INTERNATIONAL JOURNAL OF CIRCUIT THEORY AND APPLICATIONS, VOL. 24, 635-644 (1996)

THE EXISTENCE AND UNIQUENESS OF NODE VOLTAGES IN A NON-LINEAR RESISTIVE TRANSFINITE ELECTRICAL NETWORK

A. H. ZEMANIAN

State Utiiversiry of New York at Stony Brook, Stony Brook, NY 11794-2350, U.S A

SUMMARY

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.

1. INTRODUCTION

An electrical network is transfinite when it has at least two nodes that are connected by an infinite path but not by an ordinary finite one. Such a network has an infinity of branches, which we take to be countable. It also has nodes of various ranks, which connect infinite subnetworks at their infinite extremities. Transfinite networks arises quite naturally when it becomes necessary to specify conditions at infinite extremities in order to get unique voltage-current regimes (Reference 1, Examples 1.6-4, 1.6-5 and 6.8-1; Reference 2, pp. 17-18). A theory for transfinite resistive networks has recently been developed (Reference 1, Chaps 3-5; Reference 2); it establishes unique voltages and currents in all branches of a transfinite network under a finite power requirement and a generalized form of Tellegen’s equation, ideas introduced by Flanders for ordinary infinite network^.^ However, it does not guarantee the existence of all node voltages. Two difficulties may intervene. In the first place a voltage at a node no transfinitely far away from a ground node ng may fail to exist because the sum of voltages along every path between no and ng may diverge. Secondly, even when such sums exist, the voltage at no may not be unique because different choices of the paths between no and ng may have different sums. Section 3 of this work presents examples demonstrating these possibilities.

These two problems have been overcome in the case of linear networks by imposing two additional condition^.^ First, it is required that there be at least one ‘permissive’ path between no and n,; that a transfinite path in a linear network is permissive means that the sum of (linear positive) resistances in the path is finite. (A finite path is always permissive.) This insures the existence of a node voltage at no. The second requirement is more complicated, but, roughly speaking, it requires that two different one-ended infinite paths that meet infinitely often must approach the same node.

The second objective of this work is to do the same thing for certain non-linear networks whose non- linearities are of a form related to modular sequence spaces.’ A theory for infinite non-linear networks of this sort was established by DeMichele and Soardi6 and extended to transfinite networks in Reference 1 (Chap. 4). To achieve our objective, we have to introduce a non-linear version of the idea of ‘permissivity’ for a transfinite path, a concept which up to now has been defined only for linear networks (see Reference 1 (pp. 92, 154) and Reference 2 (p. 24), where ‘permissivity’ is referred to by the synonym ‘perceptibility’). To achieve our objective, we establish the existence of node voltages (Lemma 2) and the validity of

CCC 0098-9886/96/060635- 10 0 1996 by John Wiley & Sons, Ltd.

Received 7 January I995 Revised 4 December 1995

636 A. H. ZEMANIAN

Kirchhoffs voltage law around transfinite loops (Theorem 2). The uniqueness of node voltages then follows as in Reference 4.

In the following we will freely use the definitions and results of References 1 and 2. We will be dealing with a transfinite, resistive, non-linear, electrical network with countably many branches. Actually, we shall use the adjective ‘non-linear’ inclusively; that is, a linear network will be taken to be a special case of a non-linear one. Indeed, the theory of non-linear transfinite networks (Reference 1, Sections 4.7 and 4.8). upon which this work is based, encompasses linear transfinite networks. In particular, the fundamental theorem (Reference 1, Theorem 4.8-4) dictating the voltage-current regime in a non-linear network implies the fundamental theorem (Reference 1, Theorem 3.3-5) for a linear network. Both these theorems impose the restriction that the sources in the network can deliver only a finite amount of power, and the former theorem restricts the kind of non-linearity that the resistances can have.

