Carina Rodrigues
email: carina.rodrigues@ispup.up.pt

Jennifer Zeitlin

Emilija Wilson

Liis Toome

Marina Cuttini

Rolf F Maier

Véronique Pierrat

Henrique Barros

G Martens

P Martens

Reempts Van

Denmark

K Boerch

A Hasselager

L D Huusom

O Pryds

T Weber

L Estonia

H Toome

Varendi

P Y Ancel

B Blondel

A Burguet

P H Jarreau

P Truffert

B Misselwitz

Saarland (L Gortner S Schmidt

Emilia Romagna

D Baronciani

G Gargano

R Agostino

I Croci

Marche (V Carnielli F Franco

D Dilallo

Eastern Central

C Koopman-Esseboom

A Van Heijst

J Nijman

Poland

Tagus Valley

L M Graça

Northern region ( M C Machado

T Rodrigues

A K Bonamy

M Norman

Wilson Uk

ES Draper, BN E Boyle

A C Fenton

Dwa Milligan

M Bonet

Managing mother's own milk for very preterm infants in neonatal units in 11 European countries

Keywords: Gestational age, HCMV, Human cytomegalovirus, MOM, Mother's own milk, PCR, Polymerase Chain Reaction, UK, United Kingdom, VPT, Very preterm

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

Feeding with human milk has been recognised as an essential component of newborn care and is especially important for infants born very preterm (VPT, below 32 weeks of gestation) who face higher risks of adverse outcomes. WHO recommends that infants who cannot be fed mother's own milk (MOM) should receive donor human milk. Direct feeding at the breast takes time to establish after VPT birth and procedures are required for expressing, collecting, storing and administering breast milk that respect microbiological safety rules and ensure nutritional and immunological quality. However, as recommendations are scarce, these procedures appear to be dependent on organizational structures and policies of the units [START_REF] Klotz | Handling of Breast Milk by Neonatal Units: Large Differences in Current Practices and Beliefs[END_REF][START_REF] Picaud | Review concludes that specific recommendations are needed to harmonise the provision of fresh mother's milk to their preterm infants[END_REF][START_REF] Bonet | Approaches to supporting lactation and breastfeeding for very preterm infants in the NICU: a qualitative study in three European regions[END_REF].

Breast milk is one of the most common modes of postnatal human cytomegalovirus (HCMV) transmission in preterm infants with rates for fresh or frozen MOM ranging from 8% to 37%, and about 10% probability of infant infection [START_REF] Picaud | Review concludes that specific recommendations are needed to harmonise the provision of fresh mother's milk to their preterm infants[END_REF]. The debate on the risk of neonatal morbidity and long-term neurodevelopmental impairment in VPT infants with HCMV infection is ongoing (4), raising additional challenges for establishing best practices for managing MOM [START_REF] Klotz | Handling of Breast Milk by Neonatal Units: Large Differences in Current Practices and Beliefs[END_REF][START_REF] Picaud | Review concludes that specific recommendations are needed to harmonise the provision of fresh mother's milk to their preterm infants[END_REF]. In 2012, the American Academy of Paediatrics stated that the benefits of fresh MOM from HCMV-seropositive mothers outweighed the risk of HCMV infection. In 2018, the working group of the French Neonatal Society on fresh human milk use in preterm infants advised pasteurisation of MOM from HCMVseropositive mothers for infants born <28 weeks or <1000 grams [START_REF] Picaud | Review concludes that specific recommendations are needed to harmonise the provision of fresh mother's milk to their preterm infants[END_REF]. In addition to HCMV transmission, also bacterial contamination of expressed milk is suspected to lead to significant problems in VPT infants [START_REF] Picaud | Review concludes that specific recommendations are needed to harmonise the provision of fresh mother's milk to their preterm infants[END_REF].

Published reports have revealed differences in MOM management among neonatal intensive care units (NICUs) and countries [START_REF] Klotz | Handling of Breast Milk by Neonatal Units: Large Differences in Current Practices and Beliefs[END_REF][START_REF] Bonet | Approaches to supporting lactation and breastfeeding for very preterm infants in the NICU: a qualitative study in three European regions[END_REF]. We aimed to compare practices for handling MOM for VPT infants using standardised questions in European NICUs.

Data were collected as part of the area-based EPICE (Effective Perinatal Intensive

Care in Europe) study set up to investigate the use of evidence-based practices for the care of VPT infants in 19 regions in 11 countries [START_REF] Zeitlin | Cohort profile: Effective Perinatal Intensive Care in Europe (EPICE) very preterm birth cohort[END_REF].

In 2012, structured questionnaires were sent to the head of all NICUs with at least 10 VPT admissions during the study period. The unit questionnaire recorded information on the units' characteristics and policies, including procedures for managing MOM. Of the 135 eligible NICUs, 134 (99.3%) replied.

Descriptive statistics are presented as numbers and valid proportions.

Ethical approval was obtained in each region from appropriate ethics committees, as required by national legislation, and for the European database by the French data protection committee.

As shown in Table 1, 72% of the 134 neonatal units were level III and 91% had a written protocol (developed at unit or regional level) for breastfeeding/human milk use; when MOM was not available, 34% used human banked milk for all VPT infants (range: 0% to 100% in Sweden) and 13% for some groups only (range: 0%-50%).

Management of MOM varied widely across the countries. Overall, 56% of units reported using fresh MOM (not frozen and unpasteurized milk) without restrictions regarding gestational age, birth weight or risk of HCMV transmission (range: 0% in Germany to 100% in Netherlands and Denmark). Different practices were also observed between units in the same country, particularly in France and Italy. In most units (70%) MOM was not pasteurised (range: 23%-100%); and in 29 (22%) units all VPT infants received pasteurised MOM, (range: 0% to 73% in France).

HCMV serology on all mothers who express their milk (or HCMV Polymerase Chain Reaction of the milk) was required in 29% of units (range: 0% in 6 countries to 93% in Germany). Wide differences were found between NICUs in France and Italy. Among these units (n=36), 47% used human bank/MOM pasteurised, 26% frozen-thawed MOM, 3% untreated fresh MOM and 24% formula in the case of HCMV-seropositive mothers.

Systematic bacteriological analyses of the fresh or frozen-thawed MOM were not performed in 76% of units (range: 29%-100%); performed in less than 10% for the first milk feeding only, and every week in 7%.

These large variations in practice between countries could reflect differences in local regulations or guidelines, as well as lack of strong recommendations at international and national level. Heterogeneity was also observed between units within the same country, revealing that different options can operate locally, such as milk bank availability which may differ between regions within countries. Study limitations include data collected in 2012, which may underestimate current human milk bank availability, and a smaller number of units in participating regions in some countries; however, no more recent comparable information on practices is available from so many European countries.

While it is recognised that MOM should be encouraged as the primary feeding method for VPT infants, this variation indicates substantial differences in attitudes about what constitutes best practices for the management of MOM among European neonatologists. To guide practice, further studies are needed to reinforce the evidencebase on ensuing outcomes of HCMV infection as well as practices for managing these risks whilst ensuring that VPT infants can benefit from MOM. No, all milk is frozen or pasteurised 2 (25.0) 0 (0.0) 0 (0.0) 4 (19.0) 0 (0.0) 4 (18.2) 0 (0.0) 0 (0.0) 1 (5.9) 0 (0.0) 0 (0.0) 11 (8.5)

Infants whose mothers express breast milk receive frozen-thawed milk

Yes, with no restrictions 6 (66.7) 8 (100.0) 1 (25.0) 3 (13.0) 2 (14. Systematic bacteriological analysis of the mother's fresh or frozen-thawed milk is performed Yes, for the first milk feeding 0 (0.0) 0 (0.0) 0 (0.0) 6 (28.6) 4 (28.6) 3 (13.6) 0 (0.0) 0 (0.0) 0 (0.0) 0 (0.0) 0 (0.0) 13 (9.8)

Yes, every week 0 (0.0) 0 (0.0) 0 (0.0) 9 (42.9) 0 (0.0) 0 (0.0) 0 (0.0) 0 (0.0) 0 (0.0) 0 (0.0) 0 (0.0) 9 (6.8)

Another frequency 1 (11.1) 1 (12.5) 2 (50.0) 0 (0.0) 1 (7.1) 3 (13.6) 0 (0.0) 0 (0.0) 1 (5.9) 0 (0.0)

4 )

 4 Type of milk given to VPT infants born to HCMV-seropositive mothers, if HCMV serology performed

Table 1 .

 1 Management of mother's own milk for very preterm infants in the neonatal unit, by country ¥ .

		Belgium	Denmark	Estonia	France	Germany	Italy	Netherlands	Poland	Portugal	Sweden	UK	Total
		(n= 9)	(n= 8)	(n= 4)	(n= 23)	(n= 14)	(n= 22)	(n= 2)	(n= 4)	(n= 17)	(n= 4)	(n= 27)	(n= 134)
		n (%)	n (%)	n (%)	n (%)	n (%)	n (%)	n (%)	n (%)	n (%)	n (%)	n (%)	n (%)
	Level of care
	Level III	8 (88.9)	2 (25.0)	3 (75.0)	22 (95.6)	13 (92.9) 16 (72.7)	2 (100.0)	1 (25.0)	15 (88.2)	2 (50.0)	12 (44.4)	96 (71.6)
	Written protocol for breastfeeding and human milk use
	Yes, regional or network protocols	0 (0.0)	0 (0.0)	3 (75.0)	3 (14.3)	0 (0.0)	3 (13.6)	0 (0.0)	3 (75.0)	5 (29.4)	1 (25.0)	1 (3.7)	19 (14.4)
	Yes, unit protocols	6 (66.7)	8 (100.0)	1 (25.0)	15 (71.4)	12 (85.7)	18 (81.8)	2 (100.0)	1 (25.0)	10 (58.8)	3 (75.0)	25 (92.6) 101 (76.5)
	No	3 (33.3)	0 (0.0)	0 (0.0)	3 (14.3)	2 (14.3)	1 (4.6)	0 (0.0)	0 (0.0)	2 (11.8)	0 (0.0)	1 (3.7)	12 (9.1)

Use of human bank milk/donor's milk to feed VPT infants whose mothers do not express their milk

	Yes, for all infants less than 32 weeks GA	0 (0.0)	7 (87.5)	1 (25.0)	19 (82.6)	0 (0.0)	9 (40.9)	0 (0.0)	0 (0.0)	1 (5.9)	4 (100.0)	4 (14.8)	45 (33.6)
	Yes, for some infants	0 (0.0)	1 (12.5)	0 (0.0)	2 (8.7)	2 (14.3)	2 (9.1)	1 (50.0)	0 (0.0)	1 (5.9)	0 (0.0)	8 (29.6)	17 (12.7)
	No	9 (100.0)	0 (0.0)	3 (75.0)	2 (8.7)	12 (85.7)	11 (50.0)	1 (50.0)	4 (100.0) 15 (88.2)	0 (0.0)	15 (55.6)	72 (53.7)

