Nonstandard Graphs and Networks

Armen H. Zemanian
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Armen H. Zemanian: State University of New York at Stony Brook, Department of Electrical Engineering

Chapter 8 in Graphs and Networks, 2004, pp 153-169 from Springer

Abstract: Abstract Our nonstandard analyses for the determination of hyperreal operating points in transfinite resistive networks in Chapter 6 and of hyperreal transients in transfinite RLC networks in Chapter 7 do not enlarge those networks in a nonstandard way. The restoration processes employed therein merely recover the original standard networks and only provide a means of constructing hyperreal currents and voltages that satisfy Kirchhoff’s laws and Ohm’s law. Another approach might start with an arbitrary sequence of conventional (finite or infinite) 0-graphs and construct from that a nonstandard 0-graph in much the same way as an internal set in the hyperreal line *ℝ is constructed from a given sequence of subsets of the real line ℝ, that is, by means of an ultrapower construction. In this case, the resulting nonstandard graph has nonstandard branches and nonstandard nodes, and it thereby is much different from what we have presented so far.

Keywords: Nonstandard Analysis; Transfer Principle; Finite Graph; Parallel Branch; Standard Branch (search for similar items in EconPapers)
Date: 2004
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-0-8176-8178-4_8

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DOI: 10.1007/978-0-8176-8178-4_8

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