Enoch Opeyemi Oluwole

From The Zeros of the Riemann Zeta Function to Its Analytical Continuation Formula

Keywords: Riemann zeta function; Non-trivial zeros, analytic continuation formula. MSC:11M06, 11M20, 11M26

The zeros of the Riemann Zeta Function are considered in this work and from the multiplication of these zeros; the analytic continuation formula of the Riemann Zeta Function is derived. The Riemann Zeta Function is eventually obtained through this. A general formula for the zeros of the analytic continuation formula of the Riemann Zeta Function is obtained and it is shown that these zeros will always be real.

INTRODUCTION

It is a common norm to obtain a desired polynomial from its known roots just as it is possible to move from the polynomial to its unknown roots. It is an established fact that the Riemann Zeta Function has the following roots, namely; = ± = -2 .

The author seeks to proof the Riemann Hypothesis by showing that if the Riemann Zeta function can be derived from its zeros then, one can proof that all the non-trivial zeros will always have their real parts as since the zeros are unique and that the zeros of the analytic continuation formula will always be real. It is good to note that the solutions to algebraic functions are algebraic numbers and that the solutions to L functions are algebraic functions. Thus, one is justified for treating the solutions of the Riemann Zeta function as we would have treated the solutions of algebraic functions, since the Riemann Zeta function is a type of L Function.

Theorem: If

1. ALL the nontrivial zeros are on the Critical Line [which is the Riemann Hypothesis, RH] and 2. ALL the trivial zeros are on the negative real axe (at even negative integer numbers) .THEN the Analytical Continuation Formula of the Riemann Zeta Function ( ) can be derived from them.

Proof:

If = ± = -2
are the zeros of the Riemann Zeta Function, then it follows that:

( ) = - 1 2 + - 1 2 - ( + 2 ) (1)
We can show (1) that

( ) = -+ 1 4 + ( + 2 ) (2)
If we choose

( ) = + ( ) = -+ ( ) ( + 2 ) (3)
One obtains:

( ) = 2 1 + 2 -+ ( ) (4)
By the process of discretization.one obtains;

( ) = 2 1 + 2 ( -1) 1 + ( ) ( -1) (5)
We write, by the method of partial summation [START_REF] Selberg | An elementary proof of the prime-number theorem[END_REF],

2 1 + 2 = 2 1 + 2 / (5 ) ( ) = ( -1) 1 + ( ) ( -1) 2 1 + 2 ; = 2 = 1 + 2 (5 )
From Riemann 1859 [1];

-log 2 = l log 1 + 2 -2 log (6) = ∞ - 1 log ∏ 2 = 1 log 1 + 2 (7)
From the above

- 1 log 2 = 1 log 1 + 2 (8) -2 Γ 2 = - 2 = 1 + 2 / (9) ( ) = -2 Γ 2 ( -1) 1 + ( ) ( -1) 2 (10)

Using Discretization Method

Let us choose = 1 then:

( ) = -2 Γ 2 ( -1) 1 + ( ) ( -1) 2 ; = 1,2,3, … (11)
It follows from [START_REF] Enoch | A new representation of the Riemann zeta functions via its functional[END_REF] that one can write

2 = 2 = 2 ( ) ( ) = -2 Γ 2 2 ( ) ( -1) 1 + ( ) ( -1) (12) ( ) = -2 Γ 2 2 1 1 ( ) ( -1) 1 + ( ) ( -1) (13)
( ) = -2 Γ 2 2 2 2 ( ) ( -1) 1 + ( ) ( -1) (14)
( ) = -2 Γ 2 2 3 3 ( ) ( -1) 1 + ( ) ( -1) (15) ⋮ ⋮ ⋮ ( ) = -2 Γ 2 2 (∞ ∞ ) ( -1) 1 + ( ) ( -1) = 0 (16)
Elegantly, if one discretizes [START_REF] Enoch | A new representation of the Riemann zeta functions via its functional[END_REF], then it implies:

( ) = ( ) = ( ) + ( ) + ⋯ + ( ) (17) ( ) = ( -1) 2 Γ -2 1 + ( ) ( -1) (17 )
Such that;

( ) = ( -1) 2 Γ -2 1 + ( ) ( -1) (18)

An Expression in Term of Prime Number

If we write and make use of the expression;

2 = 1 2 = ∝ ( 18
)
One can write (18) as;

( ) = ( -1) 2 Γ - 1 + ( ) ( -1) 1 2 ∝ (19)
Again, from (19) one obtains;

( ) = ( -1) 2 Γ -2 1 + ( ) ( -1) ∝ ∝ ( ) (20)
If Equation ( 11) is written as a product instead of discretizing [START_REF] Enoch | A new representation of the Riemann zeta functions via its functional[END_REF] then one arrivals at

( ) = ( -1) 2 Γ - 2 1 + ( ) ( -1) (21) And ( ) = ( -1) 2 Γ -2 1 + ( ) ( -1) ∝ ∝ ( ) (22)

Product over Prime Numbers

1 -1 1 = -1, 8,9 (23)
Then [START_REF] Enoch | A new representation of the Riemann zeta functions via its functional[END_REF] will become;

( ) = 2 ( -1)Γ 2 2 1 + ( ) ( -1) 1 -1 1 (24) ( ) 1 -1 = 2 ( -1)Γ 2 2 1 + ( ) ( -1) 1 ( 25
)
If the equation in ( 25) is multiplied over prime numbers, we will arrival at;

( ) 1 -1 = 2 ( -1)Γ 2 1 2 1 + ( ) ( -1) (26)
And eventually, we have a function that gives the product the Riemann zeta analytic

( ) ≅ 1 -1 1 2 ( + 2 ) ( + 2 ) + ( ) ( -1) (63)
Terence Tao in [START_REF] Tao | The Euler-Maclaurin formula, Bernoulli numbers, the zeta function, and real-variable analytic continuation[END_REF] showed how the other properties of the Zeta Function can be shown through analytical continuation method and in the same vain, it will be possible to show that (61) has all other properties of the Riemann Zeta Function.

Conclusion

Since there are no other classes of zeros that can lead to the derivation of the Riemann Zeta Function and its analytic continuation formula as the combination presented in this work, concludes the proof of the Riemann Hypothesis. With this, my next presentation will show us how to derive the Riemann operator and its connections to the Schrödinger operator.

Enoch Opeyemi Oluwole continuation formula with another function whose zeros are the non-trivial zeros of the zeta function. With this, one obtains an equation whose RHS is the analytic continuation Formula of Riemann zeta function;

Riemann gave

For equation ( 26) and ( 27) to be equal to (28), the quantity;

Equation (30) will hold and (27) will imply (28) if and only if;

The only conditions that make (26) and ( 27) to vanish are if;

This establishes the claim of Riemann that the non-trivial zeros will always have their real parts to be .

Using the Mellin's Transformation

If we make use of [17];

frac. means the fractional part.

From the zeros of the Riemann Zeta Function to its Analytical Continuation..