Infants whose mothers express breast milk receive fresh milk*

	Yes, with no restrictions	4 (50.0)	8 (100.0)	3 (75.0)	6 (28.6)	0 (0.0)	11 (50.0)	2 (100.0)	2 (66.7)	11 (64.7)	2 (50.0)	24 (88.9)	73 (56.1)
	Yes, but with restrictions	2 (25.0)	0 (0.0)	1 (25.0)	11 (52.4)	14 (100.0)	7 (31.8)	0 (0.0)	1 (33.3)	5 (29.4)	2 (50.0)	3 (11.1)	46 (35.4)

serology on all mothers of VPT infants (or HCMV PCR of the milk) is required, if they express their milk

	Infants whose mothers express breast milk receive own mother's pasteurised milk
	Yes, all babies less than 32 weeks GA	2 (22.2)	1 (12.5)	1 (25.0)	16 (72.7)	1 (7.1)	6 (27.3)	0 (0.0)	0 (0.0)	0 (0.0)	0 (0.0)	2 (7.4)	29 (21.8)
	Yes, only some babies less than 32 weeks GA	0 (0.0)	0 (0.0)	0 (0.0)	1 (4.6)	3 (21.4)	6 (27.3)	0 (0.0)	1 (25.0)	0 (0.0)	0 (0.0)	0 (0.0)	11 (8.3)
	No, own mother's milk is not pasteurised	7 (77.8)	7 (87.5)	3 (75.0)	5 (22.7)	10 (71.4)	10 (45.4)	2 (100.0)	3 (75.0)	17 (100.0)	4 (100.0)	25 (92.6)	93 (69.9)
	Human cytomegalovirus (HCMV) Yes, according to GA or weight 1 (11.1)	0 (0.0)	0 (0.0)	5 (31.2)	13 (92.9)	6 (27.3)	0 (0.0)	0 (0.0)	2 (11.8)	0 (0.0)	0 (0.0)	27 (21.4)
	Other criteria	0 (0.0)	0 (0.0)	0 (0.0)	3 (18.8)	0 (0.0)	5 (22.7)	0 (0.0)	0 (0.0)	1 (5.9)	0 (0.0)	0 (0.0)	9 (7.1)
	No	8 (88.9)	7 (100.0)	4 (100.0)	8 (50.0)	1 (7.1)	11 (50.0)	2 (100.0)	4 (100.0) 14 (82.4)	4 (100.0)	27 (100.0) 90 (71.
						3)	16 (72.7)	2 (100.0)	1 (25.0)	9 (52.9)	3 (75.0)	24 (88.9)	75 (56.0)
	Yes, but with restrictions	2 (22.2)	0 (0.0)	1 (25.0)	1 (4.4)	11 (78.6)	3 (13.6)	0 (0.0)	0 (0.0)	1 (5.9)	1 (25.0)	3 (11.1)	23 (17.2)
	No	1 (11.1)	0 (0.0)	2 (50.0)	19 (82.6)	1 (7.1)	3 (13.6)	0 (0.0)	3 (75.0)	7 (41.2)	0 (0.0)	0 (0.0)	36 (26.9)

ACKNOWLEDGEMENT

The authors are grateful to the staff from the participating neonatal units.

FUNDING

The research received funding from the European Union's Seventh Framework Programme (grant 259882) and from the Foundation for Science and Technology -FCT (Portuguese Ministry of Science, Technology and Higher Education), the Operational Programmes Competitiveness and Internationalization (COMPETE 2020) and Human Capital (POCH), Portugal 2020, under the EPIUnit (POCI-01-0145-FEDER-006862; UID/DTP/04750/2019) and the PhD grant SFRH/BD/111794/2015 (CR).

CONFLICTS OF INTEREST

None to declare.


Eva Mackamul
email: eva.mackamul@inria.fr

Improving the Discoverability of Interactions in Interactive Systems

Keywords: discoverability, learnability, conceptual framework

In an ever increasingly complex technological landscape, interaction methods need to support users in their own discovery. This is exacerbated by a general aversion to instruction manuals and a trend towards invisible controls. Consequently, a lack of support for awareness or recognition of interaction possibilities leads to inefficient usage or complete disregard of the system. However, despite standard work imploring that users should be able to "figure out what actions are possible and where and how to perform them", this problem is rarely considered in the introduction of new interaction methods. My doctoral research is focused on improving the discovery of interaction possibilities. To do so I am combining a theoretical approach, focused on identifying, defining and framing relevant considerations with a practical approach to validate those theoretical considerations and identify and formulate actionable improvement methods for researchers and designers. In the process I hope to highlight the importance of discoverability to the research community and advocate for its increased consideration.

INTRO -CONTEXT AND MOTIVATION

New technology is being introduced with ever increasing speed. This is accompanied by HCI research proposing novel interaction techniques to utilise these expanding technical capabilities. Consequently, proposed interfaces and interaction methods have expanded beyond established WIMP or touch interactions. However, this abundance of possibilities is rarely fully utilised, as users are unaware of all the available functionalities and overburdened with discovering features and interaction methods through accidents, social examples and articles about "hidden tricks" [START_REF] Nield | 10 hidden tricks to try on your Android phone[END_REF][START_REF] Nield | Hidden iPhone tricks that will change the way you use iOS[END_REF] or "things you didn't know" [4, [START_REF] Rebolini | [END_REF][START_REF] Vincent | Best Android tricks You Didn't Know About[END_REF] rather than spontaneous interaction. This is exacerbated by propositions and implementations of novel interactions lacking visual signifiers i.e. elements that convey information about possible interaction with the system [START_REF] Norman | The design of everyday things: Revised and expanded edition[END_REF]. Examples of this can be found in commercial products (iPhone 3D touch [START_REF] Goel | Apple Piles On the Features and Users Say[END_REF]) as well as research [START_REF] Roudaut | MicroRolls: Expanding Touch-Screen Input Vocabulary by Distinguishing Rolls vs. Slides of the Thumb[END_REF]. Additionally, existing systems receive constant software updates that not only update background functionality but, at times, also the user interface.

If users don't even realise they are presented with an interaction possibility, or if they don't understand how to utilise it, they will not benefit from its introduction. Consequently, the lack of discoverability of these novel interaction possibilities impedes successful usage or adoption of the technology altogether. Despite the clear problems caused by a lack of discoverability, introductions of new interactivity, both commercially and in research, rarely feature considerations of how new users are expected to become aware of and comprehend the new possibility.

I hypothesise that this absence of consideration is not only due to a lack of awareness or acknowledgement of discoverability's importance but also due to:

• unclear and overlapping terminology used to describe the problem • lack of knowledge on how discovery can be facilitated and supported • lack of knowledge of how discovery occurs (circumstance consideration) • lack of established methodology to test whether something is discoverable

My doctoral research is focused on filling these gaps in order to improve the discovery of interactions by providing a clear understanding of the problem and offering recommendations allowing its increased and systematic consideration.

BACKGROUND AND RELATED WORK

Given the objective of investigating and improving the discovery of interactions, my focus is on initial interactions which I use to describe the specific circumstance of an interaction occurring for the very first time, either intentional or by chance. This includes the first encounter with, and discovery of, an overall system as well as its individual features and input methods. I explicitly consider these separate entities as such since their recognition and usage may be supported through differing means.

In HCI these beginning phases of interaction have been primarily examined in the context of learning how to interact with the system. Consequently, learnability is the most commonly used term to describe the users' first understanding and subsequent improvement when using an interactive system.

Grossman et al. 's survey of learnability research [START_REF] Grossman | A Survey of Software Learnability: Metrics, Methodologies and Guidelines[END_REF] found that, while there was an agreement among researchers that learnability is an important aspect of usability, there was a lack of consensus in the definition, evaluation and improvement of learnability in user interfaces. They grouped definitions into two main categories: initial learnability (new users gaining proficiency with a system) and extended learnability (long-term learning and mastering of a system). As they consider initial learnability as the ability to perform something well in a single interaction, the inception of interactions is a precondition inducing initial learnability. The separation of learnability into initial and extended illustrates the broad use of the term in both time frame and scope.

Often mentioned in conjunction or synonymously with learnability is discoverability [START_REF] Furqan | Learnability through Adaptive Discovery Tools in Voice User Interfaces[END_REF][START_REF] Kirschthaler | What Can I Say? Effects of Discoverability in VUIs on Task Performance and User Experience[END_REF][START_REF] Mihajlov | Learning to Drag: The Effects of Social Interactions in Touch Gestures Learnability for Older Adults[END_REF]. The term 'discoverability' in the context of usability and human-computer interaction was popularised by Norman [START_REF] Norman | The design of everyday things: Revised and expanded edition[END_REF], who describes it through the following question:

"Is it possible to even figure out what actions are possible and where and how to perform them?" [26, p.3] As such it describes a less purpose driven aspect of initial interaction than learnability.

Other offered definitions vary widely in scope and specificity. While some offer loose explanations of discoverability such as "the effort required to locate or "discover" an application amongst others" [START_REF] Hosio | Application Discoverability on Multipurpose Public Displays: Popularity Comes at a Price[END_REF] or "the ability for users to find and execute features through a user interface" [START_REF] Kirschthaler | What Can I Say? Effects of Discoverability in VUIs on Task Performance and User Experience[END_REF], others define discoverability specifically for their context like noting how "the "invisible" nature of VUIs can challenge users' ability to discover its capabilities and limitations" [START_REF] Furqan | Learnability through Adaptive Discovery Tools in Voice User Interfaces[END_REF] or how gestural interfaces "are not discoverable in the sense that it is hard to know what gesture to do and we often cannot look it up in the way we would look at a menu to see what options are on it" [START_REF] Gillies | What is Movement Interaction in Virtual Reality For?[END_REF]. Notably many of the offered explanations of discoverability either allude to or directly reference learnability by referring to discoverability as "a means to achieve learnability" [START_REF] Furqan | Learnability through Adaptive Discovery Tools in Voice User Interfaces[END_REF] or describing discoverability as a challenge one is confronted with when trying to learn a system [START_REF] Duala | Using Structure-Based Recommendations to Facilitate Discoverability in APIs[END_REF]. Furthermore, definitions of discoverability as, for example, "the ability with which a user can find features of the system to increase proficiency over time" [START_REF] Potluri | CodeTalk: Improving Programming Environment Accessibility for Visually Impaired Developers[END_REF] resemble those offered for learnability. This validates the notion that discoverability aids learnability and is a necessary component when aiming to achieve good learnability. However, this close association may be the cause for the lack of separate consideration of the concepts.

RESEARCH GOALS AND APPROACH

The objective of my research is to establish a body of work on how to design systems that foster the discoverability of interaction possibilities.

In doing so, I aim to improve the recognition and successful adoption of novel interaction methods upon their introduction as well as lower the entry threshold for advanced interactions with complex technology.

To achieve this, the main objectives of my PhD are to (1) Clarify the concept of discoverability and highlight how semantically related concepts differ (2) Provide fundamental knowledge of factors impacting discoverability (3) Establish and test interface design guidelines that support the discovery of interaction (4) Design and test user interface components supporting independent discovery of interaction possibilities (5) Define experimental protocols for the evaluation of discoverability Through these objectives, I hope to highlight the importance and potential of increased consideration of discoverability when introducing new interaction possibilities, while providing tools and guidelines on how to do so.