2. PERMISSIVE PATHS AND NODE VOLTAGES; THE LINEAR CASE

Let N’ be a linear resistive transfinite network and let no and n, be two maximal nodes of N ’. By ‘maximal’ we mean that the node considered is not embraced by a node of higher rank (Reference 1, p. 141). The ranks of no and ng need not be the same, Furthermore, let P be a finite, infinite or transfinite path that is terminally incident to no and n,; that is, P reaches no and ng with its two terminal tips. Let II be the index set for the branches of P. By the definition of a path, I3 will be countable. P will be called permissive if C,,,r, < m, where r, is the resistance of the jth branch in P. Let us orient P from no to n8. Every branch bj of N’ also has an orientation with respect to which the branch voltage vi and the branch current i, are measured; see Figure 1. Then the algebraic sum of the branch voltages along P from no to ng is

where the plus (minus) sign is used if the branch’s orientation agrees (disagrees) with a tracing of P from no to n8. If P is permissive, (1) converges absolutely. Indeed, for j restricted to n, with g, = r;’ denoting the branch conductance, and by the Schwarz inequality, we may write

CIv,l =CIr,i,-e,I ~ ~ ~ r , I i , I ~ r , + ~ ~ r , ~ e , I ~ g , ~ ( ~ r , ~ i,2r,>1’2+(Cr,Ce12g,)1’2<a, because C i:r, and 2 e,’g, are both finite in accordance with the theory for linear transfinite networks (Reference 1, Section 3.3). Let us now refer to n8 as the ground node or simply as ground and let its node voltage be 0 V. If there truly exists a permissive path P terminally incident to no and n,, then the value (1) is defined to be the node voltage at no with respect to n, along P . It should be emphasized that, according to this definition, node voltages are assigned only along permissive paths. For instance, (1) may converge even when P is not permissive, but in any case we do not use (1) with that non-permissive P to assign a node voltage to no. On the other hand, there may be more than one permissive path between ng and no and it may even happen that the node voltages assigned to no along two such permissive paths may differ-as we shall see. We have the problem of finding conditions on the network N‘ insuring that, given any node n8 of N’ as ground, there are unique node voltages at all the nodes of N’. A solution to this problem was obtained in Reference 4 for linear networks. Herein we solve the more general problem for non-linear networks.

Figure 1. A linear branch in the Thevenin form. The branch resistance rl and branch voltage source el are real numbers with r,>O. The branch’s orientation, by which the branch cumnt il and the branch voltage u, are measured, is indicated by the arrow for i, and

also by the + to - direction for u,. Thus u, = i, r, - e,

NODE VOLTAGES IN A NON-LINEAR TRANSFINITE NETWORK 637

3. THE NON-EXISTENCE OR NON-UNIQUENESS OF NODE VOLTAGES

We now present some examples demonstrating the difficulties regarding the existence and uniqueness of node voltages. This is done with linear networks. However, a small non-linear modification of the examples given will not change the voltage-current regimes by much and in fact our conclusions will remain valid for the modified networks.

Example 1

Consider the first difficulty in which the sum of voltages along every path between a ground node and another node is divergent. Perhaps the simplest example demonstrating this possibility is the 1-network of Figure 2(a). It is simply a 1-path P' = [ ni, Po, n'}, where Po is a one-ended 0-path starting at a ground 0-node ng" and reaching a 1-node n'. Every branch b, ( k = 2,3 ,4 , ...) of Po is in the Thevenin form of a linear resistor r k = k-''' in series with a voltage source e k = k-' with polarity directed towards a' . The sources satisfy the finite total isolated power condition C;=2eig, = C;=2k-3/2, where gk = r,' = k 1 / 2 . Thus this 1-network does have a voltage-current regime dictated by the fundamental theorem (Reference 1, Theorem 3.3-5) for transfinite networks. Moreover, the branch current i, and branch voltage Vk are obviously i ,=O and v,= e, for every branch. Consequently, the sum of the branch voltages diverges: C;=2vk = C;= , ,k - '= 00. Hence n' does not possess a (finite) node voltage with respect to the ground node ng".

