Paul Marrero

Eduardo J Acuña

PRIMAL CONJECTURE IN MATRIX E 9 φ

There is a certain enigma in everything that seems obvious."

In this paper we propose a conjecture about integer solutions to any equations, based on Primal algebra specifically this conjecture is a corollary of the Acuña s Theorem in that article. Also some problems are proposed which, if the conjecture is correct, could be solved

Introduction

The approach of this article is from the mathematical philosophical position of constructivism, that is to say that we create an arithmetic system with our rules and we observe how the elements behave and what relation they have with the real mathematics. With this in mind we introduce Primordial Algebra as a system of analysis aimed at integers, especially operations between integers ±Z. This algebra analyzes the numbers under a unitary module which is the 9 in base 10, in this way the remainders of each positive or negative integer generate groups congruent to each other, although these properties have been studied in the past it had not been possible to operate between positive and negative, the primordial algebra allows to operate between both types without falling into incongruities.

-Z -9Z

Z 9Z

/ Z ∈ Z + , -Z ∈ Z -

Notation

Before beginning to understand how primordial algebra works, it is necessary to understand the notation created for it.

• Let v be any integer number

• Let Φ be a function of the digital root of any Z

• Let Φ Z be any of the class of any Z, we replace "Z" to denote the value of the class.

• Let Z φ be any Primal Number, which is the representative value of the class and can be operated with.

Reduction of integers

The First idea that we use for this matrix was the property of reduction of integer numbers by the Digital Sum, all this is mention in the article (2021) [START_REF] Tarazona | Matrize_φˆ9. clasificación de números enteros. algebra primordial[END_REF], This is congruent whit the Method of Digital Sum Propose by Jhon Eider (2015) [START_REF] López | Suma sucesiva[END_REF], and we propose the the next Equation to found any Positive Digital Root based on Knuth (1972) [START_REF] Knuth | The Art of Computer Programming[END_REF]. Also this paper is gonna drop a resume of primal algebra, with the propose of make more easy to understand the conjecture.

The Novem Digital Root:

Φ 9 = v -9 v -1 9 (1)
in the Matrix E 9 φ the equation for the Digital Sum is:

Φ j = v -9(i -1) (2)
Both of this equations are equivalent, so we conclude when the digital root is in module 9, is equal to the Digital Sum.

Theorem Marrero Criterion

The Marrero's criterion, (2021) [START_REF] Tarazona | Matrize_φˆ9. clasificación de números enteros. algebra primordial[END_REF], also mentioned in the pre-print [START_REF] Flores | Existential refinement on the search of integer solutions for the diophantine equation x 3 + y 3 + z 3 = n[END_REF] as the K-interactive of the digital Sum. This is a criterion who tell when to stop to the Digital Sum Φ, we denoted the Criterion like "M " this is the mark to any function of Digital Sum that meet the condition of the criterion φ M .

Theorem M

Let v are integer, then there is a unique integer d ∈ D such that:

φ k (v k ) -d = 0, (3)
where D of the set of the points fixed of v.

Proof:

1. (Existence). Supposed b ∈ Z and k be a integer fixed. Consider the set

S b = φ k (v k ) -b = 0 where v k ∈ Z, S is not empty, because 1 ∈ S with k = 1. Then S ⊆ Z and let d being the smallest element of S, d ∈ S, it must have the form φ k (v k ) -d = 0, therefore φ k (v k ) = d.

(Uniqueness). Let

d 1 and d 2 ∈ D such that d 1 = d 2 . supposed that φ k1 (v k ) -d 1 = 0 and φ k2 (v k ) -d 2 = 0 (using Definition 2.1), then φ k1 (v k ) -d 1 = φ k2 (v k ) -d 2 , but φ k1 (v k ) -φ k2 (v k ) = d 1 -d 2 and φ k1 (v k ) - φ k2 (v k ) = 0. therefore d 1 = d 2 ,
this constitutes a contradiction.

Elements of the Matrix

In this section we give a brief introduction to the matrix E 9 φ , it is formed by the union of two positive and negative submatrices together with its own classification system, we adopt the following convention: Primal number Z φ ,Primal class Φ Z , integer numbers v, rows (i), columns (j). See the following figures for more details.

        1 φ 2 φ 3 φ 4 φ 5 φ 6 φ 7 φ 8 φ 9 φ 1 2 3
        Table 1: Positive sub-matrix [V + ]         -1 φ -2 φ -3 φ -4 φ -5 φ -6 φ -7 φ -8 φ -9 φ -1 -2 -3 -4 -

Primal Numbers

The primal number is the result of applying the digital sum function Φ and we will denote it by Z φ . We denote the set of positive and negative primal numbers by

D = {d| -9 φ ≤ d ≤ 9 φ |}
Note that each value of D represents a relationship between and its classes, that is, for every integer v that belongs to column j, there exists a primal number Z φ such that Z φ is congruent with j module 9.

Example

1.

9 φ = 72 2. 8 φ = 179 3. 4 φ = 9211 . . . 4.
-5 φ = -14

5.

-9 φ = -927

The Null Primal class

The Null class ∅, we will denote it by 0, is the one that is constituted by a column of zeros, and establishes a numerical border between the positive and negative classes of the matrix

E 9 φ .                      -9 φ -8 φ . . . -1 φ ∅ 1 φ 2 φ . . . 9 φ -9 -8 . . . -
                     Table 3: Matrix E 9 φ 3.

Primal Class

The primal class that we denote by Φ Z represents and are sets of congruence classes modulo 9, that is: The Primal class also are class of congruence with respect to the module 9, Vinogradov (1977) [START_REF] Vinogradov | Fundamentos de la teoría de los números[END_REF], in class 9 + -the residue is 0, but we take the module as the main value of the class, either positive or negative.

Φ Z = {v ∈ Z | Z φ ≡ v (mod 9)} 3.3.1 Example: 1. Φ 1 = {1,

Method of calculation of the positive columns

Recall that the columns of the matrix E 9 φ , we denote it by j, it is worth mentioning that said Matrix E 9 φ ,consists of 18 columns, nine are for the positive part [V + ] and nine for the negative part [V -] along with its respective primal number that we write at the beginning of each column in the matrix E 9 φ . There is many methods to calculate the columns by definition we chose the Digital Sum Φ, The novem digitalΦ 9 (1) or The matrix form Φ j (2). All mentioned before are valid to the positive Part [V + ].

Method of calculation of the Negative Columns

As the same way we define the positive columns, the negative part of the matrix contains all the negative integers Z -, this calculate is possible thanks to the negative novem digital root.

Definition:

Let v be a negative integer, we will write it as -v. We define the negative digital root novem, which we denote by ℧ and is as follows:

℧ -9 (v) = ((v + 1) mod (-9)) -1 (4)

Definition 2:

We define the negative novem floor function, [START_REF] Romero | OF ITS PROPERTIES RELATED TO MODULAR ARITHMETIC[END_REF] for a negative integer, we will denote it by and it is as follows:

℧-9 (v) = v -(-9) v + 1 -9 (5)
For the Negative Digital Sum in the matrix Φ -j we propose the next equation: Let i be the set of rows of the Matrix E 9 φ , the rows of the Matrix are always positive for any number inside the matrix, no matters if is negative, but the method to calculate their i is conditional to their sing of the number.

Φ -j = v + 9(i -1) (6)
To calculate the rows of the Matrix E 9 φ we proceed to consider two cases:

• case 1: Row a positive integer.

i v =    v 9 + 1 F or a v = 9c v 9 F or a v = 9c (7)
• case 2: Row a negative integer.

i -v =    -v 9 + 1 F or a v = 9c -v 9 F or a v = 9c (8)
Where 9c are all the multiple of 9.

Other ways to calculate the i is whit this two equations:

i v = v -φ v 9 + 1 (9) i -v = v -φ -v 9 -1 (10)
7 Decompress Primal numbers

One of the advantage of putting the integers numbers in the Matrix E 9 φ and turning into Primal numbers is that every number no matter the size can be compress into one number, but whit the matrix equations we can decompress those Primal numbers and get back the integer number. This is the fundamental base for the creation of a algebra of from the matrix, because the equations to decompress are getting isolating the formulas 2 and 6:

Isolating equation 2 to get the positive number we get:

v = Φ j + 9(i -1) (11)
Isolating equation 6 to get the negative number we get:

-v = Φ -j -9(i -1) (12)
7.1 Notation for reduced numbers in the matrix.

As we know a integer number in the Matrix is a set of three elements, his Primal Number Z φ , his position in j and i, whit this we can represent a integer inside the matrix like this:

v = Z i
Where i is the row and Z is the representative number, we don't use φ because would be redundant such that φ ≡ Z ≡ j are numerically the same, so we prefer pick only one to represent all.

Example:

Now some examples of how represent numbers in the matrix.

1. v = 178, → Z i = 7 20 2. v = 157, → Z i = 4 18 3. v = 293, → Z i = 5 33 4. v = 359, → Z i = 8 40 5. v = 999, → Z i = 9 111
If we go to the Matrix and locate the Z in the columns and the i in the rows the interception would be the correspondent number. Now whit the decompress equation we can go from Z i to v using the example 4. shown previously. 8 The Acuña's Theorem

The Acuña's Theorem [START_REF] Tarazona | Matrize_φˆ9. clasificación de números enteros. algebra primordial[END_REF], is the core theorem that expose the principles of the algebra primordial, using the properties of the matrix to arithmetic between classes.

Definition:

For all operation that we denote as (⊙ = +, -, /, * ) between two or more Primal Classes, the result will be another Primal class, and this is congruent with the numbers inside the class.

