1. Introduction

Gas dynamics is the science of the flow of air and other gas and or the notion of bodies through the air and other gas and its effects on physical systems, fluid mechanics, and thermodynamics; this science considers the products of combustion and combustion. The equations of gas dynamics are mathematical expressions based on the natural laws of conservations (mass, momentum, and energy). The study of shock fronts, rare fractions, and contact discontinuities are three major nonlinear wave equations that describe ideal gas dynamics behaviors [1]. Gas dynamics is synonymous with aerodynamics when the gas field is air, and the subject of study is flight. This is a core interest in aircraft and spacecraft design with their respective propulsion systems. Study in gas dynamics coincides with the developments of transonic and supersonic flight as the aircraft began to travel faster, and the density of air began to change, considerably increasing the air resistance as the airspeed approached the speed of sound. This phenomenon was later identified in wind tunnel experiments as an effect caused by the formation of shock waves around the aircraft. Major advances were made to describe the behavior during and after World War II, and the new understandings of compressible and high-speed flows became theories of gas dynamics [2].

In this paper, we consider nonlinear gas dynamic equation in one spatial dimension of the form:

with the initial condition where \(\tau\) is a constant, \(g\left(x,t\right)\) and \(h(x)\) are smooth functions. Suppose \(g\left(x,t\right)=0\) the Eq. (1) is said to be homogenous. In last the last two decades, several researchers have proposed several analytical and numerical solutions for nonlinear gas-dynamics equation, such as the authors [3] presented analytical solution for nonlinear Gas dynamic equation by homotopy analysis method, [4] proposed a reliable technique for solving gas dynamic equation using natural homotopy perturbation method, [5] employed an efficient technique for solving Gas dynamics equation using the natural decomposition method, [6] presented and applied Elzaki transform homotopy perturbation method for solving gas dynamics equation, [7] applied decomposition method approach for the numerical solution of nonlinear gas-dynamics equation, Finite difference schemes for solving system equations of gas dynamic in a class of discontinuous functions was presented by [8], in [9] modified homotopy perturbation method (MHPM) for dynamics Gas equation was discussed and [10] applied Laplace variation iteration method to solving the nonlinear Gas dynamics equation.

In this paper, we construct a seven-step algorithm based on the modified new iterative method (MNIM) without using any transformation, linearization, and discretization for solving nonlinear gas-dynamic equation.

2. Method of solution

2.1. Modified new iterative method (MNIM)

The new iterative method (NIM) was proposed [11] and further improvement was made and applied to several ordinary differential equations, partial differential equations, and systems of partial differential equations. The MINM offers some certain advantages over routine numerical methods. For example, uses of discretization give rise to rounding off errors causing loss of accuracy and requiring considerable computational length and time. MNIM shows better performance since it does not involve discretization of the variables, is free from rounding off errors, and does not require more computational length to simplify the problems [12,13].

Consider the new iterative method (NIM) as a numerical technique for solving an functional equation of the form

where \(N\) a nonlinear operator from a Bunch space \(B\rightarrow B\) and \(f\left(\bar{x}\right)\) is a known function. \(\bar{x}=\left(x_1,\ \ x_2,\ {\ x}_3,\ ...,\ x_n\right).\) We need to obtain the solution \(\psi\left(\bar{x}\right)\) of equation (3) having the series of the form; The nonlinear operator which is on the right-hand side of Eq. (3) can be decomposed as follow: Substituting Eq. (4) and Eq. (5) into the Eq. (3) leads to: The recurrence relation is given by Then and The q-term approximate solution of equation (3) is given by; The authors [14] discussed the modification of NIM by the introduction of source terms into the integral representing \(N(\psi)\) and stated as follows:

1. Suppose \(g\left(x,t\right)\) is a smooth and nonhomogeneous function of the independent variable, \(x\) only, we include it in \(N\left(\psi\right)\).
2. Suppose \(g\left(x,t\right)\) is a smooth and nonhomogeneous function of the independent variable, \(x\) and t, we include it in \(N\left(\psi\right)\).
3. Suppose \(g\left(x,t\right)\) is a smooth and nonhomogeneous function are of \(x\),\(t\) and both \(x\) and \(t\) then we include in \(N\left(\psi\right)\) the terms involving \(t\) and both \(x\) and \(t\).
4. Thus, the MNIM can be applied to obtain the close analytical solution.

2.2. Modified new iterative algorithm (MNIA)

In order to improve and reduce simplification involve in executing MNIM discussed in Section 2, we formulate seven steps algorithm using MAPLE 18 software commands to obtain numerical solution of nonlinear Gas dynamics equation (1).

Algorithm 1. restart:

Step 1:

Digits:=35;

\(N:=R^{+};\)

\(\tau:=R^{+};\)

\(\psi\left(x,t_0\right):=h(x);\)

\(\psi\left[0\right]:=\psi (x,t_0);\)

Step 2:

\(GDPDE:=value(-\tau*\psi*diff(\psi[0],x)-\psi*(1-\psi)+g(x,t));\)

\(\psi\left[1\right]:=value(int(GDPDE,t=0...t));\)

Step 3:

for p from 1 to N do

\(\psi\left[p+1\right]:=value((int(((-\tau*(sum(\psi[n],n=0...p))*Diff(sum[n],n=0...p,x)-(sum(\psi[n],n=0...p))*\)

\((1-sum(\psi[n],n=0...p))+g(x,t))))-(int(((-\tau*(sum(\psi[n],n= 0...p-1))*Diff(sum(\psi[n],n=0...p-1,x)-(sum(\psi[n],n=0...p-1))*(1-sum(\psi[n],n=0...p-1))+g(x,t)))));\)

end do

Step 4:

\(Example:=evalf(sum(\psi[k],k=0...N+1))\)

Step 5:         (11)

for i from 0 by 0.2 to 1 do

\(\psi\left[i\right]:=evalf(eval(Example,[x=0.001,t=i]))\)

end do

Step 6:

for i from 0 by 2 to 10 do

\(N\left[0\right]:=evalExample,[x=0]);\)

\(N\left[2\right]:=eval(Example,[x=2]);\)

\(N\left[4\right]:=eval(Example,[x=4]);\)

\(N\left[6\right]:=eval(Example,[x=6]);\)

\(N\left[8\right]:=eval(Example,[x=8]);\)

\(N\left[10\right]:=eval(Example,[x=10]);\)

end do

Step 7:

\(\psi\left[3Dplot\right]:=plot3d(Example,t=-\pi...\pi,x=-\pi...\pi,grid=[100,100],color);\)

\(\psi\left[2Dplot\right]:=plot([N[0],N[2],N[4],N[6],N[8],N[10]]),t=0...40,color\)

=[red,blue,green,yellow,black,purple],axes=BOXED,title=GDPDE);

\(\psi\left[2Dlogplot\right]:=logplot([N[0],N[2],N[4],N[6],N[8],N[10]]),t=0...40,color\)

=[red,blue,green,yellow,black,purple],axes=BOXED,title=GDPDE);

\(\psi\left[Densityplot\right]:=(Example,t=-100...100,x=0...-10);\)

Output: See analytic-numeric solutions (15,19,23),Tables 1-3 and Figures 1(a), 1(b), 1(c), 1(d),..., Figures 4(a), 4(b), 4(c), where is the computational length and is a positive constant.

3. Numerical examples

Example 1. Consider the nonlinear nonhomogeneous gas dynamic equation [2,3,4,7,10,15]

with the initial condition: \( \psi\left(x,0\right)=1-e^{-x}\,. \) The analytical solution is given as: Compare and apply Eq. (11) with Algorithm 1, when \(N=2\), \(\tau=\frac{1}{2}\) , \(g\left(x,t\right)=-e^{t-x}\) and \(h\left(x\right)=1-e^{-x}.\) We obtained approximate solution as follow:

The numerical solutions for Example 1 is given in Table 1.

Table 1. Numerical solutions \(\psi(x,t) at x=0.001\).

\(\psi(x,t)\)	Solution	Example 1
(0.001, 0.00)	Analytical	0.00099950016662500833194464283234403
	MNIA	0.00099950016662500833194464283234403
(0.001, 0.20)	Analytical	-0.2201819658998724994169694170106571
	MNIA	-0.2201819658998724994169694170106571
(0.001, 0.40)	Analytical	-0.4903336186074025654679095480461317
	MNIA	-0.4903336186074025654679095480461317
(0.001, 0.60)	Analytical	-0.8202975923459081008581182027221291
	MNIA	-0.8202975923459081008581182027221291
(0.001, 0.80)	Analytical	-1.2233164999636086074556190524624456
	MNIA	-1.2233164999636086074556190524624456
(0.001, 0.10)	Analytical	-1.7155649053185666873319827333452869
	MNIA	-1.7155649053185666873319827333452869

Example 2. Consider the nonlinear nonhomogeneous gas dynamic equation [2,3,4,7,10,15]

with the initial condition: \( \psi\left(x,0\right)=1-e^{-x}\,. \) The analytical solution is given as

Table 2. Numerical solutions \(\psi(x,t) at x=0.001\).

\(\psi(x,t)\)	Solution	Example 2
(0.001, 0.00)	Analytical	0.00099950016662500833194464283234403
	MNIA	0.00099950016662500833194464283234403
(0.001, 0.20)	Analytical	-0.2201819658998724994169694170106571
	MNIA	-0.2201819658998724994169694170106571
(0.001, 0.40)	Analytical	-0.4903336186074025654679095480461317
	MNIA	-0.4903336186074025654679095480461317
(0.001, 0.60)	Analytical	-0.8202975923459081008581182027221291
	MNIA	-0.8202975923459081008581182027221291
(0.001, 0.80)	Analytical	-1.2233164999636086074556190524624456
	MNIA	-1.2233164999636086074556190524624456
(0.001, 0.10)	Analytical	- 1.7155649053185666873319827333452869
	MNIA	-1.7155649053185666873319827333452869

Compare and apply Eq. (14) with Algorithm 1, when \(N=2\), \(\tau=1\), \(g\left(x,t\right)=-e^{t-x}\) and \(h\left(x\right)=1-e^{-x}\). We obtain approximate solution as follow:

The numerical solutions for Example 2 is given in Table 2.

Example 3. Consider the nonlinear homogenous gas dynamic equation [2,3,4,7,10,15]

with the initial condition: \( \psi\left(x,0\right)=e^{-x}\,. \) The analytical solution is given as Compare and apply Eq. (15) with Algorithm 1 when \(N=15\), \(\tau=1\), \(g\left(x,t\right)=0\) and \(h\left(x\right)=e^{-x}\). We obtain approximate solution as \begin{equation*}\label{eq23} \psi\left(x,t\right)={\psi\left(x,t\right)}_0+{\psi\left(x,t\right)}_1+{\psi\left(x,t\right)}_2+\ldots+{\psi\left(x,t\right)}_{15} \psi\left(x,t\right)=e^{-x}\left(1+t+\frac{t^2}{2!}+\frac{t^3}{3!}+\frac{t^4}{4!}+\ldots+\frac{t^{15}}{15!}\right)\,. \end{equation*}

The numerical solutions for Example 3 is given in Table 3.

