Fausto Galetto
email: fausto.galetto@gmail.com

Independent Researcher

RIEMANN'S HYPOTHESIS and THE MERTENS FUNCTION for THE PROOF OF RH I SEE IT, BUT I CAN'T BELIEVE IT

Keywords: RH (the Riemann hypothesis), (s) (the Riemann zeta function), integral equations, the Mertens function

It is well known that "The Riemann hypothesis is equivalent to the statement that for every  > 0 the function M(x)/x 1/2+ approaches zero as x ", where M is the Mertens function. Starting from the definitions of M(x) and (s) we derive an integral equation about M(x): from that we derive that |M(x)|<x 1/2+ ; the proof of the Riemann hypothesis follows.

Introduction

Riemann devised the Riemann zeta function: (z), where z is a complex number z=x+iy and i is the "positive" imaginary unit such that i 2 =-1. The Riemann Hypothesis states all the nontrivial zeros are the complex numbers z=1/2 + iy, with suitable values of y [the line z=1/2 + iy in the z plane is named Critical Line]. All the "known" zeros computed up to now [up to 2018], more than 10 12 zeros have been computed, all on the Critical Line. Many people tried to prove RH and making it a theorem; the author himself, using the theory of Hilbert Spaces, tried it [START_REF] Galetto | Riemann Hypothesis proved[END_REF] after studying the book of Titchmarsh [START_REF] Titchmarsh | The Theory of the Riemann Zeta-Function[END_REF]; there one finds that the Riemann hypothesis is equivalent to the statement that for every  > 0 the Mertens function

M(x)=O(x 1/2+ ).

The function of Mertens is defined by the sum where is the Mobius function [START_REF] Titchmarsh | The Theory of the Riemann Zeta-Function[END_REF], k and n being integer numbers. For x, a real number, the Mertens function is defined by the sum [START_REF] Titchmarsh | The Theory of the Riemann Zeta-Function[END_REF] Mertens conjectured that |M(n| < ; if this were true then Riemann's hypothesis would be true. The Mertens conjecture was disproved by A. Odlyzko and H. te Riele [START_REF] Odlyzko | Disproof of the Mertens conjecture[END_REF] in 1984.

Connections between M(x) and (s)

In this section we use the standard symbols for the Riemann zeta function (s); s is the complex variable, s=+it; x is a real variable. From the definition (1) one can prove that, for 1  (2) IF RH is true formula (2) holds also for ½<.

On the other hand one can prove that [START_REF] Odlyzko | Disproof of the Mertens conjecture[END_REF] From ( 3) and ( 2), without assuming that RH is true, we derive the integral equation ( 4)

We will prove it after the introduction to the Theory of Distributions (next section).

Theory of Distributions Basics

Distribution theory [START_REF] Amerio | Analisi Matematica[END_REF] can be thought of as the completion of differential calculus, because in the theory any distribution can be differentiated any times we want (it is one of the two great revolutions in mathematical analysis in the 20th century).

It is known what a function f(x), with x real, is. The same for f(s), with s complex. We can generalise the definition of a function f(x) by computing the integral (5)

The functions (x) are named test functions and are vanishing outside a bounded subset of the space R and are such that all partial derivatives of all orders of (x) are continuous. [for the case "monodimensional"!] The integral is to be intended in terms of Lebesgue integration theory. Any function gives rise to a distribution by setting , at least if the integral can be defined. This is certainly true if f is continuous, but actually more general functions will work. The so called "Dirac -function" actually is defined only as a distribution by <,>=(0). It cannot differentiated as a function, but it can differentiated as a distribution by <,>=-(0). In the real space R we can define the Heaviside function H(x) [also named step function] which is H(x) cannot be differentiated at x=0. On the contrary, the distribution <H,> can be differentiated and we have <H,>=<,>: we indicate it with the symbol (x). There is a second type of step function, actually distribution, named as well Heaviside function H(x) which is The derivative of this distribution is <H,>=<,>=1/2(x) For distributions we can define all the operations (and others) we can define for functions: e.g. Fourier transforms, Laplace transforms, integration, derivation, .... In the next sections we will consider the Mertens function, in the sense of the Distribution Theory, in order to take advantage of the Theory.

The integral equation for M(x)

Inserting ( 2) into (3) we get [without assuming that RH is true] (6) Some manipulations of ( 6) give (7) and finally (8)

To prove (8) we need to consider first the integral, where x and y are real variables, and s the complex variable, s=+it Integrating, when x<y, one gets 0; integrating, when x=y, one gets ½; integrating, when x>y, one gets 1. This is the second type of step distribution, H(u-1) with u=x/y; when x=y H(u-1)=H(0).

The complex integral in (7) then is related to the distribution which is 0 for any u1.

That means that the integral about the "integration variable y" must be 0 above x. Therefore we get (8). M(x) is the distribution related to the Mertens function. This is the point given in the title: I SEE IT, BUT I CAN'T BELIEVE IT

The integral inequality for the absolute value |M(x)|

In order to prove the Riemann Hypothesis RH, we must find the distribution |M(x)| related to the integral equation ( 8).

For the absolute value we have, being >0,

Let's put g(t)=|M(t)|, considered as a distribution, and find its Laplace transform (p is the complex variable)

Then Therefore

If we seek a solution with  >0

we get that is (10)

By choosing =3/2 we have -1<1/2+, that is t 1/2+ as solution of the integral inequality (9), and eventually (11)

Conclusion

Since we know that "The Riemann hypothesis is equivalent to the statement that for every  > 0 the function M(x)=O(x 1/2+ )", where M is the Mertens function and we found that |M(x)|<x 1/2+ is the solution of the integral inequality about |M(x)| [derived from integral equation about M(x), found from M(x) and (s)], the proof of the Riemann hypothesis is proved. (see also [START_REF] Galetto | Riemann Hypothesis proved[END_REF])


Alexandre Denis
email: alexandre.denis@loria.fr

Guillaume Pitel

Matthieu Quignard

Patrick Blackburn

Incorporating Asymmetric and Asynchronous Evidence of Understanding in a Grounding Model

, that removes these simplifications; we do so by distinguishing the phase of interpretation from the phase of evidence extraction and introducing the notion of floating contributions.

Introduction

A dialogue is a process that presupposes the collaboration of both participants. Each speaker in turn assumes that the other will show evidence of understanding or misunderstanding of her utterance, and evidence indicating its relevance to the previous utterance. This mutual assumption is the basis of the grounding process [START_REF] Clark | Referring as collaborative process[END_REF][START_REF] Clark | Contributing to discourse[END_REF], the process by which speakers try to reach mutual understanding. Successful grounding does not guarantee mutual understanding though: it can happen that the grounding evidence leads two speakers to believe that they have achieved perfect understanding, whereas in reality they have understood two completely different things [START_REF] Cherubini | Grounding is not shared understanding: Distinguishing grounding at an utterance and knowledge level[END_REF]. But although this shows that successful grounding is not a sufficient condition for achieving mutual understanding, it does seem to be a necessary one.