8429

Such that

From (33),

Then by using (33) in ( 24), one has

And it follows that

If we equate (28) to (36) and the compare them, we see that:

And the zeta function is obtained as

By working with equation (34), one obtains;

And again if (39) is compared with (28)

From (40), we can see that,

7. Lemma:

Proof

We want to establish that:

Substitute

= ± into (42) to obtain

The above expression will give

After which (45) becomes

Taking the limit of ( )as ⟶ ∞ lim

Thus (42) Implies that;

From the zeros of the Riemann Zeta Function to its Analytical Continuation..

From (33), we can obtain a general equation for generating the zeros of the analytic continuation formula of the Riemann Zeta function as follows:

Such that

Zeros of the Analytic Continuation Formula of the Riemann Zeta Function

Enoch has shown previously that ( 38) is always real. If one takes the limit of (51) as → ∞ one obtains;

If we choose = ± (55) , we arrival at:

The RHS is shown that

9. An Equivalent Formula for the Riemann Zeta Function

If one compares Equations ( 19) and ( 21), one can deduces that

And by comparing the RHS and the LHS of (47), one obtains;

The LHS of (60) will be equal to the RHS of (60) if we substitute (23) for -1 in (61) such that;

Another way to show that the LHS and RHS of (61) are equal is that the non-trivial zeros of the Riemann Zeta function are also the zeros of the term on the LHS of (61). But it is not obvious that the LHS of (61) has the same trivial zeros as that of its RHS until the LHS is written as;


Enoch Opeyemi Oluwole

THE RIEMANN HYPOTHESIS AND THE EULER'S QUADRATIC EQUATION

Keywords: Riemann zeta function, Euler's equation, meromorphic function, nontrivial zeros, and zeros of the analytic continuation formula

We present a general formula for obtaining the zeros of the analytic continuation formula of the Riemann zeta function from Euler's quadratic equation and show that these zeros are all real. Furthermore, substituting this formula into the analytic continuation formula, the root of the Analytic Continuation Formula (ACF) of the Riemann zeta function is determined.

Introduction

We presented in [START_REF] Enoch | From the zeros of the Riemann zeta function to its analytical continuation formula[END_REF] that the Euler's quadratic equation is connected to the ACF of the Riemann Zeta Function by showing that the equation;

41 2 + + x x (1)
could be structured to give the ACF of the Riemann Zeta Function by doing the following:

• instead of the coefficients of 2 x and x, being 1, we wrote k;

Enoch Opeyemi Oluwole 2

• for the constant integral, 41, we replaced it with ( ); t G

• multiplied the resulting equation with the trivial zeros of the Riemann zeta function;

Then the resulting equation was obtained as;

( () ) ( ) . 2 2 n z t G kz kz + + - (2)
By discretizing (2), we obtained;

( ) ( ) ( ) ( () ) ∑ ∑ ∞ ≥ ∞ ≥ + - ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + = ζ = γ 1 1 2 . 2 1 2 n n E t G kz kz n z n z z (3)
By the method of partial summation [START_REF] Atle | An elementary proof of the prime-number theorem[END_REF], we [START_REF] Enoch | From the zeros of the Riemann zeta function to its analytical continuation formula[END_REF] showed that

∑ ∑ ∑ ≤ ≤ ≤ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + n dq n d d n q n z n n z n 2 1 2 2 1 2 (4)
where

n d 2 = and . 2 1 ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + = n z q
We eventually obtained (3) as

( ) ( ) ( ) ( () ) ∑ ∑ ∑ ∞ ≥ ≤ ≤ + - ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + = ζ = γ 1 2 . 2 1 2 n n d d n q E t G kz kz n z n z z ( 5
)
Such that equation (5) was written as;

( ) ( ) ( ) ( ) ( ) ∑ ≤ - ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ π ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ - + Γ π - - = γ n d z z n z t G kz z z z z . 2 1 2 1 2 2 2 (6)
By using the expression;

∑ ∞ = - = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ - 0 . 1 1 1 1 n nz z p p [1, 2]
We arrived at

The Riemann Hypothesis and the Euler's Quadratic Equation 3( ) ( ) ( )

1 2 2 1 1 1 - ≤ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ π ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ - + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ - γ ∑ n d z z n z t G kz p z ( ) ( ) ∑ ∞ = -Γ π - = 0 2 . 1 2 1 2 n nz z p z z z (7)
Then we obtained equation [START_REF] Enoch | The Eigenvalues (energy levels) of the Riemann zeta function[END_REF], by multiplying equation ( 7) over prime numbers;

( ) ( ) ( ) ∏ ∑ - ≤ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ π ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ - + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ - γ p n d z z n z t G kz p z 1 2 2 1 1 1 ( ) ( ) ∏∑ ∞ ≥ -Γ π - = p n nz z p z z z 1 2 . 1 2 1 2 (8)

Obtaining the Zero of the ACF of the Riemann Zeta

Function from ( ) t G

We went on to set the LHS of (8) as follows:

( ) ( ) ( ) . 1 2 1 1 1 2 - ∞ ≤ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ π ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ - + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ - γ = ε ∏ ∑ p n d z z e z t G kz n p z (9)
Then obtained a general formula for ( ) [START_REF] Enoch | From the zeros of the Riemann zeta function to its analytical continuation formula[END_REF] And on using the definition of ( )

t G ( ) ( ) ( ) ( ) 1 2 1 1 1 1 2 - - ε ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ - π γ - = ∏ ∑ ∞ ≥ z kz n p z z t G p n e z z
z γ
as in [START_REF] Enoch | A Validation of the Real Zeros of the Riemann Zeta Function via the Continuation Formula of the Zeta Function[END_REF], we got

( ) ( ) , 1 ϑ - σ - = t z kz t G ( 11
)
where

( ) ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ - ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ - Γ π = σ ∑ ∞ = - 0 2 1 1 1 1 2 2 n nz z z p p z z (12) ( ). 1 - ϑ = σ Such that; ( ) ∑ ∞ = - ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ - Γ π = ϑ 0 2 1 1 1 2 2 n nz z z p p z z (13)
And

( ) ( ) ( ) ∏ ∑ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ - ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ - - π + - + π + - = τ ∞ = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ - π - p z n z z n p z n z n e z z . 1 1 1 2 1 2 2 1 1 2 1 1 2 2 1 2 1 2 2 (14)
From equation ( 11), we obtained a general equation for the zeros of the Analytic Continuation formula as;

( ) ( ) ( ) ( ) . 1 1 1 1 2 ϑ - τ - ϑ - = ϑ - τ σ - = + = z kz z kz kt t G (15)
Such that;

( ) ( ) ( )( ) ( ) 4 ; 1 1 1 1 1 2 1 2 1 = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ - ϑ - τ - ϑ - = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ - ϑ - τ σ - ± = k k z z k z z t (16) 4 = k
and t is the zero of the ACF of the Riemann Zeta Function.

