Morgan Le Delliou
email: delliou@cii.fc.ul.pt

PROBING DYNAMICAL DARK ENERGY WITH PRESS-SCHECHTER MASS FUNCTIONS

Keywords: Semi-analytic modeling, Dark matter, Galaxy clusters, Dark energy theory

This project proposes to discriminate in the wealth of models for dark energy using the formation of non-linear dark matter structures. In particular,it focuses on structures traced by the mass function of dark matter haloes.

Introduction

Among the various models proposed for the explanation of the accelerated expansion in terms of dark energy (DE), distinctions can be made between static (cosmological constant) or dynamical (e.g. quintessence), coupled or uncoupled to dark matter (DM), clustering or unclustering or even unified (Chaplygin Gas) DE. This wealth of models calls for discriminating schemes. The goal of this work is to propose a unified analysis extending previous studies (see Le Delliou 2006, 1 Manera & Mota 2006 2 and 3 references therein) of DE impact on dark matter haloes on mass functions, for confrontation with other DE assessments.

Λ 4 Q (κQ -M 1 ) 2 +M 2 ) e -λκQ SUGRA Λ 4+α Q Q α e κ 2 Q 2 2 SUSY+extra dim.=SUGRA superpot. Sahni & Wang 2000 5 Λ 4 Q (cosh (λκQ) -1) α Ferreira & Joyce 1998 Λ 4 Q e -λκQ extra dim. compactification Dodelson et al. 2000 6 Λ 4 Q e -λκQ (1 + α sin (νκQ)) Steinhardt et al. 1999 Λ 4 Q e 1 κQ
= R.P. R.P.×F.J.

Λ 4+α Q Q α e -λκQ Barreiro et al. 2000 Λ 4 Q e -ακQ +e -βκQ double exponential Bertolami et al. 2004 7 V 0 e 3(α-1)φ × cosh κφ 2/m 2 α+1 + cosh κφ 2/m -2α α+1

Models and Mass functions

We model a cosmic fluid with baryons, radiation and, either uncoupled (and coupled) DM with a(n) (un)clustering scalar field DE (quintessence) Q, (non-)minimally coupled to DM, or a Chaplygin Gas (GCG) -DM/DE unified component -defined either from P ∝ -ρ -α or from a scalar field mimicking it. We restrict to flat backgrounds, a linear coupling and model clustering through energy conservation. The models are defined by their potential (Table 1). We already studied homogeneous minimal quintessences (left side, upper part 1 ) and a coupled, clustering quintessence (left side, lower part 2 ). All potentials shown will have clustering and interaction. We use the top hat spherical collapse to model non-linear structure formation as a Friedmann sphere with higher, varying curvature. We extract the linearly extrapolated overdensity as a function of non-linear collapse scale factor δ c0 (a c ). This is combined in a Press-Schechter scheme to get the mass function of large scale structures. The results obtained so far are presented in Fig. 1.

Conclusions

Extending previous evidence of DE models impact on DM mass functions, our results permit the confrontation of several homogeneous models and the examination of clustering and interacting quintessence. This have shown that more insights can be drawn from confrontation of several homogeneous models 1 and that strong effects on mass function evolution proceed from clustering and interacting quintessence. 2 Indeed, the spread of ∼10% at 10 14 h -1 M ⊙ between mass functions and the hierarchy between models on the lower (z = 1) panel of the left part of Fig. 1 shows that the method should be most discriminant on clusters scales and that the impact of ω Q dominates other effects. Moreover, its right part entails that, contrary to homogeneous models, DE clustering increases DM clustering while coupling decreases it. This motivates our extended study of DE models with mass functions. Some pending questions remain: our use of Birkhoff's theorem with spherical symmetry in cosmology may require some mass function corrections; geometric effects are argued to induce a degeneracy in angular mass functions, 8 not taking the bias-geometry dependence 9 into account. We are extending our results to other models (Table 1; Chaplygin gas), including clustering and interacting DE. Further developments are also planned.

Fig. 1 .

 1 Fig. 1. Cumulative mass functions for different models

Table 1 .

 1 Choice of scalar field's potentials

		Explored models 1,2	models under scrutiny
	Model	Potential V	Origin	Model	Potential V
	R.P.	Λ 4+α Q Q α	Historical potential; global SUSY	Albrecht & Skordis 2000 4

Acknowledgements

See acknowledgements in. 1 Current work involves J.P. Mimoso, U. Lisboa, D.F. Mota, U. Heidelberg, C. van de Bruck, U. Sheffield and O. Bertolami, IST Lisboa.


Elsa Negre
email: elsa.negre@lamsade.dauphine.fr

Towards a knowledge(experience)-based recommender system for crisis management

Keywords: Knowledge, Experience, Recommender systems, Crisis management, Early warning systems, Decision support

An early warning system can be defined as a chain of information communication systems comprising sensor, detection, decision, and broker subsystems, in the given order, working in conjunction, forecasting and signaling disturbances adversely affecting the stability of the physical world; and giving sufficient time for the response system to prepare resources and response actions to minimize the impact on the stability of the physical world.

In this paper, we present a framework for a recommender system for crisis management.

This framework uses the actions already implemented to manage former crises to enhance the management of a given crisis. The main idea is to recommend the actions already implemented in those former crises that are similar (the similarity between two crises is based on some indicators such as the gap (hurricane, tsunami, ...)) as the actions to be implemented.

Finally, this paper proposes to exploit the knowledge gained from past experiences to make the best decision (i.e. the best actions to implement) in order to better manage a crisis ready to occur.

I. INTRODUCTION

Early warning systems are contextualized and used in many fields such as school [START_REF] Heppen | Developing early warning systems to identify potential high school dropouts[END_REF], finance [START_REF] Cheang | Early warning system for financial crises[END_REF], [START_REF] Gaytàn | A review of the literature on early warning systems for banking crises[END_REF], environment [START_REF] Angermann | Communication architecture of an early warning system[END_REF], [START_REF]Humanitarian & Development Network (HDN); Platform for the Promotion of Early Warning (UNISDR PPEW)[END_REF], humanitarian [START_REF] Buchanan-Smith | Role of early warning systems in decision making processes[END_REF], [START_REF] Bakker | Early warning by ngos in conflict areas[END_REF], [START_REF]Community early warning systems: guiding principles[END_REF], biology [START_REF] Genovesi | Towards an early warning and information system for invasive alien species (ias) threatening biodiversity in europe[END_REF], ... An early warning system can be defined as a chain of information communication systems comprising sensor, detection, decision, and broker subsystems, in the given order, working in conjunction, forecasting and signaling disturbances adversely affecting the stability of the physical world; and giving sufficient time for the response system to prepare resources and response actions to minimize the impact on the stability of the physical world [START_REF] Waidyanatha | Towards a typology of integrated functional early warning systems[END_REF]. Thus it is a set of tools to predict disasters, dropouts, ... [START_REF]of Early Warning Systems: an Assessment of Capacities, Gaps and Opportunities Toward Building a Comprehensive Global Early Warning System for All Natural Hazards: a Report Prepared at the Request of the Secretary-General of the United Nations[END_REF], [START_REF] Joseph | Early warning systems: A review[END_REF].

Early warning systems face two fundamental problems. First, there is the informational problem of obtaining both the necessary quantity and quality of intelligence in a reliable and timely fashion. Second, there is the analytic problem of avoiding misperception or other faulty analysis (likelihood of diffusion and/or escalation of the conflict, potential risks, ...) [START_REF] Bakker | Early warning by ngos in conflict areas[END_REF].

In fact, [START_REF] Peri | Early warning systems -interdisciplinary observations and policies from a local government perspective -four elements of people centered early warning systems[END_REF] explains that "Natural hazards, such as storms, droughts, volcanic eruptions, or earthquakes, become disasters only if a community or population is exposed to the natural hazard and cannot cope with its effects. Torrential rain in the middle of an ocean will not cause a disaster, but the same heavy rainfall on a vulnerable population say a shanty town on the side of a hillside stripped of trees -may result in landslides and a huge loss of life. A minor drought may cause a famine if a region's agricultural production is already highly stressed by civil war. A community that lacks an early warning system for volcanic eruptions will be devastated when volcanic ash clouds bear down upon them. Vulnerability is the potential additive that mixes with natural hazards to cause disasters". [START_REF]of Early Warning Systems: an Assessment of Capacities, Gaps and Opportunities Toward Building a Comprehensive Global Early Warning System for All Natural Hazards: a Report Prepared at the Request of the Secretary-General of the United Nations[END_REF] points out the fact that former experiences have to be taken into account in order to improve the decisions to make or the actions to implement as part of the crisis management, in particular via the use of early warning systems.

Consequently, in this paper, we propose to put in place a recommender system based on former experiences / knowledge in order to guide the decision-makers when warnings appear, i.e., to recommend actions to implement / decisions to make using the ones implemented / made during former similar situations (circumstances).

