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István Ecsedi Institute of Applied Mechanics, Faculty of Mechanical Engineering and Informatics, University of Miskolc, H-3515 Miskolc-Egyetemváros, Hungary

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Attila Baksa Institute of Applied Mechanics, Faculty of Mechanical Engineering and Informatics, University of Miskolc, H-3515 Miskolc-Egyetemváros, Hungary

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## Abstract

A mathematical model is developed to determine the steady-state electric current flow through in non-homogeneous isotropic conductor whose shape has a three-dimensional hollow body. The equations of the Maxwell's theory of electric current flow in a non-homogeneous isotropic solid conductor body are used to formulate the corresponding electric boundary value problem. The determination of the steady motion of charges is based on the concept of the electrical conductance. The derivation of the upper and lower bound formulae for the electrical conductance is based on Cauchy-Schwarz inequality. Two numerical examples illustrate the applications of the derived upper and lower bound formulae.

## Abstract

A mathematical model is developed to determine the steady-state electric current flow through in non-homogeneous isotropic conductor whose shape has a three-dimensional hollow body. The equations of the Maxwell's theory of electric current flow in a non-homogeneous isotropic solid conductor body are used to formulate the corresponding electric boundary value problem. The determination of the steady motion of charges is based on the concept of the electrical conductance. The derivation of the upper and lower bound formulae for the electrical conductance is based on Cauchy-Schwarz inequality. Two numerical examples illustrate the applications of the derived upper and lower bound formulae.

## 1 Introduction

Electrical resistance of an electrical conductor is a measure of the difficulty to pass a steady electric current through the conductor. The well-known elementary form of Ohm's law states that when the conductor carries a current I from a point P 1 at potential U 1 to a point P 2 at potential U 2 then U 1U 2 = RI, where R is the resistance of the conductor between points P 1 and P 2, it depends only on the shape and temperature and the material of the conductor. The inverse of electric resistance is the electric conductance G = 1/R. This paper deals with the electric resistance of a three-dimensional non-homogeneous conductor body. Examination of non-homogeneous structural elements is a very important task. Maróti's study [1] deals with the bending vibration of axially non-homogeneous beams. The buckling problem of axially functionally graded beams is considered in paper [2]. For prescribed frequency and buckling loads Maróti and Elishakoff [2] determined the Young's modulus in axial direction as a function of axial coordinate. The non-homogeneous isotropic hollow conductor is bounded by two closed surfaces $∂ V 1$ and $∂ V 2$ , which have no common point. The current flows inside the conductor from inner boundary surface $∂ V 1$ whose potential is $U 1$ to the outer boundary surface $∂ V 2$ whose potential is $U 2$ , $U 1 > U 2$ . Two-side estimation will be proven for the electrical conductance of non-homogeneous isotropic hollow three-dimensional conductor. The mathematical formalism follows the methods, which were used in papers [3–5]. In paper [3] upper and lower bounds are proven for the electrical resistance of homogeneous isotropic ring like axisymmetric conductor. In paper [4] the capacitance of two-dimensional cylindrical capacitor, which consists of non-homogeneous dielectric materials is studied. Examples illustrate the applications of the derived bounding formulae of capacitance [4]. A mathematical heat transfer model is developed for the steady-state heat transfer problem for homogeneous isotropic body of rotation in [5] and it is used to obtain estimations of thermal heat transfer conductance.