Table 3. Numerical solutions \(\psi(x,t) at x=0.001\).

\(\psi(x,t)\)	Solution	Example  3
(0.001, 0.00)	Analytical	0.99900049983337499166805535716765597
	MNIA	0.99900049983337499166805535716765597
(0.001, 0.20)	Analytical	1.22018196589987249941696941701065712
	MNIA	1.22018196589987249941696941328796801
(0.001, 0.40)	Analytical	1.49033361860740256546790954804613177
	MNIA	1.49033361860740256546741607307804137
(0.001, 0.60)	Analytical	1.82029759234590810085811820272212919
	MNIA	1.82029759234590810036634516523928928
(0.001, 0.80)	Analytical	2.22331649996360860745561905246244569
	MNIA	2.22331649996360854127633798922881128
(0.001, 0.10)	Analytical	2.71556490531856668733198273334528698
	MNIA	2.71556490531856371400530255152074189

4. Conclusion

In this paper, the modified new iterative algorithm (MNIA) was formulated and applied to obtain analytic-numeric solutions to the homogeneous and nonhomogeneous nonlinear gas-dynamics equation. The proposed algorithm gave analytic-numeric solutions of the three examples considered with high accuracy and good agreement with analytical solutions. The Figures 1(a), 1(b), 1(c), 1(d) depicts numerical solutions on a 3Dplot pertain to gas dynamics flow in fluid mechanics and thermodynamics, Figures 2(a), 2(b), 2(c) depict the 2D plot of gas flow in a given time interval, while 2D Logarithm-plots are presented in Figures 3(a), 3(b), 3(c) and Figures 4(a), 4(b), 4(c) demonstrated the density-plots direction of the gas flow in a gas chamber. Moreover, from the computational point of view, the proposed algorithm is faster in convergence rate, powerful and efficient in finding analytical and approximate solutions for similar nonlinear differential equations arising in applied sciences and engineering.

Conflicts of Interest:

''The author declares no conflict of interest.''

1. Introduction

Encryption is the technique of encoding messages (information) to prevent eavesdroppers or hackers from decrypting them but allowing authorized parties to do so. The message or information (commonly referred to as plaintext) is encrypted using an encryption technique, resulting in the unreadable ciphertext. Several cryptography techniques exist; including the geometric cryptographic schemes [1,2].

Based on the symmetric cryptography algorithm used to encrypt input data, this paper establishes the incompatibility between the geometric trisectioning of angles and the halting problem in recursion theory. We will in specific, focus on a form of a cryptographic technique called symmetric encryption. In this method, a similar key is utilized for both encryption and decryption procedures, and it is shared between the two communicating parties. It is a technique in which the information source encodes the plaintext and sends the ciphertext to the recipient using the key (or some set of restrictions). To decode the message and recover the plaintext, the receiver uses the same key (a set of predetermined rules). As a result, it is a type of cryptography in which both the information source and the recipient must know the key [3]. We will then employ the developed workflow for generalized geometric cryptographic schemes.

The "angles trisection-halting problem impossibility" analogy has been employed as a handy genetic example of symmetric encryption systems. The "angles trisection impossibility-halting problem impossibility" analogy is a geometric encryption method in which communications and ciphertexts are geometrically represented by angles or intervals of other geometric quantities (that may represent angles such as curves or straight-line segments). Computations are genetically based on straightedge and compass construction rules in this method [4]. The difficulty and impossibility assertions associated with solving some geometric problems, such as the trisection of an arbitrary angle using a straightedge and compass operations, serve as the foundation for the many geometric cryptography schemes [1,5]. Across the cryptographic historical frameworks, the first geometric cryptographic scheme was presented by [3], and the field has seen consistent growth in scientific research for decades [6]. Even though cryptography methods based on the plane Euclidean geometry system have practically limited physical applications (the geometric cryptographic schemes do not concern the classical straightedge and compass constructions), they are mostly utilized as pedagogic machinery for explaining more advanced cryptographic protocols [3,6].

The focus of this paper is to show that although the "angles trisectability-halting problem" impossibility analogy has frequently been utilized in the design of cryptographic techniques, its fundamental building blocks are geometrically misconstructed and it provides a weak framework for the application in cryptographic pedagogies and practices. We investigate the genetic characteristics of the two problems, the trisection of an arbitrary angle and the halting problem (based on the symmetric encryption method), and show the characteristic contrasts between the two problems in demonstrating their inherent incompatibility. Furthermore, we believe that for one to break the "angle trisectioning-halting problem" impossibility scheme; any such assertions should objectively be based on either, having sought the trisection of an arbitrary angle, or, by solving the halting problem or solving both of the problems. In this case, we use the "AG-Algorithm" [7] approach, which, when combined with the genetic definition of the two problems, invalidates the geometric cryptography analogy.

The rest of the paper's workflow is organized as follows: §2 deals with halting problem characterization, §3 concerns the angle trisection problem characterization, §4 deals with the characterization of the "angle trisection impossibility-halting problem" analogy and its limitations, §5 provides detailed discussions of the developed issues though the paper, and, §6 concludes the workflow.

2. Characterization of the Halting Problem

The halting problem is a decision problem in which you must determine if a computer program will eventually halt when run with the provided input given a computer program and an input to the program. The halting problem is a conceptual model in which there is no memory or time constraint on program execution, and so it can take an arbitrary amount of time and use a large amount of system storage space before halting.

In pseudo-code, for example, a program may run indefinitely in an infinite loop or may come to a halt relatively quickly. As a result, the halting problem has been determined to be algorithmically unsolvable [3,8]. This means that no method can be applied randomly to a program as input to determine if it will stop when run with a specific input. A generic technique to solve the halting problem for all potential program-input sets cannot exist, according to the Turing machine [3]. Turing machines are abstract computational devices that have been presented objectively to aid in examining the breadth and limitations of computational capabilities. Turing machines are the most powerful computational machines, and they form the theoretical basis for modern computers [8,9]. The next §2.1 presents a common understanding of the halting problem's operational principles.

2.1. A characteristic proof of the Halting impossibility

We set a straightforward interpretation of the problem as, given a random Turing machine \(M\) over the expression \(\Sigma = \{x, y\}\), and a random string \(w\) over \(\Sigma\), does \(M\) halt when it is given \(w\) as an input? The goal of this section is to characterize and show that the halting problem is not decidable, hence unsolvable.

Claim 1. The halting problem is undecidable and hence, unsolvable.

A Contradiction Assertion [10]: We establish an elementary proof by contradiction. We suppose that the halting problem is decidable, implying that there is a Turing machine operational named \(T_M\) that solves the halting problem. That is, given a description of a Turing machine \(M\) (expressible over \(\Sigma\)) and a string \(w\), \(T_M\) writes "True" if \(M\) halts on \(w\) and "False" if \(M\) does not halt on \(w\), and then \(T_M\) halts. Consider Figure 1;

Let us now build the new Turing machine \(T_{M_{a}}\) shown in Figure 2. First, we create a Turing machine \(T_{M_{m}}\) by adjusting the \(T_M\) depicted in Figure 1 so that if \(T_M\) accepts a string and halts, then \(T_{M_{m}}\) goes into an infinite loop (\(T_{M_{m}}\) halts if the original \(T_M\) rejects a string and halts).

Then following (Figure 2 (\(T_{M_{m}}\))) we create another Turing machine \(T_{M_{a}}\) scheme shown in Figure 3 as: recall that a \(T_{M_{a}}\) takes as input a description of a Turing machine \(T_{M}\) expressed by \(d(M)\), copies it to acquire the string \(d(M) \ast d(M)\), where \(\ast\) is a symbol that separates the two copies of \(d(M)\) and then supplies \(d(M) \ast d(M)\) to the Turing machine \(T_{M_{m}}\).

We examine how \(T_{M_{m}}\) operates when provided with a string describing \(T_{M_{m}}\) itself is given as input. Considerably, when \(T_{M_{m}}\) gets the input \(d(T_{M_{m}}\)), it makes a copy, constructs the string \(d(T_{M_{m}}) \ast d(T_{M_{m}})\) and gives it to the revised \(T_{M}\). Thus the modified \(T_{M}\) is described as the Turing machine \(T_{M_{m}}\) and the string \(d(T_{M_{m}}\)) as depicted in Figure 4.

From Figure 2, the modified \(T_{M}\) would execute an infinite loop if \(T_{M_{a}}\) halts on \(d(T_{M_{a}}\)), and on the opposite, it halts if \(T_{M_{a}}\) does not halt on \(d(T_{M_{a}}\)). In other terms, \(T_{M_{a}}\) goes into an infinite loop if \(T_{M_{a}}\) halts on \(d(T_{M_{a}}\)) and it halts if \(T_{M_{a}}\) does not halt on \(d(T_{M_{a}}\)). This is a contradiction. The contradiction has been assumed from the hypothesis that there is a Turing machine that solves the halting problem. Hence such a supposition must be wrong. The following pseudo-proof is aimed at clarifying the described hypothesis.

Elaborate Proof: The aim here is to establish an operational model fleshing out the exposed halting impossibility characteristic proof. We start by assuming that the halting problem is solvable, which implies that a solution algorithm exists. [6] states that a program \(X\) can be written as input to any program \(P\) with data \(D\), and the program will determine whether \(P\) initiated on \(D\) ultimately halts. To make a new program \(Y\), we append some commands (strictly, mathematical expressions) to \(X\). We let \(Y\) modify \(X_{s}\) so that if \(X\) halts due to a decision that \(P\) started on \(D\) halts, \(Y\) goes into an infinite loop.

If \(X\) comes to a halt with the decision that \(P\) initiated on \(D\), then \(Y\) comes to a halt as well. Finally, we make a new program called \(Z\) with the input \(P\). \(Z\) is defined so that it raises \(Y\) on program \(P\) with an input corresponding to \(P\) (that is, the input data for \(Z\) is a program). There are two probable prospects for explaining what happens when we run \(Z\) on \(Z\).