The Realness of the Zeros of

The Analytic Continuation Formula, ( )

t ε Theorem 1. Let ( ) t G
be defined as it is in equation [START_REF] Enoch | From the zeros of the Riemann zeta function to its analytical continuation formula[END_REF] then it can be shown that the LHS RHS = of [START_REF] Tao | The Euler-Maclaurin formula[END_REF] and that the zeros of the analytic continuation formula, ( ),

t ε
of the Riemann Zeta Function will always be real as

. ∞ → n Proof. Let ( ) ( ) ( ) ( ). 1 2 1 1 1 1 2 2 - - ε - π γ - = + ∏ ∑ ∞ ≥ - z kz n P z z kt p n e z z (16) ( ) ( ) ( ) ( ) ; 1 1 2 1 1 2 1 1 1 2 ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ - - - ε - π γ - ± = ∑ ∏ ∞ ≥ - - - k z z n k P z z t n e P z z such . 4 = k (17)
If we take the limit of (17) as

∞ → n and , 2 1 it z + =
we obtain;

( ) ( ) ( ) ( ) 2 1 1 1 2 1 1 2 1 1 lim lim ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ - - - ε - π γ - ± = ∑ ∏ ∞ ≥ - - - ∞ → ∞ → k z z n k P z z t n e P z z n n (18) ( ) . 1 lim 1 lim 0 lim 2 1 ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ - - - ± = ∞ → ∞ → ∞ → k z z t n n n (19) If we choose it z ± = 2 1 and 4 = k in (19), we arrival at: [ ] . , lim lim 2 1 2 R ∈ ± = ± = ∞ → ∞ → t t t t n n (20)

The Analytic Continuation Formula of the Riemann Zeta Function

Theorem 2. Let the Analytical continuation formula of the Riemann Zeta function be given as:

( ) ( ) ∫ ∞ - ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ψ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + - = ε 1 4 3 2 ; log 2 cos 4 1 2 1 dx x t x x t t ( ) ∑ ∞ = π - = ψ 1 . 2 n x n e x ( 21
)
If t is the zero of ( )

t ε then it can be shown that ( ) 0 = ε t anytime ( ) ( ) ( ) ( ) . 4 1 1 2 4 1 1 2 1 1 2 1 ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ - - - ε π - γ - ± = ∑ ∏ ∞ ≥ - - z z n P z z t n e z P z Proof. We show that ( ) ( ) ( ) ( ) ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ - ε - π γ - - = ε ∑ ∏ ∞ ≥ - - - z n P z z t n e P z z 1 1 2 2 4 1 1 2 1 ( ) ∫ ∑ ∞ ∞ = - π - ϕ × 1 1 4 3 , log cos 2 n x n dx x x e (22)
where

( ) ( ) ( ) 2 1 1 1 2 4 1 2 4 1 1 2 1 2 ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ - ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ - ε - π γ - = = ϕ ∑ ∏ ∞ ≥ - - - z n P z z t n e P z z (23) ( ) . 0 = ε t Implies that; ( ) ( ) ( ) ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ - ε - π γ - = ∑ ∏ ∞ ≥ - - - z n P z z n e P z z 1 1 2 2 4 1 1 2 1 ( ) ∫ ∑ ∞ ∞ = - π - ϕ × 1 1 4 3 . log cos 2 n x n dx x x e ( 24
)
It follows that on further simplification of the RHS of (24), we obtain [START_REF] Patterson | An introduction to the theory of the Riemann zeta function[END_REF];

( ) ( ) . 2 ; 2 1 log cos 2 1 2 1 t x x x z z = ϕ + = ϕ - - (25)
This will mean [START_REF] Patterson | An introduction to the theory of the Riemann zeta function[END_REF];

. 2

1 2 1 ϕ + = ⇒ ϕ = - i z i z (26) That is . 2 2 1 2 2 1 t i z t i z + = ⇒ = - (27)
Such that it follows that;

( ) ( ) ( ). 2 1 2 1 log cos 2 2 t i t i i i x x x x x - ϕ - ϕ + = + = ϕ (28)
Because, if we have [START_REF] Patterson | An introduction to the theory of the Riemann zeta function[END_REF];

( )

x r log cos such that ; 2 1 ir z = - then ( ) ( ). 2 1 log cos 2 1 2 1 z z x x x r - -+ =
One can see that the intergrade in equation ( 24) is the same as;

( ) ∫ ∑ ∞ ∞ = - π - ϕ 1 1 4 3 log cos 2 n x n dx x x e ( ) ∫ ∑ ∞ ∞ = ϕ - ϕ - π - + = 1 1 4 3 2 2 1 n i i x n dx x x x e (29) ∫ ∑ ∫ ∑ ∞ ∞ = ∞ ∞ = ϕ - - π - ϕ + - π - + = 1 1 1 1 4 3 4 3 . 2 1 2 1 2 2 n n i x n i x n dx x e dx x e (30)
The equation was solved and the resulting solution is given as:

⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ϕ + π + + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ϕ - π + = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ - ϕ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ϕ + - 4 3 2 4 3 2 1 4 3 1 4 3 2 1 i i i n i n . 4 3 2 2 2 1 2 ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ ϕ + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ π + × ∑ ∞ = π - n e n n (31)
By using some transformations in complex analysis such that

i n e π = 2 1 in
Enoch Opeyemi Oluwole 8 (31), we obtain;

( )

, 4 3 4 3 2 2 2 2 1 2 2 3 2 ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ ϕ + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ π + ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ϕ + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ π + + ∑ ∞ = π - πϕ - πϕ π - n e i n e e e n n n n i n (32)
where

⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ ϕ + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ π + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ π + = β ∑ ∞ = π -
1 2 ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ ϕ + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ π + ϕ = α ∑ ∞ = π - n e n n (34) Such that; ( ) ( ) ( ) . 4 3 4 1 2 4 1 1 2 1 2 2 2 1 2 1 1 2 1 2 ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ ϕ + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ π + ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ - ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ - π ε - γ - = α ∑ ∑ ∏ ∞ = π - ∞ ≥ - -
3 cos sinh ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ π α - π β - ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ π α + π β πϕ = n n i n n n (37)
By substituting (37) into equation ( 22), we will obtain;

( ) ( ) ( ) ( ) πϕ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ - π ε - γ - - ∑ ∏ ∞ ≥ - - n z n P z z n z e P z sinh 2 4 1 1 2 1 1 2 1 . 0 2 3 cos 2 3 sin 2 3 sin 2 3 cos = ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ π α - π β - ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ π α + π β ⋅ n n i n n (38)

Conclusion

We conclude by considering the analytical continuation formula as follows:

If ( ) , 0 = ε t then in equation (38) the term ( ) ( ) ( ) ( ) πϕ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ - π ε - γ - ∑ ∏ ∞ ≥ - - n z n P z z n z e P z sinh 2 2 1 2 1 1 2 1 2 1 2 3 cos 2 3 sin 2 3 sin 2 3 cos = ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ π α - π β - ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ π α + π β ⋅ n n i n n (39) ( ) ( ) ( ) ( ) πϕ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ - π ε - γ - ∑ ∏ ∞ ≥ - - n z n P z z n z e P z sinh 2 2 1 1 1 2 1 . 1 2 3 cos 2 3 sin 2 3 sin 2 3 cos = ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ π α - π β - ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ π α + π β ⋅ n n i n n ( 40
)
Such that (39) implies; If we rationalize the RHS of (41), it will result into;

⎟ ⎠ ⎞ ⎜ ⎝ ⎛ π α - π β + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ π α + π β ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ π α - π β + n n n n n n i ( 42
)
And with this reality, equation (38) will always be equal to zero anytime t as defined in (16).

  can write the RHS of (29) as;

Acknowledgement

The author thanks the anonymous referees for their valuable suggestions and comments which improved the presentation of the paper.


