Pierre Granjon
email: pierre.granjon@grenoble-inp.fr

Condition monitoring of motor-operated valves in nuclear power plants

This article deals with a new motor-operated valves (MOVs) condition monitoring method. MOVs mainly consist of an induction machine driving a valve through a mechanical actuator. Such systems need important maintenance policies since they are critical concerning the operational readiness of the safety related systems of nuclear power plants. Unfortunately, today's policies present a major drawback. Indeed, these monitoring techniques rely on the measurement of an internal mechanical quantity and require the presence of human operators in dangerous power plants areas. The aim of this research work is to develop a monitoring method relying exclusively on remote electrical measurements (MOV supply voltages and currents), without the need of any internal MOV quantity. In this article, the instantaneous active power absorbed by the induction machine is first shown to be an interesting electrical quantity to elaborate mechanical failure indicators. An efficient real-time algorithm developed to estimate this physical quantity is then detailed and mechanical failure indicators based on the estimated electrical power are proposed. Finally, their performance is illustrated through experimental data.

Introduction

Nuclear power provides about 14% of the world's electricity, with the United States of America, France and Japan together accounting for about 50% of nuclear generated electricity [START_REF]Wikipedia, The Free Encyclopedia[END_REF] . This important production is realized through complex systems subject to drastic safety constraints: nuclear power plants. Key components of such systems are motor-operated valves (MOVs), whose condition has to be continuously monitored in order to ensure power plants security, safety and performance requirements. Nowadays, condition monitoring of MOVs is usually based on the joint analysis of electrical supply and internal mechanical quantities. Unfortunately, these methods require access to the valve and increases radiation exposure of operators, leading to long overhaul outage and high maintenance costs. A possible solution is to realise MOVs condition monitoring by using supply voltages and currents only, without any mechanical measurements. Indeed, electrical quantities are accessible outside radioactively contaminated areas and allow remote and nonintrusive MOV testing. However, necessary mechanical information initially given by mechanical measurements has now to be extracted from MOVs electrical quantities. One possibility is to use the instantaneous fundamental positive-sequence active power delivered to the MOV. Indeed, this electrical quantity is closely related to the instantaneous mechanical power produced by the induction motor of the MOV, and is therefore an image of internal MOVs mechanical phenomena. This paper describes a new MOVs condition monitoring method based exclusively on the analysis of the fundamental positive-sequence active power absorbed by these systems. Section 2 justifies this approach through a general description of MOVs and of their usual monitoring techniques. The real-time active power estimation algorithm developed for this specific application is described in section 3, followed by experimental results in section 4 and some concluding remarks in section 5.

Motor-operated valves condition monitoring and problem statement

Motor-operated valves description

The purpose of motor-operated valves (MOVs) encountered in all nuclear power plants is to control the flow of fluids in the whole plant fluid system. Their typical constitution is illustrated in Fig. 1, and consists of three main parts:

-The first element is a three-phase induction machine that drives the system for opening or closing, depending on its rotation direction.

-The second part is a mechanical actuator whose input is the shaft driven by the induction machine, and whose output is the valve stem. Its purpose is to transform the output torque of the induction machine into a vertical translation force applied to the valve stem, the stem thrust. Indeed, the electrical motor drives a worm gear through a worm fixed on the machine shaft. The stem nut, driven by the worm gear, then raises or lowers the valve stem depending on its rotation direction.

-The third and last part of the MOV is a valve which is closed when the valve stem is lowered and opened in the opposite case.

Figure 1. Schematic representation of a motor-operated valve

The time evolution of some MOV's physical quantities can help to understand the different physical phenomena occurring in such a system. Fig. 2 shows the waveforms of the stem thrust and the electrical apparent power absorbed by the induction machine during opening and closing strokes. At the beginning of the opening stroke, the stem valve is under pressure in order to maintain the valve correctly closed, the measured thrust is then strongly negative. Next, the electrical power transient induced by the induction machine starting is clearly visible around t = 2 s. Then, the motor drives the mechanical actuator, cancels the pressure in the stem valve (the stem thrust tends to 0 kN just before t = 3 s) and unseats the valve (positive transient on stem thrust just after t = 3 s). Afterwards, the valve stem rises up with a constant speed and the whole MOV is in steady-state operation. The stem thrust and the electrical apparent power are approximately constant during this long period of time. Just after t = 18 s, the valve stem reaches its maximal height in the MOV and a sensor stops the induction machine and the opening stroke ends. The closing stroke contains similar events. Indeed, the electrical power transient generated by the motor starting occurs just after t = 1 s and is followed by the long steady-state behaviour when the stem lowers at constant speed. The valve finally seats around t = 17 s, and the valve stem is put under pressure (the stem thrust becomes strongly negative) in order to correctly close the valve. This last event necessitates an important motor torque, and implies an increase of the electrical power visible on the corresponding curve. The closing stroke is finally stopped by a torque switch mounted on the motor shaft. The previous description of MOVs brings to light two types of behaviour for these systems:

-short dynamic behaviours at the beginning and at the end of strokes (motor starting, actuator locking/unlocking, valve seating/unseating), -long steady-state behaviours in the middle of strokes (stem rising and lowering).

The above information will be useful to understand MOVs condition monitoring techniques described in the following paragraph.

Classical monitoring methods

Most of usual MOVs condition monitoring methods need at the same time stem thrust and electrical supply quantities measurements in order to elaborate mechanical fault indicators (2)(3) . Unfortunately, stem thrust measurement requires access to the valve for installation of mechanical transducers. Therefore, these 'at-the-valve testing methods' lead to strong human problems (because of operators radiation exposure) and strong economic problems (because of maintenance outage and costs). Other existing methods only use electrical supply quantities to elaborate mechanical fault indicators (4)(5) [START_REF] Chai | Non-invasive diagnostics of motor-operated valves[END_REF] , but they rely on an electrical model of the induction machine present in the MOV. Consequently, such 'modelbased methods' lead to problems in terms of model validity, model identification and model update usually connected to such approaches. Moreover, all these methods are based on measurements acquired during dynamic behaviours of the MOVs because their mechanical parts are then strongly excited. The previous paragraph has shown that such dynamic behaviour events have very short duration, which most often leads to fault indicators with strong variability. Finally, these drawbacks limit the performance of the MOVs condition monitoring methods usually employed in nuclear power plants.

Principle of the proposed method

In order to avoid the problems of the previously described classical methods, the method proposed in this article has the following principles. First, fault indicators are calculated by using exclusively electrical supply voltages and currents of the MOV. This approach avoids any at-the-valve measurement and allows remote testing since such quantities are accessible outside radioactively contaminated areas. Second, mechanical fault indicators are obtained through the analysis of the instantaneous fundamental positive-sequence active power absorbed by the induction machine contained by the MOV. Indeed, the usage of this particular electrical quantity presents numerous advantages:

-Apart from losses, it is equal to the mechanical power produced at the induction machine output [START_REF] Alger | Induction machines[END_REF] . Therefore, this electrical quantity is closely related to internal mechanical phenomena occurring in the MOV.

-It is estimated by using the three voltages and currents of the induction machine, and therefore realizes some kind of data fusion of the whole information contained in this threephase system.

-Its estimator is based on a physical definition and not on any induction machine model contrary to classical methods. Last, measurement and analysis of electrical supply quantities are done during steady-state operations of the MOV. As shown in paragraph 2.1, such behaviours have long duration (for example more than 10 s on Fig. 2) and lead to fault indicators with small variability. Moreover, the measured physical quantities are stationary during these operations, and classical signal processing tools such as spectral analysis can be used without any restriction. Therefore, the proposed method consists of two main steps:

-the real-time estimation of the instantaneous fundamental positive-sequence active power absorbed by the MOV, -the analysis of this instantaneous electrical quantity in order to propose efficient mechanical fault indicators. These two points are further detailed in the following two sections.

Active power estimation algorithm

This section is dedicated to the real-time estimation algorithm of the instantaneous fundamental positive-sequence active power in three-phase systems. The first paragraph deals with the physical definition of this quantity, and the second paragraph describes the proposed algorithm.

Power definition

A general expression for a three-phase voltage system is:

€ v a t ( ) v b t ( ) v c t ( ) ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ = V a cos 2πf 1 t + ϕ a ( ) + ˜ v a t ( ) V b cos 2πf 1 t + ϕ b ( ) + ˜ v b t ( ) V c cos 2πf 1 t + ϕ c ( ) + ˜ v c t ( ) ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ , ( 1 )
where € f 1 is the fundamental frequency and

€ ˜ v t
( ) are voltage components with frequency different from € f 1 . Such a system is completely characterized at the fundamental frequency € f 1 by the following vector containing three complex phasors:

€ V a V b V c ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ = V a e jϕ a V b e jϕ b V c e jϕ c ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ . ( 2
)
Since Fortescue [START_REF] Fortescue | Method of symmetrical co-ordinates applied to the solution of polyphase networks[END_REF] , it is known that such a system is equivalent to its 'symmetrical components':

€ V + V - V 0 ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ = 1 3 V a + aV b + a 2 V c
( )

1 3 V a + a 2 V b + aV c
( )

1 3 V a + V b + V c ( ) ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ = 1 3 1 a a 2 1 a 2 a 1 1 1 ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ V a V b V c ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ , ( 3 )
where

€ a = e j 2π
3 is a complex number, and

€ V + , € V -and

€

V 0 refer respectively to fundamental positive-, negative-and zero-sequence components. This transform is commonly used to analyse three-phase quantities in power networks because it allows simple and intuitive physical interpretations. Indeed, the positive-sequence component represents for example the amount of voltage contributing to the power flow from generators to loads [START_REF] Bollen | Signal processing of power quality disturbances[END_REF] . Another useful interpretation is that the torque produced by an induction motor is determined by the positive-sequence component of its supply voltages [START_REF] Bollen | Signal processing of power quality disturbances[END_REF] . The three-phase voltage system given by Eq. ( 1) is now supposed to feed a three-phase induction machine. Such a load absorbs a three-phase current system and transforms electrical input power into mechanical output power. Several electrical powers have been defined for three-phase systems [START_REF] Emanuel | Summary of IEEE standard 1459: definitions for the measurement of electric power quantities under sinusoidal, nonsinusoidal, balanced or unbalanced conditions[END_REF] , but the fundamental positive-sequence active power plays a very important role. Indeed, it is known [START_REF] Orts-Grau | Discussion on useless active dans reactive powers contained in the IEEE standard 1459[END_REF] that this power is the only one converted into mechanical power by a three-phase motor. It is definition, given thanks to symmetrical components of three-phase voltage and current systems, is the following one:

€ P + = 3 2 Re V + I + * { } = 3 2 V + I + cos θ + ( ) , ( 4 )
where € Re { } is the real part and € * the complex conjugate.