Let us consider the steady motion of charges in the non-homogeneous hollow conductor shown in Fig. 1. The conductor body occupies the space domain V and its boundary surfaces are $∂ V 1$ and $∂ V 2$ . The electric potential U on the boundary surfaces $∂ V 1$ and $∂ V 2$ are prescribed, so the following boundary conditions are valid [6–8],
$U ( r ) = U i = c o n s t a n t , r ∈ ∂ V i ( i = 1,2 ) ,$
where $r$ denotes the position vector (Fig. 1). According to Maxwell's theory [6–8] the steady motion of charges is described by the next equations:
Differential form of Ohm's law formulates that at constant temperature in isotropic conductor the current density vector $j$ is proportional to the electric field vector $E .$ Here $σ = σ ( r )$ is the conductivity of the non-homogenous hollow conductor. In Eq. (2) $∇$ is the del operator and the dot between two vectors denotes the scalar product [9]. From the above equations it follows that
Introducing a new function $u = u ( r )$ by the next definition,
$U ( r ) = ( U 1 − U 2 ) u ( r ) + U 2 U 1 ≠ U 2 .$
It is evident that $u = u ( r )$ satisfies the following boundary value problem,
$σ ( r ) Δ u + ∇ σ ⋅ ∇ u = 0 , r ∈ V , u = 1 , r ∈ ∂ V 1 , u = 0 , r ∈ ∂ V 2 .$
The function $u = u ( x , y )$ plays crucial role in the expressions of electrical resistance and electrical conductance. An electric current in the conductor is the continuous passage of the current along that conductor. The constant potential difference between the closed surfaces $∂ V 1$ and $∂ V 2$ maintains the steady flow of the electric current. The amount of charge flowing through surface $∂ V 1$ per unit time is I. The determination of I is based on the next equation
$I = − ∫ ∂ V 1 j ⋅ n d A = ( U 1 − U 2 ) ∫ ∂ V 1 σ ( r ) n ⋅ ∇ u d A = ( U 1 − U 2 ) ∫ ∂ V 1 σ ( r ) ∂ u ∂ n d A .$
In Eq. (6), n is the outer normal unit vector of the inner boundary surface $∂ V 1$ and $d A$ is the area element of $∂ V 1$ . The electrical resistance R and the conductance G of the hollow conductor is defined as [6, 8],
$R = U 1 − U 2 I = 1 ∫ ∂ V 1 σ ( r ) ∂ u ∂ n d A , G = I U 1 − U 2 = ∫ ∂ V 1 σ ( r ) ∂ u ∂ n d A .$
From Eq. (5) it follows that
$∫ V u [ σ ( r ) Δ u + ∇ σ ⋅ ∇ u ] d V = ∫ ∂ V 1 u σ ( r ) n ⋅ ∇ u d A − ∫ V σ ( r ) | ∇ u | 2 d V = 0 ,$
$G = ∫ V σ ( r ) | ∇ u | 2 d V , R = 1 ∫ V σ ( r ) | ∇ u | 2 d V .$
Note that if
$∇ σ ⋅ ∇ u = 0 , r ∈ V ,$
then $u ( r ) = u 0 ( r ) ,$ where $u 0 ( r )$ is a unique solution of the following Dirichlet type boundary-value problem
In this case
$G = ∫ V σ ( r ) | ∇ u 0 | 2 d V .$

There are several approximation methods to get the solution of the boundary-value problem Eq. (5), most of which use the results of variational calculus for example as Ritz method, finite element method [8, 9]. Other methods are also known and they used, for example finite difference methods, method of weighted residuals, boundary element method [10]. It must be mentioned that, many numerical-analytical method are used R-functions to solve the boundary value problems of electrodynamics [11–13]. The efficiency of the R-Function Method (RFM) to solving the boundary value problems of electrostatics in very complicated domain is illustrated in paper by Kravchenko and Basarab [14]. They considered a boundary-value problem of electrodynamics in the fractal regions of the Sierpiski carpet and the Koch island types [14]. Iványi solved a number of two-dimensional boundary value problems of static and stationary electromagnetisms by variational method connecting of them with the use of R-functions [12, 13, 15, 16]. It is not the aim of this paper is to give a detailed list of different analytical and numerical methods, which are used widespread in electrical engineering calculations.

## 2 Upper bound for G and lower bound for R

If the function $F = F ( r )$ which is continuously differentiable in $V ∪ ∂ V$ satisfies the boundary conditions (13) then the inequality relation (14) is valid

$F ( r ) = 1 , r ∈ ∂ V 1 , F ( r ) = 0 , r ∈ ∂ V 2 ,$
$G ≤ ∫ V σ ( r ) | ∇ F | 2 d V .$

The proof of inequality (14) 3 can be derived by the Cauchy-Schwarz inequality relation (15),