To achieve these goals, I first concentrated on the theoretical and conceptual work required to accomplish the first two objectives. This is primarily to clearly define the focus of my work and establish a conceptual basis of related concepts and relevant factors to base my practical work on. Additionally this fit the restrictions of beginning my PhD during the Covid 19 pandemic.

RESEARCH PROGRESS AND CONTRIBUTION TO DATE 4.1 Terminology Clarification

Following an extensive literature review concerning the usage and definition of discoverability, I identified three focus areas of discoverability in HCI: System, Interaction and Feature. System discoverability is focused on whether potential users notice the overall system and recognise it as something they can interact with. Interaction discoverability is concerned with whether users can identify and successfully use novel interaction and input methods. Finally, feature discoverability covers the user's ability to discover features or functionality they were previously unaware of while using a system. While this separation between different discoverabilities has been implicit with research specifying a focus on interaction discoverability [START_REF] Cheung | Improving Interaction Discoverability in Large Public Interactive Displays[END_REF] or feature discoverability [START_REF] Shannon | Software considerations for automotive pervasive systems[END_REF], these focus areas have not been collated and considered in relation to one another. The clarification of what needs to be discovered is essential when examining the connection of different discoveries since they may not always be facilitated through the same measures. My understanding of discoverability is informed by this separation and a combination of comprehensive definitions from Cardello and Srinivasan. Cardello describes discoverability as users encountering "new content or functionality that they were not aware of previously" [START_REF] Cardello | Low findability and discoverability: Four testing methods to identify the causes[END_REF]. Srinivasan et al. consider discoverability as entailing "(1) awareness -making users aware of the operations that can be performed using speech; and (2) understanding -educating users on how requests should be phrased so the system can interpret them correctly" [START_REF] Srinivasan | Discovering Natural Language Commands in Multimodal Interfaces[END_REF]. Combining these approaches, I consider discoverability as the ability for users to perceive and comprehend a system, function or input method as such when encountering it for the first time despite a lack of previous awareness or knowledge. This may be through intentional effort or serendipitously.

To highlight the differences between learnability and discoverability, figure 1 shows the approximate timeline of an isolated initial interaction. This isolation is for clarification purposes, however initial interactions can happen continuously and concurrently as users encounter new interactivity within a system and during other interactions.

Discoverability, since it covers users that may not have a particular aim to interact and yet unaware of an interaction possibility, precedes learnability and triggers initial interaction. If a potential user encounters functionality but does not notice or comprehend it as such, the initial interaction is unlikely to continue. The initial interaction ends once the user first successfully utilises the interaction possibility they are presented with. Therefore, the mere encounter with an interaction possibility is not yet an initial interaction if the user fails to successfully utilise it.

Learnability exceeds discoverability both in scope and time frame. Even when examining just initial learnability, it extends past discoverability by incorporating the user being able to successfully interact with something. As Berkun notes, "discoverability does not guarantee success" [START_REF] Berkun | The myth of discoverability[END_REF]. He argues that simply being able to discover

Intentional Interaction

Serendipitous Interaction

Time

Initial Interaction

Encounter Interaction

Discoverability

Comprehension

Extended Learning Initial Learning Successful Interaction

Figure 1: Timeline of progressing initial interaction. Both discoverability and learnability apply during this timeframe and influence the interaction's occurrence and success. While the diagram suggests a strict separation between serendipitous and intentional interaction, these commonly overlap. Although discoverability and learnability are each more applicable to the circumstance they are placed in, they are both applicable to either.

something, does not necessarily mean people can, or know how to, use it. Thus, discoverability is not a measure of efficiency or improvement. In contrast learnability is often considered in the context of performance. While discoverability undoubtedly is a factor contributing to learnability, and both concepts influence one another, these distinctions show that they should be considered independently before any conjoined examinations. Nonetheless, both are a phenomenon occurring during initial interactions and therefore influenced by similar factors.

Conceptual Framework of Impacting Factors

Investigating the factors influencing and facilitating initial interaction is crucial to enable and improve discoverability, which often triggers these interactions. I examined literature focused on the first stages of interaction and extracted relevant factors. To determine their impact on interaction and relation to one another I then reviewed literature specific to these factors. The main concepts identified can be categorised into three main actors: the system, the user and the environment. Through organising these factors and examining how factors within individual actors interrelate, I constructed a conceptual framework which is considered in depth in an ongoing submission. Figure 2 shows the separation of actors and some examples of factors that are considered within the context of that actor.

System.

The factors considered within the system are separated into system characteristics and system qualities. The characteristics considered are visibility, consistency, constraints, signifiers, feedback and feedforward as well as affordances. The first three influence the success of signifiers and feedback/feedforward, while all of them convey the presence of affordances. System qualities are considered as such as they are inherently qualities of the system but describe interactivity and are therefore part of the interaction space. The considered qualities are communicability [START_REF] Sieckenius | The semiotic engineering of human-computer interaction[END_REF][START_REF] Raquel O Prates | A case study for evaluating interface design through communicability[END_REF], navigability, findability [START_REF] Cardello | Low findability and discoverability: Four testing methods to identify the causes[END_REF][START_REF] Morville | Ambient findability: What we find changes who we become[END_REF] and guessability [START_REF] Wobbrock | Maximizing the Guessability of Symbolic Input[END_REF]. The framework collates definitions for these, semantically related, terms and highlights differences before considering how these concepts impact initial interaction. 4.2.2 User. Factors considered as part of the user include various dependencies. Familiarity enables both perceived affordances and expectations and assumptions, while goals and objectives enable motivation. The relation between attention and awareness is contentious with some arguing attention is necessary for awareness [6,[START_REF] Mack | Inattentional blindness[END_REF] while others question the necessity of attention for awareness [16,[START_REF] Victor Af Lamme | Why visual attention and awareness are different[END_REF] arguing that enhanced attention to specific stimuli can come at the expense of overall awareness. Additionally to identifying these factors, putting them in relation to one another and initial interactions and considering their impact, the framework also highlights how these factors contribute to endogenous triggers of interaction. Endogenous triggers focus on the users by themselves, and without outside influence, discovering unfamiliar functionality or interactivity through a set of different circumstances. Familiarity can trigger interaction through transfer. Similarly, the presence of motivation influences how interactions are triggered. Motivation can lead to intentional or curiosity driven interaction while in the absence of motivation interactions are only triggered through the user serendipitously or accidentally.

Environment.

The environment in this context encapsulates the physical environment in which both, the system and the user, are placed, as well as the social context of this environment. The factors considered here are the social context, as well as examples/instructions, distractions/interruptions and constraints. While independently influencing interactions (social learning [START_REF] Heyes | What's social about social learning[END_REF][START_REF] Li | Exploring an Augmented Reality Social Learning Game for Elementary School Students[END_REF], technology adaptation [START_REF] Gantt | Gardeners and Gurus: Patterns of Cooperation among CAD Users[END_REF][START_REF] Vannoy | The Social Influence Model of Technology Adoption[END_REF]), the social context can also impact the other factors. Examples The actors influencing the initial interaction as well as examples of the factors placed within them. Since the interaction happens between the system and the user they impact it while they, as well as the interaction, are influenced by the environment in which they are placed. The listed factors act as examples and do not cover all those considered in the full framework.

social context and can influence user behaviour. This is not only effective in co-located settings but also remote work settings [START_REF] Murphy | Peer Interaction Effectively, yet Infrequently, Enables Programmers to Discover New Tools[END_REF] and advertisements such as keynotes [START_REF] Mayer | How to Communicate New Input Techniques[END_REF]. Distractions and Interruptions can stem from people in the space as well as general noise, light or other technology. Constraints can be environmental (semantic and physical constraints e.g. bright light on a screen) as well as imposed through the social context (social and cultural constraints). Finally, the social context can be the trigger of initial interaction through examples, word of mouth or explicit instruction.

NEXT STEPS

Having investigated the first 2 research objectives through theoretical and conceptual work, the aim is to now validate the conceptual framework and extracted recommendations (Research Objective 3) in practical application. To do so, I aim to design and test various approaches of creating user interface components that support independent discovery of their interaction possibilities. These approaches will be informed by the considered factors and extracted guidelines. I then plan to consider their success or shortcomings strategically through the framework to identify which factors were leveraged to improve discovery and which oversights might have hindered it. The process of iteratively designing and testing differing improvement approaches will inform the correction and expansion of guidelines provided for improvement of discovery of interactions. While, due to time constraints, objective 5 will not be addressed through a specific scientific contribution, projects aimed to address the other objectives will include evaluation for which I will iterate on existing study designs. My current work is focused on the first identified approach.

My current ongoing work, the Touch Affordances project, is focused on investigating whether real world metaphors can be used to communicate the presence of multiple input possibilities on one soft button.The aim is to bring attention to otherwise signifier-less activity without modifying the properties identifying the element as a button and communicating its purpose.

Plans for following approaches to test, such as leveraging animated transitions to convey otherwise hidden functionality, exist, but are not explicitly formulated in order to remain agile by allowing the inclusion of considerations arising through the current work, research and commercial releases or external feedback.

RESEARCH SITUATION

Figure 2 :

 2 Figure2: The actors influencing the initial interaction as well as examples of the factors placed within them. Since the interaction happens between the system and the user they impact it while they, as well as the interaction, are influenced by the environment in which they are placed. The listed factors act as examples and do not cover all those considered in the full framework.

I

  am pursuing my doctorate under the supervision of Sylvain Malacria and Géry Casiez as a part of LOKI, an Inria project-team in partnership with the Université de Lille within the Joint Research Unit UMR 9189 CNRS-Centrale Lille-Université de Lille. Having started my PhD in October 2020, I am in my second year and projected to complete my studies at the end of the standard three years of a PhD program in France, in September 2023.

  and Instructions originate through the

	System		User
	Affordances		Familiarity
	Signifiers	Initial	Motivation
	Feedback/Feedforward	Interaction	Awareness
	Navigability		Perceived Affordances
	...		...
		Environment
	Social Context	Distractions/Interruptions	...

ACKNOWLEDGMENTS

I would like to thank my supervisors Sylvain Malacria and Géry Casiez for their continued support and guidance. This work was supported by the Agence Nationale de la Recherche (Discovery, ANR-19-CE33-0006)


Laouès Guendouz
email: laoues.guendouz@univ-lorraine.fr

Sébastien Leclerc

Pierre-Louis Marande

Alain Retournard

Daniel Canet

Design of a three-loop asymmetric coil producing a homogeneous radiofrequency B 1 field gradient along the axis of a vertical sample tube

Keywords: radio-frequency field gradient, dedicated radio-frequency coils, diffusion measurements, 1D profiles

A coil system generating a vertical radio-frequency (rf) field gradient ( gradient) has been built for surrounding, in a horizontal magnet, a vertical sample (object) of axial symmetry. The system comprises three coaxial loops with an overall shape either spherical or ellipsoidal. The geometry has been theoretically and experimentally devised for producing a very uniform gradient (cancellation of derivatives from second order up to sixth order) in the central region where a vertical receiver/transmitter coil is installed. The latter is of the saddle-shaped type and is geometrically and electrically decoupled from the gradient coil system. This receiver/transmitter coil not only ensures an optimal signal reception but, in addition, is able to deliver perfectly homogeneous rf hard pulses which are mandatory in most NMR experiments. In its present design, the system delivers a uniform gradient in a limited region but could be extended at will. Its main advantages over static field gradients ( gradients) appear clearly in the case of very short transverse relaxation times. This property has been emphasized in the case of experiments leading to the measurement of diffusion coefficients. Also, this system would be suitable for chemical shift imaging (CSI) experiments as confirmed by a preliminary test experiment.