∀ Z φa ⊙ Z φb ≡ v a ⊙ v b (mod 9)
This is a extension of of class of residues by Vinogradov (1977) [START_REF] Vinogradov | Fundamentos de la teoría de los números[END_REF], but this work in both positive and negative integers.

Proof:

If:

v a , Z φa ∈ φ a → v a ≡ Z φa (mod 9) v b , Z φb ∈ φ b → v b ≡ Z φb (mod 9) v c , Z φc ∈ φ c → v c ≡ Z φc (mod 9)
Then by definition:

(φ ⊙ φ) a+b= = {x ∈ Z | x ≡ Z φa + Z φb = Z φc (mod 9)}
From where:

x -

(Z φa + Z φb = Z φc ) = 9t, f or t ∈ Z x = 9t + (Z φa + Z φb = Z φc ) x = 9t + [(v a -9q) + (v b -9r) = (v c -9s)] f or q, r, s ∈ Z x = 9t + [(v a + v b ) -9(q + r) = v c -9s] x = 9t + [(v a + v b ) -9(q + r) + 9s = v c ] x = 9t + [9(-q -r + s) + (v a + v b ) = v c ] x = 9(-q -r + s + t) + (v a + v b ) = v c → x ≡ v a + v b = v c (mod 9) ∴ x ∈ (φ ⊙ φ) a+b=c .
9 Group Macro Primal (ζ φ , ⊚)

We define the group Macro Primal is not empty and have the next properties:

Proof:

Let Macro Primal be:

{ζ φ } = Φ Z | Z ∈ D ∧ {0}
9.1.1 Is closed for operation:

φ a ⊚ φ b = (φ ⊚ φ) (a⊚b) Let's suppose (φ ⊚ φ) (a⊚b) = (φ ′ ⊚ φ ′ ) (a ′ ⊚b ′ ) Then if Z φa , Z ′ φa ∈ φ a → Z φa ≡ Z ′ φa (mod 9) Z φb , Z ′ φb ∈ φ b → Z φb ≡ Z ′ φb (mod 9) ∴ 9 | Z φa -Z ′ φa → Z φa -Z ′ φa = 9q/q ∈ Z 9 | Z φb -Z ′ φb -Z φa -Z ′ φb → Z φb -Z ′ φb = 9s
/s ∈ Z By definition we have to:

(φ ⊚ φ) (a⊚b) = {v ∈ Z | (Z φa ⊚ Z φb ) (mod 9)} Then we verified that 9 | v -(Z φa ⊚ Z φb ) v -(Z φa ⊚ Z φb ) = 9t/t ∈ Z then λ)v = 9t + (Z φa ⊚ Z φb )
Starting from λ we will continue the proof for the addition operation and later we will continue from it for multiplication.

v = 9t + (Z φa + Z φb ) v = 9t + (9q + Z ′ φa + 9s + Z ′ φb ) v = 9(t + q + s) + (Z ′ φa + Z ′ φb ) How (t + q + s) ∈ Z, we finally get that: v = 9(t + q + s) + (Z ′ φa + Z ′ φb ) → v ≡ (Z ′ φa + Z ′ φb ) (mod 9) ∴ v ∈ (Z ′ φa + Z ′ φb )
Now we continue the proof from λ for the multiplication:

v = 9t + (Z ′ φa * Z ′ φb ) v = 9t + [(9q + Z ′ φa )(9s + Z ′ φb )] v = 9t + [(81qs + 9qZ ′ φb + 9sZ ′ φa + Z ′ φa * Z ′ φb )] v = (9t + 81qs + 9qZ ′ φb + 9sZ ′ φa ) + (Z ′ φa * Z ′ φb ) v = 9(t + 9qs + 9qZ ′ φb + 9sZ ′ φa + (Z ′ φa * Z ′ φb ) How (t + 9qs + 9qZ ′ φb + 9sZ ′ φa ) ∈ Z then v = 9(t + 9qs + 9qZ ′ φb + 9sZ ′ φa ) + (t + 9qs + 9qZ ′ φb + 9sZ ′ φa ) → v ≡ (Z ′ φa * Z ′ φb ) (mod 9) ∴ v ∈ (Z ′ φa * Z ′ φb ) 9.1.2 Associative property for the sum Let φ a , φ b , φ c ∈ ζ φ → [φ a + φ b ] + φ c = φ a + [φ b + φ c ] Let's supposed Z φa ∈ φ a , Z φb ∈ φ b , Z φc ∈ φ c Then [φ a + φ b ] + φ c = (Z φa + Z φb ) + φ c = [(Z φa + Z φb + Z φc ) a+b+c ] = [φ a + (Z φb + Z φc ) b+c ] = φ a + [Z φb + Z φc ] b+c = φ a + [φ b + φ c ]
9.2 Property of existence of a neutral element φ ±9 ∈ {ζ φ } is the neutral element. 9.2.1 Proof:

φ a + φ ±9 = {v ∈ Z | v ≡ (Z φa + 9) (mod 9)} Then v -(Z φa + 9) = 9t, t ∈ Z v = 9t + (Z φa + 9) v = 9(t + 1) + Z φa → v ≡ Z φa (mod 9) ∴ v ∈ (φ ⊚ φ) a+9
10 Group {ζ φ } + Proposition of elements 10.1 Opposite element φ -a ∈ {ζ φ } is the opposite element 10.1.1 Proof:

(-Z φa ) ∈ φ (-a)
Then by definition we can affirm that: As we know by definition:

φ a + φ (-a) = {v ∈ Z | v ≡ [Z φa + (-Z φa ) (mod 9)]} v = 9t + [Z φa + (-Z φa )], t ∈ Z v = 9t + 0 → v ≡ 0(mod 9) → v ≡ 9 (mod 9) Ergo v ∈ φ 9 Finally φ a + φ (-a) = φ ±9
(φ + φ) a+c = (φ + φ) b+c = {v ∈ Z | v ≡ Z φa + Z φc = Z φb + Z φc (mod 9)} From where: v -[(Z φa + Z φc ) = (Z φb + Z φc )] = 9t, t ∈ Z v = 9t + [(Z φa + Z φc ) = (Z φb + Z φc )] v = 9t + [(Z φa + Z φc ) + (-Z φc ) = (Z φb + Z φc ) + (-Z φc )] v = 9t + [(Z φa + Z φc -Z φc ) = (Z φb + Z φc -Z φc )] v = 9t + [Z φa = Z φb ] → v ≡ Z φa = Z φb (mod 9) ∴ v ∈ φ (a=b)
11 Conjecture Primal

As we mentioned before, the Primal conjecture is a corollary of the "Acuña Theorem" in chapter 8, and goes in this way:

φ a ⊙ φ b ≡ φ c (mod 9) ⇒ ∃ Z a , Z b / Z a ⊙ Z b = Z c
This conjecture says, that any operations have a class or primal number as result, that means we have a solution that is congruent mod 9, but ¿if the congruence exist that means all member of the class are solutions?, or ¿One or more numbers are solution to that problem?.

In the case that the resulting class is congruent but not all its components are results, we can deduce that at least there are numbers within these classes that satisfy all the obtained congruence.

φ a ⊙ φ b ≡ φ c (mod 9) ⇒ ¬∀ Z a , Z b / Z a ⊙ Z b = Z c 12 Arithmetic Primal
All the early theorems are the base for the arithmetic tables presented in the paper, Acuña and Marrero (2021) [START_REF] Tarazona | Matrize_φˆ9. clasificación de números enteros. algebra primordial[END_REF] So we gonna show again and add some properties and new tables.

12.1 Primal additive Table

∀ φ a + φ b = φ c + φ 1 φ 2 φ 3 φ 4 φ 5 φ 6 φ 7 φ 8 φ 9 φ 1 2 φ 3 φ 4 φ 5 φ 6 φ 7 φ 8 φ 9 φ 1 φ φ 2 3 φ 4 φ 5 φ 6 φ 7 φ 8 φ 9 φ 1 φ 2 φ φ 3 4 φ 5 φ 6 φ 7 φ 8 φ 9 φ 1 φ 2 φ 3 φ φ 4 5 φ 6 φ 7 φ 8 φ 9 φ 1 φ 2 φ 3 φ 4 φ φ 5 6 φ 7 φ 8 φ 9 φ 1 φ 2 φ 3 φ 4 φ 5 φ φ 6 7 φ 8 φ 9 φ 1 φ 2 φ 3 φ 4 φ 5 φ 6 φ φ 7 8 φ 9 φ 1 φ 2 φ 3 φ 4 φ 5 φ 6 φ 7 φ φ 8 9 φ 1 φ 2 φ 3 φ 4 φ 5 φ 6 φ 7 φ 8 φ φ 9 1 φ 2 φ 3 φ 4 φ 5 φ 6 φ 7 φ 8 φ 9 φ Table 4.
In this table, the cycle of infinite repetitions of the primal values can be clearly appreciated.

12.2 Primal subtraction table

∀ φ a -φ b = φ c
Like the subtraction is not commutative we have two tables, the first to positive results and the second for the negative results.

-

φ 1 φ 2 φ 3 φ 4 φ 5 φ 6 φ 7 φ 8 φ 9 φ -1 9 φ 1 φ 2 φ 3 φ 4 φ 5 φ 6 φ 7 φ 8 φ φ -2 8 φ 9 φ 1 φ 2 φ 3 φ 4 φ 5 φ 6 φ 7 φ φ -3 7 φ 8 φ 9 φ 1 φ 2 φ 3 φ 4 φ 5 φ 6 φ φ -4 6 φ 7 φ 8 φ 9 φ 1 φ 2 φ 3 φ 4 φ 5 φ φ -5 5 φ 6 φ 7 φ 8 φ 9 φ 1 φ 2 φ 3 φ 4 φ φ -6 4 φ 5 φ 6 φ 7 φ 8 φ 9 φ 1 φ 2 φ 3 φ φ -7 3 φ 4 φ 5 φ 6 φ 7 φ 8 φ 9 φ 1 φ 2 φ φ -8 2 φ 3 φ 4 φ 5 φ 6 φ 7 φ 8 φ 9 φ 1 φ φ -9 1 φ 2 φ 3 φ 4 φ 5 φ 6 φ 7 φ 8 φ 9 φ
Table 5.

