Eﬀective operators and their variational principles for di screte electrical network problems

2025-08-18 1 0 666.9KB 33 页 10玖币

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arXiv:2210.05761v1 [math-ph] 11 Oct 2022

Effective operators and variational principles for discrete networks

Eﬀective operators and their variational principles for discrete electrical

network problems

K. Beard,1A. Stefan,2R. Viator, Jr.,3and A. Welters2

1)Louisiana State University, Baton Rouge, LA USA

2)Florida Institute of Technology, Melbourne, FL USA

3)Swarthmore College, Swarthmore, PA USA

(*Electronic mail: awelters@ﬁt.edu)

(Dated: 13 October 2022)

Using a Hilbert space framework inspired by the methods of orthogonal projections and Hodge decompositions, we

study a general class of problems (called Z-problems) that arise in effective media theory, especially within the theory

of composites, for deﬁning the effective operator. A new and uniﬁed approach is developed, based on block operator

methods, for obtaining solutions of the Z-problem, formulas for the effective operator in terms of the Schur complement,

and associated variational principles (e.g., the Dirichlet and Thomson minimization principles) that lead to upper and

lower bounds on the effective operator. In the case of ﬁnite-dimensional Hilbert spaces, this allows for a relaxation of

the standard hypotheses on positivity and invertibility for the classes of operators usually considered in such problems,

by replacing inverses with the Moore-Penrose pseudoinverse. As we develop the theory, we show how it applies to the

classical example from the theory of composites on the effective conductivity in the periodic conductivity problem in

the continuum (2dand 3d) under the standard hypotheses. After that, we consider the following three important and

diverse examples (increasing in complexity) of discrete electrical network problems in which our theory applies under

the relaxed hypotheses. First, an operator-theoretic reformulation of the discrete Dirichlet-to-Neumann (DtN) map for

an electrical network on a ﬁnite linear graph is given and used to relate the DtN map to the effective operator of an

associated Z-problem. Second, we show how the classical effective conductivity of an electrical network on a ﬁnite

linear graph is essentially the effective operator of an associated Z-problem. Finally, we consider electrical networks

on periodic linear graphs and develop a discrete analog to classical example of the periodic conductivity equation and

effective conductivity in the continuum.a

I. INTRODUCTION

Broadly speaking, effective media (or medium) theory is an

analytical or theoretical model which describes the effective

(apparent, overall, or aggregate) behavior of a complex, mul-

ticomponent system, usually having a constitutive relation, in

terms of a simpler system with an effective constitutive rela-

tion that deﬁnes the effective parameter, moduli, coefﬁcient,

tensor, matrix, operator, etc.2–6, which we will denote here by

∗. The effective properties of the original system are then

often described in terms of this

∗.

For example, in a continuum electrical conductivity prob-

lem with a conductivity tensor ﬁeld

(x)in a periodic

medium, the constitutive relation governing the relation be-

tween the electric ﬁeld E=E(x)and the current density

J=J(x)is Ohm’s law J=

Eand the effective conductiv-

ity tensor

∗is deﬁned by the relation hJi=

∗hEi, where h·i

denotes the periodic average3,7,8.

Similar to this example, there are a common set of fun-

damental problems one wants to solve such as: Find sufﬁ-

cient conditions on the constitutive relations that guarantee

∗

is uniquely deﬁned by the effective constitutive relation and

then give an explicit formula for

∗. In addition, if possible,

give variational principles for

∗and derive upper and lower

bounds on it.

a)This work is based in part on the M.S. thesis1of the ﬁrst author whose

advisor was the fourth author.

To treat these fundamental problems, a mathematical struc-

ture has emerged from within the theory of composites3,7,8,

that has provided a common systematic method for deﬁning

the effective operator

∗starting from a “Z-problem" (aka, a

generalized “ﬁeld equation" or “cell problem"), which repre-

sents the constitutive relation, and based on a Hilbert space

framework in which orthogonal projections associated with a

Hodge decomposition plays a key role. Originally, part of the

framework arose out of homogenization theory9–13 and then

was further developed within the theory of composites3,7,14–17

(see also Chap. 2 in Ref. 8). But now the abstract frame-

work has been shown to be useful more generally8,18–21 and

leads to several important open problems22 that is fueling

research23–25. As such, this framework deserves further study

and should include new compelling examples that ﬁt within

the abstract framework. This provides one motivation for our

paper.

