Selma Belhadj Amor

Samir M Perlaza
email: samir.perlaza@inria.fr

H Vincent Poor
email: poor@princeton.edu

Decentralized Simultaneous Energy and Information Transmission in Multiple Access Channels

Keywords: Gaussian multiple access channel, simultaneous information and energy transmission, RF harvesting, information-energy capacity region, Nash equilibrium, stability

In this report, the fundamental limits of decentralized simultaneous information and energy transmission in the two-user Gaussian multiple access channel (G-MAC) are fully characterized for the case in which a minimum energy transmission rate b is required for successful decoding. All the achievable and stable information-energy transmission rate triplets (R 1 , R 2 , B) are identified. R 1 and R 2 are in bits per channel use measured at the receiver and B is in energy units per channel use measured at an energy-harvester (EH). Stability is considered in the sense of an η-Nash equilibrium (NE), with η 0 arbitrarily small. The main result consists of the full characterization of the η-NE information-energy region, i.e., the set of information-energy rate triplets (R 1 , R 2 , B) that are achievable and stable in the G-MAC when: (a) both transmitters autonomously and independently tune their own transmit configurations seeking to maximize their own information transmission rates, R 1 and R 2 respectively; (b) both transmitters jointly guarantee an energy transmission rate B at the EH, such that B b. Therefore, any rate triplet outside the η-NE region is not stable as there always exists one transmitter able to increase by at least η bits per channel use its own information transmission rate by updating its own transmit configuration.

Par conséquent, tout triplet en dehors de la région η-NE n'est pas stable car il doit toujours y avoir un transmetteur qui soit capable d'augmenter son débit d'information par au moins η bits par utilisation canal en ajustant sa propre configuration d'émission.

Mots-clés : Canal Gaussien à accès multiple (G-MAC), transmission simultanée d'information et d'énergie, collecte d'énergie RF, région de capacité d'information-énergie, équilibre de Nash, stabilité.

Introduction

In wireless communication networks, energy and information can be simultaneously transmitted [START_REF] Varshney | On energy/information cross-layer architectures[END_REF]. From the perspective of information transmission in point-to-point channels, the fundamental limit on the information rate (in bits per channel use) is given by the information capacity [START_REF] Shannon | A mathematical theory of communication[END_REF]. Information capacity is defined as the supremum over all information rates for which it is possible to reliably transmit information. Alternatively, from the perspective of simultaneous energy and information transmission (SEIT), a trade-off between the information transmission rate and the energy transmission rate (in energy units per channel use) is usually observed. For instance, consider a binary symmetric channel (BSC) with cross-over probability p, i.e., P (1|1) = P (0|0) = 1p and P (1|0) = P (0|1) = p. Assume also that the symbol '1' provides 1 energy unit whereas the symbol '0' provides 0 energy units. The capacity of this channel is 1 -H 2 (p) bits per channel use and is achieved by equiprobable inputs. Here H 2 (•) denotes the binary entropy function. Equiprobable inputs induce an energy rate of 1 2 energy units per channel use. The maximum energy rate is 1p energy units per channel use, when '1' is always sent. Let b, with 0 b 1p, denote the required minimum energy rate. If b > 1 2 , then equiprobable capacity-achieving inputs are not sufficient to achieve the minimum energy rate and the transmitter is forced to use the symbol '1' more frequently than the symbol '0', which induces an information rate loss. In this case, the maximum information rate which can be achieved is H 2 (b) -H 2 (p) and is strictly smaller than the capacity and is decreasing in b. The fundamental limit on the information rate for a minimum energy rate in point-to-point channels is given by the information-energy capacity function derived by Varshney [START_REF] Varshney | Transporting information and energy simultaneously[END_REF].

In the context of multi-user channels the information-energy fundamental limits are fully described by the information-energy capacity region. That is, the set of all achievable informationenergy tuples at which energy and information can be reliably transmitted. The informationenergy capacity region in the discrete memoryless multi-access channel (MAC) and multi-hop networks was studied by Fouladgar et al. [START_REF] Fouladgar | On the transfer of information and energy in multi-user systems[END_REF]. Recently, Belhadj Amor et al. [START_REF] Belhadj Amor | Feedback enhances simultaneous information and energy transmission in multiple access channels[END_REF][START_REF] Belhadj Amor | Simultaneous energy and information transmission in Gaussian multiple access channels[END_REF] derived the information-energy capacity region of the Gaussian MAC (G-MAC) with and without feedback where there exists an energy harvester (EH), possibly non-colocated with the main receiver. Analogously to the point-to-point case, these works show that there exist two energy regimes: One in which the energy rate constraint does not have any impact and thus, the set of achievable information rate tuples are those of the classical G-MAC. Conversely, in the other regime, increasing the information rate implies reducing the energy rate and vice-versa. An object of central interest regarding the results in [START_REF] Fouladgar | On the transfer of information and energy in multi-user systems[END_REF], [START_REF] Belhadj Amor | Feedback enhances simultaneous information and energy transmission in multiple access channels[END_REF], and [START_REF] Belhadj Amor | Simultaneous energy and information transmission in Gaussian multiple access channels[END_REF] is that the achievability of these informationenergy rate tuples is subject to the existence of a central controller that decides an operating point and indicates to all network components the corresponding transmit-receive configuration that should be used. Unfortunately, this assumption does not hold in networks in which a central controller is not feasible. This is typically the case of decentralized or ad hoc networks such as sensor networks, body area networks, among others. In this type of multi-user channels, both the transmitters and the receivers are assumed to be autonomous and to be able of unilaterally choosing their own transmit-receive configurations aiming to maximize their individual benefit, e.g., individual information rate, individual energy rate or a combination of both. Hence, from this perspective, the notion of information-energy capacity does not properly model the fundamental limits of simultaneous energy and information transmission in decentralized networks. To tackle this anarchical behavior observed in decentralized networks, the notion of stability is introduced and a new notion is presented: the energy-information η-Nash region. This region is the set of all information-energy rate tuples that are achievable and stable. In this case, stability is considered in the sense of an η-Nash equilibrium (NE) [START_REF] Nash | Equilibrium points in n-person games[END_REF], with η arbitrarily small. A multi-user channel is stable in the sense of an η-NE if none of the transmitters or receivers

RR n°8847

is able to increase its own individual benefit by more than η units by unilaterally changing its transmit-receive configuration.

The remaining of this report focuses exclusively on the analysis of the decentralized G-MAC with an external EH. Yet very simple, this channel model captures the key aspects of the intrinsic information-energy trade-off in the context of multi-user channels. This analysis strongly relies on tools brought from information theory [START_REF] Gamal | Network Information Theory[END_REF] and game theory [START_REF] Alpcan | Mechanisms and Games for Dynamic Spectrum Allocation[END_REF], [START_REF] Lasaulce | Game Theory and Learning for Wireless Networks: Fundamentals and Applications[END_REF], and [START_REF] Han | Game Theory in Wireless and Communication Networks: Theory, Models and Applications[END_REF]. Previous works have studied decentralized MACs using game-theoretic tools when the aim of each transmitter is limited to exclusively transmitting information. For instance, Lai and El Gamal [START_REF] Lai | The water-filling game in fading multiple-access channels[END_REF] proposed a framework to study the power allocation problem in fading decentralized MACs when the transmitters aim to maximize their own individual transmission rate. Gajic and Rimoldi [START_REF] Gajic | Game theoretic considerations for the Gaussian multiple access channel[END_REF] considered a similar scenario with time-invariant channels in which the transmitters have the choice of adopting any possible transmit configuration and determined the subregion of the information capacity region that is achievable at an NE. Varan and Yener [START_REF] Varan | Incentivizing signal and energy cooperation in wireless networks[END_REF] studied two-hop networks in which the source(s) is (are) incentivized to perform energy and signal cooperation to maximize the amount of its (their) own data that is reliably delivered to the destination. A review of the state of the art on fundamental limits of SEIT in point-to-point and multi-user channels is provided in [START_REF] Belhadj Amor | Fundamental limits of simultaneous energy and information transmission[END_REF].

This report studies the fundamental limits of decentralized SEIT in the two-user G-MAC when a minimum energy rate is required for successful decoding. More specifically, each transmitter chooses its own transmit configuration aiming to maximize its individual information rate to the receiver/information decoder while it guarantees an energy transmission rate higher than a given predefined threshold at a given EH. The receiver is assumed to adopt a fixed configuration that can be either single-user decoding (SUD), successive interference cancellation (SIC) or any time-sharing configuration of the previous decoding techniques. This report provides a game formulation of this problem. The main contribution is the full characterization of the η-NE information-energy region of this game, with η 0 arbitrarily small. Note that when there is no energy rate requirements, the competition between the players (the transmitters) is only through interference because it is the unique source of interdependence among the different players. However, when a minimum energy rate constraint is required for successful decoding, it clearly creates an additional interdependence among the transmitters. In fact, consider a multi-access scenario in which a given transmitter simultaneously transmits energy to an EH and information to a receiver. If this transmitter is required to deliver an energy rate that is less than what it is able to deliver by only transmitting information, it is able to fulfill the task independently of the behavior of the other transmitters since it can use all its power budget to maximize its information transmission rate and it is still able to meet the energy rate constraint. In this case, the minimum energy constraint does not play a fundamental role. Alternatively, when the requested energy rate is higher than what it is able to deliver by only transmitting information, its behavior is totally dependent on the behavior of the other transmitters since its decodable information rate depends on whether or not the other transmitters are sending messages using an average power for which the minimum energy rate constraint at the EH is satisfied.In this case, the minimum energy rate constraint drastically affects the way that the transmitters interact with each other. More critical scenarios are the case in which the requested energy rate is more than what all transmitters are able to deliver by simultaneously transmitting information using all the available individual power budgets. In these cases, none of them can unilaterally ensure reliable energy transmission at the requested rate. Hence, the transmitters must engage in an energy cooperation mechanism through which they reduce their information rate to be able to send an energy rate that is higher than the minimum required energy rate at the EH. In a decentralized setup, such a cooperation is not natural and this shows how the energy rate constraint can clearly change the behavior of the players. The remainder of the report is structured as follows. Section 2 describes the channel model and provides a game-theoretic formulation of decentralized SEIT in the G-MAC. Section 3 shows the main results of this paper and reports important observations. In Section 4 and Section 5, the proofs are provided. Finally, Section 6 concludes the paper.