+ φ -1 φ -2 φ -3 φ -4 φ -5 φ -6 φ -7 φ -8 φ -9 φ 1 -9 φ -8 φ -7 φ -6 φ -5 φ -4 φ -3 φ -2 φ -1 φ φ 2 -1 φ -9 φ -8 φ -7 φ -6 φ -5 φ -4 φ -3 φ -2 φ φ 3 -2 φ -1 φ -9 φ -8 φ -7 φ -6 φ -5 φ -4 φ -3 φ φ 4 -3 φ -2 φ -1 φ -9 φ -8 φ -7 φ -6 φ -5 φ -4 φ φ 5 -4 φ -3 φ -2 φ -1 φ -9 φ -8 φ -7 φ -6 φ -5 φ φ 6 -5 φ -4 φ -3 φ -2 φ -1 φ -9 φ -8 φ -7 φ -6 φ φ 7 -6 φ -5 φ -4 φ -3 φ -2 φ -1 φ -9 φ -8 φ -7 φ φ 8 -7 φ -6 φ -5 φ -4 φ -3 φ -2 φ -1 φ -9 φ -8 φ φ 9 -8 φ -7 φ -6 φ -5 φ -4 φ -3 φ -2 φ -1 φ -9 φ Table 6.

Also we can obtain 0 when we subtract the same class, if and only if the number they are representing is the same. These two tables in particular are complex and less intuitive, since it is an operation that depends on the size of the number and the signs to determine if the result will be positive or negative, but it also affects the result in a modular way.

Primal Division

∀ φ a φ b = φ c → φ a φ c = φ b → φ b * φ c = φ a

  Where equation 5 and 6 are congruent with each other. 6 Method of Calculation of the rows of the Matrix.

8

  40 = 8 + 9(40 -1) → 8 40 = 8 + 9(39) 8 40 = 8 + 351 → 8 40 = 359

10. 2

 2 Cancellation law of the sum Let φ a , φ b , φ c ∈ ζ φ / φ a + φ c = φ b + φ c → φ a = φ b 10.2.1 Proof:

φ

  aφ b = 0 ≡ v av b = 0 | v a = a b → φ a = φ b

Table 2 :

 2 Negative sub-matrix [V -]

						
	5	-6	-7	-8	-9	
	-10 -11 -12 -13 -14 -15 -16 -17 -18 -19 -20 -21 -22 -23 -24 -25 -26 -27 -28 -29 -30 -31 -32 -33 -34 -35 -36 . . . . . . . . . . . . . . . . . . . . . . . . . . .	     

  -19 0 19 20 . . . 27 -36 -35 . . . -28 0 28 29 . . . 36 -45 -44 . . . -37 0 37 38 . . . 45 -54 -53 . . . -46 0 46 47 . . . 54 -63 -62 . . . -55 0 55 56 . . . 63 -72 -71 . . . -64 0 64 65 . . . 72 -81 -80 . . . -73 0 73 72 . . . 81 -90 -89 . . . -82 0 82 83 . . . 90 -99 -98 . . . -91 0 91 92 . . . 99

	. . .	. . .	. . .	. . .	. . .	. . .	. . .	. . .	. . .

1 0 1 2 . . . 9 -18 -17 . . . -10 0 10 11 . . . 18 -27 -26 . . .

Table ÷

 ÷ ∅ ∅ 4 φ ; 1 φ ; 7 φ ∅ ∅ 5 φ ; 2 φ ; 8 φ ∅ ∅ 3 φ ; 9 φ ; 6 φ φ 4 7 φ 5 φ ∅ ∅ 5 φ ; 2 φ ; 8 φ ∅ ∅ 4 φ ; 1 φ ; 7 φ ∅ ∅ 3 φ ; 9 φ ; 6 φ φ 7 4 φ 8 φ

	φ 1 φ 2	φ 3	φ 4 φ 5	φ 6	φ 7 φ 8	φ 9
	φ 1 1 φ 2 φ	3 φ	4 φ 5 φ	6 φ	7 φ 8 φ	9 φ
	φ 2 5 φ 1 φ	6 φ	2 φ 7 φ	3 φ	8 φ 7 φ	9 φ
	φ 3 3 φ	1 φ 8 φ	6 φ	4 φ 2 φ	9 φ
	φ 5 2 φ 4 φ	6 φ	8 φ 1 φ	3 φ	5 φ 7 φ	9 φ
	φ 6 3 φ	7 φ 2 φ	6 φ	1 φ 5 φ	9 φ
	φ 8 8 φ 7 φ	6 φ	5 φ 4 φ	3 φ	2 φ 1 φ	9 φ
	φ 9 ∅ ∅	∅	∅ ∅	∅	∅ ∅	Z φ

Table 7 .

 7 Also we know by the Acuña's Theorem, That:

Primal multiplication table

∀ φ a * φ b = φ c * φ 1 φ 2 φ 3 φ 4 φ 5 φ 6 φ 7 φ 8 φ 9 φ 1 1 φ 2 φ 3 φ 4 φ 5 φ 6 φ 7 φ 8 φ 9 φ φ 2 2 φ 4 φ 6 φ 8 φ 1 φ 3 φ 5 φ 7 φ 9 φ φ 3 3 φ 6 φ 9 φ 3 φ 6 φ 9 φ 3 φ 6 φ 9 φ φ 4 4 φ 8 φ 3 φ 7 φ 2 φ 6 φ 1 φ 5 φ 9 φ φ 5 5 φ 1 φ 6 φ 2 φ 7 φ 3 φ 8 φ 4 φ 9 φ φ 6 6 φ 3 φ 9 φ 6 φ 3 φ 9 φ 6 φ 3 φ 9 φ φ 7 7 φ 5 φ 3 φ 1 φ 8 φ 6 φ 4 φ 2 φ 9 φ φ 8 8 φ 7 φ 6 φ 5 φ 4 φ 3 φ 2 φ 1 φ 9 φ φ 9 9 φ 9 φ 9 φ 9 φ 9 φ 9 φ 9 φ 9 φ 9 φ Table 8.

12.5 Primal power table

8 φ φ 9 9 φ 9 φ 9 φ 9 φ 9 φ 9 φ 9 φ 9 φ 9 φ 9 φ 9 φ 9 φ 9 φ 9 φ Table 9.

In this table we can see all the results for the powers from the square to the power 15, looks like a cycle appears when we reach (φ Z ) 8 and reboot in (φ Z ) 14 , so this is a open problem. 12.5.1 Proposal Conjecture 2:"infinite cycle of powers"

This means that there is also an infinite cycle for the powers, which is symmetric and constant, this cycle would demonstrate the existence of congruence in the powers, allowing us to group polynomial equations so that they have the same modular congruence or in this case the same primal class.

Corollary of Problems

All this problem are in (mod 9)


B Somon

R N Roy

I Simonetti

F Dehais

Ecological measures of cognitive impairments in aeronautics: theory and application

Keywords: Neuroergonomics, Neuroadaptive Technology, Mobile electroencephalography, Flight Simulation, Signal Processing, Inattentional deafness, cEEGrid, ASR, ERP, Spectrum i

The objective of this chapter is to focus on the use of unobtrusive easy-to-use electrophysiological systems for neuroergonomic research. In a first section, we describe the challenges and limits related to the use of such systems. Electrode localization and signal processing solutions are then proposed to overcome some of the raised issues. In a second section, we explore the feasibility to measure pilots' auditory attention in a flight simulator using an unobtrusive EEG system [1] on a small number of participants. This study aims at measuring the cerebral activity associated with inattentional deafness in an ecological context with varying degrees of workload. The end goal was to assess the possibilities of such a system to be transferred to real-flight conditions. Our results illustrate this paradox: we were able to reproduce some of the results observed in the literature, but we also faced difficulties in terms of signal processing and measure identifications. We show that despite the lower signal-to-noise ratio observed with this kind of devices, we are able to detect event-related potentials (ERPs) and frequency features. In a last part we discuss our results with regards to Neuroadaptive Systems challenges, and how we were able to overcome some of the current limits in neuroergonomics.

INTRODUCTION

Neuroergonomics is a field of research that emerged in the early XX th century with the ambition to "study the brain at work and in everyday life" by combining theoretical advances in cognitive neuroscience, bio-engineering, computer science and human factors [START_REF] Parasuraman | Neuroergonomics: Research and practice[END_REF]. From a methodological point of view, neuroergonomics proposes to use cutting-edge brain imaging to infer the neural correlates underlying performance under realistic settings. One ultimate goal of neuroergonomics is to design neuroadaptive systems that are dedicated to optimize human(s)-machine(s) interactions [START_REF] Ayaz | Neuroergonomics: the brain at work and in everyday life[END_REF]. This can be done by dynamically adapting the user interface, the task and level of automation or by "stimulating" the end-users' brain (see [START_REF] Dehais | A Neuroergonomics Approach to Mental Workload, Engagement and Human Performance[END_REF] for a review). Technically, the development of neuroadaptive technology (defined in [START_REF] Frey | Physiologically Driven Storytelling: Concept and Software Tool[END_REF] as systems with "explicit input but unaware output") presupposes to address several challenges [START_REF] Hettinger | Neuroadaptive technologies: applying neuroergonomics to the design of advanced interfaces[END_REF] amongst which the design of the neurophysiological sensors is of paramount importance. Indeed, these sensors have to fulfill some requirements before being considered as usable for the implementation of neuroadaptive technologies:

• Their ease of use and comfort: sensors have to be usable in real-life conditions (sometimes for many hours), they must be fast and easy to put on and off and must be portable, non-invasive and, ideally, unobtrusive for the user . • Their sensitivity and specificity: as mentioned amongst others by [START_REF] Hettinger | Neuroadaptive technologies: applying neuroergonomics to the design of advanced interfaces[END_REF], neuroadaptive technologies should account for a mental phenomenon specifically, precisely and repeatedly in a reliable manner. • Their technical specifications: signal quality is a very important aspect of daily-life measures as the signal is very disturbed by noise coming from everywhere (e.g., physiological artefacts, electromagnetic noise, etc). In addition, recording and processing the signal must be very fast in order to maintain real-time measures.