INTRODUCTION

The use of magnetic field gradients in NMR is ubiquitous. It is mandatory in MRI. Most of the time, gradients of the static magnetic field (also called gradients) are employed essentially because they can be applied indifferently in the three spatial directions and because, thanks to a highly developed technology, they can be easily manipulated. Although possessing some distinct advantages (to be discussed below) [START_REF] Canet | Radiofrequency field gradient experiments[END_REF], the gradients of the radiofrequency magnetic field (also called gradients) have enjoyed far less success. It can be thought that the main reason for this disaffection lies in the difficulty to apply these gradients in the three spatial directions due, inter alia, to the fact that the field is necessarily perpendicular to the direction. Nevertheless, gradients possess some unique properties which are in contrast with gradients: i) their rise and fall times are negligibly small, ii) they are immune to magnetic susceptibility effects, iii) they can act on any component of the nuclear magnetization. This has constituted the incentive to develop gradient systems devised for spatial localization and subsequent specific applications. Often, these have been single-sided flat coil(s) used to produce a gradient which is possibly perpendicular to the sample [START_REF] Haase | Spatial localization of high resolution 31 P spectra with a surface coil[END_REF][START_REF] Karczmar | Detection of motion using B 1 gradients[END_REF][START_REF] Canet | Self-diffusion measurements using radio-frequency field gradients[END_REF][START_REF] Boudot | Spatial labeling by a radio-frequency field gradient. DANTE-Z profile, probed by one-dimensional nutation imaging[END_REF][START_REF] Bourgeois | A gradient method for the detection of slow coherent motion[END_REF][START_REF] Kimmich | Magnetization-grid rotating-frame imaging technique for diffusion and flow measurements[END_REF][START_REF] Humbert | NMR microscopy by strong radio-frequency field gradients with spatial resolution better than five micrometers[END_REF][START_REF] Malveau | Chemical shift imaging in rotating solids by radio-frequency field gradients[END_REF][START_REF] Kuntz | Diffusive diffraction phenomenon in a porous polymer material observed by NMR using radio-frequency field gradients[END_REF]. However, with these arrangements, the gradient may not be uniform in a sufficiently large region (with respect to the object to be investigated). Furthermore, the uniformity of the field within planes perpendicular to the gradient direction is far from being warranted. Relatively recently [START_REF] Tijssen | Spatially resolved spectroscopy using tapered stripline NMR[END_REF], a tapered strapline device has been shown to deliver a very uniform (and possibly intense) gradient still in a single-sided way. This device, quite appealing, is however used as transmitter and receiver which implies receptivity corrections and precludes its use for generating classical hard pulses generally employed in NMR or MRI along with gradient methodology. By contrast, some time ago [START_REF] Guendouz | Single-sided radiofrequency gradient with two unsymmetrical loops: application to nuclear magnetic resonance[END_REF], we proposed a single-sided two-coil system able to deliver a very uniform gradient, as well as a uniform field in planes perpendicular to the gradient direction. This experimental configuration involves also a conventional saddle-shaped coil for usual transmit-receive NMR/MRI operations. Other devices for producing gradients include straddle coil (i.e.

anti-Helmoltz coils) with the inconvenience of zero rf field at the center of the coil assembly [START_REF] Friedrich | a "straddle coil" for in vivo NMR[END_REF][START_REF] Friedrich | Spatial localization using a "straddle coil[END_REF] and a toroid cavity by which strong but non-uniform gradients are obtained [START_REF] Woelk | Imaging diffusion in toroid cavity probes[END_REF]. Also, conic coils have been widely employed although they generally deliver an approximately uniform gradient with strong variations of the rf field in planes perpendicular to the main gradient direction. This type of coil can be positioned around the sample [START_REF] Boehmer | The cone coil, a RF gradient coil for spatial encoding along the B 0 axis in rotating-frame imaging experiments[END_REF][START_REF] Ardelean | Two-pulse nutation echoes generated by gradients of the radiofrequency amplitude and of the main magnetic field[END_REF][START_REF] Ardelean | The nutation spin echo and its use for localized NMR[END_REF][START_REF] Sakellariou | Variable rotation composite pulses for high resolution nuclear magnetic resonance using inhomogeneous magnetic and radiofrequency fields[END_REF][START_REF] Farrher | Probing four orders of magnitude of the diffusion time in porous silica glass with unconventional NMR techniques[END_REF] or remotely along the sample axis [START_REF] Topgaard | Self-diffusion measurements with chemical shift resolution in inhomogeneous magnetic fields[END_REF][START_REF] Topgaard | NMR spectroscopy in inhomogeneous B 0 and B 1 fields with non-linear correlation[END_REF].

The aim of the present work is the construction of a high-performance gradient system, able to surround a cylindrical sample/object so that the latter could be investigated along the symmetry axis direction. The probe design is along the same lines as the single-sided probe previously constructed [START_REF] Guendouz | Single-sided radiofrequency gradient with two unsymmetrical loops: application to nuclear magnetic resonance[END_REF] but, this time, it involves three loops (instead of two) still with a standard receive/transmit coil in the center of the assembly. The latter is geometrically and electrically decoupled from the three-loop gradient system and, in addition to acquire reliably the NMR signal, serves for any pulse sequence requiring homogeneous hard pulses. The three-loop gradient system has been carefully calculated and designed for delivering a very uniform gradient together with a very uniform field in planes perpendicular to the gradient direction. This device can be used in horizontal and vertical magnets, the gradient being in each instance perpendicular to the static magnetic field . The advantages of the three-coil system in comparison with the two-coil system will be presented here. For convenience, because we shall be dealing with samples containing fluids, tests were performed with a horizontal magnet. These tests will include the measurement of diffusion coefficients and a prospective experiment aiming at Chemical Shift Imaging. Although it could presumably be enlarged, the zone where the gradient is uniform is presently of the order of some centimeters which is nevertheless sufficient for standard NMR tubes.

GRADIENT DESIGN AND CALCULATIONS

First, it can be recalled that, in the case of a simple circular loop (denoted ) of radius located at an abscissa with respect to the origin O (see Fig. 1), the magnetic field along the x-axis perpendicular to the loop and positioned at its center can be easily calculated from the Biot-Savart law and approximated as follows (with rounded coefficients)

] ( 1
)
where is the current intensity in the loop and the air permeability.

From the derivative (with respect to ) of ( 1), we obtain the expression of the gradient, only homogeneous up to the second order. ]

From Eq. ( 2), it appears that the gradient is uniform (to within 1%) in a relatively small interval: [ ]. Although the single loop configuration is easy to implement, its modest performance in terms of useful volume and efficiency ( at the origin O) is noteworthy.

Fig. 1. Geometry of a general current loop carrying a current , with radius and symmetry axis Ox (at a radial distance from the origin and subtending an angle with x-axis).

Although the gradient characteristics can be improved by a single-sided two-coil system [START_REF] Guendouz | Single-sided radiofrequency gradient with two unsymmetrical loops: application to nuclear magnetic resonance[END_REF],

it seems advisable to go to a three-loop system not only to improve the gradient but also to give up with the single-sided feature in such a way that the coil assembly can surround the object under investigation (Fig. 2). Fig. 2. A simple sketch of a three-loop system Evidently, this may require further investigations and somewhat lengthy calculations which are detailed in the Appendix. From these calculations, we can derive the expression of the field for a configuration such that three loops are positioned within a sphere (configuration denoted ) and for an optimized configuration where an ellipsoid replaces the sphere (denoted ):

(

) (4) 3
where is the radius of the largest coil.

We turn now to the gradient of along the direction, having in mind that its quality rests on two main features: its strength and its uniformity. Concerning configuration , the gradient is homogeneous up to fourth order and can be expressed as follows (with rounded coefficients) .

(5)

On the other hand, for the configuration , the gradient is homogeneous up to sixth order and can be written:

.

In any case, using closed-form expression, it appears that the gradient is uniform (in the limit On the whole, although all these uniformity intervals are not rigorously centered on the origin O, the three coil systems are seen to be noticeably advantageous. In this respect, the gradient uniformity (along the symmetry axis) is visualized in Fig. 3 for the probes considered in this work.

Fig. 3. Computed normalized rf field gradient : profiles along the symmetry axis for the three-coil systems (solid line in blue) and (dashed line in red), for an optimized two-loop system [START_REF] Guendouz | Single-sided radiofrequency gradient with two unsymmetrical loops: application to nuclear magnetic resonance[END_REF] (dotted line in purple) and for a single loop (dotted line in brown). For the four systems, is the radius of the outer coil.

It must be noted that these calculations performed for a direct current remain valid in the radio-frequency domain provided that the overall dimension of the coil system and of all elements constituting this device remain small with respect to the wavelength associated with the working frequency. This is further discussed in forthcoming section "Prototype".

PROPERTIES OF THE MAGNETIC FIELD AND OF ITS ASSOCIATED GRADIENT FOR THE THREE-COIL ARRANGEMENTS TL S AND TL E

In the following, to facilitate comparisons, we shall consider devices which occupy the same space with respect to the axial symmetry axis. In practice, this amounts to take as reference the outer loop of radius denoted by as before. The dimensions of the different loop configurations studied in this work are reported in Table A2 of the Appendix. The axial component of the rf magnetic field [START_REF] Smythe | Static and Dynamic Electricity[END_REF], in a plane containing the symmetry axis, is displayed by means of a 3D plot in Fig. 5 for the three usual configurations , and (same radius and same current ). These plots allow one to appreciate the uniformity of in planes perpendicular to the gradient direction (the direction).

Configuration appears undoubtedly to be optimal.

From the analytical expression of the components of the magnetic field produced by a coil [START_REF] Smythe | Static and Dynamic Electricity[END_REF], it is possible to calculate the axial component of the field gradient at any point. Contour plots showing the deviation of the value of this component with respect to its value at the origin are displayed in Fig. 6 for our usual three systems. It can be observed that, in the Ox direction, the gradient tends to decrease when the distance to the origin increases whereas it is the opposite behavior in a direction perpendicular to Ox. Fig. 6. Contour plots of the deviation of the axial gradient from the value at center represented in an arbitrary axial symmetry plane (containing the gradient direction). Successive iso-gradient contours correspond to deviations of ±1% (close to center), ±5%, ±10%, ±15%, ±20% (far from center). These contour plots are displayed for the usual configurations: , and .. The geometry of each configuration is schematized by color dots indicating the location of the loops (see fig. 4a). Both axes have the same scale (relative to the radius of the main loop). In the last case, the thin inner curve (purple) corresponding to the ±1% contour plot for the optimal two-coil (with outer diameter ) [START_REF] Guendouz | Single-sided radiofrequency gradient with two unsymmetrical loops: application to nuclear magnetic resonance[END_REF] is added for comparison.

