Bridging knowing and proving in mathematics

An essay from a didactical perspective Nicolas Balacheff CNRS Laboratoire d'informatique de Grenoble (France) To Adrien Douady 1 An ad hoc epistemology for a didactical gap

The didactical gap

More often than not, the problem of teaching mathematical proof has been addressed almost independently from the teaching of mathematical "content" itself. Some curricula have exposed learners to a significant amount of mathematics without learning about mathematical proof as such (Herbst, 2002, p.288); others teaching This is a preprint of: Balacheff N. (2010) Bridging knowing and proving in mathematics An essay from a didactical perspective. In: Hanna G., Jahnke H. N., Pulte H. (eds.) Explanation and Proof in . Heidelberg: Springer. mathematical proof as a subject in itself without significantly relating it to concrete practical examples (cf. Usiskin, 2007, p.75). The most common didactical tradition chooses to introduce proof in the context of geometry-usually at the turn of the 8 th grade-while completely ignoring it in algebra or arithmetic, where things seem to be reduced to 'mere' computations. This orientation has changed slightly in the past decade with an increasing emphasis on the teaching of proof. However, an implicit distinction between form and content has lead to references to teaching 'mathematical reasoning' (e.g., NCTM standards) or 'deductive reasoning' (e.g., French national programs) instead of mathematical proof as such which would have moved "form" much more to the forefront of the didactical scene.

Nevertheless, it is generally acknowledged that mathematical proof has specific characteristics, among them a formal type of text (the US vocabulary often refers to "formal proof"), a specific organisation and an undisputable robustness once syntactically correct. These characteristics have given mathematics the reputation of having exceptionally stringent practices as compared to other disciplines, practices that are not socially determined but inherent to the nature of mathematics itself.

Hence, the answer to the question: "Can one learn mathematics without learning what a mathematical proof is and how to build one?" is "No". But now one can observe a double didactical gap: (i) mathematical proof creates a rupture between mathematics and other disciplines (even the 'exact sciences') and (ii) a divide in the course of mathematical teaching during the (almost) standard first 12 years of education (into an era before the teaching of proof and one after).

The origin of these gaps lies at the crosspoint of several lines of tension: rigor versus meaning, internal development versus application-oriented development of mathematics, ideal objects defined and manipulated by symbolic representations versus experience-based empirical evidence. I do not analyse these tensions here; I mention them to evoke the complexity of the epistemological and didactical problems which confront us.

One source of the didactical problems is that teaching must take into account the learners' initial understanding and competence: We can teach only to ones who know… The learners' existing knowledge often proves resistant, especially because the learners may have proven its efficiency, as in the case of their argumentative skills. In order to overcome this difficulty, teachers organize situations, mises en scène and discourses in order to "convince" or "persuade" learners (in the vocabulary of [START_REF] Harel | Students' proof schemes: Results from exploratory studies[END_REF]. Argumentation seems the best means to this end. It works both as a tool for teaching and as a tool for doing mathematics for a long while. But then learners suddenly face an unexpected revelation 1 : In mathematics you don't argue, you prove… Looking to bridge this transition, mathematics educators have searched for ideas in psychology. In the middle of the 20 th century, the success of Piaget's 'stage theory' of development suggested that proof could be taught only after the required level of development had been reached 2 . As a result, mathematical proof was introduced suddenly in curricula (if at all) in the 9 th grade -generally, the year that students have their 13 th birthday. However, this strategy has not worked so well, suggesting to some that Piaget may have been wrong.

1 Argumentation means here "verbal, social and rational activity aimed at convincing a reasonable critic of the acceptability of a standpoint by putting forward a constellation of one or more propositions to justify this standpoint" (van Eemeren et al., 2002, p.xii). "In argumentative discussion there is, by definition, an explicit or implicit appeal to reasonableness, but in practice the argumentation can, in all kinds of respects, be lacking of reasonableness. Certain moves can be made in the discussion that are not really helpful to resolving the difference of opinion concerned. Before a well-considered judgment can be given as to the quality of an argumentative discussion, a careful analysis as to be carried out that reveals those aspects of the discourse that are pertinent to making such a judgment concerning it reasonableness." (ibid., p.4) 2 See e.g. Piaget J. (1969) p. 239: "L'enfant n'est guère capable, avant 10-11 ans, de raisonnement formel, c'est-àdire de déducation portant sur des données simplement assumées, et non pas sur de vérités observées". More precisely, For more, c.f. Piaget J. (1967) Le jugement et le raisonnement chez l'enfant. Delachaux et Niestlé. Some mathematics educators then turned to psychologies of discourse and learning, feeling that the followers of Piaget had not paid enough attention to language and social interaction. Some suggested the ideas of Vygotsky and the socio-constructivists could have provided a solution (e.g. [START_REF] Forman | Contexts for Learning: Sociocultural Dynamics in Children's Development[END_REF]. However, this line of thought did not appear to be the panacea either. Then Lakatos' work seemed to suggest that a solution might be found in the epistemology of mathematics itself (e.g. [START_REF] Reichel | Lakatos and aspects of mathematical education[END_REF]; however, such attempts also failed amid scepticism from mathematicians and researchers.

The responsibility for all these failures does not belong to the theories which supposedly underlie the educational designs, but to naive or simplifying readers who have assumed that concepts and models from psychology can be freely transferred to education. In particular, they rarely take into account the nature of mathematics as content (while often emphasizing the nature of the perceived practice of mathematicians).

My objective here is then to question the constraints mathematics imposes on teaching and learning, postulating that, as for any other domain, learning and understanding mathematics cannot be separated from understanding its intrinsic means for validation: mathematical proof. First, I address the epistemology of proof, on which we could base our efforts to manage or bridge the didactical gap discussed above.

The need to revisit the epistemology of proof

Although apparently a bit simplistic, it may be good to start from the recognition that mathematical ideas are not a matter of feeling, opinion or belief. They are of the order of 'knowing' in the Popperian sense3 , by virtue of their very specific relation to proof (and proving). They provide tools to address concrete, materialistic or social problems, but they are not about the "real" world. To some extent, mathematical ideas are about mathematical ideas; they exist in a closed 'world' difficult to accept but difficult to escape. For this reason, mathematical ideas do not exist as plain facts but as statements which are accepted only once they have been proved explicitly; before that, they cannot be4 instrumental either within mathematics or for any application.

However, despite this emphasis on the key role of proof in mathematics, it must be remembered that at stake is not truth but the validity of a statement within a welldefined theoretical context (cf. [START_REF] Habermas | Wahrheit und rechtfertigung[END_REF]. For example, Euclidean geometry is no truer than Riemannian geometry. This shift from the vocabulary of truth to the vocabulary of validity, which suggests a shift from proof to validation, is more important than we may have realized. Validation refers to constructing reasons to accept a specific statement, within an accepted framework shaped by accepted rules and other previously accepted statements. From this perspective, mathematical validation searches for an absolute proof in an explicit context; it can thus claim certainty as a foundational principle. This view of validity and proof is antiauthoritarian (Hanna & Janke, 1996, p.891), insofar as it assumes a common agreement about a collective and well-understood effort. It thus fits the classical conception of what a scientific proof should be, since such a proof must clearly not depend on specific individual or social interests. In this lies the democratic aspect of mathematical proof, as noted by Hanna. Proving is an example of an intellectual enterprise that allows a minority to overcome the opinion of an established majority, according to shared rules. This is related to an ancient meaning of the word "demonstration" in English (e.g., Herbst, 2002, p.287).

So the concept of proof is not a stand-alone concept; it goes with the concepts of "validity of a statement" and "theory". This has been well explained and illustrated by the Italian school, especially Alessandra [START_REF] Mariotti | Justifying and Proving in Geometry: the mediation of a microworld[END_REF]. However, the word "theory" is the most difficult for learners. No such thing is available to learners a priori, and to understand what the word means seems out of reach. Nevertheless, learners have ideas about mathematics and about mathematical facts. They also have experience in arguing about the "truth" of a claim or the "falsity" of a statement they reject; but this is experience in argumentation in contexts that are not framed by a theory in scientific terms. To construct a proof requires an essential shift in the learner's epistemological position: passing from a practical position (ruled by a kind of logic of practice) to a theoretical position (ruled by the intrinsic specificity of a theory).

In addition, we cannot engage in the validation of 'anything' that has not been first expressed in a language. This principle applies across disciplines [START_REF] Habermas | Wahrheit und rechtfertigung[END_REF], but plays a special role in mathematics, where the access to 'mathematical objects' depends in the first place on their semiotic availability [START_REF] Duval | Sémiosis et pensée humaine[END_REF].

In other words, the teaching and learning of mathematical proof requires mastery of the relationships among knowing, representing and proving mathematically. ------------------------------------------------------------------------------------------------- - ---------------------------------------------------------------------------------------------------Constructing the diagram (Fig. 1) with dynamic geometry software6 , one can easily notice that the point I does not move when one manipulates the point P. This fact seems surprising; the crux of the situation is to propose an explanation.

A model to bridge knowing and proving

Short story 1: Fabien and Isabelle misunderstanding

Let us examine the interaction between a teacher and a student, Fabien, about this problem 7 . Fabien has observed the fact but he has no insight about the reason: "The point I does not move, but so what..." However, he noticed and proved that ABCI is a parallelogram. At this stage, from the point of view of geometry (and of the tutor), the reason I stands immobile while P moves should be obvious. The tutor then provides Fabien with several hints but with no results. After a while she desperately insists: "The others, they do not move. You see what I mean? Then how could you define the point I, finally, without using the points P, P1, P2, P3?" Throughout the interaction, the tutor is moved by one concern which can be summarized by the question: "Don't you see what I see?" But Fabien does not see the 'obvious'; it is only when she tells him the mathematical reasons for the immobility of I that the tutor provokes a genuine "Aha!" effect... In order to explain the immobility of I, the teacher had get the student to construct a link between a mechanical world-that of the interface of the software8 -and a theoretical world-the world of geometry. Only this link can turn the observed fact (the immobility of I) into a phenomenon (the invariance of I). But the construction of this link is not straightforward; it is a process of modelling.