2 Gaussian MAC with Minimum Energy Rate b

Channel Model

Consider the two-user memoryless G-MAC with an EH as shown in Fig. 1. At each channel use t ∈ N, X 1,t and X 2,t denote the real symbols sent by transmitters 1 and 2, respectively. Let n ∈ N be the blocklength. The symbols X i,1 , . . . , X i,n satisfy an expected average input power constraint

P i = 1 n n t=1 E X 2 i,t P i,max , (1)
where the expectation is over the message indices and where P i and P i,max denote respectively the average transmit power and the maximum average power of transmitter i in energy units per channel use, for i ∈ {1, 2}. The receiver observes the real channel output

Y 1,t = h 11 X 1,t + h 12 X 2,t + Z t , (2)
and the EH observes

Y 2,t = h 21 X 1,t + h 22 X 2,t + Q t , (3)
where h 1i and h 2i are the corresponding constant non-negative channel coefficients from transmitter i to the receiver and EH, respectively. The channel coefficients satisfy the following

L 2 -norm condition: ∀j ∈ {1, 2}, h j 2 1, (4)
with h j (h j1 , h j2 ) T in order to meet the energy conservation principle. The noise terms Z t and Q t are realizations of two identically distributed zero-mean unit-variance real Gaussian random variables. In the following, there is no particular assumption on the joint distribution of Q t and Z t . The signal to noise ratios (SNRs): SNR ji , with ∀(i, j) ∈ {1, 2} 2 are defined as follows

SNR ji |h ji | 2 P i,max , (5)
given the following normalization over the noise power. Within this context, two main tasks are to be simultaneously accomplished: information transmission and energy transmission.

Information Transmission

The goal of the communication is to convey the independent messages M 1 and M 2 from transmitter 1 and transmitter 2 to the common receiver. The message indices M 1 and M 2 are independent of the noise terms Z 1 , . . . , Z n , Q 1 , . . . , Q n and uniformly distributed over the sets M 1 {1, . . . , 2 nR1 } and M 2 {1, . . . , 2 nR2 }, where R 1 and R 2 denote the information transmission rates.

At each time t, the t-th symbol of transmitter i, for i ∈ {1, 2}, depends solely on its message index M i and a randomly generated index Ω ∈ {1, . . . , 2 nRr }, with R r 0, that is independent of both M 1 and M 2 and assumed to be known by all transmitters and the receiver, i.e.,

X i,t = f (n) i,t (M i , Ω), t ∈ {1, . . . , n}, (6)
for some encoding functions f

(n) i,t : M i ×N → R. The receiver produces an estimate ( M (n) 1 , M (n) 2 ) = Φ (n) (Y n ) of the message-pair (M 1 , M 2 ) via a decoding function Φ (n) : R n → M 1 × M 2 ,
and the average probability of error is given by

P (n) error (R 1 , R 2 ) Pr ( M (n) 1 , M (n) 2 ) = (M 1 , M 2 ) . (7)

Energy Transmission

Let b 0 denote the minimum energy rate that must be guaranteed at the input of the EH in the G-MAC. The minimum energy rate b must satisfy

0 b 1 + SNR 21 + SNR 22 + 2 SNR 21 SNR 22 , (8)
for the problem to be feasible. This is mainly because 1 + SNR 21 + SNR 22 + 2 √ SNR 21 SNR 22 is the maximum energy rate that can be achieved at the input of the EH given the input power constraints. This maximum energy rate can be achieved when the transmitters use all their power budgets to send fully correlated channel inputs.

The average energy transmission rate (in energy units per channel use) induced by the sequence (Y 2,1 , . . . , Y 2,n ) at the input of the EH is

B (n) 1 n n t=1 E Y 2 2,t , (9)
where the expectation is over the message indices M 1 and M 2 .

The goal of the energy transmission is to guarantee that the expected energy rate B (n) is not less than a given target energy transmission rate B that must satisfy

b B 1 + SNR 21 + SNR 22 + 2 SNR 21 SNR 22 . (10)
Hence, the probability of energy outage is defined as follows:

P (n) outage (B) = Pr B (n) < B -, (11)
for some > 0 arbitrarily small.

Simultaneous Energy and Information Transmission

The G-MAC in Fig. 1 is said to operate at the information-energy rate triplet (R 1 , R 2 , B) ∈ R 3 + when both transmitters and the receiver use a transmit-receive configuration such that: (i)

RR n°8847

information transmission occurs at rates R 1 and R 2 with probability of error arbitrarily close to zero; and (ii) energy transmission occurs at a rate not smaller than B with energy-outage probability arbitrarily close zero. Under these conditions, the information-energy rate triplet (R 1 , R 2 , B) is said to be achievable in the G-MAC with minimum energy rate constraint b.

Definition 1 (Achievable Rates). The triplet (R 1 , R 2 , B) ∈ R 3
+ is achievable if there exists a sequence of encoding and decoding functions {f

(n) 1,t } n t=1 , {f (n) 2,t } n t=1 , Φ (n) ∞
n=1 such that both the average error probability and the energy-outage probability tend to zero as the blocklength n tends to infinity. That is,

lim sup n→∞ P (n) error (R 1 , R 2 )=0, and (12)
lim sup n→∞ P (n) outage (B) =0. ( 13
)
Note that the minimum energy rate constraint b requires in particular that:

lim sup n→∞ P (n) outage (b) = 0. (14)
Often, increasing the energy transmission rate implies decreasing the information transmission rates and vice versa. An important notion to characterize the fundamental limits on this information-energy trade-off is the information-energy capacity region defined as follows:

Definition 2 (Information-Energy Capacity Region).

(R 1 , R 2 , B) that satisfy 0 R 1 1 2 log 2 (1 + β 1 SNR 11 ) , (15a) 0 R 2 1 2 log 2 (1 + β 2 SNR 12 ) , (15b) 0 R 1 + R 2 1 2 log 2 1 + β 1 SNR 11 + β 2 SNR 12 , (15c) b
B 1 + SNR 21 + SNR 22 + 2 (1 -β 1 )SNR 21 (1 -β 2 )SNR 22 , (15d)
with (β 1 , β 2 ) ∈ [0, 1] 2 .
Proof: The proof of Theorem 1 follows immediately from [5, Proposition 1] and [5, Theorem 2].

Comments and Observations: In the constraints [START_REF] Belhadj Amor | Fundamental limits of simultaneous energy and information transmission[END_REF], when the parameters β 1 and β 2 are such that

β 1 = β 2 = 1, the corresponding region is characterized by 0 R 1 1 2 log 2 (1 + SNR 11 ) , (16a) 0 R 2 1 2 log 2 (1 + SNR 12 ) , (16b) 0 R 1 + R 2 1 2 log 2 1 + SNR 11 + SNR 12 , (16c) b
B 1 + SNR 21 + SNR 22 , (16d)
RR n°8847

where the information rate constraints describe the capacity region of the G-MAC and the upper bound on the energy rate constraint corresponds to the maximum energy rate that can be achieved using independent channel inputs. On the other hand, when the parameters β 1 and β 2 are such that β 1 = β 2 = 0, the information rates are R 1 = R 2 = 0 and the energy rate

B satisfies b B 1 + SNR 21 + SNR 22 + 2 SNR 21 SNR 22 . (17)
Here the upper bound equals the maximum feasible energy rate. Hence, from this constructive viewpoint, the terms β 1 and β 2 in (15) allow the transmitters to trade-off between information and energy rates. These parameters might be interpreted as the fractions of power that transmitter 1 and transmitter 2, respectively, allocate for information transmission. The remaining fraction of power (1β i ) is allocated by transmitter i for exclusively transmitting energy to the EH. More specifically, to achieve any rate triplet in this region, at each time t, transmitter i's channel input can be written as:

X i,t = (1 -β i )P i W t + U i,t , i ∈ {1, 2}, (18)
for some independent zero-mean Gaussian information-carrying symbols U 1,t and U 2,t with variances β 1 P 1 and β 2 P 2 , respectively, and independent thereof W t are zero-mean unit-variance Gaussian energy-carrying symbols known non-causally to all terminals. The codebook and the encoding-decoding schemes for the information-carrying signals can be those described in [START_REF] Cover | Some Advances in Broadcast Channels[END_REF] or [START_REF] Wyner | Recent results in the Shannon theory[END_REF].

Note that the information-carrying signals carry both energy and information. These signals are useful to both the EH and the information decoder, whereas the other signals are only energy-carrying and are useful only to the EH. These energy-carrying signals carry only common randomness that allows the creation of correlated signals to increase the energy rate. The common randomness is known to the information decoder and does not produce any interference to the information-carrying signals as its effect can be suppressed using classical successive interference cancellation.

Note that an inherent assumption here is the existence of a central controller that determines an operating point and imposes the transmit or receive configuration to be be adopted by each network component. From this global or centralized perspective all points inside the informationenergy capacity region are possible operating points. However, in a decentralized network, each network component is an autonomous decision maker that aims to maximize its own individual reward by appropriately choosing a particular transmit or receive configuration. From this perspective, only the information-energy rate tuples that are stable can be possible asymptotic operating points. The following subsection describes decentralized simultaneous energy and information transmission over the G-MAC with minimum energy rate constraint b.

Decentralized Simultaneous Energy and Information Transmission

In a decentralized G-MAC, the aim of transmitter i, for i ∈ {1, 2}, is to autonomously choose its transmit configuration s i in order to maximize its information rate R i , while guaranteeing a minimum energy rate b at the EH. In particular, the transmit configuration s i can be described in terms of the information rates R i , the block-length n, the channel input alphabet X i , the encoding functions f

(1) i , . . . , f (n)
i , the common randomness, the power dedicated to information and energy transmission, etc. The receiver is assumed to adopt a fixed decoding strategy that is known in advance to all transmitters.

Note that if the aim of each transmitter, say transmitter i, is to maximize its own individual information rate R i subject to the minimum energy rate b at the EH, it is clear from [START_REF] Belhadj Amor | Fundamental limits of simultaneous energy and information transmission[END_REF] that one option should be using a power-split in which the component dedicated to the transmission of information β i is as high as possible. However, its power-split β i must also be chosen such that the energy-outage probability ( 14) can be made arbitrarily close to zero.