As shown previously [START_REF] Guendouz | Single-sided radiofrequency gradient with two unsymmetrical loops: application to nuclear magnetic resonance[END_REF], the other two components of the field (namely and ) can be neglected in the region surrounded by the additional coil used for the NMR signal detection (and for delivering homogeneous rf pulses). As a matter of fact, these non-axial components become significant far outside the region occupied by the three-coil system.

Thus, from an NMR point of view, the above discussion is perfectly valid.

At this point, a more in-depth comparison between the optimum two-loop system (dubbed in figures 6 and 7) described in our previous paper [START_REF] Guendouz | Single-sided radiofrequency gradient with two unsymmetrical loops: application to nuclear magnetic resonance[END_REF] and the present system is necessary. On a quantitative point of view, figure 7

Moreover, given the common variation in (

) of the gradient field intensities (see Eq. ( 6))

comparisons can be made between the two coil systems when carrying the same current .The optimum three-loop coil presents significant advantages: i) a 30 percent higher gradient field intensity ii) a 60 percent wider on-axis interval of gradient field uniformity at 1% iii) a gradient field uniformity area more centered to the coil system. Now, for the same area of gradient field uniformity at 1% along the principal axis, the coil should be 37.5% smaller and would produce an approximately 234% higher gradient field intensity.

RF PROTOTYPE

For diffusion measurements and with the prospect of Chemical Shift Imaging on standard NMR samples, a prototype of the rf gradient system has been designed to operate in a Bruker Biospec mini-imager equipped with a 2.34 T horizontal magnet featuring a cylindrical working space of 20 cm diameter. The proton resonance frequency being of 100.3 MHz, the corresponding wavelength is about 3 m. A good compromise between the available dimensions of the working space and a realistic construction of our gradient system has been found for an outer coil of diameter (see Table A2 of the Appendix for the dimensions of the other two coils). These dimensions afford a uniform gradient (in the 5% limit) over a length of about 2 cm. For transposing the results of the previous sections (strictly valid for direct current) to the radio-frequency case, the wires constituting the coil system must have a length less than (if possible, less than ). According to this constraint, we have chosen a single series resonant circuit (same current in all wires) with two identical tuning capacitors and separating symmetrically the outer loop from the other two loops (see Fig. 8a). Overview of the deviation of the gradient field in an axial section of the saddle-shaped coil: iso-gradient contours are extracted from Fig. 6. The deviation from the maximum gradient value is less than 10% in a cylindrical volume with 1 cm in diameter and 2.4 cm in height (represented by a green filled rectangle). This probe has been designed for accommodating objects (e.g. NMR tubes) positioned vertically. In order to obtain a good quality factor, the wire of the smallest coil has a diameter of 2 mm while a diameter of 4 mm has been retained for the other two coils. Matching is achieved by a coupling loop identical to the smallest loop of the gradient system, located axially and on the same side as the latter. This additional coil is tuned to the working frequency by a series capacitor. In that way, the gradient system is electrically balanced [START_REF] Terman | Radio Engineers' Handbook[END_REF], as shown by the electrical scheme of Fig. 9. Fig. 9. Equivalent circuit model of the gradient three-coil system with its self-inductances ( ), the mutual inductances ( ) and its symmetric tuning capacitors and . The resistor accounts for losses in the gradient coil system. Matching is achieved by a moveable tuned coupling loop. Both primary and secondary circuits are tuned to the Larmor frequency. Magnetic couplings are indicated by dashed lines.

Normal transmit-receive NMR operations are made possible by a saddle-shaped coil of 1.65 cm diameter and 3.3 cm height. It is made with a wire of 1.5 mm diameter. This coil is positioned vertically as a function of the gradient uniformity. As can be seen from Fig. 8(b), the gradient is quasi-uniform (in the limit of 1%) in a large central region and the uniformity remains in a limit of 5% in a region of 1 cm diameter and 2cm height. For obtaining the best gradient uniformity in the sensitive region of the saddle-shaped coil, the center of the latter has been shifted by 2 mm with respect to the origin O of the gradient system. Owing to these considerations, the diameter of the sample tube must not exceed 14 mm. Finally, the transmitreceive coil is classically tuned and matched by capacitors including a leg capacitor ensuring the electrical symmetry and thus a very good decoupling from the coils of the gradient system. Fig. 10. Photograph of the rf gradient coil prototype operating in a horizontal magnet at 100.3 MHz. The upper part of the cylindrical shield was removed for observation. On either side of the three-loop coil (type with a diameter ) are mounted the tuning capacitors: a fixed ceramic capacitors assembly in parallel with a high voltage cylindrical tunable capacitor. A moveable tuned coupling loop is placed above the gradient coil for matching and a transmit-receive saddle shape coil is visible on the inside.

A photograph of the prototype is shown in Fig. 10. It is surrounded by a cylindrical copper shield (height: 11.5 cm, diameter: 14 cm), the top of which being removable for performing different operations such as decoupling (or taking photographs). This probe has been constructed with several guiding Teflon and glass pieces to facilitate its disassembly.

As an indication, according to the legend of figure 9, the inductance of the three-loop coil is given by: [START_REF] Kimmich | Magnetization-grid rotating-frame imaging technique for diffusion and flow measurements[END_REF] By using standard formulas for (rf) inductances and mutuals [START_REF] Terman | Radio Engineers' Handbook[END_REF], the total inductance reduces to: [START_REF] Humbert | NMR microscopy by strong radio-frequency field gradients with spatial resolution better than five micrometers[END_REF] where is the wire radius of loop ( Thus, without taking into account junction wires, possible couplings and stray capacitances, we find 265 nH for the theoretical inductance of our three-loop system and predict an approximate value for the tuning capacitors and , corresponding to .

More generally, when considering a ratio of about , the inductance can be written simply which is about 67% higher of the value for the two-loop system (see Eq. ( 15) in ref. [START_REF] Guendouz | Single-sided radiofrequency gradient with two unsymmetrical loops: application to nuclear magnetic resonance[END_REF]). Note that, for ratios ( of about , the increase is almost the same. This leads for the system to weaker tuning capacitors, and possibly a higher Q factor or a longer response time (coil ringing). Taking into account the discussion in the previous section, if necessary, it is possible to reduce the inductance of the system by reducing the size of the loops while maintaining good performance in terms of gradient field uniformity and intensity, when compared to the two-loop system . Thus, a good compromise between electrical performance and gradient field quality can be found.

Finally, in practice, the probe being inside the shield, tuning could be achieved with test capacitors of about 15pF. As shown in Fig. 10, the tuning capacitors are an assembly of fixed non-magnetic ceramic capacitors of 6.8pF (100E series, American Technical Ceramics, Huntingdon Station, NY ; Peak working volts 3.6kV) and of a 5-25pF variable capacitor (RP series, Polyflon, Norwalk, CT ; Peak working volts 10kV). In this way, the system can cope with high over-voltage arising from high input power (typically 300W-1kW). The role of fixed capacitors is to reduce the current in the variable capacitors. Concerning the coupling coil, tuning is carried out with a 0.5-10 pF variable capacitor (RP series, Polyflon). The quality factor, measured in unloaded conditions by means of a network analyzer at a frequency of 100.3 MHz, is of 512 for the three-loop system and 254 for the saddled-shape coil.

In theory, the magnetic coupling between these two "orthogonal" coil systems is zero, however, in practice, due to small misalignments and connection wires a weak coupling is observed. Rather than the mutual inductance which is not a significant feature, the isolation factor between the gradient system and the transmit-receive coil has been measured with a vectorial network analyzer, it remains greater than 35 dB. Thus, for a given RF power on one of the coils, the other one will receive less than three thousandth of this RF power. This good decoupling was obtained by a fine geometric adjustment of the two RF coil systems.

For checking the rf field (and its gradient) produced by the three-loop system, we have measured in free space the voltage induced in a pick-up coil that we displaced along the symmetry axis of the system. In a compromise between detection and decoupling, the pick-up coil consisted of a 5 mm circular loop of enameled copper wire with diameter 0.8 mm, the wires at the ends of the loop were twisted over 10 cm. A network analyzer, in the transmission mode, was inserted between the gradient coil and the pick-up coil. Thus, the measured quantity represents essentially the mean magnetic field across the section of an axial cylinder with diameter corresponding to one-tenth the outer diameter of the gradient coil. The result, shown in Fig. 11, is seen to be consistent with the theoretical on-axis curve of the magnetic field.

Fig. 11. Measured profile of the on-axis magnetic field produced by the three-coil prototype. Measurements were carried out in free space using the pick-up coil described in the text. Dashed line represents the normalized theoretical curve (see Fig. 4b).

Of course, this measurement does not take into account the actual environment of the probe, namely, its shield, the bore magnet or even the sample itself. In any case, NMR diffusion measurements (of large samples) should confirm the quality of the gradient field.

Note that, for our prototype, we have considered rather large dimensions. However, for increasing the gradient field intensity, it would be possible to reduce the dimensions by a factor of 2 or even 3 and obtain, for the same carrying current , a gradient intensity 4 to 9 times greater and moreover be free from segmentation problems and alternatively achieve matching and tuning by direct capacitive coupling.

NMR EXPERIMENTS

Here, our objective is simply to go beyond methodological tests for assessing the properties of our new device through "real" classical NMR measurements. Possibly, we shall emphasize the ease of the gradient method.

Diffusion measurements

A first test is to perform a translational diffusion measurement according to the gradient methodology [START_REF] Canet | Radiofrequency field gradient experiments[END_REF]. This simple experiment consists of applying two gradient pulses of duration separated by a diffusion interval (from the end of the first gradient pulse till the beginning of the second gradient pulse), followed by a classical 90° observing pulse and data acquisition. The signal amplitude is given by [START_REF] Malveau | Chemical shift imaging in rotating solids by radio-frequency field gradients[END_REF] where is the classical longitudinal relaxation time, is the transverse relaxation time in the rotating frame ( being the classical transverse relaxation time), the gyromagnetic ratio and the gradient strength. An example of such a measurement, obtained by varying , is given in Fig. 12. The perfect decay of the diffusion curve is a clear indication of the gradient quality. Fig. 12. A typical diffusion experiment performed on a water sample with the gradient system described above. The horizontal scale corresponds to the values in ms. is set to 0.4 s. The dead time of the three-coil system is about 6.5 μs , and the RF amplifier power used for the experiments was about 150 W. With a diffusion coefficient of 2 10 -5 cm 2 s -1 , we obtain a gradient value of 3.8 G cm -1 . Owing to the large difference between and values, was considered as constant and was assimilated to .