Teacher and student did share representations, words, and arguments so that they could communicate and collaborate; however, this did not guarantee that they shared understanding. Educators have made considerable efforts to develop representations that could make the nature and the properties of mathematical concepts more tangible.

But these remain just representations with no visible referent; manipulating them and sharing factual experience does not guarantee shared meaning. Nevertheless, they are the only means of communication, since in mathematics the referent, in a semiotic sense, is itself a representation (i.e., a tangible entity produced on purpose).

In the next section, I will explore this issue of representation and its relation with the learners' building of meaning, and then take up the challenge of defining "knowing" in a way that may not solve the old epistemological problem but will provide some grounds to build a link between knowing and proving.

Trust, doubt and representations

The fascination for proof without words 9 , which would give access to the very meaning of the validity of a mathematical statement without the burden of sophisticated and complicated discourses, is a symptom of the expectations mathematics educators have attached to the use of non-verbal representations in mathematics teaching. The development of multimedia software, advanced graphical interfaces and access to 'direct manipulation' of the represented 'mathematical objects' has even strengthened these expectations. The above story of the Fabien and his tutor misunderstandings is initial evidence that things might be slightly more difficult. I will explore this difficulty now, starting with an example coming from professional mathematics.

In 1979, Benoit Mandelbrot noticed in a picture produced by a computer and a printer that the Mandelbrot set 10 -as it is now known, following a suggestion of Adrien Douady-was not connected. "A striking fact, which I think is new" Mandelbrot 11 remarked. John Hubbard, a former PhD student of Adrien Douady's who became his well known collaborator, reported that: 9 See Claudi Alsina and Roger B. Nelsen (2006), Math Made Visual: Creating Images for Understanding Mathematics, published by MAA, and a good example in Roger B. Nelsen (1993), Proofs without words: exercises in visual thinking, published by MAA. See Hanna (2000, esp. pp.15-18) for an analysis. 10 Considering the sequence of complex numbers z n+1 = z n 2 + c, the Mandelbrot set (or set M) is obtained by fixing z 0 =0 and varying the complex parameter c. 11 Quotation from p.250 of Mendelbrot (1980) Fractal aspects of the iteration of zλz(1-z) for complex λ and z. Annals of the New York Academy of Sciences. 357 (1) 249 -259 Mandelbrot had sent [them] a copy of his paper, in which he announced the appearance of islands off the mainland of the Mandelbrot set M. Incidentally, these islands were in fact not there in the published paper: apparently the printer had taken them for dirt on the originals and erased them. (At that time, a printer was a human being, not a machine). Mandelbrot had penciled them in, more or less randomly, in the copy [they] had. (Hubbard 2000 pp.3-4) This anecdote reflects two things: first, the efficiency and strength of the computerbased picture in supporting a conjecture; second, the fragility of this same picture, which depends on both the algorithmic and technical conditions of its production.

Then, Hubbard reported: One afternoon, Douady and I had been looking at this picture, and wondering what happened to the image of the critical point by a high iterate of the polynomial z 2 + c as c takes a walk around an island. This was difficult to imagine, and we had started to suspect that there should be filaments of M connecting the islands to the mainland. (ibid.) Soon, Adrien Douady realized that this meant that the set M is connected12 , but "the proof of this fact is by no means obvious," he remarked (Douady, 1986, p.162). The proof followed after a long process of writing, initiated by a Note aux Comptes-rendus in 1982. After the discovery of the connectedness, images of the set M got transformed, offering a more beautiful picture full of colours which, so to speak, 'displayed' the connectivity of M (Fig. 2).

- --------------------------------------------------------------------------------------------------The Mandelbrot set for z→z 2 +c before and after the Douady and Hubbard discovery Figure 2.

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This case supports the idea of complex relations between representation and mathematical objects-or, more precisely, the role of representations as mediators for the conceptualisation of mathematical objects. It invites more caution in considering evidence in a non-verbal representation. Not to say that non-verbal representations or expressions of an argument are of no value; rather, I emphasize that the frequent claim in education that, "A picture is worth a thousand words" has limits and cannot be accepted without further examination.

For example, graphic calculators are widely used by students. They provide students with efficient tools for calculus, blending graphical and symbolic representations. The use of this technology has led to new problem-solving strategies that take advantage of the low cost of exploring of graphical representations. Among them is what Joel Hillel (1993, p.29) called "window shopping," which consists of playing with the various possibilities offered by the display. The diagrams (below) reproduce two appearances of the graph of the same function, f(x) = x 4 -5x²+x+4. As one can 'see', these pictures can induce different conjectures about, for example, the numbers of zeros of the polynomial or its behaviour within the interval [-2, +2] - --------------------------------------------------------------------------------------------------Figure 3.

- --------------------------------------------------------------------------------------------------It is now common for teachers to warn students and teach them strategies to ensure reliable, optimal use of their calculators. Still, the problem of knowing how to balance trust and doubt when using these machines and looking for conjectures has no straightforward answer. Part of achieving this balance depends not only on how learners critically organize their explorations but also on the reliability of the embedded software. Consider the case of the function g(x)=sin(e x ). Most students are prepared to study this function without a priori foreseeing difficulties; that is, until their machine displays something like the following picture:

- --------------------------------------------------------------------------------------------------Figure 4.

- --------------------------------------------------------------------------------------------------'Window shopping' will not help to answer the questions this display raises. An algebraic study will just leave students with a question they probably cannot solve with their knowledge of mathematics and computer science. This picture results from the interference between the computation of the coordinates of each point to be displayed and the choice of which pixel to turn black on the screen. In the end, it is the product of a kind of stroboscopic effect, as suggested by Adrien Douady 13 . Producing a 'correct' figure would be a matter of first mathematically notating both the capabilities and the limitations of the drawing instrument and then using sophisticated computational strategies to decide on the intervals at which to plot an 'informative' graph.

The problem of how students can decide to trust or doubt mathematical representations goes beyond graphical representations to include any representation.

A last example, taken from Luc Trouche work (2003) on computer algebra systems demonstrates this. Consider the equation [Ln(e x -1)=x]: One can use a pocket graphical 13 Personal communication calculator to solve it algebraically or to graph it; the two pictures below (from Trouche, 2003, p.27) show the respective results.

- --------------------------------------------------------------------------------------------------Figure 5 -

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The results speak for themselves. The optimal treatment leading to a solution -in this case, that this equation has no solution -consists of a formal transformation of the algebraic expression, producing [e x -1= e x ].

The difficulty students may have relates not to their lack of mathematical knowledge but to a general human inclination not to question their knowledge and their environment unless there is a tangible contradiction between what is expected after a given action and what is obtained, as my final example will demonstrate.

In this case, upper secondary students were asked to tell what is the limit at +∞ of the function [f(x)=ln(x)+10sin(x)]. Without a graphic calculator, only five percent of the students answered wrongly; with a graphic calculator, which displayed the window reproduced below, this number grew to 25 percent (Guin & Trouch, 2001, p.65).

- --------------------------------------------------------------------------------------------------Figure 6.

- --------------------------------------------------------------------------------------------------Given such cases of error, teachers and mathematics educators might have to consider whether graphic calculators contribute positively to mathematics learning or whether students have difficulty shifting from one semiotic context to another. (Other examples of common errors include: the value of  is exactly 3.14, or a convergent series reaches its limit, or the Fibonacci series U 0 =1, U 1 =(1+√5)/2, U n =U n-1 +U n-2 is divergent). Most such errors, or "misconceptions" to use the 1980s term, are probably symptomatic of the students' knowledge, which can be legitimate in certain contexts although possibly wrong mathematically. To analyse this issue further, we must have a conceptualization of the students' knowledge which (i) allows us to make sense of it from a mathematical perspective; (ii) is relevant from a cognitive perspective; and (iii) opens the possibility of didactical solutions.

A phenomenological definition of knowing

Studying students' productions that were mathematically incorrect, the mathematics educators of the 1980s usually chose to use the word "misconception". As noted by Jere [START_REF] Confrey | A review of the research on students conceptions in mathematics, science, and programming[END_REF], such student errors should be first considered as indications of what they know. Comfrey used the generic word "conception" to refer to the rationale of students' answers to a given problem or question. I postulate that such conceptions result from the learner's interactions with the environment, and that learning is both a process and an outcome of the learner's adaptation to this environment. By "environment", I refer to a physical setting, a social context or even a symbolic system (especially now that the latter can be depicted by a technology which dynamically materialises it).

However, only some characteristics of the environment are relevant from the point of view of learning. Educators do not deal with the learner in all his or her social, emotional, physiological and psychological complexity, but from a knowledge perspective: as the epistemic subject. The same principle applies to the environment, which we restrict to the milieu defined as the subject's antagonist system in the learning process (Brousseau, 1997, p.57); that is, we only consider those features of the environment that are relevant from the epistemic perspective. This means that our characterizations of the (epistemic) subject and of the milieu are interdependent systemically (and dynamically, since both will evolve during the learning process).

Pragmatically, the only accessible evidences of a conception are behaviours and their outcomes. Our problem is to interpret these in terms of indicators of strategies the adapted nature of which must be demonstrated in a model or representation attributed to the student (Brousseau, 1997, p.215) 14 . The formalisation of a conception I propose below aims at providing such a model. Recognizing this interdependence, expressed by Noss and Hoyles 15 (1996, p.122) as situated abstraction, accepts that people could demonstrate different and possibly contradictory conceptions depending on circumstances, although knowledgeable observers may ascribe them to the same source concept.