In order to better understand the structure of a Z-problem,

let us consider the example above in more detail.

Example 1 Consider the Hilbert space of periodic square-

integrable vector-valued functions L2

#(Ω)d(d =2or d =3,

over the ﬁeld K=Ror K=C) with unit cell Ω(e.g., Ω=

[0,2

]d) and the Hodge decomposition:

L2

#(Ω)d=H=U⊥

⊕E⊥

⊕J,(1)

U={U∈H:hUi=U},(2)

E={E∈H:∇×E=0,hEi=0},(3)

J={J∈H:∇·J=0,hJi=0},(4)

Effective operators and variational principles for discrete networks 2

with the inner product and (cell) average

(E,F)H=1

|Ω|R

Ω

E(x)TF(x)dx,hFi=1

|Ω|R

Ω

F(x)dx,(5)

respectively, for all E,F∈H, where (·)Tand (·)denote

the transpose and complex conjugation, respectively. In par-

ticular, Uis the d-dimensional space of uniform (constant)

vector functions; Eis the inﬁnite-dimensional space of all

Ω-periodic ﬁelds E characterized by E =∇u for some Ω-

periodic function u; U⊥

⊕Eand U⊥

⊕Jare the spaces of

periodic vector ﬁelds which are the gradient ∇of a potential

and divergence (∇·)-free, respectively.

The Z-problem in this example is the constitutive relation,

namely, Ohm’s law (formulated as the ﬁeld equation or cell

problem3,7,8) with a measurable, bounded, periodic conduc-

tivity tensor

(x)[more precisely, it can be scalar-,

tensor-, or d ×d matrix-valued as a function of the spa-

tial variable x, but the key point is that as a left multiplica-

tion operator,

is a bounded linear operator on H, i.e.,

∈L(H)] or, more generally, with any bounded linear

operator

∈L(H)(in particular,

need not be a local

operator16,17):

J0+J=

(E0+E),(6)

where E0,J0∈U,E∈E, and J ∈J. Then effective conduc-

tivity

∗is deﬁned in terms of this Z-problem as the bounded

linear operator on U[i.e.,

∗∈L(U)] satisfying the effec-

tive constitutive relation

J0=

∗E0.(7)

Equivalently, in terms of averages between the periodic elec-

tric ﬁeld E0+E∈U⊥

⊕Eand the periodic current density

J0+J∈U⊥

⊕Jrelated through Ohm’s law (6), the effective

constitutive relation (7) is the usual relation deﬁning the ef-

fective conductivity:

hJ0+Ji=

∗hE0+Ei,(8)

since hE0+Ei=E0and hJ0+Ji=J0.

Remark 2 In this paper we will treat Hilbert spaces Hover

a ﬁeld K, where K=R, the real numbers, or K=C, the com-

plex numbers. In either case, the abstract framework and our

results are formulated in a general form regardless of which

ﬁeld is used. One motivation for including both real and

complex ﬁelds comes from the theory of composites3,7,8,17: In

static problems, usually one deals with real ﬁelds and hence a

real Hilbert space H. But for quasistatic problems, one also

deals with complex ﬁelds and hence a complex Hilbert space

Remark 3 For the Hodge decomposition above, there is an

alternative characterization (see pp. 1487–1488 and Ap-

pendix A in Ref. 26 on the Helmholtz decomposition) of these

spaces (which can be proved using Fourier series analysis, for

instance) that will be useful when we consider discrete exam-

ples in Sec. VI:L2

#(Ω)ddenotes the Hilbert space (over the

ﬁeld K) of all periodic (on Ω) vector-valued functions belong-

ing to L2

loc(Rd,Kd)with inner product (5),

U={U∈L2

#(Ω)d:U is a constant function},(9)

E={∇u:u∈L2

#(Ω)and u ∈H1

loc(Rd,K)},(10)

J={J∈L2

#(Ω)d:J∈Hloc(div,Rd,Kd),∇·J=0,hJi=0},

(11)

where ∇:H1

loc(Rd,K)→L2

loc(Rd,Kd)is the gradient oper-

ator and ∇·:Hloc(div,Rd,Kd)→L2

loc(Rd,Kd)is the diver-

gence operator. Then

J={∇×F:F∈L2

#(Ω)d,∇×F∈L2

#(Ω)d},

where ∇×:Hloc(curl,Rd,Kd)→L2

loc(Rd,Kd)is the curl op-

erator, and

L2

#(Ω)d=U⊥

⊕E⊥

⊕J,(12)

ker(∇·)∩L2

#(Ω)d=U⊥

⊕J,ran(∇)∩L2

#(Ω)d=U⊥

⊕E.