In this definition, € V + and € I + refer respectively to the positive-sequence components of voltage and current systems, and € θ + is the phase difference between them or equivalently the phase angle of the complex number

€ V + I + * .
This electrical power being closely related to the mechanical power produced by the induction machine of the MOV, its instantaneous value

€ P + t
( ) can be seen as the image of the internal mechanical phenomena generated at each time in this system. This instantaneous electrical quantity is clearly interesting to monitor mechanical phenomena and create efficient mechanical fault indicators.

Power estimation algorithm

An elegant way to obtain a real-time estimation of

€ P + t
( ) is through the instantaneous symmetrical components [START_REF] Lyon | Transient analysis of alternating-current machinery[END_REF] [START_REF] Paap | Symmetrical components in the time domain and their application to power network calculations[END_REF] . They are obtained thanks to a transformation similar to Eq. ( 3), but directly applied to time-domain waveforms instead of the fundamental phasors. For example, the instantaneous symmetrical components of the previous three-phase voltage system are given by:

€ v + t ( ) v -t ( ) v 0 t ( ) ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ = 2 3 v a t ( ) + av b t ( ) + a 2 v c t ( ) ( ) 2 3 v a t ( ) + a 2 v b t ( ) + av c t ( ) ( ) 2 3 v a t ( ) + v b t ( ) + v c t ( ) ( ) ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ = 2 3 1 a a 2 1 a 2 a 1 1 1 ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ v a t ( ) v b t ( ) v c t ( ) ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ . ( 5
)
By analogy with the symmetrical components defined in Eq. ( 3),

€ v + t ( ) € v -t ( ) and
€ v 0 t ( ) is real-valued.
The nature of such signals can be understood thanks to their frequency contents. Indeed, under the assumption that all Fourier transforms exist, the Fourier transform of Eq. ( 5) leads to:

€ V + f ( ) V -f ( ) V 0 f ( ) ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ = 2 3 V a f ( ) + aV b f ( ) + a 2 V c f ( ) ( ) 2 3 V a f ( ) + a 2 V b f ( ) + aV c f ( ) ( ) 2 3 V a f ( ) + V b f ( ) + V c f ( ) ( ) ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ . ( 6
)
More particularly, at the fundamental frequency € f 1 , this relation becomes:

€ V + f 1 ( ) V -f 1 ( ) V 0 f 1 ( ) ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ = 1 3 2V a f 1 ( ) + a × 2V b f 1 ( ) + a 2 × 2V c f 1 ( ) ( ) 1 3 2V a f 1 ( ) + a 2 × 2V b f 1 ( ) + a × 2V c f 1 ( ) ( ) 1 3 2V a f 1 ( ) + 2V b f 1 ( ) + 2V c f 1 ( ) ( ) ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ = 1 3 V a + aV b + a 2 V c ( ) 1 3 V a + a 2 V b + aV c ( ) 1 3 V a + V b + V c ( ) ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ = V + V - V 0 ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ . ( 7 )
Therefore, the spectral contents of the instantaneous symmetrical components at the fundamental frequency € f 1 consist of the classical symmetrical components of the three-phase system. More particularly, the classical fundamental positive-sequence component

€ V + is contained by the instantaneous positive-sequence component € v + t ( ) at frequency € f 1 .
Following this result, the output of a selective bandpass filter applied to

€ v + t ( ) whose only passband is centered on the fundamental frequency € + f 1 is given by € v + 1 t ( ) = V + e j2πf 1 t
. It should be noticed that this filter only passes spectral components located around € + f 1 , its transfer function being different from zero only around € + f 1 and not around € -f 1 . This transfer function does not verify the so-called hermitian symmetry, and can only be obtained with complexvalued filter coefficients. Such 'complex filters' are very useful for real-time demodulation [START_REF] Sheen | A complex filter for vibration signal demodulation in bearing defect diagnosis[END_REF] [START_REF] Granjon | Surveillance du désalignement d'un moteur asynchrone par analyse du vecteur d'espace courant[END_REF] or real-time Hilbert transform [START_REF] Reilly | Analytic signal generation -tips and traps[END_REF] applications. Similarly, the same filter applied to the instantaneous positive-sequence component

€ i + t ( ) of the three-phase current system gives € i + 1 t ( ) = I + e j2πf 1 t .
A real-time estimator

€ ˆ P + t
( ) of the fundamental positive-sequence active power € P + defined by Eq. ( 4) is finally obtained by using the two previous filtered signals in the following way:

€ ˆ P + t ( ) = 3 2 Re v + 1 t ( ) × i + 1 t ( ) * { } . ( 8 )
Figure 3 gives a schematic representation of the proposed algorithm.

Figure 3. Real-time estimation of the fundamental positive-sequence active power

Clearly, each operation appearing in Fig. 3 (the linear combination of Eq. ( 5), the complex bandpass filter or the product and real part of Eq. ( 8)) can be easily real-time implemented.

In the next section, this algorithm is applied to realize MOVs condition monitoring.

Experimental results of MOVs condition monitoring

The algorithm described in the previous section is now applied to the electrical supply quantities of a MOV. The aim is to estimate the instantaneous value of the fundamental positive-sequence active power, and next to employ this estimate to monitor the mechanical condition of the MOV.

Active power waveform

The proposed algorithm is applied to the same data used to obtain the curves shown in Fig. 2 and consisting of an opening and a closing stroke. The estimated value of the fundamental positive-sequence active power

€ ˆ P + t
( ) appears in red in Fig. 4, while the apparent power first shown in Fig. 2 now appears in black. ( ) remains lower than the apparent power absorbed by the induction machine. However, as expected, important mechanical events appear much more clearly in the active power. This is verified for the hammer blow and the valve unseating (the two events following the starting transient at the beginning of the opening stroke), and for the valve seating (at the end of the closing stroke). These results validate the idea that

€ ˆ P + t ( ) is
closely related to internal mechanical phenomena occurring into the MOV. Moreover, dynamic and steady-state behaviours of the MOV are clearly visible in Fig. 4. In what follows, the estimated active power is only analyzed during steady-state behaviours. The zoom realized in Fig. 5 shows that during these particular periods, the active power is mainly constituted by periodic components. Following previous considerations, these periodic variations can be considered as the image of mechanical phenomena generated into the MOV during steady-state behaviours, such as the rotation of the worm attached to the induction machine shaft, the rotation of the stem nut, … ( ) during the steady-state behaviours of the MOV allows to quantify the importance of each periodic components, and to link them to specific known mechanical phenomena. Fig. 6 represents the power spectral densities obtained for the corresponding signals of Fig. 5.

Figure 6. Power spectral densities of MOV active power during steady-state operations

These results show that the active power is constituted of the same periodic components for the opening and closing stroke. A first group of important components is located around the rotation frequency of the induction machine, just below 25 Hz. Therefore, the power of these components represents the quantity of mechanical phenomena applied to the shaft of the induction machine at each revolution. A second group of components is located in the low frequency band with a fundamental frequency of 0.82 Hz, the rotation frequency of the stem nut. Therefore, the power of these low frequency components is directly related to the mechanical efforts generated in the stem nut, and can be used as an indicator of the stem nut mechanical condition. Finally, these results show that the spectral contents of € ˆ P + t ( ) can be used to monitor the mechanical condition of different parts of the MOVs.

Fault indicators and experimental results

These results encourage the creation of mechanical fault indicators based on the analysis of the estimated active power. One example is the condition monitoring of the stem nut through the low frequency contents of € ˆ P + t ( ) . In that case, the indicator is simply obtained by summing the power spectral density of € ˆ P + t ( ) from 0 to 10 Hz, and represents the quantity of low frequency phenomena in € ˆ P + t ( ) , and therefore the stem nut mechanical condition. Figure 7 shows the results obtained with this particular indicator applied to 14 similar MOVs in different French nuclear power plants. This indicator reaches a high value for MOVs number 2 and 12. These particular MOVs have been controlled, and it has been noticed that the corresponding stem nuts present an important lack of grease. This first result is encouraging and other efficient mechanical fault indicators should be proposed in the future to monitor MOVs condition through electrical active power.

Conclusion

In this paper, a new condition monitoring technique dedicated to motor-operated valves in nuclear power plants has been proposed. This method is based on the idea that the fundamental positive-sequence active power absorbed by the MOV can be used to monitor the mechanical condition of this system. As a consequence, a new real-time estimation algorithm of this electrical quantity has been presented in section 3 (see Fig. 3). Moreover, an example of one mechanical failure indicator elaborated from the previous estimated active power has been described in section 4. Obtained results, shown in Fig. 4, are very encouraging and allows to monitor the mechanical condition of one particular part of the MOV. Futur works need to be addressed, however. For example, others fault indicators are needed to monitor the condition of each part of the MOV independently.

Figure 2 .

 2 Time waveforms of MOV stem thrust and electrical apparent power

  positive-, negative-and zero-sequence components. This transform thus leads to three time-domain signals among which €

Figure 4 .