The paper is organized as follows: Section 2 presents related work, Section 3 introduces our generic framework. We discuss future work in Section 4.

II. RELATED WORK

Early warning systems are specific to the context / field for the one they are implemented. However, eight early warning system characteristics have been defined [START_REF] Glantz | Early Warning Systems: Do's and Don'ts, ser. Usable science[END_REF]: continuity in operations, timely warnings, transparency, integration, human capacity, flexibility, catalysts, apolitical. The ones that are interesting in our context are:

• flexibility: this characteristic can be improved,

• integration: an early warning system is a part of a whole, some other parts can be added, for example, a recommender system, • apolitical position: we propose to recommend former actions that have maybe been implemented by different political parties. According to [START_REF]of Early Warning Systems: an Assessment of Capacities, Gaps and Opportunities Toward Building a Comprehensive Global Early Warning System for All Natural Hazards: a Report Prepared at the Request of the Secretary-General of the United Nations[END_REF] and [START_REF] Peri | Early warning systems -interdisciplinary observations and policies from a local government perspective -four elements of people centered early warning systems[END_REF], a complete and effective early warning system comprises four elements: risk knowledge, monitoring and warning service, dissemination and communication, and response capability. Among these four elements, three are more interesting in our context:

• Risk knowledge: [START_REF]of Early Warning Systems: an Assessment of Capacities, Gaps and Opportunities Toward Building a Comprehensive Global Early Warning System for All Natural Hazards: a Report Prepared at the Request of the Secretary-General of the United Nations[END_REF] indicates that there exists a problem of safeguarding and use of data, especially for old data. Having a tool integrating former experiences / knowledge can be an improvement. • Monitoring: [START_REF]of Early Warning Systems: an Assessment of Capacities, Gaps and Opportunities Toward Building a Comprehensive Global Early Warning System for All Natural Hazards: a Report Prepared at the Request of the Secretary-General of the United Nations[END_REF] indicates that there is a lack of data exchange and procedure sharing. Having a tool integrating former experiences / knowledge coming from different sources can be an improvement. • Communication: [START_REF]of Early Warning Systems: an Assessment of Capacities, Gaps and Opportunities Toward Building a Comprehensive Global Early Warning System for All Natural Hazards: a Report Prepared at the Request of the Secretary-General of the United Nations[END_REF] indicates that it is difficult to integrate lessons from the past (experiences or warnings). Having a tool integrating former experiences / knowledge coming from different sources and using what happens in the past to make a decision in the present can be an improvement. Moreover, [START_REF]Ecosystems and human well-being: A framework for assessment[END_REF] and [START_REF] Schrodt | The impact of early warning on institutional responses to complex humanitarian crises[END_REF] highlight the importance of taking into account current knowledge [START_REF]Ecosystems and human well-being: A framework for assessment[END_REF] but also past knowledge [START_REF] Schrodt | The impact of early warning on institutional responses to complex humanitarian crises[END_REF].

Additionally, [START_REF] Hall | Early warning systems: Reframing the discussion[END_REF] and [START_REF]2005-2015: Building the resilience of nations and communities to disasters[END_REF] emphasize the need of putting in place early warning systems that are efficient, relevant and user-centered and that allow interactions between all tools, persons and other available actors. As a result, international collaborations [START_REF] Joseph | Early warning systems: A review[END_REF] and information sharing mechanisms [START_REF] Hall | Early warning systems: Reframing the discussion[END_REF] have to be set up.

Finally, early warning systems produce warnings. But, a simple warning is often not enough, in particular for a politician. In case of crisis, one wants solutions, i.e., actions to implement / decisions to make [START_REF]of Early Warning Systems: an Assessment of Capacities, Gaps and Opportunities Toward Building a Comprehensive Global Early Warning System for All Natural Hazards: a Report Prepared at the Request of the Secretary-General of the United Nations[END_REF], [START_REF] Lavoix | Developing an early warning system for crises. From early warning to early action? The debate on the enhancement of the EUs crisis response capability continues[END_REF].

To overcome these shortcomings, we propose to use a recommender system (see [START_REF] Adomavicius | Toward the next generation of recommender systems: A survey of the state-of-the-art and possible extensions[END_REF] for a survey). Recommender systems are a particular form of information filtering designed to present information items (movies, music, books, images, web pages, ...) that may interest the user.

Recommender systems have been studied in many fields, cognitive science, information retrieval [START_REF] Baeza-Yates | Modern Information Retrieval[END_REF], [START_REF] Salton | Introduction to Modern Information Retrieval[END_REF], web [START_REF] Baeza-Yates | Query Recommendation Using Query Logs in Search Engines[END_REF], [START_REF] White | Studying the use of popular destinations to enhance web search interaction[END_REF], e-commerce [START_REF] Schafer | E-commerce Recommendation Applications[END_REF], web usage mining [START_REF] Srivastava | Web Usage Mining: Discovery and Applications of Usage Patterns from Web data[END_REF], [START_REF] Baeza-Yates | Modeling user search behavior[END_REF] and many others. The problem of recommendation can be summarized by the problem of estimating scores for items that have not been seen by a user. Indeed, the number of items and the number of users of the system can be very important, it is, therefore, difficult for each user to see all items or that each item is evaluated by all users. It is therefore necessary to estimate the scores for items not yet evaluated.

Intuitively, this valuation is usually based on the scores given by a user to other items and other information that will be formally described below. When it is possible to estimate the scores for items not yet evaluated, then the items with the highest scores may be recommended to the user. More formally, [START_REF] Adomavicius | Toward the Next Generation of Recommender Systems: A Survey of the State-of-the-Art and Possible Extensions[END_REF] formulates the problem of recommendation in the field of e-commerce as follows.

Definition: Recommendation for e-commerce Given P the set of all users and M the set of all possible items that can be recommended (such as books, movies, restaurants, ...). Given u a function measuring the utility of an item m for a user p, i.e., u : P × M → R. Then, for each user p ∈ P , we want to choose the item m ∈ M that maximizes the utility for the user: ∀p ∈ P, m p = argmax m∈M u(p, m).

In recommender systems, the utility of an item is usually represented by a score that indicates how a particular user liked a particular item. For example, the user M ichel gave the score 3 (the maximum score being 10) to the movie "Harry Potter". Example: Note that a cell (i, j) of this matrix corresponds to the utility score given to the movie j by the user i.

In
The central problem of recommender systems is that the utility u is not usually defined on the full P × M space, but only on a subset of it. This means that u must be extrapolated to the entire P × M space.

In recommendation systems, the utility is typically represented by the scores and is first defined over the items previously rated by users. Therefore, the recommendation engine should be able to estimate / predict the scores of item / user unevaluated combinations and to propose relevant recommendations based on these forecasts.

Once the unknown scores are estimated, actual recommendations of an item to a user are proposed by choosing the highest score among all scores provided for the user, according to the formula given in the previous definition (Recommendation for e-commerce).

A recommendation in e-commerce, as defined previously, is the item m ∈ M (set of all items (movies, books, ...)) such as the utility for a user p ∈ P (set of all users) is maximum.

To the best of our knowledge, this is the first work dealing with the problem of recommending actions in crisis management and early warning systems context. The idea of using what the other users (called, collaborative approach) did to generate recommendations is very popular in Information Retrieval [START_REF] Adomavicius | Toward the next generation of recommender systems: A survey of the state-of-the-art and possible extensions[END_REF], and Web Usage Mining [START_REF] Srivastava | Web usage mining: discovery and applications of usage patterns from web data[END_REF]. Our contribution is to adapt these existing techniques to our context. By analogy, we can define a recommendation for warnings as an action a ∈ A (set of all possible actions) to implement such as its utility for a warning w ∈ W (set of all possible warnings) is maximum.

Definition: Recommendation for warnings

Given A the set of all possible actions and W the set of all warnings, given a log of warnings and the corresponding indicators and a current triggered warning and given u a function measuring the utility of an action a for a warning w, i.e., u : A×W → R. Then, for each warning w ∈ W , the recommended action a ∈ A is the one that maximizes the utility for the warning: ∀w ∈ W, a w = argmax a∈A u(w, a).

Example:

In this example, we just illustrate the obtained matrix W ×A where a score indicates that the action has been implemented and if this action has been considered as efficient: Note that a cell (i, j) of this matrix corresponds to the utility score given to the action j by the warning i.

III. RECOMMENDATION PROCESS

In this section, we describe our generic framework for recommending actions. The framework uses both the characteristics of the current warnings / crises, and the log of former warnings / crises, i.e., the actions implemented during each former crisis. It consists of the three following steps, as illustrated in Figure 1: 1) The first step consists in identifying / selecting into the log, warnings similar to the current triggered warning, 2) The second step consists in extracting the actions that were implemented to manage the former similar warnings,

3) The last step consists in ranking the candidate recommended actions. The simplified algorithm 1 represents this process. Each step of this process can be parametrized with one or more functions. By changing these parameters, the way how recommendations are computing changes.