1. [6] \(Z\) started on input \(Z\) halts: If \(Z\) run on \(Z\) halts, then \(Y\) run on \(Z\) with input \(Z\) halts. If \(Y\) executed on \(Z\) with input \(Z\) halts, then \(X\) decided that \(Z\) started on \(Z\) never halts! Therefore, \(Z\) started on input \(Z\) halts implying that \(Z\) executed on input \(Z\) does not halt. This is a contradiction.
2. \(Z\) started on input \(Z\) does not halt: If \(Z\) executed on \(Z\) does not halt, then \(Y\) run on \(Z\) with input \(Z\) does not halt. If \(Y\) run on \(Z\) with input \(Z\) does not halt, then \(X\) decided that \(Z\) started on \(Z\) halts! Therefore, \(Z\) run on input \(Z\) never halts implying that \(Z\) started on input \(Z\) halts. This leads to a contradiction again.

Either alternative yields a contradiction, so the assumption that the halting problem is solvable must be incorrect.

2.1.1. A genetic empirical characterization of the Halting problem

This section presents both the genetic and the physical characteristics of the halting problem by examining the relationship between Turing Machines and modern-day general-purpose computers. In specific, we will be focusing on the machinery functioning frameworks. A Turing Machine is similar to a general-purpose computer in the following ways:

• a) A Turing machine is made up of a "tape" of cells and a single active cell called the "head". The tape's cells can be any color, and the head can be in any condition. A rule specifies what the head should do at each step for each Turing machine. The rule considers the state of the head as well as the color of the cell in which the head is located. Then it describes what the head's new state should be, the color it should "write" into the tape, and the direction it should move in (either to the left or to the right). The initial color arrangement of cells on the tape corresponds to the computer's input. Both "program" and "data" can be included in this input. The Turing machine's steps correlate to the computer's operation.
• b) The Turing machine's rules are analogous to computer machine-code instructions. Each section of the rule indicates what "operation" the computer should execute in response to a specific input. Turing Machines are distinguished from modern general-purpose computers by their storage capacities. However, because Turing machines have unlimited storage, we investigate the technical notion that the general-purpose computer is closer to a "finite state machine (FSM)" than a Turing machine, based on the proven characteristic similarities between the two machines. The presence of micro-transistors, which comprise the computer's major functional components, including computer memory, is important to the computer as an FSM.

2.1.2. Functional characterization of computer Halting operations

As mentioned in the previous section, general-purpose computers are fundamentally binary systems whose functioning principle is based on the working mechanisms of the constituent micro-transistors. All information in a computer is conveniently binary coded (both numerical data and text data are binary-coded).

The general-purpose computers use the micro-transistors states of either "High (True)" or "Low (False)" which genetically correspond to the binaries "1" or "0" respectively, in controlling the events between the background of the computer and the physical world. For instance;

• i) in a color image, every pixel is represented by three binary sequences that correspond to the three-pixel primary colors (green, red, and blue). Each sequence encodes an expression (could be mainly numerical valued expressions) that represents the intensity of the corresponding particular color in the pixel. These sequences are then transmitted through a video driver program to millions of liquid crystals on the computer screen, and translated into the hue that the eye observes.
• ii) Sound signals are also stored in binary form with help of a technique called pulse code modulation in which continuous sound waves are digitized by taking snapshots of their respective amplitudes every few milliseconds. The amplitudes are recorded as binary numbers which when read by the computer audio firmware, respective numbers determine how quickly the coils in the speakers should vibrate thus creating sounds of different frequencies.

From examples (i) and (ii), we deduce that physical activity such as recording and playing audio signals from a computer is genetically based on the binary functional characteristics of the computer. So such an activity is typically inherent, and not predetermined. This feature implies that the amplitudes of the signals correspond to a sequence of \(1_{s}\) and \(0_{s}\) if the signal is not continuum, and otherwise if continuum. We consider that for a program or a signal to be continuum, the computer operations must only be based on a single characteristic invariant binary-based state (either \(1_{s}\) or \(0_{s}\)) and not both states. Computer programs are naturally designed to operate with respect to binary characteristics. Considerably, the computer programs are distinct from an inherent mathematical function in that they cannot be characterized as total mathematical functions from an initial state to a final state without considerations of a time element. So inherently, computer programs have been characterized as functions or relations with a time component. The characterization based on the time element is responsible for non-termination in a computer program augmented as either a total function or as a partial function. The time element may be as simple as a Boolean variable that distinguishes finite from infinite time [5]. However, for physical interpretation, it can be a numeric variable extended with an infinite value to account for the non-termination [1]. We proceed by exploring the working principle of a (FSM) based on the binary system. We consider that given an arbitrary program and an input, for the input to decide with success if the given program halts, both programs must have a similar initial state (true or false). Otherwise, the input decides a non-terminating program. This conclusion is based on the genetic understanding that the states \(1_{s}\) or \(0_{s}\) cannot represent similar computation in a single event. For this reason, the halting problem would be impossible.

3. Establishing Euclidean geometric problems

The Euclidean geometric system (here we refer to the model established in the first six books of Euclid's Elements [11]) accentuates formalizing geometry using axioms, which appeal directly to basic concepts of geometry such as points and lines as opposed to a background for objects such as Cartesian spaces. For this paper, we in specific focus on the trisectability of angles posed by the ancient Greeks. The problem of "angles trisection" asks that: given an arbitrary angle (specifically a plane angle [12]), conceive a straightedge-compass scheme for sectioning the angle into three "equal" fractions. The inherent angles trisectability statement asks for a fixed straightedge and compass scheme that works for all plane angles. Further, the problem is restricted to some fundamental geometric conditions which most people inadvertently violate in their attempts to trisect angles (we suppose that the illustrated incompatibility schemes show the faults in employing analytic techniques as authoritative means for proving the non-trisectability of angles. It is upon the non-trisectability proofs where the geometric cryptographic protocols are established). It is expected that to resolve the trisection of angles one must use straightedge compass techniques, the solution (proof) must not involve measurements, an attempt at such a solution should not involve arithmetic, and that the geometric scheme must have a finite construction step. All these restrictions are inherently in agreement with the arbitrary nature of Euclidean geometric constructions. We consider that in solving the trisection of an angle, the size of the given angle should not be known [7].

This observation rules out the validity of applying real numbers as geometric magnitudes (a common practice characterizing the non-Euclidean geometric proofs). For centuries, mathematicians, philosophers, and science scholars have wrestled with the trisection of angles, till 1836 when first, the problem was vaguely stated as an impossible straightedgecompass problem; using ideas from the Galoi's fields [13]. However, we retrofit that the stated nontrisectability proof for any angle is flawed and it does not exactly show the trisection problem as an impossible problem. This work will employ a trisection scheme developed by [7]; to show the incompatibility between the angle trisection problem and the halting problem. The following §3.1 involves the genetic characterization of the angle trisection problem.

3.1. A characteristic proof of the angle trisection problem

A handy example to consider in this case is the geometric proof for the Pythagorean theorem (illustrated by [7]). The genetic proof for the Pythagorean theorem typically involves establishing the geometric relationship between magnitudes of similar kinds. In the proof, the is no application of real numbers as a substitute for geometrically constructed magnitudes. We think the reason for this was to preserve the purity of the Euclidean geometric system (in which real numbers play significant roles that which is not a representation of geometric magnitudes). A characteristic proof of trisecting any given angle must equally, follow the genetic rigor for establishing Euclidean geometric proofs. We consider that, any proof based on the constructability of only those magnitudes expressible as relative primes of the form \(a/b\) suggest an inherently non-Euclidean geometric generic property that given straightedge and compass, any constructible magnitude should be expressible as a rational fraction. Throughout §2, attempts have been made to characterize the existing geometric cryptographic principles. We assert that neither of the available geometric cryptographic models suits the inherent requirements for a Euclidean geometric proof. This is considered due to the nature of the proof for the halting problem. In both theory and practice, the proof for the halting problem is shy off the rigor required for typical straightedge and compass operations proofs. We thus carry on with the analytic examination of the angles trisection scheme established in [7]. The focus here is to establish the characteristic relations between the halting problem and the analytic nature of the angles nontrisectability proofs.

3.1.1. Characteristic analysis of the "AG-algorithm" solution

The "AG-Algorithm" provides a fixed system that works for any given angle. The scheme presents a "mothoddiversity" framework in which results at any desired accuracy are possible. The principle goal of this section is to provide a review characteristic of the "AG-Algorithm" algorithm using purely algebraic approaches.

3.1.2. Algebraic equivalence of the "AG-algorithm" solution

We consider Figure 5 adapted from [7] obtained following application of the "AG-Algorithm" on an arbitrarily small-angle \(\angle JAB\) constructed in a unit circle with; \(A = (0,0), B = (1,0),\) and, \(J =(\cos\theta, \sin\theta)\).

Let the points \(O\) and \(N\) trisect the chord \(\overline{JB}\) such that \(|BO| = |ON| = |NJ|\). We set points \(P\) and \(Q\) on the unit circle defined by the following equations: \(P =\) Intersection of the line \(\overline{D'N}\) and the arc \(\overline{JB}\). \(Q =\) Intersection of the line \(\overline{OA}\) and the arc \(\overline{JB}\).

Appendix (2(a)) [7] we assume a point of convergence upon which the trisection error the of chord \(\overline{JB}\) (although for unlike in appendix 1 [7] which is constructed upon a more complex geometric scheme appendix (2(a)) offers a simple analytic scheme for examining a specified angle which is not inherently obtained through complex operations as established in appendix 1 [7]) corresponds to the trisection error. Therefore, we reduce the trisection scheme depicted in Figure 5 (b)) to the assumption based on Figure 5 (a)) on the trisectability of the curve \(\widehat{JB}\) so that \(\angle PAQ = \angle JAB/3\), implying \(\overline{PQ} = \overline{BR} = \overline{RS} = \overline{SJ}\) and \(\widehat{PQ} = \widehat{BR} = \widehat{RS} = \widehat{SJ}\). Thorough analysis is done with the aid of a MATLAB code to show that as in the case of \(n\)-sectioning of a straight line segment, the "AG-Algorithm" is an exact trisection [7]. The subsequent argument concerns investigating if \(\angle PAQ = \theta/3\), where \(\theta=\angle JAB\). We initiate the investigation with the trisection of chord \(\overline{JB}\), which aids us in determining the positions \(O\) and \(N\).

Claim 2. Chord \(\overline{JB}\) is exactly trisected such that \(\overline{BO}\) = \(\overline{ON}\) = \(\overline{NJ}\).