 4 Time waveforms of MOV stem thrust, apparent and active powersNaturally, the estimated active power€ ˆ P + t

Figure 5 .

 5 Figure 5. Time waveforms of MOV active power during steady-state operations

Figure 7 .

 7 Figure 7. Stem nut fault indicator based on low frequency components of

Acknowledgements

The author would like to thank EDF-DTG for his assistance and financial support.


L Abdallah

M Haddou

AN EXACT PENALTY APPROACH FOR MATHEMATICAL PROGRAMS WITH EQUILIBRIUM CONSTRAINTS

Keywords: Optimization, nonlinear programming, exact penalty function

We propose an exact penalty approach to solve the mathematical problems with equilibrium constraints (MPECs). This work is based on the smoothing functions introduced in [3] but we do not need any complicate updating rule for the penalty parameter. We present some numerical results to prove the viability of the approach. We consider two generic applications : the binary quadratic programs and simple number partitioning problems.

Introduction

Mathematical programs with equilibrium constraints (MPECs) represent an optimization problem including a set of parametric variational inequality or complementary constraints. In this paper, we consider optimization problems with complementary constraints, in the following form [START_REF] Ampl | A Modeling Language for Mathematical Programming[END_REF] (P )

   f * = min f (x, y) < x.y >= 0 (x, y) ∈ D
where f : R 2n → R is continuously differentiable and D = [0, v] 2n . < . > denotes the inner product on R n . We made this choice for D only to simplify the exposition. We can use only compact set.

Due to the presence of complementarity constraints, there is no feasible point satisfying all inequality constraints strictly which implies that the usual nonlinear programming constraint qualification such as Mangasarian-Fromovitz constraint qualification (MFCQ) is violated at any feasible point of MPECs. Many regularization and relaxations techniques have already been proposed in the literature [START_REF] Birbil | An entropic approach for mathematical programs with equilibrium constraints[END_REF][START_REF] Facchinei | A smoothing method for mathematical programs with equilibrium constraints[END_REF][START_REF] Lin | Some exact penalty results for nonlinear programs and mathematical programs with equilibrium constraints[END_REF][START_REF] Liu | Partial exact penalty for mathematical programs with equilibrium constraints[END_REF][START_REF] Mangasarian | Exact penalty functions for mathematical programs with linear complementarity constraints[END_REF]. In this study, we proposed smoothing technique to regularize the complementary constraints based on [START_REF] Haddou | A new class of smoothing methods for mathematical programs with equilibrium constraints[END_REF], we replace each constraint

x i y i = 0 by θ ε (x i ) + θ ε (y i ) ≤ 1 where the parameterized functions θ ε : R + → [0, 1] are at least C2 and satisfy :

θ ε (x) ≃ 1 if x = 0 = 0 if x = 0
Then we define a penalty function to solve the problem. To avoid the updating parameter ε, we will consider it as some new optimization variable. This paper is organized as follows. In section 2, we present some preliminaries and assumptions on the smoothing functions. In Section 3, we consider a smooth constrained optimization problem and introduce a penalty function. We prove under some mild assumptions an existence result for the approximate problem and an exact penalty property. In section 4, we present numerical experiments concerning academic MPECs of small sizes. The last section presents a large set of numerical experiments considering binary quadratic programs and simple number partitioning problems.

Preliminaries

In this section, we present some preliminaries concerning the regularization and approximation process. We consider functions θ ε (ε > 0) with the following properties:

(1) θ ε is nondecreasing, strictly concave and continuously differentiable, (

) ∀ε > 0, θ ε (0) = 0, ( 2
) ∀x > 0, lim ε→0 θ ε (x) = 1, 3
lim ε→0 θ ′ ε (0) > 0, (5) ∃ m > 0, ∃ε 0 > 0 ∀x ∈ [0, v], ∀ε ∈]0, ε 0 ], |∂ ε θ ε (x)| ≤ m ε 2 For ε = 0, we set θ 0 (0) = 0 and θ 0 (x) = 1, ∀ x = 0. (4)
Examples of such functions are:

(θ 1 ε ) : θ ε (x) = x x + ε (θ w1 ε ) : θ ε (x) = (1 -e -( x ε ) ) k , for k ≤ 1 (θ log ε ) : θ ε (x) = log(1 + x) log(1 + x + ε)
Using function θ ε , we obtain the relaxed following problem :

(2)

(P ε )    f * ε = min f (x, y) θ ε (x i ) + θ ε (y i ) ≤ 1, i = 1, . . . , n (x, y) ∈ D Remark 2.1. < x.y >= 0 ⇒ ∀ε > 0, θ ε (x i ) + θ ε (y i ) ≤ 1.
Thus any feasible point for (P ε ) is also feasible for (P ) and then ∀ε > 0, f * ε ≤ f * . We first transform the inequality constraints into equality constraints, by introducing some slacks variables e i :

(3)

θ ε (x i ) + θ ε (y i ) + e i -1 = 0, e i ≥ 0 i = 1, . . . , n.
The problem (P ε ) becomes:

(4)

( P ε )    min f (x, y) θ ε (x i ) + θ ε (y i ) + e i -1 = 0 i = 1, . . . , n (x, y, e) ∈ D × [0, 1] n
Indeed each e i can not exceed 1.

The limit problem ( P ε ) for ε = 0

(5) ( P )    min f (x, y) θ 0 (x i ) + θ 0 (y i ) + e i = 1, i = 1, . . . , n e i ∈ [0, 1], i = 1, . . . , n
which is equivalent to (P ).

Until now, this relaxation process was introduced in [START_REF] Haddou | A new class of smoothing methods for mathematical programs with equilibrium constraints[END_REF]. To avoid the updating of parameters problem, we define the penalty functions

f σ on D × [0, 1] × [0, ε]: f σ (x, y, e, ε) =      f (x, y) if ε = ∆(x, y, e, ε) = 0; f (x, y) + 1 2ε ∆(x, y, e, ε) + σβ(ε) if ε > 0, +∞ if ε = 0 and ∆(x, y, e, ε) = 0
where ∆ measures the feasibility violation and the function β

: [0, ε] → [0, ∞) is continuously differentiable on (0, ε] with β(0) = 0. ∆(z, ε) = G ε (z) 2 where (G ε (z)) i = (θ ε (x) + θ ε (y) + e -1) i and z = (x, y, e).
Remark 2.2. ∀z ∈ D ′ , ∆(z, 0) = 0 ⇔ z feasible for P ⇔ (x, y) feasible for (P ).

Then we consider the following problem:

(6) (P σ ) min f σ (x, y, e, ε) (x, y, e, ε) ∈ D × [0, 1] n × [0, ε]
From now on, we will denote ( 7)

D ′ = D × [0, 1] n Definition 2.1.
We say that the Mangasarian-Fromovitz condition [START_REF] Mangasarian | The Fritz John necessary optimality conditions in the presence of equality and inequality constraints[END_REF] for

P σ holds at z ∈ D ′ if G ′ ε (z) has full rank and there exists a vector p ∈ R n such that G ′ ε (z)p = 0 and p i > 0 if z i = 0 < 0 if z i = w i with w i = v if i ∈ {1 . . . 2n} 1 if i ∈ {2n + 1 . . . 3n}
Remark 2.3. This regularity condition can be replaced by one of those proposed in [START_REF] Ralph | Some properties of regularization and penalization schemee for MPECS[END_REF].

The smoothing technique

The following theorem yields a condition to find a solution for (P σ ). It also proves a direct link to (P ):

Theorem 3.1. we suppose that z ∈ D ′ satisfies the Mangasarian-Fromovitz condition, and that

β ′ (ε) ≥ β 1 > 0 for 0 < ε < ε. i) If σ is sufficiently large, there is no KKT point of P σ with ε > 0.
ii) For σ sufficiently large, every local minimizer (z * , ε * ), (z * = (x * , y * , e * )) of the problem (P σ ) has the form (z * , 0), where (x * , y * ) is a local minimizer of the problem (P ).

Proof: i) Let (z, ε) a Kuhn Tucker point of P σ , then there exist λ and µ ∈ R 3n+1 such that:

(8) (i) ∇ℓ(z, ε) = ∇f σ (z, ε) + λ -µ = 0 (ii) min(λ, z i ) = min(µ, w i -z i ) = 0, i = 1 . . . 3n (iii) µ 3n+1 = min(λ 3n+1 , ε -ε) = 0,
where ∇f σ is the gradient of f σ with respect to (z, ε). Assume that there exists a sequence of KKT points (z k , ε k ) of P σ k with ε k = 0, ∀k and lim k→+∞ σ k = +∞.

Since D

′ is bounded and closed, up to a subsequence, we have

lim k→+∞ ε k = ε * lim k→+∞ z k = z * (9.i) yields to ∂ ε f σ k (z k , ε k ) + λ 3n+1 -µ 3n+1 = 0. So that ∂ ε f σ k (z k , ε k ) ≤ 0. Then, if we denote ∆ k = ∆(z k , ε k ), we have ∂ ε f σ k = - 1 4ε 2 k ∆ k + 1 2ε k ∂ ε ∆ k + σ k β ′ (ε k ) = - 1 4ε 2 k ∆ k + 1 ε k (θ ε (x k ) + θ ε (y k ) + e k + 1)(∂ ε θ ε (x k ) + ∂ ε θ ε (y k )) + σ k β ′ (ε k ) ≤ 0
Multiplying by 4ε 3 , we obtain

4ε 2 k (θ ε (x k ) + θ ε (y k ) + e k -1)(∂ ε θ ε (x k ) + ∂ ε θ ε (y k )) + 4ε 3 k σ k β ′ (ε k ) ≤ ε k ∆ k Since ∆ k , θ ε and ε 2 ∂ ε θ ε are bounded (by definition (v)), σ k → ∞ when k → ∞.
We have ε * = 0.

