D Goss

B Angl Ès

T Ngo Dac

F Pellarin

F Tavares Ribeiro

THE DIGIT PRINCIPLE AND DERIVATIVES OF CERTAIN L-SERIES

We discuss a digit principle for derivatives of certain ζ-values in Tate algebras of positive characteristic discovered by David Goss. Contents 1. Introduction 1 2. Notation 3 3. The digit Principle 3 4. Carlitz log-algebraic result and its ramifications 5 5. The one digit case 11 6. The several digit case 12 7. Some non vanishing results 12 8. Taylor expansions in the neighborhood of (θ, . . . , θ) 17

Introduction

The present paper was initially conceived as an appendix of the paper of [START_REF] Anglès | Exceptional zeros of L-series and Bernoulli-Carlitz numbers[END_REF], and the main result, essentially due to David Goss, is Theorem 3.1, a new kind of digit principle for certain derivatives of ζ-values in Tate algebras, generalizing the so-called Carlitz zeta values. Later, David Goss and us, the other authors, decided to make it into an independent article, but this plan was interrupted because David Goss suddenly died on April, 4, 2017. In the present newer version, the paper also reflects the mathematical exchanges between us and him. We would like to dedicate it to his memory.

1.1. Derivatives of Riemann's zeta function and Goss' zeta function. The functional equation of Riemann's zeta function ζ : C → P 1 (C) induces, as it is well known, trivial zeroes at the negative even integers. These zeroes are simple, and we have the following identities for the first derivatives:

(1. [START_REF] Anderson | Rank one elliptic A-modules and A-harmonic series[END_REF] ζ (-2n) = (-1) n (2n)! 2(2π) 2n ζ(2n + 1), n > 0.

Moreover, the function ζ(z) has no zero at z = 0, but we have the classical formula

(1.2) ζ (0) = - 1 2 ln(2π),
which is again a consequence of the functional equation.

Let now F q be the finite field having q elements and let θ be an indeterminate over F q . We consider the local field K ∞ = F q ((θ -1 )), which is the completion of the field K = F q (θ) for the valuation at infinity v ∞ (with v ∞ (θ) = -1), as an analogue of the real line. We observe indeed that A = F q [θ] is discrete and co-compact in K ∞ .

In the years 1980, David Goss introduced a theory of global zeta functions in the setting of function fields of positive characteristic. His program was strongly motivated also by several signs going toward the possible existence of a functional equation, and one among them was the phenomenon of trivial zeroes. Indeed, in the above setting, defining, following Goss:

ζ A (-n, z) = P 1 -z deg θ (P ) P n -1 = d≥0 z d a∈A +,d a n ∈ 1 + zA[[z]], n ≥ 0
where A + (resp. A +,d ) denotes the multiplicative monoid of monic polynomials (resp. monic polynomials of degree d) and with the product running over the irreducible polynomials of A + , one sees that ζ A (-n, z) ∈ A [z]. It is also quite easy to show that ζ A (-n, 1) = 0 if and only if n > 0 and n ≡ 0 (mod q -1). Moreover, in this case, the first derivative ζ A (-n, z) in z does not vanish at z = 1, so the trivial zeroes are in this way simple, just as those of Riemann zeta function. The polynomials ζ A (-n, z) and certain natural generalizations, have been the object of extensive investigations by several authors. Nevertheless, no analytic reason has been found, such as the poles of a gamma factor, to justify the above properties, and no relationship connecting these first derivatives to the positive values of Goss' zeta functions has been clearly recognized.

T s (K ∞ ) = K ∞ [t s
] is an ultrametric Banach algebra, the standard s-dimensional Tate algebra over K ∞ . In [START_REF] Anglès | Functional identities for L-series values in positive characteristic[END_REF][START_REF] Pellarin | Values of certain L-series in positive characteristic[END_REF], the following functions

ζ A (n; s)(t s ) = d≥0 a∈A +,d a(t 1 ) • • • a(t s ) a n ∈ T s (K ∞ ) × , n > 0, s ≥ 0
have been introduced and studied. By [START_REF] Anglès | Functional identities for L-series values in positive characteristic[END_REF]Proposition 6] we know that, for all n > 0 and s ≥ 0, ζ A (n; s) defines an entire function in s variables. By [7, Theorem 1], if n ≡ s (mod q -1), there exists λ n,s ∈ K(t s ) ∩ T s (K ∞ ) × such that:

(1.3) ζ A (n; s) = λ n,s π n ω(t 1 ) • • • ω(t s )
,

where

π := θ(-θ) 1 q-1 i>0 1 - θ θ q i -1 ∈ θ(-θ) 1 q-1 1 + θ -1 F q [[θ -1 ]]
is a fundamental period of the Carlitz exponential (see [21, §3.1]) and ω(t) := (-θ)

1 q-1 i≥0 1 - t θ q i -1 ∈ (-θ) 1 q-1 1 + θ -1 F q [t][[θ -1 ]]
is Anderson-Thakur's function (see §4). The above definitions depend on a common choice of q -1-th root of -θ (see [START_REF] Anderson | Tensor powers of the Carlitz module and zeta values[END_REF]), but the ratio

π n ω(t1)•••ω(ts) does not.
In fact, ω is the inverse of an entire function in the variable t, and its poles determine analytically, trivial zeroes of the functions ζ A (n; s), from which arises naturally the idea of studying the Taylor expansion of the functions ζ A (n; s) in the neighborhood of these trivial zeroes. In particular, if s > 1 and s ≡ 1 (mod q -1), the function ζ A (1; s) vanishes at the point t s = (t 1 , . . . , t s ) = (θ, . . . , θ). In this paper, we will study the values

(1.4) δ s := d dt 1 • • • d dt s (ζ A (1; s)) t1=•••=ts=θ ∈ K ∞ ,
and we will show, in Theorem 3.1 that a sort of digit principle holds for them, first highlighted by David Goss.

Notation

In this paper, we will use the following notation.

• N: the set of non-negative integers.

• N * = N \ {0}: the set of positive integers.

• Z: the set of integers.

• F q : a finite field having q elements.

• p: the characteristic of F q .

• θ: an indeterminate over F q .

• A: the polynomial ring F q [θ].

• A + : the set of monic elements in A.

• For d ∈ N, A +,d denotes the set of monic elements in A of degree d.

• K = F q (θ): the fraction field of A.

• ∞: the unique place of K which is a pole of θ.

• v ∞ : the discrete valuation on K corresponding to the place ∞ normalized such that v ∞ (θ) = -1. • K ∞ = F q (( 1 θ )): the completion of K at ∞. • C ∞ : the completion of a fixed algebraic closure of K ∞ . The unique valua- tion of C ∞ which extends v ∞ will still be denoted by v ∞ . • λ θ = (-θ) 1 q-1 a fixed (q -1)th-root of -θ in C ∞ .
• For s ∈ N, {t 1 , t 2 , . . . , t s } denotes a set of s variables and we will also denote it by t s .

The digit Principle

Let N be a positive integer. We consider its base-q expansion (3.1) N = k i=0 n i q i , so that n i ∈ {0, . . . , q -1} for all i. We recall that q (N ) = k i=0 n i and the definition of the Carlitz factorial:

Π(N ) = i≥0 D ni i ∈ A + , where [i] = θ q i -θ if i > 0 and D j = [j][j -1] q • • • [1] q j-1 for j > 0, while we set D 0 = 1.
It is easy to see (the details are in §4, 5 and 6) that, if we denote by a the derivative d dθ a of a ∈ A with respect to θ, the series

d≥1 a∈A +,d
a N a converges in K ∞ to a limit that we denote by δ N . This limit is easily seen to be equal to the evaluation of entire function of the variables t N

δ N := d dt 1 • • • d dt N (ζ A (1; N )) t1=•••=t N =θ
in compatibility with (1.4).