It can be remembered that diffusion measurements by gradients are formally equivalent to the widely used stimulated spin echo sequence by gradients. But this is just a formal equivalence. While the gradient method involves two gradient pulses separated by the diffusion interval followed by a observing pulse, with the method, each gradient pulse must be bracketed by two pulses. Moreover, for avoiding the effect of internal gradients, the gradient pulse is generally replaced by the so-called bipolar pulses (two gradient pulses of opposite sign separated by a hard pulse). This does not occur with gradients which are totally immune to variations of magnetic susceptibility within the sample (responsible for internal gradients). At the outcome, the gradient method generally ends up with a time interval called "led" for avoiding effects of eddy currents generated by the fall of the gradient. This is not evidently the case for gradients whose rise and fall times are negligibly short. As a matter of fact, a failure of gradients may occur in particular situations while gradients still fit the bill [START_REF] Leclerc | NMR diffusion measurements under chemical exchange involving a large chemical shift difference[END_REF]. The advantages of the gradient methodology have been well documented until now with however a gradient perpendicular to the sample tube. It was worth, within the present context, to verify that these properties still hold with a gradient along the vertical axis of the sample tube. Furthermore, for assessing the properties of gradients, diffusion measurements by incrementation of the gradient were carried out in the case of a sample for which gradients totally failed (at least with gradients available with our spectrometer: 20 G/cm). In the case of very short T 2 's (see below), the NMR signal disappears because it is destroyed by transverse relaxation during the fall time of the gradient pulses. We were dealing with a water sample containing 4mM of MnCl 2 providing a T 1 value of 50ms and a T 2 value of 3ms. This short value of T 2 explains the failure of gradient experiments for which the NMR signal is so deteriorated that any diffusion measurement turns out to be impossible. By contrast, since the rise and fall times of gradient pulses are totally negligible, diffusion measurements become feasible by this method. The experiment was performed with gradient pulses of duration 2 ms and a diffusion interval = 50 ms. Nine gradient values ranging from 0.5 G/cm up to 3.3 G/cm were used for obtaining a diffusion curve compatible which the usual value of the water diffusion coefficient.

One-dimensional Chemical Shift Imaging (CSI)

Another way for assessing the gradient quality is the sequence of figure 13, which is very simple and leads to the result given in figure 14.

Fig. 13. Basic sequence for a two-dimensional experiment involving a gradient ( ). Phases of the gradient pulse and of the pulse (optional) must be identical. is incremented so as to form a 2D map ( , ) which is further subjected to a double Fourier transform. This pulse leads to a cosine modulation in the spatial dimension (perhaps more convenient for data treatment). Without it, one would have a sine modulation.

The sequence of figure 13

The double Fourier transform with respect to and provides the spatial information in the dimension versus the chemical shift information in the dimension. The profile shown in figure 14 is deduced from the column (in the two-dimensional map) at the frequency corresponding to the water resonance. Would the sample contain other chemical species, Fourier transform of the columns corresponding to their respective resonance frequencies would yield their spatial localization. This is the basis of CSI (Chemical Shift Imaging) by gradients, much simpler to implement than the conventional methods using gradients [START_REF] Pohmann | Theoretical evaluation and comparison of fast chemical shift imaging domain[END_REF].

The latter imply first an rf pulse to take magnetization in the measuring plane and after the evolution period spin echo detection for avoiding the effects of the unavoidable fall time of gradients. Figure 14 can be considered as a significant result for gradient CSI methodology, which can be run with a single gradient period followed immediately by data acquisition.

CONCLUSION

The aim of this work was the design and construction of a novel coil system capable of producing a high quality gradient in the direction of the symmetry axis of this coil system and thus able to accommodate usual vertical NMR sample tubes. Although this represents a relatively modest volume, it can be anticipated that this limitation could be lifted and that a larger volume could be easily reached by a homothetic transformation of the present setup.

This device has first been successfully checked by diffusion measurements and then by a sequence aiming at chemical shift imaging. In each case, one can take advantage of the specific features of a gradient which involves negligible rise and fall times and which is capable on acting on any component of the nuclear magnetization. These features lead invariably to experimental conditions which are simpler and, maybe, more efficient than their counterparts with gradients. In particular, it can be anticipated that the gradient method could be quite suitable for Chemical Shift Imaging (CSI) experiments.

Fig. 4 .

 4 Fig. 4. (a) Cross-sectional view, through an axial symmetry plane, of the coil systems (upper curve, in red) and (lower curve, in blue). Loop positions are marked by bold dots and dashed lines indicate the surface surrounded the coils (spherical or ellipsoidal). (b) Curves of the on-axis magnetic field for the coil systems (solid line in blue), (dashed line in red) and a single loop as defined in the text (dotted line in brown), for comparison. All the coils have the same outer radius and are driven by the same current .

Fig. 4 (Fig. 5 .

 45 Fig. 4(a) depicts the geometry of the two configurations and for the same value of

  (a) shows the geometries of the two gradient coils with same outer diameter, and figure7(b) the corresponding on-axis magnetic field which is in favor of the system. Moreover, the gradient field uniformity area as well as the gradient value are displayed in figure7(c), and demonstrate definitely the superiority of the system.

Fig. 7 .

 7 Fig. 7. (a) Cross-sectional view, through an axial symmetry plane, of the coil systems (upper curve, in purple) [12] and (lower curve, in blue). Loop positions are marked by bold dots and dashed lines indicate the surface surrounded the coils (ellipsoidal). (b) Curves of the on-axis magnetic field (normalized to ) for the coil systems (solid line in blue), (dashed line in magenta). The coils have the same outer radius and are driven by the current . (c) Computed field gradient (normalized to ): profiles along the symmetry axis for the coil systems (solid line in blue) and (dashed line in purple). The vertical bars delimit the 1% uniformity interval.

Fig. 8 .

 8 Fig. 8. (a) Schematic diagram of the rf gradient coil prototype (type ) with balanced tuning capacitors and , and a tuned matching coupling loop. A saddle-shaped coil (diameter: 1.65cm, height: 3.3cm) is used for normal transmit-receive NMR operations. (b)Overview of the deviation of the gradient field in an axial section of the saddle-shaped coil: iso-gradient contours are extracted from Fig.6. The deviation from the maximum gradient value is less than 10% in a cylindrical volume with 1 cm in diameter and 2.4 cm in height (represented by a green filled rectangle).

  involves as many experiments (with incrementation of the time interval ) as necessary for the required spatial resolution. The signal obtained by this sequence has the form (quadrature detection assumed, : resonance frequency in the chemical shift dimension; : free induction decay (fid) of the corresponding NMR signal):.

Fig. 14 .

 14 Fig. 14. Left: A spatial profile as obtained by the sequence of figure 13 for a phantom made of a 10 mm o.d. NMR tube containing water sequestered in two compartments of about 0.5 cm length separated by a spacer of the same length. Notice the sharp edges corresponding to the two compartments indicating an exceptional spatial resolution. Center: expansion of the left spectrum. Right: the pseudo-fid obtained at the frequency of water NMR signal and leading to the left spectrum after Fourier transform (see text), () corresponds to the duration of the gradient pulses.

Funding: This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

Graphical abstract

APPENDIX : Detailed calculations for a three-loop system In the general case of a circular loop of radius and at an arbitrary distance from the origin O (which is located on its symmetry axis ), the magnetic field along the Ox axis can be calculated by means of the following expansion [START_REF] Smythe | Static and Dynamic Electricity[END_REF] (A1) where is the current intensity in the loop and the air permeability.

The various quantities in (A1) are defined in Fig. 1 of the main text; is the associated Legendre polynomial.

Consider now the system of three coaxial circular loops in the co-current mode (Fig. 2 The classical method for producing a uniform gradient consists in the cancellation of derivatives calculated from the expression, starting from order 2 up to the maximum order 24 which can lead to a derivative equal to zero. Equivalently, for an axial magnetic field and referring to Eq. (A2), this amounts to solve the following equations .

(A4)

For our three-coil system, leads to the following equation.

= 0 (A5)

In order to minimize the number of parameters, we have introduced the following quantities and .

Eqs. (A6) and (A7) below correspond to the third and fourth orders, respectively. , (A6)

In a first attempt to solve eqs. (A5-A7), we can consider that the three loops are positioned within a sphere so that . In spite of many solutions in terms of , and , it turns out that, owing to the restrictions on the ( ), only two configurations are possible. In fact, for one of these two configurations, terms of order higher than four are much lower than for the other configuration. This configuration denoted leads, in practice, to a perfectly uniform gradient and corresponds to , , . Using Biot-Savart Law and the data of Table A2 below, the expression of the relevant rf field along the axis common to the three coils is as follows (with rounded coefficients)

where is the radius of the largest coil (the third coil). Note that with the considered accuracy for the roots, the zero coefficients of the expansion remain less than 10 -6 .

Going


Yassine Hmidy

Agnès Rico

Olivier Strauss

Extending the macsum aggregation to interval-valued inputs

Keywords: Interval system, Choquet integral, Macsum aggregation

. In this article, we show that this extension can be easily computed thanks to some properties of the macsum parametric set function.

Introduction

As mentioned by Grabisch in [START_REF] Grabisch | Modelling data by the Choquet integral[END_REF], the linear model is the main tool for modeling dependencies among data. The Choquet integral can be viewed as a generalization of this modeling. It is widely used as a flexible aggregation function in many fields, such as multiple criteria decision making [START_REF] Grabisch | The application of fuzzy integrals in multicriteria decision making[END_REF], classification [START_REF] Fallah Tehrani | Learning monotone nonlinear models using the Choquet integral[END_REF], regression [START_REF] Havens | Machine learning of Choquet integral regression with respect to a bounded capacity (or non-monotonic fuzzy measure)[END_REF] or data fusion [START_REF] Abichou | Choquet integral capacities-based data fusion for system health monitoring[END_REF]. Its main advantage is that it allows to quantify the importance of each variable and the interaction between groups of variables. Therefore it is an aggregation function that offers interpretability on the top of flexibility.

In a recent paper, Strauss et al. [START_REF] Strauss | Macsum: a new interval-valued linear operator[END_REF] were interested in the use of the Choquet integral to model incoherent systems. An incoherent system is a system which, for the same inputs, can have different outputs. This kind of behavior can be due, for example, to unmeasured inputs, to a partially random behavior of the modeled system or to partial lack of knowledge on the system.

There are many works interested in this type of modeling [START_REF] Kieffer | Guaranteed recursive nonlinear state estimation using interval analysis[END_REF][START_REF] Milan | Interval regression by tolerance analysis approach[END_REF][START_REF] Troffaes | Learning and optimal control of imprecise markov decision processes by dynamic programming using the imprecise dirichlet model[END_REF][START_REF] Strauss | Macsum: a new interval-valued linear operator[END_REF]. The originality of the approach proposed by Strauss et al. is that it aims at providing the convex envelope of all the values that should have been outputted by the considered incoherent system for the same inputs. The modeling they propose, under the name of macsum, is an aggregation function that maps a vector of precise inputs onto an interval-valued output. The upper and lower bounds of this interval is computed respectively by the Choquet integral w.r.t. a particular submodular set function, called the macsum operator, and its conjugate.