One might object that this is a trivial example since every branch current in P' is zero, making the branch voltages equal to the non-summable branch source voltages. However, P' can be imbedded into a larger network in such a fashion that every one of its branches has a non-zero current and yet its branch voltages still sum to infinity. This is so for the 1-network of Figure 2(b). In that network, two replicates of Po are connected at their corresponding nodes by resistors, except at the first and last nodes where they are connected by a source branch on the left and by a source branch in parallel with another resistor on the right. All these (vertically drawn) resistors are 1 R in value. With eo+O and E + O , every branch of this network carries a non-zero current. Indeed, since we are dealing with a linear network governed by the

Figure 2. (a) A 1-network consisting of a single 1-path. Here rl = k-"' and e, = k - ' for k = 2 , 3 , 4 , .... All branch currents are zero and the 1-node n' does not have a node voltage with respect to the ground 0-node ny. fb) Another 1-network in the form of a transfinite ladder. All vertical resistances have the value 1 0. The horizontal voltage sources and resistances are as in (a). With e,, # 0

and E + O every branch has a non-zero current and yet neither ni nor n! has a node voltage with respect to nj

638 A. H. ZEMANIAN

fundamental theorem (Reference 1, Theorem 3.3-5), we can use superposition to determine the branch currents and branch voltages.

With e,, = F = 0 the branch currents are induced by all the horizontal sources and they are all zero because of the balance between the upper and lower paths. It follows as before that the node voltages at n: and n: are both infinite; that is, they fail to exist as finite voltages.

As the next case let e ,= 1 and let all the other voltage sources be zero. Now every branch to the left of the two 1-nodes carries a non-zero current. However, the two branches connected to those I-nodes have zero currents because there is no permissive 1-loop that passes through those 1 -nodes; indeed, each such 1 - loop must pass through an infinity of contiguous horizontal resistances and these will sum to infinity. Because the total power dissipated in all the resistances is finite, we can conclude that the voltages on the vertical 1 C2 resistors tend to zero as the 1-nodes are approached. In fact, the node voltages along the upper (lower) horizontal path decrease (increase) to the common voltage ( e , - i,)/2 at the 1-nodes. In short, n', and ni now have the same (finite) node voltage.

As the last case let c = 1 and let all the other voltage sources be zero. Again because of the absence of permissive I-loops, all the branches to the left of the 1-nodes carry zero currents and the two branches incident to the 1-nodes carry non-zero currents, namely ~ / 2 .

Upon adding these voltage-current regimes, we see that every branch of the 1-network of Figure 2(b) has a non-zero current and that the 1-nodes n: and n\ do not have node voltages with respect to ground.

Example 2

Let us now illustrate the second difficulty wherein node voltages exist throughout a transfinite network but at least some of them are not unique because they depend upon the choices of paths from ground to the nodes in question. Consider the 1-network of Figure 3 consisting of a sourceless lattice network and two appended source branches p, and p2, each of which is a series connection of a 1 V source and a 1 R resistor. The resistance values in the lattice are a, = bk = 1 R and ck = d, = 2 - k SZ for k = 1,2,3, . . .. Let t",, tl) and t:' be three 0-tips, each having a representative consisting only of a, branches, ck branches and d, branches respectively. Then let n),= [ t'l),n: = (n:, tz ) and n l= (n:, r,"IL where n:, n: and n: are three singleton 0-nodes containing respectively a 0-tip of P I , a 6-tip of p2 and a 0-tip of p2. The other 6-tip of PI is contained in the 0-node 11'1 as shown.

Since a, = 1 R for all k , there is no permissive 1-loop that passes through P I and t:; thus the current in P I is zero. The current in p2 is zero as well, but for a different reason. Even though each representative of t: and f : is permissive, there is no I-loop that passes through t z , b2 and t," because tl) and r: are non- disconnectable; that is, any representative of t l and any representative of r: meet (infinitely often) and therefore cannot be parts of the same loop. Since there is no source within the lattice network, we can conclude that all the branches carry zero currents.

