Jordan normal form

In linear algebra, a Jordan normal form (often called Jordan canonical form) of a linear operator on a finite-dimensional vector space is an upper triangular matrix of a particular form called a Jordan matrix, representing the operator with respect to some basis. Such a matrix has each non-zero off-diagonal entry equal to 1, immediately above the main diagonal (on the superdiagonal), and with identical diagonal entries to the left and below them.

If the vector space is over a field K, then a basis with respect to which the matrix has the required form exists if and only if all eigenvalues of the matrix lie in K, or equivalently if the characteristic polynomial of the operator splits into linear factors over K. This condition is always satisfied if K is the field of complex numbers. The diagonal entries of the normal form are the eigenvalues (of the operator), and the number of times each eigenvalue occurs is called the algebraic multiplicity of the eigenvalue.

If the operator is originally given by a square matrix M, then its Jordan normal form is also called the Jordan normal form of M. Any square matrix has a Jordan normal form if the field of coefficients is extended to one containing all the eigenvalues of the matrix. In spite of its name, the normal form for a given M is not entirely unique, as it is a block diagonal matrix formed of Jordan blocks, the order of which is not fixed; it is conventional to group blocks for the same eigenvalue together, but no ordering is imposed among the eigenvalues, nor among the blocks for a given eigenvalue, although the latter could for instance be ordered by weakly decreasing size.

Normal form

Normal form may refer to:

In formal language theory:

In logic:

In lambda calculus:

See also

Set (music)

A set (pitch set, pitch-class set, set class, set form, set genus, pitch collection) in music theory, as in mathematics and general parlance, is a collection of objects. In musical contexts the term is traditionally applied most often to collections of pitches or pitch-classes, but theorists have extended its use to other types of musical entities, so that one may speak of sets of durations or timbres, for example.

A set by itself does not necessarily possess any additional structure, such as an ordering. Nevertheless, it is often musically important to consider sets that are equipped with an order relation (called segments); in such contexts, bare sets are often referred to as "unordered", for the sake of emphasis.

Two-element sets are called dyads, three-element sets trichords (occasionally "triads", though this is easily confused with the traditional meaning of the word triad). Sets of higher cardinalities are called tetrachords (or tetrads), pentachords (or pentads), hexachords (or hexads), heptachords (heptads or, sometimes, mixing Latin and Greek roots, "septachords"—e.g.,), octachords (octads), nonachords (nonads), decachords (decads), undecachords, and, finally, the dodecachord.

Normal form (bifurcation theory)

In mathematics, the normal form of a bifurcation is a simple dynamical system which is equivalent to all systems exhibiting this bifurcation.

Normal forms are defined for local bifurcations. It is assumed that the system is first reduced to the center manifold of the equilibrium point at which the bifurcation takes place. The reduced system is then locally (around the equilibrium) topologically equivalent to the normal form of the bifurcation.

For example, the normal form of a saddle-node bifurcation is where is the bifurcation parameter.

References

Jordan Village, Connecticut

Jordan is a village in the town of Waterford, Connecticut, and the historic center of the town. It was named from the Jordan River. The village was listed on the National Register of Historic Places as Jordan Village Historic District in 1990. The district includes 58 contributing buildings and one other contributing site over an area of 57 acres (23 ha). It includes examples of Greek Revival and Queen Anne architectural styles.

Location

Jordan Village is located on land known historically as Jordan Plain, a flat land area at the head of Jordan Cove, an estuary off Long Island Sound. The historic district surrounds the intersection of Rope Ferry Road and North Road.

See also

References

Jordan (Bishop of Poland)

Jordan (died in 982 or 984) was the first Bishop of Poland from 968 with his seat, most probably, in Poznań. He was an Italian or German.

Most evidence shows that he was missionary bishop subordinate directly to the Pope. He arrived in Poland, probably from Italy or the Rhineland, in 966 with Doubravka of Bohemia to baptise Mieszko I of Poland. After the death of Jordan until 992 the throne of the Bishop of Poland was vacant, or there was a bishop of unknown name (the first theory is more probable). His successor, from 992, was Unger.

References

Jordan 198

The Jordan 198 was the car with which the Jordan Formula One team used to compete in the 1998 Formula One season. It was driven by 1996 World Champion Damon Hill, who had moved from Arrows, and Ralf Schumacher, who was in his second season with the team.

After a dismal start which saw the team fail to score a single championship point in the first half of the season, numerous improvements to the car and tyre development by Goodyear enabled Jordan to climb back into the top teams. At the 1998 Belgian Grand Prix, the team scored a historic first victory with Hill, with Schumacher finishing behind him in second place. The team finished fourth in the Constructors' Championship, one ahead of Benetton.

Complete Formula One results

(key) (results in bold indicate pole position)

References