This reveals that the choice of the transmit configuration of each transmitter depends on the choice of the other transmitter as they must guarantee the minimum energy constraint required at the EH. At the same time, depending on the decoding scheme at the receiver, the informationcarrying signal of one transmitter is interference to the other. This reasoning implies that the rate achieved by transmitter i depends on both configurations s 1 and s 2 as well as the configuration of the receiver, even if it is assumed to be fixed. This justifies the analysis of this scenario using tools from game theory.

Game Formulation

The competitive interaction of the two transmitters and the receiver in the decentralized G-MAC with minimum energy constraint b can be modeled by the following game in normal form:

G(b) = K, {A k } k∈K , {u k } k∈K , ( 19
)
where b is a parameter of the game that represents the minimum energy-rate that must be guaranteed at the EH (see ( 14)). The set K = {1, 2} is the set of players, that is, transmitter 1 and transmitter 2. The sets A 1 and A 2 are the sets of actions of players 1 and 2, respectively. An action of a player i ∈ K, which is denoted by s i ∈ A i , is basically its transmit configuration as described above. The utility function of transmitter i,

for i ∈ {1, 2}, is u i : A 1 × A 2 → R +
and it is defined as its own information rate,

ui(s1, s2) = Ri(s1, s2), if P (n)
error(R1, R2) < and P

(n) outage (b) < δ -1, otherwise, (20)
where > 0 and δ > 0 are arbitrarily small numbers and R i (s 1 , s 2 ) denotes an information transmission rate achievable (Def. 1) with the configurations s 1 , and s 2 . Note that the utility is -1 when either the error probability or the energy outage probability is not arbitrarily small. This is meant to favor the action profiles in which there is no information transmission (information rate and error probability are zero) but there is energy transmission (probability of energy outage can be made arbitrarily close to zero), over the actions in which the information rate is zero but the energy constraint is not satisfied. Often, the information rate R i (s 1 , s 2 ) is written as R i for simplicity. However, every non-negative achievable information rate is associated with a particular transmit-receive configuration pair (s 1 , s 2 ) that achieves it. It is worth noting that there might exist several transmit-receive configurations that achieve the same triplet (R 1 , R 2 , B) and distinction between the different transmit-receive configurations is made only when needed. A class of transmit-receive configurations s * = (s * 1 , s * 2 ) ∈ A 1 × A 2 that are particularly important in the analysis of this game are referred to as η-Nash equilibria (η-NE).

η-Nash Equilibrium

A transmit-receive configuration s * = (s * 1 , s * 2 ) ∈ A 1 × A 2 that is an η-NE satisfies the following conditions: Definition 3 (η-NE [START_REF] Nisan | Algorithmic game theory[END_REF]). In the game

G(b) = K, {A k } k∈K , {u k } k∈K , an action profile (s * 1 , s * 2 ) is an η-NE if for all (s 1 , s 2 ) ∈ A 1 × A 2 , it holds that u 1 (s 1 , s * 2 ) u 1 (s * 1 , s * 2 ) + η, and (21)
u 2 (s * 1 , s 2 ) u 2 (s * 1 , s * 2 ) + η. ( 22
)
RR n°8847

From Def. 3, it becomes clear that if (s * 1 , s * 2 ) is an η-NE, then none of the transmitters can increase its own information transmission rate by more than η bits per channel use by changing its own transmit-receive configuration and keeping the average error probability and the energy outage probability arbitrarily close to zero. Thus, at a given η-NE, every player achieves a utility that is η-close to its maximum achievable information rate given the transmit-receive configuration of the other players. Note that if η = 0, then the definition of NE is obtained [START_REF] Nash | Equilibrium points in n-person games[END_REF]. The following investigates the set of information and energy rate triplets that can be achieved at an η-NE. This set of rate triplets is known as the η-NE information-energy region.

Definition 4 (η-NE Region). Let η

0. An achievable information-energy rate triplet

(R 1 , R 2 , B) ∈ E b (SNR 11 , SNR 12 , SNR 21 , SNR 22 ) is said to be in the η-NE region of the game G(b) = K, {A k } k∈K , {u k } k∈K if there exists a pair (s * 1 , s * 2 ) ∈ A 1 × A 2
that is an η-NE and the following holds:

u 1 (s * 1 , s * 2 )=R 1 and (23)
u 2 (s * 1 , s * 2 )=R 2 . ( 24
)
The following section studies the η-NE region of the game G(b), with η 0 arbitrarily small, for several decoding strategies adopted by the receiver.

Main Results

This section describes the η-NE region of the game G(b) under a fixed decoding strategy. Three cases are examined: single-user decoding (SUD), successive interference cancellation (SUD), and any time-sharing between the previous decoding strategies.

η-NE Region with Single User Decoding

The η-NE region of the game G(b) when the receiver uses single-user decoding (SUD), denoted by N SUD (b), is described by the following theorem.

0 R 1 = 1 2 log 2 1 + β 1 SNR 11 1 + β 2 SNR 12 , ( 25a
) 0 R 2 = 1 2 log 2 1 + β 2 SNR 12 1 + β 1 SNR 11 , ( 25b
) b B 1 + SNR 21 + SNR 22 + 2 (1 -β 1 )SNR 21 (1 -β 2 )SNR 22 , (25c)
where

β 1 = β 2 = 1 when b ∈ [0, 1 + SNR 21 + SNR 22 ] and (β 1 , β 2 ) satisfy b = 1 + SNR 21 + SNR 22 + 2 (1 -β 1 )SNR 21 (1 -β 2 )SNR 22 ( 26
)
when b ∈ (1 + SNR 21 + SNR 22 , 1 + SNR 21 + SNR 22 + 2 √ SNR 21 SNR 22 ].
Proof: The proof of Theorem 2 is provided in Section 4.

RR n°8847

That is, when b ∈ [0, 1 + SNR 21 + SNR 22 ], N SUD (b) contains all information-energy rate triplets (R 1 , R 2 , B) such that 26). For a given choice of (β 1 , β 2 ), the constraints (25) reduce to:

R 1 = 1 2 log 2 1 + SNR 11 1 + SNR 12 , (27a)
R 2 = 1 2 log 2 1 + SNR 12 1 + SNR 11 , (27b
R 1 = 1 2 log 2 1 + β 1 SNR 11 1 + β 2 SNR 12 , (28a)
R 2 = 1 2 log 2 1 + β 2 SNR 12 1 + β 1 SNR 11 , (28b)
B = b. (28c)
Hence, at any η-NE the energy rate must be satisfied with equality in order to maximize the information rates (See Fig. 3).

η-NE Region with Successive Interference Cancellation

Let SIC(i → j) denote the case in which the receiver uses successive interference cancellation (SIC) with decoding order: transmitter i before transmitter j, with i ∈ {1, 2}. Then, the η-NE region of the game G(b) when the receiver uses SIC(i → j), denoted by N SIC(i→j) (b), is described by the following theorem.

0 R i = 1 2 log 2 1 + β i SNR 1i 1 + β j SNR 1j , ( 29a
) 0 R j = 1 2 log 2 (1 + β j SNR 1j ) , ( 29b
) b B 1 + SNR 21 + SNR 22 + 2 (1 -β 1 )SNR 21 (1 -β 2 )SNR 22 , (29c)
where

β 1 = β 2 = 1 when b ∈ [0, 1 + SNR 21 + SNR 22 ] and (β 1 , β 2 ) satisfy b = 1 + SNR 21 + SNR 22 + 2 (1 -β 1 )SNR 21 (1 -β 2 )SNR 22 ( 30
)
when b ∈ (1 + SNR 21 + SNR 22 , 1 + SNR 21 + SNR 22 + 2 √ SNR 21 SNR 22 ].
Proof: The proof of Theorem 3 is provided in Section 5. The observations in the previous case continue to hold here. The only difference with respect to the previous case is in the information rate constraint at transmitter j. Interference cancellation allows in particular the achievability of information sum-rate optimal points at an η-NE (points on the boundary of the information-energy capacity region) as shown in Fig. 2 and Fig. 3.

RR n°8847

η-NE Region with Time-Sharing

Let N (b) denote the η-NE region of the game G(b) when the receiver uses any time-sharing between the previous decoding techniques. This region is described by the following theorem.

N (b)=Convex hull N SUD (b) ∪ N SIC(1→2) (b) ∪ N SIC(2→1) (b) . ( 31
)
That is, if the receiver performs any time-sharing combination between SUD, SIC(1 → 2), and SIC(2 → 1) then the transmitters can use the same time-sharing combination between their corresponding η-Nash equilibria strategies to achieve any point inside N (b).

Proof: The proof is based on Theorem 2, Theorem 3, and a time-sharing argument. The details are omitted. 1 + SNR 21 + SNR 22 , all transmitters use the whole available average power for information transmission as shown in Fig. 2. Alternatively, when b > 1+SNR 21 +SNR 22 , both transmitters use the minimum energy needed to make the energy-outage probability arbitrarily close to zero and seek for the largest possible information transmission rate (See Fig. 3).

Example and Observations

Q 3 Q 2 Q 1 Q 4 Q 5 Q 6 B[

Existence of an η-NE

The first observation is that the existence of an η-NE, with η arbitrarily small, is always guaranteed as long as the SEIT problem is feasible, i. √ SNR 21 SNR 22 , the energy transmission cannot be performed reliably and thus, the information-energy capacity region is empty and so is the η-NE region. In this particular case, the problem is not well-posed since such an energy rate is outside the information-energy capacity region.

Remark: Note that for any given b 0, the sets N SUD (b), N SIC (b), and N (b) include only the information-energy triplets (R 1 , R 2 , B) that satisfy B b. That is, the η-NEs at which the energy constraint can be satisfied. However, this suggests that there might exists other η-NE that are not in these sets at which either one of the conditions, [START_REF] Lai | The water-filling game in fading multiple-access channels[END_REF] or [START_REF] Gajic | Game theoretic considerations for the Gaussian multiple access channel[END_REF], is not met. Consider for instance a case in which b 1 + max(SNR 21 , SNR 22 ) and both transmitters decide to use the strategies s 1 and s 2 at which none of the transmitters actually transmits, e.g., standby mode. Hence, none of the transmitters can unilaterally deviate and achieve a utility other than u 1 (s 1 , s 2 ) = u 2 (s 1 , s 2 ) = -1 which translates into an information-energy triplet (0, 0, 0) which is also an η-NE but is not in any of the sets N SUD (b) or N SIC(i→j) (b) as the energy constraint cannot be satisfied (Def. 4). More specifically, (0, 0, 0) / ∈ E b (SNR 11 , SNR 12 , SNR 21 , SNR 22 ) for all b > 0.

