lower deck); the Japanese abacus is "1/4".
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Neils Abel
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Abel, Henrik Neils
(born 1802)
Norwegian algebraist who proved the unsolvability of the general quintic (fifth degree) equation by radicals. He published this landmark result when he was 22, but had great difficulty in getting contemporary establishment mathematicians (including Gauss and Cauchy) to notice his work. Suffering from tuberculosis, he returned to Norway from Paris, but continued to publish papers in Crelle’s Journal. Just when the mathematical world was beginning to wake up to his achievements, when he was 26 years old, Abel succumbed to his illness.
Abelian group
A group in which the group operation is commutative.
abscissa
The first element of an ordered pair. Compare: ordinate.
absolute geometry
Euclidean geometry without the parallel postulate. Also called "neutral geometry."
absolutely continuous
Of measures: Given a measurable space (X, M) and two signed measures m and n on X, then n is said to be absolutely continuous with respect to m, denoted by
if for any E in M such that m(E) = 0 we have n(E) = 0.
Of functions: If f is a function with domain the set of real numbers and range the set of complex numbers, then f is absolutely continuous if for any positive e there is some d greater than zero such that for any finite set of disjoint intervals we have:
It is a theorem that f is absolutely continuous if and only if its derivative is integrable and the fundamental theorem of calculus holds for f.
abundant number
A natural number whose distinct divisors, including 1 but not including itself, have a sum greater than the number. For example, the proper factors of 12 are 1, 2, 3, 4, and 6, which sum to 16, so 12 is abundant. A number whose proper factors sum to less than the number is called deficient, and one whose factors sum to precisely the number is called perfect.
Cf. perfect number.
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.
acos
Abbreviation of arc-cosine.
actual infinite
An infinite set considered as a completed whole, e.g., the natural numbers in modern set theory. Aristotle distinguished the actually infinite from the potential infinite, and considered that only the latter was permissible to thought.
Related MiniText: Infinity -- You Can't Get There From Here...
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."
algebraically closed
A field F is algebraically closed if every polynomial with coefficients in F has a root in F. Neither the field of rational numbers nor the field of real numbers is algebraically closed, but the field of complex numbers is algebraically closed.
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 lattice
A lattice L is said to be algebraic if every element of L is a join of compact elements of L.
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.
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