Infinite Electrical Networks

Zemanian, A. H.

doi:10.1017/cbo9780511895432

Cited by 54 publications

(36 citation statements)

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“…A similar result for the usual scalar electrical networks was established by Flanders [13,14]. The proof of the theorem, which is inspired by [13], is provided in Appendix I.…”

Section: Approximating Infinite Network Currentsmentioning

confidence: 53%

“…This result is known as Thomson's Minimum Energy Principle [12,14]. Theorem 2 shows that generalized currents also obey Thomson's Principle in both finite and infinite networks.…”

Section: Theorem 2 (Generalized Current)mentioning

confidence: 91%

“…Existence and uniqueness of scalar currents in infinite networks has been examined in [13,14]. It was shown by Flanders that, unlike in finite networks, in an infinite electrical network the current is not uniquely determined by Kirchoff's laws and Ohm's law [13].…”

Section: A Existence and Uniqueness Of Generalized Currentmentioning

confidence: 99%

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Estimation from Relative Measurements: Electrical Analogy and Large Graphs

Barooah¹,

Hespanha²

2007

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Abstract-We examine the problem of estimating vectorvalued variables from noisy measurements of the difference between certain pairs of them. This problem, which is naturally posed in terms of a measurement graph, arises in applications such as sensor network localization, time synchronization, and motion consensus.We obtain a characterization on the minimum possible covariance of the estimation error when an arbitrarily large number of measurements are available. This covariance is shown to be equal to a matrix-valued effective resistance in an infinite electrical network. Covariance in large finite graphs converges to this effective resistance as the size of the graphs increases. This convergence result provides the formal justification for regarding large finite graphs as infinite graphs, which can be exploited to determine scaling laws for the estimation error in large finite graphs. Furthermore, these results indicate that in large networks, estimation algorithms that use small subsets of all the available measurements can still obtain accurate estimates.

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“…A similar result for the usual scalar electrical networks was established by Flanders [13,14]. The proof of the theorem, which is inspired by [13], is provided in Appendix I.…”

Section: Approximating Infinite Network Currentsmentioning

confidence: 53%

“…This result is known as Thomson's Minimum Energy Principle [12,14]. Theorem 2 shows that generalized currents also obey Thomson's Principle in both finite and infinite networks.…”

Section: Theorem 2 (Generalized Current)mentioning

confidence: 91%

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Estimation from Relative Measurements: Electrical Analogy and Large Graphs

Barooah¹,

Hespanha²

2007

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“…We will say that a one-ended 0-path p0 (whether permissive or not) converges to a O-terminal T O if, for each E > 0, the node sequence of pO is eventually in the ball/3~ ~ E We can relate these permissive 1-nodes to the 1-nodes in [2] and [3] as follows. Each 0-terminal can be identified with the set of all permissive one-ended 0-paths that converge to that 0-terminal.…”

Section: Condition 22 N O Has At Least One O-section Having Infinitmentioning

confidence: 99%

“…The idea of a transfinite graph or network was introduced in [2] and more thoroughly investigated in [3]. The transfinite structure was obtained by joining together various infinite extremities of infinite graphs, but the choices of those connections were quite arbitrary.…”

Section: Introductionmentioning

confidence: 99%

Permissively structured transfinite electrical networks

Zemanian

1999

Circuits Systems and Signal Process

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Abstract. Transfinite electrical networks of ranks larger than 1 have previously been defined by arbitrarily joining together various infinite extremities through transfinite nodes that are independent of the networks' resistance values. Thus, some or all of those transfinite nodes may remain ineffective in transmitting current "through infinity." In this paper, transfinite nodes are defined in terms of the paths that permit currents to "reach infinity." This is accomplished by defining a suitable metric d v on the node set AfS~ of each v-section S v, a u-section being a maximal subnetwork whose nodes are connected by twoended paths of ranks no larger than v. Upon taking the completion of A/'S~ under that metric d v, we identify those extremities (now called v-terminals) that are accessible to current flows. These are used to define transfinite nodes that combine such extremities. The construction is recursive and is carried out through all the natural number ranks, and then through the first arrow rank ~ and the first limit-ordinal rank m. The recursion can be carried still further. All this provides a more natural development of transfinite networks and indeed simplifies the theory of electrical behavior for such networks.

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The existence and uniqueness of node voltages in a non‐linear resistive transfinite electrical network

Zemanian

1996

Circuit Theory & Apps

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Transfinite resistive electrical networks have unique voltage‐current regimes under very broad conditions on their resistances and source values; this has been established previously. However, as is shown by example in this paper, such networks need not have unique node voltages with respect to a chosen ground node either because the voltages along every path from ground to another chosen node may not be summable or because the voltages along two such paths to the same node may be summable but with different sums. A second result of this work is the establishment of a sufficient set of conditions for the existence and uniqueness of node voltages in a non‐linear transfinite network. The theory of modular sequence spaces is used for this purpose.

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Infinite Electrical Networks

Cited by 54 publications

References 0 publications

Estimation from Relative Measurements: Electrical Analogy and Large Graphs

Estimation from Relative Measurements: Electrical Analogy and Large Graphs

Permissively structured transfinite electrical networks

The existence and uniqueness of node voltages in a non‐linear resistive transfinite electrical network

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