Cardinality of the set of η-NE equilibria

The unicity of a given η-NE of the game G(b) is not ensured even in the case in which the cardinality of the η-NE information-energy region is one. Consider the case in which η = 0 RR n°8847 )} and for instance, using all the power budget for sending common randomness is an NE action profile. However, there is an infinite number of possible common random sequences that can be adopted by both transmitters independently of the action taken by the receiver as in this case R 1 = R 2 = 0. The cardinality of the set of η-NEs is an acceptable lower-bound for the number of equilibria. This suggests that if the cardinality of the η-NE set is infinity. Thus, the number of η-NE is also infinity as every information-energy rate triplet in N (b) is associated with at least one achievability scheme that is an η-NE (Def. 4).

Optimality of the η-NE

Probably the most interesting observation regarding Theorem 2, Theorem 3, and Theorem 4 is that some of the sum-rate optimal triples (R 1 , R 2 , B) given a minimum energy-rate b required at the EH are achievable at an η-NE. These η-NE sum-rate optimal triplets are Pareto optimal points of the information-energy capacity region E b (SNR 11 , SNR 12 , SNR 21 , SNR 22 ). This suggests that, under the assumption that the players are able to properly choose the operating η-NE for instance via learning algorithms, there is no loss of performance in the decentralized SEIT case with respect to the fully centralized SEIT case.

Proof of Theorem 2

Consider the set of information-energy rate-triplets that can be achieved under the assumption that the receiver performs SUD to recover the messages M 1 and M 2 . This set is denoted by C SUD (b) and is defined by the following lemma.

Lemma 1 (Achievable Information-Energy Region with SUD). The set C SUD (b) contains all non-negative information-energy rate-triplets (R 1 , R 2 , B) that satisfy

0 R 1 1 2 log 2 1 + β 1 SNR 11 1 + β 2 SNR 12 , ( 32a
) 0 R 2 1 2 log 2 1 + β 2 SNR 12 1 + β 1 SNR 11 , (32b) b
B 1 + SNR 21 + SNR 22 + 2 (1 -β 1 )SNR 21 (1 -β 2 )SNR 22 , (32c)
where

(β 1 , β 2 ) ∈ [0, 1] 2 .
Proof: The proof of achievability follows similar arguments to those in the proof of Theorem 1 when the decoder is restricted to use SUD to recover the messages M 1 and M 2 .

Let the subset

U SUD (b) ⊆ C SUD (b) contain all information-energy rate-triplets (R 1 , R 2 , B) ∈ C SUD (b) satisfying 0 R 1 = 1 2 log 2 1 + β 1 SNR 11 1 + β 2 SNR 12 , ( 33a
) 0 R 2 = 1 2 log 2 1 + β 2 SNR 12 1 + β 1 SNR 11 , ( 33b
) b B 1 + SNR 21 + SNR 22 + 2 (1 -β 1 )SNR 21 (1 -β 2 )SNR 22 , (33c)
where To prove Theorem 2, the first step is to show that

β 1 = β 2 =
N SUD (b) ⊆ U SUD (b). (35)
That is, any achievable information-energy rate-triplet (R 1 , R 2 , B) ∈ V SUD (b) cannot be an η-NE with η arbitrarily small, i.e., V SUD (b) ∩ N SUD (b) = ∅. This is proved by Proposition 1.

Proposition 1. Any information-energy rate-triplet (R 1 , R 2 , B) ∈ V SUD (b) is not an η-NE with η arbitrarily small. That is, N SUD (b) ⊆ U SUD (b). (36)
Proof: The proof of Proposition 1 is provided in Section 4.1. The second step consists of showing that:

U SUD (b) ⊆ N SUD (b). (37)
That is, all information-energy rate-triplets in U SUD (b) are achievable at at least one η-NE with η 0 arbitrarily small. This is proved by Proposition 2.

Proposition 2. Any information-energy rate-triplet (R 1 , R 2 , B) ∈ U SUD (b) is achievable at an η-NE with an arbitrarily small η 0. That is,

U SUD (b) ⊆ N SUD (b). (38)
Proof: The proof of Proposition 2 is provided in Section 4.2. This completes the proof of Theorem 2.

Proof of Proposition 1

Any information-energy triplet (R 1 , R 2 , B) ∈ V SUD (b) satisfies at least one of the following conditions:

R 1 < 1 2 log 2 1 + β 1 SNR 11 1 + β 2 SNR 12 , (39)
R 2 < 1 2 log 2 1 + β 2 SNR 12 1 + β 1 SNR 11 , (40)
B<b,

B>1 + SNR 21 + SNR 22 + 2 (1 -β 1 )SNR 21 (1 -β 2 )SNR 22 , (41)
where

β 1 = β 2 = 1 when b ∈ [0,
Since the only source of correlation is common randomness whose effect is canceled before decoding the messages (See Remark 1), for any values of the utililities (R * 1 , R * 2 ), one can always find β 1 and

β 2 satisfying λ = (1 -β 1 )(1 -β 2 ) (68)
such that the utilities at the equilibrium (Recall the necessary condition at any η-NE with η arbitrarily small) can be written as

u 1 (s * 1 , s * 2 )=R * 1 = 1 2 log 2 1 + β 1 SNR 11 1 + β 2 SNR 12 -1 , (69)
u 2 (s * 1 , s * 2 )=R * 2 = 1 2 log 2 1 + β 2 SNR 12 1 + β 1 SNR 11 -2 , (70)
with 1 , 2 > 0 arbitrarily small. The corresponding energy rate is given by

B * = 1 + SNR 21 + SNR 22 + 2 (1 -β 1 )SNR 21 (1 -β 2 )SNR 22 . (71)
One way to construct the channel inputs X * i,t is

X * i,t = (1 -β i )P i,max U * t + β i P i,max V * i,t , i ∈ {1, 2}, (72)
where U * , V * 1 , and V * 2 are zero-mean unit-variance Gaussian RVs that are mutually independent. The variable U * depends on the common randomness Ω and the variable V * i depends on the message M i for i ∈ {1, 2}. In this case, the lower bound (67) is simply λ > 0 which implies that β 1 < 1 and β 2 < 1 in (69) and (70), respectively.

Without loss of generality consider transmitter 1 and assume that in the action s * 1 , transmitter 1 uses a power-split β 1 < 1, which achieves a utility as in (69). Regardless of the action s * 2 , transmitter 1 can use an action s 1 in which it uses a power-split β 1 = 1 and which achieves

R 1 = 1 2 log 2 1 + SNR 11 1 + β 2 SNR 12 , (73a)
B =1 + SNR 21 + SNR 22 b, (73b)
and thus, the resulting utility of transmitter 1 is

u 1 (s 1 , s * 2 )= 1 2 log 2 1 + SNR 11 1 + β 2 SNR 12 . (74)
From ( 69) and (74), it follows that

u 1 (s 1 , s * 2 ) -u 1 (s * 1 , s * 2 ) = 1 2 log 2 1 + (1 -β 1 )SNR 11 1 + β 1 SNR 11 + β 2 SNR 12 + 1 > 0, (75)
which contradicts the assumption that (s * 1 , s * 2 ) is an η-NE (Def. 3), with an η 0 arbitrarily small. Hence, a necessary condition for an action profile to be an η-NE with η 0 is β 1 = 1.

A similar argument applies to transmitter 2, i.e., and thus any action profile for which β 1 = 1 or β 2 = 1 cannot be an η-NE with η 0. As a consequence, any action profile with

B = b, (80)
5 Proof of Theorem 3

The proof of Theorem 3 follows the same lines as the proof of Theorem 2 when considering the set of information-energy rate-triplets which can be achieved if the receiver performs SIC(i → j) to recover the messages M 1 and M 2 . This set is denoted by C SIC(i→j) (b) and is defined as the set of (R 1 , R 2 , B) that satisfy

0 R i 1 2 log 2 1 + β i SNR 1i 1 + β j SNR 1j , ( 93a
) 0 R j 1 2 log 2 (1 + β j SNR 1j ) , (93b)

Conclusion

In this paper, the fundamental limits of decentralized SEIT in the two-user G-MAC with minimum received energy rate constraint were derived in terms of η-NE regions, with η 0 arbitrarily small. The results presented here for the two-user G-MAC can be easily extended to an arbitrary number of users. A key observation in this work is the fact the decentralization induces no loss of performance for SEIT as long as the players are able to properly choose the operating η-NE for instance via learning algorithms. Recently, Belhadj Amor et al. have shown that channel output feedback enhances SEIT as it provides additional cooperation among the users. An interesting question to look at is whether feedback may help in the decentralized case. Also, the knowledge given to each player and the order in which actions can be taken substantially change the nature of the game and the corresponding stable region. Furthermore, such a region varies depending on the associated notion of equilibrium, e.g., η-Nash equilibrium (η-NE) [START_REF] Nash | Equilibrium points in n-person games[END_REF],

Stackelberg equilibrium [START_REF] Stackelberg | Marketform und Gleichgewicht[END_REF], correlated equilibrium [START_REF] Aumann | Subjectivity and correlation in randomized strategies[END_REF], satisfaction equilibrium [START_REF] Perlaza | Quality-of-service provisioning in decentralized networks: A satisfaction equilibrium approach[END_REF], etc.