Different models of the grounding process define when (and, sometimes, how) an utterance is added to the common ground (a representation of what is believed to have been mutually accepted). In the Contribution Model [START_REF] Clark | Contributing to discourse[END_REF] the grounding process results in a recursively structured directed acyclic graph representation of the dialogue grounding structure, the basic unit of which is the contribution. Contributions are twofold units consisting of: (1) an utterance called the presentation (or Pr) and (2) an acceptance linked to a sequence of contributions or a single utterance. The acceptance (or Ac) contains the negotiation of the understanding of the presentation in order to reach the grounding criterion. The grounding criterion is a threshold defined by [START_REF] Clark | Referring as collaborative process[END_REF] to represent the level of understanding required by the contributor; we shall use the expression grounding status to mean the current state of the believed mutual understanding of an utterance. When the grounding criterion holds for a contribution, that is, when its status is grounded, both speakers consider it closed and can choose whether or not to integrate its semantic content as a mutual belief. The grounding status is established via simple evidence of understanding and relevance. The Contribution Model was the pioneering approach to the modeling of grounding and its insights influenced the subsequent development of formal models intended for computational applications.

Probably the best known of these subsequent models is the Grounding Acts Model [START_REF] Traum | A "speech acts" approach to grounding in conversation[END_REF][START_REF] Traum | A Computational Theory of Grounding in Natural Language Conversation[END_REF][START_REF] Traum | Computational models of grounding in collaborative systems[END_REF]. This model is based on the notion of grounding acts, low level communicative acts whose goal is to ground content at the utterance level. The basic unit of analysis provided by the Grounding Acts Model is a non-recursive sequence of utterances called a Discourse Unit (DU). The grounding pro- cess is modelled by an update of the state of a Discourse Unit by a grounding act; this makes the approach particularly suitable for integration into information-state based models of dialogue (such as [START_REF] Matheson | Modelling grounding and discourse obligations using update rules[END_REF]); transitions between states are modelled in [START_REF] Traum | A "speech acts" approach to grounding in conversation[END_REF] and many subsequent papers using finite state automata triggered by the various grounding acts. For example, a RequestRepair act by participant A would send a Discourse Unit into a state where a Repair by participant B and a subsequent Acknowledge by A would be needed to ground it.

The Grounding Acts Model makes the assumption that the grounding level can be distinguished from the intentional level. However, as was noted by [START_REF] Stirling | The nature of common ground units: an empirical analysis using map task dialogues[END_REF], it is often not easy to delineate DUs, which makes it difficult to clearly distinguish the grounding level from deeper levels of understanding that emerge via complex exchanges. Hence, as our primary motivation is to explore ways of uniformly integrating grounding at the utterance level with complex negotiations of understanding, we have not taken the Grounding Acts Model as our point of departure.

Instead we have chosen to develop the Exchange Model approach presented in [START_REF] Cahn | A computational architecture for the progression of mutual understanding in dialog[END_REF][START_REF] Cahn | A psychological model of grounding and repair in dialog[END_REF], which are more directly based on the original Contribution Model. The central innovation provided by Exchange Models is a level of exchange that is higher than the level of contributions (this central notion is very much in the spirit of the implicit adjacency pairs used in [START_REF] Clark | Contributing to discourse[END_REF]). Like work based on the Grounding Acts Model, these Exchange Models have a formal definition and provide online models of grounding. What makes them particularly useful for our purposes, however, is that they follow the Contribution Model in producing graph-like representations of the dialogue grounding structure; in our view, this makes them particularly well-suited for modeling more complex negotiations of understanding.

Nonetheless, different as these three types of model are, they share a common deficiency: they cannot deal with wrongly recognized or unrecognized grounding acts or evidence of understanding. In the original Exchange Models, the evidence is always assumed to be symmetric and synchronous-that is correctly and immediately understood by the hearer. In the Grounding Acts Model, matters are a little more subtle. There an unrecognized grounding act would initiate a new Discourse Unit, and hence the model might be said to handle asymmetric grounding. Nonetheless, it is not obvious how, once grounded, this Discourse Unit should be reintegrated, nor how the effects of the newly understood grounding act could be taken into account with respect to previous Discourse Units. It may be the case that the Grounding Acts Model and related information states approaches (such as [START_REF] Matheson | Modelling grounding and discourse obligations using update rules[END_REF][START_REF] Larsson | Information state and dialogue management in the trindi dialogue move engine toolkit[END_REF]) could be extended to handle this kind of reintegration, perhaps by providing additional update rules. But we have found that the (recursive) graph-like representations used by Exchange Models provides a particularly perspicuous setting for a preliminary explorations of the issues involved.

Accordingly, we shall proceed as follows. In Section 2 we discuss existing Exchange Models and their shortcomings in more detail. In Section 3, we present an augmented Exchange Model, inspired by [START_REF] Cahn | A computational architecture for the progression of mutual understanding in dialog[END_REF], which repairs these deficiencies. In Section 4, with the help of an example, we show in detail how the model works. Section 5 concludes.

A closer look at Exchange Models

The Exchange Models proposed in [START_REF] Cahn | A computational architecture for the progression of mutual understanding in dialog[END_REF][START_REF] Cahn | A psychological model of grounding and repair in dialog[END_REF] are intended to formalize the Contribution Model to enable it to be embedded in dialogue systems. Like the Contribution Model, they are based on (recursive) graphlike structures, but they add a level of exchange above the Contribution/Presentation/Acceptance levels present in the Contribution Model. An exchange is a pair of contributions defined relative to a task: the first contribution proposes a task while the second contribution executes the task. The grounding process itself is modeled by a decision table based on two features: (1) the evidence of understanding manifested in an utterance and (2) the role of the current utterance in an exchange (i.e. in a dialogue task). In these models, the grounded dialogue structure is represented from each speakers' individual point of view, and "all contribution graphs are private models, and can represent the perspective of only one agent" [START_REF] Cahn | A psychological model of grounding and repair in dialog[END_REF].