Thus, a conception is attached neither to the subject nor to the milieu, but exists as a property of the interaction between the subject and the milieu-its antagonist system (Brousseau, 1997, p.57). The objective of this interaction is to maintain the viability of the subject/milieu system (or [S↔M] system) by returning it to a safe equilibrium after some perturbation (i.e., the tangible materialization of a problem). This implies that the subject recognizes the perturbation (e.g., a contradiction or uncertainty) and that the milieu has features which make the perturbation tangible (since otherwise, the milieu may "absorb" or "tolerate" errors or dysfunctions).

- --------------------------------------------------------------------------------------------------A conception is the state of dynamical equilibrium of an action/feedback loop between a subject and a milieu under proscriptive constraints of viability. 16 Figure 7 -

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From this definition of conception, I can derive a definition of knowing as the characterization of a dynamic set of conceptions. This definition has the advantage of being in line with our usual use of the word "knowing" while providing grounds to understand the possible contradictions evidenced by learners' behaviours and their variable mathematical development. A conception is a situated knowing; in other words, it is the instantiation of a knowing in a specific situation detailed by the properties of the milieu and the constraints on the relations (action/feedback) between this milieu and the subject. 16 These constraints do not address how the equilibrium is recovered but the criteria of this equilibrium. Following Stewart (1994, pp. 25-26), I argue that these constraints are proscriptive -they express necessary conditions to ensure the system's viability -and not prescriptive, since they do not tell in detail how equilibrium must be reconstructed. This definition of conception provides a starting point but still has to be refined in order to make it relevant to our research. To do so, I will now introduce the model cK¢17 , in order to provide an effective tool to concretely represent and analyze the corpus of data obtained from the observation of students' activities. This model aims to establish a necessary bridge between knowing and proving by providing a more balanced role to control structures with respect to the role usually assigned to actions and representations.

A model to bridge knowing and proving: cK¢

That validation plays a key role in the emergence of 'knowing' has been established at least since Popper proposed the criterion of falsification and Piaget introduced the process of cognitive disequilibrium. This principle is also inherent in a "conception" as we define it, adding the explicit condition that a conception is not selfcontradictory.

"Proving" is the most visible part of the intellectual activity related to validation.

However, as the Italian school has clearly demonstrated (Boero et al. 1996), proving cannot be separated from the on-going controlling activity involved in solving a problem or achieving a task. To some extent, "proving" can be seen as an ultimate achievement of controlling and validating. No one can claim to know without a commitment to and a responsibility for the validity of the claimed knowledge. In return, this knowledge functions as a means to establish the validity of a decision in the course of performing a task and even in the process of building new knowledgeespecially in the learning process. In this sense, knowing and proving are tightly related. Hence, a conception is validation dependent: In other words, we can diagnose the existence of a conception because there is an observable domain in which "it works", in which there are means to validate it and to challenge possible falsifications. This is the essence of Vergnaud's (1981, p.220) statement that problems are the sources and criteria of concepts.

Vergnaud demonstrated that we could characterize students' conceptions with three components: problems, representation systems and invariant operators (1991, p.145) 18 . I take this model as a starting point, with the addition of the related control structure.

Then, I can characterize a conception by a quadruplet (P, R, L, ∑) in which:

-P is a set of problems;

This set corresponds to the class of the disequilibria the considered subject/milieu [S↔M] system can recognize; in mathematical terms: P is the set of problems which can be solved-in pragmatic terms, P is the conception's sphere of practice.

-R is a set of operators; -L is a representation system; R and L describe the feedback loop relating the subject and the milieu, namely the actions, feedbacks and outcomes.

-∑ is a control structure;

The control structure describes the components that support the monitoring of the equilibrium of the [S↔M] system. This structure ensures the conception's coherence; it includes the tools needed to take 18 Vergnaud in fact proposed this definition at the beginning of the 1980's. decisions, make choices, and express judgement on the use of an operator or on the state of a problem (i.e., solved or not).

This model aims at accounting for the [S↔M] system and is not restricted to one of its components19 . The representation system allows the formulation and the manipulation of the operators by the active subject as well as by the reactive milieu.

The control structure allows expression and discussion of the subject's means for deciding the adequacy and validity of his or her action as well as the milieu's criteria for selecting a feedback. This symmetry allows us both to take the subject's perspective when evaluating his or her knowing and the milieu's perspective when designing the best conditions to stimulate and support learning. Moreover, it gives us a framework in which to describe, analyze and understand the didactical complexity of learning proof by taking into account the interrelated relevant dimensions: the subject, the milieu and the problem.

In the next section I will give an illustration of this distinctive role of the control structure and the light it sheds on the learners' behaviors we observe and aim at understanding. I will then summarize the proposed framework discussing the relations we must establish between action, formulation and validation in order to understand the didactical complexity of learning and teaching mathematical proof. These three dimensions provide the means we need to build a bridge between knowing and proving.

Proving from a learning perspective

Short story 2: Vincent and Ludovic mismatch

Vincent and Ludovic are two middle school students who had no specific difficulties with mathematics. They volunteered to participate in an experiment that Bettina [START_REF] Pedemonte | Etude didactique et cognitive des rapports de l'argumentation et de la démonstration dans l'apprentissage des mathématiques[END_REF] was carrying out to study the cognitive unity between problem solving and proof. The problem was the following:

- ------------------------------------------------------------------------------------------------------Construct a circle with AB as a diameter. Split AB in two equal parts, AC and CB. Then construct the two circles of diameter AC and CB… and so on.

How does the perimeter vary at each stage?

How does the area vary? ------------------------------------------------------------------------------------------------------With no hesitation, the two students expressed -with the formulas they knew wellthe perimeter and the area of the first steps in the series of drawings. Their letters represent quantities and the formulas are another description of the reality the drawing factually displays. The students conjectured that the perimeter will be constant and that the area decreases to zero. But Vincent noticed that "the area is always divided by 2…so, at the limit? The limit is a line, the segment from which we started …". The discussion then continued: 41. Vincent: It falls in the segment… the circle are so small. 42. Ludovic: Hmm… but it is always 2πr.

43. Vincent: Yes, but when the area tends to 0 it will be almost equal… 44. Ludovic: No, I don't think so.

45. Vincent: If the area tends to 0, then the perimeter also… I don't know… 46. Ludovic: I will finish writing the proof.

Although Vincent and Ludovic collaborate well and seem to share the mathematics involved, the types of control they have on their problem-solving activity differ.

Ludovic is working in the algebraic setting (c.f., [START_REF] Douady | The interplay between different settings: Tool object dialectic in the extension of mathematical ability[END_REF]; the control is provided by his ensuring the correctness of the symbolic manipulation and his knowledge of elementary algebra. Vincent is working in a symbolic-arithmetic setting; the control comes from a constant confrontation between what the formula "tells" and what is displayed in the drawings. Both students understood the initial situation in the "same" way, both syntactically manipulated the symbolic representations (i.e., the formulas of the perimeter and of the area), but their controls on what they performed were different, revealing that the conceptions they mobilized were also significantly different. I deduce that the operators they manipulated (algebraic writings, sketching diagrams, etc.), although they coincided from the behavioural perspective, were semantically different. Moreover, from this evidence, an observer could argue that the students were not addressing the same "problem";

Vincent was "baffled" by the gap between what he saw and what he computed, while Ludovic was "blind" to this gap. (Actually, Ludovic's knowledge of calculus would not have been sufficient to provide any relevant explanation).

The symbolic representation plays the role of a semiotic mediator between the two students' different conceptions. It allows communication between the students and is instrumental for each in controlling the problem-solving process and building a proof.

We know that two different representations may demonstrate two different understandings; however, here one given representation also supports different understandings and hence different proofs.

The complex nature of proof

Many theorists have attempted to answer the question of what counts as a proof, from either an epistemological or an educational point of view. However, there is no single, final answer. The Vincent and Ludovic discussion above confirms that sheer formal computation is not enough. As in one of the best previous anecdotes in the history of mathematics20 , Vincent could well say to Ludovic: I see it, but I don't believe it. As several authors have emphasised, a proof should be able to fulfil the need for an explanation; however the explanatory nature of a proof may become the object of an even more irreconcilable disagreement than was its rigor. Consider the simple mathematical statement: The sum of two even numbers is itself even. The following figures provide a sample of proofs of this statement. A discussion of these proofs by mathematicians, mathematics teachers and learners provokes very different responses from each.

- ----------------------------------------------------------------------------------------------------Example adapted from Healey and Hoyles, (2000 p.400) Figure 9 -----------------------------------------------------------------------------------------------------The arguments in such a discussion involve three types of critical considerations: the search for certainty, the search for understanding and the requirements for a successful communication. The complex nature of proof lies in the fact that any effort to improve a candidate-proof on one of these dimensions may change its value on the other two. There is no clear standard to decide on the correct balance. Restricting the evaluation to the "certainty" side is playing safe, as this side is compulsory for the transformation of mathematical ideas. However, such reductionism is not viable from a learning perspective, especially when students are first introduced to mathematical proof; their control structures are not appropriately evolved. Educators at this point need to give academic status to activities that may not lead to what would be a proof for professional mathematicians but that still make sense as mathematical activities.

Hence, my proposal to structure the relations between explanation, proof and mathematical proof as I did to ground my own work [START_REF] Balacheff | Une étude des processus de preuve en mathématique chez des éléves de Collège[END_REF]. This structuration distinguished between pragmatic and intellectual proof, and within both it identified categories related first to the nature of the student knowing and his or her available means of representation.

The rationale for this organisation (sketched below in figure 10) is the postulate that the explaining power of a text (or non-textual "discourse") is directly related to the quality and density of its roots in the learner's (or even mathematician's) knowing.

What is produced first is an "explanation" of the validity of a statement from the subject's own perspective. This text can achieve the status of proof if it gets enough support from a community that accepts and values it as such. Finally, it can be claimed as mathematical proof if it meets the current standards of mathematical practice. So, the keystone of a problématiques of proof in mathematics (and possibly any field) is the nature of the relation between the subject's knowing and what is involved in the 'proof'.