Proof Proof of Claim 2. Applying vector notation based analysis, we set:

From Eq. (1), when \(t = 0\) we get \(B = (1,0)\), and when \(t = 1\) we get \(J = (\cos\theta, \sin\theta)\). We preserve the positions \(B\) and \(J\) for later use. Assuming that chord \(\overline{JB}\) is exactly trisected, we now set \(t = \frac{2}{3}\) and solve for point \(N\) as: \begin{align*} &N = (1,0) + 2/3 (\cos\theta - 1, \sin\theta)\,,\\ &N = (1 + 2/3 \cos\theta - 2/3 , 2/3 \sin\theta)\,,\\ &N = (1/3 + 2/3 \cos\theta, 2/3 \sin\theta)\,. \end{align*} Again from Eq. (1), we put \(t = 1/3\) and solve for point \(O\) as follows: \begin{align*} &O = (1,0) + 1/3 (\cos\theta - 1, \sin\theta)\,,\\ &O = (1 + 1/3 \cos\theta - 1/3 , 1/3 \sin\theta)\,,\\ &O = (2/3 + 1/3 \cos\theta, 1/3 \sin\theta)\,. \end{align*} This proof assumes that since \(\overline{D'N}\) and \(\overline{OA}\) are parallel as established by [7] and that since points \(O\) and \(N\) were obtained using similar plane geometric configurations, chord \(\overline{JB}\) is exactly trisected. Now to compute the angle trisection errors, we have to determine the points \(\overline{D'N}\) and \(\overline{OA}\) on the circumference of a unit circle. We proceed using points \(O\) and \(N\) in writing the line equations \(\overline{D'N}\) and \(\overline{OA}\) as follows: Eq. (2) further implies that Eq. (4) further implies that To find the coordinates of the points \(P\) and \(Q\) we determine the points of intersections between lines \(\overline{D'N}\) and \(\overline{OA}\) with the unit circle defined as: Using the equation for \(\overline{D'N}\) in (6) we get; Simplifying (7) we obtain Eqs (8) and (9) as follows; \begin{align*} &x^{2} + a^{2}\left(x^{2} + x + 1/4\right) = 1\,,\\ &(1 + a^{2})x^{2} + a^{2}x + a^{2}/4 - 1 = 0\,,\\ &x=\frac{-a^{2}+\sqrt{a^{4}-4(1+a^{2})\left(\frac{a^{2}}{4}-1\right)}}{2(1+a^{2})}\,, \end{align*} \begin{align} \notag&x=\frac{-a^{2}+\sqrt{a^{4}-4\left(\frac{a^2}{4}+\frac{a^4}{4}-1-a^2\right)}}{2(1+a^{2})}\,,\\ \end{align} Consequently, Eqs (8) and (9) give us the point \(P\), Eq. (10) Now we find the point of intersection between the line \(\overline{OA}\) and the unit circle circumference as follows: From Eq. (11) we make: So that the point \(Q\) becomes: which lies on the unit circle.

From Figure 5, we apply the dot product in determining the angle between \(AQ\) and \(AP\) as follows:

Since \(P\) and \(P\) are on the unit circle we get: To determine the dot product, we consider the trigonometric functions (cosine and sine) for an angle of some magnitude (say \( \angle PAQ= 0.75^{\circ})\) at 15 decimals accuracy and the slopes \(a\) and \(b\) as follows; Using Eqs (17) and (19) in Eqs (10) and (13), we obtain the coordinates of points \(P\) and \(Q\) respectively. We then use in the cosine triple angle formula \(\cos(3\theta) = 4\cos^{3}(\theta) - 3\cos(\theta);\) the variables \(a\) and \(b\) and the points \(P\) ad \(Q\) in determining the resulting angle trisection error. For simplistic analysis, we employ the MATLAB script shown in (appendix 2 (a) [7]) which is based on the condition: \begin{equation*} pi/(\theta \times 1000000000) \  with \  0 > \theta = pi/240. \end{equation*} Under this condition, the "AG-Algorithm" projects the most accurate to exact results depending on the required accuracies limits.

4. The angle trisection-Halting problem analogy

The "angle trisection-halting problem" analogy of geometric cryptography is based on the notion that it is easy to triple a given angle than it is to trisect such an angle. The geometric cryptography common operations in the realm of symmetric cryptography functions assign the triple of a magnitude (say angle) to a given angle. Hence, symmetric cryptography can be thought of as a one-way function in which the only constructions permissible are those mimicking the ruler and compass constructions workflows. This section aims at constructing a one-way geometric identification scheme that is then used to characterize the "angle trisection-halting problem" impossibility analogy.

4.1. One-way geometric cryptography identification scheme

This section adopts the identification scheme suggested by [6,10]. The identification model is probabilistic based on the tossing of a coin. Depending on the number the coin is flipped, the outcome is "true" if and only if all the chances are "heads" and otherwise, if "tails" appear. We assume two parties; Alex and June where Alex wishes to establish a means of proving his identity later to June.

4.1.1. Initialization of the identification scheme

Alex publishes a copy of an angle \(Y_{A}\) which is constructed by Alex as the triple of an angle \(X_{A}\) he has constructed at random. Because trisecting an angle is impossible Alex is confident that he is the only one who knows \(X_{A}\).

4.1.2. Working of the identification scheme

Alex gives June a copy of an angle \(R\) which he has constructed as the triple of an angle \(K\) that he has selected at random. June flips a coin and tells Alex the result. If June says "heads" Alex gives June a copy of the angle \(K\) and June checks that \(3 \ast K = R\). If June says "tails" Alex gives June a copy of the angle \(L = K + X_{A}\) and June checks that \(3\ast L = R + Y_{A}\). The four steps are repeated \(t\) times independently. June accepts Alex's proof of identity only if all \(t\) checks are successful. This protocol is an interactive proof of knowledge of the angle \(X_{A}\) (the identity of Alex) with error \(2^{-t}\).

4.2. Characteristic limitations of the "Angle trisection-Halting impossibility" analogy

Through §4.1 we note that although the "angle trisection-halting problem" analogy is typically basic, this paper asserts that the concept is built upon a misconception. Consider the following statements which expose the limitations exhibited by the "angle trisection-halting problem" analogy.

• a The "angle trisection-halting problem" analogy assumes that the trisectability of an arbitrary angle using compass and straightedge is inherently probabilistic. This perspective is incorrect as it is built on the assumption of prior knowledge of the size of the resulting angles that trisect a given angle (as well, a known angle) thus it is either "true" or the binary 1 when the trisection is achieved, and "false" or the binary 0 otherwise. In contrast, the problem of angles trisection requires no predetermined results, neither is the size of the trisectioned angle necessary in the construction (established in §3)
• b As illustrated using the (FSM), the halting problem is inherently characterized by the two binary states (\(1_{s}\) and \(0_{s}\)) which determines the nature of a program (either halting or continuum) and so the problem can never be solved since arbitrarily, computing events based on either (\(1_{s}\) or \(0_{s}\)) cannot be inherently similar as \(1\neq 0\). From a geometric perspective, the use of the states (\(1_{s}\) and \(0_{s}\)) assumes that binary 1 is geometrically reducible to binary 0. This is an inconsistent cognitive meaning of the analogy which implies that a quantity of magnitude 1 is reducible to another quantity of magnitude 0 via compass and straightedge, thus deciding the halting problem. The Euclidean geometric system forbids the application of real numbers as geometric magnitudes, and thus sensible, being a practical model, the number 0 is not allowed in straightedge and compass schemes.
• c From §4.1.2, the error expression \(2^{-t}\) suffer two main limitations: (i) it assumes that an angle could be trisected if the error is infinitely small because the more iterations there is, the small the error (that is, the error decrease with increase in \(t\) values); (ii) it also assumes that the exact trisection of an angle corresponds to infinitely repeated bisection steps in the interface between the alternatives of the states (\(1_{s}\) and \(0_{s}\)). Contrary, it is possible to have a small angle exactly trisected at \(t = 1\) (following the "AG-Algorithm" [7]), and this angle multiplied to the desired fraction that leads to the trisection of a larger angle (this is an attribute of the "AGAlgorithm"). These limitations are further illustrated using the MATLAB code provided in appendix (2 (a) [7]), where the trisection of the given angle is exact for the values of \(t\) with respect to the sizes of the angles.
• d Further, we consider from §4.1.2, the error expression \(2^{-t}\). The expression implies that the geometric trisection of an angle corresponds to repeated bisections in the sense that if we apply the \(t\) values sequentially in according to the number of iterations such that \(t = 1, 2, 3, 4, 5, 6, 7;\) we obtain the corresponding decreasing error results as; \(t = 0.5 0.25 0.125 0.0625 0.03125 0.015625 0.0078125\) (the resulting angle is half of the previous angle). This is the misconception exposed by relating the trisection of an angle and solving the halting problem. The problem of angles trisection does not ask for approximate solutions regarding repeated bisections operations.

5. Discussion

Regarding the physical significance of the problems; the halting problem is characterized by a time component which may be based on a Boolean variable that distinguishes finite from infinite time [10], or it can be based on a numeric variable (such as integers) extended with an infinite value to account for non-termination [1]. On the other hand, the trisection of an arbitrary angle problem requires typically, a plane geometric scheme that sections any given angle into three equal fractions. It has been established that the Euclidean geometric system allows only those operations mimicking straightedge and compass rigor of constructions. Although in cryptographic schemes the ordinary practical straightedge and compass constructions are not exhibited, here we concern ourselves with the principles behind the "angles trisectability-halting problem" analogy. We begin by asserting that strictly, Euclidean geometric problems are not time-bound, and so is the trisectability of angles. The notion that the trisection of an angle in general requires infinite construction steps is a misconstruction of the inherent characteristics of the problem. Indeed, the notion suggests approximate solutions for angle trisection based on multiple bisections of a given angle. This view completely violates the standard notion of Euclidean geometric exactness as provided by [14].

Further, the characteristic non-Euclidean proofs (as modeled in cryptographic applications) introduce time and resource requirements for the problem of angles trisection. Indeed, it is one of the fundamental reasons that the practical trisectability of an angle via straightedge and compass operations correspond to the infinite geometric bisection of a plane angle. This notion is an evident characteristic feature of the elliptic curve cryptographic (ECC) applications [1,15]. For instance, in §4.1.2 the iteration error expression \(2^{-t}\) is signify that no single instance the trisection of an angle could be numerically exact, or that the trisection accuracy of an angle increase with the increase in the number of iterations via repeated geometric operations of bisecting a magnitude. Nonetheless, we treat such reasoning as elusive and an invalid notion about the trisectability of angles. Further, the anticipation that a trisection is achievable if all results are "heads" impose a genetic Euclidean restriction which Euclid himself has not demonstrated through the Elements. §4.1.1 has outlined some of the possible limitations of the "angle trisection-halting problem" impossibility analogy which makes it an invalid concept for application of cryptographic principles in information processing. Further, the pseudo-codes in appendix (1 [7]), appendix (2 (a) [7]), and appendix (2 (b) [7]) expose that the trisection of any angle is possible given the treatment suggested is using the probabilistic error \(2^{-t}\) for \(t\) iterations of a program; in which the error \(2^{-t}\) corresponds to some degree of accuracy. To this effect, we observe that the genetic differences between the two problems ("angle trisection-halting problem") make them incompatible and thus invalidating the geometric cryptography concepts.