(13)

It is these representations, speciﬁcally, (9)– (13), that we will

consider for an analogy in the periodic lattice example in Sec.

VI D, but we will ﬁnd that there are nuances which lead us

to conclude that the analogy is not a perfect one (see Sec.

VI D 3).

A. What is the Z-problem and eﬀective operator?

The above discussion motivations the following precise

deﬁnition of the Z-problem (a term coined in Ref. 8, see

Chaps. 7 & 10, that also may be called a generalized “cell

problem" as coined in Ref. 18 or “ﬁeld equation" following

3, 8, 14–17) associated with a Hilbert space having an orthog-

onal triple decomposition [or, in light of (1)-(4) and Ref. 13, a

“generalized Hodge decomposition" as coined in Ref. 18] and

effective operator (or “Z-operator," see Chaps. 7 & 10 in Ref.

8) that comes from the abstract theory of composites3,8.

Deﬁnition 4 (Z-problem and effective operator) The

Z-problem

(H,U,E,J,

),(14)

is the following problem associated with a Hilbert space H,

an orthogonal triple decomposition of Has

H=U⊥

⊕E⊥

⊕J,(15)

and a (bounded) linear operator

∈L(H): given E0∈U,

ﬁnd triples (J0,E,J)∈U×E×Jsatisfying

J0+J=

(E0+E),(16)

Effective operators and variational principles for discrete networks 3

such a triple (J0,E,J)is called a solution of the Z-problem at

E0. If there exists a (bounded) linear operator

∗∈L(U)

such that

J0=

∗E0,(17)

whenever E0∈Uand (J0,E,J)is a solution of the Z-problem

at E0, then

∗is called an effective operator of the Z-problem.

For instance, it follows from this deﬁnition that Exam-

ple 1deﬁnes a Z-problem (H,U,E,J,

), where H=

L2

#(Ω)dhas the orthogonal triple decomposition (1) with

U,E,Jgiven by (2), (3), (4) (which is a special case of

a Hodge decomposition) and the effective conductivity

∗is

an effective operator of that Z-problem according to our deﬁ-

nition above.

B. Overview of the paper

To every Z-problem (H,U,E,J,

)(e.g., Example 1),

there are three important sets of problems that naturally arise:

(i) (Solvability of the Z-problem) Under what conditions

does the Z-problem (16) have a solution; a unique so-

lution? If possible, ﬁnd a formula for all solutions in

terms of

and parameterized by those E0∈Ufor

which solutions exist.

(ii) (Existence, uniqueness, and representation formulas for

the effective operator) Under what conditions does the

effective operator

∗exist; is unique? If it exists, ﬁnd

representation formulas for it in terms of

(iii) (Variational principles & bounds) If

∗=

(i.e., self-

adjoint) and

≥0 (i.e., positive semideﬁnite), are there

variational principles: (1)for the solutions of the Z-

problem? (2)that deﬁne the effective operator

∗?(3)

that can be used to derive upper and lower bounds on an

effective operator

∗in terms of

In this paper, we will focus on answering these questions.

To do so, we will ﬁrst consider classical results in Sec. II.

Here, certain “strong hypotheses" will be used such as

∗=

≥0 and

invertible (the “classical" hypotheses). Often the

latter will be too restrictive though and the invertibility of the

subblock

11 of

[see (18)-(20)] will be enough. Second, in

Sec. III, we discuss how we plan to relax or weaken the hy-

potheses and then give insight into the reason these “weaker

hypotheses" [see (H1)-(H4)] will naturally occur. Next, in or-

der to extend the results in Sec. II under the weaker hypothe-

ses, we introduce a uniﬁed framework for treating the varia-

tional principles associated to constrained linear equations in

Sec. IV. Then, in Sec. V, we use this framework to prove the

main results of this paper on the Z-problem and effective oper-

ator which answer the sets of questions (i),(ii), and (iii) above

under the weaker hypotheses. After this, we will develop in

Sec. VI, three important examples of discrete electrical net-

work problems for which our abstract framework and results

apply. These examples provide far more than just evidence

that our abstract framework can be used to solve for their asso-

ciated effective operators but also, equally as important, there

exists a connecting relationship between all of them, which

becomes apparent from the results in this paper. In Appen-

dices Aand B, we provide some fundamental results on the

abstract theory of composites, effective operators, and Hodge

decompositions that are needed in this paper and interesting

in their own right, but are best placed in these appendices.