In particular, if n = q j with j > 0, we will see (Proposition 5.1) that

δ 1 = - k≥1 1 [k] and δ q j = D j [j] π 1-q j .
Let N ≥ 1, q (N ) ≥ 2 and N ≡ 1 (mod q -1). We set:

(3.2) B N (t, θ) = (-1) q (N )-1 q-1 L N (t) k i=0 ω(t q i ) ni π -1 ,
where N has base q expansion (3.1), L N (t) = a∈A+ a(t) N a , and ω(t) is the Anderson-Thakur special function (see §4). By [START_REF] Anglès | Arithmetic of positive characteristic L-series values in Tate algebras[END_REF], Lemma 7.6, we have:

B N (t, θ) ∈ A[t].
We will prove the following:

Theorem 3.1. If N ≥ q is such that N ≡ 1 (mod q -1) and q (N ) ≥ q, then δ N π = β N Π(N ) Π([ N q ]) q k i=1 δ q i π ni ,
where for x ∈ R, [x] denotes the integer part of x, and where

β N = (-1) q (N )-1 q-1 B N (θ, θ).
Theorem 3.1 can be viewed as a kind of digit principle for the values δ j in the sense of [START_REF] Conrad | The digit principle[END_REF].

In §4, using a log-algebraic result which was originally discovered by Leonard Carlitz in 1942, we give the first properties of Anderson and Thakur function ω(t). In §5 we discuss the one-digit case of our Theorem, while the general case is discussed in §6. In §8 we also give some complements on these problems.

Carlitz log-algebraic result and its ramifications

This section is an elementary introduction to some of the recent developments on the arithmetic of special values of certain L-functions introduced by David Goss in 1979 ( [START_REF] Goss | v-adic zeta functions, L-series and measures for function fields[END_REF]). We have tried to keep this paragraph as self-contained as possible. All the results contained in this section are well-known but some of their proofs are new.

Lemma 4.1. Let X 1 , . . . , X m be m ≥ 1 indeterminates over C ∞ . Let d ∈ N be an integer such that (q -1)d > m. Then: a∈A +,d a(X 1 ) • • • a(X m ) = 0.
Proof. This Lemma is a special case of [START_REF] Goss | Basic Structures of Function Field Arithmetic[END_REF], Lemma 8.8.1. We have:

a∈A +,d a(X 1 ) • • • a(X m ) = ζ1,...,ζ d ∈Fq m k=1 (X d k + d l=1 ζ l X l-1 k ).
If we develop the right side of the above equality and we use that ζ∈Fq ζ n = 0 if n ≡ 0 (mod q -1), we get the assertion of the Lemma.

Lemma 4.2. Let d ≥ 1 be an integer. Then:

a∈A +,d 1 a = 1 l d ,
where

l d = d k=1 (θ -θ q k ). Proof. Let us set: e d (X) = a∈A,deg θ a<d (X -a).
Then one can show by induction on d (see [START_REF] Goss | Basic Structures of Function Field Arithmetic[END_REF], pages 46 and 47) the following identity due to Leonard Carlitz:

e d (X) = d k=0 D d D k d-k X q k ,
where 0 = 1. Taking the logarithmic derivative, we get:

D d l d e d (X) = a∈A,deg θ a<d 1 X -a .
Evaluating the above equality at θ d and using the fact that e d (θ d ) = D d ([21], Proposition 3.1.6), we get the desired result.

Let t be an indeterminate over C ∞ . Leonard Carlitz also obtained the following remarkable result ( [START_REF] Carlitz | Some topics in the arithmetic of polynomials[END_REF], formula (5.8)):

Proposition 4.3. Let d ∈ N, d ≥ 1. Then: a∈A +,d a(t) a = 1 l d d-1 k=0 (t -θ q k ).
Proof. Let us set:

F (t) = a∈A +,d a(t) a ∈ K[t].
Then for k ∈ {1, . . . , d -1}, we have by Lemma 4.1:

F (θ q k ) = a∈A +,d a q k -1 = a∈A +,d a(θ) q-1 a(θ q ) q-1 • • • a(θ q k-1 ) q-1 = 0.
One also observes that F (θ) = 0 since d ≥ 1. Therefore:

F (t) =   a∈A +,d 1 a   d-1 k=0 (t -θ q k ).
It remains to apply Lemma 4.2.

Let τ : C ∞ [[t]] → C ∞ [[t]] be the homomorphism of F q [[t]
]-algebras such that:

τ ( n≥0 α n t n ) = n≥0 α q n t n , α n ∈ C ∞ .
We denote by

T t ⊂ C ∞ [[t]] the Tate algebra in the variable t with coefficients in C ∞ , which is the completion C ∞ [t] v∞ for the Gauss valuation at infinity v ∞ . Observe that: {f ∈ C ∞ [[t]], τ (f ) = f } = F q [[t]], from which one deduces easily that {f ∈ T t , τ (f ) = f } = F q [t].
Let φ : A → A[t]{τ } be the homomorphism of F q -algebras such that:

φ θ = θ + (t -θ)τ.
We refer the reader to [START_REF] Anglès | Arithmetic of positive characteristic L-series values in Tate algebras[END_REF] for a detailed study of such objects that we may call Drinfeld modules over Tate algebras. Let log φ be the unique element in 1+τ C ∞ [[t]]{{τ }} such that: log φ φ θ = θ log φ . We have:

Lemma 4.4. log φ = 1 + d≥1 1 l d d-1 k=0 (t -θ q k )τ d . Proof. Write log φ = n≥0 l n (φ)τ n , l n (φ) ∈ C ∞ [[t]], with l 0 (φ) = 1. From the equation log φ φ θ = θ log φ , we get for n ≥ 1: (θ -θ q n )l n (φ) = l n-1 (φ)(t -θ q n-1
).

The Lemma follows.

Let us observe that log

φ converges on {f ∈ T t , v ∞ (f ) > -1} since for all d ≥ 0, v ∞ ( 1 l d d-1 k=0 (t -θ q k )) = q d -1 where v ∞ is the ∞-adic Gauss valuation on T t . We set L(t) = ζ A (1; 1) = d≥0 a∈A +,d a(t) a = P monic prime of A 1 - P (t) P -1 ∈ T × t .
Then, Proposition 4.3 implies immediately the following log-algebraic result in the sense of Anderson ([1], [START_REF] Anderson | Log-Algebraicity of Twisted A-Harmonic Series and Special Values of L-series in Characteristic p[END_REF]):

Corollary 4.5. We have the following equality in T t :

L(t) = log φ (1).
We refer the interested reader to [START_REF] Anglès | Stark units in positive characteristic[END_REF][START_REF] Anglès | Anderson-Stark Units for Fq[END_REF][START_REF] Anglès | Arithmetic of function fields units[END_REF][START_REF] Green | Special L-values and shtuka functions for Drinfeld modules on elliptic curves[END_REF][START_REF] Papanikolas | Log-Algebraicity on Tensor Powers of the Carlitz Module and Special Values of Goss L-Functions[END_REF] for the recent developments around Anderson's log-algebraicity Theorem.

