Diophantine equation ellipse
A polynomial equation with integer coefficients. (Named after the 3rd century Greek mathematician Diophantus of Alexandria.)
Cf. Hilbert's Problems (the tenth problem), Fermat's Last Theorem.
A graph whose edges are directed, i.e. have distinguished ends. One end of every directed edge is called the head and the other is called the tail, and the edge is said to be from the tail to the head. In pictorial representations of graphs, directed edges are drawn to end with arrows, pointing to the head. The i, j entry in the adjacency matrix of a directed graph is the number of edges from vertex i to vertex j.
General: So-called “Discrete Mathematics” consists of those branches of mathematics which are concerned with the relations among fixed rather than continuously varying quantities, e.g., combinatorics and probability.
Topology: A topology on a set X is discrete if every subset of X is open, or equivalently if every one-point set of X is open.
Two sets are disjoint if they have empty intersection.
A union of sets which are disjoint.
A set of points consisting of a circle together with its interior points. The set consisting only of the interior points of a circle is called an open disk.
The distance between two points in a space is given by the length of the geodesic joining those two points. In Euclidean space, the geodesic is given by a straight line, and the distance between two points is the length of this line. The distance between two points a and b on a real number line is the absolute value of their difference, i.e., d(a, b) = |a - b|. In two (or more) dimensions, the distance is given by the (generalized) Pythagorean theorem, i.e., in a Cartesian coordinate system of n dimensions, where a = (a1, ... ,an) and b = (b1, ... ,bn), the distance d(a, b) is given by
The concept of distance may be generalized to more abstract spaces – such a distance concept is referred to as a metric.
Graph Theory: The length of the shortest path between two vertices of a graph. If there is no path between two vertices, their distance is defined to be infinite. The distance between two vertices v and u is denoted by d(v, u). In a connected graph, distance is a metric.
See distributive property.
A lattice is called distributive if for all elements x, y, and z of the lattice we have x (y z) = (x y) (x z) and x (y z) = (x y) (x z).
An algebraic property of numbers which states that for all numbers a, b, and c, a(b + c) = ab + ac.
Cf. commutative, associative.
To divide a number a by another number b is to find a third number c such that the product of b and c is a, that is, b × c = a. The number a is called the dividend, the number b is called the divisor, and the number c is called the quotient. The operation of dividing may be denoted by a horizontal or diagonal slash separating the dividend and divisor (with the dividend on top), or by a horizontal dash with a dot above and below it placed between the dividend and divisor.
In the case of whole numbers a and b there may not be a whole number quotient; however, there are always unique whole numbers q and r such that a = b × q + r, with r < b. In this case q is called the quotient and r is called the remainder. If in a particular case r = 0, we say that b divides a, and this is often denoted by b|a.
A number that is being divided.
A number that is dividing another.
A polyhedron having twelve faces.
The faces of a regular dodecahedron are regular pentagons.
Cf. Platonic solid.
Dodgson, Charles Lutwidge
Oxford mathematician most famous for the books Alice in Wonderland and Through the Looking Glass, which he wrote under the pseudonym Lewis Carroll. However, he also wrote several mathematics textbooks, and delighted in inventing bizarre and humorous syllogisms for exercises in Aristotelian logic. These syllogisms commonly appear in introductory texts on logic to this day. He also formulated what is now called Carroll's Paradox, which shows the need for formal rules of inference in any system of logic.
General: A universe of discourse, that is, the class of objects under consideration. Functions and relations: The domain of a function (relation) is the set of elements which the function (relation) maps to its range set.
See scalar product.
See Euler number.
General: A line formed by the intersection of two planes. In a 3-dimensional figure (such as a polyhedron), the line or curve where two faces or surfaces meet.
Graph Theory: One of two kinds of entities in a graph. Restricted to being incident on exactly two vertices.
The set of edges of some graph. For a graph G, the edge set of G is denoted by E(G), or, if there is no ambiguity as to the graph in question, simply by E.
The locus of points in the plane, the sum of whose distances from two fixed points, called the foci, remains constant.
Like the hyperbola and parabola, the ellipse is a conic section.