1v /branch P I P

V

la

\

/ branch p2

Figure 3. The 1-network discussed in Example 2

NODE VOLTAGES IN A NON-LINEAR TRANSFINITE NETWORK 639

Now consider any possible node voltage at n:. There are many permissive paths connecting ground n: and nt, but they all pass through the source branch PI and are otherwise confined to the lattice. The voltage u:, assigned to nz along any one of these paths is 1 V. To be sure, there are other paths that reach nl through rt, but all of them are non-permissive. (Had we allowed node voltages to be assigned along non-permissive paths, ub would be 0 V for those paths.) In short, rz; has the unique node voltage u:= 1 V because of the requirement that node voltages be assigned only along permissive paths.

As for the node voltages u j and Z I ~ at nr and nl respectively, there is a permissive path starting at n:, passing along the c, branches and reaching n,' through f:. Thus v: = 0 V for that path. There is another permissive path starting at n; and passing through the d, branches, through n: and through the source branch p2 to reach n,' again. For that path, uf = 1 V. We thus have non-uniqueness for the node voltage at n:. The same conclusion can be drawn for n:.

Here too we can construct another example, wherein the lattice branches carry non-zero currents, simply by inserting some voltage sources in some of the lattice branches. The source branches P, and p2 would again have zero currents. Superposition would show that nb has a unique node voltage but nj and nl have non-unique node voltages.

4. A CLASS OF NON-LINEAR TRANSFINITE NETWORKS

We turn now to our second objective of finding conditions on our transfinite network insuring that it has unique node voltages at all nodes once a ground has been selected. We shall do this for all ranks of transfiniteness up to and including w , the first limit ordinal: thus 0s v d w. Our conclusions also hold for many ranks larger than w. This can be shown recursively simply by repeating the arguments used for the lower ranks v. The only difference would be that our notation would become more cumbersome.

Let N " (0 d v d w ) denote a non-linear v-network,Iv2 every branch of which is in the Norton form, as shown in Figure 4. (We choose the Norton form only for convenience; under the conditions imposed below, the Thevenin and Norton forms are entirely equivalent.) Thus the jth branch is a parallel connection of a pure current source h, and a non-linear resistor R,(.) carrying a current f,. In accordance with the polarity conventions shown in Figure 4, the branch voltage v, equals R,(f,) = RJ(i, + hJ), where i, is the branch current. Thus there is no coupling between branches. As always, we take the branch's orientation to be that indicated by the arrow for i, in Figure 4. In the special case of a linear resistor the constant branch resistance r, is the slope of the straight line f,H R,( f , ) , i.e. rJ = dR,/df,.

Condition I

N " (0 6 v d w ) is a non-linear v-network whose countably many branches are in the Norton form, shown in Figure 4. Every R, is a continuous, strictly increasing, odd mapping of the real line R' into R' with R,(f)+mas f+m.

R,(x) dx and M:(u) = /: G,( y ) dy where GI: ZIH G,(v) = f is the inverse function of R,: f H R,(B = u. Given that N" satisfies Condition 1, the modular sequence space 1, is defined

For f , u E [w' we set M,(fl=

Figure 4. The Norton form of a non-linear resistive branch with an independent current source. Here f, = i, + h, and v, = u, = I?,(&). The branch's orientation is indicated by the current arrows and also by the + to - direction of the voltage

640 A. H. ZEMANIAN

as the set of all sequences f = ( f l , f2 , ...) of real numbers f j such that C M , ( f , / t ) <m for some t >0, and C,

is the subspace of I, for which C M j ( f i / t ) < - for all t > O . (Here c denotes a summation over all branch indices j . ) f, and cM are Banach spaces with the norm

We assume henceforth the following.

