Discrete Extremal Length
Suppose that is some graph and is a collection of paths in . There are two variants of extremal length in this setting. To define the edge extremal length, originally introduced by R. J. Duffin, consider a function . The -length of a path is defined as the sum of over all edges in the path, counted with multiplicity. The "area" is defined as . The extremal length of is then defined as before. If is interpreted as a resistor network, where each edge has unit resistance, then the effective resistance between two sets of veritces is precisely the edge extremal length of the collection of paths with one endpoint in one set and the other endpoint in the other set. Thus, discrete extremal length is useful for estimates in discrete potential theory.
Another notion of discrete extremal length that is appropriate in other contexts is vertex extremal length, where, the area is, and the length of a path is the sum of over the vertices visited by the path, with multiplicity.
Read more about this topic: Extremal Length
Famous quotes containing the words discrete and/or length:
“We have good reason to believe that memories of early childhood do not persist in consciousness because of the absence or fragmentary character of language covering this period. Words serve as fixatives for mental images. . . . Even at the end of the second year of life when word tags exist for a number of objects in the childs life, these words are discrete and do not yet bind together the parts of an experience or organize them in a way that can produce a coherent memory.”
—Selma H. Fraiberg (20th century)
“At length he would call to let us know where he was waiting for us with his canoe, when, on account of the windings of the stream, we did not know where the shore was, but he did not call often enough, forgetting that we were not Indians.... This was not because he was unaccommodating, but a proof of superior manners. Indians like to get along with the least possible communication and ado. He was really paying us a great compliment all the while, thinking that we preferred a hint to a kick.”
—Henry David Thoreau (18171862)