Gelfond-Schneider Theorem Heine-Borel property
If a is a complex number that is also algebraic, and if b is an irrational number, then a b is a transcendental number.
generalized continuum hypothesis
The claim that there is no set of intermediate cardinality between any given set and its power set. The GCH is independent of (i.e., cannot be proved true or false by) the known axioms of set theory.
Cf. continuum hypothesis.
Topology: Given a topological surface S, the genus of S is the ordered pair (h, c) in which h is the number of handles on the surface and c is the number of cross-caps.
Graph Theory: The genus of a graph is the minimum genus of the topological surface on which the graph can be drawn without crossovers. Intuitively, the minumum number of handles one must attach to a sphere to draw the graph on it.
A curve on a space is a geodesic if, given any two points on the curve, the segment of the curve joining the points is the shortest such curve. In Euclidean space, geodesics are straight lines. On a sphere, the geodesics are the great circles of the sphere.
Graph Theory: A shortest path between two vertices is sometimes called a geodesic.
An infinite series such that the ratio of consecutive terms is constant. Such a series may be denoted
where r is the common ratio. A geometric series converges if and only if the common ratio is less than 1, in which case the sum of the series is given by
Cf. arithmetic series, harmonic series.
Abbreviation for greatest lower bound.
The foremost logician of the 20th century. He is famous for many deep results in the foundations of mathematics, including the Godel Completeness Theorem, the Godel Incompleteness Theorem, and that the generalized continuum hypothesis and the axiom of choice are consistent with the other axioms of set theory. He moved from Austria to the Institute for Advanced Study at Princeton, New Jersey, in 1940. He was a lifelong friend of Albert Einstein, and in the 1950’s formulated versions of Einstein’s relativity theories that permitted the logical possibility of time travel.
General: A picture or diagram representing one or more relationships among certain objects or quantities.
Functions: The collection of ordered tuples (x,y), consisting of domain elements x (possibly also tuples) and corresponding range elements y. Although informally we often mean the picture when we speak of the graph of a function, the graph should always be formally thought of as a set of tuples.
Graph Theory: A collection of nodes (also called vertices), together with a set of edges joining the nodes. Formally, a graph G is a set of vertices V, a set of edges E, and a symmetric relation between them specifying incidence. Every edge must be incident on exactly two (not necessarily distinct) vertices, called its ends. Two vertices incident on the same edge are called adjacent. Graphs are often represented pictorially, with the vertices being points (or small circles or squares or some such), and the edges being curves (often line segments), with the endpoints of each such curve being the ends of that edge in the graph.
Cf. Konigsberg Bridge Problem.
On a sphere, a great circle is a circle containing the endpoints of a diameter of the sphere, that is, a circle of maximal size. A geodesic on a sphere always lies on a great circle.
greatest common divisor
The greatest common divisor of two natural numbers a and b is the largest natural number c such that c divides a and c divides b.
greatest lower bound
A lower bound which is greater than or equal to every other lower bound.
Cf. Least upper bound.
Letters of the Greek alphabet are commonly used in mathematics. See the article for a full description.
A set together with a binary operation defined on its elements, satisfying
If the group operation is commutative, the group is called Abelian. If the condition that every element have an inverse is dropped, the set is called a monoid. If in addition the requirement that an identity element exist is dropped, the set is called a semigroup. Finally, if the condition that the group operation be associative is dropped, the set is called a groupoid.
- The group operation is associative,
- There is an identity element, i.e., an element e of the group so that for any other element a of the group we have ea = ae = a, and
- for every element a of the group there is an element a´, called the inverse of a, so that aa´ = a´a = e.
Cf. field, module, ring
A group isomorphism from a group to itself. That is, a bijective function from a group to itself that preserves the group operation, i.e. f(ab) = f(a)f(b) for all a, b.
Cf. group homomorphism, group isomorphism.
A function from one group to another that preserves the group operation, i.e., f(ab) = f(a)f(b) for all a, b.
Cf. group automorphism, group isomorphism.
A group homomorphism that is both “one-to-one” and “onto.” That is, a bijective function from one group to another that preserves the group operation, i.e. f(ab) = f(a)f(b) for all a, b.
Cf. group homomorphism, group automorphism.
The infinite series whose terms are the reciprocals of the natural numbers
This series diverges, i.e., the sum does not exist.
A topological space is Hausdorff if every pair of distinct points have disjoint neighborhoods. Two disjoint compact sets in a Hausdorff space have disjoint neighborhoods. A compact subset of a Hausdorff space is closed.
Heawood Map Coloring Theorem
For every positive integer n, the chromatic number of the orientable surface of genus n (sphere with n handles) is given by
For n = 0, this statement is exactly the Four Color Theorem, which was proven later, and does not bear Heawood's name.
Every closed and bounded subset of the real numbers is compact.