A. Data sources

First of all, we give an overview of what we consider as our sources.

Each time a warning is set off by the early warning system, then the decision-makers have first to decide if this warning is sufficient major to be taken into account. If this warning is major, the decision-makers have to set up some actions in order to face the crisis / problem pointed out by the early warning system. Thus, the warning triggered by the early warning system can generate, or not, the implementation of a certain number of actions. From our point of view, in both cases, the triggered warning is important and has to be logged.

Each warning is associated to a set of indicators. These indicators have also to be logged. In fact, these indicators are thresholds, values intervals, characteristics, ... For example, the Stockholm International Peace Research Institute (SIPRI) identifies 1260 indicators of early warning ([30]) that can be divided into indicator categories (and subcategories) such as justice and human rights, socio-cultural factors, geopolitical setting, military and security, ... (see [31] for more details). Unfortunately, it seems that indicators are specific to each kind of risk / crisis.

In [START_REF] Barton | Early warning? a review of conflict prediction models and systems[END_REF], the authors try to organize the large volume of indicators present in 30 different models and show the diversity of indicators and the special repartition among the different categories. Their review can be helpful in answering questions about: (i) What information the models use to anticipate future events, (ii) How are the early warning models alike, (iii) How are they different, ... Thus, the authors highlight that comparing indicators and so, warnings from different early warning systems is an hard task.

Our work is a first step towards a recommender system for early warning systems. So, this problem is not the topic of our proposition but it will be a challenge to rise in future work. Finally, in addition to the warning and the corresponding indicators (and values), we consider that the actions implemented to answer to the triggered warning have also to be logged.

More formally, the warning triggered by the early warning system can be seen as a 3-uple containing the warning description, the set of pairs of an indicator and its value and the set of implemented actions. Note that the set of implemented actions can be empty if the decision-maker do not decide to implement some actions when a warning is triggered by the early warning system. So, we have, for each warning W i , ∀i ∈ N + * :

W i = Description i , { Indicator 1 i , V alue 1 i , ..., Indicator n i , V alue n i } , Action 1 i , ..., Action m i
where m ∈ N + and n ∈ N + * .

Note that, in this paper, we consider that indicators are the same for each warning (only the corresponding values are different). Finally, our data are composed of a log of former triggered warnings and the current triggered warning.

B. Identifying similar crises

Using the log of former triggered warnings and the current triggered warning, this first step consists in identifying similar warnings. We propose to use a match function. This function is used to find a set of warnings matching a given warning. In fact, this match function is used to search among the set of former warnings which ones are matching the current triggered warning. This function outputs a set of warnings similar to the current warning. Note that this returned set can be empty.

A first example of the match function simply consists in comparing, for each logged warning, each pair Indicator l , V alue l to each pair Indicator c , V alue c of the current warning where Indicator l = Indicator c . It comes to compare V alue l and V alue c .

Because the values of the indicators can be quantitative or qualitative, some similarity measures can be used and / or combined. One purpose easy to implement is to use the cosine similarity which is a (well-known and often used) measure of similarity between two vectors of an inner product space that measures the cosine of the angle between them. One of the advantages of this measure is that it is normalized, i.e., the returnined similarity measure between the two vectors is bounded between 0 and 1.

In fact, for each warning, the indicator vector w i is composed by the indicator values (V alue k i , ∀k ∈ [1.

.n]) such as (according to the notation given Section III-A):

w i = (V alue 1 i , V alue 2 i , ..., V alue n i )
The formula of cosine similarity is:

sim( w 1 , w 2 ) = <w1,w2> w1 w2 = k w k 1 w k 2 k w k 2 1 k w k 2 2
Some more sophisticated and more appropriated similarity measures have to be proposed in future work such as, for example, semantic similarity for qualitative indicators.

C. Extracting actions to recommend

The match function of the previous step outputs a set of similar logged warnings, SimW , matching the current triggered warning. The goal of this step is to extract a set of actions that will be the basis for the recommendation. Our purpose consists in extracting for each warning of SimW , the corresponding set of actions, without duplicates. The obtained set of actions, CandAct, is the set of unordered recommendations which is returned.

D. Ranking the recommended actions

In the previous step, a set of recommendations is obtained. The purpose of this next step is to select the most relevant one w.r.t a satisfaction criterion expressed by the user. To this end, an action ranking is needed, that orders the candidate recommendations. Again, there are many ways of ranking the candidates, from very basic to sophisticated ones. We list here just few:

• Ranking the candidates according to their number of occurrences • Ranking the candidates according to a user profile. For example, some actions sometimes can not be implemented in a given context. • Ranking the candidates according to the order of the corresponding warning into the log, i.e., the most recent implemented actions will be rank first. For example, diffusion channels evolve over time.

E. Default recommendation

As previously noted, the set of candidate recommendations can be empty: when no similar warning is founded or when the set of similar warnings is empty. In such a case, some recommender systems are able to provide the user with a default recommendation. Of course, various default recommendations can be proposed to the user. Unfortunately, considering our context of crisis management and the impacts and stakes of such a context, and according to the status of this work, we consider that it is better to recommend anything as default recommendation. Proposing an efficient default recommendation is an interesting research issue.

IV. EXAMPLE OF INSTANTIATION

In this section, we display a possible instantiation of our framework to illustrate the applicability of our algorithm.

Suppose the early warning system triggers a warning, W c and our log L of former triggered warnings is composed by three warnings W 1 , W 2 , W 3 such as:

• Wc = Descc, { I 1 , V 1 , I 2 , V 2 , I 3 , V 3 , I 4 , V 4 } , ∅ • W 1 = Desc 1 , { I 1 , V 1 , I 2 , V 2 , I 3 , V 3 , I 4 , V 5 } , {A 1 , A 2 , A 3 } • W 2 = Desc 2 , { I 1 , V 6 , I 2 , V 7 , I 3 , V 8 , I 4 , V 9 } , {A 4 , A 5 } • W 3 = Desc 3 , { I 1 , V 10 , I 2 , V 2 , I 3 , V 3 , I 4 , V 4 } , {A 1 , A 6 }
The corresponding utility matrix could be:

u(w,a) A1 A2 A3 A4 A5 A6 W 1 6 5 6 W 2 7 8 W 3 9 6 Wc
First, by representing each warning with an indicator vector, we obtain:

Wc = (V1, V2, V3, V4) W1 = (V1, V2, V3, V5) W2 = (V6, V7, V8, V9) W3 = (V10, V2, V3, V4)
Suppose now computing the cosine similarity shows that sim( W c , W 1 ) = sim( W c , W 3 ) and sim( W c , W 2 ) >> sim( W c , W 1 ). So, the first step of our process returns SimW = {W 1 , W 3 }.

Then, our extract function returns the actions corresponding to each warning of SimW , i.e. CandAct = {A 1 , A 2 , A 3 , A 1 , A 6 }. This set is then cleaned. So, the second step of our process returns CandAct = {A 1 , A 2 , A 3 , A 6 }.

Finally, our rank function orders the candidate actions. Here, we combine the two of the three options we propose Section III-D. In this toy example, from our point of view, the most relevant actions are the ones that frequently occur, then the ones which warnings are more recent. Compare to the other actions, A 1 occurs twice. And W 3 is the more recent warning of the log. So, the last step of our process returns the ordered set of actions: {A 1 , A 6 , A 2 , A 3 }.

The corresponding utility matrix could become:

V. CONCLUSION AND PERSPECTIVES

In this position paper, we propose challenges and research issues for recommending actions to implement when a warning is triggered by an early warning system for crisis management. We turn our framework towards a generic framework, in the sense that it can be instantiated to change the way recommendations are computed. We give few examples of how it can be instantiated.

Challenges and research issues to rise include (but are not limited to):

• The implementation of our framework and the conduction of experiments in order to better assess the quality of recommended actions, as well as the assessment of various instantiations of our framework in order to determine to what context they are better adapted. • The problem of the indicators has been mentioned in this article. Indeed, a large number of indicators exists. They are different depending on the type of risk but also they can be different for the same type of risk. However, no standard exists. A standardization of indicators, for example, by type of risk (or other), seems to be a promising research issue. • The investigation of other instantiations of our framework. For example, we would like to investigate how to compute the more efficient distance between indicators.

As another example, we would like to propose other Match and Rank functions. More precisely, the Match function which is at the heart of the candidate generation, could be instantiated with more sophisticated distances. • When no recommendation can be computed, considering our context of crisis management and the impacts and the stakes of such a context, the purpose of a smart default recommendation is an interesting research issue. • The integration of the user knowledge has to be taken into account in our framework. Finding a way in that sense, for example, with the user profile, is an other research issue. • The longer term objective is to provide a generic platform of recommendations that can adapt to the needs of users, types of crises and many other parameters. This work in progress, which uses the knowledge gained from past experiences to make the best decision in the present, is a first step towards the enhancement of early warning systems and crisis management.