We denote by (-θ)

1 q-1 a fixed q -1th root of -θ in C ∞ , and we recall:

π = (-θ) 1 q-1 θ i≥1 (1 -θ 1-q i ) -1 ∈ C × ∞ , ω(t) = (-θ) 1 q-1 i≥0 (1 - t θ q i ) -1 ∈ T × t .
The following result is due to F. Pellarin ([24], Theorem 1):

Theorem 4.6. We have the following equality in T t :

L(t)ω(t) π = 1 θ -t .
Proof. We give a new proof of this result by using Proposition 4.3. Let d ≥ 1 be an integer. By Carlitz formula (Proposition 4.3):

θ -d t d a∈A +,d a( 1 t ) a( 1 θ ) = d k=1 (1 -θ 1-q k ) -1 d-1 k=0 (1 - t θ q k ). Now: θ -d t d a∈A +,d a( 1 t ) a( 1 θ ) = a∈A,a(0)=1,deg θ a≤d a(t) a .
Furthermore:

a∈A,a(0)=1,deg θ a≤d a(t) a = - a∈A,a(0) =0,deg θ a≤d a(t) a .
Letting d tend to +∞, we get:

(1 - t θ ) a∈A\{0} a(t) a = - π θ ω(t) -1 .
Finally observe that:

a∈A\{0} a(t) a = - a∈A+ a(t) a .
The Theorem follows.

The function ω(t) was introduced by G. Anderson and D. Thakur in [START_REF] Anderson | Tensor powers of the Carlitz module and zeta values[END_REF] (see [START_REF] Green | Special L-values and shtuka functions for Drinfeld modules on elliptic curves[END_REF], [START_REF] Anglès | Special functions and twisted L-series[END_REF] for generalizations of this special function). The Anderson-Thakur special function is intimately connected to Gauss-Thakur sums as it was highlighted in [START_REF] Anglès | Universal Gauss-Thakur sums and L-series[END_REF].

Let C : A → A{τ } be the Carlitz module ( [START_REF] Goss | Basic Structures of Function Field Arithmetic[END_REF], chapter 3), in other words, C is the homomorphism of F q -algebras given by C θ = τ + θ. Let us set:

exp C = i≥0 1 D i τ i ∈ T t {{τ }}.
exp C is called the Carlitz exponential. We have the following equality in T t {{τ }} :

exp C θ = C θ exp C .
Let us observe that exp C converges on T t .

Lemma 4.7. We have:

ker exp C | C∞ = πA.
Proof. Note that the edges of the Newton polygon of exp C (X)

X = i≥0 1 Di X q i -1 are (q i -1, iq i ), i ≥ 0. Since ker exp C | C∞ is an A-module, we deduce that there exists η ∈ C ∞ , v ∞ (η) = -q q-1 such that: ker exp C | C∞ = ηA.
Since exp C defines an entire function on C ∞ , we deduce that:

exp C (X) = i≥0 1 D i X q i = X a∈A\{0} (1 - X ηa ).
Recall that, for n ∈ N, ζ∈F × q ζ n = -1 if n ≥ 1, n ≡ 0 (mod q -1) and ζ∈F × q ζ n = 0 otherwise. We deduce:

X exp C (X) = 1 - n≡0 (mod q-1),n≥1 η -n ( a∈A+ 1 a n )X n .
We therefore get:

-η 1-q a∈A+ 1 a q-1 = 1 θ q -θ .
Now, a simple computation shows that τ (ω(t)) = (t -θ)ω(t). Thus, by Theorem 4.6, we get:

( d≥0 a∈A +,d a(t) a q )(t -θ)ω(t) π q = 1 θ q -t .
We evaluate t at θ to obtain:

-π 1-q a∈A+ 1 a q-1 = 1 θ q -θ .

Thus:

η π ∈ F × q .
We will need the following crucial result in the sequel:

Proposition 4.8. We have the following equality in T t :

ω(t) = exp C π θ -t .
Proof. This result is a consequence of the formulas established in [START_REF] Pellarin | Values of certain L-series in positive characteristic[END_REF]. We give a detailed proof for the convenience of the reader.

Recall that ω(t) ∈ T × t . Let us set

F (t) = exp C π θ -t .
By Lemma 4.7, we observe that:

C θ (F (t)) = exp C θ π θ -t = exp C (θ -t + t) π θ -t = exp C ( π) + exp C t π θ -t = t exp C π θ -t = tF (t).
Therefore:

τ (F (t)) = (t -θ)F (t). Since τ (ω(t)) = (t -θ)ω(t), we get: τ F (t) ω(t) = F (t) ω(t) .
We have then:

F (t) ω(t) ∈ F q [t].

Now observe that

F (t) = exp C   j≥0 π θ j+1 t j   = j≥0 λ θ j+1 t j , where λ θ j+1 = exp C ( π θ j+1 ). Note that λ θ = (-θ) 1 q-1 .
We also observe that for all

j ≥ 0, v ∞ (λ θ j+1 ) = j + 1 -q q-1 . This implies v ∞ ( F (t) λ θ -1) > 0. By the definition of ω(t), we also have v ∞ ( ω(t) λ θ -1) > 0. Thus: v ∞ F (t) ω(t) -1 > 0.
Since F (t) ω(t) ∈ F q [t], we get ω(t) = F (t).

Notice that ω(t) defines a meromorphic function on C ∞ without zeroes. Its only poles, simple, are located at t = θ, θ q , θ q 2 , . . .. As an immediate consequence of Proposition 4.8, we get: Corollary 4.9. For all j ≥ 0, we have:

(t -θ q j
)ω(t) | t=θ q j = -π q j D j .

Let exp φ ∈ 1 + τ T t {{τ }} be such that:

exp φ θ = φ θ exp φ .
By the same argument as that of the proof of Lemma 4.4, we have:

exp φ = 1 + i≥1 i-1 k=0 (t -θ q k ) D i τ i .
Observe that exp φ converges on T t .

Lemma 4.10. The exponential series exp φ induces an exact sequence of F q [t]modules:

0 → π ω(t) A[t] → T t → T t → 0.
Proof. Let us observe that in T t {{τ }}:

exp C ω(t) = ω(t) exp φ .
Thus exp C defines an entire function on C ∞ and thus exp

C (C ∞ ) = C ∞ . Therefore: exp C (T t ) = T t .
Since ω(t) ∈ T × t , we get:

exp φ (T t ) = T t , ker exp φ = 1 ω(t) ker exp C .
Now, Lemma 4.7 implies:

ker exp C = πA[t].
Following L. Taelman ([25]), we introduce the module of "units" associated to

φ/A[t]: U (φ/A[t]) = {f ∈ T t ∩ K ∞ [[t]] | exp φ (f ) ∈ A[t]}.
Observe that U (φ/A[t]) is an A[t]-module. We have:

Proposition 4.11. U (φ/A[t]) = L(t)A[t].
Proof. By Carlitz log-algebraic result (Corollary 4.5):

exp φ (L(t)) = 1.
Thus:

L(t)A[t] ⊂ U (φ/A[t]).
Now, let us set:

M = {f ∈ T t ∩ K ∞ [[t]] | v ∞ (f ) > 0}.
We have:

M ∩ A[t] = {0}, exp φ (M) = M, T t ∩ K ∞ [[t]] = A[t] ⊕ M. Since L(t) ∈ (T t ∩ K ∞ [[t]]) × and v ∞ (L(t)) = 0, we get: T t ∩ K ∞ [[t]] = L(t)A[t] ⊕ M. Thus: exp φ (T t ∩ K ∞ [[t]]) ∩ A[t] = exp φ (L(t)A[t]).
We deduce that:

U (φ/A[t]) ⊂ L(t)A[t] + ker exp φ .
The Proposition is then a consequence of Lemma 4.10 and Theorem 4.6.