$( ∫ V σ ( r ) ∇ F ⋅ ∇ u d V ) 2 ≤ ∫ V σ ( r ) | ∇ F | 2 d V ⋅ ∫ V σ ( r ) | ∇ u | 2 d V .$
A simple computation leads to the result
$∫ V σ ( r ) ∇ F ⋅ ∇ u d V = ∫ ∂ V 1 σ ( r ) n ⋅ ∇ u d A = ∫ V σ ( r ) | ∇ u | 2 d V .$

The combination of the inequality relation (15) with Eq. (16) and using formula (9) gives (14). A brief discussion shows that the sign of equality in relation (14) is valid only if $F ( r ) ≡ u ( r ) .$

## 3 Lower bound for G, upper bound for R

Let $q = q ( r )$ be a vector field defined in the hollow space domain $V ∪ ∂ V$ , which satisfies the following equations

$∇ ⋅ q = 0 , r ∈ V , n ⋅ q = 0 , r ∈ ∂ V 1 ,$
in this case
$G ≥ ( ∫ ∂ V 1 σ ( r ) n ⋅ q d A ) 2 ∫ V σ ( r ) q 2 d V , ∫ V q 2 d V ≠ 0 .$

In lower bound formula (18) equality is reached only if $q ≡ λ ∇ u ,$ where $λ$ is an arbitrary constant which is different from zero.

The proof of lower bound formula (18) is based on the Cauchy-Schwarz inequality relation (19)

$( ∫ V σ ( r ) p ⋅ q d V ) 2 ≤ ∫ V σ ( r ) p 2 d V ∫ V σ ( r ) q 2 d V .$
Let
$p = ∇ u$
be in inequality relation (19). A simple calculation yields the result
$∫ V σ ( r ) ∇ u ⋅ q d V = ∫ ∂ V u σ ( r ) n ⋅ q d A − ∫ V u ∇ ⋅ σ ( r ) q d V = ∫ ∂ V 1 σ ( r ) n ⋅ q d A .$
Substitution Eq. (21) into Cauchy-Schwarz inequality (19) gives
$( ∫ ∂ V 1 σ ( r ) n ⋅ q d A ) 2 ≤ ∫ V σ ( r ) | ∇ u | 2 d V ∫ V σ ( r ) q 2 d V .$

From inequality relation (22) the proof of lower bound formula, (18) can be obtained immediately.

Let $f = f ( r )$ be non-identically constant function in $V ∪ ∂ V$ , which satisfies the Laplace equation in  $V$

$∇ ⋅ ∇ f = Δ f = 0 , r ∈ V .$
The following lower bound formula is valid for G
$G ≥ ( ∫ ∂ V 1 ∂ f ∂ n d A ) 2 ∫ V | ∇ f | 2 σ ( r ) d V .$

The proof of (24) can be obtained from (18) with under-mentioned $q ( r )$

$q ( r ) = ∇ f σ ( r ) , r ∈ V ∪ ∂ V .$

## 4 Numerical examples

In the numerical examples the spherical coordinate system is used. The connection between the Cartesian coordinates $x , y , z$ and the spherical coordinates $r , ϕ , ϑ$ is $x = r cos ϕ sin ϑ , y = r sin ϕ sin ϑ , z = r cos ϑ .$ Developed numerical example relate to axisymmetric electrical problems.

The meridian section of hollow spherical domain is shown in Fig. 2. The specific conductivity is a given function of the radial coordinate
$σ ( r , α ) = σ 0 exp ( α r a 1 ) .$