G Bourda
email: bourda@obs.u-bordeaux1.fr

P Charlot

R Porcas

S Garrington

VLBI observations of weak extragalactic radio sources for the alignment of the future GAIA frame with the ICRF

Keywords: astrometry, reference systems, quasars

The space astrometry mission GAIA will construct a dense optical QSO-based celestial reference frame. For consistency between the optical and radio positions, it will be important to align the GAIA frame and the International Celestial Reference Frame (ICRF) with the highest accuracy. Currently, it is found that only 10% of the ICRF sources are suitable to establish this link, either because they are not bright enough at optical wavelengths or because they have significant extended radio emission which precludes reaching the highest astrometric accuracy. In order to improve the situation, we have initiated a VLBI survey dedicated to finding additional high-quality radio sources for aligning the two frames. The sample consists of about 450 sources, typically 20 times weaker than the current ICRF sources, which have been selected by cross-correlating optical and radio catalogues. This paper presents the observing strategy and includes preliminary results of observation of 224 of these sources with the European VLBI Network in June 2007.

The ICRF (International Celestial Reference Frame) is the fundamental celestial reference frame adopted by the International Astronomical Union (IAU) in August 1997 (Ma et al., 1998;Fey et al., 2004), currently based on the VLBI (Very Long Baseline Interferometry) positions of 717 extragalactic radio sources. The European space astrometry mission GAIA, to be launched by 2011, will survey about one billion stars in our Galaxy and 500 000 Quasi Stellar Objects (QSOs), brighter than magnitude 20 (Perryman et al., 2001). Unlike Hipparcos, GAIA will construct a dense optical celestial reference frame directly in the visible, based on the QSOs with the most accurate positions (i.e. with magnitude V 18; [START_REF] Mignard | IAU 25 Joint Discussion 16: The International Celestial Reference System, Maintenance and Future Realizations[END_REF]). In the future, the alignment of the ICRF and the GAIA frame will be crucial for ensuring consistency between the measured radio and optical positions. This alignment, to be determined with the highest accuracy, requires several hundreds of common sources, with a uniform sky coverage and very accurate radio and optical positions. Obtaining such accurate positions implies that the link sources must have V 18 and no extended VLBI structures. In a previous study, we investigated the current status of this link based on the present list of ICRF sources [START_REF] Bourda | A&A in preparation Condon[END_REF]. We found that although about 30% of the ICRF sources have an optical counterpart with V 18, only one third of these are compact enough on VLBI scales for the highest astrometric accuracy. Overall, only 10% of the current ICRF sources (≃70 sources) are thus available today for the alignment of the GAIA frame. This highlights the need to identify additional suitable radio sources, which is the purpose of the project described here.

Searching for additional sources suitable for aligning accurately the ICRF and the GAIA frame implies going to weaker radio sources having flux densities typically be- . Flux density distribution (units in mJy) for the sources detected in our initial experiment EC025A (June 2007). On average, the flux density of these sources is 20 times and 7 times weaker than that of the ICRF and VCS sources (Kovalev et al., 2007 and references therein). low 100 mJy. This can now be realized owing to recent increases in the VLBI network sensitivity (e.g. recording now possible at 1Gb/s) and by using a network with large antennas like the EVN (European VLBI Network). A sample of about 450 radio sources, for which there are no published VLBI observations, was selected for this purpose by cross-identifying the NRAO VLA Sky Survey (NVSS; Condon et al., 1998), a deep radio survey (complete to the 2.5 mJy level) that covers the entire sky north of -40 • , with the Véron- Cetty & Véron (2006) optical catalogue. This sample is based on the following criteria: (i) V 18 (to ensure very accurate positions with GAIA), (ii) δ -10 • (for possible observing with northern VLBI arrays), and (iii) NVSS flux density 20 mJy (for possible VLBI detection). The observing strategy to identify appropriate link sources in the sample includes three successive steps: (1) To determine the VLBI detectability of these weak radio sources; (2) To image and determine an accurate astrometric position for the sources detected in the previous step; and (3) To refine the astrometry for the most compact sources of the sample.

The first observations for this project (experiment EC025A) were made in June 2007, with a network of four EVN telescopes. Half of our sample (i.e. 224 target sources, most of which belonging to the CLASS catalogue; [START_REF] Myers | [END_REF] was observed during this experiment to determine their VLBI detectability (step 1 described above). The rest of the sources will be observed in October 2007. Our results for EC025A indicate excellent detection rates of 99% at X band and 95% at S band, with 222 sources and 211 sources detected at X and S bands, respectively. The mean correlated flux densities have a median value of 32 mJy at X band and 55 mJy at S band (see Figure 1). The spectral index α of the 211 radio sources detected at both frequencies was also investigated. Its median value is -0.3 and most of the sources have α > -0.5, hence indicating that they must have a dominating core component, which is very promising for the future stages of this project.

  Figure1. Flux density distribution (units in mJy) for the sources detected in our initial experiment EC025A (June 2007). On average, the flux density of these sources is 20 times and 7 times weaker than that of the ICRF and VCS sources(Kovalev et al., 2007 and references therein).


Béatrice Linot
email: beatrice.linot@loria.fr

Trust in Computer-Supported crisis management information sharing

My Doctoral research aims to identify the psychological and social factors that influence trust and determine the information sharing behavior of professional participants in the crisis response system. Building on the idea that the computer disrupts these factors, our aim is to design tools that restore the conditions of trust in a framework of collaborative information sharing. I combine theory and methods used in psychology and human factors, with computer science to determine how and why trust is degraded in relation to civil security operations. I propose to (1) identify the multi-level factors influencing trust during collaborative activities supported by computers (e.g., contextual factors, organizational factors, individual factors,); and (2) identify data-based design guidelines for digital devices that promote the sharing of information related to civil security and thereby develop and maintain shared situational awareness during collaborative activities.

Related Work

France has experienced several disasters in the last decade: Floods (Var, Alpes Maritimes (2015), Seine basin & Loire (2016); Storms Lothar, Martin (1999), Klaus (2009), Xynthia (2010); Terrorist attacks, Charlie hebdo and Bataclan (2015), Nice (2016); the Germanwings crash (2015), explosion of AZF plan in Toulouse (2001). The resulting disorganization and puts a premium on communication between different specialties (e.g. Police, firefighters, medical technicians etc.). Communication between specialties is essential [START_REF] Quarantelli | Research Findings on Organizational Behavior in Disasters and implications for disaster planning[END_REF]. [START_REF] Lagadec | Cellules de crise », les conditions d'une conduite efficace[END_REF] and [START_REF] Dautun | Contribution à l'étude des crises de grande ampleur : connaissance et aide à la décision pour la Sécurité Civile[END_REF] agree with this emphasis on communication, but enrich our understanding of the problem. Because crisis events are unexpected and stress local resources, they often require cooperation among team members who do not know each other and bring different technical expertise, experiences, culture, and organizations.