The above Proposition reflects a class formula similar to that obtained in [START_REF] Taelman | Special L-values of Drinfeld modules[END_REF]. We refer the interested reader to the references [START_REF] Anglès | Stark units in positive characteristic[END_REF][START_REF] Anglès | Arithmetic of positive characteristic L-series values in Tate algebras[END_REF][START_REF] Anglès | Anderson-Stark Units for Fq[END_REF][START_REF] Anglès | with an appendix by V. Bosser, Arithmetic of characteristic p special L-values[END_REF][START_REF] Debry | Towards a class number formula for Drinfeld modules[END_REF][START_REF] Demeslay | A class formula for L-series in positive characteristic[END_REF][START_REF] Fang | Special L-values of abelian t-modules[END_REF][START_REF] Fang | Equivariant Special L-values of abelian t-modules[END_REF][START_REF] Fang | Equivariant trace formula mod p[END_REF].

The one digit case

Recall that:

L(t) = ζ A (1; 1) = d≥0 a∈A +,d a(t) a ∈ T t .
Furthermore, we recall that we have the following equality in T t (Theorem 4.6):

L(t)ω(t) π = 1 θ -t .
This implies that L(t) extends to an entire function on C ∞ (see also [START_REF] Anglès | Functional identities for L-series values in positive characteristic[END_REF], Proposition 6). We set:

L (t) = d≥0 a∈A +,d a (t) a ∈ T t ,
where a (t) denotes the derivative d dt a(t) of a(t) with respect to t. The derivative d dt induces a continuous endomorphism of the algebra of entire functions on C ∞ , and therefore L (t) extends to an entire function on C ∞ . Thus, for j ≥ 0 an integer,

d≥1 a∈A +,d a q j
a converges in K ∞ and we have:

δ q j = d≥1 a∈A +,d a q j a = L (t) | t=θ q j .
Proposition 5.1. The following properties hold:

(1) We have:

δ 1 = - k≥1 1 [k] .
(2) Let j ≥ 1 be an integer, then:

δ q j = Π(q j ) [j] π 1-q j .
Proof.

(1) It is well known that, for n > 0, D n = a∈A+,n a [21, Proposition 3.1.6].

Therefore, a∈A+,n a a = -1

[n] from which the first formula follows. (2) By [START_REF] Goss | Basic Structures of Function Field Arithmetic[END_REF], Remark 8.13.10, we have:

L(t) | t=θ q j = 0.
Thus:

δ q j = L (t) | t=θ q j = L(t) t -θ q j | t=θ q j . But, L(t) t-θ q j (t -θ q j )ω(t) π = 1 θ -t .
It remains to apply Corollary 4.9.

Remark 5.2. The transcendence over K of the "bracket series" δ 1 = i≥1 1

[i] was first obtained by Wade [START_REF] Wade | Certain quantities transcendental over GF(p n , x)[END_REF]. The transcendence of δ 1 directly implies the transcendence of π.

The several digit case

As a consequence of [START_REF] Anglès | Arithmetic of positive characteristic L-series values in Tate algebras[END_REF], Lemma 7.6 (see also [START_REF] Anglès | Functional identities for L-series values in positive characteristic[END_REF], Corollary 21), the series L N (t) = d≥0 a∈A +,d a(t) N a has a zero of order at least N at t = θ. Furthermore,

L N (t) = d≥1 a∈A +,d a (t) N a
defines an entire function on C ∞ such that

δ N = L N (θ).
Proof of Theorem 3.1. Recall that N = k i=0 n i q i is the q-expansion of N . We set s = q (N ). Recall moreover Equation (3.2):

(-1)

s-1 q-1 B N (t, θ) = L N (t) k i=0 ω(t q i ) ni π -1 ∈ A[t].
Observe that:

L N (t) = a∈A+ k i=0 a(t q i ) ni a .
Let s = k i=0 n i and let t 1 , . . . , t s be s indeterminates over C ∞ . We set:

L s (t) = a∈A+ a(t 1 ) • • • a(t s ) a .
Since δ N = L N (θ), it is the evaluation at t 1 = . . . = t n0 = θ, t n0+1 = . . . = t n0+n1 = θ q , . . . ,

t n0+•••+n k-1 +1 = . . . = t n0+•••+n k-1 +n k = θ q k of the function L s (t) k i=0 ni j=1 (t n0+...+ni-1+j -θ q i )
.

We obtain, by [START_REF] Anglès | Functional identities for L-series values in positive characteristic[END_REF] Theorem 1, by Corollary 4.9 and our previous discussions:

β N = δ N k i=0 ( -π q i Di ) ni π .
Now, by Proposition 5.1, we have, for all i ≥ 1, D i = [i]δ q i π q i -1 . We obtain the Theorem by using the fact that:

Π(N ) Π([ N q ]) q = i≥1 [i] ni .

Some non vanishing results

Proposition 7.1. Let N ≥ 1 be an integer such that N ≡ 1 (mod q -1). Then δ N = 0.

Proof. It follows from Theorem 3.1 and the fact that B N (t, θ) | t=θ = 0 ( [START_REF] Anglès | Exceptional zeros of L-series and Bernoulli-Carlitz numbers[END_REF]).

The aim of this section is to prove that the series δ N do not vanish for other values of N : Theorem 7.2. Suppose that q > 2. Let N ≥ 1 be a positive integer such that 2 ≤ s := q (N ) ≤ q -1. Then δ N = 0. 7.1. Decomposition of series in K ∞ . Let i be an integer, 0 ≤ i ≤ q -2. We set:

K (i) ∞ := θ i F q ((θ 1-q )) =        n≤n0 n≡i mod q-1 α n θ n ; n 0 ∈ Z; α n ∈ F q        ⊂ K ∞ .
Then, we have the obvious decomposition:

K ∞ = 0≤i≤q-1 K (i)
∞ and the characterization:

K (i) ∞ = f (θ) ∈ K ∞ ; ∀λ ∈ F * q f (λθ) = λ i f (θ) .
For simplicity, if f ∈ K ∞ , we will note f |θ →λθ for the image of f under the substi-

tution θ → λθ, so that if f ∈ K ∞ , then f ∈ K (i)
∞ if, and only if f |θ →λθ = λ i f for all λ ∈ F * q . Consider now for an N ≥ 1, and d ≥ 0,

f = a∈A +,d a N a then if λ ∈ F * q , f |θ →λθ = λ N (d-1)-d f . Thus, f ∈ K (d(N -1)-N mod q-1) ∞ .
Proposition 7.3. Let N ≥ 1 be an integer, and m = q-1 (N -1,q-1) . Then, δ N = 0 if, and only if, for all 0 ≤ j ≤ m -1,

d≡j (mod m) a∈A +,d a N a = 0.
Proof. Write for all d ≥ 0

f d = a∈A +,d a N a .
Then,

f d ∈ K (d(N -1)-N mod m) ∞
, and if, d, d ≥ 0,

d(N -1) -N ≡ d (N -1) -N mod q -1 if and only if d ≡ d mod m.
Remark 7.4. The "worst" case in the above proposition occurs when m = 1, so that the proposition is empty. But this is equivalent to N ≡ 1 mod q -1 and we already know by Proposition 7.1 that δ N does not vanish. Otherwise, the vanishing of δ N is equivalent to the vanishing of at least two series. The worst remaining case is then when m = 2, that is, N ≡ q+1 2 mod q -1.