Theorem 2

 2 (η-NE Region of the Game G(b) with SUD). Let b ∈ [0, 1 + SNR 21 + SNR 22 + 2 √ SNR 21 SNR 22 ] and η 0 arbitrarily small. Then, the set N SUD (b) of η-NEs of the game G(b) contains all information-energy rate-triplets (R 1 , R 2 , B) which satisfy:

) b B 1 +

 1 SNR 21 + SNR 22 . (27c) Thus, any projection of N SUD (b) over a plane B = b 1 , with b b 1 1 + SNR 21 + SNR 22 reduces to a unique information rate point (See Fig. 2). When b ∈ (1 + SNR 21 + SNR 22 , 1 + SNR 21 + SNR 22 + 2 √ SNR 21 SNR 22 ], there is an infinity of possible pairs (β 1 , β 2 ) satisfying (

Theorem 3

 3 (η-NE Region of the Game G(b) with SIC). Let b ∈ [0, 1 + SNR 21 + SNR 22 + 2 √ SNR 21 SNR 22 ] and η 0 arbitrarily small. Then, the set N SIC(i→j) (b) contains all informationenergy rate-triplets (R 1 , R 2 , B) satisfying:

Theorem 4

 4 (η-NE Region of the Game G(b)). Let b ∈ [0, 1 + SNR 21 + SNR 22 + 2 √ SNR 21 SNR 22 ] and η 0 arbitrarily small. Then, the set N (b) is defined as:

  Consider a symmetric G-MAC with SNR 11 = SNR 12 = SNR 21 = SNR 22 = 10 (EH and receiver are co-located). Note that for all b

Figure 2 :

 2 Figure 2: Left figure depicts the the projection of the sets N SUD (b) (square point), N SIC(i→j) (b) (round points), and N (b) (blue region) over the R 1 -R 2 plane for b 1 + SNR 21 + SNR 22 . The information capacity region is also plotted as a reference (white region inside solid lines) for SNR 11 = SNR 12 = SNR 21 = SNR 22 = 10. Note that the information capacity region with and without energy transmission rate constraint are identical in this case. Right figure is a 3-D representation of N (b) (blue volume). The information-energy capacity region E 0 (SNR 11 , SNR 12 , SNR 21 , SNR 22 ) is also plotted as a reference (white volume inside solid lines) for SNR 11 = SNR 12 = SNR 21 = SNR 22 = 10.Three main observations arising from Theorem 2, Theorem 3, and Theorem 4 are described in the sequel.

BFigure 3 :

 3 Figure 3: Left figure depicts the projection of the sets N SUD (b) (dotted line), N SIC(i→j) (b) (dashed lines), and N (b) (blue region) over the R 1 -R 2 plane for b = 0.7B max > 1 + SNR 21 + SNR 22 . The information capacity region without energy transmission constraints (region inside solid lines) is plotted for SNR 11 = SNR 12 = SNR 21 = SNR 22 = 10 (Note that B max 1 + SNR 21 + SNR 22 + 2 √ SNR 21 SNR 22 ). Right figure is a 3-D representation of N (b) (blue volume). The informationenergy capacity region E 0 (SNR 11 , SNR 12 , SNR 21 , SNR 22 ) is also plotted as a reference (white volume inside solid lines) for SNR 11 = SNR 12 = SNR 21 = SNR 22 = 10.

  e., as long as b 1 + SNR 21 + SNR 22 + 2 √ SNR 21 SNR 22 . This statement follows immediately from the fact that N SUD (b) = ∅, N SIC (b) = ∅, and thus N (b) = ∅, which ensures the existence of at least one action profile (s * 1 , s * 2 ) that is an η-NE. Interestingly, when b > 1 + SNR 21 + SNR 22 + 2

1

 1 when b ∈ [0, 1 + SNR 21 + SNR 22 ] and (β 1 , β 2 ) ∈ [0, 1] 2 are chosen to satisfy the following equality b = 1 + SNR 21 + SNR 22 + 2 (1β 1 )SNR 21 (1β 2 )SNR 22 (34) RR n°8847 when b ∈ (1 + SNR 21 + SNR 22 , 1 + SNR 21 + SNR 22 + 2 √ SNR 21 SNR 22 ]. Let also the subset V SUD (b) ⊆ C SUD (b) be defined by V SUD (b) C SUD (b) \ U SUD (b). Note that for any b ∈ [0, 1 + SNR 21 + SNR 22 + 2 √ SNR 21 SNR 22 ], the sets V SUD (b) and U SUD (b) form a partition of C SUD (b).

Case 1 :

 1 b ∈ [0, 1 + SNR 21 + SNR 22 ]

  Transmission décentralisée et simultanée d'information et d'énergie dans les canaux à accès multiple Résumé : Dans le présent-rapport, les limites fondamentales de la transmission décentralisée et simultanée de l'information et de l'énergie dans les canaux Gaussiens à accès multiple à deux utilisateurs (G-MAC) sont déterminées dans le cas où un débit minimal b de transmission d'énergie est requis pour un décodage réussi. Tous les triplets de débits atteignables et stables de transmission d'énergie et d'information (R 1 , R 2 , B) sont identifiés. Les débits d'information R 1 et R 2 en bits par utilisation canal sont mesurés au niveau du récepteur et le débit d'énergie B en unités d'énergie par utilisation canal est mesuré au niveau d'un collecteur d'énergie. La stabilité est considérée au sens d'un η-équilibre de Nash (η-NE), avec η 0 arbitrairement petit.

Le résultat principal est la caractérisation complète de la région η-NE d'information-énergie, i.e., l'ensemble des triplets d'information-énergie (R 1 , R 2 , B) qui sont atteignables et stable dans le G-MAC quand: (a) les deux transmetteurs règlent leurs configurations d'émission d'une manière autonome et indépendante dans le but de maximiser leurs débits individuels de transmission d'information R 1 et R 2 , respectivement; (b) les deux transmetteurs garantissent conjointement un débit de transmission d'énergie B au niveau du collecteur d'énergie tel que B b.

  Theorem 1 (Information-Energy Capacity Region with Minimum Energy Constraint b). The information-energy capacity region E b (SNR 11 , SNR 12 , SNR 21 , SNR 22 ) is the set of all informationenergy rate triplets

The information-energy capacity region E b (SNR 11 , SNR 12 , SNR 21 , SNR 22 ) of the G-MAC with minimum energy rate constraint b is the closure of all achievable information-energy rate triplets (R 1 , R 2 , B). The information-energy capacity region E b (SNR 11 , SNR 12 , SNR 21 , SNR 22 ) is described by the following theorem.

  and b = 1 + SNR 21 + SNR 22 + 2 √ SNR 21 SNR 22 . In this case, N (b) = {(0, 0, 1 + SNR 21 + SNR 22 + 2 √ SNR 21 SNR 22

  1 + SNR 21 + SNR 22 ] and (β 1 , β 2 ) are chosen to satisfy the following equality b = 1 + SNR 21 + SNR 22 + 2 (1β 1 )SNR 21 (1β 2 )SNR 22 (43) when b ∈ (1 + SNR 21 + SNR 22 , 1 + SNR 21 + SNR 22 + 2 √ SNR 21 SNR 22]. Before introducing the proof of Proposition 1, some necessary conditions for NE-action profiles are identified. These conditions are provided by Lemma 2 and Lemma 3. Under these necessary conditions, it is later shown that any rate triplet (R 1 , R 2 , B) that satisfies at least one of the conditions (39)-(42) is not an η-NE, with η arbitrarily small. This establishes the proof of Proposition 1.The fact that the energy rate cannot exceed the maximum feasible value given the constrained power budget at the transmitters, together with assumption (62) lead the following constraints on B * :max{b, 1 + SNR 21 + SNR 22 } < B * 1 + SNR 21 + SNR 22 + 2 SNR 21 SNR 22 . (63)Using continuity arguments, the energy rate B * can be written as:B * = 1 + SNR 21 + SNR 22 + 2 SNR 21 SNR 22 λ.Assumption (62) requires in particular that the energy rate is larger than 1 + SNR 21 + SNR 22 which is the maximum energy rate that can be achieved at the input of the EH when the channel inputs X * 1,t and X * 2,t are independent. The strict inequality in (62) implies that X * 1,t and X * 2,t must be correlated, i.e., λ > 0. The parameter λ can thus be interpreted as the Pearson correlation factor between X * 1,t and X * 2,t which is the same at each time t since the inputs are i.i.d. (Lemma 3), i.e., ∀t ∈ {1, . . . , n},

									(64)
	where λ ∈ [0, 1] is given by
				λ	B * -(1 + SNR 21 + SNR 22 ) 2 √ SNR 21 SNR 22	.	(65)
		λ	1 n	n t=1	E X * 1,t X * 2,t P 1,max P 2,max	=	E X * 1,t X * 2,t P 1,max P 2,max	.	(66)
	Assumption (62) implies the following lower bound on λ:
	λ >	max{b, 1 + SNR 21 + SNR 22 } -(1 + SNR 21 + SNR 22 ) 2 √ SNR 21 SNR 22	.

  B * > 1 + SNR 21 + SNR 22 . (76) cannot be an η-NE with η 0. Case 2: b ∈ (1 + SNR 21 + SNR 22 , 1 + SNR 21 + SNR 22 + 2 √ SNR 21 SNR 22 ] In this case, the lower bound (67) is SNR 21 + SNR 22 + 2 (1β 1 )SNR 21 (1β 2 )SNR 22 , (78) with strict inequality. Without loss of generality, consider transmitter 1 whose utility is given by (69) where (β 1 , β 2 ) satisfy (78). Independently of the action s * 2 of transmitter 2, transmitter 1 can use an alternative action s 1 in which it uses more correlation by choosing a parameter β 1 that satisfies b = 1 + SNR 21 + SNR 22 + 2 (1β 1 )SNR 21 (1β 2 )SNR 22 ,

	λ >	b -(1 + SNR 21 + SNR 22 ) 2 √ SNR 21 SNR 22	> 0,	(77)
	and thus (β 1 , β 2 ) must satisfy
	b < 1 + (79)
	and achieves
	R 1 =	1 2	log 2 1 +	β 1 SNR 11 1 + β 2 SNR 12	-,

  b B 1 + SNR 2i + SNR 2j + 2 (1β i )SNR 2i (1β j )SNR 2j ,(93c)with (β i , β j ) ∈ [0, 1] 2 are feasible power-splits, i.e.,b 1 + SNR 2i + SNR 2j + 2 (1β i )SNR 2i (1β j )SNR 2j .

	(94)

RR n°8847

This research is supported in part by the European Commission under Marie Skłodowska-Curie Individual Fellowship No. 659316 (CYBERNETS). H.