(ii) Let σ sufficiently large and (z * , ε * ) a local minimizer for (P σ ). If (z * , ε * ) satisfies the Magasarian-Fromovitz condition, then (z * , ε * ) is a Kuhn-Tucker points for f σ . By (i), we conclude that ε * = 0.

Let V a neighborhood of (z * , 0), for any z feasible for P such that (z, 0) ∈ V we have

(9) f σ (z * , 0) ≤ f σ (z, 0) = f (x, y) < +∞ (since ∆(z, 0) = 0).
We can conclude that ∆(z * , 0) = 0, otherwise f σ (z * , 0) would be +∞. So that < x * , y * >= 0 and (x * , y * ) is a feasible point of (P ).

Back to (9) f (x * , y * ) = f σ (z * , 0) ≤ f σ (z, 0) = f (x, y). Therefore (x * , y * ) is a local minimizer for (P ).

Remark 3.1. The previous theorem is still valid if we consider penalty functions of the form

(10) f σ (x, y, e, σ) = f (x, y) + α(ε)∆(x, y, e, ε) + σβ(ε) with α(ε) > 1 2ε .

Numerical results

In this section we consider some preliminary results obtained with the approach described in the previous section. We used the SNOPT solver [START_REF] Gill | SNOPT, A large-scale smooth optimization problems having linear or nonlinear objectives and constraints[END_REF] for the solution on the AMPL optimization platform [START_REF] Ampl | A Modeling Language for Mathematical Programming[END_REF]. In all our tests, we take β(ε) := √ ε. We consider various MPECs where the optimal value is know [START_REF] Macmpec | Ampl collection of Mathematical Programs with Equilibrium Constraints[END_REF]. Tables 1 and2 summarizes our different informations concerning the computational effort of the SNOPT, by using respectively θ w1 and θ 1 function:

• Obj.value : is the optimal value • it : correspond to the total number of iterations • (Obj.) and (grad.) : correspond to the total number of objective function evaluations and objective function gradient evaluation • (constr.) and (jac.) : give respectively the total number of constraints and constraints gradient evaluation. We remark that by considering θ w1 or θ 1 we obtain the optimal know value in almost all the considered test problems.

Application to simple partitioning problem and binary quadratic problems

In this section, we consider two real applications : the simple number partitioning and binary quadratic problems. These two classes of problems are know to be NP hard. We propose here a simple heuristic to obtain local solutions. 5.1. Application to simple partitioning problem. The number partitioning problem can be stated as a quadratic binary problem. We model this problem as follows.

We consider a set of numbers S = {s 1 , s 2 , s 3 , . . . , s m }. The goal is to partition S into two subsets such that the subset sums are as close to each other as possible. Let x j = 1 if s j is assigned to subset 1, 0 otherwise. Then sum 1 , subset 1's sum, We will minimize the square of this difference

diff 2 := {c -2 m j=1 s j x j } 2 ,
We can rewrite diff 2 as diff 2 = c 2 + 4x T Qx, where q ii = s i (s ic), q ij = s i s j . Dropping the additive and multiplicative constants, our optimization problem becomes simply

U QP min x T Qx x ∈ {0, 1} n
We rewrite the problem as the follows:

U QP min x T Qx x.(1x) = 0 We can now, use the proposed algorithm to get some local solutions for (UQP).

The results reported here on modest-sized random problems of size m = 25 and m = 75. An instance of each size are considered with the element drawn randomly from the interval (50, 100).

Each of the problems was solved by our approach, using the two functions θ 1 ε and θ w ε . We present in the table 3 the number of optimal solution obtained with 100 different initial points generated randomly from the interval [0, 1]:

• Best sum diff : corresponds to the best value of | 100 i=1 (Q * round(x[i])-0.5 * c| • Integrality measure : correspond to the max i |round(x i ) -x i |
• Nb: correspond to the number of tests such that the best sum is satisfied.

• N b 10 : correspond to the number of tests such that the sum : Table 3. using the θ 1 and θ w1 function 5.2. Application to binary quadratic problems. We consider some test problems from the Biq Mac Library [START_REF] Wiegele | Biq Mac Library -A collection of Max-Cut and quadratic 0-1 programming instances of medium size[END_REF]. These problems are written in the simple following formulation:

| 100 i=1 (Q * round(x[i]) -0.5 * c| ≤ 10 Problem Best sum diff Nb Integrality measure N b 10 (θ 1 , θ w1 ) (θ 1 , θ w1 ) (θ 1 , θ w1 ) (θ 1 , θ w1 ) N P 25.1 (1, 0) (1,
min y T Qy y ∈ {0, 1} n
where Q is a symmetric matrix of order n. For the Q matrix, ten instances have been generated. The parameters are the following:

• diagonal coefficients in the range [-100, 100]

• off-diagonal coefficients in the range [-50, 50],

• seeds 1, 2, . . . , 10. We apply the technique described in section 2. We present in the table 4 the number of optimal solution obtained with 100 different initial points (Nbop) generated randomly from the interval [0, 1], and for a size matrix equal 100. The fourth column precise the obtained value when different to know optimal value. Problem Know. value N bop (θ 1 , θ w1 ) Found value (θ 1 , θ w1 ) be100. 1 -19412 (17, 14) be100. [START_REF] Birbil | An entropic approach for mathematical programs with equilibrium constraints[END_REF] -17290 (14, 12) be100. [START_REF] Haddou | A new class of smoothing methods for mathematical programs with equilibrium constraints[END_REF] -17565 (9, 13) be100. [START_REF] Macmpec | Ampl collection of Mathematical Programs with Equilibrium Constraints[END_REF] -19125 (9, 14) be100. [START_REF] Facchinei | A smoothing method for mathematical programs with equilibrium constraints[END_REF] -15868 (2, 2) be100. [START_REF] Gill | SNOPT, A large-scale smooth optimization problems having linear or nonlinear objectives and constraints[END_REF] -17368 (31, 31) be100. [START_REF] Lin | Some exact penalty results for nonlinear programs and mathematical programs with equilibrium constraints[END_REF] -18629 (0, 0) (-18473, -18475) be100. [START_REF] Liu | Partial exact penalty for mathematical programs with equilibrium constraints[END_REF] -18649

(1, 1) be100.9

-13294 (0, 0) (-13248, -13248) be100. [START_REF] Mangasarian | Exact penalty functions for mathematical programs with linear complementarity constraints[END_REF] -15352 [START_REF] Ralph | Some properties of regularization and penalization schemee for MPECS[END_REF][START_REF] Macmpec | Ampl collection of Mathematical Programs with Equilibrium Constraints[END_REF] Table 4. using the θ 1 and θ w1 functions Using θ w1 or θ 1 we obtain the optimal know value in almost of our tests. We obtain a local solutions for only two examples. For each instance, the algorithm found an optimal solution and needs < 1 s for the resolution.

conclusion

In this paper, we have introduced an exact penalty approach to solve the mathematical program with equilibrium constraints.

is sum 1

 1 = m j=1 s j x j and the sum for subset 2 is sum 2 =

Table 1 .

 1 using the θ w1 function

	Problem		Data		Obj.val.		it	Obj. grad constr. Jac
	bard1		no		17		331	193	192
	desilva		(0, 0)		-1		892	655	656	655	656
			(2, 2)		-1		448	416	415	416	415
	Df1		no	3.26e -12		3	28	27	28	27
	Bilevel1		(25, 25)		5		470	214	213
			(50, 50)		5		295	168	169
	Bilevel2		(0, 0, 0, 0)		-6600		232	55	54
			(0, 5, 0, 20)		-6600		180	56	55
			(5, 0, 15, 10)		-6600	-6599.9	331	97	96
	flp4		flp4-1.dat		1.9e -29		66	9	8
			flp4-2.dat	3.08e -29		66	9	8
			flp4-3.dat		1.1e -28		66	9	8
	gauvin		no		0		184	71	70
	jr1		no		0.5		1175	814	813
	scholtes1		1		2		12	10	9	10	9
	hs044		no		14.97		375	101	100
	nash1		(0, 0)		0		0	2	1
			(5, 5)	1.72e -17		13	13	12
			(10, 10)	2.24e -12		12	12	11
			(10, 0)	4.29e -12		12	11	10
			(0, 10)	1.46e -13		13	12	11
	qpec1		no		80		1249	443	442
	liswet1-inv		liswet1-050	0.0929361		215	126	125
	Stack1		0		-3266.67		27	26
			100		-3266.67		7	17	16
			200		-3266.67		7	17	16
	Water-net	Water-net.dat		931.1		2070	886	885	886	885
	Problem		Data		Obj.val.		it	Obj. grad constr. Jac
	bard1		no		17		433	248	247
	desilva		(0, 0)		-1		7	255	254	255	254
	Df1		no		0		657	961	960	961	960
	gauvin		no		9.5e -05		164	82	81	82	81
	Bilevel1		(25, 25)		5		401	190	198
			(50, 50)		5		391	183	182
	Bilevel2		(0, 0, 0, 0)		-6600		2458	487	486	487	486
			(0, 5, 0, 20)		-6600		2391	727	721
			(5, 0, 15, 10)		-6600		2391	727	721
	hs044		no		17.08		617	261	260	261	260
	jr1		no		0.5		67	54	53
	nash1		(0, 0)		3.35e -13	203	111	110
			(5, 5)		6.7e -24		146	71	70
			(10, 10)		2.3e -17		133	85	84
			(10, 0)		8.1e -16		379	238	237
			(0, 10)		2.37e -18 1228	848	847
	qpec1		no		80		1895	518	517
	liswet1-inv	liswet1-050		0.028		3559	462	461
	scholtes1		1		2		51	106	105	106	105
	Stack1		0		-3266.67		64	58	57
			100		-3266.67		30	32	31
			200		-3266.67		30	32	31
	Water-net	Water-net.dat	931.369		919	282	281	282	281

Table 2 .