- ---------------------------------------------------------------------------------------------------Figure 10 ----------------------------------------------------------------------------------------------------This recognition of a proof's roots in knowing may justify a statement as strong as

Harel & Sowder's that "one's proof scheme is idiosyncratic and may vary from field to field, and even within mathematics itself," (1998, p.275). However, this view misses the social dimension of proof, which transcends an entirely subjective feeling of understanding (as well as "ascertaining" or "persuading"; Harel & Sowder,ibid.,p. 242). From a didactical perspective, the issue is not psychological but epistemological, being directly related to the role a proof plays in building links between a theory that provides its framework and means and a statement that it aims to validate. The transcendence of a proof, proposed by [START_REF] Habermas | Wahrheit und rechtfertigung[END_REF] as a requirement for a problématique of truth and justification, is a dimension too often forgotten in favour of a psychological or sociological analysis of proving. This transcendence is not a dogmatic but a pragmatic position which allows the construction of knowledge as a collective asset which can be shared and be sustainable without depending on its author(s) and circumstance(s) of birth.

The technicalities of mathematical proof are then essential, and can be accepted as the price for a viable construction of mathematics. In this respect, formal rigour is a weapon against the biases that "idiosyncratic proof schemes" may produce.

Knowing and proving in the didactical genesis of proof

Learning mathematics starts with the first years of schooling, at least from an institutional point of view. As is well documented, learners at this elementary level depend as much on their experience as on the teacher as a reference to distinguish between their opinions, their beliefs and their actual knowledge. The criterion for assessing this difference rests either in the tangible efficiency of the knowledge at stake or in ad hoc validation by the teacher. But the teacher has to rely on knowledge, demonstrating that authority is not the ultimate reference. Hence, efficiency and tangible evidence are the supports for the validity of a statement: It's true because we verify that it works. Mathematical learners are first of all practical persons; to enter mathematics they have to change their intellectual posture and become a theoretician. This shift can easily be seen in the passage from practical geometry (the geometry of drawings and shapes) to theoretical geometry (the deductive or axiomatic geometry), or from symbolic arithmetic (computation of quantities using letters) to algebra. A learner making the transition from the practical to the theoretical has to face the epistemological difficulty of a transition from knowing in action to knowing in discourse: The origin of knowing is in action but the achievement of mathematical proof is in language (see below figure 12).

Again, the tight relationship among action, formulation (semiotic system) and validation (control structure) imposes itself [START_REF] Brousseau | Theory of didactical situations in mathematics[END_REF]. This trilogy which defines a conception, also shapes didactical situations 21 ; there is no validation possible if a claim has not been explicitly expressed and shared; and there is no representation without a semantic which emerges from the activity (i.e., from the interaction of the learner with the mathematical milieu).

Indeed, this passage from mathematics as a tool whose rationale is 'transparent', to mathematics as a theoretically-grounded means for the production and evaluation of explicit validation has a key stepping stone: language; as a symbolic technology (Bishop 1991 p.82), not just a means for social interaction and communication.

Language allows learners to understand and appropriate the value of mathematical proof compared with the pragmatic proof they were used to. Now, this language could be of lower levels than the naïve formalism mathematicians use; the level of language will bind the level of the proof learners can produce and/or understand. However, there is room for genuine mathematical activity at all these levels, provided that the learners have moved beyond empiricism and have seen the added value of the theoretical posture (see figure 12 below).

21 figure 11 below sketches the interactions between these three poles This figure illustrates the approximate mapping between the critical categories in each of the three dimensions (action, formulation and validation). It points the most difficult problem for teacher, that is to provide students with the means to switch from a pragmatic approach of truth to a theoretical approach of validity based on mathematical proof. Realising that language is a tool is a critical milestone on this move.

Figure 12 -----------------------------------------------------------------------------------------------------

Still an open problem: the situations…

After a few decades, researchers have now reached a consensus on the variety of meanings that proof may have for learners (if not for teachers). Several classifications and analyses of the complexity of the different aspects of mathematical proof have been extensively reported. Although they still express significant differences [START_REF] Balacheff | The role of the researcher's epistemology in mathematics education: an essay on the case of proof[END_REF], researchers have converged on considering mathematical proof as a core issue in the challenge of learning and teaching mathematics; mathematical knowing and proving cannot be separated. In other words an educational problématique of proof cannot be separated from that of constructing mathematical knowledge.

This challenge is well understood from an epistemological perspective. However, it is far from clear from a didactical perspective. A lot of effort has gone into proposing problems and mathematical activities which could facilitate the learning of mathematical proof. At the turn of the 20 th century, computer science and humancomputer interaction research have made so much progress that it is possible to provide learners and teachers with environments able to provide much more mathematically relevant feedback on users' activities. Especially, dynamic geometry environments and computer algebra systems allow learners to experience conjecturing and refuting in a manner never available before, hence giving them access to a dialectic necessary to ground the learning of mathematical proof.

However, there is some evidence that learners can remain in a pragmatic intellectual posture, not catching the value of mathematical proof.

Prompting the ultimate move from pragmatic to theoretic knowing requires designing situations so that the pragmatic posture is no longer safe or economical for the learners, while the theoretical posture demonstrates all its advantages. The resultant social and situational challenges are levers which one can use to modify the nature of the learners' commitment to proving. Such design is possible if solving a problem is no longer the main issue and fades away behind the issue of being "sure" of the validity of the solution. We already have some examples which witness the possibility of designing such situations (e.g., [START_REF] Bussi | Mathematical discussion and perspective drawing in primary school[END_REF], Boero et al. 1996, Arsac and Mantes 1997, etc.). The scientific challenge is now to better understand the

  figure when P3 and P are coincident? Construct the point I, the midpoint of

  Figure 1.

  Figure 8.

Popper (1959) proposed falsification as the the empirical criterion of demarcation of knowledge, scientific theories or models.

Or should not be...

From Capponi, 1995, Cabri-classe, sheet 4-10.

e.g. Cabri-geometry (here used for the drawing), or Geometer Sketchpad; or Geogebra or one of the several others now available sometimes open access.

A more detailed analysis can be found in[START_REF] Balacheff | Analyse du rôle de l'enseignant dans une situation de préceptorat à distance : TéléCabri[END_REF],[START_REF] Sutherland | Didactical complexity of computational environments for the learning of mathematics[END_REF].

Another student's search for an explanation illustrates well what is meant here by mechanical world: "So... I have said... But is not very clear... That when, for example, we put P to the left, then P3 compensates to the right. If it goes up, then the other goes down..."(Sébatien,).

Régine Douady remembers that Adrien had been quickly convinced of the connectivity of M, thanks to the theoretical argument which convinced him in an astonishingly "simple" way. However, to complete the explicit proof took some time(2008, personal communication).

For the convenience of the English-speaking reader, I take all the references to Brousseau's contributions to mathematics education from Kluwer, 1997 butBrousseau's work was primarily published between 1970 and[START_REF] Confrey | A review of the research on students conceptions in mathematics, science, and programming[END_REF]

This proposition should be understood in the light of the development of the 'situated learning paradigm' of Jeane Lave and Etienne Wenger, whose work was published in the early 1990s.

The letters cK¢ stand for : "conception", "knowing" and "concept"; more about this model is presented and discussed on [http://ckc.imag.fr]

By extension, one can often refer to students' conceptions as acceptable given that one can account precisely for the circumstances, which are the milieu and the constraints within which [S↔M] functioned.

"Je le vois, mais je ne le crois pas", wrote Cantor to Dedeking, in 1877, after having proved that for any integer n, there exists a bijection between the points on the unit line segment and all of the points in an ndimensional space.


F Blondelle

A Sultan

• E Collin

H Godfrin
email: henri.godfrin@grenoble.cnrs.fr

E Collin

Electrical conductance of bolted copper joints for cryogenic applications

Keywords: copper joints, conductance, low temperature, cryogenics PACS 84.32.Ff, 07.20.Mc, 72.15.Eb, 72.15.Eb

We present the results of electric contact resistance measurements at low temperatures on copper-to-copper bolted joints. Our accurate and systematic data display a rather small dispersion, and may be a useful tool for cryogenic applications like pulse-tubes, dilution refrigerators and nuclear refrigerators.

Introduction

The design of low temperature equipment involves thermally connecting parts without introducing significant temperature gradients under the nominal heat flux. Copper is often used for this purpose due to its good thermal conductivity and low cost [START_REF] Lounasmaa | Experimental principles and methods below 1 K[END_REF][START_REF] Pobell | Matter and methods at low temperatures[END_REF][START_REF] Richardson | Experimental techniques in condensed matter physics at low temperatures[END_REF][START_REF] Flynn | Cryogenic engineering[END_REF]. Since its low temperature thermal conductivity is due to electron transport limited by impurity scattering, substantial work has been performed to obtain high conductivity copper, developing purification and annealing methods [START_REF] Martin | [END_REF][6][7]. As a result, samples with a very large electronic mean free path are obtained, providing large values of the thermal conductivity at sub-Kelvin temperatures [START_REF] Ho | Thermal Conductivity of the Elements: A Comprehensive Review, supplement no. 1[END_REF][START_REF]Cryogenic Properties of Copper, information compiled for the International Copper Association[END_REF][START_REF] Hust | Thermal conductivity of aluminium, copper, iron and tungsten from 1 K to the melting point[END_REF][START_REF]Copper and dilute copper alloy thermal conductivity calculator[END_REF]. It is desirable to introduce joints in the cryogenic system at different places, either for convenience in the original mounting, or in order to be able to change samples, sample holders, experimental cells or nuclear stages in ultra-low temperature research. The joints introduce contact thermal resistances in series with the thermal links, thus degrading the systems performance. Contact resistances are known to depend on the geometry, the surface conditions, and the applied pressure. The literature provides typical values for different types of joints, measured in different ways, in different conditions. The dispersion of published results, however, is large. We therefore undertook a systematic study of contact resistances in simple copper-to-copper bolted joints, used in our fast-cooling pulse-tube dilution refrigerator equipped with a small copper nuclear stage. This work was developed in the framework of the EU Microkelvin program (www.microkelvin.eu), with the objective of providing more reliable tools for cryogenic design.

