In this paper, we study the properties of effective impedances of finite electrical networks, considering them as weighted graphs over an ordered field. We prove that a star-mesh transform of finite network does not change its effective impedance. Moreover, we consider two particular examples of infinite ladder networks (Feynman’s network or *LC*-network and *CL*-network, both with zero at infinity) as networks over the ordered Levi–Cività field $R$. We show that the sequence of effective impedances of finite *LC*-networks converges to the limit in the order topology of $R$, but the sequence of effective impedances of finite *CL*-networks does not converge in the same topology. We calculate an effective impedance of a finite ladder network as an auxiliary result.

## I. INTRODUCTION

It is known that electrical networks with resistances are related to weighted graphs, where weights of edges are positive real numbers (see, e.g., Refs. 9, 19, and 22). Moreover, it is known that an effective resistance of the finite network does not change under some transforms of a network [star-mesh transform, parallel and series laws, etc. (see, e.g., Refs. 14 and 19)]. In Refs. 9 and 14, the notion of effective resistance for the infinite network is stated. An effective resistance is tightly related to the random walk and Dirichlet problem on a graph, which are described in many papers and books (e.g., Refs. 2, 13, 22, and 26). In Ref. 20, a finite electrical network with alternating current and passive elements is considered as a generalization of the electrical network with resistances. It is shown there that such a network is related to weighted graphs over non-Archimedean ordered fields of rational functions $R(\lambda )$. The generalization of effective resistance for this case is called *effective impedance*. The inverse of effective impedance is called *effective admittance*. The most known physics infinite network with passive elements is Feynman’s ladder network (*LC*-network, see Ref. 12). In Ref. 27, the effective impedances of the *LC*-network and *CL*-network are considered as limits of complex-valued effective impedances of exhausted sequences of finite networks. Moreover, in Refs. 1 and 7, electrical networks on fractals are considered. In the present paper, we bring together the theory of random walks on graphs (where weights are always positive, see, e.g., Ref. 9) and electrical engineering, where electrical networks with impedances are widely considered. Moreover, we present a new application of the Levi–Cività fields whose other applications to mathematical physics are described in Refs. 3 and 4.

This paper is organized as follows: In Sec. II, we describe some properties of an electrical network over an ordered field. We consider an electrical network with passive elements as a weighted graph whose weights are positive elements of any ordered field. This is a natural generalization of weighted graphs over $R$. Moreover, in the previous paper by the same author,^{20} some statements about physical voltage and physical effective impedance can be shown using a theory of networks over ordered fields. The main result of Sec. II is given in Theorem 4, which gives the mathematical description of well-known in electrical engineering star-mesh transform (see, e.g., Ref. 16). The mathematical conceptions of parallel and series laws, as well as Δ − *Y* and *Y* − Δ transform, follow from Theorem 4 as corollaries.

In Sec. III, we discuss the question whether one can generalize the notion of effective resistance for infinite networks with zero on infinity (see, e.g., Refs. 14 and 22) for the case of non-Archimedean weighted graphs. The main theorem of this section is Theorem 15. It shows that a sequence of effective admittances of finite networks, exhausted a given infinite network, decreases. Unfortunately, it does not give a convergence over a non-Archimedean field. First, we present the general calculation of admittance of a finite ladder network (see Fig. 7) over an ordered field. Then, we consider the *LC*-network and *CL*-network (with zero on infinity) as electrical networks over ordered Levi–Cività field $R$, which contains a subfield isomorphic to $R(\lambda )$ (see Refs. 3, 4, and 15). The ordered Levi–Cività field (which is non-Archimedean) is widely used in different branches of mathematics, mathematical physics, and programming. This field $R$ has “infinitely small” elements, which qualitatively differ it from the field of real numbers $R$, where all elements are finite. The concept of “infinitely small” element allows more precise computations, particularly in computational differentiation (see Refs. 3, 4, and 17, p. 373–383). In Ref. 4, how techniques of the Levi–Cività field can be used for the solving of ordinary differential equations (ODE) and partial differential equations (PDE) over real numbers is described. All the above show the importance of the $R$ as an extension of $R$. Indeed, considering networks with weights from $R$, we show more interesting and smaller results for infinite networks in the case of real-weights. A Cauchy completeness of the Levi–Cività field in order topology (Refs. 3 and 21) gives us an opportunity to arise the question whether effective admittance of the infinite network could be defined as the limit of effective admittances of corresponding finite electrical networks in this case. We show that in the case of the *LC*-network, the sequence of effective admittances of finite networks converges in ordered topology of the Levi–Cività field (Theorem 20). The result is similar to the classical Feynman’s result, which was stated in Ref. 12, but is not carefully explained there. Other explanations for this result are given in Refs. 10, 18, 23, 24, and 27. Moreover, we show that admittances of the finite *CL*-network do not converge in the same topology (example 23). The latest example shows that, in general, it is not possible to generalize the notion of effective resistance for infinite networks for the case of non-Archimedean weights, or, more precisely, in the case of $R$-weights, there is some other possibility for a network, apart from being transient or recurrent. This again shows that computations over the Levi–Cività field are more precise than over the real numbers. Moreover, we give a sufficient condition on an infinite ladder network for convergence of the sequence of admittances (Remark 22).