Application of Theorem 1 to the function $F ( r ) = F ( r ) = 1 − 1 / r − 1 / a 1 1 / a 2 − 1 / a 1$ gives
$G ( α ) ≤ G U ( α ) = − 1 ( a 1 − a 2 ) 2 { 4 π σ 0 a 1 a 2 [ a 1 exp ( α a 2 a 1 ) + E i ( 1 , − α a 2 a 1 ) α a 2 − a 2 exp ( α ) − a 2 E i ( 1 , − α ) α ] } ,$
here $E i ( 1 , x )$ is the exponential integral [17, 18]. Putting the following function $f ( r ) = f ( r ) = r − 1$ in the lower bound formula (24) gives
$G ( α ) ≥ G L ( α ) = 4 π σ 1 a 1 a 2 − a 1 exp ( − α a 2 a 1 ) + E i ( 1 , α a 2 a 1 ) α a 2 + a 2 exp ( − α ) − a 2 E i ( 1 , α ) α .$
Lengthy, but elementary calculations shows that, the exact value of electrical conductivity which is obtained from the solution of boundary value problem (5) is
$G ( α ) = 4 π σ 1 a 1 a 2 − a 1 exp ( − α a 2 a 1 ) + a 2 exp ( − α ) + α a 2 E i ( 1 , α a 2 a 1 ) − α a 2 E i ( 1 , α )$
that in this case is $G ( α ) = G L ( α )$ . The validity of upper bound (14) for $α = − 2.5$ and $α = 2.5$ examplifies as follows

Figure 3 shows the upper and the lower bounds of the conductance G as a function of α for $− 2 ≤ α ≤ 2 .$ In this example

Figure 4 shows the plot of function $F = F ( r )$ , the plot of exact analytical solution $u = u ( r )$ and the plot of $u F = u F ( r )$ which is obtained from Finite Element (FE) approximation in the case of Example 1 for $α = 2.5$ . The FE model is developed in ABAQUS with DC3D8 elements (node numbers are $706 356$ ) and for definition of the nonlinear material a special user subroutine usdfld() is applied.

The meridian section of axisymmetric hollow domain bounded by two spherical surfaces as it is shown in Fig. 5. The following data are used $a 1 = 0.3 m , a 2 = 0.5 m , b = 0.025 , σ 1 = 7.69 × 10 6 1 / m Ω , σ ( r , n ) = σ 1 ( r a 1 ) n .$

Let $F ( r , ϑ ) = ln R ( ϑ ) r ( ln R ( ϑ ) a 1 ) − 1$ be in Theorem 1 and in Theorem 3 $f ( r ) = r − 1 .$

Figure 6 shows the upper and the lower bounds as a function of power index for $− 2 ≤ n ≤ 2 .$

## 5 Conclusions

A mathematical model is developed to determine the steady-state electric current flow through in non-homogeneous isotropic conductor whose shape is a three-dimensional hollow body. The hollow body considered is bounded by two closed surfaces which have no common points. The derivation of the upper and lower bound formulae for the electrical conductance is based on the two types of Cauchy-Schwarz inequality. Two numerical examples illustrate the applications of the derived upper and lower bounds for the conductance. The derived upper and lower bound formulae of electric conductance can be used to check the results of numerical computations obtained by finite element method, boundary element method and by any other numerical methods.

## References

• [1]

Gy. Maróti , “Finding closed-form solutions of beam vibration,” Pollack Period., vol. 6, no. 1, pp. 141154, 2011.

• [2]

Gy. Maróti and I. Elishakoff , “On buckling of axially functionally graded beams,” Pollack Period., vol. 7, no. 1, pp. 313, 2012.

• [3]

I. Ecsedi , Á. J. Lengyel , A. Baksa , and D. Gönczi , “Bounds for the electrical resistance for homogenous conducting body of rotation,” Multidiszciplináris tudományok, vol. 11, no. 5, pp. 104122, 2021.

• Crossref
• Search Google Scholar
• Export Citation
• [4]

I. Ecsedi and Á. J. Lengyel , “Bounding formulae for capacitance of cylindrical capacitor with non-homogeneous material,” WSEAS Trans. Electron., vol. 13, pp. 125131, 2021.

• Crossref
• Search Google Scholar
• Export Citation
• [5]

I. Ecsedi , Á. J. Lengyel , and D. Gönczi , “Bounds for the thermal conductance of body of rotation,” Int. Rev. Model. Simulations, vol. 13, no. 3, pp. 185193, 2020.