Figure 1 .

 1 Figure 1. Overview of the recommendation framework

Algorithm 1

 1 RecoEWS(L, W c , Match, Extract, Rank, Default, Clean, ≺) Require: L: The log of former triggered warnings, Wc: The current triggered warning, Match: A match function between two warnings, Extract: A function extracting actions, Rank: A function ranking actions, Default: A function returning a default recommendation, Clean: A function deleting duplicates in a given set, ≺: An action ranking. Ensure: An ordered set of recommendations SimW ← ∅ // f or the similar warnings CandAct ← ∅ // f or the candidate actions for each warning wi ∈ L do SimW ← SimW ∪ M atch(wi, Wc) end for for each warning wj ∈ SimW do CandAct ← CandAct ∪ Extract(wj) end for Clean(CandAct) if CandAct = ∅ then return Rank(CandAct, ≺) else return Def ault(L) end if

  this example, items are movies that the users Elsa, Camille, Michel and Nicolas have given a score. We obtain the matrix P × M :

	u(p,m)	Harry Potter Ice Age	Ice Age 2 Hulk	Transformers
	Elsa		8		7
	Camille	9	8		6
	Michel	3	5	5	5
	Nicolas	5	3	3	3


Michel Morange
email: morange@biologie.ens.fr

What history tells us XXXVIII. Resurrection of a transient forgotten model of gene action

Keywords: embryonic induction, hormone, one geneone enzyme relation, organizer

An incoherent historical description of the path to the one geneone enzyme relation

The one geneone enzyme relation of George Beadle and Edward Tatum is one of the pillars of molecular biology. The numerous steps that led to its emergence have been well described by historians. Alfred Sturtevant observed the partial correction of the vermilion mutation in Drosophila gynandromorphs. He interpreted it as the result of the diffusion of a substance present in the wild-type tissues surrounding the eye bud (Sturtevant 1920 and1932). Under his influence, Beadle and Boris Ephrussi initiated their experiments on the genetic determinism of eye colour in Drosophila, using the embryological technique of eye bud transplantation (Beadle and[START_REF] Beadle | Transplantation in Drosophila[END_REF][START_REF] Beadle | Development of eye colors in Drosophila: diffusible substances and their interrelations[END_REF]. They spent long years characterizing the chemical nature of the diffusible substances involved (Thimann and [START_REF] Beadle | Development of eye colors in Drosophila: diffusible substances and their interrelations[END_REF]Tatum and Beadle 1940). This work was a failure, since the chemical nature of these substances was first described by Adolf Butenandt in Germany (Butenandt et al. 1940). Finally, Beadle and Tatum successfully reoriented their work by reversing the experimental approach, and looking at the genetic control of well-known metabolic pathways in a new model organism, Neurospora (Beadle and Tatum 1941 Historians (and biographers) have already noticed these "abnormalities". Some present them as oddities without significance, the use of "loosely defined" terms: the word "hormone" is often placed between inverted commas when this early work is described (Horowitz 1990, 2;Kay 1993;Berg and Singer 2003). Gayon has particularly emphasized the recurrent and late use of the word "hormone" to designate the substances involved in the coloration of the eyes, and shown that the model that Ephrussi had in mind was clearly different from the one geneone enzyme relation (Burian et al. 1991;Gayon 1994). Rheinberger has highlighted Kühn's hormonal interpretation of his results (Rheinberger 2000). Nevertheless, I think that it is possible to go further: there was a model of gene action that was shared by most of the participants. It was a tacit, rarely fully explicit model. It also included evidence obtained by Hans Spemann on the existence of embryonic inducers and of an organization centre, as well as the recent efforts initiated by Joseph Needham to characterize the chemical nature of the latter.

The model and its supports

The model was very simple. It proposed that one of the functions of genes was to control the production or action of hormones. These hormones behaved as embryonic inducers, which explained how genes were able to control development.

The model found strong support in the importance attributed to hormones in physiological processes, and in the rapid progress made in their characterization. It was obvious in the 1930s that hormones controlled differentiation and development: such was the case of the growth and thyroid hormones (the latter being involved in the metamorphosis of amphibians), but also of steroid and in particular sexual hormones. Interestingly, vitamin D was also a steroid derived from cholesterol, and this observation established a link between hormones and other important objects of research in biochemistry, vitamins: vitamin D was also important for bone formation. Observations of hormone action on development, and of their control by genes, were obtained in plants (van Overbeek 1935) The link between embryonic inducers and hormones was not introduced by Spemann, but proposed in 1931 by Joseph Needham in his monumental book on chemical embryogenesis (Needham 1931). He suggested that the substance responsible for the effects of the organizer was a steroid, and the first results of the purification of the active principle present in the organizer seemed to support this view [START_REF] Armon | Between biochemists and embryologists -the biochemical study of embryonic induction in the 1930s[END_REF].

To consider that the products of genes were hormones fully agreed with the vision of genes as "controllers". Slightly later Conrad Waddington, faced with the paradox that artificial substances were able to mimic the action of the organizer, proposed the idea that the substance present in the organizing centre acted as an evocator, switching on a response that was already predetermined in the recipient tissue (Waddington 1936). This idea was not incompatible with this conception of gene action, considered as essential but limited. We need to take off our present-day information glasses in order to understand how the function of genes was viewed in the 1930s!

Evidence for the existence of this model

One difficulty is that the model was never fully and clearly made explicit. There are probably two reasons for this cautious attitude of biologists. The first was that there were other models of gene action, which had never been cast aside: in particular, the case of the relation between genes and enzymes. Most of the mutations controlling eye development were autonomous, demonstrating the existence of mechanisms of gene action other than the hormonal one (see later). In addition, the way genes might control the production or action of hormones was totally unknown. The second reason was that the model was at the crossroads of different disciplines and attempts to establish relations between three highly different areas of research: biochemistry (including the study of hormones and vitamins), embryology, and genetics. To present the model explicitly generated the risk of being harshly criticized by specialists of other disciplines.

A second difficulty is the huge effort necessary to abandon our current vision and to read the publications of Beadle, Ephrussi and Kühn without interpreting their results in terms of presentday understanding. When they describe the successive roles of genes in the production of substances, we must not immediately substitute a model in which genes give rise to enzymes that control the successive steps of a metabolic pathway. This interpretation progressively emerged from the experimental observations, both in the case of Beadle and Ephrussi and of Kühn: it was not the interpretation that guided the initial work, in contradiction with the authors' own retrospective reading of their work [START_REF] Beadle | Recollections[END_REF]. Despite a lack of a direct evidence for the existence and importance of the hormonal model, I believe that the strongest argument in its favour is that it provides an explanation for the puzzling observations that I outlined before. The use of the term "hormone" by the protagonists was recurrent, and not justified: it was not an inappropriate use of a word, but it was a central part of the model that guided the experiments. The exclusive attention paid to the non-autonomous characters, vermilion and cinnabar, also becomes obvious: if genes act through the production of hormones, from the very definition of hormone action the characters had to be non-autonomous. and Haldane 1939;Sapp 1990). In his work, Moewus claimed to have demonstrated that the differentiation of the sex types was due to a cascade of hormones of the -carotenoid type. The synthesis of these hormones was the result of enzyme activities controlled by genes. Interestingly, the model proposed by Moewus at a time (1938)(1939)(1940) when the hormonal model was facing increasing difficulties mixed the role of hormones and the gradually increasing place of enzymes in gene action.

Some conclusions about this historical episode

The hormonal model of gene action was the result of a process of coalescence of results and concepts from three rapidly growing disciplines: biochemistry (endocrinology), genetics, and embryology. It has been already mentioned by historians that the period 1930-1940 was a time of unification in biology (Smocovitis 1992). The emergence of the modern evolutionary synthesis was the most obvious sign of this unification. The hormonal model of gene action might have been another example of unification, supported by the rapid progress made in the characterization of hormones and of the mechanisms of embryogenesis, and by the increasingly central role of genes in biological explanations. But the model remained in an embryonic form, and never reached the adult stage, where it would have been described in articles and books, and discussed at scientific meetings. Scientific developments also follow rapid rhythms and are subject to fashions. 1933-1935, the years crucial for the experiments on Drosophila and Ephestia, as well as for the chemical characterization of the organizer (Waddington et al. 1933;Witkowski 1985), were those when the Nobel Prize in Physiology or Medicine was awarded to Morgan (1933) andto Spemann (1935). No doubt that, during these years, genes and embryonic induction were present in the minds of many biologists! A last lesson from this historical episode is that science functions well. The hormonal model did not find experimental support, whereas a relation between genes and enzymes was progressively strengthened. The first model was rapidly abandoned without resistance, and the transition occurred smoothly, so smoothly that it became invisible to those who retrospectively studied these experiments. The fact that the hormonal model was never made fully explicit probably facilitated its disappearance.