Among neuroimaging techniques, although functional Magnetic Resonance Imaging (fMRI) and magnetoencephalography (MEG) provide relevant insights to identify the neural mechanisms underlying human performance in real-life situations [START_REF] Durantin | Neural signature of inattentional deafness[END_REF][START_REF] Desmet | Observing accidental and intentional unusual actions is associated with different subregions of the medial frontal cortex[END_REF][START_REF] Desmet | How social is error observation? The neural mechanisms underlying the observation of human and machine errors[END_REF], these devices can't be applied to mobile neuroergonomics research. Two other techniques, namely electroencephalography (EEG) and functional Near Infra-Red Spectroscopy (fNIRS), open promising prospects. These two techniques are fast and easy to set-up, barely invasive and have shown to have complementary benefits to measure cognition out of the lab [START_REF] Dehais | Monitoring pilot's mental workload using ERPs and spectral power with a six-dryelectrode EEG System in real flight conditions[END_REF][START_REF] Gateau | In silico vs. over the clouds: on-the-fly mental state estimation of aircraft pilots, using a functional near infrared spectroscopy based passive-BCI[END_REF]. In this chapter, we will mainly focus on ultra-mobile EEG for neuroadaptive purposes. We will first detail some challenges and issues related to the use of such sensors and propose solutions to overcome these limitations. A use case in aviation, involving seven participants in a flight simulator, will be taken as a concrete illustration of the implementation of neuroadaptive technologies with mobile EEG (namely a cEEGrid [START_REF] Debener | Unobtrusive ambulatory EEG using a smartphone and flexible printed electrodes around the ear[END_REF] system).

METHODOLOGICAL CONSIDERATIONS

Since the discovery of EEG by Hans Berger, many technical advances have led to improve this technique from a hardware, but also software solutions point of view. One major breakthrough for neuroergonomics was the development of mobile, dry and more or less low cost systems [12]. They allow for a decreased preparation time, less annoyance of the participant over time, and the possibility to record data over long periods of time without any questions about the gel drying or creating bridges between various electrodes. Many different dry sensors have been commercialized to various aims: recreational use (e.g., MindWave, NeuroSky Inc., San Jose, CA [START_REF]NeuroSky -Body and Mind. Quantified[END_REF]), neurofeedback/meditation (e.g., Muse, InteraXon Inc., Toronto, Canada [START_REF]Muse -Meditation Made Easy[END_REF]), BCI in general (e.g., Enobio, Neuroelectrics, Barcelona, Spain [START_REF]Enobio EEG systems[END_REF]), etc. Most of these systems have also been tested in medical or research conditions as they can provide fully operational mobile EEG recordings allowing for BCI studies or studies in real-life settings [START_REF] Katona | Speed control of Festo Robotino mobile robot using NeuroSky MindWave EEG headset based brain-computer interface[END_REF][START_REF] Dehais | EEG-engagement index and auditory alarm misperception: an inattentional deafness study in actual flight condition[END_REF]. Likewise other systems like in-ear EEG, i.e. very unobtrusive systems, have been used for a few years now for research and medical purposes (see Fig. 7.1a) [START_REF] Bleichner | Exploring miniaturized EEG electrodes for brain-computer interfaces. An EEG you do not see?[END_REF][START_REF] Fiedler | Single-channel in-ear-EEG detects the focus of auditory attention to concurrent tone streams and mixed speech[END_REF][START_REF] Goverdovsky | In-ear EEG from viscoelastic generic earpieces: Robust and unobtrusive 24/7 monitoring[END_REF]. Even though they are very useful to allow new insights in the underlying cognitive processes that take place in various situations, several papers have raised issues concerning the signal quality and acquisition/comparison difficulty with usual EEG recording systems [12]. In the case of neuroadaptive systems using new EEG recording tools, we would like to raise two practical difficulties: i) the localization of the electrodes which can be very different from the usual 10-20 system [START_REF] Jasper | Report of the committee on methods of clinical examination in electroencephalography[END_REF], and ii) the type of signal processing analyses which can be performed with those systems.

Electrode location

Electrode location is a very important matter in EEG acquisition. Standards have been defined, covering the whole scalp, for comparison and reproductibility reasons [START_REF] Jasper | Report of the committee on methods of clinical examination in electroencephalography[END_REF][START_REF] Keil | Committee report: publication guidelines and recommendations for studies using electroencephalography and magnetoencephalography[END_REF]. As the activity recorded under each electrode corresponds to a mixture of the activity emerging from several neural sources, knowing the exact location of every electrode is very important to characterize the exact source of a specific signal [START_REF] Homölle | Using a structured-light 3D scanner to improve EEG source modeling with more accurate electrode positions[END_REF][START_REF] Scherg | Separation and identification of event-related potential components by brain electric source analysis[END_REF]. Many signal processing tools currently allow to have a better estimation of source localization (e.g., based on blind source separation [START_REF] Makeig | Blind separation of auditory event-related brain responses into independent components[END_REF] and filtering [START_REF] Nunez | A theoretical and experimental study of high resolution EEG based on surface Laplacians and cortical imaging[END_REF][START_REF] Lotte | Regularizing common spatial patterns to improve BCI designs: unified theory and new algorithms[END_REF]). However, most of these tools are based on the initial or theoretical definition of the location of the electrodes (predefined models can be used). Wet cap-EEG methods, as well as a few dry ones, allow to have a proper definition of the location of the electrodes based on a fixed distance between the different sites according to the size of the head of participants. Even though this doesn't individualize the locations at the participant level, it allows for consistency across participants, thus for comparable recordings across participants or sessions.

Yet, with many compact and/or mobile EEG systems, one problem arises: they don't always follow the exact 10/20 locations. For example, the "original Muse" system [START_REF]Muse -Meditation Made Easy[END_REF][START_REF] Krigolson | Choosing MUSE: Validation of a Low-Cost, Portable EEG System for ERP Research[END_REF] (see Fig. 7.1b) only bears four recording electrodes: two large electrodes located behind each ear associated to TP9 and TP10 10-20 system electrodes, and two frontal electrodes associated to AF7 and AF8 standard locations. Three other electrodes located at frontal sites are used as reference electrodes. However, as mentioned by [START_REF] Krigolson | Choosing MUSE: Validation of a Low-Cost, Portable EEG System for ERP Research[END_REF]: "One thing we note here, was that certain head shapes, head sizes, and hair styles made data collection difficult [. . . ]". [START_REF] Bleichner | Exploring miniaturized EEG electrodes for brain-computer interfaces. An EEG you do not see?[END_REF]. In-ear EEG system with five electrodes: three recording electrodes located in the high concha (HC), the low concha (LC) and the outer ear canal (IE); and two reference electrodes located above the ear (AE) and on the mastoid behind the right ear (MA). (b) Original Muse with its recording (TP9/10, AF7/8) and reference (FPz, blue) electrodes. (c) cEEGrid EEG recording system (white C-shape around the ear) as observed with the get_chanlocs toolbox [START_REF] Homölle | Using a structured-light 3D scanner to improve EEG source modeling with more accurate electrode positions[END_REF] that allows to localize electrodes (black crosses) at the participant level.

It follows that the electrodes may not always be located at precise standard sites. Going further, if we take the example of another mobile EEG recording system (the cEEGrid [START_REF] Debener | Unobtrusive ambulatory EEG using a smartphone and flexible printed electrodes around the ear[END_REF]), the electrodes situated all around the ears cannot be linked to standard locations (except for the ones on the mastoids being linked to TP9 and TP10). This system is very interesting as it has wet electrodes, with only a small quantity of gel, and it is very unobtrusive, portable, not invasive and harmless for the participant. Moreover, the electrodes are all sticking to the skin of the participant, allowing for the gel not to dry over long periods of time [START_REF] Debener | Unobtrusive ambulatory EEG using a smartphone and flexible printed electrodes around the ear[END_REF]. Yet, one major drawback of this system is the inability to recover a standard electrode localization file, and difficulties to compare the results with usual cap-EEG results (as demonstrated in Section 7.3 of this chapter). Besides the conceptual problem of not knowing the exact location of each electrode, getting the right electrode position has an effect on both preprocessing and analysis. Finally, the small number of electrodes on many mobile EEG systems prevents us from using some preprocessing or processing functions, and from performing some analyses. An interesting signal processing tool for example is the Laplacian transformation which handles volume conduction issues. It can be used both as a very effective preprocessing technique to remove noise due to scalp conduction, but also an analysis technique to isolate current source densities. However, it requires both a good localization of the recording electrodes in space, and a good coverage of the surface for the model to work properly [START_REF] Kayser | Issues and considerations for using the scalp surface Laplacian in EEG/ERP research: A tutorial review[END_REF]. Here, we are reviewing the use of a few of these techniques, according to the literature, on the above-mentioned systems.