Necessary Conditions on NE-Actions

Let (R * 1 , R * 2 , B * ) be an η-NE for any η 0 arbitrarily small, achievable by an action profile (s * 1 , s * 2 ) ∈ A 1 × A 2 . Denote by X * i,1 , . . . , X * i,n the channel inputs generated by transmitter i, for i ∈ {1, 2}, with the equilibrium action s * i and denote by P * i their average power, that is

From the assumption that (R * 1 , R * 2 , B * ) is achievable, P

error (R * 1 , R * 2 ) and P

(n)

outage (B * ) can be made arbitrarily small. Thus, from [START_REF] Aumann | Subjectivity and correlation in randomized strategies[END_REF] it follows that

Using this notation, the following can be stated.

Lemma 2 (Common Randomness). A necessary condition for the action profile (s * 1 , s * 2 ) to be an η-NE action is that, if the channel inputs X * i,t are of the form X * i,t = X * i,1,t + X * i,2,t where X * i,1,t and X * i,2,t are an information-carrying component and a non-information-carrying component, respectively, then, X * i,2,t must exclusively be common randomness that is known to the receiver, for i ∈ {1, 2}.

Proof: Without loss of generality, consider transmitter 1 whose utility is given by:

From the assumptions of the lemma, component X * i,2,t does not increase the information rate. Let R 1 denote the information rate that can be achieved by transmitter 1 if the interference caused by its component X * i,2,t can be completely canceled at the receiver before decoding the messages M 1 and M 2 .

Assume that in the action s * 1 , the component X * i,2,t does not exclusively carry common randomness that is known to the receiver. Hence, the receiver is not able to cancel the energycarrying component before decoding it. This additional interference reduces the information rate of transmitter 1. Let δ > 0 denote the penalty on the information rate of transmitter 1 that is caused by this additional interference. That is,

Regardless of the action s * 2 , transmitter 1 can use an alternative action s 1 in which the component X * i,2,t carries only common randomness known to the receiver. Thus, its effect can be completely canceled and the information transmission can be performed at rate R 1 . The corresponding utility is

From ( 46), (47), and (48), it holds that

The utility improvement is bounded away from zero, thus the action profile (s * 1 , s * 2 ) cannot be η-NE (Def. 3), with an arbitrarily small η 0.

Remark 1. Since the messages M 1 and M 2 are independent, the only possible source of correlation between the time-t channel inputs X 1,t and X 2,t is the common randomness that is known non-causally to the transmitters and to the receiver. Lemma 3 (IID Gaussian Inputs with Maximum Power). A necessary condition for the action profile (s * 1 , s * 2 ) to be an η-NE action is that the input symbolsX * i,t , with i ∈ {1, 2}, must be generated i.i.d. following a zero-mean Gaussian distribution with variance P * i = P i,max . Proof: Without loss of generality, consider transmitter 1 and let R1 denote the information rate that can be achieved by transmitter 1 when the input symbols are generated i.i.d. following a Gaussian distribution with maximum power P 1,max and where the information-carrying components of both transmitters are uncorrelated.

Assume that in the action s * 1 , the input symbols are not generated i.i.d. following a Gaussian distribution with variance P * 1 . Since Gaussian distribution maximizes the entropy and since the information rates are increasing in the input power, using non-Gaussian inputs or using less power result in a loss in the achievable information rate. Thus in the action s * 1 the utility of transmitter 1 is

where ζ > 0 quantifies the loss in information rate.

From the assumption that the receiver implements SUD, independently of the action s * 2 of transmitter 2, there always exists an alternative action s 1 in which transmitter 1 uses i.i.d. Gaussian codebooks with variance P * 1 = P 1,max , which achieves an information rate (and thus a utility)

From ( 50) and ( 51), it follows that

The utility improvement is bounded away from zero, thus the action profile (s * 1 , s * 2 ) cannot be η-NE (Def. 3), with an arbitrarily small η 0.

Proof of (39) and (40)

Without loss of generality, consider transmitter 1 and assume that in the action profile (s * 1 , s * 2 ), the information rate R * 1 satisfies (39). From Lemmas 2 and 3, a necessary condition for the action s * 1 to be an η-NE action is to have i.i.d. Gaussian channel inputs with maximum power P 1,max in which the energy-carrying component exclusively carries common randomness known to the receiver. This condition implies that any information rate R 1 satisfying

can be achieved with arbitrarily small probability of error. Assume that in the action s * 1 transmitter 1's utility (information rate) satisfies

with ξ > 0. Regardless of the action of transmitter 2, transmitter 1 can always choose an alternative action s1 in which it has a utility

RR n°8847

From (54) and (55), it holds that

The utility improvement is bounded away from zero, thus the action profile (s * 1 , s * 2 ) cannot be η-NE (Def. 3), with an arbitrarily small η 0.

The same reasoning holds for transmitter 2 and thus an action profile (s * 1 , s * 2 ) which induces an information-energy rate triplet (R * 1 , R * 2 , B * ) for which either

cannot an η-NE, with η arbitrarily small.

Proof of (41)

This is trivial since, B * < b implies that the triplet (R * 1 , R * 2 , B * ) is not achievable (Def. 1).

Proof of (42)

Assume that there exists an energy-information triplet (R * 1 , R * 2 , B * ) that is achievable at an η-NE by the action profile

where

From the previous parts of the proof (Lemma 2 and Lemma 3), a necessary condition at any η-NE with η arbitrarily small is to have the transmitters use Gaussian codebooks in which the channel inputs {(X * 1,t , X * 2,t )} n t=1 are generated i.i.d. according to a Gaussian distribution with maximum powers P 1,max and P 2,max , respectively. with 0 < < 1 arbitrarily small. Hence its utility is given by

From ( 78) and (79), it holds that β 1 > β 1 . Hence, the utility improvement (( 69) and ( 82)) induced by the alternative action s 1 is given by

which contradicts the assumption that (s * 1 , s * 2 ) is an η-NE (Def. 3), with an η 0 arbitrarily small. This completes the proof of Proposition 1.

Proof of Proposition 2

Let η 0 be arbitrarily small and assume that the decoder performs SUD.

Case 1:

The targeted energy rate b is less than what is strictly necessary to guarantee reliable communication at maximum information sum-rate. Thus, the energy rate constraint is vacuous and the transmitters can exclusively use all their available power budget to send information, i.e.,

To achieve (R * 1 , R * 2 , B * ), transmitters 1 and 2 can use the action profile (s * 1 , s * 2 ) described in the sequel. Transmitters 1 and 2 use independent Gaussian codebooks with powers P 1,max and P 2,max , respectively, as in [START_REF] Cover | Some Advances in Broadcast Channels[END_REF] or [START_REF] Wyner | Recent results in the Shannon theory[END_REF]. The messages M 1 and M 2 are encoded at the information rates R * 1 and R * 2 , respectively. The resulting average energy rate at the input of the EH is given by B (n) = 1 + SNR 21 + SNR 22 , which ensures that the energy outage probability P (n) outage (B * ) can be made arbitrarily small as the blocklength tends to infinity. From the assumption that the receiver performs SUD, probability of error

) can be made arbitrarily small as the blocklength tends to infinity. Hence the resulting utilities are given by:

Assume that the action profile (s * 1 , s * 2 ) is not an η-NE. Then, from Def. 3, there exist at least one player i ∈ {1, 2} and at least one strategy si ∈ A i such that the utility u i is improved by at least η bits per channel use when player i deviates from s * i to si . Without loss of generality, let transmitter 1 be the deviating player and denote by R1 its new information rate. Hence,

From (84a), (85a), and (86), it holds that

RR n°8847

Since the information rate-pair (R * 1 , R * 2 ) already saturates the decoding capability of the receiver, the new information rate pair ( R1 , R * 2 ) cannot be achieved and will result in a probability of error bounded away from zero and consequently the corresponding utility will be:

which contradicts the initial assumption (86) and establishes that the action profile (s

where

To achieve (R * 1 , R * 2 , B * ), transmitters 1 and 2 can use the action profile (s * 1 , s * 2 ) described in the sequel. Transmitters 1 and 2 use power fractions β * 1 and β * 2 of their power budgets P 1,max and P 2,max to send information using independent Gaussian codebooks as in [START_REF] Cover | Some Advances in Broadcast Channels[END_REF] or [START_REF] Wyner | Recent results in the Shannon theory[END_REF], and use the remaining power ((1β * 1 )P 1,max and (1β * 2 )P 2,max ) to send common randomness that is known to both transmitters and the receiver. This common randomness does not carry any information and does not produce any interference to the information-carrying signals. The messages M 1 and M 2 are encoded at the information rates R * 1 and R * 2 chosen by transmitters 1 and 2, respectively. The receiver first subtracts the common randomness and then performs SUD to recover the messages M 1 and M 2 .

The resulting average energy rate at the input of the EH is given by B (n) = 1 + SNR 21 + SNR 22 + 2 (1β * 1 )SNR 21 (1β * 2 )SNR 22 . This ensures that B * b and that the energy outage probability P (n) outage (B * ) can be made arbitrarily small as the blocklength tends to infinity. Assume that the action profile (s * 1 , s * 2 ) is not an η-NE. Then, from Def. 3, there exist at least one player i ∈ {1, 2} and at least one strategy si ∈ A i such that the utility u i is improved by at least η bits per channel use when player i deviates from s * i to si . Without loss of generality, let transmitter 1 be the deviating player and denote by R1 its new information rate. Hence,

The new information-energy rate-triplet ( R1 , R * 2 , B * ) is outside the information-energy capacity region and will result in a utility

which contradicts the assumption (91) and establishes that (s * 1 , s * 2 ) is an η-NE.

RR n°8847


Michael A Sherbon
email: michael.sherbon@case.edu

Fundamental Physics and the Fine-Structure Constant

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Leonhard Euler was one of the greatest mathematicians of the eighteenth century. In The Legacy of Leonhard Euler mathematician Lokenath Debnath says, "It is remarkable that Euler discovered two elegant and most beautiful formulas in mathematics." [START_REF] Debnath | The Legacy of Leonhard Euler: A Tricentennial Tribute[END_REF].

e iπ + 1 = 0 and e 2πi -1 = 0.