• Crossref
• Search Google Scholar
• Export Citation
• [6]

J. D. Jackson , Classical Electrodynamics. New York: Wiley and Sons, 1988.

• [7]

P. P. Silvester and R. L. Ferrari , Finite Elements for Electrical Engineers. Cambridge Univeristy Press, 1983.

• [8]

P. Hammond , Energy Method in Electromagnetism. Oxford: Clarendon Press, 1997.

• [9]

G. A. Korn and T. M. Korn , Handbook for Scientists and Engineers. New York: D. von Nosrand, 1961.

• [10]

A. P. Boresi , K. P. Chong , and S. Saigal , Approximate Solution Methods in Engineering Mechanics. John Willey & Sons, Inc, 2003.

• [11]

V. L. Rvachev and T. I. Sheiko , “R-functions in boundary value problems in mechanics,” Appl. Mech. Rev., vol. 48, no. 4, pp. 151188, 1995.

• [12]

A. Iványi , Continuous and Discrete Simulations in Electrodynamics (in Hungarian). Budapest: Akadémiai Kiadó, 2003.

• [13]

A. Iványi , “R-functions in electromagnetism,” Technical Report No. TUB-TR-93-EE08, Budapest, 1993.

• [14]

V. F. Kravchenko and M. A. Basarab , “Solving the boundary value problems of electrodynamics in the regions of fractal geometry by the method of R-functions” (in Russian), Tech. Phys. Lett., vol. 29, no. 12, pp. 10551057, 2003.

• Crossref
• Search Google Scholar
• Export Citation
• [15]

A. Iványi , “Variational methods for static electric field” (in Hungarian), Elektrotechnika, vol. 71, pp. 2125, 1978.

• [16]

A. Iványi , “Determination of static and stationary electromagnetic fields by variational calculus,” Period. Polytech. Electr. Eng., vol. 23, pp. 201208, 1979.

• Search Google Scholar
• Export Citation
• [17]

E. Masina , “A review on the exponential-integral special function and other strictly related special functions,” Lecture Notes, University of Bolona, Italy, 2019.

• Search Google Scholar
• Export Citation
• [18]

The exponential integral, Wolfram Mathworld. [Online]. Available: https://mathworld.wolfram.com/ExponentialIntegral.html. Accessed: Jan. 21, 2021.

• Search Google Scholar
• Export Citation
• [1]

Gy. Maróti , “Finding closed-form solutions of beam vibration,” Pollack Period., vol. 6, no. 1, pp. 141154, 2011.

• [2]

Gy. Maróti and I. Elishakoff , “On buckling of axially functionally graded beams,” Pollack Period., vol. 7, no. 1, pp. 313, 2012.

• [3]

I. Ecsedi , Á. J. Lengyel , A. Baksa , and D. Gönczi , “Bounds for the electrical resistance for homogenous conducting body of rotation,” Multidiszciplináris tudományok, vol. 11, no. 5, pp. 104122, 2021.

• Crossref
• Search Google Scholar
• Export Citation
• [4]

I. Ecsedi and Á. J. Lengyel , “Bounding formulae for capacitance of cylindrical capacitor with non-homogeneous material,” WSEAS Trans. Electron., vol. 13, pp. 125131, 2021.

• Crossref
• Search Google Scholar
• Export Citation
• [5]

I. Ecsedi , Á. J. Lengyel , and D. Gönczi , “Bounds for the thermal conductance of body of rotation,” Int. Rev. Model. Simulations, vol. 13, no. 3, pp. 185193, 2020.

• Crossref
• Search Google Scholar
• Export Citation
• [6]

J. D. Jackson , Classical Electrodynamics. New York: Wiley and Sons, 1988.

• [7]

P. P. Silvester and R. L. Ferrari , Finite Elements for Electrical Engineers. Cambridge Univeristy Press, 1983.

• [8]

P. Hammond , Energy Method in Electromagnetism. Oxford: Clarendon Press, 1997.

• [9]

G. A. Korn and T. M. Korn , Handbook for Scientists and Engineers. New York: D. von Nosrand, 1961.

• [10]

A. P. Boresi , K. P. Chong , and S. Saigal , Approximate Solution Methods in Engineering Mechanics. John Willey & Sons, Inc, 2003.

• [11]

V. L. Rvachev and T. I. Sheiko , “R-functions in boundary value problems in mechanics,” Appl. Mech. Rev., vol. 48, no. 4, pp. 151188, 1995.