Preprocessing

One difficulty of EEG recording in everyday-life conditions is a lower data quality and more noisy channels. The reasons for this decreased signal to noise ratio are, most of the time, poorer electrode quality compared with in-lab recording systems, and artifacts due to the surrounding environment (especially in aeronautics because of the electromagnetic noise in the cockpit) or to the tasks at hand (motion artefacts). In this case, it is necessary to remove the noise through signal processing as it is the case for example with blind source separation techniques. Here, we will describe quickly a few of these techniques and assess their applicability to some mobile EEG systems according to the literature and to their characteristics (see Table 7.1). One of the very often used BSS technique in EEG is independent component analysis (ICA) [START_REF] Delorme | EEGLAB: an open source toolbox for analysis of single-trial EEG dynamics including independent component analysis[END_REF], and another is principal component analysis (PCA). ICA separates linearly mixed sources based on their independence. Likewise, PCA decomposes data, based on their relative variance. PCA is one of the main processing tools used for the artifact subspace reconstruction algorithms which allows to clean EEG data in real-time and is wildly used for neuroergonomics studies [START_REF] Mullen | Real-time modeling and 3D visualization of source dynamics and connectivity using wearable EEG[END_REF] (as well as its Riemannian adaptation [START_REF] Blum | A Riemannian modification of artifact subspace reconstruction for EEG artifact handling[END_REF]). Both techniques (ICA and PCA) require no a priori knowledge on electrode locations per se, but a large amount of data (in terms of time points and electrodes) is best for a better decomposition of the various sources. They are not easily applicable to a small amount of electrodes in a recording, and to our knowledge no study has been published on data acquired with systems that have less than 8 electrodes. Consequently, at some point, removing entire channels from the recordings becomes a necessity. This step is very often followed by a channel interpolation step of the removed channels in order to avoid biases between the right and left hemispheres during analysis. However, the interpolation of channels is, this time, a step which requires not only the electrode localization files, but also covering a good surface around the missing electrodes. These two assumptions might not be met for many mobile EEG systems.

System

Preprocessing ICA [START_REF] Delorme | EEGLAB: an open source toolbox for analysis of single-trial EEG dynamics including independent component analysis[END_REF] ASR/rASR [START_REF] Mullen | Real-time modeling and 3D visualization of source dynamics and connectivity using wearable EEG[END_REF][START_REF] Blum | A Riemannian modification of artifact subspace reconstruction for EEG artifact handling[END_REF]

Further processing and analyses

Several questions are raised for the use of unobtrusive EEG in real-life conditions when it comes to perform classical EEG data analyses (i.e., event-related potentials [START_REF] Luck | An introduction to the event-related potential technique[END_REF] -ERPsspectral decomposition [START_REF] Cohen | Analyzing neural time series data: theory and practice[END_REF], event-related spectral perturbations and inter-trial coherence [START_REF] Makeig | Mining event-related brain dynamics[END_REF][START_REF] Makeig | Auditory event-related dynamics of the EEG spectrum and effects of exposure to tones[END_REF] -ERSPs/ITCs -and source localization [START_REF] Jas | A reproducible MEG/EEG group study with the MNE software: recommendations, quality assessments, and good practices[END_REF]). First, because of the nature of ERPs and the fact that those evoked activities have a small amplitude compared with the global level of "noise" in EEG activity [START_REF] Luck | An introduction to the event-related potential technique[END_REF], it is necessary to have a good signal to noise ratio with unobtrusive EEG in order to properly detect ERPs through averages in a context where trials are not repeatable hundreds of times (see Table 7.2). Fortunately spectral domain analyses are far less sensitive to this problem and seem to be applicable to many EEG recording systems (even with only one or two electrodes [START_REF] Kumar | Feature extraction from EEG signal through one electrode device for medical application[END_REF]). For that, they can be applicable (and have been applied) to all the EEG systems (see Table 7.2). However, as mentioned in the previous paragraph, problems with electrode localization can also raise several questions regarding data analyses. The defined sources can be different or corrupted because of a bad location definition. Moreover, comparisons between participants or across sessions for one participant both at the source or at the electrode site level can be erroneous (e.g. one participant's T7 electrode actually corresponds to another participant's FTT7h in a 96-electrode layout).

System Analyzes

ERPs Spectra ERSP/ITC Source localization

Muse original [START_REF] Krigolson | Choosing MUSE: Validation of a Low-Cost, Portable EEG System for ERP Research[END_REF] [42]

∅ not enough electrodes and no standard locations cEEGrid [START_REF] Debener | Unobtrusive ambulatory EEG using a smartphone and flexible printed electrodes around the ear[END_REF] [1]

[43] [START_REF] Bleichner | Concealed, unobtrusive ear-centered EEG acquisition: cEEGrids for transparent EEG[END_REF] In-ear [START_REF] Kidmose | A study of evoked potentials from ear-EEG[END_REF] [20] ∅ [START_REF] Kappel | Ear-EEG Forward Models: Improved Head-Models for Ear-EEG[END_REF] Dry electrode [START_REF] Dehais | Monitoring pilot's mental workload using ERPs and spectral power with a six-dryelectrode EEG System in real flight conditions[END_REF] [10]

[31] ∅ TABLE 7.2 Evaluation of various EEG analyses in the time, frequency, time-frequency and spatial domains, and their applicability to several unobtrusive portable EEG systems which can be used for neuroergonomic studies. : has already been used in the literature and demonstrated its applicability; : cannot be used due to technical limitations; ∅: usage not tested/not reported.

Overcoming limitations

Several issues of mobile EEG have been raised in this section. Two main problems stand out in our opinion: (i) the one of electrode locations, which is the starting point of several difficulties in recording, processing and analyzing the data; and (ii) denoising data for more accurate online and offline analyses. Concerning the electrode locations, new very effective tools have been developed and can be used easily and fastly when recording in ecological conditions. Here, we show the example of the Structure sensor 3D scanning device (Occipital Inc., Boulder, CA) adapted on an Apple IPad (Apple, Cupertino, CA). This device has been proven to be very useful to improve the accuracy of EEG electrode positions [START_REF] Homölle | Using a structured-light 3D scanner to improve EEG source modeling with more accurate electrode positions[END_REF]. It can localize and export electrode localization according to the usual polar and/or Cartesian coordinates systems. After a quick head scanning step, the recorded object files can be loaded in EEGlab [START_REF] Delorme | EEGLAB: an open source toolbox for analysis of single-trial EEG dynamics including independent component analysis[END_REF] thanks to FieldTrip [START_REF] Oostenveld | FieldTrip: open source software for advanced analysis of MEG, EEG, and invasive electrophysiological data[END_REF] functions. Then the experimenter can select three fiducials which will allow the proper computation of the electrode locations. These fiducials, defined as the less varying across participants, are the nasion (called NAS), left helix-tragus junction (LHT) and right helix-tragus junction (RHT). For accuracy purposes, this selection has to be done twice with less than 2 distance between repetitions. Depending on the quality of the scanning, two approaches have came up for fiducials definitions. One is recommended on the get_chanlocs wiki page [START_REF] Lee | get chanlocs -SCCN[END_REF], the other one has been tested in our lab due to varying quality of our 3D scanning.

The first approach is to stick very visible targets on the nasion, LHT and RHT of the participant before scanning his/her head. Unfortunately, the ecological conditions that we were in, i.e. in the flight simulator room in our lab with low light, low contrast and mainly artificial light, created a bad environment for high accuracy recording of these target points. Consequently, it is recommended, for higher accuracy and reproductibility of the measures, to pinpoint the fiducials directly on the JPG file extracted with each object file from the scanner. Following the fiducials localization, a template can be loaded in order to have a first a priori on electrode locations and distance from a reference. Then, the experimenter is asked to locate precisely each electrode on a 3D image in a specific order (see Fig. 7.1c). Alike for the fiducials pinpointing the electrodes directly on the JPG image before loading the object file can be relevant. This allows to repeat the electrode localization measures more easily. Thanks to this easy-to-use 3D scanning method, the various preprocessing and analysis tools can be performed on mobile, non-standard EEG recording systems, but can also improve usual cap-EEG recording accuracy. More advice and tips about 3D scanning and get_chanlocs can be found on the get_chanlocs wiki page [START_REF] Lee | get chanlocs -SCCN[END_REF]. This improvement leads to the ability of using most of the preprocessing and analysis tools described above. Thus, it raises the second issue, which is data denoising. Concerning this matter, our experience on data recorded in very noisy everyday-life conditions has shown that adapting the parameters of several preprocessing tools was necessary for artifact removal. Indeed, several tools and functions are very strict and remove a lot of data. Adapting parameters allow to keep as much data as possible, but still remove the major artifacts and sources of noise. This is the case, for example, for the ASR function. Parameters for this function have been extensively studied [START_REF] Chang | Evaluation of Artifact Subspace Reconstruction for Automatic EEG Artifact Removal[END_REF][START_REF] Blum | Evaluation of Riemannian ASR on cEEGrid data: an artifact correction method for BCIs[END_REF]. Yet, using it on noisy data can become tricky as it removes part of the data, thus we can be led to remove a lot of it. Adapting the data removal criteria in those cases can be interesting. Likewise, artifact removal through ICA can be useful for noisy everyday-life EEG data. But selection criteria or thresholds for noisy components have to be adapted (especially with ICLabel for example). When few channels are available, the exclusion of too many "noisy" component can lead to remove brain activity without noticing it.