Groupware systems aim to provide participants with common awareness, i.e., information about the presence, activities, and availability of the other participants in the same system [START_REF] Bente | Avatar-mediated networking: Increasing social presence and interpersonal trust in net-based collaborations[END_REF][START_REF] Dunaway | Research Agenda in Intelligent Infrastructure to Enhance Disaster Management, Community Resilience and Public Safety[END_REF][START_REF] Xiu | A Generic Trust Framework for Large-Scale Open Systems using Machine Learning[END_REF]). Yet empirical studies in several domains (e.g., e-banking, civil security, healthcare, military, industries, etc.) reveal low participant confidence in these systems. Additionally, low confidence generates inappropriate behavior (e.g., altering and degrading performance of users technology) reducing use, thereby affecting efficiency. The French tool CRISORSEC is intended to support information sharing among crisis actors. Yet, users question its utility, its form, its uses, its limits and its possible perverse effects. Laurence Créton-Cazanave (a sociologist-geographer), studied CRISORSEC difficulties in French metropolitan areas. Créton-Cazanave, reinforces the link between trust and communication [START_REF] Créton-Cazanave | L'application Crisorsec, un dispositif numérique au coeur des enjeux de la gestion des crises en milieu métropolitain[END_REF]. Rapport GéNéPi, (2015) echoes the same problems: the technical and tools issues, the communication issues and the organizational issues during crisis situation management. One of the limitations of these studies is the absence of performance data either with or without the assistance of communication tools. I suggest that understanding performance in these different situations informs design requirements.

My pilot data (including 4 visits, 4 observations and 18 interviews) revealed several issues in crisis management collaboration. In particular, several different tools complement CRISORSEC. This strongly suggests that CRISORSEC does not support communication as intended, potentially breaking the link between the source and recipient of information that is maintained in the chosen tools. Identifying (and compensating for) the cause of this drift will improve the design of next generation communication tools. Across the visits, observations and interviews) I noted incidents and malfunctions related to the notion of trust: -To the tools due mainly to technical malfunctions (18), usability (11) and security (1).

-The data due to characteristics such as credibility and relevance (7), and recency (9).

-The person due to the skills (credibility, experience) of the person you trust (4), and the nature of the relationships, the well-being, and the previous experiences with the other (4). Technical malfunctions and usability issues will yield to more rigorous engineering. However, the other categories suggest more subtle issues of design and functionality.

Research questions

A psychological model of communication, including trust, is key to the design of computersupported crisis management communication tools. My research questions are:

1) Does mediation by computer for information sharing tools break the adaptive link in communication and thus reduce trust?

The use of alternative communication tools such as e-mail and telephone aims potentially recreates a missing adaptive link in existing tools such as CRISORSEC. In conventional, unmediated communication, the sender shapes the message, taking into account the specific needs of recipient with respect to his tasks in a global plan. Participants trust other participants to provide important information. If observers omit detail, recipients are justified in assuming that the omitted detail is not relevant. The sender may specify schedule, data characteristics, and situation that determine the activity of the recipient. I will examine the use of alternative means of communication (phone calls, sms, secure email, etc.) despite tool availability:

-Presence, number and objectives of parallel exchanges (e.g., verification, crossreferencing, questions of relevance, recency, reliability of information, interpretation); -The lack of use of information (Not taken into account and retention of information); -The lack of sharing (on the sharing tool) of known information (sharing or not information, validation of information by the hierarchy); -Development and use of a parallel tool in lieu of the common tool of sharing.

2) What is the role of context in establishing trust? During crisis response trust is a building process, depends mostly on contextual than individual and organizational factors. To support this claim, I must examine all three potential influences: -Individual (propensity to trust, experience, domain expertise, perceived risk, task goals).

-Local context (the seriousness of the situation, the level of risk, distributed and mediated work? update and relevance data, context situation, etc).

-Organizational context (formal responsibilities, management structure, existing communication tools and practices).

Methods

The goals is to obtain performance data concerning the factors that influence the decision to accept or distribute information. I seek convergent, ideally quantifiable evidence to address the above research questions. Psychology and ergonomics provides three general methods to gather data while minimizing experimenter bias: observation, interviews/surveys and experimental tasks. The study participants staff Crisis intervention, in crisis cells triggered for a given event (e.g., from CODIS, CORGN, CRRA, prefecture, CIC, COZ and COGIC); and includes professionals such as (fire-brigade, civil security associations, staff of the SAMU, gendarmerie, police, prefecture, military, and other partners). The proposed work will be conducted in two phases: Phase 1 largely concerns realistic crisis response activities, including observation and interviews. Phase 2 concerns largely experimentally contrived tasks, which provide both quantitative and qualitative data.

Phase 1-step 1: "Crisis exercises" Observation: Observations will be conducted in simulated crisis response in the French territory. Data collection includes participant written responses, video/audio recording, photographs and/or experimenter notes. In contrast to a traditional ethnographic exercise, I seek evidence regarding the ebb and flow of trust.

Phase 1 Step 2: Self-confrontation interviews (post-observation): Purely observational data may not reveal the intentionality considerations behind the observed information sharing. The goal of follow-up interviews is to obtain explanation concerning the factors that influence the decision to accept or distribute information. Phase 1 Step 3: Critical Incident interviews: One of the problems with observational study is that the conclusions are dependent upon the particular sample observed. In complex domains, this sample is highly likely to be biased. The critical incident technique is designed to facilitate the study of unobserved events, incidents or processes that the subject has previously experienced as significant, to clarify how they have been managed and the resulting effects.

Phase 2 Step 1: Incident Sorting: One of the limitations of Phase 1 activity is the confounding of data with particular participants and the absence of a domain analysis that integrates the data. Card sorting allows participants to group incidents according to an abstraction hierarchy of similarity. Phase 2 will use Phase 1 data develop a systematic set of scenarios concerning the sharing of information. Analysis identifies the terms, concepts, key words and actions inherent in the characteristics that distinguish those situations that foster trust in the sharing and exchange of data between different civil security services.

Phase 2 Step 2: Situational Judgment test (survey): My final method provides the best opportunity to obtain large scale evidence regarding a standardized set of operational conditions. A variety of Situational judgement tests concerning information sharing scenarios, informed by all of the above resources, will be distributed in the form of a questionnaire to professional civilian security on a larger scale. Of particular interest is the consistency or variability in response.

Expected contributions -Examine the extent to which trust issues pervade current communication tools.

-Establish the main factors that build trust in the information sharing activity.

-Specify the essential contextual factors that favor conditions of trust and more particularly for the design of collaborative sharing tools. Understanding mediated and unmediated communication will contribute design requirements for the next generation of crisis communication tools and more generally groupware systems.


Béatrice Linot
email: beatrice.linot@univ-lorraine.fr

Jérôme Dinet

François Charoy

Valerie L Shalin

Temporal Dispersion In Distributed Work

Keywords: Communication, information sharing, temporal dispersion, distributed work

Effective communication is critical to sociotechnical systems. Such systems may entail distributed authority and responsibility, and physical resources that are not collocated. This is certainly the case in crisis management. To improve communication during crisis management, researchers have tried to enhance the communication technology. However, communication problems persist even in the presence of robust technology. Several studies of communication have examined the consequences of the spatial distribution of operators working on the achievement of a common goal. Yet, few have researchers have investigated the effects of temporal distribution. Drawing on documentation of the Deepwater Horizon Accident in 2010, we select examples to reveal the temporal distribution of events that affect the availability of contextual information across a socio-technical system. We claim that temporal distance, like spatial distance, reduces global awareness and the opportunity to observe the incidental behaviour that renders activities mutually accessible.

