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In mathematics, and especially in algebraic geometry, the intersection number generalizes the intuitive notion of counting the number of times two curves intersect to higher dimensions, multiple (more than 2) curves, and accounting properly for tangency. One needs a definition of intersection number in order to state results like Bézout's theorem.
An alternative definition of the intersection number of a graph is that it is the smallest number of cliques in (complete subgraphs of ) that together cover all of the edges of . [ 1 ] [ 12 ] A set of cliques with this property is known as a clique edge cover or edge clique cover , and for this reason the intersection number is also sometimes ...
So the intersection of the empty family should be the universal set (the identity element for the operation of intersection), [4] but in standard set theory, the universal set does not exist. However, when restricted to the context of subsets of a given fixed set X {\displaystyle X} , the notion of the intersection of an empty collection of ...
Differential equations are an important area of mathematical analysis with many applications in science and engineering. Analysis is the branch of mathematics dealing with continuous functions , limits , and related theories, such as differentiation , integration , measure , infinite sequences , series , and analytic functions .
The intersection of A with any of B, C, D, or E is the empty set. In mathematics, the intersection of two or more objects is another object consisting of everything that is contained in all of the objects simultaneously. For example, in Euclidean geometry, when two lines in a plane are not parallel, their intersection is the point at
To give a definition, in the general case, of the intersection multiplicity was the major concern of André Weil's 1946 book Foundations of Algebraic Geometry. Work in the 1920s of B. L. van der Waerden had already addressed the question; in the Italian school of algebraic geometry the ideas were well known, but foundational questions were not ...
Moreover, the clique number ω(G) of a graph G is the number of vertices in a maximum clique in G. The intersection number of G is the smallest number of cliques that together cover all edges of G. The clique cover number of a graph G is the smallest number of cliques of G whose union covers the set of vertices V of the graph.
Since André Weil's initial definition of intersection numbers, around 1949, there had been a question of how to provide a more flexible and computable theory, which Serre sought to address. In 1958, Serre realized that classical algebraic-geometric ideas of multiplicity could be generalized using the concepts of homological algebra .