In regard to the approach we use in this paper to treat the

problems (i)-(iii), it is new, even in the classical setting. It is

based on block operator methods and Schur complement the-

ory. As such, we will review the classical results and include

proofs using this new approach, before generalizing them later

in the paper.

Finally, although our focus is on weakening the hypotheses

on the operator

to achieve comparable results in the classi-

cal setting, we will still assume, in general, some form of self-

adjoint hypotheses, e.g.,

∗=

, when we state and prove our

results. The reason for this is two fold. The ﬁrst is that

∗=

is the most common hypothesis in the examples we consider

in this paper. The second reason is that the other typical hy-

potheses on

, that it has positive imaginary or positive real

part (see, for instance, Refs. 3, 8, 17, and 27), is important

enough to deserve a separate study of which the current paper

will be useful.

II. CLASSICAL RESULTS

In the classical setting, i.e., with the strong assumptions

that

∗=

≥0 and

invertible, all these problems can

solved3,8,16,17. We will review this now and include proofs

using a new approach to solving these problems which

emphasizes block operator methods and the use of Schur

complements28 (although the importance of it in the theory

of composites was originally recognized in Ref. 29). Later

in the paper we generalize our approach when we relax the

invertibility hypothesis.

Let (H,U,E,J,

)be a Z-problem (as deﬁned in Def.

4). Then we can write the operator

= [

i j]i,j=0,1,2∈L(H)(18)

as a 3 ×3 block operator matrix with respect to the orthog-

onal triple decomposition (15) of the Hilbert space H=

U⊥

⊕E⊥

⊕J. More precisely, we introduce the orthogonal

projections Γ0,Γ1,Γ2of Honto H0=U,H1=E,H2=J,

respectively, and deﬁne

i j ∈L(Hj,Hi),

i j =Γi

Γj:Hj→Hi,(19)

for i,j=0,1,2. In particular, for i=j=1,

11 is the com-

pression of

to E, that is,

11 =Γ1

Γ1|E,(20)

i.e., the restriction of the operator Γ1

Γ1on Hto the closed

subspace E. Then the Z-problem (16) is equivalent to the sys-

Effective operators and variational principles for discrete networks 4

tem

00E0+

01E=J0,(21)

10E0+

11E=0,(22)

20E0+

21E=J.(23)

Finally, from this and assuming

11 is invertible, we get the

classical formulas for the solution of the Z-problem and the

effective operator as a Schur complement:

J0=

∗E0,E=−

−1

10E0,J=

20E0+

21E,(24)

∗=

11/

11 =

00 −

−1

10.(25)

This proves the following theorem which answers the ques-

tions (i) and (ii) above in the case

11 is invertible.

Theorem 5 If (H,U,E,J,

)is a Z-problem (as in Def. 4)

and

11 [as deﬁned by (20)] is invertible then the Z-problem

(16) has a unique solution for each E0∈Uand it is given

by the formulas (24), (25). Moreover, the effective operator

of the Z-problem exists, is unique, and is given by the Schur

complement formula (25).

The Schur complement formula (25) for the effective oper-

ator

∗is useful for deriving some of it’s basic properties as

we shall now see.

Corollary 6 If

∈L(H)and

11 is invertible then

(

∗)∗= (

∗)∗.(26)

In particular, if

∗=

then

(

∗)∗=

∗.(27)

Proof. Assume the hypotheses. Then since (

i j)∗= (

∗)ji

for each i,j=0,1,2 it follows that (

11)∗= (

∗)11 is invert-

ible so by Theorem 5applied to the adjoint Z-problem, i.e.,

to the Z-problem (H,U,E,J,

∗), we have following for-

mula for the associated effective operator (

∗)∗:

(

∗)∗= (

∗)00 −(

∗)01(

∗)−1

11 (

∗)10

= (

00)∗−(

10)∗(

−1

11 )∗(

01)∗

= [

00 −

01(

11)−1

10]∗

= (

∗)∗.