The model defined in [START_REF] Cahn | A computational architecture for the progression of mutual understanding in dialog[END_REF]) (henceforth EM92) uses three categories of evidence: UN-DERSTOODRELEVANT, NOTUNDERSTOOD and u 1 : Where does Dan work ? s 2 : In the natural language group u 3 : No, I meant his office Figure 1: UNDERSTOODNOTRELEVANT evidence example (dialogue 6.3.3) UNDERSTOODNOTRELEVANT. However it considers the integration of the utterances of the speaker based on the evidence she intended to produce, and does not take into account how the hearer actually interprets this evidence. This model can thus only render the perspective of the speaker. The model defined in [START_REF] Cahn | A psychological model of grounding and repair in dialog[END_REF] (henceforth EM99) is limited to the system's point of view, unlike EM92 which is participantagnostic. It is based on two categories of evidence: ACCEPTABLE and NOTACCEPTABLE for the user. This is not enough to cover all cases; for example, it is not possible for the system to warn that its utterance was misunderstood (though the user can do this). The main improvement of EM99 over EM92 is that it characterizes grounding from the hearer's point of view (in this case the system) and integrates the utterances of the user as well as those of the system.

So what is wrong with these models? An example should make matters clear. Consider Figure 1, which shows an example dialogue from [START_REF] Cahn | A computational architecture for the progression of mutual understanding in dialog[END_REF]. When S utters s 2 , she believes her utterance is a relevant answer to u 1 . That is, she appends s 2 as the second contribution of the exchange initiated by u 1 (see Figure 2).1 When S receives u 3 and extracts the UNDER-STOODNOTRELEVANT evidence it conveys, she has to restructure her view of the dialogue to take into account the fact that s 2 is not relevant with respect to u 1 . That is, she must move s 2 into a new exchange in the acceptance phase of u 1 and append u 3 as the second contribution of this exchange (see Figure 3). But now take the point of view of U on the situation: she interprets s 2 as not relevant with respect to u 1 as soon as she receives it, and thus does not have to revise her model. Instead she integrates s 2 directly into the acceptance phase of her own utterance u 1 . To properly model such situations from both points of view, we need to consider not merely the intended evidence of understanding, but also the actual interpretation of evidence.

However EM92 cannot handle this behavior because it does not consider the interpretation of the hearer: it is focused on how integrate an utterance according to the evidence of understanding the speaker intended it to convey, and not on how the hearer actually interpreted the utterance. The model EM99, on the other hand, partially takes into account the hearer's interpretation in one case, namely when the user gives ACCEPTABLE evidence and does not propose a task. For example, consider the exchange shown in Figure 4. EM99 handles such examples well. If the system believes it does not understand u 2 , its next utterance would initiate a new exchange in the acceptance phase of u 2 . On the other hand, if it believes it understands u 2 , its next utterance would initiate a new exchange at the dialogue level. This is a step in the right direction, but it does not go far enough: the interpretation of the hearer needs to be taken into account in all cases, for both speakers, whatever the evidence conveyed by the interpreted utterance. Only if the hearer understands can she extract the evidence of understanding. If the interpretation of the hearer is not taken into account, it is impossible to consider the non-understanding (or the misunderstanding) of the evidence of understanding. As a consequence, models that make this simplification can only deal with symmetric and synchronous evi-u 1 : Where's Dan ? s 2 : Dan Smith or Dan Jones ? (+ noise) u 3 : Uh, what did you say ?

Figure 5: Not understanding a NOTUNDERSTOOD evidence of understanding dence of understanding. Symmetric means that the presented evidence of understanding is always understood by the hearer as expected by the speaker. Synchronous means that it is also understood as soon as the utterance is emitted. Neither EM92 nor EM99 can handle the fact that the acceptance function of an utterance is not always played at the same moment for the two speakers. In the example in Figure 5, the user does not know that the utterance s 2 manifests NOTUNDERSTOOD evidence because she herself has not understood s 2 sufficiently to extract the evidence of understanding.

An extended Exchange Model

The aim of our work is to specify a grounding model that handles asymmetric and asynchronous evidence of understanding. We do so in a way that retains the advantages of both EM92 and EM99, and follow these models in using a small collection of sharp understanding categories. We found EM92, which does not distinguish the user from the system and has better categories of understanding, to be a better starting point, and thus have reworked the key insights of EM99 in the setting of EM92. We introduce asymmetry and asynchronicity into this model by distinguishing two steps in the interpretative process: the understanding of an utterance, and the extraction of the evidence of understanding it conveys. Our model is defined from the point of view of the hearer (who we will refer to by self ). It considers how self understands an utterance before taking into account the evidence of understanding it shows. If the utterance is UN-DERSTOODRELEVANT, the evidence it shows is extracted and is integrated as in EM99. If the utterance is UNDERSTOODNOTRELEVANT, it is integrated as initiating a new exchange under the acceptance phase of the previous contribution without considering the evidence of understanding it shows. If the utterance is NOTUNDERSTOOD, the contribution is introduced as floating, waiting for later integration in the graph.2 Its integration into the main dialogue structure is possible only when its acceptance phase shows what the evidence of understanding was. Then the utterance has to be reinterpreted, taking into account the newly understood evidence. However the grounding status of floating items could remain pending and never be solved, for example if the acceptance phase is abandoned.

Do floating contributions have an analog in the Grounding Acts Model? We don't believe so. One could try comparing the collection of ungrounded states of a Discourse Unit to the acceptance phase of a contribution. That is, the open state of the acceptance phase of a contribution in an Exchange Model could be regarded as the analog of the ungrounded states of a Discourse Unit in the Grounding Acts Model. But the notion of a floating contribution is stronger than the notion of ungrounded states: when a contribution is floating, it means not only that it is not grounded yet, but also that the evidence it manifests is not known either. Furthermore, floating status isn't correctly captured by the ungroundable state used in the Grounding Acts Model either. The ungroundable state is a terminal state, reached by canceling the grounding process. However the floating status that some contributions acquire in our approach is intended to be temporary-if the evidence conveyed by the contribution comes to be understood, its floating status is cancelled and the contribution is integrated into the main dialogue structure.

So: how can we augment the Exchange Model to handle asymmetric and asynchronous evidence of understanding? The main additions we shall make are the following. First, in order to keep track of the floating contributions, we have to maintain another structure which contains the sequence of pending contributions. Second, we have to handle reinterpretation and specify how the newly acquired evidence of understanding is used to integrate a floating contribution. Third, an utterance can now give evidence of understanding concerning many previous utterances, because an utterance which closes an acceptance phase and gives evidence as such can now reveal how to interpret the accepted contribution too.