 2 using the θ 1 function

Université Libanaise, Laboratoire de Mathématiques et Applications (LaMA), Tripoli, Liban. E-mail: lina_abdallah@hotmail.fr

MAPMO, UMR 6628, Univ. Orléans, France. E-mail: mounir.haddou@univ-orleans.fr 1

We have proved a convergence result under suitable constraint qualification conditions. We performed a numerical computation by applying our approach to some tests from the library MacMPEC. Then, we considered some examples from the Biq Mac Library and some randomly generated partitioning problems. We used two different smoothing functions and our limited numerical tests gave almost the same result for each one.


Elemér E Rosinger
email: eerosinger@hotmail.com

Marie-Louise Nykamp

Microscopes and Telescopes for Theoretical Physics : How Rich Locally and Large Globally is the Geometric Straight Line ?

One is reminded in this paper of the often overlooked fact that the geometric straight line, or GSL, of Euclidean geometry is not necessarily identical with its usual Cartesian coordinatisation given by the real numbers in R. Indeed, the GSL is an abstract idea, while the Cartesian, or for that matter, any other specific coordinatisation of it is but one of the possible mathematical models chosen upon certain reasons. And as is known, there are a a variety of mathematical models of GSL, among them given by nonstandard analysis, reduced power algebras, the topological long line, or the surreal numbers, among others. As shown in this paper, the GSL can allow coordinatisations which are arbitrarily more rich locally and also more large globally, being given by corresponding linearly ordered sets of no matter how large cardinal. Thus one can obtain in relatively simple ways structures which are more rich locally and large globally than in nonstandard analy-

sis, or in various reduced power algebras. Furthermore, vector space structures can be defined in such coordinatisations. Consequently, one can define an extension of the usual Differential Calculus. This fact can have a major importance in physics, since such locally more rich and globally more large coordinatisations of the GSL do allow new physical insights, just as the introduction of various microscopes and telescopes have done. Among others, it and general can reassess special relativity with respect to its independence of the mathematical models used for the GSL. Also, it can allow the more appropriate modelling of certain physical phenomena. One of the long vexing issue of so called "infinities in physics" can obtain a clarifying reconsideration. It indeed all comes down to looking at the GSL with suitably constructed microscopes and telescopes, and apply the resulted new modelling possibilities in theoretical physics. One may as well consider that in string theory, for instance, where several dimensions are supposed to be compact to the extent of not being observable on classical scales, their mathematical modelling may benefit from the presence of infinitesimals in the mathematical models of the GSL presented here. However, beyond all such particular considerations, and not unlikely also above them, is the following one : theories of physics should be not only background independent, but quite likely, should also be independent of the specific mathematical models used when representing geometry, numbers, and in particular, the GSL. One of the consequences of considering the essential difference between the GSL and its various mathematical models is that what appears to be the definitive answer is given to the intriguing question raised by Penrose : "Why is it that physics never uses spaces with a cardinal larger than that of the continuum ?".

"History is written with the feet ..." Ex-Chairman Mao, of the Long March fame ... "Science is nowadays not done scientifically, since it is mostly done by non-scientists ..."

Anonymous

A "mathematical problem" ? For sometime by now, American mathematicians have decided to hide their date of birth and not to mention it in their academic CV-s. Why ? Amusingly, Hollywood actors and actresses have their birth date easily available on Wikipedia. Can one, therefore, trust American mathematicians ? Why are they so blatantly against transparency ? By the way, Hollywood movies have also for long been hiding the date of their production . Recently, [START_REF] Ya | Arithmetic of Infinity[END_REF][START_REF] Ya | [END_REF][START_REF] Ya | A new applied approach for executing computations with infinite and infinitesimal quantities[END_REF][START_REF] Ya | Numerical computations and mathematical modelling with infinite and infinitesimal numbers[END_REF][START_REF] Ya | Numerical point of view on Calculus for functions assuming finite, infinite, and infinitesimal values over finite, infinite, and infinitesimal domains[END_REF][START_REF] Ya | Counting systems and the First Hilbert problem[END_REF][START_REF] Ya | Lagrange lecture: Methodology of numerical computations with infinities and infinitesimals[END_REF][START_REF] Ya | Higher order numerical differentiation on the Infinity Computer[END_REF][START_REF] Ya | On accuracy of mathematical languages used to deal with the Riemann zeta function and the Dirichlet eta function[END_REF], a good illustration of that fundamental difference between an abstract idea and its various mathematical models was given with respect to the abstract idea of numbers, and on the other hand, what is suggested to be called as one or another specific numeral system which is but one of the many particular ways to model mathematically numbers.

As far as the GSL is concerned a variety of rather different mathematical models are known for it, among them those given by nonstandard analysis, [START_REF] Keisler | Elementary Calculus: An Approach Using Infinitesimals[END_REF][START_REF] Robinson | Non-standard analysis[END_REF], reduced power algebras, [START_REF] Rosinger | From Reference Frame Relativity to Relativity of Mathematical Models : Relativity Formulas in a Variety of Non-Archimedean Setups[END_REF][START_REF] Rosinger | Mathematics and[END_REF][START_REF] Rosinger | How Far Should the Principle of Relativity Go[END_REF][START_REF] Rosinger | Two Essays on the Archimedean versus Non-Archimedean Debate[END_REF][START_REF] Rosinger | Heisenberg Uncertainty in Reduced Power Algebras[END_REF][START_REF] Rosinger | No-Cloning in Reduced Power Algebras[END_REF][START_REF] Rosinger | Special Relativity in Reduced Power Algebras[END_REF][START_REF] Rosinger | Surprising Properties of Non-Archimedean Field Extensions of the Real Numbers[END_REF][START_REF] Rosinger | Four Departures in Mathematics and Physics[END_REF], the topological long line, [START_REF] Steen | Conterexamples in Topology[END_REF], or the surreal numbers, [START_REF] Conway | On Numbers and Games[END_REF], the latter giving in fact a GSL which is no longer a set but much larger still, being a proper class. And obviously, the abstract idea of the GSL is a particular instance of a mixture between the abstract ideas of geometry and numbers.

Since, however, it plays such a fundamental role in the construction of a large number of theories of physics, it may indeed be useful to focus on the issue of the GSL all on its own, as done in the sequel.

Amusingly, that fundamental difference between an abstract idea and its possible various mathematical models is on occasion known and accepted by various users of mathematics, among them physicists, for instance. Such is the case, among others, with the abstract idea of geometry which is by now well understood -and not only in general relativity -not to refer to, and not to be identical with any one single mathematical model, but to have a rather large number of quite different such mathematical models.

However, it should be recalled that less than two centuries ago, following a philosophy of the calibre of that of Kant, the abstract idea of geometry was seen as being reduced to, and perfectly identical with, one single mathematical model, namely, that of Euclidean geometry. And as if to aggravate the error in such a view, Kant considered Euclidean geometry as an a priori concept.

As it happened nevertheless in the 1820s, non-Euclidean geometry was introduced by Lobachevski and Bolyai, followed not much later by Riemannian geometry, and more near to our times, by the more or less explicit recognition of the immense difference between the abstract idea of geometry, and on the other hand, any of its mathematical models.

And still, the errors in approaching the concept of geometry were to continue for a while longer. The "Erlangen Program", for instance, published by Felix Klein in 1872, saw geometry as reduced to the rather narrow framework of the study of properties invariant under certain group transformations ...

In this regard, one may simply say that one does not -and in fact, can never ever -know what geometry, or for that matter, numbers are. And all one can know instead are merely various mathematical models of geometry, or of numbers.

Here, in the above spirit, we highlight the fundamental difference between the abstract idea of the GSL, and on the other hand, its various mathematical models.

In other words, we can never ever really know what the GSL may be. Instead, we may only try to represent it with one or another mathematical model ...

The importance of stressing this fundamental difference is to make theoretical physicists aware of the most simple and elementary -yet so widely missed -fact that confining all mathematical models used in physics to those based on the usual Cartesian coordinatisation of the GSL, a coordonatisation given by the customary field R of real numbers, is a highly dubious and detrimental approach. Indeed, as seen in the sequel, the GSL can admit other mathematical models which are far more rich locally, as well as far more large globally. Thus they allow the modelling of realms of physics which have so far escaped attention, or have been modelled inappropriately due to the relative poverty of the usual Cartesian coordinatisation given by R.

The coordinatisations constructed in the sequel are not only linear orders, but can also have useful commutative group, and in fact, vector space structures. Consequently, they allow the extension of the usual Differential Calculus. Thus one can obtain in relatively simple ways commutative group and vector space structures, together with a Differential Calculus, which may be arbitrarily more rich locally, and also arbitrarily more large globally than those constructed earlier in the literature, [START_REF] Conway | On Numbers and Games[END_REF][START_REF] Keisler | Elementary Calculus: An Approach Using Infinitesimals[END_REF][START_REF] Robinson | Non-standard analysis[END_REF][START_REF] Steen | Conterexamples in Topology[END_REF].

1. The View of the GSL through Microscopes and Telescopes ...

The GSL, that is, the straight line in Euclidean geometry, has since Descartes been coordinatised by the usual field R of real numbers. This association -so critically useful in modern physics, starting with Newtonian mechanics -has ended up with a tacit and rather universal identification between the GSL and the field R of usual real numbers. And as it happens, that situation is further entrenched by the rather simple mathematical fact that R is the only field which is linearly ordered, complete and Archimedean. As for the Archimedean Axiom, it again happens to be another quite universally and tacitly entrenched assumption, ever since the time of Euclid in ancient Egypt, more than two millennia ago, an assumption which, however, does not seem to have any known motivation in modern physics, [START_REF] Rosinger | From Reference Frame Relativity to Relativity of Mathematical Models : Relativity Formulas in a Variety of Non-Archimedean Setups[END_REF][START_REF] Rosinger | Mathematics and[END_REF][START_REF] Rosinger | How Far Should the Principle of Relativity Go[END_REF][START_REF] Rosinger | Two Essays on the Archimedean versus Non-Archimedean Debate[END_REF][START_REF] Rosinger | Heisenberg Uncertainty in Reduced Power Algebras[END_REF][START_REF] Rosinger | No-Cloning in Reduced Power Algebras[END_REF][START_REF] Rosinger | Special Relativity in Reduced Power Algebras[END_REF][START_REF] Rosinger | Surprising Properties of Non-Archimedean Field Extensions of the Real Numbers[END_REF][START_REF] Rosinger | Four Departures in Mathematics and Physics[END_REF].