Condition 2

The non-linear v-network N” satisfies Condition 1. Moreover, h = ( h l , h2, . . .) E I,. Furthermore, there are a positive integer j , and two positive numbers a and B, both greater than unity, such that for every j 2;”

(i) fR,(fl d a M , ( f ) for O s f d u,, where a, is the unique positive number for which M,(u ) = 1 (ii) uG,(u) s pM:(u) for 0 s u s d,, where d, is the unique positive number for which M, (d,) = 1.

Let f; be the dual of fM and let c i b e the dual of c,. Also, let e denote an isomorphism between Banach

*,

spaces. Under Condition 2 we have the following relations (Reference 1, pp. 129-130):

fM = CM (2)

(4) 1; = c ; z 1,. (3)

* * * (fM) (fM*) I(,*)* = fM

Thus lM is reflexive; that is, the dual of the dual of lM is isomorphic to lM. A basic current in the transfinite network N” is defined in Reference 1 (p. 154) and Reference 2 (p. 20).

It is a countable superposition of (generally transfinite) loop currents satisfying certain conditions. Lo is the span of all basic currents in f, and L is the closure of Lo in 1,. L is a linear subspace of and is a Banach space in itself when supplied with the norm of f,.

The next theorem (Reference 1, Theorem 5.5-7) is an adaptation to transfinite networks of a part of the fundamental theory established by DeMichele and Soardi6 for ordinary infinite non-linear networks. HereR denotes the resistance operator R ( f ) = ( R l ( f l ) , R 2 ( f 2 , ...), where f = (fl , f2 , ...). R maps 1, into 1,. (Reference 1, Lemma 4.7-2). According to the polarity conventions of Figure 4, the branch voltage vector v = ( v l , v2, .. .) is related to the branch current vector i = ( i l , i2, ...) and the branch current source vector h = ( h l , h,, .. .) through the equation v = R(i + h). Furthermore, for v E fM* and s E f,, (v, s) will denote the pairing (v, s) = Cv,s, E R 1 (Reference 1, page 126).

Theorem 1

Under Condition 2, there exists a unique i E L such that

(R(i + h), s) = 0

for all s E L. Furthermore, i satisfies Kirchhoffs current law at every finite maximal 0-node and v = R(i + h) satisfies Kirchhoff’s voltage law around every 0-loop and also around every 5-loop (1 s 5 6 Y) having a unit flow in L.

We assume henceforth that the voltage-current regime in N’ is the one dictated by Theorem 1.

5. PERMISSIVE PATHS AND NODE VOLTAGES

In this section we establish sufficient conditions for the non-linear v-network N“ to have node voltages. Our first task is to extend the idea of ‘permissivity’ to the paths, loops and tips in our non-linear network N‘. In

NODE VOLTAGES IN A NON-LINEAR TRANSFINITE NETWORK 641

the following, p and q are real numbers such that p - ' + q - ' = 1 and 1 < p < w . Thus = > q > 1. The inequality (6) below is illustrated in Figure 5.

Lemma I Assume that a branch resistance function Rj is bounded according to

(f)" d Rj(f) d pifP-'

for all f > 0, where yj and pi are positive numbers with 1/ y7-I Q pi. Then both restrictions (i) and (ii) of Condition 2 are satisfied by that RP

Proof. By (6) and for all f 5 0

Since p yj'-'pj > 1, we have obtained a sufficient condition for the satisfaction of (i) in Condition 2.

of Rj, we have from (6) that As for (ii), first note that q - 1 = (p - l)- ' . Hence, with u = R i ( n and with Gi being the inverse function

(j9-' Q Gj(u) d p9-'

for all u 3 0. Consequently

Now 1 < q c OQ and yjpp-' 5 1. Thus here too we have Condition 2.