  ). Hans-Jörg Rheinberger has shown how the German biologists Alfred Kühn and Ernst Caspari, before Beadle and Tatum, developed a similar and independent study on the flour moth Ephestia. They obtained identical results, and proposed similar hypotheses(Rheinberger 2000). The work was partially interrupted by war and has not been fully recognized by the scientific community and by historians.

	A last surprising observation is that Beadle introduced the one
	gene -one enzyme relation belatedly (Beadle 1945), and
	marginally. Obviously, the establishment of this relation was
	not what guided the choice of Neurospora nor does it explain
	the importance attributed to the first results obtained on this
	system.
	Beadle and Ephrussi used the word "hormone" to designate
	the nature of the substances involved, and this habit persisted
	up to 1940 for Beadle (Tatum and Beadle 1940) and 1944 for
	Ephrussi (Ephrussi and Herold 1944) -the same word was
	used by Kühn. To characterize the substances involved,
	Beadle recruited a specialist of auxin, a plant hormone, to help
	him in the work!

Even complemented by the contribution of Kühn and Caspari, this historical presentation remains puzzling. What Beadle and Tatum, as well as Kühn and Caspari, initially aimed at were the mechanisms of gene action in development, whereas the relation finally established between genes and enzymes was general and, moreover, not new: Archibald Garrod, L. T. Troland, John Haldane and Muriel Wheldale among others had already proposed a similar relation

(Olby 1974)

. Additional difficulties emerge when the initial work of Beadle and Ephrussi is closely scrutinized. Among the more than twenty mutations that affected eye colour, they focused their attention on two non-autonomous mutations, i.e. mutations that could be corrected by surrounding wild-type tissues in mosaic animals. Apart from a reference to Sturtevant's earlier observations, the reasons for focusing, from the outset, on non-autonomous characters are not obvious. In addition,

  The abandonment of this "hormonal" model was slow and progressive, which also explains why the relation one geneone enzyme was not immediately placed at the pinnacle by Beadle. The experiments on Neurospora were initiated as a return to an empirical strategy to determine experimentally the products of gene action, a reorientation of the work triggered by the disappointment of not having found the hormones involved in development.Other indirect observations are also in favour of the existence and influence of this model. I will briefly describe two of them. The first is the representation of gene structure proposed by the Russian biologist Koltzoff(Koltzoff 1939; Morange 2011). Often mentioned as the first model of gene structure, it is also criticized for its naivety, and the heterogeneity of the chemical molecules supposed to form the genetic material. To a long polypeptide chain, Koltzoff attached amino acids and hormones. The presence of hormones in the genetic material was retrospectively considered as absurd, whereas it had its full significance in a hormonal model of gene action. Frauds and/or non-reproducible experiments can also be used to gain access to the range of expectations dominant at a given time. The results of experiments will be accepted too rapidly (or invented in the worst case) if they support the non-explicit model that researchers have in mind. The experiments performed by Franz Moewus in Richard Kuhn's lab on the hormonal control of sex-types in the alga Chlamydomonas between 1938 and 1940 belong to this category of experiments that were received with suspicion, and never confirmed (Philip

  The path to the double helix (London: MacMillan) Philip U and Haldane JBS 1939 Relative sexuality in unicellular algae. Nature 143 334 Rheinberger H-J 2000 Ephestia: the experimental design of Alfred Kühn's physiological developmental genetics. J. Hist. Biol. 33 535-576 Sapp J 1990 Where the truth lies: Franz Moewus and the origins of molecular biology (Cambridge: Cambridge University Press) Smocovitis VB 1992 Unifying biology; the evolutionary synthesis and evolutionary biology. J. Hist. Biol. 25 1-65 Sturtevant AH 1920 The vermilion gene and gynandromorphism. Proc. Soc. Exp. Biol. Med. 51 325-327 Sturtevant AH 1932 The use of mosaics in the study of the developmental effects of genes. Proc. Sixth Int. Cong. Genetics 1 304-307 Tatum EL and Beadle GW 1940 Crystalline Drosophila eyecolor hormone. Science 91 458 Thimann KV and Beadle GW 1937 Development of eye colors in Drosophila: extraction of the diffusible substances concerned. Proc. Natl. Acad. Sci. USA 23 143-146 Van Overbeek J 1935 The growth hormone and the dwarf type of growth in corn. Proc. Natl. Acad. Sci. USA 21 292-299 Waddington CH 1936 The problems of embryology. Br. Med. J. 2 862-864 Waddington CH, Needham J and Needham DM 1933 Physico-chemical experiments on the amphibian organiser. Nature 132 239 Witkowski J 1985 The hunting of the organizer: an episode in biochemical embryology. TIBS 10 379-381

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	Press), 207-227
	Butenandt A, Weidel W and Becker E 1940 Kynurenin als
	Augenpigmentbildung auslosendes Agens bei Insekten.
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	Ephrussi B and Herold JL 1944 Studies of eye pigments of
	Drosophila. I. Methods of extraction and quantitative
	estimation of the pigment components. Genetics 29 148-175
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	drosophile: la contribution spécifique de Boris Ephrussi in
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	Horowitz NH 1990 George Wells Beadle (1903-1989).
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Acknowledgements

I am indebted to David Marsh for critical reading of the manuscript.


Giuseppe Iurato
email: giuseppe.iurato@community.unipa.it

Eye movement pre-algebra and visual semantic algebra Their possible links within denotational mathematics framework, Husserlian phenomenological theory, and structuralism

Keywords: Eye movements, Visual-Spatial Skills, Algebra, Denotational Mathematics, Husserl's Phenomenology, Structuralism

published or not. The documents may come

Eye movement pre-algebra and visual semantic algebra

I. INTRODUCTION

Basically, human visual function, the chief one among the physiological functions of human perception, has four main abilities: fixing eyes upon an object; pursuing a moving object; responding to a stimulus which appears in the external neighbourhood of visual field, through a movement of eyes toward the direction along which such a stimulus manifests; exploring visually environmental space in searching of objects and their details. These basic four skills spring out from the general cognitive capability of visual-spatial attention (VSAt), one of the most important cognitive function of human psyche. Neurophysiology has showed that VSAt always precedes any next eye movement, that is to say, the former implicitly guides and directs the latter in its explicit action of orientation, predisposing the related visual field, so influencing and conditioning many other higher cognitive functions (like memory, voluntary judgement, etc.) [START_REF] Facoetti | Reading and selective spatial attention: evidence from behavioral studies in dyslexic children[END_REF].

Human eye movements are classified into two main types, namely, fixations and saccades, respectively when eyes stop in a certain position, and when they suddenly and fastly move towards another position. The resulting sequence of fixations and saccades, is called a scanpath. A smooth pursuit refers instead to eyes slowly following an object in movement. The set of fixational eye movements includes the so-called microsaccades, which are nothing but small, involuntary (i.e., unconscious) saccades that occur during attempted fixation of an object. From much time, it has been deemed that most information by eyes come mainly from fixations or smooth pursuits, but not by saccades. Instead, recent neurophysiology research has shown what primary role play saccades, included microsaccades, for the general visual perception [START_REF] Martinez-Conde | Microsaccades Counteract Visual Fading during Fixation[END_REF].

In any case, the class of eye movements comprehends the following ones: saccades, smooth pursuit movements, vergence movements, and vestibulo-ocular movements, which will be briefly described, in their physiological essence, in the next section. Now, from a mathematical viewpoint, in this note, just we would like to consider a possible mathematical structure of universal algebra, that we shall call eye movement pre-algebra (in short, EMpA), formalizing these eye movements, in such a way to be closely related to another formal structure belonging to Yingxu Wang's denotational mathematics framework, called visual semantic algebra (in short, VSA), through a suitable conceptual bridge casted by Husserlian phenomenology.

II. TYPES OF EYE MOVEMENTS AND THEIR FUNCTIONS:

A BRIEF SURVEY Human beings are epi-ontogenetically prepared, since their childbirth, to develop eye movements, in particular saccades. To be more precise, there are four basic types of human eye movements, i.e., saccades, smooth pursuit movements, vergence movements, and vestibulo-ocular movements. The functions of each type of eye movement are briefly introduced herein, closely following [3], [START_REF] Harris | Developmental Psychology. A Student's Handbook[END_REF].

Saccades (in short, ) are rapid, ballistic movements of the eyes that abruptly change the point of fixation. They range in amplitude from the small movements, made while reading for example, to much larger movements, made while gazing around a room for example. Saccades can be elicited voluntarily, but they also occur reflexively whenever the eyes are open, even when are fixed on a target. For instance, the rapid eye movements that occur during an important phase of sleep are also saccades. After the onset of a target for a saccade (as, for example, the stimulus in the case of the movement of an already fixated target), it takes about 200 ms for eye movement to begin. During this delay, the position of the target with respect to the fovea is computed (that is, how far the eye has to move), and the difference between the initial and intended position, or "motor error", is converted into a motor command that activates the extraocular muscles to move the eyes the correct distance in the appropriate direction.