6. Conclusion

Throughout the provided review, we note that there is no genetic and physical relationship between the geometric trisection of an arbitrary angle and the halting problem. Thus, the focus of the paper has been achieved; to create an inherent demarcation between the theory of Euclidean plane geometry and the computability theory by breaking the ("angle trisection-halting problem") impossibility analogy. Since the trisection of any angle or any curve that subtends a given angle is reasonably, geometrically possible [7], the ("angle trisection-halting problem") analogy is fully invalidated. We, therefore, assert that the concept of geometric cryptography is genetically faulty and thus it should not be used as an important subject for cryptographic techniques as it provides weak geometric frameworks that can be easily broken during information transmissions between entities.

Author Contributions:

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Conflicts of Interest:

''The authors declare no conflict of interest.''

1. Introduction

Of common knowledge, a battery, such as a lithium-ion battery (LIB), consists of two opposite poles, an anode and a cathode. Much research is dedicated into seeking ways to improve the energy density and energy storage capacity of LIBs, and silicon (Si) has emerged as one promising element for creating a better anode [3]. It has been claimed that Si-based anodes of LIBs have an 11-fold higher energy capacity than graphite-based LIB anodes [4,5]. The ability to improve the performance of a LIB, for example via the Si-based anode, would have obvious important applications for the future of sustainable and renewable energy sources. One way to achieve this is to combine Si with heteroatom-doped graphene [6].

This Si and graphene-linked possibility was emphasized in a recent review by Li et al., [7] in Energy Storage Materials, where the authors stated the following: ''The modified Si could first assemble with graphene oxide (GO) through van der Waals interactions, which was then added to the graphite suspension dropwise for further electrostatic assembly with the help of mutual attraction of heterosexual charges''. Of note is the latter part of their proposed mechanism, namely the need for ''heterosexual charges'' in order to achieve this goal. The reference they cited was a paper by Hu et al., [2], who stated the following: ''Therefore, the GO/Si suspension could be added dropwise to the G suspension for electrostatic assembly by mutual attraction of heterosexual charges.'' (p. 889). The source of the term ''heterosexual charges'' thus seems to be Hu et al., [2].

2. The non-existence of ''heterosexual charges''

There is a problem with this claim, however, namely that LIBs, or any battery for that matter, do not have multiple charges, as suggested by the prefix "hetero", and much less sexuality. Therefore, this is an erroneous term that has been to describe a non-existent object or phenomenon. Given that this is a review that has already cited this nonsense term from a 2019 paper, the greater risk is that this non-existent concept may be propagated into the future literature, through citation, especially if one considers the high profile and popularity of the journal, Energy Storage Materials. The [7] paper appears to have blindly cited [2] without questioning the claim of "heterosexual charges", at least according to the journal's co-editors-in-chief. The [7] and [2] papers have already been cited 22 and seven times, respectively according to the papers' websites.

Two possible explanations for the existence of this nonsensical term, or tortured phrase [8], may be the use of reverse translation software and/or text thesauruses to avoid plagiarism detection, or mistranslation of a Chinese term into its English equivalent. In these cases, this might arise if the authors used a weak or unspecific translation tool with erroneous output, or if an editing, English revision or translation company was involved, having provided poor and erroneous editorial services. Of relevance to this point, whereas the [2] paper does not contain any acknowledgement or declaration of interests statement, the [7] paper does, neither paper indicating that any language, translation or editing service was involved. On this topic, it is important to note that it is unethical to not disclose the use of third party services, including language, translation or editing companies, in a scientific paper [9].

The corresponding authors of both papers were contacted (October 11-31, 2021) with a request for a published PDF copy of the papers, as well as queries regarding the source of this 'tortured phrase' and to appreciate if any third party service was employed. Neither authors provided a PDF file of their published paper. Only the corresponding author of Hu et al., responded by email, indicating that the nonsensical neologism was entirely created by the authors, and not by any third party service, and that this term was accidentally created by an imperfect Chinese-English translation introduced by a student. The intended phrase was supposed to indicate "opposite charges". The possibility of the existence of other nonsensical neologisms in these papers was not explored, but merits additional scrutiny.

Most importantly, this nonsensical term does not exist, nor does the concept of the sexuality of a charge, either in LIBs, or in the field of electrical engineering. Consequently, this term is a factual and conceptual error. This is similar to a tortured phrase that claimed the heterosexuality of the carbon structure [10]. Ideally, all errors should be corrected [11], although it is easy to appreciate that total chaos may enshroud the scientific literature if individual errors were all to be corrected, even more so if several errors were detected at different moments in time for the same paper. However, despite the inconvenience, should such errors be left uncorrected? The answer may lie in a mix between the authors' voluntary desire to correct this (and other) errors, combined with the editors' insistence on doing so and the publisher's correction policy. Yet, if left uncorrected, the greater risk is the promulgation of this error into downstream literature, as has already occurred once from [2] to [1], and indirectly to another six papers that have cited the Hu et al., [2] paper and to another 22 papers that cited the Li et al., [1].

Although an earlier version of this paper, which was submitted as a letter to the editor, was desk rejected by Energy Storage Materials and within October, 2021, for being "out of scope" and due to "too many submissions" to those journals, the co-editors-in-chief of Energy Storage Materials and the corresponding author of Hu et al., (2019) affirmed that errata would be published. Bibliometrically speaking, at least to some, the citation of errors could be interpreted as the reward of error, in the form of the gain of citations, as well as recognition for a non-existent term. It costs US\(\$ 31.50\) and US\(\$53.00\) to access the PDF file of the [7] and [2] papers, respectively. The sale of, or benefit derived from erroneous or fraudulent literature, is not often discussed but merits greater awareness and debate [3].

3. Conclusion

Finally, it is important to discuss the issue of publication of critique that may emerge during post-publication peer review. Although it is possible to critique publications on websites and blogs, or even on social media such as Twitter, these are not academically the ideal locations for debates to occur. What this case and experience revealed was the poor publishing infrastructure in place by these two journals to accommodate letters to the editor or evidence-based commentaries [10]. Considering that the overall tone and message was critical, not allowing academics to debate papers precisely in the journals where errors are detected, amplifies the culture of publishing bias, which over-emphasizes positive results at the expense of negative ones [12].

Acknowledgments :

The author thanks the hat tips provided at PubPeer by Mikhail V. Simkin (Department of Electrical and Computer Engineering, University of California - Los Angeles, CA, USA) and Nicholas (Nick) Wise (Department of Engineering, University of Cambridge, UK).

Conflicts of Interest:

''The author declares no conflict of interest.''

1. Introduction

The 5G simply stands for the fifth generation and refers to the next and newest mobile wireless standard based on the IEEE 802.11 ac standard of broadband technology [1], however technologies used in 5G are still being defined. Fifth generation (5G) is not as previous generations, an evolution of the existing, but it is rather considered as a cellular network revolution that builds on the evolution of existing technologies. These technologies are complemented by new radio concepts that are designed to meet the new and challenging requirements of some use cases today's radio access networks cannot support [2]. The anticipated capabilities of 5G will include massive system capacity, very high data rates everywhere, very low latency, ultra-high reliability and availability, very low device cost and energy consumption, and energy-efficient networks.

2. 5G requirements and capabilities

According to the Groupe Speciale Mobile Association (GSMA) [2], to qualify for 5G a connection should meet most of these criteria on the Figure 1.

Because these capabilities are specified from different perspectives, they do not make an entirely coherent list - it is difficult to achieve a new technology that could meet all of these conditions simultaneously.

Equally, whilst these eight requirements are often depicted as a single list, no use case, service or application has been identified that requires all eight performance attributes across an entire network simultaneously

Very high data rates available everywhere and global coverage

Every generation of mobile communications has been associated with higher data rates compared to the previous generation. In the past, the development much focused on the peak data rate delivered under the ideal condition. However, future applications demand strictly on the actual data rate that the network can provide under real-life conditions in different scenarios [3].

• With 5G, it should support data rates exceeding 10Gbps in specific scenarios such as indoor and dense outdoor environments.
• Data rates of several 100Mbps should generally be achievable in urban and suburban environments.
• Data rates of at least 10Mbps should be accessible almost everywhere, including sparsely populated rural areas in both developed and developing countries.

The global coverage is achievable using any of existing wireless technologies, as well as 5G, and it depends on how network operators establish their own cells.

Very low latency and 99.999% availability

Some subjected 5G use cases, such as autonomous driving, public safety and control of critical infrastructure and industry processes, may require much lower latency compared with what is possible with the mobile-communication systems of today.

To support such latency-critical applications, 5G should allow for an application end-to-end latency of 1ms or less, although application-level framing requirements and codec limitations for media may lead to higher latencies in practice. Many services will distribute computational capacity and storage close to the air interface. This will create new capabilities for real-time communication and will allow ultra-high service reliability in a variety of scenarios, ranging from entertainment to industrial process control [4].

In addition to very low latency, 5G should also enable connectivity with ultra- high reliability and ultra-high availability. For critical services, such as control of critical infrastructure and autonomous driving, connectivity with certain characteristics, such as a specific maximum latency, should not merely be 'typically available.' Rather, loss of connectivity and deviation from quality of service requirements must be extremely rare. For example, some industrial applications might need to guarantee successful packet delivery within 1 ms with a probability higher than 99.999%.

Lower device and network energy consumption

The reduction of power consumption by networks and devices is fundamentally important to the economic and ecological sustainability of the industry.

Energy-per-bit usage should be reduced by a factor of 1,000 to improve upon connected device battery life. Since 5G will be the key enabler of IoT, with billions of wirelessly connected devices such as sensors, actuators, and most of them are required to have the operation life time up to 10 years, 5G should provide an appropriate scheme where connectivity is occasional, and the amount of throughput is minimal [5].

While energy consumption on device side has always been prioritized, energy efficiency on the network side has recently emerged as an additional KPI (Key Performance Indicator). It is an important component in decreasing operational cost, as well as a driver for better dimensioned nodes, leading to lower total cost of ownership. Energy efficiency enables off-grid network deployments that rely on medium-sized solar panels as power supplies, thereby enabling wireless connectivity to reach even the most remote areas. Energy efficiency is a key aspect to realize operators' ambition of providing wireless access in a sustainable and more resource- efficient way. In 5G, 90% of network energy usage will be reduced compared to previous generation.

Massive connectivity

5G differs from 4G LTE in that it will be planned not for one traffic type, but multiple types.