## II. PROPERTIES OF EFFECTIVE IMPEDANCE FOR FINITE NETWORK OVER ORDERED FIELD

We consider locally finite connected graphs whose weights are positive elements of some fixed ordered field. Moreover, at each graph, we fix a vertex with potential 1 and a set of vertices with potential 0. Such a structure we call a network. Let us give a precise definition.

*K*, ≻) is a structure,

(

*V*,*E*) is a locally finite connected graph without loops (*V*is a set of vertices, |*V*| ≥ 2,*E*is a set of edges),*ρ*:*V*×*V*→*K*is a non-negative function (*ρ*_{xy}⪰ 0) such that*ρ*_{xy}= 0 if and only if*xy*∉*E*,*a*_{0}∈*V*is a fixed vertex, and*B*= {*a*_{1}, …,*a*_{|B|}} ∈*V*\{*a*_{0}} is a fixed non-empty subset of vertices.

The function *ρ* is called *admittance*. We denote by *B*_{0} = *B* ∪ {*a*_{0}} *the set of boundary vertices* and by $z=1\rho $ a positive function of *impedance*, defined on *E*. Since the function *ρ* uniquely determines the set of edges, we will omit edges in the notation of a network and write Γ = (*V*, *ρ*, *a*_{0}, *B*).

For any *x*, *y* ∈ *V*, we write *x* ∼ *y* if *xy* ∈ *E*, and *x*, *y* ∉ *V* otherwise.

The network is called *finite* if |*V*| < *∞*. Otherwise, it is called *infinite*.

Since the graph is locally finite, the weight *ρ*_{xy} gives rise to a function on vertices as follows:

where here and further in notations $\u2211y$ means $\u2211y\u2208V$. Then, *ρ*(*x*) is called the *weight of a vertex**x*. We have 0 < *ρ*(*x*) ∈ *K* for any vertex *x* of a network.

Let us consider the following system of linear equations on a given finite network Γ = (*V*, *ρ*, *a*_{0}, *B*):

where $\Delta \rho v(x)=\u2211y(v(y)\u2212v(x))\rho xy$.

We consider (2) as a discrete boundary value *Dirichlet problem*. This Dirichlet problem arises from electrical engineering. Indeed, by Ohm’s complex law (see, e.g., Refs. 8 and 11), an impedance of any edge *xy*, containing passive elements (resistor, coil, and capacitor), can be written in the following form:

where *R*_{xy} > 0 is a resistance, *L*_{xy} > 0 is an inductance, *D*_{xy} > 0 is the inverse of a capacitance ($Dxy=1Cxy$), *ω* is a frequency of the alternating voltage, and *i* is the imaginary unit. Then, denoting *λ* = *iω* (this idea goes back to Ref. 6), we have for an admittance of an edge,

where *L*_{xy} = 0, *R*_{xy} = 0, or *D*_{xy} = 0 means the lack of the corresponding element on the edge.