Saccadic eye movements are said to be "ballistic" because the saccade-generating system cannot respond to subsequent changes in the position of the target during the course of the eye movement. If the target moves again during this time (which has the order of about 15-100 ms), the saccade will miss the target, and a second saccade must be made to correct the error. Saccades can also be voluntary, but are mainly made unconsciously.

Smooth pursuit movements (in short,

) are much slower tracking movements of the eyes designed to keep a moving stimulus on the fovea. Such movements are under voluntary control in the sense that the observer can choose whether or not to track a moving stimulus. Surprisingly, however, only highly trained observers can make a smooth pursuit movement in the absence of a moving target. Most people who try to move their eyes in a smooth fashion without a moving target simply make a saccade. The smooth pursuit system can be tested by placing a subject inside a rotating cylinder with vertical stripes. In practice, the subject is more often seated in front of a screen on which a series of horizontally moving vertical bars is presented to conduct this "optokinetic test". The eyes automatically follow a stripe until they reach the end of their excursion. There is then a quick saccade in the direction opposite to the movement, followed once again by smooth pursuit of a stripe. This alternating slow and fast movement of the eyes in response to such stimuli is called optokinetic nystagmus. Optokinetic nystagmus is a normal reflexive response of the eyes in response to large-scale movements of the visual scene and should not be confused with the pathological nystagmus that can result from certain kinds of brain injury (for example, damage to the vestibular system or the cerebellum).

Vergence movements (in short,

) align the fovea of each eye with targets located at different distances from the observer. Unlike other types of eye movements in which the two eyes move in the same direction (conjugate eye movements), vergence movements are disconjugate (or disjunctive). They involve either a convergence or divergence of the lines of sight of each eye to see an object that is nearer or farther away. Convergence is one of the three reflexive visual responses elicited by interest in a near object. The other components of the so-called near reflex triad are accommodation of the lens, which brings the object into focus, and pupillary constriction, which increases the depth of field and sharpens the image on the retina.

Vestibulo-ocular movements (in short,

) stabilize the eyes relative to the external world, thus compensating for head movements. These reflex responses prevent visual images from "slipping" on the surface of the retina as head position varies. The action of vestibulo-ocular movements can be appreciated by fixating an object and moving the head from side to side; the eyes automatically compensate for the head movement by moving the same distance but in the opposite direction, thus keeping the image of the object at more or less the same place on the retina. The vestibular system detects brief, transient changes in head position and produces rapid corrective eye movements.

Sensory information from the semicircular canals directs the eyes to move in a direction opposite to the head movement. While the vestibular system operates effectively to counteract rapid movements of the head, it is relatively insensitive to slow movements or to persistent rotation of the head. For example, if the vestibulo-ocular reflex is tested with continuous rotation and without visual cues about the movement of the image (i.e., with eyes closed or in the dark), the compensatory eye movements cease after only about 30 seconds of rotation. However, if the same test is performed with visual cues, eye movements persist. The compensatory eye movements in this case are due to the activation of the smooth pursuit system, which relies not on vestibular information but on visual cues indicating motion of the visual field.

Human visual attention is a cognitive process deployed to reduce the wide complexity of external visual scene analysis. To this purpose, a subset of the available visual information is selected by shifting the focus of attention across the visual scene towards the most salient objects, locations, and features. Visual attention is driven by two main mechanisms, said to be overt and covert attention. Overt attention relates to the process of actively directing a sense towards an object or an event. In terms of overt visual attention, this is mainly exhibited as moving the eye gaze towards an object or location in the visual space. Covert attention, instead, describes the mechanism of mentally shifting the focus of attention without moving the eyes 1 . Overt visual search has been well explored, while covert visual attention search is a largely unexplored topic [START_REF] Engelke | Cognitive Correlates of Overt and Covert Attention in Visual Search[END_REF].

On the other hand, covert attention is closely related to cognitive unconscious realm. Indeed, recent researches 2show that in the first milliseconds of incipient perception, humans apprehend (above all, in covert attention) only in an unconscious fashion the given intentioned object, also providing (pre-)judgements on it [START_REF]The New Unconscious[END_REF]. So, we would like to formalize (through the so-called EMpA) these latter types of mental processeswhich neurophysiologically are mainly supported by the above eye movementsinvolved in such an unconscious acquisition of the intentioned object, which will give rise then the ordinary visual, conscious perception just formalized by VSA.

III. THE VISUAL SEMANTIC ALGEBRA (VSA)

The cognitive processes of pattern recognition may be modelled, within Y. Wang's denotational mathematical framework, through a new mathematical structure known as Visual Semantic Algebra (VSA), where the cognitive informatics theories for pattern recognition are explored, such as cognitive principles of visual perception and the basic mechanisms of object and pattern recognition, as falling into the overt attention realm. Such a denotational mathematical means, that is to say, VSA, is developed to manipulate basic geometric shapes and figures, as well as their compositions, by a set of algebraic operations [START_REF] Wang | On Visual Semantic Algebra (VSA): A Denotational Mathematical Structure for Modeling and Manipulating Visual Objects and Patterns[END_REF], [START_REF] Wang | In Search of Denotational Mathematics: Novel Mathematical Means for Contemporary Intelligence, Brain, and Knowledge Sciences[END_REF].

The physical or environmental space can be formally modelled by Euclidean space, that is to say, the collection of all abstract points in the three Cartesian coordinates, equipped with the usual Pythagorean distance. Therefore, the spatial properties of abstract objects, shapes, and their interrelations, can be formally studied by Euclidean geometry, which can be classified into plane geometry and solid geometry: the former studies the geometric figures in a plane, while the latter studies the geometric objects known as solids in the three-dimensional space [START_REF] Wang | On Visual Semantic Algebra (VSA): A Denotational Mathematical Structure for Modeling and Manipulating Visual Objects and Patterns[END_REF], [START_REF] Wang | In Search of Denotational Mathematics: Novel Mathematical Means for Contemporary Intelligence, Brain, and Knowledge Sciences[END_REF].

In order to efficiently and formally model the abstract visual objects, their semantic representations, as well as their rigorous compositions and manipulations, a new denotational mathematics structurethe VSAhas just been introduced to this end, as well as for moulding pattern recognition and processing. VSA is a denotational mathematical structure which formally manipulates visual objects by algebraic operations on symbolic or semantic objects in geometric analyses and compositions. To be precise, if the basic geometric shapes (2-D), solids (3-D), figures (F) and spatial reference limits (L), are collectively called visual objects, or geons, let be a finite non-empty set of geons, while let be a finite non-empty set of spatial relational operations, then the universal visual object environment, say , which forms the discourse's environment of the abstract visual objects, is a formal structure of the type . Then, a visual semantic algebra (VSA) in the universal visual object environment , is a formal structure of the type VSA , where is a non-void finite set of geons, while is a nonempty finite set of algebraic operations on , to be more precise, a set of relational and behavioural operators on [START_REF] Wang | On Visual Semantic Algebra (VSA): A Denotational Mathematical Structure for Modeling and Manipulating Visual Objects and Patterns[END_REF], [START_REF] Wang | In Search of Denotational Mathematics: Novel Mathematical Means for Contemporary Intelligence, Brain, and Knowledge Sciences[END_REF].

IV. VISUAL SEMANTIC ALGEBRA (VSA) AND EYE MOVEMENT PRE-ALGEBRA (EMPA) IN THE HUSSERLIAN THEORY AND STRUCTURALISTIC FRAMEWORK Let VSA be a visual semantic algebra in the universal visual object environment . On the other hand, from a philosophical viewpoint, the term object, or thing, has a larger meaning. Indeed, from Kant onwards, we mainly distinguish a thing in itself (or the general thing) and a sensible thing of the external spacetime. Then, a basic issue arose in regard to give or not a proper meaning to the being of the thing. Husserl gives a proper or intrinsic sense to the being of the thing, which is given to us with a transcendental modality going beyond the whole infinite set of apparitions of it, which is a unit that goes beyond these latter. So, the being of the thing is opposed to the being of the lived experiences of consciousness. Just in this regard, Husserl distinguishes an immanent perception and a transcendental perception of the thing [START_REF] Costa | La fenomenologia[END_REF], [START_REF] Zahavi | Husserl's Phenomenology[END_REF].

Husserl starts from the notion of intentionality means as the typical feature of human consciousness to relate with something which is other of the consciousness' act itself, i.e., with something which goes beyond this latter. The former belongs to the transcendental perception, while the latter belongs to the immanent perception and gives rise to the set of all possible representations (imagines) of the object of transcendental realm. Therefore, since this typical relationship with the object does not fall into the consciousness field (due to the fact that it goes beyondi.e., transcendsany act of consciousness directed on it), this relationship is outside every psychological discipline, that is to say, it does not have psychological nature but rather it has a logical-transcendental nature, in agreement with Kant according to whom the object gives itself as such, to the consciousness, according to another modality of being of consciousness (i.e., the logical-transcendental one) which is different from the psychological one. This modality of being may be explicated by means of the socalled phenomenological reduction (made by epoché plus suspension of judgement), which led to the absolute ego, neglecting every reality's consideration [START_REF] Costa | La fenomenologia[END_REF], [START_REF] Zahavi | Husserl's Phenomenology[END_REF].