Introduction 1.Spatial Dispersion in Work

Successful socio-technical systems depend on communication and situation awareness, for example, during response to natural disasters [START_REF] Quarantelli | Disaster Crisis Management (Preliminary paper #113). Disaster Research Center[END_REF]. Communication issues persist even in the presence of functioning communication equipment [START_REF] Lagadec | Le dirigeant et les crises[END_REF][START_REF] Dautun | Contribution à l'étude des crises de grande ampleur : connaissance et aide à la décision pour la Sécurité Civile[END_REF]. Classical ergonomics has recognized the spatial properties of the work environment [START_REF] Van Cott | Human engineering guide to equipment design[END_REF], including the more recent exploration of communication issues. We claim temporal dispersion has the same effect on communication as spatial dispersion.

Work is often distributed, either because of excessive task demands or separable tasks that require special skills. Each team member holds unique situation awareness [START_REF] Endsley | Design and evaluation for situation awareness enhancement[END_REF],

which may yet be compatible with the other agents [START_REF] Stanton | Genotype and phenotype schemata and their role in distributed situation awareness in collaborative systems[END_REF]. Salas et al. [START_REF] Salas | On teams, teamwork, and team performance: Discoveries and developments[END_REF] suggest that teams perform best with a "shared mental model". Shared awareness of the overall ! 2

work process provides context for individual activities [START_REF] Dourish | Awareness and Coordination in Shared Workspaces[END_REF]. Team member spatial proximity fosters information sharing and friendship [START_REF] Monge | The dynamics of organizational proximity[END_REF][START_REF] Van Den Bulte | The effects of R&D team co-location on communication patterns among R&D, marketing, and manufacturing[END_REF].

Certainly, spatial dispersion interferes with general factors pertaining to group dynamics, such as trust and norms [START_REF] Forsyth | Methodological advances in the study of group dynamics[END_REF]. Spatially proximal team members use voice and behavior to render their current task-specific activities accessible to others [START_REF] Luff | The collaborative production of computer commands in command and control[END_REF]. Several negative consequences arise without spatial proximity, such as the reduction of mutual knowledge, difficulty in work coordination and a decline in communication [START_REF] Scott | Collaboration Technology in Military Team Operations: Lessons Learned from the Corporate Domain[END_REF][START_REF] Allen | Managing the flow of technology[END_REF][START_REF] Cramton | The mutual knowledge problem and its consequences in dispersed collaboration[END_REF][START_REF] Herbsleb | Distance, dependencies, and delay in a global collaboration[END_REF][START_REF] Kraut | Tasks and relationships in scientific research collaborations[END_REF][START_REF] Zahn | Face-to-face communication in an office setting: The effects of position, proximity, and exposure[END_REF].

To compensate for the spatial distance that occurs in domains such as crisis management, researchers hope to employ technology [START_REF] Ernstsen | Situation Awareness in Disaster Management: A Study of a Norwegian Collaboration Exercise[END_REF]. However, Olson & Olson [START_REF] Olson | Distance matters[END_REF] note that communication technology fails to compensate for the effect of spatial distance. Karsenty [START_REF] Karsenty | Cooperative work and shared visual context: An empirical study of comprehension problems in side-by-side and remote help dialogues[END_REF] describes the compensatory verbal behavior in the absence of shared visual context in computer supported collaborative work. Dourish & Belloti [START_REF] Dourish | Awareness and Coordination in Shared Workspaces[END_REF] note that the provider of information is not the direct beneficiary, but the beneficiary depends on the sender's intuitions regarding the recipients' context and resulting information requirements. Moreover, the sender controls delivery, restricting recipient access, although the sender may not know what information is required and when.

Temporal Dispersion in Work

To the extent that temporal properties of work have been an ergonomic concern, having enough time to accomplish work is the typical focus, often because subtasks have temporal dependencies [START_REF] Franke | Collaborative Coordination of Activities with Temporal Dependencies[END_REF]. In dynamic situations, when rapid decision-making is required, Endsley [START_REF] Endsley | Design and evaluation for situation awareness enhancement[END_REF] notes that situation awareness is "the perception of the elements in the environment within a volume of time and space, the comprehension of their meaning and the projection of their status in the near future", (p. 97). Social psychology portends the challenge of temporally dispersed events, which weakens salience [START_REF] Liberman | The effect of temporal distance on level of mental construal[END_REF]. With increasing temporal distance, people frequently take more risk and feel more confident, [START_REF] Gilovich | Effect of temporal perspective on subjective confidence[END_REF][START_REF] Nisan | Dimension of time in relation to choice behavior and achievement orientation[END_REF]. Distant events (both past and future) seem more abstract and stereotypical [START_REF] Liberman | The effect of temporal distance on level of mental construal[END_REF], lacking in the very detail that is required to assure situation awareness. Thus, we claim that with temporal distance, like spatial distance, participants lose global awareness.

Methodological Considerations

The effect of temporal dispersion is difficult and costly to simulate, hampering the effort to obtain relevant performance data [START_REF] Liu | Spatiotemporal Mashups: A Survey of Current Tools to Inform Next Generation Crisis Support[END_REF]. Researchers are rarely present during the response to an unanticipated crisis. However, sometimes a government initiates a post-hoc analysis. This is the case in the 2010 Deepwater Horizon accident involving the explosion of a mobile oil drilling platform at the Macondo site in the Gulf of Mexico [27,28]. While response to the unfolding disaster is surely relevant to the communication problem, relevant data also result from investigation of the cause of the accident, yielding extensive video-taped testimony. These provide an unusually detailed, publicly accessible account of the work practice in complex temporally distributed socio-technical systems. In this paper, we use select illustrations of human ! 3

behavior from the Deepwater Horizon accident to reveal a daunting communication task. We argue that temporally distributed work impacts the construction of a global representation of the situation; the actors are not aware of the interdependent relations between their respective activities and therefore do not necessarily "see" the importance of communicating apparently innocuous information.

Deepwater Horizon Explosion

Fig. 1. Deepwater Horizon event chronology (adapted from Wikipedia [START_REF]Timeline of the Deepwater Horizon oil spill[END_REF]).

The investigative commission only notes temporal considerations with respect to cost management. To be sure, serious engineering deficiencies surrounded the blow out preventer, inaccurate diagrams, and ineffectively designed hydraulic shear rams. An automatic shutdown system that should have cut power and hydraulic lines was broken. Though tragic, these outright engineering failures do not concern us here. They are clearly both wrong and remedied with prescribed procedure and organizational culture [START_REF] Sharit | Human error and human reliability analysis[END_REF].