Hence, defined *ρ*_{xy} is a positive element in the ordered field of rational functions with real coefficients $R(\lambda )$ with the order, defined as follows [e.g., Ref. 5 (p. A.VI.21) and Ref. 25 (p. 234)]: for any rational function,

with *b*_{k} ≠ 0, *d*_{m} ≠ 0; write

and

Afterward, we assume that we keep potential (voltage) 1 at the vertex *a*_{0} and ground the set *B* (i.e., keep potential zero). Then, all the other vertices should be considered as inner nodes, and by Kirchhoff’s law, Δ_{ρ}*v*(*x*) = 0 for them (a sum of incoming flows is equal to the sum of outgoing flows, see, e.g., Refs. 8, 11, and 16). Therefore, the Dirichlet problem (2) appears.

The Dirichlet problem (2) is a *n* × *n* system of linear equations over the field *K*, where *n* = |*V*|. It can also be written in a matrix form [here, we already have substituted $v(a0)=1,v(ai)=0,i=1,|B|\u0304$ in the first (*n* − |*B*| − 1) equations],

where *k* = |*B*|, *A* is a symmetric matrix (*A* = *A*^{T}), and $v\u0302,b$ are vector-columns of length (*n* − *k* − 1),

In Ref. 20, it is proved that the Dirichlet problem (2) has a unique solution for any finite network over an ordered field in the case of |*B*| = 1. Moreover, exactly this fact, applied to the ordered field of rational functions, allows for the strict definition of the physical effective impedance (as a complex number), which shows a deep relation of the present theory to mathematical physics (see Ref. 20). In the case of |*B*| > 1, the proof follows exactly the same outline; therefore, we do not present it here.

*effective admittance*of a finite network Γ as

*v*is the solution of the Dirichlet problem (2).

*effective impedance*is defined by

*Z*

_{eff}(Γ) ∈

*K*∪ {

*∞*}.

*For the solution*

*v*

*of the Dirichlet problem (2), we have*

*where*∇

_{xy}

*v*=

*v*(

*y*) −

*v*(

*x*)

*.*

The proof of this result follows the same outline as the proof of the similar result in Ref. 20.

There are transforms of electrical networks, which are widely used in electrical engineering for calculation of effective impedances. The most known transforms are star-mesh transform, *Y* − Δ and Δ − *Y* transform, and parallel and series laws (Figs. 1–5). A mathematical justification of these transforms in the case of weighted graphs over $R$ is known (see, e.g., Ref. 19). We justify these transforms in the case of arbitrary ordered field. Therefore, we can also apply our theorems to $R$ in order to get classical results about these transforms. The main statement is Theorem 4 (star-mesh transform), since other transform happens to be its corollaries. Multigraphs are not allowed in this paper; therefore, we use a modification of the parallel law and refer to it as a *parallel-series law*.

(star-mesh transform). *Let* Γ = (*V*, *ρ*, *a*_{0}, *B*) *be a finite network,* |*V*| = *n**,* *B*_{0} = *B* ∪ {*a*_{0}}*, and* *x*_{1}, …, *x*_{m} ∈ *V**,* 3 ≤ *m* ≤ *n**, are such that*

*x*_{1}∉*B*_{0}*and**y*≁*x*_{1}*for all**y*∈*V*\{*x*_{2}, …,*x*_{m}}*.*

*If one removes the vertex*

*x*

_{1}

*, edges*$x1xi,i=2,m\u0304$

*and changes the admittances of the edges*$xixj,i,j=2,m\u0304,i\u2260j$

*as follows:*

*not changing the other admittances, then for the new network, the solution of the Dirichlet problem (2) for all the vertices will be the same as the solution of the Dirichlet problem (2) on the original network at the corresponding vertices.*

*x*

_{1}, …,

*x*

_{l}∉

*B*

_{0}, where

*l*=

*m*− |{

*x*

_{1}, …,

*x*

_{m}} ∩

*B*

_{0}|. Writing equations for

*x*

_{1}, …,

*x*

_{l}as the first ones and denoting

*k*= |

*B*|, we have

*y*≁

*x*

_{1}for all

*y*∈

*V*\{

*x*

_{2}, …,

*x*

_{m}}, and

*ρ*(

*x*

_{1}) ≠ 0, where

*v*(

*x*

_{1}) from the Dirichlet problem, changing admittances as in the statement of the theorem.□

*Under the star-mesh transform of a network, the effective impedance and the effective admittance do not change.*

In the proof, we are using notations from the proof of Theorem 4.