7.2.

Proof of Theorem 7.2. For d ≥ 0, we set:

b d (X) = d-1 l=0 (X -θ q l )
and recall that:

l d = (θ -θ q d )(θ -θ q d-1 ) . . . (θ -θ q ).
Observe that:

v ∞ (l d ) = - q d+1 -q q -1 .
Recall that we have expanded N in base q: N = q e1 + . . . + q es with 0 ≤ e 1 ≤ . . . ≤ e s and 2 ≤ s ≤ q -1. Since 2 ≤ s ≤ q -1, the log-algebraicity result [START_REF] Anglès | Anderson-Stark Units for Fq[END_REF], Proposition 5.6. (see also Example 5.7) gives another expression for δ N :

a∈A +,d a(t 1 ) • • • a(t s ) a = s i=1 b d (X) l d so that S d := a∈A +,d a N a = s i=1 d dX b d (X) |X=θ q e i l d and δ N = d≥1 S d .
Let e ≥ 0 be an integer. We define the function f e : N * → N as follows:

f e (d) =     
-(e -1)q e + q d -q e q-1

if d ≥ e + 1, -(d -1)q e if 1 ≤ d ≤ e et d ≡ 0 (mod p), -(d -2)q e -q d-1 if 1 ≤ d ≤ e et d ≡ 0 (mod p).
This function is strictly decreasing.

Lemma 7.5. Let d ≥ 1 and e ≥ 0 be integers. Then, we have:

v ∞ d dX b d (X)
X=θ q e

= f e (d).

Proof. Write:

d dX b d (X) = d-1 l=0 (X -θ q l ) d-1 l=0 1 X -θ q l .
The Lemma follows by direct calculations. Lemme 7.5 implies that for d ≥ 1,

(7.1) v ∞ (S d ) = -v ∞ (l d ) + s i=1 f ei (d) = q d+1 -q q -1 + s i=1 f ei (d).
We will distinguish two cases: e s ≡ 0 (mod p) et e s ≡ 0 (mod p).

Proposition 7.6. Suppose that e s ≡ 0 (mod p). Let d ≥ 1 be an integer such that d = e s . Then:

v ∞ (S d ) > v ∞ (S es ). In particular, δ N = 0.
Proof. Since e s ≡ 0 (mod p), Equation (7.1) implies:

v ∞ (S es ) = q es+1 -q q -1 + s i=1 f ei (e s ) = q es+1 -q q -1 - s i=1
(e i -1)q ei + q es -q ei q -1 .

We will distinguish three cases:

Case 1: d ≥ e s + 1. By (7.1), we have:

v ∞ (S d ) = q d+1 -q q -1 + s i=1 f ei (d) = q d+1 -q q -1 - s i=1
(e i -1)q ei + q d -q ei q -1 .

Since s ≤ q -1, we obtain:

v ∞ (S d ) -v ∞ (S es ) = q d+1 -q es+1 q -1 - s i=1 q d -q es q -1 = (q -s) q d -q es q -1 > 0. Thus, v ∞ (S d ) > v ∞ (S es ) for d ≥ e s + 1. Case 2: d ≤ e s -2.
Since the functions f ei are strictly decreasing, it follows that:

v ∞ (S d ) -v ∞ (S es ) = q d+1 -q es+1 q -1 + s i=1 (f ei (d) -f ei (e s ))
≥ q d+1 -q es+1 q -1 + (f es (e s -2) -f es (e s ))

≥ q d+1 -q es+1 q -1 + 2q es > -q es+1 q -1 + 2q es > 0. Thus, v ∞ (S d ) > v ∞ (S es ) for d ≤ e s -2.
Case 3: d = e s -1.

Since the functions f ei are strictly decreasing, we obtain:

v ∞ (S es-1 ) -v ∞ (S es ) = q es -q es+1 q -1 + s i=1 (f ei (e s -1) -f ei (e s )) ≥ q es -q es+1 q -1 + (f e1 (e s -1) -f e1 (e s )) + (f es (e s -1) -f es (e s ))
≥ q es -q es+1 q -1 + q es-1 + q es = q es-1 > 0.

Thus,

v ∞ (S es-1 ) > v ∞ (S es ).
The proof is finished.

Proposition 7.7. Suppose that e s ≡ 0 (mod p). Let d ≥ 1 be an integer such that d / ∈ {e s -1, e s , e s + 1}. Then:

v ∞ (S d ) > v ∞ (S es ) > v ∞ (S es-1 ).
Proof. Let t be the integer such that 0 ≤ t ≤ s -1 and e t < e t+1 = . . . = e s . Since e s ≡ 0 (mod p), Equation (7.1) implies:

v ∞ (S es ) = q es+1 -q q -1 + s i=1 f ei (e s ) = q es+1 -q q -1 - t i=1
(e i -1)q ei + q es -q ei q -1 -(s -t)((e s -2)q es + q es-1 ) = q es+1 -q q -1 -s i=1

(e i -1)q ei + q es -q ei q -1 + (s -t)(q es -q es-1 ).

We will distinguish three cases:

Case 1: d ≥ e s + 2. By (7.1),

v ∞ (S d ) = q d+1 -q q -1 + s i=1 f ei (d) = q d+1 -q q -1 - s i=1
(e i -1)q ei + q d -q ei q -1 .

Since s ≤ q -1 and d ≥ e s + 2, we get:

v ∞ (S d ) -v ∞ (S es ) = q d+1 -q es+1 q -1 - s i=1 q d -q es q -1 -(s -t)(q es -q es-1 )
= (q -s) q d -q es q -1 -(s -t)(q es -q es-1 ) ≥ q es+2 -q es q -1 -(q -1)(q es -q es-1 ) > 0.

Thus,

v ∞ (S d ) > v ∞ (S es ) for d ≥ e s + 2.
Case 2: d ≤ e s -2.

We have:

v ∞ (S d ) -v ∞ (S es ) = q d+1 -q es+1 q -1 + s i=1 (f ei (d) -f ei (e s ))
≥ q d+1 -q es+1 q -1 + (f e1 (e s -2) -f e1 (e s )) + (f es (e s -2) -f es (e s ))

≥ q d+1 -q es+1 q -1 + (q es-1 + q es-2 ) + (q es + q es-1 ) ≥ -q es+1 q -1 + (q es + q es-1 ) + (q es-1 + q es-2 )

> 0. It follows that v ∞ (S d ) > v ∞ (S es ) for d ≤ e s -2.
Case 3: d = e s -1.

Since s ≥ q -1, it follows that:

v ∞ (S es-1 ) -v ∞ (S es ) = q es -q es+1 q -1 + s i=1 (f ei (e s -1) -f ei (e s ))
= q es -q es+1 q -1 + sq es-1 = (-q + s)q es-1 < 0 Thus, v ∞ (S es ) > v ∞ (S es-1 ).

The proof is finished.

Suppose now that e s ≡ 0 (mod p). Set m = q-1 (q-1,N -1) Then m ≥ 2, and by the above proposition, the series d≡es (mod m) S d does not vanish. By Proposition 7.3, we obtain that δ N = 0. This completes the proof of Theorem 7.2.

8.