A formalization 3 is presented in Table 1 and Tastanding are possible though [START_REF] Brennan | Interaction and feedback in a spoken language system: a theoretical framework[END_REF]. 3 The tables are simplifi ed. We do not discuss here dialogue beginning and ending nor the necessity of having a three-fold context Si, Oj , S k to manage reinterpretation. The actual implementation also deals with evaluation utter-

Self interpretation of Oi

Integration of Oi

Integration of Si+1 UNDERSTOODRELEVANT w/r Si-1

integrate Oi according to the evidences of understanding it shows: if Oi shows evidences about another utterance Sj , call ble 2. It covers the aforementioned cases with delays in the integration of utterances. We use the following notation: O i stands for the utterance produced by the other speaker (O) at time i, and S i+1 stands for the utterance produced by self (S) at time i + 1. An utterance is said to initiate an exchange if it is the presentation of the first contribution of this exchange. An exchange is said to be open if its first contribution is set while its second contribution is not. The main dialogue structure is called D and the floating structure F . Finally, "integrate u i after u j " is a shorthand for:

• if u j initiated an exchange, append u i as the second contribution of this exchange;

• else append u i as the second contribution of the closest upper level open exchange, if there is one;

• else (all exchanges are closed), u i initiates a new exchange at the dialogue level.

• in all cases u i closes the acceptance phase of u j .

Reinterpretation of an utterance consists of calling the algorithm again with a new interpretation and new evidence of understanding. The only difference is that the contribution presented by this utterance does not have to be created because it already exists in the floating structure. If the utterance is eventually understood (relevant or not) it can be moved in the dialogue structure in accordance with its new interpetation, and the new evidence of understanding it shows. This evidence of understanding is consequently acquired asynchronously by the two participants.

The main simplifying assumptions made by our algorithm are the following: ances and abandons. For a complete description, please refer to http://www.loria.fr/˜denis

• We suppose a direct correlation between the result of an interpretation of O i and the evidence of understanding conveyed by S i+1 . How S interprets O i is manifested in the utterance she produces in turn S i+1 . That is, if an utterance is not understood or not relevant, one has to clarify the situation. This simplification is based on the collaborative dialogue hypothesis.

• In the version of the algorithm presented above, the evidence of understanding is either understood or not. That is, the asymmetry is binary and there cannot be any misunderstanding of the evidence of understanding. Such misunderstandings would be more complex to handle because of the increased divergence between the participants dialogue representation structures. But systems can be mistaken when extracting the evidence of understanding, and we think it will be necessary for dialogue systems to represent this.

• Contributions always alternate. The present algorithm does not actually manage several contributions in one speech turn because this would mean taking into account interleaved evidence of understanding. But, once again, we feel that this extension will be necessary to handle more realistic dialogues.

A tool illustrating our model has been implemented in Java. This takes as input a dialogue where each utterance is annotated by the evidence of understanding its speaker believes it to convey. The resulting output is the dialogue structure and the floating structure for both speakers at different steps. The tool was used to generate the diagrams used in this paper, and in particular, the diagrams in the example to which we now turn.

Detailed example

The example in Table 3 is a modification of the example 6.3.3 in [START_REF] Cahn | A computational architecture for the progression of mutual understanding in dialog[END_REF], in which the second utterance is not understood by the user (called A whereas the other participant, the system, is called B). This dialogue illustrates the asymmetry and asynchronicity of the learning of the evidence of understanding showed by b 2 . The left column presents the utterances and the evidence of understanding showed by them from their speaker's point of view. The two other columns present the dialogue structure according to each point of view.

The first utterance a 1 is believed UNDER-STOODRELEVANT 4 by B and is integrated normally as initiating an exchange at the dialogue level. The dialogue viewed by A is the same.

The second utterance b 2 shows a divergence. B believes that b 2 presents an UNDERSTOODREL-EVANT evidence of understanding and thus integrates it as the second contribution of the first exchange. However this evidence is not shared by A, who does not understand b 2 and therefore cannot integrate it. She just keeps the contribution floating, awaiting to be integrated when it is sufficiently understood (see F a in Table 3).

Utterance a 3 shows that b 2 was NOTUNDER-STOOD by A and that she requires clarification. Because a 3 is understood by B, the evidence of understanding it contains is used to integrate it as the initiator of a new exchange under the acceptance phase of b 2 contribution.

Utterance b 4 shows that a 3 was interpreted UN-DERSTOODRELEVANT by B. Therefore it is integrated by both speakers as the second contribution of the clarification exchange. However there is a new divergence when processing the utterance b 4 . For B, b 4 is only an answer to the clarification request. But with b 4 , A can now interpret b 2 . As b 2 is now understood by A, she can extract the evidence of understanding it showed, and act according to her own interpretation. In this case, because b 2 is UNDERSTOODNOTRELEVANT by A, she won't take into account the evidence of understanding showed by b 2 . The acquisition of the evidence of understanding showed by b 2 is asynchronous but not taken into account; see Table 1. The reinterpretation of b 2 , according to the UN-DERSTOODNOTRELEVANT rule, leads to the b 2 contribution being embedded as initiating a new exchange under the acceptance phase of a 1 .

The utterance a 5 is crucial for reaching the grounding criterion. It makes available two pieces of evidence of understanding: first it shows that b 4 is an UNDERSTOODRELEVANT reply to a 3 and second it shows that b 2 is UNDERSTOOD-NOTRELEVANT with respect to a 1 . Its effect is to revise the B view on the dialogue to create a new exchange in the acceptance phase of a 1 . Doing this means that the structures of the grounding model converge for both speakers; they now agree on the current view of dialogue.

The utterance b 6 is the final answer to the first question. It shows that a 5 was UNDER-STOODRELEVANT by B and UNDERSTOODREL-EVANT by A. It is integrated as a relevant reply to the first contribution of the upper level exchange.

Discussion and further work

This paper discusses the problems posed by asymmetric and asynchronous evidence of understanding, and gives a preliminary model of how such evidence could be handled. It does so by distinguishing the phase of interpretation from the phase of evidence extraction and introducing the notion of floating contributions into the Exchange Model. Such contributions cannot be immediately attached to the dialogue structure because the evidence of understanding they show is not known. When these contributions are accepted, they have to be reinterpreted in order to extract the evidence of understanding they manifest.

A side effect of our model is that it provides a novel solution to the recursive acceptance problem defined in [START_REF] Traum | A Computational Theory of Grounding in Natural Language Conversation[END_REF][START_REF] Traum | Computational models of grounding in collaborative systems[END_REF]: if an acceptance utterance needs to be accepted before it can play its acceptance function, then no contribution would ever be complete. To solve the problem, we make the assumption that a participant may form the belief that she has understood (or not) an utterance as soon as she receives it; she does not have to subordinate her belief to further acceptance (we believe that this assumption can be motivated by the ideas on timing in joint actions in Chapter 3 of [START_REF] Herbert | Using Language[END_REF]). The acceptance function of an utterance can be played, from the hearer's point of view, as soon as she understands the utterance. On the other hand, to check whether what she said successfully played its intended acceptance role, the speaker of the utterance has to wait for the hearer's response. However, as soon as the hearer responds, the appropriate acceptance function may be played. But when misunderstanding occurs, the acceptance role of an utterance is delayed up to the moment it is sufficiently understood to be integrated into the common ground.