The case {*x*_{1}, …, *x*_{m}} ∩ *B*_{0} = ∅ is trivial. The cases, when {*x*_{1}, …, *x*_{m}} ∩ *B* = ∅ or {*x*_{1}, …, *x*_{m}} ∩ {*a*_{0}} = ∅ are obvious due to (5).

*x*

_{m}=

*a*

_{0}. Then, if we denote the new network by Γ′ we have, by (5),

*v*(

*x*

_{j}) = 0 for all $j=j+1,m\u22121\u0304$ and

*a*

_{0}=

*x*

_{m}].□

(series law). *Let* Γ = (*V*, *ρ*, *a*_{0}, *B*) *be a finite network,* *B*_{0} = *B* ∪ {*a*_{0}}*. Let* *a*, *b*, *c* ∈ *V* *are such that*

*b*∉*B*_{0}*,**a*≁*c**,**a*∼*b**,**b*∼*c**, and**b*≁*x**for all**x*∉ {*a*,*c*}*.*

*If one removes the vertex*

*b*

*and edges*

*ab*,

*bc*

*and adds the edge*

*ac*

*with the admittance,*

*not changing other admittances, then for the new network, the solution of the Dirichlet problem (2) for all the vertices will be the same as the solution of the Dirichlet problem (2) on the original network at the corresponding vertices. The effective impedance (admittance) of the new network coincides with the effective impedance (admittance) of the original one.*

Apply Theorem 4 and Corollary 5 (*x*_{1} = *b*) for the case *m* = 3 and *ρ*_{ac} = 0.□

(parallel-series law). *Let* Γ = (*V*, *ρ*, *a*_{0}, *B*) *be a finite network,* *B*_{0} = *B* ∪ {*a*_{0}}*.*

*Let* *a*, *b*, *c* ∈ *V* *are such that*

*b*∉*B*_{0}*,**a*∼*b*,*b*∼*c*,*a*∼*c**, and**b*≁*x**for all**x*∉ {*a*,*c*}*.*

*If one removes the vertex*

*b*

*and edges*

*ab*,

*bc*

*and adds the edge*

*ac*

*with the admittance,*

*not changing other admittances, then for the new network, the solution of the Dirichlet problem (2) for all the vertices will be the same as the solution of the Dirichlet problem (2) on the original network for the corresponding vertices. The effective impedance (admittance) of the new network coincides with the effective impedance (admittance) of the original one.*

Apply Theorem 4 and Corollary 5 (*x*_{1} = *b*) for the case *m* = 3.□

(*Y* − Δ transform). *Let* Γ = (*V*, *ρ*, *a*_{0}, *B*) *be a finite network,* *B*_{0} = *B* ∪ {*a*_{0}}*. Let* *a*, *b*, *c*, *d* ∈ *V* *are such that*

*d*∉*B*_{0}*,**d*∼*a*,*d*∼*b*,*d*∼*c**, and**d*≁*x**for all**x*∉ {*a*,*b*,*c*}*.*

*If one removes the vertex*

*d*

*and edges*

*da*,

*db*,

*dc*

*and sets*

*not changing other admittances, then for the new network, the solution of the Dirichlet problem (2) for all the vertices will be the same as the solution of the Dirichlet problem (2) on the original network for the corresponding vertices. The effective impedance (admittance) of the new network coincides with the effective impedance (admittance) of the original one.*

*ρ*

_{ab},

*ρ*

_{bc},

*ρ*

_{ac}are all equal to zero, then it is an

*Y*− Δ transform; otherwise, it is an

*Y*− Δ transform and a parallel law.