Electrical and thermal conductance: Wiedemann-Franz law

The thermal conductance of a system is defined as K = P /δT , the ratio of the heat flux (or power) P and the resulting temperature difference δT = T H -T C . For a homogeneous sample of length L and cross section A, K = κA/L, where κ is the thermal conductivity, a characteristic of the material. The quantity of interest in the heat transport through thermal links and joints connected in series, is the inverse of the thermal conductance, the thermal resistance R th expressed in units of K/W.

The physics of the electronic thermal and electrical conductivities of metals is described in detail in condensed matter books [START_REF] Kittel | Introduction to Solid State Physics[END_REF]. At room temperature, the collisions with thermal phonons dominate the electron scattering processes, and the corresponding electrical conductance is independent on the impurity concentration. At low temperatures, below about 10 K, the thermal conductance of a metal is proportional to the temperature, while the electrical conductance is constant: the low temperature or "residual" electrical resistance is determined by the collisions of the electron gas with the impurities.

When the temperature difference across the sample is large with respect to the average temperature, a quadratic temperature dependence of the heat flow is obtained by integration due to the linear temperature dependence of κ:

T 2 H -T 2 C = 2L/A κ/T P (1)
where κ/T is a constant. The electrical resistance of a metallic sample is determined by its resistivity (for copper, 1.68 10 -8 Ω m at 20 C) and geometrical factors. At low temperatures, the resistance is determined by the impurity concentration and the same geometrical factors. The ratio r (called RRR, the residual resistivity ratio) of the electrical resistance at room temperature to that at 4.2 K is therefore a measure of the purity of the sample:

RRR = r = R(T = 20 C) R(T = 4.2 K) (2)
According to the Wiedemann-Franz law, for a metal at low temperatures the ratio of the electronic contribution of the thermal conductivity κ to the electrical conductivity σ is proportional to the temperature T and to the Lorentz number L:

κ σ = L T (3)
where L = 2.44 10 -8 WΩ/ K 2 ; the value determined experimentally for copper is L = 2.2 10 -8 WΩ/ K 2 [START_REF] Kittel | Introduction to Solid State Physics[END_REF].

Although departures from the Wiedemann-Franz law have been observed in special cases, it is verified with reasonable accuracy in the context of copper samples and joints, which also display a metallic behaviour.

Measuring thermal conductances at low temperatures is not an easy task. It is thus common practice in low temperature work to measure the electrical resistance of metallic samples and make use of the Wiedemann-Franz law to estimate their thermal resistance.

R th = R L T ( 4
)
where R is the electrical resistance of the sample. When the temperature difference across the sample is large with respect to the temperature, the quadratic expression is found :

T 2 H -T 2 C = 2R P L ( 5
)
where P is the heat flux (power) across the sample.

Conductance values required in typical applications

Pulse-tube cryocoolers are presently used below 4K in a large number of cryogenic devices. We have developed (CNRS/Air Liquide) since 1999 pulse-tube based dry dilution refrigerators, with a commercial unit delievered in 2003 [START_REF] Prouvé | [END_REF]. In these machines, it is important that the condensation stage remains at the lowest temperature, i.e., avoiding thermal gradients at the level of the cold head. Since the cooling power is on the order of 1 W, and the temperature increase around 4 K should be less than 0.1 K, the thermal resistance should be lower than 0.1 K/W. Using the Wiedemann-Franz relation, the maximum contact electrical resistance is 10 nΩ.

Cooling samples or a nuclear demagnetization stage with a dilution refrigerator is often performed by bolting copper parts to the bottom of the mixing chamber. At very low temperatures, on the order of 10 mK, the cooling power is typically 1 µW. The thermal contact resistance should therefore be lower than 10 3 K/W. The corresponding electrical resistance is 200 nΩ. The cooling power P of a dilution refrigerator is given by the expression [1, 2]

P = 84 ṅ3 T 2 (6)
where the 3 He flow rate ṅ3 is expressed in moles/sec, P in Watts and T in Kelvin.

The dilution refrigerator can thus be represented by an equivalent electrical resistor connected to a bath at zero temperature, as seen from expressions (4) and ( 5):

R = L 168 ṅ3 = 1.45 µΩ 100 µmol/sec ṅ3 (7)
Contact resistances below 100 nΩ will be sufficient for connecting samples to typical dilution units with flow rates on the order of 200 µmol/sec.

Cooling nuclear stages to less than 10 mK using a dilution refrigerator is achieved through metallic thermal links including a heat switch [START_REF] Lounasmaa | Experimental principles and methods below 1 K[END_REF][START_REF] Pobell | Matter and methods at low temperatures[END_REF]. The time needed for cooling a copper nuclear stage of about 1 kg through the thermal link is on the order of one day, it is therefore desirable to reduce the thermal resistance of the link to a value much lower than the effective resistance of the dilution refrigerator, i.e. to about 100 nΩ. Below about 10 mK one must consider the thermal (Kapitza) resistance between the sintered silver heat exchanger and the helium mixture in the mixing chamber. The Kapitza resistance is in series with the metallic links' resistances. The Kapitza resistivity is ρ K = 14/T 2 in K 2 m 2 /W for the silver powder we use [START_REF] Lounasmaa | Experimental principles and methods below 1 K[END_REF][START_REF] Pobell | Matter and methods at low temperatures[END_REF]14]. For 100 m 2 of surface area the Kapitza resistance R K is 1.4 10 3 K/W at 10 mK, equivalent to an electrical resistance of 340 nΩ. This effective resistance depends on temperature as T -2 , it is the main limitation for cooling nuclear stages to temperatures below 10 mK.

At ultralow temperatures, an electrical contact resistance of 100 nΩ will cause a temperature difference of 40 µK at 100 µK for a heat leak of 1 nW.

Finally, the thermal links themselves may have a non-negligible resistance. For a copper block of 1 cm 2 section and 1 cm length, with a RRR of 100, the low temperature electrical resistance is low, 17 nΩ, but a wire made of similar copper of 1 mm 2 section and 10 cm length has a low temperature resistance of 17 µΩ, clearly unacceptable.

We conclude that contact resistances for the devices considerd here should be in the range 10 to 100 nΩ. The most delicate situations are encountered around 4 K where large power is applied at the level of the pulse-tube. The samples are representative of the copper purchased by low temperature laboratories. We did not use Cu-b1 or Cu-b2 phosphorous copper, a poor thermal conductivity material often used for manufacturing copper plates.

Low resistance measurements

The resistance measurements were done using a resistance bridge TRMC2, designed in our laboratory and commercialised by Air Liquide. The low-impedance board has an offset of less than 3 nΩ, and the noise is less than 10 nΩ. The bridge has 28 current (square wave) ranges (1 nA to 8 mA), 9 voltage ranges (5 to 800 µV), 63 resistance ranges (625 µΩ to 800 kΩ). The resolution is 0.002% (16 bits).

Four-wires measurements were performed using a well-shielded cryogenic set-up, first at room temperature, and then at 4.2 K in a liquid helium Dewar. Averaging the output signal and checking the offset allowed us to measure resistances with a sensitivity better that 10 nΩ. The absolute accuracy of the measurements, after removing a sample and modifying the wiring configuration, is estimated to be better that 50 nΩ (error bars indicated in the graphs).

Residual Resistivity Ratio measurements

The table 1 gives the values of the RRR (Residual Resistivity Ratio) determined for the samples of section 4.1. A major surprize was the observation of very low RRR values (in the range 5 to 10) for standard copper plates used in mechanical workshops (several samples from different manufacturers). RRRs on the order of 100 are quoted for Cu-a1 copper in Cryogenics books, values which have been verified in our laboratory several times. We also measured copper samples from old Cu-a1 (before 1980) pieces of equipment, obtaining the values 80-120 given in the table. We conclude that the purity of standard commercial Cu-a1 and also Cu-c1 copper available today is, probably because of recycling, not as good as generally believed. It is therefore necessary to check carefully the RRR of copper material to be used in cryogenic construction.

Contact resistance measurements

The set-up shown in figure 1 was used in the measurements of contact resistances. Two copper plates of about 4 cm length, 1 cm width and 1 mm thickness were bolted using stainless steel screws of different diameters, with diffferent steel washers. Current leads were bolted at the opposite ends of the plates, while thin copper ribbons (thickness 70 µm, width of about 7 mm) were placed as shown for the voltage measurement. The measurements were performed at 4.2 K, using the techniques described in section 4.2. The results shown in the following paragraphs, unless stated otherwise, correspond to type 4 samples: OFHC copper plates, 1 mm thick (see section 4).