The implemented model we have presented still suffers from a number of limitations; for example it does not deal with misunderstanding of the evidence of understanding. Planned future work will cover these more complex divergences in dialogue structure in addition to multi-contributions, that is, when several contributions by the same speaker in the same turn. We hope that this model and its implementation will be the first stage of a larger enterprise: specifying the grounding status of the contents of a contribution in terms of dialogue structure.

Decalog 2007 :

 2007 Proceedings of the 11th Workshop on the Semantics and Pragmatics of Dialogue, pages 33-40. Trento, Italy, 30 May -1 June 2007. Edited by Ron Artstein and Laure Vieu.

Figure 2 :

 2 Figure 2: Dialogue 6.3.3 after s 2

Figure 3 :

 3 Figure 3: Dialogue 6.3.3 after u 3

  Figure 4: ACCEPTABLE evidence example

Table 2 with

 2 Sj else call it with Si-1

	integrate Si+1 after Oi

Table 1 :

 1 Integration of both utterances (O i , S i+1 )

Table 2 :

 2 Integration of O i when it is understood and thought relevant

	Utterance and evidence of understanding it shows	Point of view of A	Point of view of B
	a1: Where does Dan work ?
	b2: In the natural language group
	UNDERSTOODRELEVANT(a1)
	a3: What did you say ?
	NOTUNDERSTOOD(b2)
	b4: I said: in the natural language group
	UNDERSTOODRELEVANT(a3)
	a5: No, I meant his offi ce
	UNDERSTOODRELEVANT(b4)
	UNDERSTOODNOTRELEVANT(b2)
	b6: Near post H33
	UNDERSTOODRELEVANT(a5)

Table 3 :

 3 Detailed example

In the paper we sometimes use the term utterance instead of contribution, or an utterance symbol to denote a contribution; in these case we always mean "the contribution presented by this utterance".

In this version, NOTUNDERSTOOD means that the evidence is not understood either. Multiple degrees of under-

The fi rst utterance is assumed relevant when it is understood

Acknowledgements

We would like to thank TALARIS members Luciana Benotti and Daniel Coulon for feedback and discussion during the writing of this paper.


Keywords: Eddy currents, finite element method, perturbation method, skin and proximity effects

Skin and proximity effects are calculated in both active and passive conductors via a subproblem finite element method based on a perturbation technique. A reference limit problem is first solved by considering either perfect conductors or magnetic materials via appropriate boundary conditions. Its solution gives then the source for the eddy current perturbation subproblems in each conductor with its actual conductivity or permeability and its own mesh. The proposed method accurately determines the current density distributions and ensuing losses in conductors of any shape in both frequency and time domains, which overcomes the limitations of the impedance boundary condition technique.

I. INTRODUCTION

precise consideration of the skin and proximity effects in active and passive conductors is important for an accurate calculation of their field distribution and the ensuing Joule losses. Calculating these effects with the classical finite element (FE) method usually presents difficulties. The mesh must be fine enough with respect to the skin depth in all the materials, which then leads to a large system of equations.

Impedance boundary conditions (IBCs) [START_REF] Krähenbühl | Thin layers in electrical engineering. Example of shell models in analyzing eddy-currents by boundary and finite element methods[END_REF] defined on the conductor boundaries are an alternative to avoid meshing the conductor interior. Such boundary conditions (BCs) are nevertheless generally based on analytical solutions and in practice only valid far from any geometrical discontinuities, e.g. edges and corners. They are also generally restricted to frequency domain and linear analyses.

In this contribution, a method is developed to overcome the limitations of IBCs, considering conductors of any shape not only in the frequency domain but also in the time domain. The magnetic vector potential FE magnetodynamic formulation is used. The developed method is based on the coupling of reference and perturbation solutions [START_REF] Badics | An effective 3-D finite element scheme for computing electromagnetic field distorsions due to defects in eddy-current nondestructive evaluation[END_REF]- [START_REF] Dular | Sub-domain finite element method for efficiently considering strong skin and proximity effects[END_REF], each of these being calculated in distinct meshes. A reference limit eddy current FE problem is first solved by considering either perfect conductive or magnetic properties, via appropriate BCs on the conductor boundaries. The solution of the limit problem gives the sources for FE perturbation subproblems in each conductor then considered with a finite conductivity or permeability. Each of these problems requires an appropriate Manuscript received June 24, 2007. This work was partly supported by the Belgian Science Policy (IAP P6/21) and the Belgian French Community (Research Concerted Action ARC 03/08-298).

P. Dular and R. V. Sabariego are with the University of Liège, Dept. of Electrical Engineering and Computer Science, B28, B-4000 Liège, Belgium (e-mail's: Patrick.Dular@ulg.ac.be, r.sabariego@ulg.ac.be). P. Dular is also with the Belgian National Fund for Scientific Research (F.N.R.S.). L. Krähenbühl is with the Université de Lyon, F-69003 Lyon, France; he is also with the Laboratoire Ampère, UMR CNRS 5005, École Centrale de Lyon, F-69134 Écully Cedex, France (e-mail: Laurent.Krahenbuhl@ec-lyon.fr). volume mesh of the associated conductor and its surrounding region. The developed technique is validated on an application example. Its main advantages versus the IBC technique are pointed out.