This proves the corollary.

Now under the self-adjoint assumptions that

∈L(H)

and

∗=

, the next two theorems answer the questions

(iii).(2)under slightly weaker hypotheses, namely, in the case

11 ≥0 and

11 is invertible and all of (iii) in the case

≥0

and

is invertible in which the notion of duality plays a key

role.

Theorem 7 (Dirichlet minimization principle) If

∈L(H),

∗=

11 ≥0, and

11 is invertible

then the effective operator

∗is the unique self-adjoint

operator satisfying the minimization principle:

(E0,

∗E0) = min

E∈E(E0+E,

(E0+E)),∀E0∈U,(28)

and, for each E0∈U, the minimizer is unique and given by

E=−

−1

10E0.(29)

Moreover, we have the following upper bound on the effective

operator:

∗≤

00,(30)

where

00 =Γ0

Γ0|U.(31)

In fact, as our proof of this theorem below shows, the theorem

is still true if we drop the self-adjoint hypothesis

∗=

and

replace it with the weaker hypothesis:



11∗

=

11,(32)

where



11= (Γ0+Γ1)

(Γ0+Γ1)|U⊥

⊕E.(33)

In other words, the weaker hypothesis (32) is equivalent to the

hypothesis that the compression of

to U⊥

⊕Eis self-adjoint.

We can use Theorem 7to derive some additional properties

of the effective operator

∗.

Corollary 8 Suppose

∈L(H),

∗=

≥0. Then the fol-

lowing are true: (a)If

11 is invertible then (

∗)∗=

∗≥0.

(b)If

is invertible then 0≤(

j j)∗=

j j are invertible for

each j =0,1,2and

∗is invertible.

Proof. Suppose

∈L(H),

∗=

≥0. (a): If

11 is

invertible then by Corollary 6we know (

∗)∗=

∗and by (28)

in Theorem 7together with the hypothesis

≥0, it follows

immediately that

∗≥0. (b): Suppose

is invertible. Then

it follows from the hypotheses that

≥

IHfor some scalar

>0,where IHdenotes the identity operator on H. From

this it follows that (

j j)∗=

j j =Γj

Γj|Hj≥

IHj, where IHj

is the identity operator on the Hilbert space Hjfor each j=

0,1,2 with H0=U,H1=E,H2=J. And this implies

j j

is invertible for each j=0,1,2. Now it follows immediately

from (28) in Theorem 7that (

∗)∗=

∗≥

IUwhich implies

∗is invertible.

Theorem 9 (Thomson minimization principle) If

∈L(H),

∗=

≥0,and

is invertible then (

∗)−1is

the unique self-adjoint operator satisfying the minimization

principle:

(J0,(

∗)−1J0) = min

J∈J(J0+J,

−1(J0+J)),∀J0∈U,(34)

and, for each J0∈U, the minimizer is unique and given by

J=−

−1

20J0.(35)

Moreover, we have the upper and lower bounds on the effec-

tive operator:

0≤[(

−1)00]−1]≤

∗≤

00,(36)

where

(

−1)00 =Γ0

−1Γ0|U.(37)

Effective operators and variational principles for discrete networks 5

We will give a proof of these two theorems which is surpris-

ingly simple now and based on the following well-known min-

imization principle for Schur complements28,30 and the notion

of duality between the direct and dual Z-problems as deﬁned

next. In this regard, the Dirichlet variational principle, i.e.,

Theorem 7, is called the direct variational principle and the

Thomson (or Thompson) variational principle, i.e., Theorem

9, is called the dual (or complementary) variational principle

(see, for instance, Refs. 3, 8, and 17).

We will need the following well-known result28 (the proof

is omitted as it is trivial).