Bolts, washers, torque and force

The experiments have been performed using stainless steel hexagonal (Allen) socket head cap bolts, of ISO metric thread M3, M4 or M5. The holes in the copper plates were drilled to the corresponding size (3, 4 and 5 mm), increasing the hole diameter as needed. Stainless steel washers were used: either standard (ISO 7089) flat washers, or Belleville (spring) washers, or both (in this case, the flat washer was in contact with the copper surface). The bolts were tightened using a torque-limiting wrench. The table 2 gives the maximum tightening torque for very high quality class stainless steel bolts and the corresponding applied force for different diameters. Note that for usual stainless steel bolts the maximum torque is lower by a factor of two (see the data-sheets provided by the manufacturers). The recommended tightening torque determined for our bolted copper contacts (a torque which does not deform permanently the copper or a good quality stainless steel bolt), is also given; it corresponds to forces of about 3000 N. The force (fig. 2) is calculated using Kellerman-Klein's expression [START_REF] Kellermann | Untersuchungen ber den Einfluss der Reibung auf Vorspannung und Anzugsmoment von Schraubenverbindungen[END_REF][START_REF] Guillot | Modélisation et calcul des assemblages vissés[END_REF]:

T = (p/2π + 0, 583dµ t + Dµ h ) F 10 -3 ( 8
)
where T is the applied torque in Nm, p the pitch in mm/turn, d the average diameter in mm at the bolt flanks, D the average diameter in mm under the bolt head, µ t and µ h the friction coefficients at the threads and under the head, respectively, and F the force in N. The force is the important parameter, which determines the contact resistance: increasing the contact area does not reduce the contact resistance. The friction coefficients are assumed to be on the order of 0.12; typical values are 0.1 for greased bolts, and 0.15 as received.

Influence of the surface conditions

The measurements have been performed for different surface conditions. The surfaces were flat (plates), they were either cleaned with emery paper (400grit), and/or in a standard workshop acid bath. In some of the measurements, Apiezon N vacuum grease was applied as a very thin layer, and the excess was removed with the fingers or a cleaning paper.

As seen in figure 3, the cleaning procedure improves the contact, although much less than commonly believed. Adding a thin layer of grease does neither improve, nor degrade the contact substantially. However, we have observed that thick layers of grease can yield much higher values of the resistance and highly irreproducible results (not shown).

Influence of the bolt size

The data shown in figure 4 show that M3 bolts give relatively high contact resistances. Torques exceeding the values given in the table 2 clearly damage the bolt, matting the bolt threads, thus leading to the observed flattening of Fig. 3 Influence of the surface conditions on the contact resistance of copper plates bolted using M4 bolts and flat washers. Data: copper plates as received, after cleaning (acid+400grit emery), and with a thin layer of grease.

the resistance vs pressure curve. A substantial improvement is obtained using M4 or M5 bolts. For these, the maximum torque is not determined by the bolt properties, but by the plastic deformation of the copper plates, in particular for annealed samples. The largest applied torques shown in figure 4 correspond to the limit where the washers started imprinting a clear trace on the copper; the contact does not improve further at our highest torque (see figures 4 and5) and becomes irreproducible. The results obtained with M3 bolts are inconsistent with the assumption that the force is the only relevant physical parameter. One would expect that for small torques,i.e. that do not damage the bolt, the resistance should be comparable to that obtained with larger bolts for the same force. We do not have a quantitative explanation for this effect, but M3 bolts clearly work near their mechanical limit in this context, while M4 bolts work in their optimal range and larger bolts are oversized for this application. As seen comparing figure 4 to figure 5, plating the contacts with a thin layer of Apiezon N grease does not change significantly the electrical resistance. A thick layer of grease degrades the contact substantially, see section 5.2.

Influence of the Belleville washers

Spring washers (also named Belleville washers) are often used to maintain the pressure when cooling, compensating the effect of thermal contraction of the materials. The results of measurements performed using different combinations of standard (flat) and Belleville washers are shown in figure 6.

Influence of other parameters

The higher resistance data correspond to copper joints with Belleville washers only. In this case, the edge of the conical washers locally exceeds the elastic limit of the copper. As expected, this is not a convenient set-up, but it has the merit of showing clearly the negative effect of localized stress on the quality of copper joints. The measurements display a high irreproducibility (two examples are shown in figure 6). In order to protect the copper and better distribute the pressure, flat washers have been intercalated between the Belleville washers and the copper surfaces (sample 2F+2B). A substantial improvement of the contact is achieved; however, similar results are obtained using only flat washers: the compensation of thermal contraction expected from the Belleville washers, does not play a role.

Other parameters influence the electrical conductance of bolted contacts. Different surface cleaning procedures have been tried. Using 400-grit emery, which creates asperities on the order of 0.4 µm, seems to be the best method. Sand-blasting the copper parts may seem interesting, but it gave worse contacts. Gold or silver plating improve the contact conductance, and this was verified for some of our samples, silver-plated using commercial solutions. We obtained in this case contact resistances on the order of 50 nΩ for torques of 2 to 2.5 Nm, that is, a factor of 4 better than those obtained for bare copper. Intercalating indium also improved the electrical contact (but indium becomes superconducting at low temperatures). The conductance is larger, but still within the same order of magnitude. We also made measurements using brass bolts; we find that they give contact resistances substantially higher than stainless steel bolts, because lower torques must be used.

Data summary and conclusions

We present in figure 7 a representative sample of our numerous contact resistance measurements. Our data correspond to copper joints, cleaned in acid and/or with 400-grit emery, bolted using stainless steel M4 bolts and flat washers (we showed that smaller or larger sizes are not particularly useful). The most striking features are the little dispersion of the data for samples of different purity and annealing conditions, and the good proportionality of the electrical conductance to the force. The graph includes results obtained by other groups measuring thermal or electrical resistance. The data in the upper part of the graph correspond to thermal conductance measurements by Berman et al. [17] and Salerno et al. [18][19][20]. They are clearly well above our data, while following closely, too, a 1/P pressure dependence. In this case, pressure was applied by mechanical devices at low temperatures. Copper demountable joints investigated by Didschuns et al. [21] also displayed the same pressure dependence; they were gold plated, and hence not quantitatively comparable to the samples considered here. Electrical resistance measurements were performed by several groups. The experimental conditions were often different from the simple bolted-plates geometry, but representative results [22][23][24][25] have been plotted in figure 7. Most of the data (for very different geometries) seem to form a coherent set with ours. A notable exception is that of Okamoto et al. [26], who obtain resistance values an order of magnitude lower, in the 10 nΩ range. The measuring method (current-decay) is different, but the sample conditions are rather similar, we have no explanation for this discrepancy. It would be desirable to perform a simultaneous measurement of the thermal and electrical conductivity on well characterized samples; this is by no means a trivial task, since both measurements are difficult, and some statistics are certainly needed.

The results presented here have been obtained using plates of thickness on the order of 1 mm, but we also measured several samples in the range 0.070 to 6 mm, obtaining similar results within a factor of two. We expect thicker plates to behave differently, because the differential thermal expansion between the bolts and the copper.

We conclude that joints made out of clean (acid bath and/or 400-grit emery paper) millimeter-thick copper plates, using M4 stainless steel bolts and standard flat washers, eventually coated by a protective very thin layer of Apiezon N grease, provide a simple thermal contact adequate for low temperature work. The applied force is in the 3000 N range with a tightening torque of about 2.5 Nm (lower than 2 Nm for highly annealed copper samples), sufficient to provide a good contact without damaging the copper parts. The resulting contact resistance is on the order of 200 nΩ. This value can be reduced by a factor on the order of 4, by gold or silver plating. Using several bolted contacts in parallel (optimizing the distance between bolts may be necessary) will be sufficient for many cryogenic applications.

4

 4 Experimental details4.1 Copper samples: Residual Resistivity RatioWe have used several copper samples, grouped in the following classes (the type of copper and the date of purchase are indicated): 1) Standard electrolytic copper (Cu-a1) plates (2013) 2) Standard electrolytic copper (Cu-a1) plates (1977) 3) Standard OHFC (Cu-c1) copper (2013) 4) OFHC 1 mm copper plates >99.9% (1996), Goodfellow CV000747/15 5) OFHC 1 mm copper plates, as above (annealed in 2013)

Fig. 1

 1 Fig. 1 Schematic drawing of the sample set-up for contact resistance measurements.

Fig. 2

 2 Fig.2Force vs. torque relations for different sizes of bolts, drawn up to the limit corresponding to high quality class stainless steel bolts.

Fig. 4 Fig. 5

 45 Fig. 4 Influence of the bolt size (M3, M4 or M5; a and b indicate different runs) on the contact resistance of bolted copper plates, using flat washers. The surfaces were cleaned, no grease was used. Dashed line: 1/F dependence. The force is calculated from the applied torque (see text).

Fig. 6

 6 Fig. 6 Contact resistance of copper plates bolted with M4 bolts, using on each side of the sample a flat washer (2F), or only a Belleville washer (2B), or a Belleville and a flat washer on each side (2F+2B), the flat one in contact with the copper. No grease was used. Dashed line: 1/F dependence.

Table 1

 1 Residual Resistivity Ratio of copper plates

	Stand.(1) 2013 Stand.(2) 1977 OFHC Cu-c1 (3) OFHC (4) OFHC (5)Ann.
	5	80-120	50	142	480

Table 2

 2 Maximum tightening torque for high quality class stainless steel bolts, resulting forces, and the adequate torque for bolted copper plates.

	Bolt	M3	M4	M5
	Max. Torque (Nm)	2.5	5.8	12
	Max. Force (N)	4100 7300 12400
	Recommended Torque (Nm) 1-2	2.5	3

  Summary of the contact resistance results. The upper points correspond to thermal resistance measurements obtained in contacts under mechanical pressure applied at low temperatures; the electrical resistance is deduced using the Wiedemann-Franz law. All other data correspond to electrical resistance measurements. The solid line shows the 1/F dependence obeyed by the data.

		100 000
				Berman (1956)
				Manninen	(1977)
		10 000		Deutsch (1978)
	Resistance (nOhms)	1 000		Lau and Zimmermann (1979) Salerno (1984) Okamoto (1990) Gmelin (1999) Belleville and no grease
		100		Grease and no Belleville
				Grease and Belleville
				No grease and no Belleville
		10
		100	1000	10000
			Force (N)
	Fig. 7

Acknowledgements This work was supported by the European Community Research Infrastructures under the FP7 Capacities Specific Programme, MICROKELVIN project number 228464, and by the grant High-Q Fermions (ANR-2010-INTB-403-01).