• [12]

A. Iványi , Continuous and Discrete Simulations in Electrodynamics (in Hungarian). Budapest: Akadémiai Kiadó, 2003.

• [13]

A. Iványi , “R-functions in electromagnetism,” Technical Report No. TUB-TR-93-EE08, Budapest, 1993.

• [14]

V. F. Kravchenko and M. A. Basarab , “Solving the boundary value problems of electrodynamics in the regions of fractal geometry by the method of R-functions” (in Russian), Tech. Phys. Lett., vol. 29, no. 12, pp. 10551057, 2003.

• Crossref
• Search Google Scholar
• Export Citation
• [15]

A. Iványi , “Variational methods for static electric field” (in Hungarian), Elektrotechnika, vol. 71, pp. 2125, 1978.

• [16]

A. Iványi , “Determination of static and stationary electromagnetic fields by variational calculus,” Period. Polytech. Electr. Eng., vol. 23, pp. 201208, 1979.

• Search Google Scholar
• Export Citation
• [17]

E. Masina , “A review on the exponential-integral special function and other strictly related special functions,” Lecture Notes, University of Bolona, Italy, 2019.

• Search Google Scholar
• Export Citation
• [18]

The exponential integral, Wolfram Mathworld. [Online]. Available: https://mathworld.wolfram.com/ExponentialIntegral.html. Accessed: Jan. 21, 2021.

• Search Google Scholar
• Export Citation
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• Expand

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Editor(s)-in-Chief: Iványi, Amália

Editor(s)-in-Chief: Iványi, Péter

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Miklós M. Iványi

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Scimago
Quartile Score
Civil and Structural Engineering Q3
Computer Science Applications Q3
Materials Science (miscellaneous) Q3
Modeling and Simulation Q3
Software Q3
Scopus
Cite Score
340/243=1,4
Scopus
Cite Score Rank
Civil and Structural Engineering 219/318 (Q3)
Computer Science Applications 487/693 (Q3)
General Materials Science 316/455 (Q3)
Modeling and Simulation 217/290 (Q4)
Software 307/389 (Q4)
Scopus
SNIP
1,09
Scopus
Cites
321
Scopus
Documents
67
Days from submission to acceptance 136
Days from acceptance to publication 239
Acceptance
Rate
48%

2019
Scimago
H-index
10
Scimago
Journal Rank
0,262
Scimago
Quartile Score
Civil and Structural Engineering Q3
Computer Science Applications Q3
Materials Science (miscellaneous) Q3
Modeling and Simulation Q3
Software Q3
Scopus
Cite Score
269/220=1,2
Scopus
Cite Score Rank
Civil and Structural Engineering 206/310 (Q3)
Computer Science Applications 445/636 (Q3)
General Materials Science 295/460 (Q3)
Modeling and Simulation 212/274 (Q4)
Software 304/373 (Q4)
Scopus
SNIP
0,933
Scopus
Cites
290
Scopus
Documents
68
Acceptance
Rate
67%

Pollack Periodica
Publication Model Hybrid
Submission Fee none
Article Processing Charge 900 EUR/article
Printed Color Illustrations 40 EUR (or 10 000 HUF) + VAT / piece
Regional discounts on country of the funding agency World Bank Lower-middle-income economies: 50%
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Further Discounts Editorial Board / Advisory Board members: 50%
Corresponding authors, affiliated to an EISZ member institution subscribing to the journal package of Akadémiai Kiadó: 100%
Subscription fee 2022 Online subsscription: 327 EUR / 411 USD 321
Print + online subscription: 393 EUR / 492 USD
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Print + online subscription: 405 EUR / 492 USD
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Pollack Periodica
Language English
Size A4
Year of
Foundation
2006
Volumes
per Year
1
Issues
per Year
3
Founder's
H-1117 Budapest, Hungary 1516 Budapest, PO Box 245.
Publisher's
H-1117 Budapest, Hungary 1516 Budapest, PO Box 245.
Responsible
Publisher
ISSN 1788-1994 (Print)
ISSN 1788-3911 (Online)

### Monthly Content Usage

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Oct 2022 0 39 23
Nov 2022 0 34 18
Dec 2022 0 1 1

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