Taylor expansions in the neighborhood of (θ, . . . , θ)

We shall first analyze the function ω in the neighborhood of θ.

8.1.

Laurent expansion of ω. The Tate algebra T t := C ∞ [t] v∞ is endowed with the family of continuous C ∞ -linear endomorphisms (D n ) n≥0 where D n is the n-th higher derivative in t defined by D n (t m ) = m n t m-n . Let us suppose that a sequence of elements (A i ) i>0 of T × t is given, so that for all i, A i is a 1-unit, that is, v ∞ (A i -1) > v ∞ (A i ) = 0, and lim i→∞ (A i -1) = 0 in T t . Then, the product i>0 A i converges to an element F ∈ T × t . We easily obtain, from Leibniz's rule, and for n > 0, the formula:

D n (F ) = p1,p2,•••∈N, i≥1 pi=n i≥1 D pi (A i ).
Here, by definition, a composition of a positive integer n is an s-tuple (with unrestricted s) of positive integers (n 1 , . . . , n s ) such that i n i = n. We deduce, for n > 0:

D p1 (A i1 ) A i1 • • • D ps (A is ) A is .
We now choose F = (t -θ)ω, and we recall that the function omega of Anderson and Thakur has the following product expansion:

ω(t) = (-θ) 1 q-1 i≥0 1 - t θ q i -1 .
hence, setting

A i = 1 - t θ q i -1 , i ≥ 0 we have F = -θ(-θ) 1 q-1 i>0 A i .
Observe now that, for all j > 0 and i ≥ 0:

D j (A i ) A i = (θ q i -t) -j .
We deduce

D n (F ) F = s≥1 P 0<i1<•••<is (θ q i 1 -t) -p1 • • • (θ q is -t) -ps
where the second sum is over the compositions P = (p 1 , . . . , p s ) of n. Define η(∅) := 1 and, for (p 1 , . . . , p s ) as above,

η(p 1 , . . . , p s ) = 0<i1<•••<is [i 1 ] -p1 • • • [i s ] -ps ∈ K ∞ ,
where [i] = θ q i -θ. We deduce, taking into account the above computations:

Lemma 8.1. The following formula holds:

(8.2) (t -θ)ω(t) = -π n≥0      s≥1 p=(p1,...,ps) i pi=n η(p 1 , . . . , p s )      (t -θ) n .
Note that, by the usual conventions, the coefficient corresponding to n = 0 is -π.

We now compute the higher derivatives of another infinite product. Let us consider:

G = i>0 1 -t θ q i i>0 1 -θ θ q i = i>0 B i ,
where

B i = θ q i -t θ q i -θ .
We know from the formula (4.6) that this coincides with L(t). Since D n (B i )/B i equals 1, 1 t-θ q i or 0 depending on whether n = 0, 1 or n > 1, we have the series expansion:

D n (G) G = 0<i1<•••<in (t -θ q i 1 ) -1 • • • (t -θ q in ) -1 .
We deduce:

Lemma 8.2. The following formula holds:

L(t) = n≥0 (-1) n η(1, . . . , 1 n times )(t -θ) n .
The above formula of course agrees with (1) of Proposition 5.1. Note also that both entire functions (t -θ)ω and L(t) define invertible formal series in K

∞ [[t -θ]].
8.2. The s variable case. We work in T s and we suppose that n ≡ s (mod q -1). We recall that

a∈A + a(t 1 ) • • • a(t s ) a n ∈ T s (K ∞ ) ×
represents an entire function in the variables t s . We can expand in series

ζ A (n; s)(t s ) = i c i (t 1 -θ) i1 • • • (t s -θ) is ∈ K ∞ [[t 1 -θ, . . . , t s -θ]].
The Newton polyhedron of this series is likely to be interesting. We consider the case n = 1, s > 1. Then, in [START_REF] Anglès | Functional identities for L-series values in positive characteristic[END_REF], it is proved that

L(t s ) = ζ A (1; s)(t s ) = πλ 1,s ω(t 1 ) • • • ω(t s ) , with λ 1,s ∈ A[t s ].
It is easy to show that, for all i = 1, . . . , s with s > 1, L(t s )| ti=θ = 0. This means that

µ 1,s := λ 1,s (t 1 -θ) • • • (t s -θ) ∈ A[t s ].
The main conjecture in [START_REF] Anglès | Functional identities for L-series values in positive characteristic[END_REF], proved in the paper [START_REF] Anglès | Exceptional zeros of L-series and Bernoulli-Carlitz numbers[END_REF], asserts that if s > 1, then µ 1,s is a unit of K[[t 1 -θ, . . . , t s -θ]]; it does not vanish at t i = θ. This implies (it is in fact equivalent to) the non-vanishing of δ s . In [START_REF] Anglès | Functional identities for L-series values in positive characteristic[END_REF] the following result is proved (see Theorem 1):

Theorem 8.3. For n ≥ 1, n ≡ s (mod q -1), V n,s := π -n ζ A (n; s)ω(t 1 ) • • • ω(t s ) s i=1 δ-1 j=0 1 - t i θ q j ∈ K[t s ],
where δ is the smallest non-negative integer such that at once q δ -n ≥ 0 and s + q (q δ -n) ≥ 2.

8.2.1. Some examples with s = n. If n = s = 1, Theorem 8.3 is sharp by the explicit formula L(χ t , 1) = -π (t-θ)ω . We take as another example the case t = s = q for which δ = 1. We have ζ A (q; s) = τ (ζ A (1; s)) = -π q (t 1 -θ) • • • (t q -θ)ω(t 1 ) • • • ω(t q ) so, again, the result is sharp. 8.2.2. More about the polynomials V n,s . We have observed that V 1,s is a unit of K[[t 1 -θ, . . . , t s -θ]] for all s > 1, s ≡ 1 (mod q-1) (recall that V 1,s is a polynomial) and the case s = n = 1 is clear too. What about the more general case of V n,s with n ≡ s (mod q -1) and n > 1? We shall set, for commodity,

V * n,s = π -n ζ A (n; s)ω(t 1 ) • • • ω(t s ) s i=1 (θ -t i ).
If n ≥ 2, then δ ≥ 1 in Theorem 8.3 and V n,s ∈ V * n,s K[[t 1 -θ, . . . , t s -θ]] × . From now on, we assume that n ≥ 2. We want to analyze the singularity of V * n,s at t i = θ for all i which is the same as that of V n,s . 8.2.3. Case s ≤ n. This is the simplest case. Indeed, since ev θ (ζ A (n; s)), the evaluation at t 1 = • • • = t s = θ of ζ A (n; s), equals the Carlitz zeta value ζ A (n -s) which is non-zero, we see that V n,s (or equivalently, V * n,s ) are units of K[[t 1 -θ, . . . , t s -θ]].

8.2.4. Case s > n. We shall prove:

Theorem 8.4. Let us consider integers n, s such that n ≡ s (mod q-1), s > n ≥ 2. For all choices of (k 1 , . . . , k s ) ∈ Z s , (t 1 -θ) k1 • • • (t s -θ) ks V n,s , an element of K((t 1 -θ, . . . , t s -θ)), is not a unit of K[[t 1 -θ, . . . , t s -θ]].

