Permissive Transfinite Networks

Armen H. Zemanian
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Armen H. Zemanian: University at Stony Brook, Electrical Engineering Department

Chapter 4 in Pristine Transfinite Graphs and Permissive Electrical Networks, 2001, pp 53-80 from Springer

Abstract: Abstract The transfinite graphs constructed in Chapter 2 and examined in Chapter 3 were quite arbitrary in the sense that the choices of which ρ-tips would comprise which (ρ + l)-nodes were not restricted in any way. With regard to transfinite electrical networks, those choices need not reflect the ways in which currents can or cannot flow. Consider, for example, the one-way infinite ladder within the network of Figure 4.1 driven at its input by a 1 V source and having at its infinite extremity two 1-nodes, $$ n_2^1 $$ and $$ n_2^1 $$ containing, respectively, the 0-tips of the one-ended paths along its upper and lower horizontal branches. Let all of the ladder’s horizontal resistances be 1 Ω. Let the vertical resistances be arbitrary. Also, another resistor, R, is connected to those 1-nodes in a pristine way (i.e., through two extraction paths of shorts — see Section 1.4). The infinite total resistance of the horizontal resistors of the ladder prevents current from flowing through the 1-nodes and thereby through R; we restate this by saying that the 0-tips of those two horizontal one-ended paths of the ladder are “nonpermissive.” Thus, this choice of nonsingleton 1-nodes is pointless so far as the transmission of electricity is concerned.

Keywords: Limit Point; Characteristic Line; Recursive Construction; Vertical Resistance; Unit Symbol (search for similar items in EconPapers)
Date: 2001
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DOI: 10.1007/978-1-4612-0163-2_4

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