II. REFERENCE AND MODIFIED EDDY CURRENT PROBLEMS

A. Reference problem and its strong formulation

A given problem p consists in solving the magnetodynamic equations in a bounded domain Ω p , with boundary Γ p = Γ h,p ∪ Γ b,p = ∂Ω p (possibly at infinity), of the 2-D or 3-D Euclidean space. The eddy current conducting part of Ω p is denoted Ω c,p and the non-conducting one Ω c,p C , with

Ω p = Ω c,p ∪ Ω c,p C . Massive conductors belong to Ω c,p .
A problem, defined as problem p = 1, is first considered. Its equations and material relations in Ω 1 , and boundary conditions (BCs) and interface conditions (ICs), are

curl h 1 = j 1 , curl e 1 = -∂ t b 1 , div b 1 = 0 , (1a-b-c) b 1 = μ 1 h 1 , j 1 = σ 1 e 1 , (1d-e) n × h 1 | Γ h,1 = 0 , n × e 1 | Γ e,1 ⊂ Γ b,1 = 0 , n ⋅ b 1 | Γ b,1 = 0 , (1f-g-h) [n × h 1 ] γ 1 = j su,1 , [n × e 1 ] γ 1 = k su,1 , [n ⋅ b 1 ] γ 1 = b su,1 , (1i-j-k)
where h is the magnetic field, b is the magnetic flux density, e is the electric field, j is the electric current density (including source and eddy currents), μ is the magnetic permeability, σ is the electric conductivity and n is the external unit normal to a boundary. Note that (1b) is only expressed in Ω c,1 , whereas it is reduced to the form (1c) in Ω c,1 C . Also (1g) is more restrictive than (1h). The notation [ ⋅ ] γ = ⋅ | γ + -⋅ | γ -expresses the discontinuity of a quantity through any interface γ (of both sides γ + and γ -), which is allowed to be non-zero (the associated surface fields j su,1 , k su,1 and b su,1 are usually unknown, i.e., parts of the solution). The subscript of each quantity refers to the associated problem p. As it will be shown, it is intended to solve successive problems, the solutions of which being added to get the solution of a complete problem. At this first step, the solution p = 1 is a solution as a whole, called reference or source solution.

B. Modified problem defining perturbations

A modification of the problem p = 1 due to a change of permeability or conductivity in some subregions leads to a perturbation of each field quantity. Both large and small perturbations are considered. This can result from the change of properties of existing materials [START_REF] Badics | An effective 3-D finite element scheme for computing electromagnetic field distorsions due to defects in eddy-current nondestructive evaluation[END_REF] or from the addition of new materials in the ambient region [START_REF] Dular | A perturbation method for computing field distortions due to conductive regions with h-conform magnetodynamic finite element formulations[END_REF], which actually also amounts to changing some material properties. The governing equations and relations in another domain Ω 2 , i.e. a modified form of Ω 1 , and the BCs and ICs, are still of the form (1) with all the involved quantities relative to the new solution and the Subdomain Perturbation Finite Element Method for Skin and Proximity Effects Patrick Dular, Ruth V. Sabariego and Laurent Krähenbühl A involved boundaries relative to Ω 2 (also with the subscript 2).

To point out the decomposition of the new solution as a perturbation of the reference one, these quantities are written as

h = h 1 + h 2 , b = b 1 + b 2 , j = j 1 + j 2 , e = e 1 + e 2 , j su = j su,1 + j su,2 , k su = k su,1 + k su,2 , b su = b su,1 + b su,2 .
(

Subtracting each equation of ( 1) from its counterpart in the so-modified problem, the perturbation equations that define the problem p = 2, are

curl h 2 = j 2 , curl e 2 = -∂ t b 2 , div b 2 = 0 , (3a-b-c) b 2 = μ 2 h 2 + b s,2 , j 2 = σ 2 e 2 + j s,2 , (3d-e) n × h 2 | Γ h,2 = 0 , n × e 2 | Γ e,2 ⊂ Γ b,2 = 0 , n ⋅ b 2 | Γ b,2 = 0 , (3f-g-h) [n×h 2 ] γ 2 = j su,2 , [n×e 2 ] γ 2 = k su,2 , [n⋅b 2 ] γ 2 = b su,2 , (3i-j-k)
where the so-defined volume sources b s,2 and j s,2 are obtained from the reference solution as

b s,2 = (μ 2 -μ 1 ) h 1 , j s,2 = (σ 2 -σ 1 ) e 1 . (4-5)
The perturbation fields are still governed by the Maxwell equations in their classical forms (3a-b-c) whereas their associated material relations now include additional sources (4) and ( 5). These sources only occur in the modified regions.

Solving the perturbation problem (3) instead of the modified one enables to avoid operations already performed in the reference problem [START_REF] Krähenbühl | Thin layers in electrical engineering. Example of shell models in analyzing eddy-currents by boundary and finite element methods[END_REF]. At the discrete level, the meshes of both reference and perturbation problems can be significantly simplified, each problem asking for mesh refinement of different regions.

By construction, the summation of the solutions of ( 1) and (3), i.e. (2), gives exactly the solution of the modified problem under the condition that all the materials are linear and that the wished changes in domain Ω 2 with reference to domain Ω 1 are expressed by ( 4) and ( 5) in (3d-e).

C. Possible approximations in the perturbation problem

For the sake of simplicity at the discrete level, domain Ω 2 can neglect some materials initially present in domain Ω 1 , while these must be present in the considered complete problem. Any intersection of non-material regions of Ω 2 with the material regions of Ω 1 is thus allowed.

The corrections (4) and ( 5) are therefore not applicable in the omitted material regions. The material relations for the complete fields in these regions are thus approximated as

b 1 + b 2 = μ 1 h 1 + μ 2 h 2 , j 1 + j 2 = σ 1 e 1 + σ 2 e 2 , (6a-b)
which result from the summation of (1d-e) with their counterparts in (3d-e) without volume sources (4) and ( 5). These relations can be transformed, with (2), as

b = μ 1 h + (μ 2 -μ 1 ) h 2 , j = σ 1 e + (σ 2 -σ 1 ) e 2 , (7a-b)
to point out the error made when their second terms are not negligible, which is the case when the material properties differ too much and the perturbation fields are too large compared to the reference fields. The correct relations, b = μ 1 h and j = σ 1 e, only occur in the regions where μ and σ are unchanged.

For large perturbations, an iterative procedure between the problems is required to obtain an accurate solution. Each subproblem is responsible for giving the suitable correction, as a perturbation. Such iterations also allow nonlinear analyses.