(1) William Eisen's description and interpretation of Euler's equation e iπ +1 = 0 in relation to the Great Pyramid design shows four curves of e x from x = 0 to x = π, one curve on each side. Dividing the sides by π lengths results in a small square in the center called the Golden Apex A, the geometry and symmetry thought to generate the four fundamental forces of nature [START_REF] Eisen | The Essence of the Cabalah[END_REF]. Eisen then asks the obvious question about the exponential function and Golden Apex interpretation, "Just how could the builders of the Great Pyramid have been so knowledgeable of the mathematics of the universe ...?"

A = e π -7π -1 ≃ √ 2/3π ≃ 0.1495. (2)
A is the side of the Golden Apex square. √ A ≃ e/7 and A + 1 = e π -7π ≃ R ≃ 1.152, radius of the regular heptagon with side one. The sin(2πA) ≃ φ/2, where φ is the golden ratio, 1/2φ ≃ φ √ A. The tan(2πA) ≃ 1 + √ A ≃ K/2π, see Eq. ( 7) [START_REF] Sherbon | Quintessential Nature of the Fine-Structure Constant[END_REF]. The polygon circumscribing constant is K ≃ 2 tan(3π/7) ≃ φ 2 /2A, see Eq. ( 7) and discussion [START_REF] Sherbon | Quintessential Nature of the Fine-Structure Constant[END_REF]. A is also the reciprocal harmonic of the gravitational constant. Also, A ≃ tan 2 e -1 and A -1 ≃ cosh 2 ( 7/e) ≃ φ √ 2πe. ln(A -1 ) ≃ π/ √ e ≃ 6/π, with the cube-sphere proportion.

The regular heptagon radius,

R = csc(π/7)/2 ≃ φ/ √ 2 ≃ cot 2 α -1 and 2πR ≃ 1/φ √ α.
Also, RA ≃ 2 √ α, where alpha α is the fine-structure constant, see the Eq. ( 7) discussion.

RA ≃ φ/e 2 ≃ ln(π/ √ 7) ≃ 7/π/K. (3) R -1 ≃ √ φ sin α -1 with e π -π e ≃ sin α -1 [3]. The cosh 2 ( √ A) ≃ e A -A e ≃ π/e.
With the fine-structure constant, 2πα is equal to the electron Compton wavelength divided by the Bohr radius and πα is the percentage of light absorbed by graphene [START_REF] Nair | âĂIJFine Structure Constant Defines Visual Transparency of Graphene[END_REF].

4/π ≃ √ A/2A ≃ S/

The Nature of the Fine-Structure Constant

Introduced by Arnold Sommerfeld, the fine-structure constant determines the strength of the electromagnetic interaction. Alpha, the fine-structure constant is α = e 2 / c in cgs units [START_REF] Sherbon | Quintessential Nature of the Fine-Structure Constant[END_REF]. The fine-structure constant related to the Golden Apex of the Great Pyramid:

2A ≃ 2 √ πα ≃ 4m e /m p α ≃ φ 2 /K. (5) Also, 2A ≃ tanh S -1 ≃ tan 2 (1/2) ≃ √ K/π 2 and √ 2A ≃ √ π/S.
When substituting the fine-structure constant value and approximate value for the proton-electron mass ratio αm p /4m e ≃ φ + √ 3 and ln(m p /m e )/ ln(α -1 ) ≃ 2πφA, see the discussion of Eq. ( 14) [START_REF] Sherbon | Quintessential Nature of the Fine-Structure Constant[END_REF]. The Wilbraham-Gibbs constant is G w and the sinc function is the sinc x = sin x/x [START_REF] Zi-Xiang | An Observation of Relationship Between the Fine Structure Constant and the Gibbs Phenomenon in Fourier Analysis[END_REF]:

G w = ˆπ 0 sinc x dx ≃ e sin α -1 ≃ K/ √ 7π. (6)
The Wilbraham-Gibbs constant G w ≃ φ ln π ≃ φ 2 / √ 2 ≃ 1.852. The Wilbraham-Gibbs constant is related to the overshoot of Fourier sums in the Gibbs phenomena [START_REF] Zi-Xiang | An Observation of Relationship Between the Fine Structure Constant and the Gibbs Phenomenon in Fourier Analysis[END_REF] and other approximations: G w ≃ sec(1) ≃ exp(φ -1 ) ≃ 7A √ π ≃ 5/7 √ A, Eq. ( 17) discussion. The inverse Kepler-Bouwkamp constant is the polygon circumscribing constant K [3]:

K = π 2 ∞ n=1 sinc 2π 2n + 1 = ∞ n=3 sec π n . (7)
Also, the polygon circumscribing constant

K ≃ 2π(1 + √ A) ≃ 7/4πα ≃ √ 11φ 2 and K ≃ φ 2 /2A ≃ 5A/ √ α ≃ 4π √ SA ≃ 3/2RA ≃ 2 tan(3π/7) ≃ 8.7. KA ≃ e/φ ≃ R + A.

Half the face apex angle of the Great Pyramid plus half the apex angle is approximately

70 • and sin α -1 ≃ 2 cos 70 • [3]. csc α -1 ≃ R √ φ ≃ 7/S ≃ √ 85/2π
≃ ln(372/85), see below. 528/504 ≃ 7A, see discussion of Eq. ( 17) [START_REF] Sherbon | Quintessential Nature of the Fine-Structure Constant[END_REF]. First level sum of Teleois proportions is 85, foundational in Great Pyramid design [START_REF] Sherbon | Quintessential Nature of the Fine-Structure Constant[END_REF]. 85/11 ≃ R/A and 528/85 ≃ 2π, see Eq. ( 17). The latest value for the inverse fine-structure constant by Aoyama et al, α -1 ≃ 137.035 999 157 (41), from the most recent experimental results and quantum electrodynamics [START_REF] Aoyama | Tenth-Order Electron Anomalous Magnetic Moment: Contribution of Diagrams without Closed Lepton Loops[END_REF]. Eq. ( 8) derives the approximate value for α -1 ≃ 137.035 999 168 [START_REF] Sherbon | Fundamental Nature of the Fine-Structure Constant[END_REF]:

sin α -1 ≃ 504/85K = 7!/(713 + 137)K. (8)
Sum of the eight main resonant nodes on the Turenne Rule is 372 ≃ e/α, part of a spectrum analysis also related to crystallographic groups [8].

K

The Four Fundamental Forces of Nature

The heptagon is a feature of the Cosmological Circle, geometric template for many ancient architectural designs; related to the cycloid curve and history of the least action principle [START_REF] Sherbon | Fundamental Nature of the Fine-Structure Constant[END_REF]. Golden Apex A, silver constant S and fine-structure constant of Eq. ( 4) discussion.

α E ≃ exp(-2/Ae) ≃ A/2πS ≃ 7.29 × 10 -3 . (9)
The electromagnetic coupling constant is α E = α = e 2 / c in cgs units [START_REF] Sherbon | Fundamental Nature of the Fine-Structure Constant[END_REF]. Together with Eq. ( 12) the ratio

α W /α E ≃ Aπ 3 ≃ √ πφ 2 . √ α ≃ Ae/φ √ K ≃ √ R/4π and 2πφR ≃ 1/ √ α.
Gravitational constant G = c/m 2 P l ≃ 6.67191(99)ÃŮ10 -11 m 3 kg -1 s -2 and Planck mass m P [START_REF] Rosi | Precision Measurement of the Newtonian Gravitational Constant Using Cold Atoms[END_REF][START_REF] Burrows | Astronomical Reach of Fundamental Physics[END_REF]

. Gravitational coupling α G = Gm 2 e / c = (m e /m P ) 2 , Golden Apex A: α G ≃ exp(-K √ π/A) ≃ 1.752 × 10 -45 . (10)
Also, -ln(α G ) ≃ 2R/A 2 ≃ exp(Aπ 3 ) ≃ 2πR d / √ α, where dodecahedron circumradius

R d = φ √ 3/2. Aπ 3 ≃ K sin θ C
, where θ C ≃ 32 • , half the Great Pyramid face apex angle. An approximation with the Golden Apex A for the strong force coupling constant [START_REF] Enterria | High-Precision α S Measurements from LHC to FCC-ee[END_REF]:

α S ≃ exp(A - √ 2φ) ≃ A/ φ ≃ 1.177 × 10 -1 . (11)
The tetrahedral angle θ t ≃ 109.5 • , csc 2 (θ t ) = 9/8, the whole tone.

α S ≃ ln(9/8) ≃ √ α tan(2πA) ≃ 2/A/π 3 ≃ K √ α/2π, √ 2φ -A ≃ e/ √ φ and α -1 S ≃ 7π √ A ≃ π 2 /R [7]
. Approximation with the Golden Apex for the coupling constant of the weak force [START_REF] Bodek | Precision Measurements of Electroweak Parameters with Z Bosons at the Tevatron[END_REF]:

α W ≃ exp(-1/2A) ≃ 2A/K ≃ 3.4 × 10 -2 . ( 12
) Also, α W /α E ≃ Aπ 3 ≃ √ πφ 2 . Coupling constant α W = g 2 w /4π ≃ A/ √ 2π
, where g w is the coupling constant mediating the weak interaction. The Fermi coupling constant G F determines the strength of Fermi's interaction that explains the beta decay caused by the weak nuclear force.

G F = g 2 w /4 √ 2m 2 W = α W π/ √ 2m 2 W [13]
, m W is the mass of the W boson. The weak interaction is mediated by the exchange of W and Z gauge bosons. Weinberg angle θ W , sin 2 θ W ≃ √ αφ 2 ≃ 3A/2 [START_REF] Bodek | Precision Measurements of Electroweak Parameters with Z Bosons at the Tevatron[END_REF][START_REF] Olive | The Review of Particle Physics[END_REF], Eq. ( 17)

; cos θ W = m W /m Z ≃ √ 2/φ. K 2 ≃ cos(2θ W )/α ≃ A 6/π. The pyramid base angle θ B , θ W /θ B ≃ √ 2A ≃ √ π/S.