Lemma 10 (Aitken block-diagonalization formula) If H =

⊥

⊕H1is a Hilbert space, A = [Ai,j]i,j=0,1∈L(H), and A11

is invertible then

A00 A01

A10 A11=IH0A01A−1

0IH1A/A11 0

0A11 IH00

A−1

11 A10 IH1,

(38)

IH0A01A−1

0IH1−1

=IH0−A01A−1

0IH1,(39)

IH00

A−1

11 A10 IH1−1

=IH00

−A−1

11 A10 IH1.(40)

Theorem 11 (Schur complement: minimization principle)

If H =H0

⊥

⊕H1is a Hilbert space, A = [Ai,j]i,j=0,1∈L(H),

A∗=A,A11 is invertible and A11 ≥0then the Schur comple-

ment of A with respect to A11, i.e., A/A11 =A00 −A01A−1

11 A10,

is the unique self-adjoint operator satisfying the minimization

principle:

(x,A/A11x) = min

y∈H1

(x+y,A(x+y)),∀x∈H0,(41)

and, for each x ∈H0, the minimizer is unique and given by

y=−A−1

11 A10x∈H1.(42)

Moreover,

A/A11 ≤A00.(43)

Proof. Assume the hypotheses. Then (A11)∗=A11 ≥

I,for

some scalar

>0. Next, it follows by Lemma 10 that Ahas

the block factorization (38) with

IH0A01A−1

0IH1∗

=IH00

A−1

11 A10 IH1.

Hence, for any x∈H0and y∈H1,

(x+y,A(x+y)) = x

y,A00 A01

A10 A11x

y

= I0

A−1

11 A10 Ix

y,A/A11 0

0A11 I0

A−1

11 A10 Ix

y

= (x,A/A11x) + A−1

11 A10x+y,A11 A−1

11 A10x+y

≥(x,A/A11x),

with equality iff A−1

11 A10x+y,A11 A−1

11 A10x+y =0 iff

A−1

11 A10x+y=0 iff y=−A−1

11 A10x. This proves the equal-

ity (41) and the uniqueness of the minimizer which is given

by (42). The proof of the theorem now follows from all this

and the fact that A/A11 ∈L(H0)is self-adjoint so is uniquely

determined by its quadratic form (see, for instance, Theorem

4.4 in Ref. 31) and the upper bound (43) follows from consid-

ering the RHS of (41) with y=0∈H1.

Deﬁnition 12 (Dual Z-problem) Given a Z-problem

(H,U,E,J,

)with invertible

(the direct Z-problem),

the corresponding dual Z-problem is the Z-problem

(H,U,J,E,

−1)associated with the orthogonal de-

composition

H=U⊥

⊕J⊥

⊕E(44)

and operator

−1∈L(H), i.e., the problem: given J0∈U,

ﬁnd vectors (E0,J,E)∈U×J×Esatisfying

E0+E=

−1(J0+J).(45)

An effective operator for this Z-problem will be denoted by



−1∗′. In other words, if there exists an operator 

−1∗′∈

L(U)such that

E0=

−1∗′J0(46)

whenever (E0,J,E)is a solution of the dual Z-problem at J0

then 

−1∗′is an effective operator for the dual Z-problem.

Proof of Theorems 7and 9.Assume the hypotheses of

Theorem 7. Then the proof of Theorem 7follows immediately

from Theorem 5with the Schur complement formula (25) for

∗and by Theorem 11 with

∗=A/A11,(47)

where

H=H0

⊥

⊕H1,H0=U,H1=E,(48)

A= [Ai j]i,j=0,1= [

i j]i,j=0,1.(49)

Now notice it follows from this proof that we can replace the

hypothesis

∗=

in Theorem 7with the weaker hypothesis

32 and the conclusions of Theorem 7are still true.

Next, assume the hypotheses in Theorem 9. Then by

Corollary 8we know that 0 ≤(

j j)∗=

j j are invertible

for each j=0,1,2 and similarly for

−1since by hypothe-

ses it follows that (

−1) =

−1≥0. Thus, by Theorem 5

(in particular the uniqueness of the effective operator) it fol-

lows that the effective operator (

−1)∗′of the dual problem

(H,U,J,E,

−1)satisﬁes

(

−1)∗′= (

∗)−1,(50)

and by Corollary 8we also know that (

−1)∗′≥0 and

∗≥0.

The result now follows immediately from this by Theorem 7

applied to the dual Z-problem. In particular, we have from

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arXiv:2210.05761v1[math-ph]11Oct2022EffectiveoperatorsandvariationalprinciplesfordiscretenetworksEﬀectiveoperatorsandtheirvariationalprinciplesfordiscreteelectricalnetworkproblemsK.Beard,1A.Stefan,2R.Viator,Jr.,3andA.Welters21)LouisianaStateUniversity,BatonRouge,LAUSA2)FloridaInstituteofTechnology,M...

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Eﬀective operators and their variational principles for di screte electrical network problems

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