Ultimately, investigators associate the failures with communication (see Fig. 2). We note the underlying temporal dispersion which, like spatial dispersion, compromises the human awareness that may compensate for the observed analytic deficiencies. However, these communication failures may resist proceduralized remedy [START_REF] Perrow | Normal Accidents: Living with High Risk Technologies[END_REF]. They may be the unavoidable result of spatio-temporally cross-disciplinary work that is always dependent upon numerous risky judgments and impromptu changes to a previously established plan. Temporally distant operations provide context that is particularly likely to be sequestered. But constant skepticism between experts on an extremely large base rate of non-consequential judgment increases false alarms. The result of false alarms is not merely wasted money; as Endsley [START_REF] Endsley | Design and evaluation for situation awareness enhancement[END_REF] noted, dynamical systems, in particular, require timely response. April 20, 2010 -The "Ops Note" plan prevails: the crew would remove 3,300 feet of mud from below the mud line and set the cement plug after the riser had been displaced.

April 20, 2010 -Explosion following misinterpreted pressure testing.

"BP, Transocean, and Halliburton failed to communicate adequately. Information appears to have been excessively compartmentalized at Macondo as a result of poor communication. BP did not share important information with its contractors, or sometimes internally even with members of its own team. Contractors did not share important information with BP or each other. As a result, individuals often found themselves making critical decisions without a full appreciation for the context in which they were being made (or even without recognition that the decisions were critical)" [27] p. 123.

"BP personnel on the rig were not properly trained and supported, and all three companies failed to communicate key information to people who could have made a difference" [28] p. x.

"Finally, due to poor communication, it does not appear that the men performing and interpreting the test had a full appreciation of the context in which they were performing it. Such an appreciation might have increased their willingness to believe the well was flowing" [27] p. 119.

"Insufficient communication, both prior to and during the final displacement, affected risk awareness and well monitoring on the Deepwater Horizon. BP did not adequately inform Transocean about the risks at the Macondo well, particularly the risks of a poor bottomhole cement job. ! 5

We have selected a few examples from investigation reports and publicly available testimony of the Deepwater Horizon investigation. Although our methods do not follow anthropologically rigorous grounded theory (i.e., raising an emerging problem by systematic analysis of all the data), we suggest that the illustrations we provide are representative of nominal operations.

The Deviations from the Nominal Plan

The Nominal Plan

Below we first identify important features of the nominal plan to drill the well and then hand it over to a less costly rig, a process known as abandonment. Then we identify deviations from the nominal plan, which were not appreciated or communicated and combined to create the Deepwater disaster.

Planned operations at the Macondo site included [27]: 1) Centralizers (BP). "When a casing string hangs in the center of the wellbore, cement pumped down the casing will flow evenly back up the annulus, displacing any mud and debris that were previously in that space and leaving a clean column of cement. If the casing is not centered (with centralizers), the cement will flow up the path of least resistance-the larger spaces in the annulus-and slowly or not at all in the narrower annular space. Remaining channels of drilling mud create paths and gaps through which pressurized hydrocarbons can flow", (p. 96).

2) Well drilling and mud circulation (Transocean). Transocean carried out drilling in parallel with the data collection activity carried out by (Sperry Sun). A series of steel tubes protect more fragile sections of the well structure and prevent the entrance of high-pressure fluids (like hydrocarbons) from flowing up the well. Drilling mud cools the drill bit during drilling and plays a critical role in controlling the hydrocarbon pressure in a well. An insufficiently low mud weight allows fluids such as oil and gas to enter the well, causing a "kick." But excessive mud can lead to the leakage of mud into the formation resulting in "lost returns". The rig crew must therefore monitor and adjust the weight of the drilling mud. A proper drilling operation includes a 2,760barrel circulation of drilling mud prior to cementing. Sperry Sun was responsible for the mud data logging at Macondo, (p. 92).

3) Cementing (Halliburton). Crews pump in a specialized cement blend to seal the wellbore. Cement guarantees the watertightness and therefore safety of the well. Proper cement positioning and avoidance of contamination is critical. Drilling mud, cement, and casing strings work together to enable the crew to control wellbore pressure. But the crew cannot directly observe success. Because of the inherent risk, a cement slurry is tested before it is used in a cement job, (p. 99). 4) Negative pressure test (BP & Transocean). Pressure tests provide indirect indicators of cement integrity and whether it has sealed off the distant well. Pressure and volume readings evaluate whether each barrel of cement pumped into the well displaces an equal volume of drilling mud-producing "full returns." While they suggest generally that the job has gone as planned, they do not assure job quality, (p. 99).

! 6

Critical, Temporally Dispersed Deviations

Given shared general knowledge, awareness of the specific status of the nominal plan, and especially deviations from this plan, constitute the other part of the shared situation awareness concern. But, with Note Deepwater provided spatial proximity: without considering drilling depth where people were not located or multiple rig decks where fewer than 150 people were located, the entire event unfolded within a footprint comparable to a standard soccer field (112 x 78 m). But, Deepwater lacked temporal proximity: deviations from the nominal plan occurred not just hours, but some more than weeks before their dire consequence resulted. Had these deviations been communicated, operators might not have missed the warning signals of impending doom. On the other hand, we suspect that changes to the nominal plan always occur because the world never unfolds as planned. The technical problem is identifying a priori those changes that only have consequence in combination with other unanticipated changes.

1) Centralizers (April 1). As shown in Fig. 3, BP used a fewer number of ultimately suboptimal centralizers due to procurement oversights. BP did not inform Halliburton of the number of centralizers it eventually used. 2) Well drilling and mud circulation (April 9). BP circulated approximately 350 barrels of mud before cementing, in contrast to the recommended 2,760 barrels [28] p. 200. BP engineers feared increased mud circulation risked another lost-return event.

"But on April 1, team member Brian Morel learned that BP's supplier (Weatherford) had in stock only six "subs"15centralizers designed to screw securely into place between sections of casing. The alternative was to use "slip-on" centralizers-devices that slide onto the exterior of a piece of casing where they are normally secured in place by mechanical "stop collars" on either side. These collars can either be welded directly to the centralizers or supplied as separate pieces. The BP team-and Wells Team Leader John Guide in particular-distrusted slip-on centralizers with separate stop collars because the pieces can slide out of position or, worse, catch on other equipment as the casing is lowered", (p. 96).

"Gagliano's calculations (based on computer simulations using proprietary software called OptiCem), suggested that the Macondo production casing would need more than six centralizers to avoid channeling, (p. 97).

"When Guide learned the next day of the decision to add more centralizers, he initially deferred, but then challenged the decision, p. 97).

"Walz had earlier assured Guide that the 15 additional centralizers would be custom-designed one-piece units that BP had used on a prior well and would limit the potential for centralizer "hang up." But when the centralizers arrived, BP engineer Brian Morel, who happened to be out on the rig, reported that the centralizers were of conventional design with separate stop collars", (p. 97).

"Morel e-mailed BP drilling engineer Brett Cocales to question the need for additional centralizers. Cocales responded that the team would "probably be fine" even without the additional centralizers and that "Guide is right on the risk/ reward equation. Guide pointed out to Walz that the new centralizers were not custom-made as specified. "Also," he noted, "it will take 10 hrs to install them." He complained that the "last minute addition" of centralizers would add 45 pieces of equipment to the casing that could come off during installation, and concluded by saying that he was "very concerned." In the end, Guide's view prevailed; BP installed only the six centralizer subs on the Macondo production casing", (p. 97).