III. THE REFERENCE PROBLEM AS A LIMIT CASE

A. Reference problem with perfect conductors

A first reference problem considers conductors Ω cpe,1 ⊂ Ω c,1 ⊂ Ω 1 , of boundary Γ cpe,1 = ∂Ω cpe,1 , as perfect, i.e. with σ 1 → ∞. This results in a zero skin depth and thus surface currents. The surface currents are considered to flow between Γ cpe,1 + and Γ cpe,1 -, the inner and outer sides of Γ cpe,1 with regard to Ω cpe,1 . The domain Ω cpe,1 can be extracted from Ω 1 in (1) and treated via a BC of zero normal magnetic flux density on its boundary Γ cpe,1 + . Given that only zero fields exist in Ω cpe,1 , the same BC appears on Γ cpe,1 -. One thus has

n ⋅ b 1 | Γ cpe,1 + = 0 , n ⋅ b 1 | Γ cpe,1 -= 0 . (8a-b)
Further, the trace of the magnetic field is unknown on Γ cpe,1 + and vanishes on Γ cpe,1 -, i.e. with (1i), [START_REF] Badics | An effective 3-D finite element scheme for computing electromagnetic field distorsions due to defects in eddy-current nondestructive evaluation[END_REF] with its finite conductivity σ 2 , which alters the distribution of the eddy current density and the other fields. The fields in Ω cpe,2 are not surface fields anymore but penetrate the conductors. They are solutions of problem (3) with Ω 2 now including Ω cpe,2 but with particular ICs (3i-k) through Γ cpe,2 = ∂Ω cpe,2 .

n × h 1 | Γ cpe,1 + = j su,1 , n × h 1 | Γ cpe,1 -= 0 . (9a-b) The modified problem (3) now considers Ω cpe,2 = Ω cpe,1 ⊂ Ω c,
On the one hand, (3k) with ( 2) and (1k) leads to

[n ⋅ b 2 ] Γ cpe,2 = b su,2 = b su -b su,1 = [n ⋅ b] Γ cpe,2 -b su,1 = 0, (10)
due to the continuity of n⋅b in the complete solution (2) and the zero value of b su,1 via (8a-b).

On the other hand, (3i) with ( 2) and (1i) leads to

[n×h 2 ] Γ cpe,2 = j su,2 = [n×h] Γ cpe,2 -j su,1 = -n×h 1 | Γ cpe,1 +, ( 11
)
due to the continuity of n×h in (2) and relation (9a).

B. Reference problem with perfect magnetic materials

Another reference problem considers now conductors Ω cpm,1 ⊂ Ω c,1 ⊂ Ω 1 , of boundary Γ cpm,1 = ∂Ω cpm,1 , as perfect magnetic materials, i.e. with μ 1 → ∞. The domain Ω cpm,1 can thus be extracted from Ω 1 in (1) and treated via a BC fixing a zero tangential magnetic field on its boundary Γ cpm,1 + . Because only zero fields exist in Ω cpm,1 , the same BC appear on Γ cpm,1 -. One thus has

n × h 1 | Γ cpm,1 + = 0 , n × h 1 | Γ cpm,1 -= 0 . (12a-b)
Also, the trace of the magnetic flux density is unknown on Γ cpm,1 + and vanishes on Γ cpm,1 -, i.e. with (1k),

n ⋅ b 1 | Γ cpm,1 + = b su,1 , n ⋅ b 1 | Γ cpm,1 -= 0 . (13a-b)
The modified problem (3) now considers Ω cpm,2 = Ω cpm,1 with its finite permeability μ 2 . The perturbation fields are solutions of problem (3) with Ω 2 now including Ω cpm,2 but with particular ICs (3i-k) through Γ cpm,2 = ∂Ω cpm,2 .

On the one hand, (3k) with ( 2) and (1k) leads to

[n⋅b 2 ] Γ cpm,2 = b su,2 = [n⋅b] Γ cpm,2 -b su,1 = -n⋅b 1 | Γ cpm,1 +, ( 14
)
due to the continuity of n⋅b in (2) and relation (13a). On the other hand, (3i) with ( 2) and (1i) leads to

[ n×h 2 ] Γ cpm,2 = j su,2 = [n×h] Γ cpm,2 -j su,1 = 0, (15)
due to the continuity of n×h in (2) and j su,1 = 0 via (12a-b).

IV. FINITE ELEMENT WEAK FORMULATIONS

A. b-conform weak magnetodynamic formulations

The eddy current problems p are defined in Ω p with the magnetic vector potential formulation [START_REF] Dular | Dual magnetodynamic formulations and their source fields associated with massive and stranded inductors[END_REF], expressing the electric field e p in Ω c,p via a magnetic vector potential a p together with the gradient of an electric scalar potential v p , and the magnetic flux density b p in Ω p as the curl of a p .

The resulting a-v magnetodynamic formulation of problem p = 1 is obtained from the weak form of the Ampère equation (1a), i.e. [START_REF] Dular | Dual magnetodynamic formulations and their source fields associated with massive and stranded inductors[END_REF],

1 1 1 1 ( curl ,curl ') μ - Ω a a ,1 ,1 1 1 1 1 ( , ') ( grad , ') c c t v σ σ Ω Ω + ∂ + a a a h b Γ Γ + < × > + < × > = n h a n h a , 1 1 1 ' ( ) F ∀ ∈ Ω a , (16)
where

F 1 1 (Ω 1 )
cp cp t v σ σ Ω Ω + ∂ + a a a ,2 2 , ' h Γ +< × > n h a ,2 ,2 2 [ ] , ' 0, cp cp Γ Γ +< × > = n h a 1 2 2 ' ( ) F ∀ ∈ Ω a , ( 17
)
where Ω cp, 2 stands for the perturbed conducting region, i.e. either Ω cpe, 2 or Ω cpm, 2 . A major consequence of the bconform formulation used is that ICs (10) and (11) are to be defined respectively in strong and weak senses, i.e. (10) in F 2 1 (Ω 2 ) and (11) in a surface integral term. Note that the approximation (6b) amounts to neglecting the second and third terms of (17), while (6a) acts on its first term.

B. Perfect conductors perturbed to real ones

For a perfect conductor domain Ω cpe, 1 , the BC (8a) leads to an essential BC on the primary unknown a 1 that can be expressed via the definition of a surface scalar potential u 1 (in general single valued, if no net magnetic flux flows in Ω cpe, 1 ) [START_REF] Dular | Dual finite element formulations for lumped reluctances coupling[END_REF], i.e., u

Γ Γ Γ ⋅ = ⇔ × = × n a n a n . ( 18
)
In a 2D model with perpendicular currents, condition (18) amounts to defining a floating (constant) value for the perpendicular component of a 1 for each conductor.

The reference formulation is of the form (16) where the perfect conductors are extracted from Ω 1 and Ω c,1 and are only involved through their boundaries Γ cpe,1 with the condition (18). This latter condition is to be strongly defined in F 1 1 (Ω 1 ). The boundaries Γ cpe,1 are thus to be added to Γ b,1 .

The surface integral term <n×h 1 , a'> Γ cpe,1 is non-zero only for the function grad u' (from (18)), the value of which is then the total surface current I 1 flowing in Γ cpe,1 (this can be demonstrated from the general procedure developed in [START_REF] Dular | Dual finite element formulations for lumped reluctances coupling[END_REF]). It is zero for all the other local test functions (at the discrete level, for any edge not belonging to Γ cpe,1 ). This way, the circuit relation can be expressed for each conductor Ω cpe,1 and the coupling with electrical circuits is possible.