qxp;-' 1; G , ( Y ) dY = qx.pj+MT(u)

a sufficient condition for the satisfaction of (ii) of 0

"I

t

Figure 5. The bounds on Rj and G, illustrated for the case where p > 2

642 A. H. ZEMANIAN

Definition 1

Let P denote either a 5-path or a <-loop (i; s Y ) with infinitely many branches (Reference 1, pp. 144, 147, 148; Reference 2, pp. 10, 12, 13). (Our conclusions hold trivially if P has only finitely many branches.) Also, let 0 be the index set for the branches embraced by P except for possibly finitely many of them. P is called strongly permissive if 0 can be so chosen that for every j E 0 there are two positive numbers y, and p/ such that the following hold.

( i ) c = yp-'p,, where c is independent o f j E 0 and 1 < c Sm.

(ii) The bounds (6) hold for all f 5 0 and j E 0. (iii) c, p, < 00.

If P is a representative of a <-tip t (Reference 1, pp. 140, 148; Reference 2, pp. 10, 12, 13), then f c is also called strongly permissive.

The algebraic sum of the branch voltages along the oriented 5-path or <-loop P is

where II is the index set for all the branches embraced by P and the plus (minus) sign is used if the orientations of P and the jth branch agree (disagree). Kirchhoffs voltage law, when it holds for a 5-loop P , asserts that (8) equals zero.

Lemma 2

Let P be a strongly permissive c-path or c-loop. Then (8) converges absolutely.

Proof. We need merely show absolute convergence for all but possibly finitely many of the terms in (8). Thus, with 0 as before and with c denoting a summation over the j E 0, we have by Holder's inequality and the relationship q ( p - 1) = p that

c I uj I = c I R,(A) I 4 c p j I f, i p - l = c p ; / q I f j ~ p - ~ p p (c pi I f j IPPYZ p l ) l / p

The right-hand side is finite. Indeed, c p,<- by (iii) of Definition 1. Also by (7), (i) of Definition 1 and the evenness of the MI we have

c P / If, l p P C c M/(f,)

Since every branch is in the Norton form, we have u j = v, = I?,(&). It follows from Theorem 1 and equation 0 (2) that f = i + h E L C I , = c,. This insures that 1 Ad,(&) < 00.

6. THE EXISTENCE AND UNIQUENESS OF NODE VOLTAGES

Theorem 2

Under Condition 2 and the voltage-current regime dictated by Theorem 1 Kirchhoff's voltage law is satisfied around every strongly permissive [-loop ([ 4 Y), and for such a loop, (8) converges absolutely.

Proof. By virtue of Theorem 1 and Lemma 2 we need merely show that every strongly permissive 5;- loop P , where [ 2 1, has a unit flow s in L. Thus let ll be the index set for all the branches embraced by P. With 0 defined as in Definition 1, 0 is an infinite set and Il\0 is a finite set. Let 1 sI I = 1 for all j E IT and let s, = 0 for all j $? n. Then

NODE VOLTAGES IN A NON-LINEAR TRANSFINITE NETWORK 643

The first summation on the right-hand side has finitely many terms and is therefore finite. For the second summation we have from Definition 1 and the evenness of the M, that

Hence s E I,. Moreover, since s is a loop current and since a loop current is a special case of a basic 0

Let n and m be two totally disjoint nodes (Reference 1, pp. 72, 141; Reference 2, p. 8). whose ranks need not be the same. Also let P be a (-path (( G Y ) that meets n and m terminally and is oriented from n to m. If P is strongly permissive, (8) converges absolutely according to Lemma 2 and we define (8) to be the node voltage of n with respect to rn along P. Let P and Q be two strongly permissive paths (with possibly differing ranks) that meet m and n terminally. If the ranks of P and Q are both zero, then (8) is the same along both paths. However, if at least one of them has a rank larger than unity, then (8) need not be the same for both, as was shown by Example 2. An additional condition is needed to insure that the choice of the strongly permissive path between m and n does not affect (8). The one used in Reference 4 (namely Condition 8.1 therein) for a linear network also works for the non-linear network considered herein. It is given by Condition 3 below.

