Groupoid C*-algebras with Hausdorff Spectrum

... Suppose that the maps of H/Hx onto H · x are homeomorphisms for each x ∈ X . The spectrum C ∗ (H n X )∧ is Hausdorff if and only if the map x 7→ Hx is continuous with respect to the Fell topology and X /H is Hausdorff. Note: We can use [Orloff Clark; 07] and [Ramsay; 90] to show that if either X /H ...

... Suppose that the maps of H/Hx onto H · x are homeomorphisms for each x ∈ X . The spectrum C ∗ (H n X )∧ is Hausdorff if and only if the map x 7→ Hx is continuous with respect to the Fell topology and X /H is Hausdorff. Note: We can use [Orloff Clark; 07] and [Ramsay; 90] to show that if either X /H ...

[edit] Star polyhedra

... dimensionality of -1. These posets belong to the larger family of abstract polytopes in any number of dimensions. ...

... dimensionality of -1. These posets belong to the larger family of abstract polytopes in any number of dimensions. ...

MATHEMATICS MAGAZINE

... posed questions as to which collections of animals can be transformed by Mad Vet machines into other collections. Recently, such scenarios have been used as the basis of various problem solving and Math Circle activities; see, for instance, [13]. In this article we take a different approach, using M ...

... posed questions as to which collections of animals can be transformed by Mad Vet machines into other collections. Recently, such scenarios have been used as the basis of various problem solving and Math Circle activities; see, for instance, [13]. In this article we take a different approach, using M ...

Topic 3 Space, Shape and Orientation

... If two triangles are equiangular, their corresponding sides are in proportion. If two triangles have their corresponding sides in proportion, they are equiangular. 'Equiangular' means 'equal angles' which means that the three angles of the one triangle are equal in size to the three angles of ...

... If two triangles are equiangular, their corresponding sides are in proportion. If two triangles have their corresponding sides in proportion, they are equiangular. 'Equiangular' means 'equal angles' which means that the three angles of the one triangle are equal in size to the three angles of ...

Parabolas

... We saw in Section 3.1 that the graph of the equation y = ax2 + bx + c is a U-shaped curve called a parabola that opens either upward or downward—depending on whether the sign of a is positive or negative. • Here, we study parabolas from a geometric rather than an algebraic point of view. ...

... We saw in Section 3.1 that the graph of the equation y = ax2 + bx + c is a U-shaped curve called a parabola that opens either upward or downward—depending on whether the sign of a is positive or negative. • Here, we study parabolas from a geometric rather than an algebraic point of view. ...

DuarteFrancisSeville30apr13

... The fundamental fact that, in the plane, any 4-sided polygon with equal length sides must be a rhombus, leads us to consider rhombic frameworks in general. Endowed with the additional topological properties, we call them rhombic carpets. In particular, when the rhombi are squares, we shall call them ...

... The fundamental fact that, in the plane, any 4-sided polygon with equal length sides must be a rhombus, leads us to consider rhombic frameworks in general. Endowed with the additional topological properties, we call them rhombic carpets. In particular, when the rhombi are squares, we shall call them ...

# 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.