Symmetry Principles and Physical Measures

Steven Weinberg is often quoted, "... there are symmetry principles that dictate the very existence of all the known forces of nature." [START_REF] Sundermeyer | Symmetries in Fundamental Physics[END_REF]. Saul-Paul Sirag says, "By far the deepest theoretical advance afforded by the group-theory approach is the set of ADE Coxeter graphs, .... It is plausible to think of the ADE graphs as the ultimate Platonic archetypes." [START_REF] Sirag | ADEX Theory: How the ADE Coxeter Graphs Unify Mathematics and Physics[END_REF]. The discovery of symmetry principles and their application [START_REF] Jungmann | Fundamental Symmetries and Interactions[END_REF][START_REF] Quigg | Electroweak Symmetry Breaking in Historical Perspective[END_REF] has advanced to the monster group and the modular j-function (referred to as "monstrous moonshine") [START_REF] Gannon | Moonshine Beyond the Monster: The Bridge Connecting Algebra, Modular Forms and Physics[END_REF], the umbral moonshines and string theory; along with vertex operator algebras in related physical approaches with conformal field theory [START_REF] Duncan | Moonshine[END_REF]. For a half-period ratio of τ, the modular function j(τ ) with q = exp(2πiτ ) has the Fourier expansion [START_REF] Sloane | Coefficients of Modular Function j as Power Series in q[END_REF]:

j(τ ) = q -1 + 744 + 196884q + 21493760q 2 + .... ( 13
)
The ln(21493760/196884

) ≃ √ 7π ≃ √ 2/2A. The ln 196884 ≃ 2/A/2A ≃ KR d and R d = φ √ 3/2
is the circumradius of the regular dodecahedron with the side equal to one. From Grumiller et al, the chiral half of the monster group conformal field theory originally proposed by Ed Witten has a partition function given by the j-function. "The number 196884 is interpreted as one Virasoro descendant of the vacuum plus 196883 primary fields corresponding to flat space cosmology horizon microstates." The quantum correction in the respective low-energy entropy is then proportional to the ln 196883 [START_REF] Grumiller | Black Holes and Thermodynamics[END_REF].

G.f. is a Fourier series which is the convolution square root of j(τ ), see Eq. ( 13) [START_REF] Adamson | Convolution Square Root of A000521[END_REF].

G.f. = q -1 + 372q + 29250q 3 -134120q 5 + .... (14)
The ln(29250/372) ≃ φ 2 /4A ≃ K/2 and the ln(2 × 372) ≃ A -1 . The ln 372 ≃ φ 2/A ≃ √ π/2A ≃ 2K cos 70 • and sin 70 • ≃ 372/396 ≃ 2πA, with the hieroglyphic geometry for gold. An application in a special case of umbral moonshine is the Mathieu moonshine work of Eguchi, Ooguri and Tachikawa [START_REF] Eguchi | Notes on the K3 Surface and the Mathieu group M 24[END_REF] and followed by others, includes the q-series e(q) whose coefficients are then " ... twice the dimension of some irreducible representation of the Mathieu group M 24 ." Pierre-Philippe Dechant continues, "Modularity is therefore very topical, also in other areas and a Clifford perspective on holomorphic and modular functions could have profound consequences." [START_REF] Dechant | Clifford Algebra is the Natural Framework for Root Systems and Coxeter Groups. Group Theory: Coxeter, Conformal and Modular Groups[END_REF]. A normalized q-series e(q): e(q) = 90q + 462q 2 + 1540q 3 + 4554q 4 + 11592q 5 + ....

The ln(462/90) ≃ √ 7/φ ≃ √ Aφ 3 with the ln 90 ≃ √ 3φ 2 ≃ 2/3A ≃ U √ K, see Eq. ( 16). Also correlated, 90/372 ≃ φA and 462/372 ≃ φA + 1 ≃ 2/φ, recall discussion of Eq. (8).

A gold pyramid at the tip of the Great Pyramid, with octahedral and icosahedral symmetry [START_REF] Verheyen | The Icosahedral Design of the Great Pyramid[END_REF], was the "Golden Tip" described by John Michell with the support of Algernon Berriman's metrology [START_REF] Michell | The New View Over Atlantis[END_REF]. This was represented by the height of a pyramid with 5 cubic inches, 0.152 ≃ 11.7/(7 × 11), √ 137 ≃ 11.7 and 0.152 is the tenth part of the Greek cubit of 1.52

′ . This pyramidion might have been similar to the legendary Golden Sun Disc of Mu. Apex angle of the regular heptagon triangle is 3π/7 and an approximation to the apex angle of the Great Pyramid [START_REF] Sherbon | Fundamental Nature of the Fine-Structure Constant[END_REF]. The "Golden Tip" harmonic of U ≃ φ sin 70 • ≃ 2πφA ≃ ln(m p /m e )/ ln(α -1 ) ≃ 1.52 [START_REF] Sherbon | Quintessential Nature of the Fine-Structure Constant[END_REF]. From the heptagon geometry found in the Cosmological Circle, AU ≃ R/πφ ≃ 5/7π ≃ cot(3π/7) ≃ 0.227 and the tan(3π/7) ≃ eφ ≃ 4.4. The ratio 7/5 ≃ φ/R ≃ R d , the circumradius of the regular dodecahedron having the side equal to one, see the discussion following Eq. ( 13) and [START_REF] Sherbon | Fundamental Nature of the Fine-Structure Constant[END_REF].

AU ≃ 85/372 ≃ φ √ eα ≃ exp(-ARK) ≃ 2/3 √ K. (16)
The sum of A+U ≃ 1/4A, A/U ≃ R √ α and the ARK ≃ 7/π ≃ coth 2 R ≃ 3/2 ≃ 10A. Interesting parallels can be found between William Eisen's Golden Apex of the Great Pyramid design, the torus topology of Einstein's relativity and Wolfgang Pauli's World Clock Vision; with his i ring imaginal geometry of the unit circle on the complex plane and also symbolic of the unus mundus as the natural ontology of quantum theory [START_REF] Sherbon | Wolfgang Pauli and the Fine-Structure Constant[END_REF].

Pauli's World Clock also has the golden ratio geometry related to the fine-structure constant together with four men swinging pendulums [START_REF] Sherbon | Wolfgang Pauli and the Fine-Structure Constant[END_REF]. Ancient Egyptian architects converted celestial time periods into lengths that are equivalent to those of the pendulum measures as rediscovered by Galileo and explained by Sir Isaac Newton. Roger Newton states that " ... speculations concerning a long-awaited reconciliation between Einstein's general theory of relativity and the quantum, known in their various guises as superstring theory, employ as their basic element the properties of Galileo's simple pendulum ...." [START_REF] Newton | Galileo's Pendulum: From the Rhythm of Time to the Making of Matter[END_REF]. Flinders Petrie found the harmonic of the standard day when converted by the pendulum formula results in the length of the Royal Cubit, "... basis of the Egyptian land measures .... This value for the cubit is 20.617

′′ while the best examples in stone are 20.620 ′′ ± 0.005 ′′ ." The Egyptian Royal Cubit is the traditionally known measure basis of the Great Pyramid "... and its base measures 440 Royal Cubits in length." [START_REF] Petrie | Origin of the Time Pendulum[END_REF].

R.C. ≃ 144/7 ≃ A/α ≃ φ + φ/ √ α ≃ √ 7π/AU ≃ 7 √ K. (17)
The canonical Royal Cubit of 20.736 ′′ is the harmonic of 144 2 and the standard harmonic Royal Cubit R.C. ≃ 20.618 [START_REF] Michell | The New View Over Atlantis[END_REF]

Conclusion

From the exponential function of Euler's equation to the geometry of a fundamental form, the Golden Apex of the Great Pyramid was described, leading to a calculation of the fine-structure constant and its close relationship with the proton-electron mass ratio. Golden Apex related equations were then found for four fundamental forces of nature. Juan Maldacena restates it, "The forces of nature are based on symmetry principles." [START_REF] Maldacena | The Symmetry and Simplicity of the Laws of Physics and the Higgs Boson[END_REF]. These symmetry principles were then associated with traditional physical measures. And finally, here is William Eisen's quotation again, "Just how could the builders of the Great Pyramid have been so knowledgeable of the mathematics of the universe ...?" [START_REF] Eisen | The Essence of the Cabalah[END_REF].

  2, with Eq. (8) discussion. K + 2R ≃ 11 and √ e/φ ≃ 1 + αφ 2 [3].

  ≃ φ ln(372 -137), ln 372 ≃ 1 + ln 137 ≃ √ π/2A and ln(137 × 372) ≃ φ/A. 137 2 + 372 2 ≃ 396 2 and from the harmonic radii of the Cosmological Circle, 108 + 396 = 504 [3]. 504/396 = 14/11 ≃ 4/π. Pyramid base angle θ B ≃ tan -1 (4/π). The ln(4/π) ≃ Aφ and π/4 ≃ csc α -1 -sin α -1 ≃ sin θ B . The pyramid apex angle θ A , sin α -1 ≃ θ B /θ A ≃ √ 2 tan(π/7) ≃ S/7 ≃ 2π/ √ 85 [3]. 528/396 = 4/3 and 396/85 ≃ S/A. The quartz crystal harmonic is Q c = 786432 and 1/ √ α ≃ ln(AQ c ). Base octave harmonic of Q c , 192 ≃ 7πK and φ/π ≃ 192/372 [7].

  Also, 144/85 ≃ e/φ ≃ A + U [3]. Plato's "fusion number" 1746, described by John Michell, represented the apex of the Great Pyramid [27]. Fusion number 1746 ≃ 144 7π/A, 372/1746 ≃ √ 2A and AU ≃ 396/1746. 528/372 ≃ √ 2, 528/1746 ≃ 2A, 528/504 ≃ 7A, 1746/504 ≃ 2 √ 3 and R.C. ≃ 1746/85. The Great Pyramid Key is 528 ≃ ln(7/A)/α [3].

	. Also, R.C. ≃ of the Cosmological Circle is 44 and 44 2 + 137 2 ≃ 144 2 [28]. 22 = √ 7π tan(3π/7). Basic square perimeter √ 44 + 440 ≃ 7π and A ≃ π/ √ 440 [3]. 372/440 ≃ 4 √ 2A, 144/372 ≃ √ A and 504/144 = 7/2.

Acknowledgments

Special thanks to Case Western Reserve University, MathWorld and WolframAlpha.