We need the definition of 'non-disconnectable tips' for tips whose ranks are higher than zero. (This was defined only for 0-tips in Reference 1 (p. 104).) Recall that a representative of a p-tip, where p is any natural number, is a one-ended p-path which in turn is a one-way infinite alternating sequence of p-nodes n," and ( p - 1)-paths P,"-' of the form

current, we have s E L. Thus truly P has a unit flow in L.

~ " = { n ~ , P ~ - l , n ~ , P ~ - ' , n ~ , P ~ - ' , ...) (9)

where the first node nJ has a rank q d I( and certain conditions are satisfied (Reference 1, p. 144). (If p = 0, the ( p - 1)-paths embraced in (9) are replaced by branches.) Similarly, a representative of an &tip is a one- ended $-path which in turn is an alternating sequence of the form

p'J= {ni,Pp-' ,n';r,P';~- ' , n;>,pPq-' 2- ,... 1 (10)

where 7 < po < p i < p2 < . . . and again certain conditions are satisfied (Reference 1, p. 147). Now consider an infinite sequence (mi, mar m3, . . . ) of nodes m, with possibly differing ranks. We shall

say that the m, approach a p-tip t" (an &tip t') if there is a representative (9) for t" ((10) for fa) such that for each natural number i all but finitely many of the m, are shorted to nodes embraced by the members of (9) ((10)) lying to the right of nr (n").

Let t, and tb be two tips, not necessarily of the same rank. We say that t, and t, are non-disconnectable if there is an infinite sequence of nodes that approach both I, and tb.

We now restrict the transfinite graphs of the networks we consider by imposing the following condition; it is used in the proof of Theorem 3, but should be viewed as contributing to the sufficient conditions for that theorem's conclusion, not as a necessary one.

Condition 3

If two tips are non-disconnectable, then either they are shorted together or at least one of them is open. If

The next theorem and corollary comprise the conclusions of this paper.

those tips are non-disconnectable and strongly permissive, then they are shorted together.

Theorem 3

Under the hypothesis of Theorem 2, assume that the tips of ranks no larger than d in the v-network N' ( Y 4 w) satisfy Condition 3. Let ng and no be two nodes (of possibly different ranks) and let there be at least

644 A. H. ZEMANIAN

one strongly permissive path connecting ng and n,. Then no has a unique node voltage with respect to n,; that is, n,, obtains the same node voltage with respect to ng along all strongly permissive paths between ng and no.

Using now our definition of strong permissivity and the foregoing results of this section, we can prove Theorem 3 in virtually the same way as Theorem 8.2 of Reference 4 was proven. As before, the proof is based upon Kirchhoffs voltage law, which for our non-linear networks is asserted by Theorem 2.

Corollary 1

Under the hypothesis of Theorem 2 assume that the tips of all ranks no larger than 15 in the v-network N’ (v G o) satisfy Condition 3. Also assume that every two nodes of N’ are connected through at least one strongly permissive path, Choose a ground node ng in N’ arbitrarily. Then every node of N’ has a unique node voltage with respect to ng.

7. A FINAL COMMENT

All the results concerning non-linear transfinite networks that have been achieved so far deal only with resistive networks. What can be said about such networks having both resistive and reactive elements remains an open problem. In fact, this beckons as a completely new area of research in mathematical circuit theory.

ACKNOWLEDGEMENT

This work was supported by the National Science Foundation under Grants DMS-9200738 and MIP-9423732.

REFERENCES

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5 . J. Y. T. Woo, ‘On modular sequence spaces’, Stud. Math., 48,271-289 (1973). 6. L. DeMichele and P. M. Soardi, ‘A Thomson’s principle for infinite, nonlinear, resistive networks’, Proc. Am. Math. Soc., 109,

247-269 (1995).

461-468 (1990).