Theorem 4 and Corollary 5 (*x*_{1} = *d*) for the case *m* = 4.□

The *Y* − Δ transform is invertible. In general, it is not the case for a star-mesh transform.

*Y*transform).

*Let*Γ′ = (

*V*′,

*ρ*′,

*a*

_{0},

*B*)

*be a finite network, and let*

*a*,

*b*,

*c*∈

*V*

*are such that*

*a*∼

*b*,

*b*∼

*c*

*, and*

*a*∼

*c*

*. If one adds a vertex*

*d*

*and edges*

*da*,

*db*,

*dc*

*setting*

*and removes the edges*

*ab*,

*bc*,

*ac*,

*not changing other admittances, then for the new network,*

*the solution of the Dirichlet problem (2) for any vertex in*

*V*

*will be the same as the solution of the Dirichlet problem (2) on the original network for the corresponding vertex. Moreover, the effective impedance and the effective admittance do not change under this transform.*

*ρ*

_{da},

*ρ*

_{db}, and

*ρ*

_{dc}from (8), assuming

*ρ*

_{ab}= 0,

*ρ*

_{bc}= 0, and

*ρ*

_{ac}= 0. Summing up the inverses of all three equations, one obtains

*ρ*

_{da}follows. To obtain the equations for

*ρ*

_{db}and

*ρ*

_{dc}, one should multiply (10) by $1\rho ab\u2032\rho bc\u2032$ and $1\rho ac\u2032\rho bc\u2032$, respectively.

The fact that an effective impedance and an effective admittance do not change follows from Theorem 10.□

All transforms described in this section preserve the positivity of admittances and impedances on the edges.

## III. EFFECTIVE IMPEDANCE OF INFINITE NETWORKS OVER AN ORDERED FIELD

In this section, we introduce a notion of finite approximations for any infinite network over an ordered field and discuss the convergence of the sequence of effective admittances of finite approximations in the order topology. We assume that any infinite network has a potential zero at infinity. We remind that a subset *M* ⊂ *K* is open in the order topology of an ordered field (*K*, ≻) if and only if for any *k*_{0} ∈ *M*, there exists *ɛ* ≻ 0, *ɛ* ∈ *K* such that an *ɛ*-ball

is a subset of *M*.

As an important example, we calculate an effective impedance of an infinite *LC*-network over the Levi–Cività field.

### A. Infinite networks

Let Γ = (*V*, *ρ*, *a*_{0}, *B*) be an infinite network over an ordered field (*K*, ≻). Let us consider the sequence of finite graphs $(Vn,\rho Vn\xd7Vn)$, where *V*_{n} = {*x* ∈ *V*∣dist(*a*_{0}, *x*) ≤ *n*}, $n\u2208N$.

We denote by

the *boundary* of the graph $(Vn,\rho Vn\xd7Vn)$. Note that *V*_{n+1} = *∂V*_{n+1} ∪ *V*_{n}.

Let us denote *B*_{n} = *B* ∩ *V*_{n}. Then,

is a *sequence of finite networks exhausted the infinite network* Γ.

This is an analog to the classical approach to infinite networks with resistors (see, e.g., Refs. 9, 14, and 22).

Let us consider the Dirichlet problem (2) on each Γ_{n},

^{20}we have

*f*:

*V*

_{n+1}→

*K*such that $f(a0)=1,f\u2202Vn+1\u222aBn+1\u22610$.

*V*

_{n+1}\

*∂V*

_{n+1}) =

*V*

_{n}and

*B*

_{n+1}∩

*V*

_{n}=

*B*

_{n}, the inequality (13) is true for

*v*

^{(n)}is the solution of (11) for Γ

_{n}. Then,

Even in a Cauchy complete, non-Archimedean ordered field inequalities (12) for all $n\u2208N$ do not imply that the sequence ${Peff(\Gamma n)}n=1\u221e$ converges. Obviously, if the sequence of effective admittances of finite networks converges, then the corresponding sequence of the effective impedances also has a limit (finite or infinite).