Amusingly however, since the 1960s, the Nonstandard Analysis of Abraham Robinson, [START_REF] Keisler | Elementary Calculus: An Approach Using Infinitesimals[END_REF][START_REF] Robinson | Non-standard analysis[END_REF], has convincingly shown that the GSL can be endowed with a coordinatisation by a far larger field than R, namely, that given by the field * R of nonstandard real numbers. And such a coordinatisation is considerably more sophisticated than the usual one, both locally and globally, [START_REF] Conway | On Numbers and Games[END_REF][START_REF] Keisler | Elementary Calculus: An Approach Using Infinitesimals[END_REF][START_REF] Robinson | Non-standard analysis[END_REF][START_REF] Steen | Conterexamples in Topology[END_REF]. Also, it does no longer satisfy the Archimedean Axiom. And in fact, it simply cannot satisfy that axiom in view of the above remark on the unique position of R.

Yet with very few exceptions, the fact of the existence of such a considerably more sophisticated coordinatisation of the GSL seems not to have registered in the least either with mathematicians, or with physicists, in spite of its manifest advantages, among others in the study of continuous time stochastic processes which have importance also in physics.

The crucial issue -so far quite widely missed -is that in physics one operates, as with a basic theoretical instrument, not so much with the GSL itself, but rather with its specific coordinatisation which, as mentioned, is nearly universally and also quite stubbornly stuck to the field R of usual real numbers.

And to the extent that such a coordinatisation is not sophisticated enough, one can expect two major negative outcomes which, so far, have passed rather unnoticed :

• fundamental physical phenomena may escape the general awareness, just like it happened before the introduction of instruments such as microscopes and telescopes,

• fundamental physical phenomena which do not escape notice may nevertheless be thoroughly misinterpreted and misrepresented due to the coarseness of the instrument given by the usual coordinatisation employed, namely, by the field R of usual real numbers.

As far as mathematics is concerned, there is comparatively with physics a considerable freedom, since major criteria for relevance are not so much and so directly related to physical relevance. Consequently, the mathematics which obtains based on the usual Cartesian coordinatisation of the GSL need not lead to errors, and only risks to leave aside a vast realm of possibly relevant mathematics.

And then, above all for physics and physicists the following long disregarded critically important question arises :

• How far can one -and in fact, should one -go in more sophisticated coordinatisations of the Geometric Straight Line ?

One can recall related to this question the rather intriguing and highly relevant question of Penrose, [START_REF] Penrose | The Road to Reality : A Complete Guide to the Laws of the Universe[END_REF] :

"Why is it that physics never uses spaces with a cardinal larger than that of the continuum ?"

As it happens, however, there is no trace in the recent, or for that matter, earlier physics literature that this question would be considered, let alone debated ... Now, as also illustrated by nonstandard analysis or by the reduced power algebras, the concept of "more sophisticated coordinatisations" of the GSL has at least the following two aspects, [START_REF] Rosinger | From Reference Frame Relativity to Relativity of Mathematical Models : Relativity Formulas in a Variety of Non-Archimedean Setups[END_REF][START_REF] Rosinger | Mathematics and[END_REF][START_REF] Rosinger | How Far Should the Principle of Relativity Go[END_REF][START_REF] Rosinger | Two Essays on the Archimedean versus Non-Archimedean Debate[END_REF][START_REF] Rosinger | Heisenberg Uncertainty in Reduced Power Algebras[END_REF][START_REF] Rosinger | No-Cloning in Reduced Power Algebras[END_REF][START_REF] Rosinger | Special Relativity in Reduced Power Algebras[END_REF][START_REF] Rosinger | Surprising Properties of Non-Archimedean Field Extensions of the Real Numbers[END_REF][START_REF] Rosinger | Four Departures in Mathematics and Physics[END_REF] :

• a local one corresponding to the richness around each coordinate point, like for instance corresponding to the nonstandard monad of infinitesimally near points to each given coordinate point, and as such, seen by Keisler's microscope, [START_REF] Keisler | Elementary Calculus: An Approach Using Infinitesimals[END_REF],

• a global one corresponding to the largeness of the set of all coordinate points, like for instance given by the nonstandard infinitely large coordinate points, and as such, seen by Keisler's telescope, [START_REF] Keisler | Elementary Calculus: An Approach Using Infinitesimals[END_REF].

Background Independence : How Far Should the Principle of Relativity Go ?

As mentioned in [START_REF] Rosinger | From Reference Frame Relativity to Relativity of Mathematical Models : Relativity Formulas in a Variety of Non-Archimedean Setups[END_REF][START_REF] Rosinger | Mathematics and[END_REF][START_REF] Rosinger | How Far Should the Principle of Relativity Go[END_REF][START_REF] Rosinger | Two Essays on the Archimedean versus Non-Archimedean Debate[END_REF][START_REF] Rosinger | Heisenberg Uncertainty in Reduced Power Algebras[END_REF][START_REF] Rosinger | No-Cloning in Reduced Power Algebras[END_REF][START_REF] Rosinger | Special Relativity in Reduced Power Algebras[END_REF][START_REF] Rosinger | Surprising Properties of Non-Archimedean Field Extensions of the Real Numbers[END_REF][START_REF] Rosinger | Four Departures in Mathematics and Physics[END_REF], beyond, and quite likely above, the previous considerations, the study of the essential difference between the abstract idea of the GSL, and on the other hand, its various specific mathematical models may have a foundational importance in theories of physics regarding the issue of how far should the Principle of Relativity go ?, [START_REF] Rosinger | From Reference Frame Relativity to Relativity of Mathematical Models : Relativity Formulas in a Variety of Non-Archimedean Setups[END_REF][START_REF] Rosinger | How Far Should the Principle of Relativity Go[END_REF].

A remarkable fact in this regard is that the GSL can be modelled by algebras from a very large class of reduced power algebras, and in doing so, basic results from relativity, quantum mechanics and quantum computation remain valid, [START_REF] Rosinger | Heisenberg Uncertainty in Reduced Power Algebras[END_REF][START_REF] Rosinger | No-Cloning in Reduced Power Algebras[END_REF][START_REF] Rosinger | Special Relativity in Reduced Power Algebras[END_REF]. And here one should point out that such reduced power algebras are not fields, since they have zero divisors. Also, these algebras are not one dimensional, as the GSL is supposed to be, and instead, they may be infinite dimensional. Furthermore, such algebras are in general not linearly ordered or Archimedean, as the GSL is assumed to be.

Consequently, one can note that such basic physical properties as presented in [START_REF] Rosinger | Heisenberg Uncertainty in Reduced Power Algebras[END_REF][START_REF] Rosinger | No-Cloning in Reduced Power Algebras[END_REF][START_REF] Rosinger | Special Relativity in Reduced Power Algebras[END_REF], are indeed independent of a very large class of mathematical models, this fact thus being an illustration to a significant extension of the Principle of Relativity. Indeed, this time one can show that certain basic physical properties do not depend not only on reference frames, but also on a very large class of mathematical models which replace the usual model of the GSL.

A General Construction for Local Richness

We shall start with a general construction which brings about local richness, a construction which will mostly be used in some of its particular cases, such as for instance in Example 3.1. below. However, the interest in such a generality is, among others, in the fact that it clarifies what is actually involved in the issue of local richness.

Let (A, ≤) be a linearly ordered set. This set will have each of its elements a ∈ A endowed as if with a microscope which will open up, both to the left and to the right, the view from each such point a ∈ A, and do so with corresponding linearly ordered sets that are the local realms not accessible within A itself.

Such a local realm at each a ∈ A, a realm which does not exist in A itself, is given by two linearly ordered sets (B a , ≤ a ) and (C a , ≤ a ), with the first to the left of a ∈ A, and the second to the right of a ∈ A, as illustrated below

A • a ∈ A
i) d ∈ D a , d ∈ D a , a ≤ a , a = a ii) d, d ∈ B a , d ≤ a d iii) d ∈ B a , d = a iv) d ∈ B a , d ∈ C a v) d = a, d ∈ C a vi) d, d ∈ C a , d ≤ a d vii) d = d
Clearly, we have the strictly increasing mapping

(3.5) i : A a -→ a ∈ D
based on which, whenever there exists a ∈ A, such that B a ∪ C a = φ, we shall consider having the strict inclusion

(3.6) A D
which in fact can be seen as having the linearly ordered set (A, ≤) as a strict subset in (D, ≤), and endowed with the induced linear order from (D, ≤).

Example 3.1.

Let (A, ≤) = R and (B a , ≤ a ) = {a} × (-∞, 0), (C a , ≤ a ) = {a} × (0, ∞), with a ∈ A = R. Then (3.3) gives (3.7) D a = ({a} × (-∞, 0)) ∪ {a} ∪ ({a} × (0, ∞)), a ∈ A hence from (3.2) we obtain (3.8) D = (A × (-∞, 0)) ∪ A ∪ (A × (0, ∞))
therefore, the mapping (3.9)

j : D -→ R 2
given by (3.10)

j(a, x) = (a, x) a, x ∈ R, x = 0 j(a) = (a, 0) a ∈ R
is an order isomorphism between (D, ≤) and R 2 , when the latter is considered with the lexicographic order, see Appendix

(3.11) (a, x) (b, y) ⇐⇒   a < b or a = b, x ≤ y  
which obviously is a linear order on R 2 .