So, the transcendental perception concerns that object, or thing, placed into the ordinary space-time, but never presents into the consciousness in its full actuality. From this, it springs out the main feature of the thing to be in itself, which allows consciousness to return on the same object to be identified; from this, it follows a certain contingency of the thing, and not its necessity. Moreover, the object is given to the subjective phenomena only by successive approximation through a series of apparitions, adumbrations, and approaches, which refer to the overall transcendental unity (also in the sense of Gestalt theory) of the object. Then, the essence of the thing integrates in themselves and, at the same time, transcends, all the elements of this approximating series, which will led to the transcendental perception of the thing.

Instead, immanent perception is nothing but the wellknown Cartesian cogito, or the transcendental ego, i.e., the reflected and indubitable consciousness (also said to be transcendental consciousness), whose objects are the various lived experiences stored (e.g., remembrances, imagines, desires, feelings, etc.), mainly characterized by their immediacy and absoluteness, of which is impossible to deny the existence. The immanent being is therefore absolute and sure, that is to say, it exists regardless other, is moreover self-consistent and self-perceived, while the transcendental being, i.e., the world of things or objects, is necessarily related to some consciousness. Thus, the transcendental consciousness is reached on the basis of a primary opposition between the immanent perception and the transcendental perception (i.e., the perceived), that is, between the (immanent) perception of the various states of consciousness and the (transcendental) perception of the external world, giving rise to an immediate unity or totality (Gestalt) [START_REF] Costa | La generazione della forma. La fenomenologia e il problema della genesi in Husserl e Derrida[END_REF].

But, the simple perception does not entail knowledge. This takes place only when the object of perception encounters the subjective (individual) lived experience, which is something that is already present inside us at a mere original state, then becoming (mental) thought by means of symbolic marking with its main representation function, which is the diachronic result of the own lived experience meant as a connected flow, or current, of past experiences (Erlebnis), which 'catches' the object. Each representation of any consciousness' act is therefore the main feature of human thought, through which an arbitrary object (of transcendental perception) belongs to the consciousness field. So, the external world itself seems to exist in dependence on some consciousness. From this stance, it follows, among other thing, that typical idealistic view of Husserl, centred on the notion of transcendental Ego, which has an ontological priority as primary constitutive basis of lived experience [START_REF] Costa | L'estetica trascendentale fenomenologica. Sensibilità e razionalità nella filosofia di Edmund Husserl[END_REF].

Therefore, for Husserl, perception is that intentional act of consciousness in which are present, at the same time, the perceived object (of transcendental perception), and the perceiving subject (of immanent perception). In regard to immanent perception, then, we may distinguish between an objective aspect and a subjective aspect of Erlebnis. The objective aspect is said to be noema, and is the object as considered by the consciousness' reflection in its various modes of being with which it is given to consciousness itself (e.g., the perceived, the remembered, the imagined, and so forth). But, it is distinct from the object itself of transcendental perception: for example, the object of transcendental perception of a tree, is different from the corresponding object of immanent perception, as the latteri.e., the noemais the set of all the properties, predicates, and modalities of being of the former (e.g., a green tree, a bare tree, an illuminated tree, a lonely tree, etc.) for which it exists for our immanent consciousness (to give rise the regional ontologies) but still not at a representational level, as this latter is reached only by means of the basic, dialectic opposition noesisnoema of the intentional contents of Erlebnis, just with the constitution of the noesis, with its flow of contents of lived experiences.

Moreover, noema also provides the so-called giveness of the experience, that is to say, it is, so to speak, the ''threw down'' of the human (internal and external) senses, phenomenologically reduced, that is to say, the manifestation of perceptual consciousness; it is nothing but the so-called hyle, the primary outcome of (covert) visualization that, thanks to noesis, will give rise to the intentional element of Erlebnis, clear and net in its (overt) visualization to consciousness. The noema, therefore, is a kind of ''clot'' of hyletic data (sensory matter). The noesis, instead, i.e., the subjective (or psychological) aspect of Erlebnis, is the set of all those (eidetic) acts of human consciousnesslike, re-evoking, remembering, imagining, and so forthwhich are able, so to speak, ''to catch'' the object of transcendental perception as noema; it gives rise to the so-called eidetic sciences: formal ontology, material ontology, and formal logic [START_REF] Conci | Per una fenomenologia dell'originario[END_REF], [START_REF] Williford | Husserl's hyletic data and phenomenal consciousness[END_REF].

So, in the immanent perception, we have an immediate, unitary and total consciousness of a whole, a totality, fully structured, whose behavior is not determined by its composing (so-called atomic) elements (as in association psychology), but by a pre-constituted system of certain pre-existent structural laws, which are internal to the totality itself. This is one of the main principles of Gestalt theory, in its content very close to the unity of the thing of transcendental perception claimed by Husserl's theory, as the essence of the thing integrates in itself, and at the same time transcends, the totality of all its apparitions. Whereupon, immanent perception, only when is put in relationship with transcendental perception, gives rise to the basic pair noema-noesis of Erlebnis. Analogously, the transcendental consciousness is reached on the basis of a primary opposition between immanent perception and transcendental perception, where an object acquires its status of real existence only through the noetic lived experiences [START_REF] Costa | La generazione della forma. La fenomenologia e il problema della genesi in Husserl e Derrida[END_REF].

On the other hand, a structure may be defined as any set of relationships, well-defined and determined, so it may be understood either as a form in the sense of Gestalt psychology or as a system of relationships according to either Claude Lévi-Strauss' structural anthropology and mathematics. In the latter sense, a structure may be also considered as a totality self-regulating itself by means of a pre-constituted system of internal transformations or rules, always generating only elements belonging to the same structure, and running in such a manner to preserve the those basic laws of the structure, which are already implicitly (or unconsciously) present at a level of the incipit of the perception (i.e., the sensation), in the socalled consciousness field, which however spreads out of transcendental consciousness and holds together, in their synthetic totality, either immanent and transcendental perception; in it, from the hyletic data of transcendental perception, the natural object acquires its proper existential status only through noetic lived experiences, so reaching transcendental Ego, where its representations finally may take place [START_REF] Costa | La fenomenologia[END_REF][START_REF] Zahavi | Husserl's Phenomenology[END_REF][START_REF] Costa | La generazione della forma. La fenomenologia e il problema della genesi in Husserl e Derrida[END_REF][START_REF] Costa | L'estetica trascendentale fenomenologica. Sensibilità e razionalità nella filosofia di Edmund Husserl[END_REF][START_REF] Santone | Critica del Principio di Esistenza, edizione online, Narcissus-me (StreetLib Srl)[END_REF][START_REF] Gurwitsch | Phenomenology and the Theory of Science[END_REF].

Therefore, in the incipit of the perception (i.e., in the sensation), it seems that every human being, so to speak, as early owns, unconsciously, systems of relations, meant as above, which then become certain structures by means of suitable explication processes triggered by the basic, dialectic opposition between immanent and transcendent perception; to be precise, by the unconscious acquisition of an implicitly pre-existent system of laws and rules at the level of the first perception, a structure may explicitly spring out from the primary, basic opposition between immanent and transcendental perception; a relationship between objects may then arise from the consequent, opposition noesis-noema inherent transcendental Ego, from which, finally, a representation is then possible.

To be more precise, we should distinguish between the notion of system and of structure. The former refers to a synthetic view of a totality of elements, that, seen from a functional stance, tries to catch laws and rules governing it, no matter its component parts; furthermore, the system is not the result of the simple sum of its elements. This notion is however closely related with the latter notion, that of structure, as Lévi-Strauss and Jean Piaget have pointed out. Indeed, even if the structure evokes the unity and cohesion of the parts, hence refers to the system, it is not the system itself, as such, but rather is the internal order of the system together the group of transformations characterizing such a system. In this regard, the structure is still independent of the elements or constituents parts, as it is closely related with the system of relationships among them. This last standpoint, centered on the notion of structure3 , mainly arises from cultural anthropology context that, above all with the works of A.R. Radcliffe-Brown and E.E. Evans-Pritchard, basically identify the structure with the system of relations which undergo it. This viewpoint will be then retaken by functionalism and, above all, by the structural anthropology of Lévi-Strauss, until up the many and interesting applications, in a wider context, of the general structuralism as for instance in linguistics, epistemology, sociology, human sciences, by Roman Jakobson, Piaget, Roland Barthes, Jacques Lacan, Noam Chomsky, Talcott Parsons, and so on [START_REF] Boudon | Strutturalismo e scienze umane[END_REF].

