Parry-Sullivan invariant
Encyclopedia
In mathematics
, the Parry–Sullivan invariant (or Parry–Sullivan number) is a numerical quantity of interest in the study of incidence matrices
in graph theory
, and of certain one-dimensional dynamical systems. It provides a partial classification of non-trivial irreducible incidence matrices.
It is named after the English mathematician Bill Parry
and the American mathematician Dennis Sullivan
, who introduced the invariant in a joint paper published in the journal Topology
in 1975.
where I denotes the n × n identity matrix.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the Parry–Sullivan invariant (or Parry–Sullivan number) is a numerical quantity of interest in the study of incidence matrices
Incidence matrix
In mathematics, an incidence matrix is a matrix that shows the relationship between two classes of objects. If the first class is X and the second is Y, the matrix has one row for each element of X and one column for each element of Y. The entry in row x and column y is 1 if x and y are related ...
in graph theory
Graph theory
In mathematics and computer science, graph theory is the study of graphs, mathematical structures used to model pairwise relations between objects from a certain collection. A "graph" in this context refers to a collection of vertices or 'nodes' and a collection of edges that connect pairs of...
, and of certain one-dimensional dynamical systems. It provides a partial classification of non-trivial irreducible incidence matrices.
It is named after the English mathematician Bill Parry
Bill Parry (mathematician)
Professor William Parry FRS was an English mathematician. During his research career, he was highly active in the study of dynamical systems, and, in particular, ergodic theory, and made significant contributions to these fields. He is considered to have been at the forefront of the introduction...
and the American mathematician Dennis Sullivan
Dennis Sullivan
Dennis Parnell Sullivan is an American mathematician. He is known for work in topology, both algebraic and geometric, and on dynamical systems. He holds the Albert Einstein Chair at the City University of New York Graduate Center, and is a professor at Stony Brook University.-Work in topology:He...
, who introduced the invariant in a joint paper published in the journal Topology
Topology (journal)
Topology is a mathematical journal publishing scholarly articles related to topology and geometry. It was founded by J. H. C. Whitehead in 1962 and is published by Elsevier....
in 1975.
Definition
Let A be an n × n incidence matrix. Then the Parry–Sullivan number of A is defined to bewhere I denotes the n × n identity matrix.