As stated in [START_REF] Strauss | Macsum: a new interval-valued linear operator[END_REF], this modeling can be thought of as an imprecise linear model, thus representing a lack of knowledge about the input-output relationship of the modeled system. It has the major advantage over most of the other Choquet integral based approaches in that it depends only on a small number of parameters leading to very simple computation.

However, the inaccuracy may not be due solely to scant knowledge of the system. It can also be due to an imprecise measurement of the inputs [START_REF] Lamberto | Autocatalytic reactions as dynamical systems on the interval[END_REF][START_REF] Floquet | Analysis of operational heat exchanger network robustness via interval arithmetic[END_REF][START_REF] Lin | Interval-valued time series models: Estimation based on order statistics exploring the agriculture marketing service data[END_REF]. One can therefore be interested in modeling a system that is imprecisely known and whose inputs are imprecise too. For example, in [START_REF] Lacerda | Interval predictor models for data with measurement uncertainty[END_REF], Lacerda and Crespo propose to extend their interval predictor model to the case where inputs are interval-valued.

In this article, we are interested, as Lacerda and Crespo, in extending the macsum model to the case where the inputs are interval-valued. One of the difficulties of this kind of extension is that it can lead to very complex computations, thus losing the advantage of the simplicity of the macsum model. We show in this paper that extending the macsum model to interval-valued inputs leads to computation whose complexity is equivalent to that of the macsum model with precise inputs. As mentioned by Dubois and Prade [START_REF] Dubois | On the use of aggregation operations in information fusion processes[END_REF], a minimal requirement for those extensions should be to be monotonically increasing in the wide sense with respect to each argument. We propose here two extensions. The first extension is disjunctive. It aims at dealing with the case where some sets to be aggregated cannot be considered as reliable. It presents the advantage of containing all the information but can lead to too broad intervals. The second extension is conjunctive. It aims at dealing with consonant sets of information. It would lead to more narrow intervals, giving the minimal set of common information in the considered sets.

This article is organized as follows. Section 2 gives the background needed to understand the following. Section 3 is a reminder of the macsum aggregation. Section 4 introduces the disjunctive extension and Section 5 the conjunctive extension of the macsum aggregation to interval-valued inputs we propose. In section 6, an example is given to illustrate the low complexity calculation of the bounds of these extensions. We then conclude in Section 7.

Background

-Ω = {1, . . . , N } ⊂ N. -A vector is a function x : Ω → R defined by an element of R N denoted x = (x 1 , • • • , x N ) ∈ R N . -Let x ∈ R N , we define x + , x -∈ R N such that ∀i ∈ Ω, x + i = max(0, x i ) and x - i = min(0, x i ). -x = [x,
x] is a real interval whose lower bound is x and upper bound is x.

-IR is the set of real intervals.

-A vector of real intervals is an element of

IR N denoted x = (x 1 , x 2 , .., x N ). -We say that x ∈ x if ∀i ∈ Ω, x i ∈ x i .
-An aggregation function f : R N → R maps several input values onto a single real value.

-An interval aggregation function f : IR N → IR maps several interval-valued inputs onto a single interval-valued output. -A set function is a function µ : 2 Ω → R that maps any subset of Ω onto a real value. To a set function µ is associated a complementary set function

µ c : ∀A ⊆ Ω, µ c (A) = µ(Ω) -µ(A c ), with µ(∅) = 0. -A set function µ is said to be submodular if ∀A, B ⊆ Ω, µ(A∪B)+µ(A∩B) ≤ µ(A) + µ(B). -A set function µ is said to be supermodular if ∀A, B ⊆ Ω, µ(A ∪ B) + µ(A ∩ B) ≥ µ(A) + µ(B). -The asymmetric Choquet integral of x ∈ R N w.r.t. a set function µ, denoted
ȵ [START_REF] Grabisch | Fuzzy measures and integrals: theory and applications[END_REF], can be defined by:

ȵ (x) = N k=1 x (k) .(µ(A (k) ) -µ(A (k+1) )) = N k=1 (x (k) -x (k-1) ).µ(A (k) ),
where (.) is the permutation that sorts the element of x in increasing order:

x (1) ≤ x (2) ≤ • • • ≤ x (N ) with x (0) = 0 and A (i) (i ∈ Ω) being the coalition of Ω such that A (i) = {(i), . . . , (N )} with A (N +1) = ∅.
-∀x, y ∈ IR, the natural union between x and y and the natural intersection between x and y are respectively defined as:

x ∪ y = {z ∈ R| z ∈ x or z ∈ y} and x ∩ y = {z ∈ R| z ∈ x and z ∈ y}. -∀x, y ∈ IR,

The macsum aggregation

A set function is said to be parametric when its computing involves values of a predefined set of parameters. The macsum operator is a parametric set function denoted ν ϕ where ϕ stands for the vector of parameters. This set function was introduced in [START_REF] Strauss | Macsum: a new interval-valued linear operator[END_REF], and is defined as ∀ϕ ∈ R N , ∀A ⊆ Ω:

ν ϕ (A) = max i∈A ϕ + i + min i∈Ω ϕ - i -min i∈A c ϕ - i , ν c ϕ (A) = min i∈A ϕ - i + max i∈Ω ϕ + i -max i∈A c ϕ + i .
Let ϕ ∈ R N , the macsum aggregation is defined by using the macsum operator ν ϕ as:

A νϕ (x) = [ Čν c ϕ (x), Čνϕ (x)]. (1)
Let ψ ∈ R N , ∀A ⊆ Ω, we denote λ ψ the linear parametric set function defined by:

λ ψ (A) = i∈A ψ i .
Let ψ ∈ R N , the linear aggregation is defined by using the linear operator λ ψ as:

A λ ψ (x) = Čλ ψ (x) = i∈Ω ψ i .x i .
Contrarily to the macsum aggregation, the linear aggregation is precise-valued. It can easily be extended to interval valued inputs by ∀x ∈ IR:

A λ ψ (x) = { Čλ ψ (x) / x ∈ x} = inf x∈x A λ ψ (x), sup x∈x A λ ψ (x) ,
this set being convex due to the fact that x is convex and A λ ψ is linear.

The macsum core of a vector ϕ of R N is a convex subset of R N defined as:

M(ϕ) = ψ ∈ R N / ∀A ⊆ Ω, ν c ϕ (A) ≤ λ ψ (A) ≤ ν ϕ (A) .
Since the macsum operator is submodular, as proven in [START_REF] Strauss | Macsum: a new interval-valued linear operator[END_REF], the macsum core of a vector of R N is never empty. Moreover, Equation (1) can be interpreted as:

A νϕ (x) = A λ ψ (x) / ψ ∈ M(ϕ) ,
this set being convex [START_REF] Strauss | Macsum: a new interval-valued linear operator[END_REF]: ∀ψ ∈ M(ϕ), ∃y ∈ A νϕ (x) such that y = A λ ψ (x) and ∀y ∈ A νϕ (x), ∃ψ ∈ M(ϕ) such that y = A λ ψ (x).

Disjunctive extension to interval-valued inputs

Extending the macsum aggregation to interval valued inputs consists in considering aggregating the set of all A νϕ (x) with x ∈ x. In this section, we consider a very conservative aggregation consisting in keeping all the aggregated interval-valued outputs of the macsum aggregations for each x of x. This can be defined,∀x ∈ IR N , by:

D νϕ (x) = x∈x A νϕ (x) = x∈x Čν c ϕ (x), Čνϕ (x) = A λ ψ (x) / x ∈ x, ψ ∈ M(ϕ) . (2)
Remark 4. Given the Choquet integral w.r.t. the macsum operator is continuous as proven in [START_REF] Grabisch | Aggregation Functions. Encyclopedia of Mathematics and its Applications[END_REF] (Propostion 5.39) and x is a compact subspace of R N , we have

that x∈x [ Čν c ϕ (x), Čνϕ (x)] = ˜ x∈x [ Čν c ϕ (x), Čνϕ (x)]. Thus Equation (2) can be rewritten, ∀x ∈ IR N : D νϕ (x) = ˜ x∈x Čν c ϕ (x), Čνϕ (x) = inf x∈x Čν c ϕ (x), sup x∈x Čνϕ (x) . (3)
Due to the definition of the disjunctive extension given in Equation ( 2), and the fact that the A νϕ (x) are convex sets of R, the bounds of D νϕ (x) are reached, i.e. ∃y ∈ x and ∃ψ ∈ M(ϕ) such that sup x∈x Čνϕ (x) = A λ ψ (y) and ∃y ∈ x and ∃ψ ∈ M(ϕ) such that inf x∈x Čν c ϕ (x) = A λ ψ (y ).

The two following propositions are needed to compute the disjunctive extension.

Proposition 1. (Proposition 5.1 in [16]). Let ϕ ∈ R N , let ψ ∈ M(ϕ), ∀i ∈ Ω, ϕ i > 0 ⇒ ψ i ≥ 0 and ϕ i < 0 ⇒ ψ i ≤ 0. Proposition 2. Let ϕ ∈ R N , let ψ ∈ M(ϕ), ∀i ∈ Ω, ϕ i = 0 ⇒ ψ i = 0.
Proof. Let j ∈ Ω such that ϕ j = 0. We have ν ϕ ({j}) = 0 + min i∈Ω ϕ - imin i∈Ω\j ϕ - i . As ϕ j = 0, either min i∈Ω ϕ - i < 0 and then min i∈Ω ϕ - i = min i∈Ω\j ϕ - i which gives ν ϕ ({j}) = 0, or min i∈Ω ϕ - i = ϕ j = min i∈Ω\j ϕ - i = 0, which also gives ν ϕ ({j}) = 0. Furthermore, we have ν c ϕ ({j}) = 0 + max i∈Ω ϕ + i -max i∈Ω\j ϕ + i . Proving that max i∈Ω ϕ + i = max i∈Ω\j ϕ + i and consequently ν c ϕ ({j}) = 0 can be done in the same manner. As ψ ∈ M(ϕ), we have

∀A ⊆ Ω, ν c ϕ (A) ≤ λ ψ (A) ≤ ν ϕ (A), therefore ν c ϕ ({j}) ≤ λ ψ ({j}) ≤ ν ϕ ({j}), thus 0 ≤ ψ j ≤ 0 i.e. ψ j = 0.
Therefore the non-negativity (resp. non-positivity) of an element of a parameter entails the non-negativity (resp. non-positivity) of the corresponding element of any parameter of its macsum core.