Therefore, it seems [START_REF] Boudon | Strutturalismo e scienze umane[END_REF] that the link between structure and system of relations, is the central core around which revolves any other possible definition of structure. Just structural linguistics claims that, if one wishes to define a certain entity, say , then it should be considered from ''outside'' not from ''inside'', that is to say, considering all its possible relationships with the related (semantic 4 ) context in which it relies or is embedded, not considering its intrinsic properties. So, a structure may be defined as a kind of transformational syntax of the system, that is to say, with the set of all rules of relationship, combination and permutation (i.e., a group 5 of transformations) of its composing terms or elements. This mainly stems from the well-known 1940s studies of Lévi-Strauss on the systems of kinship, and was later retaken by Piaget. So, the early basis upon which to build up any structure is the preliminary acquisition of a certain system of relations, from which then to obtain a structure, as mentioned just above. It has been Lévi-Strauss to have pointed out that does not exist structure without a notion of relation or transformation, that a structure does not reduce to the system of relations to which it refers, and that it is the result of the action of the set of all the variants of such a system (or else, its representations). Now, intentionality is, so to speak, the ''objectivating'' function of consciousness, so phenomenological theory of intentionality is nothing but phenomenological theory of objectivity (and its structure and forms). Intentionality may be defined as a correlation of subjective (noetic) and extra-subjective (noematic) factors, two levels which, by themselves, make up a structured whole, i.e., a gestaltic form. Aron Gurwitsch has, for instance, associated this conception of intentionality with an equation of the type , where refers to the perception while refers to a relation between internal ( ) and external ( ) data, that is to say, to noematic properties of objects. In the Husserl's framework, refers to the immanent perception, while and refer to the transcendental perception, so a formal relation may take place only at the noetic stratum, i.e., at an higher level of abstraction, so having put into connection immanent perception with relations amongst noematic data which are already preexistent but at a proto-object level, or a pre-linguistic level of perception [START_REF] Wiegand | On Referring to Gestalts[END_REF].

Kevin Mulligan has distinguished between what he calls

external relations on the one hand, and internal relations on the other. In order to explicate the use of the notion of "external relation," we may, for example, assume that a given stone a has mass-trope α and a certain other stone b has mass-trope β. The view that is rejected by Mulligan is the view that the statement "This stone a is heavier than b" is made true by a relational trope that exists in addition to mass-tropes α and β. This external relation between a and b is a relation between things. It is supposed to be an entity that is borne by a as well as by b. Both Mulligan and Daniel Von Wachter wish to reject those theories that hold that is the truthmaker of the statement "This stone a is heavier than b". Mulligan also suggests that a given relation between things (to be more precise, an external relation, hence belonging at the structure level) obtains because a certain internal relation (hence belonging at the system level) between properties of these same things, already holds. So, the relation of "being heavier" that obtains between a and b, is made true by an internal relation between the mass-tropes α and β, undergoing the former. Furthermore, Mulligan holds too that internal relations between tropes6 are "irreducibly relational entities". Therefore, from an epistemological stance for Husserlian phenomenological theory, objectivities are the result of objectivating acts performing only at a very high layer within the (higher) stratified area of transcendental consciousness, hence belonging to transcendental Ego field, much well above perceptual proto-objects lying at the level of system of relations, pre-structural therefore [START_REF] Wiegand | On Referring to Gestalts[END_REF][START_REF] Mulligan | Relationsthrough tick and thin[END_REF]. Thus, we may identify two main levels, the first one (pre-linguistic, for proto-objects) concerning the system of relationships (Mulligan's internal relations), and the second one regarding the structure (Mulligan's external relations) built up upon this first level, coherently with above Lévi-Strauss ideas according to which context (i.e., semantics) is a basic component in determining the exact and correct setting of a structure [START_REF] Boudon | Strutturalismo e scienze umane[END_REF]. The latter level then performs between immanent and transcendental perception, mainly running from noematic level to noetic one, until up to reach the highest strata of transcendental consciousness field, with the final rising of symbolic and abstract representations by means of Erlebnis. Now, Yingxu Wang's visual semantic algebra (VSA), as it has been defined in the previous section, may be considered just as the result of a possible formalization of most of this latter process of human understanding according to Husserl's phenomenology. Indeed, it formalizes the main steps of human visual perception (overt attention), from transcendental to immanent perception of the external world objects or things, till to their geometric (i.e., formal) representation (as geons), in the transcendental consciousness field, with respect to a given visual object environment, that, providing the right semantic context (provided by Erlebnis), will allow to define other formal structures, as the VSA, in terms of (geometric) relations among geons (just the elements of the formal structure VSA).

All this, however, takes place within the transcendental consciousness field, starting from the interplay between immanent and transcendental perception, till to the basic, dialectic opposition noema-noesis which will give rise to single objects or things (whose set be ) and relations (whose set be ) amongst them, hence to the usual, many formal structures of the general type of universal algebra (and of denotational mathematics), together their representations just enabled by Erlebnis and its contents which provide the right semantic context by means of the so-called sedimentation process of meanings for ideal or abstract objects, meant as meaning-constitution acts of intentional consciousness that, being this free from any psychological or subjective aspect, makes so possible a collective understanding of such abstract entities. All this is however carried out between the strata of both transcendental and immanent perception, even inside overt attention realm, this being just coherent with what Husserl says about mathematical entities, which are considered as the main result of intentionality [START_REF] Tieszen | Mathematical Realism and Transcendental Phenomenological Idealism[END_REF][START_REF] Duke | Husserl and the problem of abstract object[END_REF]. Nevertheless, and this is just the main aim of the present paper, we wish to claim on what there might be before one reaches this main stage of knowledge; this has been accomplished just laying out our argument within a welldetermined epistemological framework, say , outlined thanks to either Husserl's phenomenological theory and structuralism, framework that has allowed, among other things, to identify a preliminary stage (i.e., the system stage) prior to what has been just said above (i.e., the structure stage).

To be precise, within this epistemological framework, we have identified an early stage, belonging to the so-called system stage, in which only relational entities exist at a pre-linguistic level for the so-called proto-objects to be acquired at a covert attention level and prior to overt attention of transcendental perception from which starts to spring out the next structure stage, as discussed just above. This early system stage, mainly acquired in an implicit fashion, is prior to the next structure stage which will be even more acquired at an explicit way until up to reach higher transcendental consciousness field, with its symbolic representations. So, what we wish to highlight in this paper, and at this point of the discussion made so far, is that just this system stage, of covert attention, is implicitly acquired thanks to the various eye movements described at the first sections of the present work, whose set therefore, say , is what precedes the incipit of the next structure stage giving rise to . So, in terms of universal algebra, we might say that , in a certain sense, generates with respect to , that is to say, Thus, we may say, in general, that is a pre-algebra of the algebra (according to universal algebra [START_REF] Cohn | Universal Algebra[END_REF]) , which, in our case, is said to be the eye movement prealgebra (EMpA), and it is such that , where , is a set defined according to what has been said in section II, about its composing elements (which are only relations). So, we intend to refer to the new fact according to which only a system of relationships, i.e., , occurring at an implicit (or unconscious) manner, is able to generate, within the framework , the structure , seen as an algebra to be explicitly meant in the sense of universal algebra [START_REF] Cohn | Universal Algebra[END_REF]. But, all this must be understood only within the epistemological framework as above worked out, with the contribution of Husserlian phenomenology and structuralism, since, in the current setting of universal algebra, every possible notion of ''generation'' of a generic algebraic structure, however does ever reference to another algebraic (sub)structure still of the type (set of elements, set of relations), not to a set of relations only, as in the case of above for . So, this new, formal notion of generation of an explicitly given structure, starting only from an implicitly given system of relations, has been possible exclusively making reference to both Husserlian phenomenology theory and structuralism, in establishing the above framework with respect to which such a notion of generation has been worked out.

Maybe, mirror neuron system might be involved in covert attention mechanism.

See, above all, the related works of John A. Bargh in[START_REF]The New Unconscious[END_REF], and references therein.

Historically, the functionalistic stance based on a certain notion of system of transformations, was already present, although in nuce, in the works of D'Arcy W. Thompson.

It should be much more complete to consider too the pragmatic context besides the semantics one; for an instance of pragmatic view of a formal structure, cf.[START_REF] Iurato | A Pragmatic Characterization of Concept Algebra. A few formal remarks on Denotational Mathematics as seen by Y. Wang[END_REF].

In the sense of Universal Algebra; see[START_REF] Cohn | Universal Algebra[END_REF].

Tropes are usually considered to have spatio-temporal nature[START_REF] Mulligan | Relationsthrough tick and thin[END_REF], so they belong to transcendental perception. They may be considered as not yet structured elementary entities of phenomenal reality.

ACKNOWLEDGMENTS

My thank to Professor Yingxu Wang, for having always given me a chance to present, in a certain manner and in some place, my research studies on the foundations and applications of denotational mathematics.