For example, a massive number of new IoT devices will be attached to a 5G network. While these devices will add little traffic compared to mobile broadband, they will require signaling to communicate with the network. Bell Labs estimates that a typical IoT device may need 2,500 transactions or connections to consume 1MB of data [5].

This explain why, 5G should provide 1000x bandwidth per unit area to support broad range of traffic types. Moreover, a single cell of 5G should be able to serve well up to 1 million devices.

It is important to highlight that not all of these above performance indicators will be required by every terminal everywhere and all the time. Each connected device will typically have its mix of latency, bandwidth and traffic intensity characteristics.

3. Use cases

5G will support countless emerging use cases with a high variety of applications and variability of their performance attributes: From delay-sensitive video applications to ultra-low latency, from high speed entertainment applications in a vehicle to mobility on demand for connected objects, and from best effort applications to ultra-reliable ones such as health and safety [6].

3.1. Mobile broadband access

This 5G segment encompasses the use of portable devices in a range of very diverse scenarios, specifically including the more challenging situations in terms of high density area, high mobility. 5G is expected to provide its service in densely-populated areas (e.g., multi-storey buildings, dense urban city centers or events), where thousands of people per square kilometer live and/or work, or in high speed vehicles (such as train, aircraft). The following are some typical use cases of this family:

• Smart office: In a future office, it is expected that most of the devices will be wirelessly connected. Users will interact through multiple and wirelessly connected devices. This suggests a scenario in which hundreds of users require ultra-high bandwidth for services that need high-speed performance of bandwidth-intensive applications, processing of an enormous amount of data in a cloud, and instant communication by video.
• o Live stream in public event: This use case is characterized by a high connection density in a large-scale event sites, such as stadiums or hall parks. Several hundred thousand users per km2 may be supported, possibly integrating physical and virtual information such as information on athletes or musicians, etc., during the event. People can watch high definition (HD) playback video, share live video or post HD photos to social networks. These applications will require a combination of ultra- high connection density, high date rate and low latency.
• o High speed transportation: Beyond 2020, all kind of transportations will be huge improved in term of speed, ensuring the persistent connectivity for traveler is an important feature. While travelling, passengers will use high quality mobile Internet for obtaining information, entertaining or working. Examples are watching a HD movie, gaming online, accessing company systems, interacting with social clouds, or having a video conference. Delivering a satisfactory service to the passengers (e.g. up to 1000) at a speed higher than 500 km/h may be a great challenge.

3.2. Critical communication

This family of use cases are usually related to ultra- reliable and ultra-latency requirements. It will consist of drones and air traffic control, cloud driven virtual reality, smart factories, collaborative robotics, public safety, connected transportation and e- health. However, each use case will have their specific demand, for instance, may require one or more attributes such as extremely high throughput, mobility, critical reliability, etc. For example, the autonomous driving use case that requires ultra-reliable communication may also require instant reaction (based on real-time interaction), to prevent road accidents. Others such as remote computing, with stringent latency requirement, may need robust communication links with high availability [7]. The following are some typical use cases of this family:

• Automotive driving: Vehicle-to-vehicle (V2V), vehicle-to-infrastructure (V2I) and other Intelligent Transport Systems (ITS) applications require very low latency - much lower than is currently offered by LTE. Driverless cars and the next generation of driver- assisted cars will need real-time safety systems that can exchange data with other vehicles and infrastructure around them.
• Collaborative robotics in industrial automation: Automation will complement human workers, not only in jobs with repetitive tasks (e.g., production, transportation, logistics, office/administrative support) but also within the services industry. As 5G networks are rolled out, traditional robots programmed to carry out specific functions will be replaced by new models connected to the cloud. These new robots will have access to almost unlimited computing power, making them more flexible, more usable and more profitable to own and operate. For many robotics scenarios in manufacturing a round-trip reaction time of less than 1ms is anticipated.
• Drones: The ultra-fast, ultra-reliable of 5G will open the huge potential of sky transportation, especially the use of drones. Governments and disaster-relief organizations will use 5G-connected drones to aid in emergency efforts. Drones in the sky will communicate and share real- time information with each other and teams on the ground, increasing the speed and effectiveness of search-and-rescue missions. Drones may be exploited for logistics such as autonomous delivery of packages on routes with no/low civil population. The network will support a multitude of autonomous drones to navigate your local skies safely, and even send notifications to your devices announcing their arrival. Another amazing drone practice in 5G can be deployed is internet drones, to provide reliable ubiquitous connectivity.
• eHealth: Extreme Life Critical: Beyond 2020, remote treatment will emerge based on the development of monitoring devices as sensors for electrocardiography (ECG), pulse, blood glucose, blood pressure, temperature. 5G specificities will make the command-response time close to zero and provide the practitioner with great operation comfort and accuracy. In the near future, a patient who needs an urgent or specific operation could be operated by a practitioner remotely located.

3.3. Massive IoT

This segment is extremely broad, containing not just M2M but consumer based services too. It is likely to consist of an ecosystem of potentially very low cost devices such as sensors or trackers. Typical use cases are highly varied and may include camera surveillance, driverless cars, smart home, some wearables and machine type communications including metering, sensors and alarms with a wide range of characteristics and demands. The following describes some typical use cases of this family:

• Smart wearables: It is predicted that the use of wearables consisting of multiple types of devices and sensors will become mainstream. For example, a number of ultra-light, low power, waterproof sensors will be put in people's clothing. These sensors can monitor various environmental and health attributes like pressure, temperature, heart rate, blood pressure, body temperature, breathing rate and volume, skin moisture, etc. A wearable device be able to communicate to other devices will lead to rich experiences for users. A key challenge for 5G to enable fully this use case is the management of the number of devices as well as the data and applications associated with these devices.
• Connected home: Our future homes will be full of connected devices, not only providing data on their environment, but also connecting with each other. A smart thermostat may "talk" to a smoke detector, so that the collective information can provide more reliable information in the event of a fire at home. In case no one is present, this information can be remotely communicated to mobile devices and bring the fire brigade to the rescue. Homes are anticipated to become enormous sources of information and data will be shifted to mobile devices for remote monitoring, control and eventual decision. 5G is expected to support such connected home scenarios, whilst bringing down the service costs.
• o Smart grids: The consumption and distribution of energy, including heat or gas, is becoming highly decentralized, making the need for automated control of an extremely distributed sensor network. A smart grid interconnects such sensors, using digital information and communications technology to gather and act on information. This information can include the behaviors of providers and consumers, allowing the smart grid to enhance the efficiency, reliability, economics and sustainability of the production and distribution of fuels such as electricity in an automated fashion. A smart grid can be seen as another sensor network with low delays. 5G is expected to handle massive amount of low cost devices communication as in smart grids.

4. Key enabling technologies

Wireless technology will be the starting point to enable novel capabilities of 5G. The following briefly introduce key technological components supporting the evolution of 5G in future:

4.1. mmWave

In 5G networks, spectrum availability is one of the key challenges of supporting the vast mobile traffic demand. Nowadays, the current spectrum is crowded already. Especially in very dense deployments it will be necessary to go higher in frequency and use larger portions of free spectrum bands. This means that 5G networks will operate in a wide spectrum range with a diverse range of characteristics, such as bandwidth and propagations conditions [8]. The mmWave bands provide ten times more bandwidth than the 4G cellular-bands. Therefore, the mmWave bands can provide the higher data rates required in future mobile broadband access networks. The main reason that mmWave spectrum lies idle is that, it had been deemed unsuitable for mobile communications because of rather hostile propagation qualities, including strong path loss, atmospheric and rain absorption, low diffraction around obstacles and penetration through objects, strong phase noise and exorbitant equipment costs. However, semiconductors are maturing, their costs and power consumption rapidly falling, and the other problems related to propagation are now considered increasingly surmountable given time and focused effort. These adaptive directional beams with large antenna array gain are key in tackling the large propagation loss in the mmWave [9].

4.2. Advanced small cell

To achieve significant throughput enhancement in a practical manner, it is necessary to place a large number of cells in a given area and to manage them intelligently. The 5G system is expected to utilize higher frequencies to take advantage of the large bandwidth in the mmWave bands. Hence, the considerably high propagation loss of mmWave makes it suitable for dense small cell deployment, which leads to higher spatial reuse. Cell shrinking has numerous advantages, the most important being the reuse of spectrum across a geographic area and the resulting reduction in the number of users competing for resources at each base station. Distributed and self-configuring network technologies will make it easy to deploy many small base stations in urban and suburban areas.

4.3. Massive MIMO

One promising technology for meeting the future demands is massive MIMO. Well-established by the time LTE was developed, MIMO was a native ingredient by using two-to-four antennas per mobile unit and as many as eight per base station sector shown in Figure 2. The proposal was to equip base stations with a number of antennas much larger than the number of active users per time frequency signaling resource, and given that under reasonable time- frequency selectivities accurate channel estimation can be conducted for at most some tens of users per resource, this condition puts the number of antennas per base station into the hundreds. This bold idea, initially termed "large-scale antenna systems" but now more popularly known as "massive MIMO", offers enticing benefits: huge enhancements in spectral efficiency, smooth out channel responses because of the vast spatial diversity, and simple transmit/receive structures. The promise of these benefits has elevated massive MIMO to a central position in preliminary discussions about 5G.

4.4. Multi RAT

An important feature therein will be increased integration between different RATs, with a typical 5G-enabled device having radios capable of supporting not only a potentially new 5G standard (e.g., at mmWave frequencies), but also 3G, numerous releases of 4G LTE including possibly LTE-Unlicensed, several types of WiFi, and perhaps direct M2M communication, all across different spectral bands. Techniques for interworking and integrating the 5G system with other RATs will be exploited based on advanced PHY/MAC/network technologies and efficient methods of cell deployment and spectrum management. By taking advantage of multiple RATs, the 5G system will be able to use the benefit of the unique characteristics of each RAT and improve the practicality of the system as a whole. For instance, the 4G system is used for transferring the control messages to maintain the connection, to perform handover, and to provide real-time services such as VoLTE. The technology operating in mmWave unlicensed frequency band would offer the gigabit data rate service. Multiple mmWave cells can be overlaid on top of the underlying 4G macro cells. Moreover, user equipment is an integrated part of the network, any change in the network would affect the operation at the UE. And it requires a corresponding change in the UE side, such as UE with multi - RAT policy. There may be a need for new security solutions for key exchange or derivation protocols in UE side upon handover or when interworking with other Radio Access Technologies (RATs).

4.5. Advanced D2D

Advanced D2D communication is an attractive technology that improves spectral efficiency and reduces end-to-end latency for 5G. Not entirely depending on the cellular network, D2D devices can communicate directly with one another when they are in close location. Hence, D2D communication will be used for offloading data from network so that the cost of processing those data and related signaling is decreased. For example, the cars can communicate with each other to share the information for safety alarm and infotainment without cellular base station. The home appliances communicate with each other for home automation service. Since the data is directly transmitted and not going through the core network, the end- to-end latency can be considerably reduced as shown in Figure 3.