If for the given infinite network Γ, the limit of effective admittances (impedances) of exhausted finite networks exists in *K*, we call this limit the *effective admittance (impedance) of the network* Γ *with zero potential at infinity* and denote it by $Peff(\Gamma )$ [*Z*_{eff}(Γ)].

### B. Examples: Ladder networks over Levi–Cività field

Ladder networks are objects of interest of mathematical physics and electrical engineering for a very long time, since Richard Feynman (in 1966) has described an infinite *LC*-network in his *Lectures on Physics* [see Ref. 12 (p. 22-13) and Refs. 10, 18, 23, 24, and 27]. In all these sources, the effective impedance was calculated over the field of real numbers. We would like to consider admittances as elements of the Levi–Cività field $R$. In this section, we show that the sequence of effective impedances of finite *LC*-networks converges to the limit in the order topology of $R$, but the sequence of effective impedances of finite *CL*-networks does not converge in the same topology. These examples show a difference with the case of real positive weights, since there any network is either transient (effective resistance is finite) or recurrent (effective resistance is infinite) (see, e.g., Refs. 9 and 26). Let us investigate the behavior of the sequence ${Peff(\Gamma n\alpha \beta )}n=1\u221e$ of effective admittances of finite networks exhausted the ladder network in Fig. 6 (*α*, *β* ∈ *K*, *α*, *β* ≻ 0). More precisely, let us consider a network Γ^{αβ} = (*V*, *ρ*, *a*_{0}, *B*), where

*V*= {*a*_{0},*a*_{1},*a*_{2}, …,*x*_{1},*x*_{2}, … },$\rho a0x1=\alpha $, $\rho xkxk+1=\alpha ,\rho akxk=\beta ,k\u2208N$, and

*ρ*_{xy}= 0, otherwise, and*B*= {*a*_{1},*a*_{2}, … }.

This network is similar to Feynman’s ladder network and *CL*-network (see Refs. 12 and 27) but has zero potential at infinity. Therefore, for any ordered field *K*, Theorem 15 is true for this network. We will refer to this network as an *αβ**-network*. We show (Theorem 20 and example 23) that whether ${Peff(\Gamma n)}n=1\u221e$ converges in a Cauchy completion of *K* depends on *α* and *β*.

#### 1. Finite ladder network over ordered field

Let Γ_{n} be a sequence of finite networks exhausted an *αβ*-network (see Fig. 7).

The Dirichlet problem (2) for this network is as follows:

Using the second line in (14), we obtain the following recurrence relation for $v(xk),k=2,n\u22121\u0304$:

since *v*(*a*_{k}) = 0 for $k=2,n\u22121\u0304$. The characteristic polynomial of (15) is

Its roots are

where

Note that *ξ* should not necessarily belong to *K*. However, it is known that any ordered field possess a real-closed extension $K\u0304$. Then, in $K\u0304$ exists exactly one positive square root of $\beta \alpha +\beta 2\alpha 2$ (see, e.g., Ref. 5). Therefore, we fix the extension $K\u0304$, denote the positive square root by *ξ*, and make all the further calculations in $K\u0304$.

The solution of the recurrence equation (15) is

where $c1,c2\u2208K\u0304$ are arbitrary constants.

We use first and third equations in (14) as boundary conditions for this recurrence equation. Substituting (18) in the boundary conditions, we obtain the following equations for the constants:

which, by (16), is equivalent to

Therefore,

since *ψ*_{1}*ψ*_{2} = 1 by (16).

Now, we can calculate the effective admittance of Γ_{n},

Since $Peff\Gamma n$ is an element of *K* as a rational function of the solution of Dirichlet problem (2) over *K*, it can also be written without *ξ*,

where in the last line, we have used binomial expansion.

#### 2. Infinite ladder networks over the Levi–Cività field

We consider two examples of the *αβ*-network over the Levi–Cività field $R$. First, let us describe the Levi–Cività field $R$. We take the definition of $R$ and theorems about its properties from Refs. 3, 4, and 21.