"BP did not inform Halliburton of the number of centralizers it eventually used, let alone request new modeling to predict the impact of using only six centralizers. Halliburton happened to find out that BP had run only six centralizers when one of its cement engineers overheard a discussion on the rig", (p. 116).

The decision compromised the integrity of the cement job, and because circulation activity was never communicated, confused the Sperry mudloggers. To compensate, BP used additional spacer material subsequently, but failed to indicate the corresponding risk increase to Transocean.

3) Cementing (April 2). Halliburton generally flies cement samples from the rig back for laboratory testing to assure compatibility with well conditions. Two tests (in February and April) both indicated that the foam cement slurry would be unstable. Halliburton sent the February 10 laboratory report to BP on March 8 as an attachment to an e-mail. The attachment included the results indicating that the cement was unstable. No evidence suggests that BP examined the attached report. The April test provided similar results, with no evidence that Halliburton specifically alerted BP. Instead, Halliburton personnel modified the pre-testing conditioning time requirements to argue for a successful result. A final test result from April 13 was e-mailed to BP, but again the indications of serious instability was not explicitly communicated [28] p. 130. 4) Negative Pressure Test (April 11). Temporary abandonment procedures were initially specified on April 11 but changed several times including changing the order of steps and at least once omitting the key negative pressure test [28] p. 154. A shoreside team had decided to "deviate" from the testing procedure in the government permit. A subsequently written detailed displacement procedure failed to specify how to interpret the sensor data that accompanied testing. Critically, the April 14 procedures directed that the negative pressure test would be conducted on the kill line. But rig personnel did not follow this instruction during the first negative pressure test. Instead, they conducted the initial negative pressure test on the drill pipe. Ad hoc modifications to the plan contributed to the misinterpretation of the negative pressure test data (see Fig. 4).

"In the nine days before BP began the temporary abandonment of the Macondo well, the company went through at least four different versions of temporary abandonment procedures. Each version switched the order of several key steps. In response to the April 12 prodding from Murry Sepulvado, Morel circulated a draft plan for upcoming operations at Macondo later that day. The draft plan included temporary abandonment procedures that instructed the rig crew to set the lockdown sleeve first and then to set a surface cement plug in seawater. The plug would be set just 933 feet below the mudline. Morel's draft did not include a negative pressure test. After reviewing it, well site leader Ronnie Sepulvado reminded Morel that he needed to include a negative pressure test", (p. 131-132).

"Reviewing it, Hafle was concerned that the Ops Note procedure was different than the procedure MMS had approved. Kaluza woke up Morel. Morel explained that the rest of the shoreside team had decided to "deviate" from the procedure in the MMS-approved permit, which called for conducting the negative pressure test before displacement. "The team in town wanted to do something different," Kaluza later explained according to notes of BP's post-blowout interviews. "They decided we could do the displacement and negative test together -don't know why -maybe trying to save time.... Anytime you get behind, they try to speed up", (p. 134).

"For instance, BP originally planned to install the lockdown sleeve at the beginning of the temporary abandonment. BP's decision to change plans and set the lockdown sleeve last triggered a cascade of other decisions that led it to severely underbalance the well while leaving the bottomhole cement as the lone physical barrier in place during displacement of the riser. [...]. Walz admitted that there was "no structured approval process" and that "changes [were] made with email and verbal discussion." Cocales stated that there was "no formal process on communicating changes to [the] well plan." Murry Sepulvado stated that it was not unusual to receive emails like the Ops. Note containing procedural changes that had not been risk assessed through a formal process. And ac- Subsequent analysis (see Fig. 5) indicates that decisions surrounding cement operations and centralizers provided context critical for the proper interpretation of an improperly interpreted negative well pressure test. The Chief Counsel's team explains that, "knowledge of ongoing rig activity -is essential to accurate interpretation of the data. Absent that knowledge, it is difficult to ascertain whether anomalous data are benign or problematic", [28] p. 201.

Conclusions

In this article, we illustrated temporal dispersion, in which several specialties performing successive activities achieve a common goal. However, a plan never unfolds as initially conceived, requiring the communication of "important" changes between specialists. This did not occur on Deepwater in several key instances. No doubt, latent organizational factors contributed to the communication failures [START_REF] Reason | Human Error[END_REF]. Our analysis is certainly not intended to justify shoddy engineering or irresponsible profit motives. However, we suspect that the risk of temporal dispersion persists without these organizational factors. A disastrous outcome merely provided an opportunity to observe the risk. The absence of temporal co-location further exemplifies the role that the environment can play in offloading controlled attentional processes [START_REF] Zhang | Representations in distributed cognitive tasks[END_REF]. Together, spatial and temporal co-location provide a relevance heuristic for capturing attention automatically, and support behavioral indicators that render activities accessible to others. Without these simultaneous conditions, effortful conceptual association is required to determine relevance.

"Beginning April 14, the procedures directed that the negative pressure test would be conducted on the kill line. But rig personnel did not follow this instruction during the first negative pressure test. Instead, they conducted the initial negative pressure test on the drill pipe. This may suggest that in addition to creating better test proce-"The pressure data were not ambiguous. Rather, they showed repeatedly that formation fluids, in this case hydrocarbons, were flowing into the well. The failure to properly conduct and interpret the negative-pressure test was a major contributing factor to the blowout. Given the risk factors surrounding the primary cement job and other prior unusual events (such as difficulty converting the float valves), the BP Well Site Leaders and, to the extent they were aware of the issues, the Transocean crew should have been particularly sensitive to anomalous pressure readings and ready to accept that the primary cement job could have failed. It appears instead they started from the assumption that the well could not be flowing, and kept running tests and coming up with vari -! 9

Fig. 2 .

 2 Fig. 2. Failure of communication between different actors at Macondo [27, 28].

Fig. 3 .

 3 Fig. 3. Rationale concerning the number of centralizers [27].

Fig. 4 .

 4 Fig. 4. Modifications to temporary abandonment procedures [28].

Fig. 5 .

 5 Fig. 5. Pervasive misinterpretation of test results [27].

  Transocean argues that if BP had done so, its crew might have demonstrated -heightened awareness" [28] p. 186. "Transocean and Sperry Drilling personnel did not communicate effectively about the displacement operation. And the BP well site leader did not play a sufficiently active role in ensuring such communication. Communication broke down between the drill crew and the mudloggers on several occasions. […]. The drill crew repeatedly failed to inform Keith of various activities that influenced well monitoring data.[…]. Keith reported after the explosion that he was concerned that simultaneous activities would complicate monitoring but never expressed those concerns to others.[…]. Even after the Transocean crew shut down the pumps to investigate an anomaly, they did not inform the Sperry Drilling mudlogger, senior Transocean personnel, or the BP well site leader of the anomaly or ask for their help in resolving it" [28] p. 187.

Acknowledgments

Valerie Shalin's participation in this project is supported by NSF Grant 1520870, Srinivasan Parthasarathy Principal Investigator.