Circular planar graphs and resistor networks

Curtis, Edward B.; Ingerman, David; Morrow, James

doi:10.1016/s0024-3795(98)10087-3

Cited by 143 publications

(304 citation statements)

References 3 publications

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“…Usually N q is called the response matrix of the network. Given A, B ⊂ δ(F ) a pair of disjoint subsets, we consider the submatrix of the response matrix In [4], we characterized those M -matrices that are the response matrix of a network, which represent an extension of the previous work by Curtis et al; see [12].…”

Section: Preliminariesmentioning

confidence: 99%

“…For instance, Sylvester and Uhlmann treated in [9,18] the uniqueness of solution; Curtis, Ingerman and Morrow have worked on critical circular planar networks conductivity reconstruction [12,13,14,16]; Borcea, Druskin, Guevara and Mamonov have gone into EIT problems in depth and their last works on the subject treat numerical conductivity reconstruction [6,7,8].…”

Section: Introductionmentioning

confidence: 99%

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Overdetermined partial boundary value problems on finite networks

Araúz

Carmona

Encinas

2015

Journal of Mathematical Analysis and Applications

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In this work we define a class of non self-adjoint boundary value problems on finite networks associated with Schrödinger operators. The novelty lies on the fact that on a part of the boundary no data is prescribed, whereas in another part of the boundary both the values of the function as of its normal derivative are given. We show that overdetermined partial boundary value problems are the key for solving inverse boundary value problems on finite networks, since they provide the theoretical foundations of the recovery algorithm. We analyze the uniqueness and existence of solution of overdetermined partial boundary value problems through the non-singularity of the partial Dirichlet-to-Neumann maps. These maps allow us to determine the value of the solution in the part of the boundary where no data was prescribed. Afterwards, we execute a full conductance recovery for spider networks.

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Section: Preliminariesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Overdetermined partial boundary value problems on finite networks

Araúz

Carmona

Encinas

2015

Journal of Mathematical Analysis and Applications

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“…In fact, f is linear (see [CIM,§1]), and we have natural bases for the spaces of applied voltages and observed currents at the boundary vertices. Thus, we can make the following definition: Definition 2.1.3.…”

Section: Circular Planar Electrical Network Up To Equivalencementioning

confidence: 99%

“…[CIM,§4]), discussed below, to form an electrical positroid S 1 . By the maximality of S 0 , S 1 = π(G 1 ) for some critical graph G 1 .…”

Section: Proof Of Theorem 517mentioning

confidence: 99%

“…n × n response matrices are characterized in [CIM,Theorem 4] as the symmetric matrices whose circular minors are non-negative, such that the sum of the entries in each row (and column) is zero. Furthermore, [CIM,Lemma 4.2] gives a combinatorial criterion, in terms of the underlying electrical network, for exactly which circular minors are strictly positive. However, until now, the combinatorial properties of these response matrices have remained largely unstudied, despite their inherently combinatorial descriptions.…”

Section: Introductionmentioning

confidence: 99%

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Circular planar electrical networks: Posets and positivity

Alman

Lian

Tran

2015

Journal of Combinatorial Theory, Series A

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Following de Verdière-Gitler-Vertigan and Curtis-Ingerman-Morrow, we prove a host of new results on circular planar electrical networks. We first construct a poset EP n of electrical networks with n boundary vertices, and prove that it is graded by number of edges of critical representatives. We then answer various enumerative questions related to EP n , adapting methods of Callan and Stein-Everett. Finally, we study certain positivity phenomena of the response matrices arising from circular planar electrical networks. In doing so, we introduce electrical positroids, extending work of Postnikov, and discuss a naturally arising example of a Laurent phenomenon algebra, as studied by Lam-Pylyavskyy.

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Pseudospectra of isospectrally reduced matrices

Vasquez

Webb

2014

Numer. Linear Algebra Appl.

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The isospectral reduction of matrix, which is closely related to its Schur complement, allows to reduce the size of a matrix while maintaining its eigenvalues up to a known set. Here we generalize this procedure by increasing the number of possible ways a matrix can be isospectrally reduced. The reduced matrix has rational functions as entries. We show that the notion of pseudospectrum can be extended to this class of matrices and that the pseudospectrum of a matrix shrinks as the matrix is reduced. Hence the eigenvalues of a reduced matrix are more robust to entry-wise perturbations than the eigenvalues of the original matrix. We also introduce the notion of inverse pseudospectrum (or pseudoresonances), which indicates how stable the poles of a matrix with rational function entries are to certain matrix perturbations. A mass spring system is used to illustrate and give a physical interpretation to both pseudospectra and inverse pseudospectra.

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Circular planar graphs and resistor networks

Cited by 143 publications

References 3 publications

Overdetermined partial boundary value problems on finite networks

Overdetermined partial boundary value problems on finite networks

Circular planar electrical networks: Posets and positivity

Pseudospectra of isospectrally reduced matrices

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