A subset *M* of the rational numbers $Q$ is called left-finite, if for every $r\u2208Q$, there are only finitely elements of *M* that are smaller than *r*.

*addition*is defined component-wise,

*multiplication*is defined as follows:

It is proved in Ref. 3 that $R$ is an ordered field with a set of positive elements,

We denote by *τ* the following element in $R$:

which plays a role of infinitesimal in the Levi–Cività field. Therefore, the Levi–Cività field is a non-Archimedean ordered field.

In Ref. 3, we can write any $\alpha \u2208R$ as

since $\alpha n=\u2211i=1n\alpha (qi)\tau qi$ converges strongly to the limit *α* in the order topology.

The set of all polynomials over real numbers $R[\tau ]={an\tau n+\u2026a1\tau +a0\u2223ai\u2208R,n\u2208N}$ is a subring of Levi–Cività field $R$ due to (22). Therefore, since $R$ is a field, the field of rational functions over real numbers

is isomorphic to a subfield of $R$.

*τ*at right hand side and left hand side, starting from the lowest power, one obtains

*i*> 2. Therefore, solving the recurrence relation for

*α*(

*q*

_{i}), we obtain

Note that the corresponding order in the field $R(\tau )$ is as follows:

Therefore, we can consider the Levi–Cività field $R$ as an ordered extension of the ordered field $R(\tau )$ with the positiveness defined as (24). The isomorphism between $R$ and the ordered extension of the ordered field $R(\lambda )$ with the positiveness defined by (3) is given by a substitution,

Consequently, we can consider electrical networks over fields of rational numbers,^{20} as networks over the Levi–Cività field, and investigate the behavior of the sequence of effective admittances of finite electrical networks. In Ref. 3, the Levi–Cività field is Cauchy complete in the order topology and real-closed.

The most known ladder network is Feynman’s ladder (see Ref. 12). Its properties from the point of view of electrical engineering are described in Refs. 10, 18, 23, 24, and 27. The other studied ladder network is the *CL*-*network* (see, e.g., Ref. 27). We consider both these networks over the Levi–Cività field, which is a completely new approach.

Let us consider Feynman’s infinite ladder *LC*-network, assuming that it has zero potential at infinity (see Fig. 8). It is an *αβ*-network with *α* = *L*^{−1}*τ* and *β* = *Cτ*^{−1}, where *L*, *C* > 0, $\alpha ,\beta \u2208R$ (substitutions *λ* = *iω*, *τ* = 1/*λ*).

*For Feynman*’

*s ladder*

*LC*

*-network (*

*α*=

*L*

^{−1}

*τ*

*,*

*β*=

*Cτ*

^{−1}

*, where*

*L*,

*C*> 0

*) with zero potential at infinity,*

*in the order topology of the Levi–Cività field*$R$

*, where*Γ

_{n}

*is the sequence of the exhausted finite networks,*

*ξ*

*is defined by (17).*

*LC*-network,

*LC*-network (see Ref. 12).

*ξ*as an element of the Levi–Cività field, i.e., as power series (22),

*A*of the last expression is

*n*→

*∞*. Indeed, for any

*ɛ*≻ 0, $\epsilon =\u2211i=1\u221e$, we have

For the *CL*-network (*α* = *Cτ*^{−1}, *β* = *L*^{−1}*τ*, *L*, *C* > 0, see Fig. 9), effective admittances of the exhausted finite networks do not converge in the Levi–Cività field $R$.

Therefore, the following question:

Under what conditions could the effective admittance of the infinite network over the non-Archimedean field be defined?

remains open. Note that Remark 22 gives some sufficient condition for the *αβ*-network.

## ACKNOWLEDGMENTS

This research was supported by IRTG 2235 Bielefeld-Seoul, “Searching for the regular in the irregular: Analysis of singular and random systems.”

The author thanks her scientific advisor, Professor Alexander Grigor’yan, for fruitful discussions on the topic.

## DATA AVAILABILITY

Data sharing is not applicable to this article as no new data were created or analyzed in this study.