To conclude, the development of highly portable measuring systems draws a very promising avenue for neuroergonomic studies. Given the beneficial characteristics of EEG for use in neuroergonomic studies (high temporal resolution and portability, low invasiveness, and ease of use), this neuroimaging technique has been in the spotlight for development of new unobtrusive, wireless systems. As mentionned by [START_REF] Kam | Systematic comparison between a wireless EEG system with dry electrodes and a wired EEG system with wet electrodes[END_REF], there is a need, however, to evaluate these systems and their performance to ensure that they are comparable to state of the art wet EEG systems commonly used in laboratories. Here, we raised some issues concerning several of these new systems (namely the Muse original [START_REF]Muse -Meditation Made Easy[END_REF] -InteraXon Inc., Toronto, Canada -the cEEGrid [START_REF] Debener | Unobtrusive ambulatory EEG using a smartphone and flexible printed electrodes around the ear[END_REF] -TMSi & Debener, Otto von Guericke University, Magdeburg, Germany -in-ear EEG in general and dry electrode systems in general). Notably, electrode recording quality and electrode locations are two major points that have to be considered when using these types of systems in real-life conditions. They can have a impact on the acquisition phase, the (pre-) processing phase, but also the analysis phase. Keeping these considerations in mind, we tested one of these systems (the cEEGrid -TMSi & Debener, Otto von Guericke University, Magdeburg, Germany) to use in a noisy, everyday-life-like environment. The idea was to assess to measure auditory attention and the neurophysiological of inattentional deafness with a cEEGrid system during flight simulation, as aviation is one very relevant domain to design neuroadaptive technologies [START_REF] Dehais | A Neuroergonomics Approach to Human Performance in Aviation, Improving Aviation Performance through Applying Engineering[END_REF].

EXPERIMENTAL DATA

The objective of this section is to illustrate the implementation of the first steps of neuroadaptive technology in aviation. In this example, we focus on the monitoring of pilot's auditory attention and the phenomenon of inattentional deafness presented in Section 7.1. This latter degraded mental state is defined as the failure to perceive unexpected but perfectly salient audible sounds [START_REF] Durantin | Neural signature of inattentional deafness[END_REF][START_REF] Lavie | Blinded by the load: attention, awareness and the role of perceptual load[END_REF]. Several experimental studies have shown that this phenomenon can take place in the cockpit leading to miss critical auditory alarms. It is a known cause of several incidents or accidents in highly demanding contexts [START_REF] Billings | Toward a human-centered aircraft automation philosophy[END_REF][START_REF] Berberian | The out-of-the-loop Brain: a neuroergonomic approach of the human automation interaction[END_REF][START_REF] Dekker | MABA-MABA or abracadabra? Progress on humanautomation co-ordination[END_REF]. The temporal resolution of EEG offers relevant prospects to account for this phenomenon that takes place at an early perceptual stage [START_REF] Dehais | Inattentional deafness to auditory alarms: Interindividual differences, electrophysiological signature and single trial classification[END_REF][START_REF] Lavie | Blinded by the load: attention, awareness and the role of perceptual load[END_REF]. For instance, neuroergonomic experiments conducted in flight simulator and in real flight conditions revealed that ERP amplitude [START_REF] Dehais | Inattentional deafness to auditory alarms: Interindividual differences, electrophysiological signature and single trial classification[END_REF][START_REF] Dehais | A pBCI to Predict Attentional Error Before it Happens in Real Flight Conditions[END_REF], PSD in the , and bands [START_REF] Dehais | EEG-engagement index and auditory alarm misperception: an inattentional deafness study in actual flight condition[END_REF] as well as inter-trial coherence [START_REF] Callan | Disruption in neural phase synchrony is related to identification of inattentional deafness in real-world setting[END_REF] are neural signatures of inattentional deafness. These studies were conducted with bulky and uncomfortable systems (32-64 dry-electrode system) that are not suited for every day aeronautical operations. It is now time to assess the use of new, small, easily concealed EEG systems, namely the cEEGrid (TMSi, Oldenzaal, Netherlands [START_REF] Bleichner | Identifying auditory attention with ear-EEG: cEEGrid versus high-density cap-EEG comparison[END_REF]), into more ecological contexts. Indeed, it was shown that this system could be used to measure everyday-like activity [START_REF] Debener | Unobtrusive ambulatory EEG using a smartphone and flexible printed electrodes around the ear[END_REF] as well as polysomnography for example [START_REF] Sterr | Sleep EEG derived from behind-the-ear electrodes (cEEGrid) compared to standard polysomnography: a proof of concept study[END_REF], over several hours. Here, we used it in a flight simulator in motion to benchmark it in a controlled risk-free yet moving environment. To be able to use this system in real-life to measure auditory attention and possibly predict inattentional deafness would be a major step forward in the neuroergonomics and neuroadaptive technologies domains.

Materials and Methods

Participants

Seven participants took part in this experiment (6 men, 6 right-handers). They were all healthy, had no visual or hearing impairment as attested by their flying certificate and were not under any medication. They were on average 23.28 ± 1.21 y.o. All the participants were familiar with piloting as they were either undergoing the French piloting license (PPL -with at least ten hours of flight experience) or had passed it already. The study was approved by the Institutional Review Board of the local French ethics committee (Comité d'Evaluation Ethique de l'Inserm, IRB00003888-18-460) and conducted according to the principles expressed in the 1964 Declaration of Helsinki. Participants provided a written informed consent prior to the experiment. One participant was removed from further analyses due to bad data quality.

Experimental task and procedure

The flying task was performed using the ISAE-SUPAERO three-axis (roll, pitch and height) motion flight simulator designed by the French Flight Test Center which simulates a twin-engine aircraft flight model. It is composed of the classical actuators (side-stick, rudder, throttle, flaps lever and autopilot), and displays the simulated environment on eight screens disposed in semi-circle outside the cockpit (see Fig. 7.2a). Inside the cockpit, the user interface is composed of a Primary Flight Display, a Navigation Display and an Electronic Central Aircraft Monitoring Display page. Two stereophonic speakers located under the displays on each side of the cabin are used to broadcast continuous engine sounds (77 dB SPL) and to trigger the oddball sounds (90 dB SPL).Finally, the flight simulator environment enables to configure the visual environment (e.g., moment of the day, visibility, wind and weather conditions).

Flying scenarii

Participants had to perform one low-workload and one high-workload scenario. Both scenarii consisted of an approach and landing on the right runway of Toulouse-Blagnac Airport (LFBO-14R). This flying scenario is an adaptation of the VOR-DME arc approach of the airport and has already been tested in a previous experiment on the audiovisual conflict and inattentional deafness [START_REF] Dehais | Inattentional deafness to auditory alarms: Interindividual differences, electrophysiological signature and single trial classification[END_REF]. The aircraft position was initialized 20 nautical miles ( ) from the airport, at an altitude of 4000 feet ( ), a heading of 345 • and a speed of 170 knots ( ). First, participants had to make a U-turn on the left to come back to a 200 • heading. When they arrived at a 14 distance from the airport, they had to take a 270 • heading. Participants then had to keep that heading until they would cross the 334 • radiant of the VOR-DME beacon. Finally, they had to take a 144 • heading allowing them to face the runway. On this final approach and landing phase, they would intercept the Instrument Landing System (ILS -a system that sends radio waves to guide the pilot to the runway) and were asked to follow it for landing. In order to induce increased workload for both tasks, we added some wind during the whole flying scenarii, making it more difficult for participants to follow an accurate heading. In this study, the workload levels were manipulated according to the weather/visibility conditions. In the high load scenario, a very thick layer of clouds was implemented to drastically decrease the visibility on the landing ground. In the low load scenario there was really good visibility with a clear sky. Both flying scenarii took roughly 15 minutes to perform.

Oddball task

Concomitantly with the flying task, participants had to respond to an auditory oddball task. Two types of auditory stimuli (chirp sounds -= 44100 , 0.1 duration) were presented to the participants: an increasing-frequency sound (up-chirp) and a decreasing one (down-chirp) [START_REF] Artieda | Potentials evoked by chirp-modulated tones: a new technique to evaluate oscillatory activity in the auditory pathway[END_REF]. Standard sounds were presented with a 75% rate and odd sounds with a 25% rate. The type of sound (i.e., up-chirp and down-chirp) associated with standard and odd was randomized across participants. The total number of sounds presented for each flying scenario varied across participants (on average 276 ± 17 for the low workload and 272 ± 13 for the high workload condition) according to the time it took them to perform the approach and landing for each scenario.

Procedure

Upon their arrival, participants were asked to read and sign the informed consent form if they were willing to take part in the experiment. Then, after a quick overview of the various tasks they would have to do, the two cEEGrids (for right and left ear) were positioned according to the manufacturer and literature recommendations [START_REF] Debener | cEEGrid -How to use[END_REF][START_REF] Mirkovic | Target speaker detection with concealed EEG around the ear[END_REF].Participants were then installed in the flight simulator and were provided the exact flying and approach procedure (i.e. the VOR-DME arc approach described in paragraph 7.3.1.2.1). They were not familiar with this approach, and had never performed it before. Consequently, they were asked to perform a first familiarization phase -without motion -during which they got used to the instruments and dynamics of the flight simulator (i.e. the PFD, ND, ILS and VOR). This phase was shorter than the experimental flying scenarii, as the aircraft's starting position was at the 200 • heading, 15 from the airport. During this phase, participants were also trained with the oddball sounds and the side-stick trigger they had to press every time they heard a target/odd sound. The order of the two workload levels was randomized across participants (4 started with the low workload scenario and 2 with the high workload one). An emphasis was made for all participants on the fact that no task should be prioritized over the other.

Measure and analysis

All the behavioral and physiological data were streamed, recorded and synchronized through Lab Streaming Layer and associated apps [START_REF] Kothe | LSL -The LabStreaming Layer[END_REF].