For the associated perturbation formulation (17), the IC (10) is strongly expressed via the continuity of the vector potential a 2 through Γ cpe,2 . The IC (11) can rather only act in a weak sense via the surface integral term related to Γ cpe,2 + in (17). Indeed, its involved quantity n × h 1 is not known in a strong sense on Γ cpe,2 + , but rather in a weak sense. One has ,

+ Γ Γ Γ < × > = < -× > n h a n h a 1 , 1 ,1 1 1 1 1 \ , ' ( curl ,curl ') cpe cpe μ + - Ω Ω Γ = <-× > = - n h a a a (19) in case no part of Ω c,1 \ Ω cpe,1 is in contact with Ω cpe,1 2 ,2 ,2 2 1 [ ] , ' , ' cpe cpe cpe
(otherwise the second and third terms of ( 16) have to be considered as well). This way, the surface integral term related to Γ cpe,2 + in ( 17) is calculated from a volume integral coming from the reference problem. Its consideration via a volume integral, limited at the discrete level to one single layer of FEs touching the boundary, is the natural way to average it as a weak quantity. At the discrete level, the reference quantity a 1 in ( 19) is initially given in the mesh of the reference problem. It has afterward to be expressed in the mesh of the perturbation problem for being used in (19). This can be done through a projection method [START_REF] Geuzaine | A Galerkin projection method for mixed finite elements[END_REF] of its curl limited to the layer of finite elements touching Ω cpe,1 .

C. Perfect magnetic materials perturbed to real ones

When perfect magnetic materials Ω cpm,1 ⊂ Ω c,1 ⊂ Ω 1 are considered, the reference formulation is of the form (16) where the perfect materials are extracted from Ω 1 and Ω c,1 and are only involved through their boundaries Γ cpm,1 added to Γ h,1 with the BC (12a), i.e. ,

' 0, ' 0

cpm h cpm + Γ ⊂Γ Γ < × > =< > = n h a a . ( 20
)
For the associated perturbation formulation (17), the IC (14) must be strongly expressed via the vector potential a 2 . It can be expressed with a known discontinuous component b d,2 of b 2 only acting outside Ω cpm,2 , with b 2 = b c,2 +b d,2 , i.e.

[ n⋅b 2 ] Γ cpm,2 = n⋅b d,2 | Γ cpm,2 + = -n⋅b 1 | Γ cpm,1 + , (21)
or with an associated known discontinuous component a d,2 of a 2 , with a 2 = a c,2 +a d,2 , also only acting outside Ω cpm,2 , i.e.

a d,2 | Γ cpm,2 + = -a 1 | Γ cpm,1 + . ( 22
)
The IC (15) cancels the surface integral term related to

Γ cpm,2 + in (17), i.e. <[n×h 2 ] Γ cpm,2 , a'> Γ cpm,2 + = <0, a'> Γ cpm,2 + .
V. APPLICATION Two core-inductor systems are considered as test problems (Figs. 12). Their five copper stranded inductors are connected in series. The core of system 1 (Fig. 1) is made of aluminium (σ Al = 2.7 10 7 Ω -1 m -1 ). The one of system 2 (Fig. 2) is made of steel (σ St = 10 6 Ω -1 m -1 and relative permeability μ r,St = 270). A 2D model with a vertical symmetry axis is considered. For a direct comparison with the IBC technique, a frequency domain analysis is done. However, the subdomain perturbation technique can be directly applied to time domain analyses without any change. The working frequencies are 5 kHz and 500 Hz for systems 1 and 2 respectively, to give equal skin depths δ = δ Al = δ St = 1.37 mm. The core half-width is 12.5 mm (9.1 δ) and its height is 50 mm (36.5 δ). Holes are considered in the core in order to point out the effect of several corners. They are non-uniformly distributed to allow for different lengths of plane portions between them (small lengths should penalize the IBC technique).

The magnetic flux lines are shown in Figs. 1 and2 for the different calculations performed, i.e. the conventional FE approach, the reference problem and the perturbation problem. Figs. 3 and4 show the eddy current and Joule power density distributions in the core, as well as the relative error on these quantities made by the IBC versus the subdomain FE approach, the results of which are checked to be very similar to those of the conventional FE approach. The error significantly increases in the vicinity of the conductor corners: it reaches 50% for the Joule power density and 30% for the current density in the smallest plane portions. This affects the total losses accuracy when the size of the conductor portions decreases. The error with the IBC is shown to be significant up to a distance of about 3 δ from each corner. A good accuracy is only obtained beyond this distance for system 1. The IBC error increases with δ with respect to the structure dimensions, whereas the developed approach successfully and accurately adapts its solution to any δ.

VI. CONCLUSIONS

The developed subdomain perturbation method offers a way to uncouple FE regions in eddy current frequency and time domain analyses, allowing the solution process to be lightened. The skin and proximity effects in conductors can be accurately determined in a wide frequency range, allowing precise Joule losses calculations.

Once calculated, the reference solution can be used in each subproblem not only for a single signal but for several signals. This allows efficient parameterized analyses on the signal form and the electric and magnetic characteristics of the conductors in a wide range, i.e. on all the parameters affecting the skin depth. Nonlinear analyses, e.g. with temperature dependent conductivities, could then benefit from this.

Fig. 1 .Fig. 2 .Fig. 3 .Fig. 4 .

 1234 Fig. 1. Magnetic flux lines for the conventional FE solution b (left), the reference solution b 1 (middle) and the perturbation solution b 2 (right); system 1 with a conductive non-magnetic core.

  is a gauged curl-conform function space defined on Ω 1 and containing the basis functions for a as well

	1 μ -2 2 ( σ + (	2 ,2 Ω c p curl ,curl ') 2 a a ,2 2 \ , ' ) c t Ω Ω ∂ a a	+	2 ( g r a d , ' ) 2 v σ a	,2 Ω Ω \ c	c p	,2
	(	2	2	, ')	,2	2 ( grad , ') 2	,2

as for the test function a' (at the discrete level, this space is defined by edge finite elements); ( • , • ) Ω and < • , • > Γ respectively denote a volume integral in Ω and a surface integral on Γ of the product of their vector field arguments. The surface integral term on Γ h,1 accounts for natural BCs of type (1f). The term on the surface Γ b,1 with essential BCs on n⋅b 1 is usually omitted because it gives no local contribution to (16). It will be shown to be the key for the post-processing of the reference solution, a part of which being n × h 1 | Γ b,1 .

The weak formulation of the perturbation problem (3) is

,1,11 1, ' , ' 0

,1 ,1 ,1 1 1 1 curl 0 grad cpe cpe cpe