Proposition 3. (Proposition 2.2 in [16]) Let ϕ ∈ R N , let ψ ∈ M(ϕ), ∀x ∈ R N we have A λ ψ (x) = Čλ ψ (x) ∈ A νϕ (x) = Čν c ϕ (x), Čνϕ (x) .
Considering that ∀x ∈ IR, Čν c ϕ (x) = inf ψ∈M(ϕ) Čψ (x) and Čνϕ (x) = sup ψ∈M(ϕ) Čψ (x) Equation ( 3) can be rewritten:

D νϕ (x) = inf x∈x inf ψ∈M(ϕ) Čλ ψ (x), sup x∈x sup ψ∈M(ϕ) Čλ ψ (x) (4) = inf ψ∈M(ϕ) inf x∈x Čλ ψ (x), sup ψ∈M(ϕ) sup x∈x Čλ ψ (x) . (5)
Now, let x ∈ IR N be a vector of real intervals and let ϕ ∈ R N . Let us define

x * ∈ R N the vector such that ∀i ∈ Ω, x * i = x i if ϕ i ≥ 0 and x * i = x i if ϕ i < 0. Let us also define x * ∈ R N the vector such that ∀i ∈ Ω, x * i = x i if ϕ i ≥ 0 and x * i = x i if ϕ i < 0. Proposition 4. ∀x ∈ IR N , ∀ϕ ∈ R N , ∀ψ ∈ M(ϕ), sup x∈x Čλ ψ (x) = Čλ ψ (x * ) and inf x∈x Čλ ψ (x) = Čλ ψ (x * ).
Proof. Given Propositions 1 and 2, we know that ∀i ∈ Ω, if ϕ i ≥ 0 then ψ i ≥ 0 and if ϕ i < 0 then ψ i ≤ 0. Furthermore, by construction Čλ ψ (x) = i∈Ω ψ + i .x i + i∈Ω ψ - i .x i . Thus ∀x ∈ x we have:

Čλ ψ (x) = i∈Ω ψ i .x i ≤ i∈Ω ψ + i .x i + i∈Ω ψ - i .x i = Čλ ψ (x * ). Consequently sup x∈x Čλ ψ (x) = Čλ ψ (x * ).
Proving that inf x∈x Čλ ψ (x) = Čλ ψ (x * ) can be done by the same way.

Proposition 5. ∀x ∈ IR N , ∀ϕ ∈ R N , D νϕ (x) = [ Čν c ϕ (x * ), Čνϕ (x * )].
Proof. We know from [START_REF] Strauss | Macsum: a new interval-valued linear operator[END_REF] that ∀x ∈ R N , ∀ϕ ∈ R N , Čνϕ (x) = sup ψ∈M(ϕ) Čλ ψ (x), and from Proposition 4 that sup x∈x sup ψ∈M(ϕ) Čλ ψ (x) = sup ψ∈M(ϕ) Čλ ψ (x * ). Thus, sup x∈x Čνϕ (x) = Čνϕ (x * ). We can prove that inf x∈x Čν c ϕ (x) = Čν c ϕ (x * ) in the same way.

Thus the relation between the parameter of the macsum operator and the vectors of its core allows to compute the macsum aggregation on interval-valued inputs with a linear complexity.

Conjunctive extension to interval-valued inputs

In this section, we consider the conjunctive aggregation that can be viewed as the antonymous version of the previously defined disjunctive aggregation. Where disjunctive aggregation is conservative and tries not to reject any information, conjunctive aggregation tries to reduce the set of values to those for which each set being aggregated agrees.

There are two possible ways to create this conjunctive aggregation: either by making the conjunction on the input vectors x belonging to the interval-valued input vector x, thus taking the conjunctive counterpart of Equation ( 4), or by making the conjunction on the vectors ψ belonging to the macsum core of the macsum parameter ϕ, thus taking the conjunctive counterpart of Equation [START_REF] Goldsztejn | Modal intervals revisited part 1: A generalized interval natural extension[END_REF].

The first approach leads to:

C νϕ (x) = x∈x A νϕ (x) = x∈x {A λ ψ (x) / ψ ∈ M(ϕ)}, (6)
while the second approach leads to:

C νϕ (x) = ψ∈M(ϕ) A λ ψ (x) = ψ∈M(ϕ) {A λ ψ (x) / x ∈ x}. (7)
Let us consider the computation of Equation ( 6). If C νϕ (x) = ∅, then:

C νϕ (x) = ˜ x∈x A νϕ (x) = ˜ x∈x Čν c ϕ (x), Čνϕ (x) (8)
= sup x∈x Čν c ϕ (x), inf x∈x Čνϕ (x) = sup x∈x inf ψ∈M(ϕ) Čλ ψ (x), inf x∈x sup ψ∈M(ϕ)
Čλ ψ (x) .

In the same way, computation of Equation ( 7), if C νϕ (x) = ∅, leads to:

C νϕ (x) = ˜ ψ∈M(ϕ) A λ ψ (x) = sup ψ∈M(ϕ) inf x∈x Čλ ψ (x), inf ψ∈M(ϕ) sup x∈x Čλ ψ (x) . (9)
Proposition 6. ∀x ∈ IR N , ∀ϕ ∈ R N , the upper bound of C νϕ (x) equals the lower bound of C νϕ (x) and vice versa.

Proof. Lets remind that ∀ϕ ∈ R N , ∀ψ ∈ M(ϕ), we have that

sup x∈x Čλ ψ (x) = Čλ ψ (x * ) and that ∀x ∈ R N , inf ψ∈M(ϕ) Čλ ψ (x) = Čν c ϕ (x). Therefore, inf ψ∈M(ϕ) sup x∈x Čλ ψ (x) = inf ψ∈M(ϕ) Čλ ψ (x * ) = Čν c ϕ (x * ). Moreover, sup x∈x inf ψ∈M(ϕ) Čλ ψ (x) = sup x∈x Čν c ϕ (x) = Čν c ϕ (x *
). Thus, inf ψ∈M(ϕ) sup x∈x Čλ ψ (x) = sup x∈x inf ψ∈M(ϕ) Čλ ψ (x), i.e. the upper bound of C νϕ (x) equals the lower bound of C νϕ (x). By the same reasoning we obtain that the lower bound of C νϕ (x) equals the upper bound of C νϕ (x).

If we refer to the theory of generalized intervals [START_REF] Goldsztejn | Modal intervals revisited part 1: A generalized interval natural extension[END_REF], we can say that if C νϕ (x) is proper then C νϕ (x) is improper and conversely. In other words,

C νϕ (x) = ∅ ⇒ C νϕ (x) = ∅ and C νϕ (x) = ∅ ⇒ C νϕ (x) = ∅.
In that sense those two approaches can be considered as dual. We thus have:

C νϕ (x) = Čν c ϕ (x * ), Čνϕ (x * ) , if Čν c ϕ (x * ) ≤ Čνϕ (x * ), and
C νϕ (x) = Čνϕ (x * ), Čν c ϕ (x * ) , if Čν c ϕ (x * ) ≥ Čνϕ (x * ).
We thus propose to consider, for the conjunctive extension of the macsum aggregation to interval-valued inputs, the following expression:

C νϕ (x) = min Čν c ϕ (x * ), Čνϕ (x * ) , max Čν c ϕ (x * ), Čνϕ (x * ) .
This formula gathers in the same expression the two conjunctive counterparts (i.e. Equations ( 8) and ( 9)) of the disjunctive extension. As in the disjunctive case, this extension only needs computation of two Choquet integrals.

Example

In this section, we give a simple example illustrating the simplicity of the computation of the macsum aggregation of interval-valued inputs. Let x = ([-2, 3], [0, 1], [-5, 8]) ∈ IR 3 and let ϕ = (-2, 3, -1) ∈ R 3 . Let us compute D νϕ (x) and C νϕ (x). We have x * = (-2, 1, -5) and x * = (3, 0, 8). Now lets sort x * in increasing order:

x * (.) = (x * 3 , x * 1 , x * 2 ) = (-5, -2, 1). Therefore ϕ (.) = (ϕ 3 , ϕ 1 , ϕ 2 ) = (-1, -2, 3). Remember that ν ϕ (A (k) ) = max i∈A (k) ϕ + i -min i∈A c (k) ϕ - i + min i∈Ω ϕ - i , with A (k) = {(k), . . . , (N )}, thus ν ϕ (A (k) ) = max N i=k ϕ + (i) -min k-1 i=1 ϕ - (i) + min i∈Ω ϕ - i (by convention min 0 i=1 ϕ - (i) = 0). Thus: Čνϕ (x * ) = 3 k=1 (x * (k) -x * (k-1) ).ν ϕ (A (k) ), = 3 k=1 (x * (k) -x * (k-1) ). 3 max i=k ϕ + (i) - k-1 min i=1 ϕ - (i) + min i∈Ω ϕ - i , = (x * (1) -x * (0) ). 3 max i=1 ϕ + (i) - 0 min i=1 ϕ - (i) + min i∈Ω ϕ - i + (x * (2) -x * (1) ). 3 max i=2 ϕ + (i) - 1 min i=1 ϕ - (i) + min i∈Ω ϕ - i + (x * (3) -x * (2) ). 3 max i=3 ϕ + (i) - 2 min i=1 ϕ - (i) + min i∈Ω ϕ - i , = -5. (3 -2) + (-2 + 5). (3 + 1 -2) + (1 + 2). (3 + 2 -2) , = -5 + 6 + 9 = 10.
Lets now sort x * in increasing order:

x * (.) = (x * 2 , x * 1 , x * 3 ) = (0, 3, 8). Therefore ϕ (.) = (ϕ 2 , ϕ 1 , ϕ 3 ) = (3, -2, -1). Remember that A c (k) = {(1), . . . (k -1)}, we have ν c ϕ (A (k) ) = min N i=k ϕ - (i) - max k-1 i=1 ϕ + (i) + max i∈Ω ϕ + i (by convention min 0 i=1 ϕ - (i) = 0). Then: Čν c ϕ (x * ) = 3 k=1 (x * (k) -x * (k-1) ).ν c ϕ (A (k) ), = 3 k=1 (x * (k) -x * (k-1) ). 3 min i=k ϕ - (i) - k-1 max i=1 ϕ + (i) + max

Discussion

The macsum aggregation, as introduced in [START_REF] Strauss | Macsum: a new interval-valued linear operator[END_REF], allows to model incoherent systems. Aggregating precise inputs within the macsum approach leads to intervalvalued outputs. In contrast to a simple linear model, it gives a convex set of linear models which makes it more flexible, while preserving the simplicity of use of linear model, as it depends only on N parameters, N being the number of inputs. A promising path would be to use this extension of the macsum aggregation as a simple and flexible learning model. This gives rise to other theoretical issues on its ability to approximate any functions that maps interval-valued inputs onto interval-valued outputs. Moreover, as the macsum operator needs the separation of the positive and negative elements of its parameter, further work could go towards the definition of a simpler submodular parametric set function in order to facilitate the computations during a learning process.

Remark 1 .Remark 3 .

 13 the extensive union between x and y and the extensive intersection between x and y are respectively defined as: x ∪y = [min(x, y), max(x, y)] and x ∩y = [max(x, y), min(x, y)]. The natural and extensive unions of two intervals x and y are equal if x∩y = ∅. Remark 2. x ∪y is the convex envelope of x ∪ y. If x∩y = ∅, then x∩y = x ∩y.

  * (1) -x * (0) ).

  * (2) -x * (1) ).

  * (3) -x * (2) ).

  (-2 + 3) + (3 -0). (-2 -3 + 3) + (8 -3). (-1 -3 + 3) , = 0 -6 -5 = -11.

Finally, D

  νϕ (([-2, 3], [0, 1], [-5, 8])) = [-11, 10]. By the same way we show that C νϕ (([-2, 3], [0, 1], [-5, 8])) = [-6, 7].