C M Fernández Peruchena

M Blanco

A Bernardos

Generation of series of high frequency DNI years consistent with annual and monthly long-term averages using measured DNI data

Keywords: DNI, high frequency, synthetic generation, enveloping, nondimensionalization, CSP, TMY

In recent years, many researchers are suggesting that statistical approaches are needed to achieve sound estimates of the economic feasibility of commercial Concentrating Solar Thermal Power (CSTP) plants. A key element of such statistical approaches is the capacity of generating large series of high-frequency years which are consistent with the estimated variability of the monthly and annual values of the relevant meteorological variables. This paper presents a method of using high-frequency solar Direct Normal Irradiance (DNI) measurements to generate high frequency DNI series consistent with the variability and distribution of natural series, and also matching monthly and annual long term averages. The method takes advantage of a novel technique for the nondimensionalization of measured high-frequency daily DNI curves. This nondimensionalization technique makes it possible to consistently use measured solar DNI data to generate new daily curves of high-frequency DNI data and the subsequent long series of high-frequency DNI years. The novel technique for the nondimensionalization of measured high-frequency daily DNI curves is based on the nondimensionalization of the temporal scale by dividing the elapsed Universal Time (UT) since sunrise by the total elapsed UT from sunrise to sunset and on the nondimensionalization of the solar DNI scale by dividing each actual solar DNI value by the corresponding DNI value of the clear-day solar DNI envelope curve.

Introduction

When analysing the economic feasibility of a commercial Concentrating Solar Thermal Power (CSTP) plant, the usual approach consist in developing an energy model of the CSTP plant and in feeding this model with a Typical Meteorological Year (TMY), which specify the instantaneous values of solar Direct Normal Irradiance (DNI) and other relevant meteorological variables at high frequency (i.e., time intervals of 10-minutes or less), in order to obtain an estimate of the annual average energy production of the CSTP plant. This annual energy production of the plant is then used, together with an estimate of the investment cost of the plant and its yearly operating and maintenance costs to compute the Internal Rate of Return, the Levelized Cost of Energy, and other economic indicators that are of interest for the economic feasibility analysis.

Strictly speaking, the construction of a TMY requires of the order of 30 years of high-frequency measured DNI and other relevant meteorological data [START_REF] Wilcox | Users manual for TMY3 data sets[END_REF]. Since such long term high frequency series of measured DNI and other meteorological data are seldom available for most locations on Earth, satellite-derived series and Numerical Weather Prediction Models (NWPM) are usually employed instead. These methods are capable of estimating the long-term average of the annual and monthly Direct Normal Insolation and of other relevant meteorological variables; however they do not provide the high frequency instantaneous value data required to simulate the energy behaviour of the CSTP plant.

To generate a high-frequency year of meteorological data, consistent with the long-term annual and monthly estimates, several approaches have been proposed in the literature. These approaches are, at best, useful to generate one high-frequency pseudo-TMY, which could then be used in conjunction with the energy model of the plant to estimate the annual long-term average electricity energy production. For economic feasibility studies and financial projections of debt servicing the TMY approach has been questioned as an appropriate methodology on the basis that it intentionally rules out the possibility of "extreme" years, and, therefore, consistently underestimates the risk of "bad meteorological years" and the effect of the associated poor annual electricity production [START_REF] Bender | The road to bankability: improving assessments for more accurate financial planing[END_REF].

In recent years, CENER, Sandia National Labs, and other research organizations are advancing the idea that more sophisticated approaches are needed to estimate the economic feasibility of a commercial CSTP plant project, which address both the variability of the solar resource and of the cost of some key investment and operation and maintenance elements of the plant as well as the uncertainty associated with the energy model estimates and cost data. In particular, there is an increasing interest in adopting probabilistic approaches to model variability and uncertainties in both electricity production and system cost to achieve sound estimates of the economic feasibility of commercial Concentrating Solar Thermal Power (CSTP) plants [START_REF] Ho | [END_REF]4].

A key element of such more sophisticated approaches to the feasibility analysis of a CSTP plant is the capacity of generating no one high-frequency TMY year, but a large series of high-frequency years, which are consistent with the estimated variability of the monthly and annual values of solar radiation and of other relevant meteorological variables. An efficient and robust general scheme for the generation of such large series of high-frequency synthetic TMYs does not exist yet. Many complex issues have to be addressed to develop such scheme. This article addresses one of those issues: the generation of synthetic typical high-frequency solar DNI years from typical daily solar DNI years and one-year of high-frequency solar DNI measurements.

Current methodologies for the generation of high-frequency direct solar radiation years

Modeling system performance and economics of solar thermal power plants has been carried out traditionally by means of deterministic analyses. Most CSTP energy simulation tools use a Typical Meteorological Year (TMY) as one of their main input to assess to assess the annual energy production of a CSTP plant [START_REF] Wilcox | Users manual for TMY3 data sets[END_REF][START_REF] Hall | Generation of Typical Meteorological Years for 26 SOLMET Stations[END_REF]. In broad terms, a TMY can be defined as a series of hourly values of the relevant meteorological variables, measured or estimated, at the plant location. Usually, these hourly values are considered "specific (point) values" rather than distributions of values that reflect the inherent uncertainty in many of the CSTP plant features and processes. As a result, the confidence and uncertainty associated with the results are unknown [START_REF] Ho | Tools for Probabilistic Modeling of Concentrating Solar Power Plants[END_REF].

Current TMY approach to solar resource assessment is based on a two-step procedure: (1) an estimation of the long term annual and monthly direct solar insolation of accumulated values, and (2) the selection of concrete days of measured direct solar irradiance data to reproduce the long term annual and monthly solar insolation accumulated behavior. For estimating the long term values of solar radiation, two main methodologies are used: the weighted mean of different databases [START_REF] Pagola | [END_REF][START_REF] Meyer | Combining solar irradiance measurements and various satellitederived products to a site-specific best estimate[END_REF], and the use of long series of solar irradiance insolation data estimated by satellite, or from Numerical Weather Prediction Models.

The needed measured data to build the high-frequency TMY is obtained from the measurement campaign carried out at the plant location. For commercial CSTP plants, the measurement campaign consist usually of one year of solar DNI and other relevant meteorological measurements taken on site at hourly or lower time intervals. Obviously, in almost all cases, the monthly and annual direct solar insolation values of the measured year do not fully agree with the corresponding long-term direct solar insolation estimates. Because of that, the measured year cannot be used as the high-frequency TMY for the site. But, the TMY for the site can be constructed from the measured year by replacing days of the measured years with other days of the same year in order to change the annual and monthly direct solar insolation of the year so that they agree with the corresponding long-term estimates.

Obviously, because the duration of the days varies with the date, the available options to replace a day by another are rather limited: Only days which are close to each other in the year, i.e. days which have relatively similar daylight duration can be interchanged. This limitation precludes the use of this approach to generate long series of direct solar radiation year which are compatible with the long-term estimates of the annual and monthly direct solar insolation values and their statistics. This is the main reason this methodology is used to generate just a representative TMY for the location under consideration.

New methodology based on the nondimensionalization of the DNI daily solar radiation curve

To overcome the limitation of the traditional approach to generate high-frequency direct solar radiation years using the solar radiation data obtained in the one-year measurement campaign, a new strategy is proposed in this article. This strategy is centered on the use of nondimensionalized high-frequency measured daily DNI curves as the basic building element of high-frequency DNI months and years compatible with the long term solar radiation behavior expected at a given site. The daily DNI curve is the representation of DNI data against Universal Time (UT). To fully nondimensionalize this curve it is necessary to non-dimensional time as well as DNI data. This can be done by defining characteristic values of time and DNI and divide the elements of each duple (time, DNI) by the corresponding characteristic values.

For a given daily curve of high-frequency DNI data, the nondimensionalization of time is achieved by counting time from sunrise and dividing the elapsed time since sunrise by the total time elapsed between sunrise and sunset for that given day. Likewise, the nondimensionalization of a DNI datum is achieved by dividing this datum by the corresponding value of the clear-day solar DNI envelope. As shown in Fig. 1, this nondimensionalization scheme transform every daily high-frequency DNI curve into a dimensionless curve where the dimensionless time scale goes from 0 to 1 and the dimensionless DNI scale goes also from 0 to 1. The clear-day solar DNI envelope is calculated by appropriately adjusting the two parameters (E , β) of the clearday ASHRAE exponential decrease model [START_REF]ASHRAE handbook of Fundamentals[END_REF] so that the curve defined by the model is the tightest possible upper boundary of the cloudless measured DNI values for the given day. The ASHRAE exponential decrease model is defined by the following expression:

( )         - ⋅ = φ β sin exp 0 E DNI (1)
Where: DNI is the solar direct normal irradiance, φ the solar elevation, E 0 the apparent extra-terrestrial irradiance, and β the overall extinction parameter.

To adjust the parameters (E 0 , β) in order to obtain the clear-day envelope of every measured high-frequency daily DNI curve a slightly modified version of the algorithm proposed for that purpose by Gómez-Camancho and Blanco [START_REF] Camacho | [END_REF] is used. The proposed full nondimensionalization of the daily solar DNI curve has several interesting properties.

For the purpose at hand, the following are of special interest:

• All daily solar DNI curves are mapped into the unit square.

• Ideal clear-days DNI curves (those which follow the clear-day DNI envelope) are mapped into upper horizontal border of the unit square. • The area under the mapped dimensionless DIN curve in the unit square is a number between 0 and 1, which equals the ratio between the actual dialy direct solar insolation and the maximum daily direct solar insolation corresponding to the ideal clear-day DNI curve. Since once they are transformed, all dimensionless daily DNI curves in a year have the same horizontal and vertical scales, any day may be replaced by any other day. Regarding time, there are 365 distinct different possibilities, since the duration of each day along the year is fixed for a given location. However, regarding the selection of clear-day envelopes the possibilities are countless, since for every day of the year the values of the tuple (E 0 , β) are not deterministically fixed. This makes it possible to use just a year of high-frequency DNI data to generate a very large number of high-frequency DNI years which are consistent with the long-term monthly and annual DNI estimates and their associated statistics, and which capture the high-frequency behaviour of the DNI at the specific site.