Behavioral data

Participants' behavioral responses were continuously streamed from the flight simulator via LSL continuously at a 100 frequency rate. Response type and reaction times were extracted and analyzed a posteriori with MATLAB (The MathWorks, Natik, USA). There were on average 210 ± 12 standard (76.5 ± 0.76%) and 65 ± 6 odd sounds (2.5 ± 0.76%) for the low workload task, and 206 ± 9 standard (75.7 ± 0.94%) and 67 ± 5 odd sounds (24.3 ± 0.94%) for the high workload task. On average, less than 2 false positive (pulling the trigger for a standard sound) and late responses (i.e., pulling the trigger for an odd sound more than 2 after sound display) were observed per participant for the whole experiment. Based on these, hit and miss rates were computed for each participant in every condition (i.e., low workload and high workload) respectively as the number of correctly detected targets and the number of targets not responded to over the total number of odd sounds. In addition, the d' coefficient was computed as a measure of sensitivity to the sounds for each experimental condition [START_REF] Swets | Decision processes in perception[END_REF]. Finally, reaction times (RT) were computed as the time delay between sound display and participant's response. Late responses ( > 2 ) were not considered in the analysis.

Miss rate, d' and RT were analyzed with MATLAB (The MathWorks, Natik, USA) with a paired t-test with workload level as within subject factor. In addition, for each condition, the difference of the d' from 0 was tested with a unilateral t-test, showing higher than chance detection of odd sounds. All results are reported as ± .

Electroencephalography

The electroencephalogram (EEG) was continuously recorded from the two 10-channel cEEGrids (TMSi, Oldenzaal, Netherlands). cEEGrids are C-shaped around the ear adhesive 10 Ag/AgCl arrays which allow to record electrical brain activity non-invasively, unobtrusively and are very portable [START_REF] Bleichner | Identifying auditory attention with ear-EEG: cEEGrid versus high-density cap-EEG comparison[END_REF].

On each grid (left and right ear), two electrodes can be used as ground (L4a/R4a) and reference (L4b/R4b). The other 8 electrodes on each grid (L1-L8 and R1-R8) are recording channels. On our set-up (see Fig. 7.2b), the L4a and L4b electrodes were the ground and reference electrodes for both grids.

After cleaning and preparing the skin around each ear with an abrasive paste and alcohol, a small amount of electrolyte gel was applied on each of the 10 electrodes of each grid (as recommended on the cEEGrid website [START_REF] Debener | cEEGrid -How to use[END_REF]). The two grids were then positioned around each ear with a double-sided adhesive and hard-wire-connected together and onto a LiveAmp Bluetooth 24-bit DC-amplifier (Brain Products GmbH, Gilching, Germany) performed with an ActiCap adaptor. As mentioned in [START_REF] Mirkovic | Target speaker detection with concealed EEG around the ear[END_REF], electrode locations can vary slightly from one participant to another. Data were collected wirelessly via Bluetooth at 1000 through the LSL LiveAmp Recorder app. cEEGrid data were analyzed using EEGLAB [START_REF] Delorme | EEGLAB: an open source toolbox for analysis of single-trial EEG dynamics including independent component analysis[END_REF] (v.2019.1) on MATLAB (The Mathworks Inc.). Raw EEG data were down-sampled to 250 , high-pass filtered (1 ; Hamming window FIR, order: 826), cleaned for the 50 line noise (CleanLine function under EEGLAB; [START_REF] Mullen | CleanLine toolbox[END_REF]) and denoised thanks to an Artifact Subspace Reconstruction (ASR; [START_REF] Miyakoshi | Artifact Subspace Reconstruction plug-in[END_REF][START_REF] Chang | Evaluation of Artifact Subspace Reconstruction for Automatic EEG Artifact Removal[END_REF][START_REF] Chang | Evaluation of Artifact Subspace Reconstruction for Automatic Artifact Components Removal in Multichannel EEG Recordings[END_REF]) which removes artifacts based on a PCA. Finally, data were re-referenced to the average of R5 and L5 electrodes (closest to the mastoids, see Fig. 7.2b) and epoched time-locked to the sound stimuli (-1.5 to 2 around standard and odd sounds). Epoched data were baseline-corrected in the -200 to 0 time-window before stimulus. Finally, data were low-pass FIR filtered at 40 .

Epoched data were averaged according to the type of stimulus and response of the participant (hit, miss and standard) but also to the workload level (high workload vs. low workload). Usual event-related potentials (ERPs, in the time domain) and spectrograms (in the frequency domain) were computed. Spectrograms were obtained with a Welch periodogram and a 1.25window and 155 points overlap. At first, we observed the data according to the stimulus-response pattern type (hit, miss, standard) to get some insight into inattentionnal deafness related activity. Then, in order to increase the SNR and to avoid difference related to the number of trials for spectral activity, data for hits and misses were averaged as odds and the same number of trials were selected for the odd and the standard condition and in the high and low workload conditions whenever necessary. Thus, the final analyses report statistical comparisons according to the type of stimulus (odd vs. standard) and the workload level (high workload vs. low workload). Permutation statistics were applied on all time and frequency data with a significance level set at = .05 to compare the various conditions (type of stimulus × workload).

Finally, according to the previous cEEGrid studies on oddball paradigms, data at the L2 and R2 electrodes were analyzed [START_REF] Debener | Unobtrusive ambulatory EEG using a smartphone and flexible printed electrodes around the ear[END_REF]. Moreover, as we didn't expect any effect of sound lateralization on the EEG activity, data at L2 and R2 electrodes were grouped. In accordance with the oddball-related and previous cEEGrid studies, visual inspection of ERP data revealed a negative component around 150 -the N1 ERP -and a broader positive one starting around 300 after sound display -the P3 component -at the L2 and R2 electrodes. The N1 amplitude was computed for each participant as the most negative peak in the 100 -200 time-window following stimulus display, and the latency of this peak was further analyzed [START_REF] Luck | An introduction to the event-related potential technique[END_REF]. The P3 amplitude was computed for each participant as the mean amplitude, but also as the most positive peak, in the 250 -450 post-stimulus time-window [START_REF] Debener | Unobtrusive ambulatory EEG using a smartphone and flexible printed electrodes around the ear[END_REF]. Topographies for the P300 component are computed with a multisubject-adapted version of the cEEGrid plugin [START_REF] Bleichner | cEEGrid plugin[END_REF] in EEGLAB [START_REF] Delorme | EEGLAB: an open source toolbox for analysis of single-trial EEG dynamics including independent component analysis[END_REF]. N1/P3 amplitudes and latencies were analyzed with a two-way repeated-measure ANOVA with MATLAB (The MathWorks, Natik, USA) with stimulus type (odd vs. standard) and workload level (high vs. low workload) as within subject factors. The statistical significance was set at = .05.

Results

Behavioral data

All participants managed to land safely on the runway in the two conditions. There were on average the same number of oddball trials in the two workload conditions (275 ± 17 in the low workload and 272 ± 13 in the high workload condition), and the same odd/standard rate (76.6% standard in the low workload and 75.7% in the high workload condition). No significant difference was found in the miss rates between the low workload (23.4±8.18%) and the high workload (27.9±5.41%) conditions. Likewise, the sensitivity to sounds ( ) was equivalent in both conditions, and was significantly greater than 0 in the low workload (3.5 ± 0.30, (5) = 11.73, < 1.10 -3 ) and in the high workload condition (3.7 ± 0.27, (5) = 13.87, < 1.10 -3 ). Concerning reaction times (RT), there was no significant difference between RT to hits in the low workload (745.8 ± 46.77ms) and the high workload (777.5 ± 46.51ms) conditions (see Fig. 7.3a).

EEG measures

ERP analysis

When comparing the three types of stimuli (hit, miss, and standard) without any concern about the workload level, we observe an N1 component for the three types of stimuli peaking around 180 post-stimulus (see Fig. 7.3b). We also observe on these ERPs a very high amplitude variability (high standard errors) for the miss condition compared with hit and standard conditions, leading us to regroup hit and miss conditions as odd.

We were able to observe the N1 component on the grand averages. It reached its maximum on average 161.17 ± 6.18 post-stimulus. The ANOVA revealed no effect of the stimulus type nor of the workload on this latency. The average peak amplitude of N1 was -9.12 ± 3.50 . The ANOVA revealed no statistical difference between the stimulus types, nor between the workload conditions. Yet, the average peak amplitude for odd sounds was -11.34±3.46 and for standard sounds -6.90±1.72 (see Fig. 7.3c).

Concerning the P300 component, it was barely observable on our datasets. Yet, topographies (see Fig. 7.3d) seem to show an overall positive activity in the P300 time window.

Spectral analysis

Spectral activity in the four conditions (types of stimulus × workload level) are presented on Fig. 7.3e. Permutation analysis revealed a main effect of the type of stimulus (odd vs. standard; see Fig. 7.3e) on the low-frequency spectral activity in the : [10 -11] and in the : [23 -24] frequency windows (both < .05). Mean normalized activity in the [10 -11] frequency window was 0.23 ± 0.05 2 . -1 for odd sounds and -0.23 ± 0.05 2 . -1 for standard sounds. Mean normalized activity in the [23 -24] frequency window was 0.143 ± 0.09 2 . -1 for odd sounds and -0.143 ± 0.09 2 . -1 for standard sounds. There was no effect of workload and no interaction effect on spectral activity.