Proof. Observe that the main result of [START_REF] Anglès | Exceptional zeros of L-series and Bernoulli-Carlitz numbers[END_REF] implies that V 1,s-n+1 is a unit of K[[t 1θ, . . . , t s-n+1 -θ]]. In this case, ev θ (V * n,s ) = 0 (indeed, n > s and s ≡ n (mod q -1) implies ev θ (V * n,s ) = ev θ (ζ A (n; s)) = 0. Let us choose I ⊂ {1, . . . , s} a subset with n -1 elements and let us denote by J the set {1, . . . , s} \ I. We have |J| = s -n + 1. We observe:

V * n,s = π -n
V * n,s = V * 1,s-n+1 (t J ) = V 1,s-n+1 (t J ) j∈J (θ -t j ).
We know that V 1,s-n+1 (t J ) is a unit of K[[t j -θ; j ∈ J]]. In particular, expanding

V * n,s = i1,...,is≥0 c i1,...,is (t 1 -θ) is • • • (t s -θ) is ∈ K[[t 1 -θ, . . . , t s -θ]],
we have c 0,...,0 = 0 and, for any subset J ⊂ {1, . . . , s} with s -n + 1 elements, by setting (i 1 , . . . , i s ) with i j = 1 for j ∈ J and i j = 0 otherwise, we have c i1,...,is = 0. This ends the proof of the Proposition.

 [START_REF] Anderson | Rank one elliptic A-modules and A-harmonic series[END_REF]The third author (T. ND.) was partially supported by ANR Grant PerCoLaTor ANR-14-CE25-0002.


Mathieu Le Goc

Charles Perin

Sean Follmer

Embedded Personal Physicalizations

Keywords: Physicalization, Personal visualization, Casual visualization, Embedded visualizations

With the emergence of quantified-self, smart devices, Internet of Things and ubiquitous robotics, we envision new opportunities to create dynamic embedded physicalizations. In particular, we see new challenges arising in the context of personal and casual physicalizations at home. In this paper, we discuss the research directions and potential benefits of dynamic embedded physicalizations in the residential context, or Embedded Personal Physicalizations.

INTRODUCTION AND MOTIVATIONS

The field of visualization is at a tipping point. As computers have transitioned from massive corporate calculators to personal computers to mobile devices, visualization is also becoming increasingly personal. Visualization has already moved from supercomputers for scientific visualization to desktop tools for information visualization, data science and data journalism.

Nowadays we generate massive amounts of personal data using devices and sensors such as "smart" meters and thermostats in our homes; personal devices such as Fitbit trackers, mobile phones, and smart watches on our bodies. However, this data is not readily available for the individuals to make sense of or get empowered for better decision making and behavior change. We argue that for visualizations to become personal, ubiquitous, and play a major role in our everyday lives, we need to explore new technologies and approaches to visualizing data, designed specifically for a personal usage context. There is a need to design interfaces so that this data is used to amplify people's cognition and help them make informed decisions.

In this paper we argue that embedded personal physicalizations, that can leverage multimodal interactions, ambient visualizations, peripheral attention, and social group discussions, are a promising approach to tackling this challenge. We start by discussing related work around ambient personal visualizations, ubiquitous interfaces and robotics, actuated physicalizations, and embedded visualization. Building on these bodies of work, we illustrate opportunities for physicalizations at home through a scenario. We further discuss the challenges and research directions of embedded personal physicalizations.

Ambient Personal Visualizations

There has been a growing interest in personal visualization and personal visual analytics, which brings forth the question: "How can the power of visualization and visual analytics be made appropriate for use in personal contexts -including for people who have little experience with data, visualization, or statistical reasoning?" [START_REF] Huang | Personal visualization and personal visual analytics[END_REF]. There is also evidence that people can engage in the act of constructing personal visualization at home, promoting self-reflection [START_REF] Thudt | Self-reflection and personal physicalization construction[END_REF]. One of the key aspects of these artifacts is their physical presence within people's intimate spaces, and how they can "serendipitously spark reflection" and facilitate "continuous awareness" [START_REF] Thudt | Self-reflection and personal physicalization construction[END_REF].

In their discussion of casual information visualization [START_REF] Pousman | Casual information visualization: Depictions of data in everyday life[END_REF], Pousman et al. argue that ambient visualizations produce awareness insightsinsights that come from maintaining awareness of a data stream. For example, the act of linking information, such as an one's Fitbit activity levels with the visible health of one's house plant, amplifies awareness of one's overall health [START_REF] Botros | Go and grow: Mapping personal data to a living plant[END_REF]. Such ambient visualizations can have a persuasive role in a personal context [START_REF] Moere | Towards designing persuasive ambient visualization[END_REF]. Displays are also becoming increasingly smaller thus better suited to personal use [START_REF] Moere | Towards designing persuasive ambient visualization[END_REF]. It is also important to convey information in a "subjectively pleasant way" in a personal context [START_REF] Moere | Towards designing persuasive ambient visualization[END_REF]. This aligns with the observation that people who craft personal visual narratives using visualization introduce subjective elements at all stages of the visualization creation process [START_REF] Thudt | Subjectivity in personal storytelling with visualization[END_REF].

Ubiquitous Interfaces and Robotics

Zooids [START_REF] Le Goc | Zooids: Building blocks for swarm user interfaces[END_REF], small wheeled-robots capable of moving on horizontal surfaces and sensing user input. Zooids enable versatile creation of ubiquitous robotic interfaces [START_REF] Kim | Ubiswarm: Ubiquitous robotic interfaces and investigation of abstract motion as a display[END_REF]. These can move, manipulate, sense, display, and interact with both user and environment thus enabling to create rich interactive spaces. Similarly, Cellulo robots [START_REF] Özgür | Haptic-enabled handheld mobile robots: Design and analysis[END_REF] allow to create tangible user interfaces for diverse learning applications. Each robot can accurately move in space and deliver haptic feedback to leverage kinesthetic learning.

Actuated Physicalizations

Data-in-Place [START_REF] Taylor | Data-in-place: Thinking through the relations between data and community[END_REF] explores ambient dynamic physicalizations to better understand their impact and how it can be used to improve the life of communities, at street level. Building on shape-changing interfaces, Taher et al. [START_REF] Taher | Exploring interactions with physically dynamic bar charts[END_REF] introduced Emerge, a dynamic physicalization of a 3D barchart alongside interaction techniques to support information exploration tasks. Dynamic composite physicalization [START_REF] Le Goc | Dynamic composite data physicalization using wheeled micro-robots[END_REF] constitute a subclass of physicalizations made of multiple elements whose topology can be reconfigured, or can reconfigure itself. We presented our implementation of dynamic composite physicalizations using Zooids [START_REF] Le Goc | Zooids: Building blocks for swarm user interfaces[END_REF].

Personal Physicalizations

Stusak et al. [START_REF] Stusak | Activity sculptures: Exploring the impact of physical visualizations on running activity[END_REF] explored physicalizations as representations of personal data such as tracked running activity, and showed that the resulting sculptures engage users in reflective processes and trigger social interactions. Huron et al. [START_REF] Huron | Constructing visual representations: Investigating the use of tangible tokens[END_REF] showed how physicalizations can help users with no prior experience in infovis create, use and explain visual representations of personal data.

