![A) Area Between Curves in [a , b]](http://s1.studyres.com/store/data/010244503_1-8d5ca7e45bea07d60196d1e4f510559f-300x300.png)
A GRAPH FROM THE VIEWPOINT OF ALGEBRAIC TOPOLOGY 1
... Given points x and y of the space X, a path in X from x to y is a continuous map f : [a, b] → X of some closed interval in the real line into X such that f (a) = x and f (b) = y. X is called path connected if every pair of points X can be joined by a path in X. X is called locally path connected if, ...
... Given points x and y of the space X, a path in X from x to y is a continuous map f : [a, b] → X of some closed interval in the real line into X such that f (a) = x and f (b) = y. X is called path connected if every pair of points X can be joined by a path in X. X is called locally path connected if, ...
17 Quadratic Functions
... In this section we look at factorisation and how this can be used to solve quadratic equations. In Unit 11 you factorised expressions; we now take this one stage further to solve equations. In Unit 11 you looked at factorizing expressions with common factors. We now develop this to solving equations ...
... In this section we look at factorisation and how this can be used to solve quadratic equations. In Unit 11 you factorised expressions; we now take this one stage further to solve equations. In Unit 11 you looked at factorizing expressions with common factors. We now develop this to solving equations ...
Discrete Mathematics Introduction
... i and in square j for i 6= j (unless it is a double headed snake!). The addition principle: Consider k sets S1 ,. . . , Sk that are pairwise disjoint. We denote by |Si | the size of set Si . The total number of elements in all sets is |S1 | + |S2 | + . . . + |Sk |. There is another way to count the ...
... i and in square j for i 6= j (unless it is a double headed snake!). The addition principle: Consider k sets S1 ,. . . , Sk that are pairwise disjoint. We denote by |Si | the size of set Si . The total number of elements in all sets is |S1 | + |S2 | + . . . + |Sk |. There is another way to count the ...
Dual graph
In the mathematical discipline of graph theory, the dual graph of a plane graph G is a graph that has a vertex for each face of G. The dual graph has an edge whenever two faces of G are separated from each other by an edge. Thus, each edge e of G has a corresponding dual edge, the edge that connects the two faces on either side of e.Graph duality is a topological generalization of the geometric concepts of dual polyhedra and dual tessellations, and is in turn generalized algebraically by the concept of a dual matroid. Variations of planar graph duality include a version of duality for directed graphs, and duality for graphs embedded onto non-planar two-dimensional surfaces.However, the notion described in this page is different from the edge-to-vertex dual (line graph) of a graph and should not be confused with it.The term ""dual"" is used because this property is symmetric, meaning that if H is a dual of G, then G is a dual of H (if G is connected). When discussing the dual of a graph G, the graph G itself may be referred to as the ""primal graph"". Many other graph properties and structures may be translated into other natural properties and structures of the dual. For instance, cycles are dual to cuts, spanning trees are dual to the complements of spanning trees, and simple graphs (without parallel edges or self-loops) are dual to 3-edge-connected graphs.Polyhedral graphs, and some other planar graphs, have unique dual graphs. However, for planar graphs more generally, there may be multiple dual graphs, depending on the choice of planar embedding of the graph. Testing whether one planar graph is dual to another is NP-complete.