4.6. Network slicing

With network slicing technology, a single physical network can be coupled into multiple virtual networks allowing the operator to offer optimal support for different types of services for different types of customer segments. The key benefit of network slicing technology is it enables operators to provide networks on an as-a-service basis, which improves operational efficiency while reducing time- to-market for new services [10].

In the future development, 5G architecture still being driven by its applications, technologies, from the various use cases to diverse types of communications (human- to-human, human-to- machine and machine-to-machine communications).

5. Conclusion

The motivations behind the development of 5G are: the explosive increase in demand for wireless broadband services needing faster, higher-capacity networks that can deliver video and other content-rich services; and the IoT that is fueling a need for massive connectivity of devices, plus a need for ultra-reliable, ultra-low-latency connectivity over Internet Protocol (IP). The current wireless networks are struggling to deal with those distinct application requirements, its limitation of performance led to the slow adaption of these new services. From that reason, the need to introduce a new advanced architectural framework for the processing and transport of information is required, which will bring new unique network services and capabilities that far beyond those previous generation of mobile networks [12]. The 5G appears to satisfy all these above service demands.

Furthermore, use cases will be delivered across a wide range of devices (e.g., smartphone, wearable, sensors) and across a fully heterogeneous environment. For the sake of simplicity, we divide 5G use cases into three main families: Mobile Broadband Access, Critical Communication and Massive IoT. Each of these families contains load of different use cases that are possible to deploy in the future

Author Contributions:

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Conflicts of Interest:

''The authors declare no conflict of interest.''

1. Introduction

Circular Economy (CE) is thought to be the definitive solution to achieve sustainability if it can be accomplished with non-toxic and natural materials. Qualitative and quantitative methods to assess the circularity of products are rife [1,2,3,4,5,6,7,8,9]. However, none of them has yet been agreed as international standard for circularity scoring. Meanwhile, Life Cycle Assessment (LCA) is a tool for product sustainability evaluation [10,11] in which carbon dioxide emissions are most in focus.

Pena et al., clarified the potentials of LCA and the need of its coherent application in the development, adoption, and implementation of CE worldwide to advance more effectively and efficiently towards environmental sustainability [12]. Ford and Fisher used primary energy analysis of the life cycle to confirm the environmental feasibility of using 100% recycled Acrylonitrile Butadiene Styrene (rABS) in the caseworks of small consumer electronic products (EEE) as a step towards more circular design and manufacturing [13]. Schulte et al., analyzed the environmental consequences of electrophysiology catheters considering two modeling perspectives, the implementation of LCA, including a cut-off approach and combining LCA and a circularity indicator measuring multiple life cycles [14]. Collection rate was found to be an important parameter for successful overall circularity [14]. The influence of collection rate was also identified for mobile phones [15]. However, the focus was on larger product systems on not on the design improvement of one product. In any case, the present research focuses more on the immediate circular eco-design and its effect on the lifetime and the carbon score. Anyway, the L.1023 standard [1] from International Telecommunication Union's branch for Standardization (ITU-T) is a qualitative scoring method by which ICT goods and other EEE can be assessed from 0% (worst) to 100% (best) for circularity in three dimensions Product Durability (PD), Ability to Recycle, Reuse, Refurbish and Upgrade, equipment level (3RUe) and Ability to Recycle, Reuse, Refurbish and Upgrade, manufacturer level (3RUm). The ability to provide business models supporting CE is included in L.1023. However, LCA is merely addressed by availability and quality of the LCA study, and not by absolute carbon and LCA values. The present research will show how L.1023 and carbon scores can be combined for low carbon circular product design.

The assessment method outlined in L.1023 consists of three steps:

1. Setting the relevance and applicability (R) of each criterion for circular product design for the ICT good at hand.
2. Assess the margin of improvement (MI) of each criterion.
3. Calculate the circularity score from 0 to 100% for the ICT good at hand for all three criteria groups (CGs) PD, 3RUm and 3RUe.

This includes:

• Using a predefined value matrix (or formula) to identify the % score from 0 to 100 for each combination of \(R\times MI\).
• Derive individual averages for the included criteria separately for all three CGs: PD, 3RUm and 3RUe.

However, for L.1023, no method for establish weighting factors for PD, 3RUe, 3RUm has been defined. As a result, single product circularity scores cannot be obtained with L.1023. Here an approach based on Analytical Hierarchy Process (AHP) is presented. The links to Life Cycle Assessment (LCA) results of a baseline and 75 redesigned generic electronic product (EEE) is also outlined.

AHP is very well known [16,17,18] as a method to derive weighting factors for multiple criteria and illustrate uncertainty trade-offs, and so is product life cycle carbon footprint (PCF) for determining relations between life cycle stages [19]. Bringing further clarity to the connections between Circularity Scoring (CS) and PCF scores for EEE is one of the goals of the present research.

For the first time the effect of product lifetime reduction is used with AHP to determine weighting factors for criteria groups within the L.1023 standard.

The scope of the present investigation is the production of one generic EEE with a lifetime of 5 years. The present research can support the application of the L.1023 circularity scoring by providing a method by which a single (%) score can be obtained instead of three different. The news value of the present research concerns the weighting factors for groups in a specific circularity scoring context and the role of PCF and related PCF scores.

2. Problem formulation

The present research focuses on finding a methodology for quantifying weighting factors for the scores for PD, 3RUe, 3RUm criteria groups of L.1023. In the present research the hypotheses are:

• AHP and product lifetime can be used to determine weighting factors for the three criteria groups of L.1023.
• The change in PCF score due to a change in weighted L.1023 score can be derived.

3. Research approach

The first step of the present research approach is to use the L.1023 standard [1] to calculate unweighted scores for PD, 3RUe and 3Rum for a baseline and a redesigned version of the EEE, respectively.

The second step is to estimate the lifetime of the \(EEE (LT_{EEE,k} )\). The third step is to estimate how much the worst criterion score (i.e., Margin of Improvement \((MI)=4\) for e.g. \(PD1\), \(MI=4\) for e.g. \(3RUe1\) etc.) individually in each criteria group (CG) would reduce the lifetime of the EEE resulting in so called individual lifetime reduction factors (\(LTRF_{CG,i,j,n}\), see Eq. (1)). The forth step is to multiply all \(LTRF\) within each criteria group (e.g. \(i=PD\)) to arrive at a new number of lifetime units (larger than one) (\(ALTRS_{CG,i}\), see Eq. (2)) which the EEE needs per lifetime for each group (\(U_{EEE,i,k}\), see Eq. (3)). The relation between the new number of lifetime units (baseline is 1 for all) is the basis for the AHP weighting factors for the Groups.

where

\(U_{EEE,i,k} =\) units of EEE of generation \(k\) required during EEE lifetime for Criteria Group \(i,\)

\(ALTRS_{CG,i} =\) accumulated lifetime reduction score for Criteria Group \(i,\)

\(LTRF_{CG,i,j,n}=\) lifetime reduction factor \(n\) for Criteria Group \(i\) and Criterion \(j,\)

\(LT_{EEE,k} =\) Lifetime EEE generation \(k,\) years,

\(LTR_{CGi,j} =\) lifetime reduction for Criteria Group \(i\) and Criterion type \(j,\) years,

\(i =\) Criteria Group type. PD, 3RUe, 3RUm,

\(j =\) Criterion type,

\(n =\) number of criteria in Criteria Group \(1,2,3,...,n\),

\(k = EEE\) generation, e.g. baseline and redesigned.

Table 1. Accumulated lifetime reduction, units per lifetime and weights for Electronic Product (EEE).

		Electronic product \((EEE) (LT_{EEE,k}=5 years)\)
Group (CG)	Code	MI	Lifetime reduction (years), \(LTR_{CGi,j}\)	\(LTRF_{CGi,j}\)
Product Durability	PD1	4	1	(5-1)/5=0.8
	PD2	4	0.05	0.99
	PD3	4	2.5	0.5
	PD4	4	2.5	0.5
	PD5	Not applicable	n.a	No. battery
	PD6	4	1	0.8
Ability to Recycle, Repair, Reuse upgrade -equipment level	3RUe1	4	1	0.98
	3RUe2	4	0.1	0.98
	3RUe3	4	0.1	1
	3RUe4	4	0	0.98
	3RUe5	4	0.1	1
	3RUe6	4	0	1
	3RUe7	4	0	1
	3RUe8	4	0	1
	3RUe9	4	0	1
Ability to Recycle, Repair, Reuse upgrade-manufacturer level	3RUm1	4	1	0.8
	3RUm2	4	2.5	0.5
	3RUm3	4	0.1	0.98
	3RUm4	4	1	0.8
	3RUm5	4	0	1
	3RUm6	4	0	1

Very few EEE would score \(MI=4\) for all criteria but it is applied here to predict the effect on lifetime. As shown in Table 1, for \(MI=4\) in PD3 it is assumed that without maintenance infrastructure and availability of wear-out parts the lifetime of the EEE would be reduced 50%.

Table 2 shows the \(ALTRS_{CG,i}\) and resulting \(U_{EEE,i,k}\) and AHP weights (w).

Table 2. Accumulated lifetime reduction, units per lifetime and weights for Electronic Product (EEE).

i	\(ALTRS_{CG,i}\)	\(U_{EEE,i,k}\)	Relative	AHP Weights (w)
PD	0.158	6.31	1.00	0.6
3RUe	0.943	1.06	0.17	0.1
3RUm	0.313	3.19	0.51	0.3

From Table 1 it is clear that a very low robustness \((PD4)\) and providing no maintenance (PD3) reduce the lifetime much more than 3Rue and 3RUm criteria, except for non availability of spare parts \((3RUm2)\). Observe that the estimation of lifetime reduction is done for the worst possible design (\(MI=4\) for all applicable criteria) of EEE.

Table 3. Explanation of codes for sub-criteria within each group.

Code	Explanation
PD1	Software and data support
PD2	Scratch resistance
PD3	Maintenance support
PD4	Robustness
PD5	Battery for portable ICT goods
PD6	Data security
3RUe1	Fasteners and connectors
3RUe2	Diagnostic support
3RUe3	Material recycling compatibility
3RUe4	Disassembly depth
3RUe5	Recycled/renewable plastics
3RUe6	Material identification
3RUe7	Hazardous substances
3RUe8	Critical Raw Materials
3RUe9	Packaging recycling
3RUm1	Service offered by manufacturer
3RUm2	Spare parts distribution
3RUm3	Spare parts availability
3RUm4	Disassembly information
3RUm5	Collection and recycling programmes
3RUm6	Environmental footprint assessment knowledge available to improve the equipment material efficiency

4. L.1023 scores for Electronics product

In this research a baseline (Table 4) and a redesigned EEE (Table 5) are evaluated with the L.1023 standard.