We can, therefore, identify (D, ≤) with (R 2 , ), modulo the above mapping j.

A Specific Method to Construct Local Richness

This method is suggested by Example 3.1. in section 3 above.

As in section 3, let (A, ≤) be a linearly ordered set which will be locally enriched.

Let (B, ≤) be any linearly ordered set which will produce the local enrichment at each a ∈ A. Namely, for each a ∈ A, we shall construct the linearly ordered set based on which one can see the linearly ordered set (A, ≤) as a strict subset in (A, ≤) (B, ≤), whenever we have A, B = φ. Furthermore, (A, ≤) is endowed with the induced linear order from (A, ≤) (B, ≤).

It is easy to see that, if the cardinals of (A, ≤) and (B, ≤) are infinite, then while with the lexicographic order, denoted by , A × B is linearly ordered.

Furthermore, see Appendix, Proposition A1.1., (A × B, +, ) is a linearly ordered commutative group. Now, similar with (3.9), (3.10), the mapping

(5.3) j : Ā -→ A × B
given by (5.4)

j(a, b) = (a, b) a ∈ A, b ∈ B, b = 0 j(a) = (a, 0) a ∈ A
where 0 is the neutral element in A, respectively B, is a strictly increasing group isomorphism.

Also, the mapping (5.5) i : A a -→ (a, 0) ∈ Ā is a strictly increasing group homomorphism.

In this way we obtain Theorem 5.1.

Let (A, +, ≤), (B, +, ≤) be linearly ordered commutative groups, and let

B + = {b ∈ B | b ≥ 0, b = 0}. Then, see (4.7) (5.6) ( Ā, ≤) = (A, ≤) (B + , ≤)
is a linearly ordered commutative group, with the strictly increasing group homomorphism (5.7) i : A a -→ (a, 0) ∈ Ā

Vector Space Structures on Locally Enriched Linearly Ordered Sets

Let us consider the commutative group of the locally enriched linearly ordered set ( Ā, ≤) = (A, ≤) (B + , ≤) in (5.6), and let us suppose that the respective commutative groups A and B are vector spaces over a linearly ordered filed F . Then one can define the scalar multiplication (6.1)

F × Ā (c, ā) -→ cā ∈ Ā by (6.2) c(a, b) = 0 ∈ A, c = 0 ∈ F, a ∈ A, b ∈ B, b = 0 c(a, b) = (ca, cb), c ∈ F, c = 0, a ∈ A, b ∈ B, b = 0 ca = ca, a ∈ A Theorem 6.1.
Let (A, +, ≤), (B, +, ≤) be linearly ordered commutative groups which are vector spaces over a linearly ordered field F . Then the action (6.1), (6.2) defines on ( Ā, ≤) = (A, ≤) (B + , ≤) a linearly ordered vector space over the linearly ordered field F .

Proof.

In view of (5.3), (5.4) and Theorem 5.1., we have the strictly increasing group isomorphism

(6.3) j : Ā -→ A × B
However, in view of Proposition A1.2., (A × B, +, ., ) is a linearly ordered vector space over F . Therefore, it only remains to show that the strictly increasing group isomorphism (5.3), (5.4) makes compatible the scalar multiplications in Ā and A × B. Finally, cj(a) = c(a, 0) = (ca, 0), and j(ca) = (ca, 0).

Iterations of the Operation

Let (A, ≤), (B, ≤) and (C, ≤) be three linearly ordered sets.

Let (φ, ≤) denote the void linearly ordered set, then

(7.1) (A, ≤) (φ, ≤) = (A, ≤), (φ, ≤) (A, ≤) = (φ, ≤)
Further, the linearly ordered sets

(7.2) (A, ≤) (B, ≤), (B, ≤) (A, ≤)
are in general not order isomorphic.

Similarly, the linearly ordered sets

(7.3) (A, ≤) ( (B, ≤) (C, ≤) ), ( (A, ≤) (B, ≤) ) (C, ≤)
are in general not order isomorphic.

We note however that

(7.4) (A, ≤) ( (B, ≤) (C, ≤) ) = (E, ≤)
where

(7.5) E = a∈A { [ ( b∈B ( ( C ×{-b})∪{b}∪( C ×{+b} ) ) )×{-a}]∪ ∪ {a} ∪ ∪ [ ( b∈B ( ( C × {-b}) ∪ {b} ∪ ( C × {+b} ) ) ) × {+a} ] } Similarly (7.6) ( (A, ≤) (B, ≤) ) (C, ≤) = (F, ≤)
where (7.7)

F = a ∈ S a∈A ( ( B×{-a})∪{a}∪( B×{+a} ) ) ( ( C × {-a }) ∪ ∪ {a } ∪ ∪ ( C × {+a } ) )
By using the above defined iterations of the operation , the results in sections 5 and 6 regarding the endowment of the locally enriched linearly ordered sets with commutative group, respectively, vector space structures, obviously allow the construction of arbitrarily large linearly ordered sets which are commutative groups or vector spaces. Indeed, let (A, +, ≤), (B, +, ≤) be linearly ordered commutative groups which are vector spaces over a linearly ordered field F . Then in view of Theorem 6.1., we obtain the following linearly ordered vector spaces over the linearly ordered field F , namely

(A, +, ≤) (B, +, ≤)

( (A, +, ≤) (B, +, ≤) ) (B, +, ≤) ( ( (A, +, ≤) (B, +, ≤) ) (B, +, ≤) ) (B, +, ≤) . . .
And in view of (4.9), it follows that in this way one can construct linearly ordered vector spaces of arbitrarily large cardinal over the linearly ordered field F .

Infinite Iterations of

Let (A, ≤) be a nonvoid linearly ordered set, and n ≥ 1. We define

(8.1) (A, ≤) n = (A, ≤) if n = 1 ((A, ≤) n-1 ) (A, ≤) if n ≥ 2
Clearly, that definition can be extended to all ordinal numbers α. In-deed, if α is not a limit ordinal, then we define (8.2) (A, ≤) α = ((A, ≤) α-1 ) (A, ≤)

while for α a limit ordinal, we define

(8.3) (A, ≤) α = β<α (A, ≤) β
We note that, in view of (4.8), we have strict inclusions

(8.4) (A, ≤) 1 (A, ≤) 2 (A, ≤) 3 . . . (A, ≤) n . . . . . . (A, ≤) ω (A, ≤) ω+1 . . .
and this justifies the definition in (8.3).

In view of (4.9), given any nonvoid linearly ordered set (A, ≤), the cardinal of the linearly ordered set in (A, ≤) α can be arbitrary large, for sufficiently large ordinal α.

Obviously, the above construction in (8.1) -(8.4) can be extended to the case when for each ordinal α, one has given a linearly ordered set (A α , ≤).

In fact, one can let the respective iterations run over all ordinals, and thus obtain linearly ordered proper classes which are no longer sets.

Furthermore, iterations such as in (7.8) can be performed for arbitrary cardinals, thus obtaining arbitrarily large linearly ordered commutative groups or vector spaces over linearly ordered fields.

9. Looking through Microscopes ...

It is easy to see that the constructions in sections 3 -7 above do the following. One takes a given nonvoid linearly ordered set (A, ≤) and constructs with the help of the same, or of any other nonvoid linearly ordered set (B, ≤) the corresponding extension ( Ā, ≤) = (A, ≤) (B, ≤) of (A, ≤), such that

• A is a strict subset of Ā,

• the linear order on Ā induces the initial linear order on A,

• each point a ∈ A obtains booth to the left and right a linearly ordered extension in Ā.

Thus the local structure of A is enriched by Ā. And this enrichment happens as follows

• at each point a ∈ A, a "microscope" given by (B, ≤) is installed with the effect that instead of the single point a ∈ A, now we can see the set ({a-} × B) ∪ {a} ∪ ({a + ×B}) which contains a ∈ A, and in addition, it extends to the left and to the right of a ∈ A with a realm twice the size of B,

• for two different points a, a ∈ A, their respective added realms do not intersect.

A General Construction for Global Largeness

Here, we present a construction similar with the above one for local enrichment which, this time, enlarges the global structure of any given linearly ordered set (A, ≤).

Let us note first that the above general construction of local enrichment in section 3 does itself give a certain kind of global enlargement since the initial linearly ordered set (A, ≤) is replaced with the larger linearly ordered set (D, ≤) and one has the strictly increasing mapping (3.5), (3.6). However, as seen next, there is also an alternative way for global enlargement of a linearly ordered set.

For that purpose, let (B, ≤), (C, ≤) be two linearly ordered sets which for convenience are supposed to be disjoint. Then we define the linearly ordered set

E = ( b∈B ({b} × A) ) ∪ A ∪ ( c∈C ({c} × A) ) = = ( B × A ) A ( C × A )
while the linear order e ≤ e is defined for e, e ∈ E according to the appropriate condition among the following seven It is easy to see that the above mapping k is in general not an order isomorphism between (E, ≤) and ( Ā, ≤).

11. Looking through Telescopes ...

A simple interpretation of the construction of global largeness in section 10 is the following one. Both at the left and right ends of the linearly ordered set (A, ≤) one respective telescope is placed. And each of these telescopes can see as many copies of (A, ≤) stretching to the left and right of (A, ≤), as many elements are in the linearly ordered sets (B, ≤) and (C, ≤), respectively.

Differential Calculus on the GSL

The developments in Appendix 2 below do obviously allow a Differential Calculus on any of the extensions in sections 3 and 10 above, as long as such extensions are vector spaces over a field. is an order isomorphic embedding, that is, it is an order isomorphism between A and the image i(A) of A in A × B.