As stated in the introduction, an important step towards defining an efficient and robust general scheme for the generation of large series of high-frequency synthetic TMY is the generation of synthetic typical high-frequency solar DNI years from series of typical daily solar DNI years and one-year of high-frequency solar DNI measurements.

In this article we are not addressing the issue of how to generate the series of typical daily solar DNI years (this a research topic in itself), but we are proposing a methodology that will allow the transformation of those series of typical daily solar DNI years into series of high-frequency solar DNI years (10-minutes or less), which can be used to feed CSTP energy models in order to get an estimate of the long-term average annual energy production of the CSTP plant at a given site and its associated probability distribution.

This methodology consist in selecting, for each one of the 365 daily direct solar radiation insolation values that compose a given typical daily solar DNI year, the nondimensionalized high-frequency daily solar DNI curve which once transformed back into a dimensional high-frequency daily curve has a direct solar daily insolation closest to the solar radiation insolation value of the given day.

Testing the validity of the new proposed methodology for generating high-frequency typical DNI years

To test the validity of the new proposed methodology for generating high-frequency typical DNI years from series of typical daily solar DNI years and one-year of high-frequency solar DNI measurements, the following steps were carried out:

1. Several typical daily years of solar DNI for a given location were obtained, 2. One year of high-frequency solar DNI measurements for that same location was also obtained.

3. The methodology presented in the previous section, based on the use of nondimensionalized highfrequency daily curves, was used to generate the corresponding synthetic high-frequency solar DNI years. 4. A statistical test was carried out to proof that the high-frequency solar DNI years generated are indeed typical of the given location, i.e., that these synthetic years cannot be statistically distinguished from real high-frequency years measured at the given location. We used solar radiation measurements from the Baseline Surface Radiation Network (BSRN) station of Carpentras (France). The BSRN is a project of the World Climate Research Program (WCRP) that aims to measure radiative fluxes at the Earth surface with the highest possible accuracy using meteorological stations with wellcalibrated state-of-the-art instrumentation at selected sites in the major climate zones [START_REF] Ohmura | [END_REF]. At the Carpentras BSRN station, as in most BSRN stations around the world, DNI is measured with a Pyrheliometer, Kipp & Zonen, CH1, and it is sampled at 1 Hz with a one-minute averaging time. BSRN data undergo rigorous quality checks [START_REF] Ohmura | [END_REF]10], to assure high accuracy as well as homogeneity in the data.

The measurement periods used to generate the series of typical daily years (step 1) and to test the validity of the methodology presented in the previous section (step 4) correspond to fourteen consecutive years, from 1998 to 2011, which include a wide range of conditions representative of the location, including composition of cloud types, the occurrence of differential aerosol loading and ranges of atmospheric humidity, and different seasonal conditions among others. The year used as one year of high-frequency solar DNI measurements for the location (step 2) was 1997.

The statistical metric used to check that the high-frequency solar DNI years generated are indeed typical of the given location (step 4) is the KSI (Kolmogorov-Smirnov test Integral) index, which is the integral of the differences between the Cumulative Distribution Functions (CDF) of two data sets [13]. The KSI is therefore calculated as follows:

max min x n x KSI D dx = ∫ (2)
In practical terms, as D n is a discrete variable and the number of integration intervals is identical in all cases, trapezoidal integration is used over the whole range of the independent variable x. A relative value of KSI is obtained by normalizing the critical area, a critical. min (%) 100

xman n x critical D dx KSI a = ∫ (3)
where a critical is calculated as:

( ) max min critical c a V x x = ⋅ - (4)
And V c can be determined from the number of values N in the sample:

35 , 63 . 1 ≥ = N N V c (5)
The minimum value of the KSI(%) index is zero, which means that the CDFs of the two sets compared are equal. Since KSI(%) is below 100% when frequency distributions are close to each other [START_REF] Pagola | [END_REF], for our purposes any value of KSI(%) below 100 is considered to be a proof that the synthetic years cannot be statistically distinguished from the real high-frequency years measured at the location.

Results

Fig. 2 shows the frequency histograms of the fifteen years of high-frequency (1-minute) solar DNI data measured at the Carpentras BSRN station used to validate the methodology for generating high-frequency typical DNI years presented in this article. It is clear from the figure that the all histograms exhibit a very similar pattern, having their peaks and valleys located at approximately the same range of DNI values. In the figure the year 1997, which was used in the validation process as the one year of high-frequency solar DNI measurements used to extract the nondimensionalized solar DNI curves is plotted in red in Fig. 2. As a result of the generation method proposed, we obtain 1-min DNI series reproducing daily DNI values imposed as input. The 1-min DNI generated series does not intend to be coincident with the measured ones, but account for its variability and distribution. This is clearly shown in Fig. 3 (a), where 3 days of measured and generated DNI values are represented. The first day series are very similar, with the exception of a passing cloud. In the second day shown in Fig. 3 (a) both series follow a similar tendency, showing also similar maximum values and variability, although they are less coincident than the ones of the first day. In the third day shown in Fig. 3 (a), on the other hand, it can be appreciated a contrary tendency between both datasets: while measured series are near zero approximately until the solar noon, the generated series follow the opposite pattern, decaying at zero after the solar noon. Notwithstanding, both series show similar values and present the similar variability, which is the goal the new methodology aims to achieve. The similitude of both datasets distributions, 1-min measured and generated DNI, can be appreciated in frequency histograms of the whole period compared (from 1998 to 2011), Fig. 3 (b). Fig. 4 shows graphically the KSI% values for measured versus generated 1-min DNI series for each of the fourteen years analyzed, as well as for the whole fourteen years period considered as a whole. As was stated in the previous section, as long as the KSI% index is below 100, it can be stated that the series being compared are statistically identical. As the figure shows, in all cases, the KSI% index is well below that threshold, which proofs that the high-frequency (1-minute) synthetic years produced using the proposed methodology cannot be statistically distinguished from real high-frequency years measured at the location.

Conclusions

A new methodology for transforming series of typical daily solar direct normal insolation values into series of typical high-frequency (10-minutes or less) solar DNI years has been developed and validated. This constitutes a relevant step in the quest for the definition of an efficient and robust general scheme for the generation of large series of high-frequency synthetic TMYs which can be entered into CSTP plant energy production models to compute not only the average long-term annual energy production of a CSTP plants but their associated variability or probability distribution, which in turn can be used in probabilistically-oriented economic feasibility studies, which will advance the state of the art of the economic assessment methodologies used to analyse commercial CSTP projects.

The developed methodology to generate high high-frequency (10-minutes or less) solar DNI years from series of typical daily solar DNI years and one-year of high-frequency solar DNI measurements is based on the full nondimensionalization of the daily solar DNI curve.

The validation of this methodology against high-quality and high-frequency (1-minute) experimental solar DNI data has provided KSI% values for DNI series similar to those found in the literature for state-of-the-art models, with one major difference regarding the temporal resolution: to the best of our knowledge, this is the first time, that 1-min solar DNI synthetic years have been synthetically produced which cannot be statistically distinguished from measured high-frequency years.

Other approaches presented in the literature shows that the KSI (%) between measured and modeled solar irradiance datasets are much higher for DNI than for GHI, indicating a worst fit in DNI [START_REF] Dumortier | Description of solar resource products Summary of benchmarking results Examples of use[END_REF]. Even in modeled DNI datasets with low bias, poor KSI values have been found, indicating that the match in the distribution functions needs further improvement [START_REF] Hoyer-Klick | MESoR -Management and exploitation of solar resource knowledge[END_REF]. If one considers that this previous work has been mostly carried out on hourly series and not 1-minute series and that it is reasonable to assume that the goodness of distribution fitting obtained in those studies will decrease at higher temporal frequencies, the relevance of the results presented in this paper are highlighted.

Future works must be directed to the improvement of the enveloping DNI curve model, for example based on enveloping measured curves. Moreover, for the application of the proposed methodology for increasing the DNI frequency, generation of a number of synthetic years must be carried out at daily DNI scale. There are existing methodologies to generate long term GHI series [START_REF] Bohlen | Time series analysis of the long term monthly horizontal solar radiation[END_REF][START_REF] Tiba | Analysis of monthly time series of solar radiation and sunshine hours in tropical cliamtes[END_REF], and also synthetic procedures to increase frequency of montly GHI series [18,19]. The combination of these synthetic generation procedures with existing models to obtain DNI values from GHI must be investigated in order to achieve a realistic daily DNI series to which the methodology presented and validated in this paper could be applied. Furthermore, the generation of high-frequency solar DNI years should be followed by the generation of the corresponding values of the other relevant meteorological variables needed to compose a typical TMY year. This, perhaps, can be approached using schemes similar to the ones proposed by Meteonorm [START_REF]Handbook II: Theory[END_REF].

Fig. 1 .

 1 Fig. 1. (a) Instantaneous DNI daily curve and its clear-sky envelope; (b) nondimensionalized DNI curve.

Fig. 2 .

 2 Fig. 2. Frequency histograms of 1-min DNI series measured at carpentras BSRN station (1997-2011). Frequency histogram corresponding to 1997 is plotted in red.

Fig. 3 .

 3 Fig. 3. (a) 1-min DNI series measured and generated; (b) frequency histogram of 1-min DNI measured and generated for 1998-2011.

Fig. 4 .

 4 Fig. 4. KSI% values for measured and generated 1-min DNI series

Acknowledgements

The authors would like to thank the manager and staff of Carpentras BSRN station for their efforts in establishing and maintaining that station.