Table 4. Baseline design of Electronic product (EEE) unweighted circularity scores.

		EEE (baseline)
Circularity Group (CG)	Code	Margin of improvement(MI)	Relevance (R)	Circularity Score (CS)	Average score
Product Durability	PD1	2	3	53	55
	PD2	2	3	53
	PD3	3	3	27
	PD4	1	3	86
	PD5	0	0	0
	PD6	2	3	53
Ability to Recycle, Repair Reuse, upgrade - equipment level	3RUe1	2	2	48	31
	3RUe2	3	3	27
	3RUe3	3	3	27
	3RUe4	2	3	53
	3RUe5	4	3	14
	3RUe6	3	3	27
	3RUe7	3	3	27
	3RUe8	3	3	27
	3RUe9	3	3	27
Ability to Recycle, Repair, Reuse, upgrade -manufacturer level	3RUm1	2	1	45	39
	3RUm2	3	2	32
	3RUm3	2	1	45
	3RUm4	3	2	32
	3RUm5	2	1	45
	3RUm6	4	1	31

Table 5. Redesigned Electronic product (EEE) unweighted circularity scores.

		EEE (baseline)			12/18/2021
Circularity Group (CG)	Code	Margin of improvement(MI)	Relevance (R)	Circularity Score (CS)	Average score
Product Durability	PD1	2	3	53	60
	PD2	2	3	53	45
	PD3	1	3	53
	PD4	0	3	86
	PD5	2	0	0
	PD6	2	3	53
Ability to Recycle, Repair Reuse, upgrade - equipment level	3RUe1	2	2	48
	3RUe2	2	3	53	52
	3RUe3	3	3	27
	3RUe4	1	3	86
	3RUe5	3	3	27
	3RUe6	2	3	53
	3RUe7	3	3	27
	3RUe8	2	3	53
	3RUe9	3	3	27
Ability to Recycle, Repair, Reuse, upgrade -manufacturer level	3RUm1	1	1	69
	3RUm2	3	2	32
	3RUm3	2	1	45
	3RUm4	2	2	48
	3RUm5	2	1	45
	3RUm6	1	1	69

In Table 6, uncertainties are expressed as orders of magnitude. As shown in Table 2, AHP weights are obtained from creating relative weights of \(U_{EEE,i}\). The \(AHP\) application method presented in [18] (§3, Table 4) is applied to the present example of Baseline and Redesigned \(EEE\) according to Eqs (4)-(26):

where

\(s_{i}=\) score of alternative j,

\(w_{i}=\) weight of indicator i,

\(\rho_{i,j}=\) relative performance of alternative j for indicator i,

\(\Delta s_{i}=\) uncertainty of score of alternative j,

\(\Delta \rho_{i,j}=\) uncertainty of relative performance of alternative j for indicator i,

\(\Delta (ln s_{baseline})=\) uncertainty of baseline alternative,

\(\Delta (ln s_{redesigned})=\) uncertainty of redesigned alternative,

\(W_{PD}=\) weight of indicator PD,

\(W_{3RUe}=\) weight of indicator 3RUe,

\(W_{3R}=\) weight of indicator 3RUm,

\(\rho_{PD,baseline}=\) relative performance of baseline alternative for PD,

\(\rho_{PD,redesigned}=\) relative performance of redesigned alternative for PD,

\(\rho_{3R,baseline}=\) relative performance of baseline alternative for 3RUe,

\(\rho_{3R,redesigned}=\) relative performance of redesigned alternative for 3RUe,

\(\rho_{3RU,baseline}=\) relative performance of baseline alternative for 3RUm,

\(\rho_{3RUm,redesigned}=\) relative performance of redesigned alternative for 3RUm,

\(s_{baseline}=\) score of baseline alternative,

\(s_{redesigned}=\) score of redesigned alternative,

\(\Delta (ln \rho_{PD,baseline})=\) uncertainty of relative performance of baseline alternative for PD,

\(\Delta (ln \rho_{PD,redesigned})=\) uncertainty of relative performance of redesigned alternative for PD,

\(\Delta (ln \rho_{3RUe,baseline})=\) uncertainty of relative performance of baseline alternative for 3RUe,

\(\Delta (ln \rho_{3RUe,redesigned})=\) uncertainty of relative performance of redesigned alternative for 3RUe,

\(\Delta (ln \rho_{3RUm,baseline})=\) uncertainty of relative performance of baseline alternative for 3RUm,

\(\Delta (ln \rho_{3RUm,redesigned})=\) uncertainty of relative performance of redesigned alternative for 3RUm,

\(\Delta (ln s_{baseline})=\) uncertainty of score of baseline alternative,

\(\Delta (ln s_{redesigned})=\) uncertainty of score of redesigned alternative.

Table 6. Decision making for L.1023.

Criteria Group (CG)	Performance		Relative Performance		Analytical Hierarchy Process(AHP) weights	Score		t	type I error  probability
	Baseline  EEE	Redesigned  EEE	Baseline  IEEE	Redesigned IEEE		Baseline  EEE	Redesigned  EEE
Product  Durability	55	60	0.48\(\pm\)0.04	0.52\(\pm\)0.04	0.60	0.29\(\pm\)0.03	0.31\(\pm\)0.03	0.98	0.33
Ability to  Recycle,  Repair, Reuse, upgrade -equipment  level	31	45	0.41\(\pm\)0.04	0.59\(\pm\)0.04	0.10	0.04\(\pm\)0.03	0.06\(\pm\)0.02	3.9	0
Ability to Recycle, Repair, Reuse, upgrade-equipment  level	39	52	0.43\(\pm\)0.04	0.57\(\pm\)0.04	0.30	0.13\(\pm\)0.03	0.17\(\pm\)0.02	3.1	0
					Total	0.46\(\pm\)0.02	0.54\(\pm\)0.02	2.8	0.006

The type I error probability that the decision-maker's requirement 257 is not met is only 0.6%. The AHP scores 0.46 and 0.54 will later be combined with the PCF score.

Table 7 shows the relative PCF scores for Global Warming Potential during 100 years (GWP100) for the baseline EEE and the individual PCF scores of the EEE with worst case PD, 3RUm and 3Rue criteria. The total carbon score is much higher for the PD worst case scenario compared to 3RUm and 3RUe as more units (6.31) need to be used during EEE lifetime compared to the other two CGs (3.19 and 1.06).

Table 7. Carbon scores for EEE.

Scenario	TOTAL CO2eq. (relative)	Manufacturing (%)	Use (%)	End-of-first-life (%)
Baseline EEE	100	79.4	21	-0.36
PD	520	96.4	4.0	-0.43
3RUm	274	92.8	7.7	-0.42
3RUe	105	80.4	20	-0.36

In Table 8 the carbon scores of the baseline EEE and redesigned EEE are presented. By using the AHP scores in Table 6 for baseline EEE and redesigned EEE, 0.46 and 0.54, the redesigned EEE eventually uses 0.84 units during 5 years and thereby has a lower CO2e score thanks to the improved circular product design measures. The circular redesign leads to 12% carbon reduction.

5. Discussion

The present research is illuminating the problem of weighting different Circular Economy criteria in the international L.1023 circular scoring standard [1] and the relation to carbon scoring for environmental impact. Effect on lifetime is chosen as basis for the weighting. Effect on recycling rate is another option. The effect on recycling rate of MI=4 may be less pronounced for several criteria than their effect on lifetime.

Table 8. Relative carbon scores for redesigned EEE as effect of changed criteria.

Scenario	Total \(Co_{2}eq\). (relative)	}Manufacturing (%)	Use (%)	End of  the  first  life (%)	Number of used EEE units during  5 years	Weighted EEE units  used during 5 years	AHP score
Baseline EEE	100	79.4	20.9	-0.36	1	\(U_{EEE,i,k}\times\) Weights =\(6.31\times0.6+1.06\times0.1\) +3.16\(\times\)0.3=4.84	0.46
Redesigned EEE	88	76.6	23.7	-0.34	4.05/4.84=0.84	4.84\(\times\)0.46/0.54=4.05	0.54
Redesigned EEE  with only M=1	68	69.3	31.1	-0.31	2.79/4.84=0.58	4.84\(\times\)0.37/0.63=2.79	0.63

In any case, the result of the AHP process shows that, when evaluated with weighting for single circularity score, the redesigned EEE scores slightly higher (that is better) than the baseline EEE. Ideally the carbon (and other indicators and single scores) result would also be better for the redesigned EEE than the baseline EEE. This is also demonstrated herein (Table 8) ibn which redesigned is 12% better than baseline. The rationale is that the redesigned EEE would require e.g. fewer EEE units used per lifetime. Likely the improvement of the criteria in Table 3 have helped increase the lifetime and lower the carbon score of the redesigned EEE. Moreover, compared to the baseline EEE, the relative carbon score for a redesigned EEE scoring MI=1 for all sub criteria in L.\(1023\) is around \(68\) compared to \(88\) in Table 8 for the mixed MI values of Table 5. This is based on AHP scores of \(0.37(\pm0.02)\) and \(0.63(\pm0.01)\) for baseline and redesigned EEEs, respectively. As not all criteria are highly relevant (\(R=4\)), a perfect Circular Score of 100% is not possible for the present example. The relevance (R) may be different for each case and determined by business model and others. The MI on the other hand can be determined objectively.

The uncertainty range for each design alternative's AHP score is assumed to be around 10% or 0.04 orders of magnitude. The uncertainty is judged to be rather small as it is rooted in the ''wrong'' choice of MI values for some criteria.

A criterion for modular design is missing from L.1023 despite being an important criterion in other circularity scoring methods [3,7]. Obviously a modular design criterion - added to the 3RUe Group with MI=4 in Table 1 - would reduce the lifetime of several EEE and increase the weight of 3RUe [20].

6. Conclusions

Using lifetime reduction, AHP can systematically be used to determine weighting factors for the three criteria groups of L.1023. For a redesign of an EEE product, the change in carbon score due to a change in weighted L.1023 score can be derived.

7. Next steps

Here the assumed effects on product lifetime of hypothesized worst MI levels are investigated for one EEE. Effect on recycling rate is another option. In general, EEE may have several special considerations and several additional criteria (and perhaps criteria groups) will have to be developed for potential updates of L.1023. Systematic uncertainty estimation of the AHP weighted scores for individual criteria groups can be improved. Another outlook is to include further indicators and single scores for full LCA combined with AHP and uncertainty analyses.

Conflicts of Interest:

''The author declares no conflict of interest.''