A1.2. Lexicographic Order and Commutative Group Structure

Let now (A, +, ≤) and (B, +, ≤) be two linearly ordered commutative groups. Namely, we assume that, for a, a , a ∈ A, the relations hold (A1.3) a ≤ a =⇒ a + a ≤ a + a (A1.4) a ≥ 0, a ≥ 0 =⇒ a + a ≥ 0 where 0 ∈ A is the neutral element in A, as well as (A1.5) a, -a ≥ 0 =⇒ a = 0

We note that (A1.3) implies (A1.4), (A1.5). Indeed, if a ≥ 0, a ≥ 0, then 0 ≤ a, 0 ≤ a , thus 0 ≤ a = 0 + a ≤ a + a . Also, if a, -a ≥ 0, then 0 ≤ a, hence -a = 0 + (-a) ≤ a + (-a) = 0, thus -a ≤ 0, which with the assumption -a ≥ 0, gives -a = 0, and finally a = 0. Let (A, +, ≤) and (B, +, ≤) be two linearly ordered commutative groups.

Then their product considered with the lexicographic order (A × B, +, ) is again a linearly ordered commutative group, and the mapping (A1.7) A a -→ (a, 0) ∈ A × B is a strictly increasing group homomorphism.

Proof.

As noted above, we only have to show that (A1.3) holds. Let therefore (x, y), (u, v), (w, z) ∈ A × B. If (x, y) (u, v), then

  c r o s c o p e" For convenience of notation, we suppose that the sets A, B a , C a are pair-wise disjoint. Then we define the linearly ordered set (3.1) (D, ≤) which is the enrichment of A with the respective local views, where (3.2) D = a∈A D a with, see Fig. 3.1. above (3.3) D a = B a ∪ {a} ∪ C a The linear order ≤ on D in (3.2) is defined as follows. For d, d ∈ D, we have d ≤ d , if and only if one of the next seven conditions holds (3.4)

2 )

 2 B a = {a-} × B and for β = (a-, b), β = (a-, b ) ∈ B a , we have β ≤ a β , if and only if (4.3) b ≤ b Similarly, we define the linearly ordered set (4.4) (C a , ≤ a ) where (4.5) C a = {a+} × B and for γ = (a+, b), γ = (a+, b ) ∈ C a , we have γ ≤ a γ , if and only if (4.6) b ≤ b Now we can perform the construction in section 3, and obtain the corresponding linearly ordered set in (3.1), which we shall denote by (4.7) (A, ≤) (B, ≤) Clearly, we have the injective strictly increasing mapping, see (3.5) (4.8) (A, ≤) a -→ a ∈ (A, ≤) (B, ≤)

(4. 9 )Example 4 . 1 .

 941 car((A, ≤) (B, ≤)) = car(A, ≤) . car(B, ≤) For the linearly ordered set (D, ≤) in Example 3.1., we obviously have (4.10) (D, ≤) = R (0, ∞) 5. Commutative Group Structures on the Locally Enriched Linearly Ordered Sets Let (A, +, ≤), (B, +, ≤) be linearly ordered commutative groups, and let B + = {b ∈ B | b ≥ 0, b = 0}. Then (4.7) gives the locally enriched linearly ordered set (5.1) ( Ā, ≤) = (A, ≤) (B + , ≤) Now let us recall that A × B is a commutative group with the binary operation (5.2) (a, b) + (a , b ) = (a + a , b + b )

  Let therefore c = 0 ∈ F and a ∈ A, b ∈ B, b = 0. Then (6.2) gives in Ā the relation c(a, b) = 0 ∈ A, while in A × B we have cj(a, b) = c(a, b) = (0, 0). On the other hand, j(0) = (0, 0). Let now c ∈ F, c = 0, a ∈ A, b ∈ B, b = 0. Then similarly c(a, b) = (ca, cb), while cj(a, b) = c(a, b) = (ca, cb). On the other hand, j(ca, cb) = (ca, cb).

20 (

 20 as the global enlargement of A, by taking(10.2)

(10. 3 )( 10

 310 i) e ∈ {b} × A, e ∈ {b } × A, b ≤ b , b = b ii) e = (b, a), e = (b, a ) ∈ {b} × A, a ≤ a iii) e ∈ {b} × A, e ∈ ∪A ∪ c∈C ({c} × A) ) iv) e = a, e = a ∈ A, a ≤ a v) e ∈ A, e ∈ c∈C ({c} × A) vi) e ∈ {c} × A, e ∈ {c } × A, c ≤ c , c = c vii) e = (c, a), e = (c, a ) ∈ {c} × A, a ≤ a We note a certain connection between the two constructions, namely, of local richness and global largeness. Indeed, in the particular case of section 4, we constructed the linearly ordered set (10.4) ( Ā, ≤) = (A, ≤) (B, ≤) in (4.7), (4.8), for any two linearly ordered sets (A, ≤) and (B, ≤). Here, Ā is the local enrichment of A. And it follows easily that (10.5) Ā = ( (A-) × B ) A ( (A+) × B ) where A-= {a -| a ∈ A} and A+ = {a + | a ∈ A}.On the other hand, the global enlargement of A in (10.1) -(10.3) is given by the linearly ordered set (E, ≤), where(10.6) E = ( B × A ) A ( C × A )Now, similar with section 4, we can particularize that construction as follows. Let (F, ≤) be a linearly ordered set, and let us take (B, ≤) = (F -, ≤) and (C, ≤) = (F +, ≤), where the linear orders on F -and F + are the same with that on F . Then(10.6) givesE = ( (F -) × A ) A ( (F +) × A )For convenience of notation, let us replace F with B, and thus we have (10.7) E = ( (B-) × A ) A ( (B+) × A ) Let us now define the bijective mapping , a) -→ (a-, b) a -→ a (b+, a) -→ (a+, b) Remark 10.1.

Appendix 1 :

 1 Lexicographic Order and Commutative Group Structures A1.1. Lexicographic Order Let (A, ≤) and (B, ≤) be two linearly ordered sets. We recall that the lexicographic order is defined on A × B by (A1.1) (a, b) (c, d) ⇐⇒   a ≤ c, a = c or a = c, b ≤ d   and it gives again a linear order on A × B. Furthermore, for every given b ∈ B, the mapping (A1.2) i : A a -→ (a, b) ∈ A × B

Further, assumptions similar

  to (A1.3) -(A1.5) are made about (B, ≤). Recalling that A × B has the commutative group structure given by (A1.6) (a, b) + (c, d) = (a + b, c + d) we obtain Proposition A1.1.

5)

The above adaptations are to be treated elsewhere.

(A1.8) (x, y) + (w, z) (u, v) + (w, z)

Here we have to consider the following situations. Case 1. x ≤ u, x = u. Then x + w ≤ u + w, x + w = u + w, thus (A1.8) indeed holds.

Case 2. x = u, y ≤ v. Then x + w = u + w, y + z ≤ v + z, hence (A1.8) again holds.

A1.3 Lexicographic Order and Vector Space Structure

Let (A, +, ≤) and (B, +, ≤) be two linearly ordered commutative groups which are vector spaces over a field F . Then their group product (A × B, +) is naturally a vector space over F , according to the action (A1.9)

Let (A, +, ≤) and (B, +, ≤) be two linearly ordered commutative groups which are vector spaces over a linearly ordered field (F, +, ., ≤). Then the lexicographically ordered group (A × B, +, ., ) is a linearly ordered vector space over (F, +, ., ≤).

Proof.

We only have to prove that, for c ∈ F, (x, y) ∈ A × B, we have (A1.10) c ≥ 0, (x, y) (0, 0) =⇒ (cx, cy) (0, 0) Case 1. c = 0. Then (cx, cy) = (0, 0) (0, 0).

Case 2. c ≥ 0, c = 0. Then x ≥ 0, x = 0 gives cx ≥ 0, cx = 0, thus indeed (cx, cy) (0, 0).

If now x = 0, then (x, y) (0, 0) yields y ≥ 0, thus cy ≥ 0. But cx = 0, hence again (cx, cy) = (0, cy) (0, 0).

Appendix 2 : Differential Calculus on Vector Spaces

Let G, H be commutative groups which are also vector spaces over a field F .

We consider a set of sequences in

which are called convergent to 0 ∈ F , and which have the property to be a vector space over F . Now we can define an operation of limit on certain sequences in F , as follows. Given (c n ) n∈N ∈ F N and c ∈ F , then we have

if and only if

Similarly, we consider a set of sequences in G

which are called convergent to 0 ∈ G, and which have the property to be a vector subspace over F .

Again, we can define an operation of limit on certain sequences in G, as follows. Given (x n ) n∈N ∈ G N and x ∈ G, then we have

Finally, we do the same in H with a respective set of sequences

which also, are a vector subspace over F . Now, for every given v ∈ G, v = 0, we can define a derivation D v on certain functions f : G -→ H as follows. Let (A2.7)

be the set of all f : G -→ H, such that the limit

exists for all g ∈ G and (c n ) n∈N ∈ C F , for which c n = 0, with n ∈ N, and for each g ∈ G this limit is unique, being independent of the respective sequences (c n ) n∈N .

Proposition A2.1.

Similarly, for c ∈ F , we have cf ∈ D v and D v (cf ) = cD v f . Remark A2.1.

1) Typical further conditions required on the sequences in C F are as follows, [START_REF] Beattie | Convergence Strutures and Applications to Functional Analysis[END_REF][START_REF] Rosinger | Beyond Topologies[END_REF] :

Similar conditions may be required on the sequences in C G or C H .

2) In case of certain groups G, H or fields F , it may be appropriate to replace the convergent sequences with filters. This is a well known procedure, [START_REF] Beattie | Convergence Strutures and Applications to Functional Analysis[END_REF][START_REF] Rosinger | Beyond Topologies[END_REF].

3) When the groups G, H or the fields F are linearly ordered, the respective concepts of convergence may be required to satisfy appropriate compatibility conditions.

4) When the groups G, H are not only vector spaces over the fields F but also algebras, then one can require the derivations D v to satisfy the classical Leibniz rule of product derivative.