DISCUSSION

The main goal of this chapter was to discuss the challenges related to the use of neurophysiological measures in real-life conditions. In a first, methodological, section we raised theoretical concerns about new mobile, unobtrusive and non-invasive EEG recording systems and provided some advice to overcome them. In a second applied section, we assessed the usability of one of these systems (the cEEGrid [START_REF] Debener | Unobtrusive ambulatory EEG using a smartphone and flexible printed electrodes around the ear[END_REF]) to measure auditory attention and detect inattentional deafness in ecological settings. To this purpose, we recorded seven participants (only six were analyzed) with two 10-electrode C-shaped ear-centered EEG recording devices. Participants were asked to perform two approaches on Toulouse-Blagnac airport, each one corresponding to a specific level of workload depending on the implemented weather and visibility. Concomitantly, ) in the 0 -40 frequency window in response to standard (blue) and odd (red) sounds in the high workload (plain line) and low workload (dashed line) conditions. Grey windows show statistically significant differences in power spectral densities between odd and standard conditions. participants had to perform a classical auditory oddball task. This experimental paradigm allowed us to successfully induce failure of auditory attention as our participants exhibited a mean miss rate around 20%. Regarding the electrophysiological results, the usual N100 ERP component was observed both after odd and standard sounds detection. This N100 is one of the two ERP components elicited by auditory oddballs. The N100, known to reflect changes in the acoustic environment, is usually observed at fronto-central sites with a peak latency around 100 [START_REF] Luck | The Oxford handbook of event-related potential components[END_REF]. In our study, we observed this ERP at a later latency than usual. Two main hypotheses can be raised concerning this latency shift. It can be due either to the flying task that might delay sensory and/or cognitive processes, or to the recording sites being further from the dipole sources. According to the literature, we believe that the latter is the cause. Indeed, a former study performing the same flying task showed no latency shift in the N100 which was observed on average 116 post-stimulus [START_REF] Dehais | Inattentional deafness to auditory alarms: Interindividual differences, electrophysiological signature and single trial classification[END_REF]. On the other hand, a slight (even though not computed) delay has been observed previously in another study with the cEEGrid system [START_REF] Debener | Unobtrusive ambulatory EEG using a smartphone and flexible printed electrodes around the ear[END_REF]. Nevertheless, the delayed N100 had a similar (even higher) amplitude in our study compared with the literature [START_REF] Dehais | Inattentional deafness to auditory alarms: Interindividual differences, electrophysiological signature and single trial classification[END_REF] and was, as expected, observed for the three types of trials (hit, miss and standard).

Even though cEEGrids show promising results consistent with the literature, we were not able to observe the usual target-related P300 component in our study contrarily to previous cEEGrid study [START_REF] Bleichner | Identifying auditory attention with ear-EEG: cEEGrid versus high-density cap-EEG comparison[END_REF][START_REF] Debener | Unobtrusive ambulatory EEG using a smartphone and flexible printed electrodes around the ear[END_REF]. This leads to difficulties in characterizing the cerebral activity related to inattentional deafness, i.e., to dissociate responses to hit and miss trials. In our opinion, the inability to detect the P300 can be related to two different problems which were raised in the methodological section of this chapter (Section 7.1). One main reason is that our data were collected in an highly ecological environment (motion flight simulator) that is prone to decrease the signal to noise ratio of the collected data. The flying task was also challenging leading our participants to several motor actions (i.e., head, eye, arm and leg movements) to face the situation. Yet, we know that high muscular activity is degrades the quality of the electrophysiological signal recordings [START_REF] Luck | An introduction to the event-related potential technique[END_REF][START_REF] Cohen | Analyzing neural time series data: theory and practice[END_REF]. Movements may also affect the skin contact with cEEGrid electrodes leading to additional signal loss issue. Similarly to the N100, another reason for the difficulty to detect the P300 could be related to the location of the electrodes. Indeed, we know that the sources that give rise to the P300 are best recorded at parietal midline sites. Studies about source localization have identified different sources with several of them being tangential to the skull or at medio-central areas, thus not well recorded at around-the-ear temporal locations [START_REF] Tarkka | Electric source localization of the auditory P300 agrees with magnetic source localization[END_REF]. But still, one study with the cEEGrid system managed to record the P300 activity at upper electrode sites (R1-R4/L1-L4) [START_REF] Debener | Unobtrusive ambulatory EEG using a smartphone and flexible printed electrodes around the ear[END_REF]. It has to be noted though that in this study, the P300 amplitude is quite low (around 3 ) compared with other oddball paradigms [START_REF] Kayser | Principal components analysis of Laplacian waveforms as a generic method for identifying ERP generator patterns: I. Evaluation with auditory oddball tasks[END_REF]. In this line, another reason that explains our difficulties in detecting the P300 can come from the signal processing tools which were used. Indeed, these tools can be more or less aggressive towards EEG data [START_REF] Robbins | How Sensitive are EEG Results to Preprocessing Methods: A Benchmarking Study[END_REF] according to the quality of the data and the parameters provided to them. With the present data, we have used for example mostly the default values of the clean_rawdata() function of EEGLAB to perform the ASR. Yet, in this context, it might have removed part of the ERP data to which the cEEGrid electrodes were not very sensitive initially. Nevertheless, the topographies allow to show some positive activity in the time-window of the P300. In order to observe more easily the usual oddball-related ERPs with the very portable, unobtrusive cEEGrid electrodes, one possibility would be to perform a bipolar analysis. One study attempted to define standards in bipolar ERP measures for cEEGrid [START_REF] Bleichner | Identifying auditory attention with ear-EEG: cEEGrid versus high-density cap-EEG comparison[END_REF], but results have to be replicated. Even though no significant effect between the types of trials was observed on the timelocked ERP data, we were able to detect an effect on the spectral activity when comparing the electrophysiological response to odd and standard sounds. This is consistent with the fact that spectral analyses are usually less sensitive to the size of datasets in EEG data [START_REF] Cohen | Analyzing neural time series data: theory and practice[END_REF].

CONCLUSION

New EEG technologies have been developed thanks to state-of-the-art tools. They open a promising avenue for neuroergonomic studies in ecological conditions, as most of them have good data quality, can be used over long periods of time (either with dry EEG or wet electrodes), are easy to set up and are more or less conspicuous. Yet, they can also be very different from usual systems in the cognitive neuroscience field in terms of electrode locations and placement, impedance measures, number of electrodes and data quality. Thus, the tools used for preprocessing, processing and analyzing the signal need to be adapted in order to (1) reproduce well-known literature results, and (2) go further in confirming these results in everyday-life settings. In this chapter, we gave a few clues on how to adapt these new systems, but there are still some missing pieces in the puzzle of neuroergonomic studies of attentional incapacitation.

FIGURE 7 . 1

 71 FIGURE 7.1 Mobile EEG systems. (a) Adapted from[START_REF] Bleichner | Exploring miniaturized EEG electrodes for brain-computer interfaces. An EEG you do not see?[END_REF]. In-ear EEG system with five electrodes: three recording electrodes located in the high concha (HC), the low concha (LC) and the outer ear canal (IE); and two reference electrodes located above the ear (AE) and on the mastoid behind the right ear (MA). (b) Original Muse with its recording (TP9/10, AF7/8) and reference (FPz, blue) electrodes. (c) cEEGrid EEG recording system (white C-shape around the ear) as observed with the get_chanlocs toolbox[START_REF] Homölle | Using a structured-light 3D scanner to improve EEG source modeling with more accurate electrode positions[END_REF] that allows to localize electrodes (black crosses) at the participant level.

FIGURE 7 . 2

 72 FIGURE 7.2 (a) Three-axis motion flight simulator at the ISAE-SUPAERO. (b) Localization of the cEEGrid electrodes for both ears with recording reference (blue) and DRL (green) electrodes indicated. Electrodes R4a and R4b on the right grid were not recorded on our set-up. Layout adapted from the cEEGrid plugin [65] in EEGLAB (v.2019.1) [30]. (c) Localization of the left ear grid when fitted around the ear of a participant after cleaning and preparing the skin of the participant.

FIGURE 7 . 3

 73 FIGURE 7.3 Results of the oddball task during flight simulation at the (a) behavioral, and (b to e) neurophysiological level. (a) Boxplots representing miss rate (left plot -percent according to odd sounds) and reaction times for hits (right plot -in ms) in the low workload (left-hand side of each plot) and high workload (right-hand side) conditions. The red line shows the median and the bottom and top of each box show the 25th and 75th percentile respectively. ERPs for averaged L2 and R2 electrodes for (b) each type of trial (hit -black-vs. miss -red-vs. standard -blue), and (c) for each type of sound (odd in red vs. standard sounds in blue) and each workload level (high workload -plain line-vs. low workload -dashed line). (d) Topographies on the cEEGrid templates showing average amplitudes in the P300 time-window (250 -450 ) in response to standard (left) and odd sounds (right) in the high workload (top) and low workload (bottom) conditions. (e) Subjectnormalized power spectral density ( 2 /) in the 0 -40 frequency window in response to standard (blue) and odd (red) sounds in the high workload (plain line) and low workload (dashed line) conditions. Grey windows show statistically significant differences in power spectral densities between odd and standard conditions.

TABLE 7 .

 7

1 Evaluation of various EEG preprocessing or processing functions and their applicability to several unobtrusive portable EEG systems which can be used for neuroergonomic studies. : has already been used in the literature and demonstrated its applicability; : cannot be used due to technical limitations; ∅: usage not tested/not reported.

(a) (b) (c)

The Very high frequency Omnidirectional Range-Distance Measuring Equipment is a radio navigation tool that allow computing a fixed position to help aircraft land.

BIBLIOGRAPHY 18 BIBLIOGRAPHY

ACKNOWLEDGEMENTS

The authors would like to acknowledge the french Agence pour l'Innovation de la Défense (AID) -Direction Générale de l'Armement (DGA) for financing the "MAIA Project" through which the data were collected. They also thank Jérôme Dartigues for his help with the flight simulator.

ETHICAL STATEMENT

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.