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The sheet of paper represents a plane in the spacetime continuum, and the two points represent a distance to be traveled, but theoretically, a wormhole could connect these two points by folding that plane (i.e. the paper) so the points are touching. In this way, it would be much easier to traverse the distance since the two points are now ...
Theorem [7] [8] — A linear map between two F-spaces (e.g. Banach spaces) is continuous if and only if its graph is closed. The theorem is a consequence of the open mapping theorem ; see § Relation to the open mapping theorem below (conversely, the open mapping theorem in turn can be deduced from the closed graph theorem).
Geometric graph theory in the broader sense is a large and amorphous subfield of graph theory, concerned with graphs defined by geometric means. In a stricter sense, geometric graph theory studies combinatorial and geometric properties of geometric graphs, meaning graphs drawn in the Euclidean plane with possibly intersecting straight-line edges, and topological graphs, where the edges are ...
Generalized Borel Graph Theorem [11] — Let : be a linear map between two locally convex Hausdorff spaces and . If X {\displaystyle X} is the inductive limit of an arbitrary family of Banach spaces, if Y {\displaystyle Y} is a K-analytic space, and if the graph of u {\displaystyle u} is closed in X × Y , {\displaystyle X\times Y,} then u ...
This is commonly represented on a graph with physical locations along the horizontal axis and time running vertically, with units of for time and ct for space. Light cones in this representation appear as lines at 45 degrees centered on the object, as light travels at c t {\displaystyle ct} per t {\displaystyle t} .
Closed graph theorems are of particular interest in functional analysis where there are many theorems giving conditions under which a linear map with a closed graph is necessarily continuous. If f : X → Y is a function between topological spaces whose graph is closed in X × Y and if Y is a compact space then f : X → Y is continuous.
An investigation by the National Transportation Safety Board later revealed that the jet was missing bolts meant to hold on its door plug, a panel that covers slots in the plane body used for ...
A geometric graph is a graph in which the vertices or edges are associated with geometric objects. Examples include Euclidean graphs, the 1-skeleton of a polyhedron or polytope, unit disk graphs, and visibility graphs. Topics in this area include: Graph drawing; Polyhedral graphs; Random geometric graphs; Voronoi diagrams and Delaunay ...