Embedded Visualizations

In his talk "Seeing Spaces" [START_REF] Victor | Seeing spaces[END_REF], Victor depicts a maker space that allows makers to prototype and design using advanced seeing tools. This space contains the technology to capture and display information so that makers can observe and understand complex behaviors, keep track of the captured data, and compare parallel design variations in the resulting implementations. By capturing and representing information concurrently in context, makers have direct and easy access to large amounts of information, which would otherwise be complex to distill and cumbersome to acquire with traditional tools. Willett et al. [START_REF] Willett | Embedded data representations[END_REF] coin these as embedded data representations, representations that "are made up of multiple physical presentations that each independently display data related to their respective physical referent". Embedded physicalizations are precised as "consisting of materials or objects that are associated with the data referents". We extend these with embedded personal physicalizations, that we define as embedded visualizations that are physical representations of data in a personal context such as a home. With embedded personal physicalizations, we aim to create pervasive and seamless representations of information in casual contexts, by leveraging the low spatial indirection of embedded physicalizations. While they come with new challenges, we envision that embedded personal physicalization will empower people by augmenting their awareness and understanding of their environment, and inform decision making.

OPPORTUNITIES FOR PHYSICALIZATIONS AT HOME

With new technology come new opportunities for personal physicalizations at the junction of ubiquitous computing, robotics, and casual and personal visualizations. Recent advances in Internet of Things and ubiquitous computing, alongside with the widespread adoption of quantified self and smart devices, now facilitate the creation of large networks of sensors that collect data in a wide range of environments. This information can then feed embedded visualizations in real-time.

Traditional sensors can be distributed across the house as well as the power grid to collect, monitor, and provide information regarding utilities consumption (electricity, water, gas) [START_REF] Quintal | Watt-i-see: A tangible visualization of energy[END_REF], ambient air quality (e.g., presence of dangerous gases such as carbon monoxide), ambient temperature, illumination conditions, or human presence.

This collection of data can already provide residents with statistics and trends, to raise awareness regarding their own behavior. Yet in this personal context, phone applications or websites can seem intrusive or patronizing and raise privacy concerns. A relatively recent study also found that both the aesthetics of the representation and the context of use need to be considered when designing visualizations for environmental and resource consumption data in people's homes [START_REF] Rodgers | Exploring ambient and artistic visualization for residential energy use feedback[END_REF].

In this section we present situations depicting the daily life of a family whose house is equipped for data collection and embedded physicalizations. We illustrate this system with a potential variation of Zooids [START_REF] Le Goc | Zooids: Building blocks for swarm user interfaces[END_REF] (and thus not yet implemented), for which each robot, equipped with a variety of sensors, could move freely within the house.

Ubiquitous Sensor Network and Embedded Physicalizations

Saturday morning. The father decides to cook his specialty for the weekend family breakfast, an omelette. Looking at the physicalizations on the refrigerator of what is available in the fridge, he knows how many eggs there are, what vegetables he can incorporate into the dish, which ingredients are going bad and need to be used right away, etc..

Having access to all this information, he is able to serve up a delectable dish whilst using urgent ingredients. Zooids keep track of what has been used and what ought to be replenished, given the family habits when grocery shopping. As the children prepare to leave the kitchen to get dressed up, the father looks at the weather forecast for the day. He likes to keep a group of Zooids on the side of the fridge to display the forecast in the kitchen (see Fig. 1). As these indicate a sunny and warm day, the father tells his children to wear T-shirts and hats.

Meanwhile, the mother wants to install a new wall light in the living room. To help her find an appropriate location for the lamp, a group of Zooids climb on the walls and start scanning the surface to find the location of wall studs and electric cables in the wall. The Zooids align to create a line following electric cables, and synchronously flash their light in yellow to depict lightning and signify danger. Meanwhile, other Zooids detect the multiple studs to indicate potential locations for the new lamp. With these Zooids representations, the mother can now see in-situ the constraints and the potential spots for her new light fixture. From the multiple choices, she decides on an optimal location where the lamp will not only be supported by a stud while remaining close to electric cables, but also near her couch where she can do her evening reading. Happy with her choice, she starts drilling away safely.

Saturday afternoon. While working, the mother doesn't notice the dust building up and air quality degrading rapidly. Some of the Zooids measure the air composition at regular intervals, to help monitor the air quality in the house. As the Zooids detect increasing amounts of dust in the air, several Zooids bring a respirator mask within reach by the mother's toolbox. Meanwhile, another Zooid draws a sign in the dust on the floor, to create an additional representation of the poor quality of the air. As the mother finishes drilling, she notices the sign drawn by the Zooid in the dust and sees the respirator, and feels that her nose is starting to be itchy. Now aware of the poor quality of the air, she opens the window, put on the respirator and starts cleaning the room.

Saturday evening. Once dinner is over and the kids have put all the dishes away into the dishwasher, the Zooids remind the family that the compost bin is almost full and that it's trash pickup day tomorrow. The daughter, seeing the level of her contribution towards household chores as indicated by the Zooids, decides to take the compost out. As she is heading out with the compost bin, the Zooids near the garbage light up to indicate that it too is almost full. The daughter decides to take care of both the compost and the garbage at the same time.

Meanwhile, the older son looks at the physicalizations near the TV to choose the family's entertainment for the evening. He looks at the representations of which shows have been watched recently, how much each family member enjoyed it, how adventurous the family is feeling about what to watch, etc. He does not consider his younger brother's preferences tonight as the Zooids on the family schedule show that the younger son is away at a sleepover at his friend Ben's house. Saturday night. After the father comes back from putting the kids to bed, both parents decide to take a closer look at the recent biometric measurements of their newborn daughter displayed on the ambient visualization in the living room (see Fig. 2). The recent evolution of the baby's weight, height, and other statistics indicate to the parents that the baby is very healthy and follows the national growth charts.

Checking the house before bed, the father finds that the compost and trash have been emptied, as indicated by the Zooids lights. The Zooids do one last final checkup of the house, and they confirm that all Fig. 2. Parents looking at biometrics and statistics of their newborn baby on the in their living room.

the windows and doors are locked, the family dog is safe and sound in his little house in the garden, and the electronic appliances are turned off. The father climbs into bed next to his wife, ready to discuss the fantastic movie his older son had played for the family tonight.

By making information more available through multiplying and diversifying representations of data within the space, embedded personal physicalizations can increase people's cognition and awareness of their surroundings, and trigger reflection and associations of ideas.

Personal Physicalizations for Collaboration and Decision Making

In our most recent paper [START_REF] Le Goc | Dynamic composite data physicalization using wheeled micro-robots[END_REF], we presented a scenario depicting how physicalizations composed of Zooids can support collaboration and decision-making. In this scenario (see Fig. 3), a family is planning their future summer holidays. They want to find a place that meets the expectations of each member of the family. To do this, they physicalize a dataset containing information about numerous attractive places. They can then explore the multiple dimensions of the dataset and isolate places of agreement, based on the result of queries. With this method, the whole family collaborates and participates in the decision-making process; they discuss and compromise to settle on a location. They also create an ambient physicalization that they attach to the kitchen wall. This physicalization represents the price of the flight for a few locations the family is interested in. Placing this on the kitchen wall allows them to keep track of the evolution of the projected costs, as they aim at buying tickets when prices would reach the threshold matching the budget set by the parents. With this scenario, we conjecture that the tangible nature of the physicalization may better support collaboration by bringing the members of the family around a physicalization and providing them with a common ground, i.e., Reference Space [START_REF] Buxton | Mediaspace-meaningspace-meetingspace[END_REF]. This allows richer communication and thus can improve collaboration. Jansen et al. [START_REF] Jansen | Opportunities and challenges for data physicalization[END_REF] speculate that physicalizations may better engage the audience, fostering our perception, skills, and experience to interact with the physical world [START_REF] Jacob | Reality-based interaction: a framework for post-wimp interfaces[END_REF]. Based on existing work in the fields of education and tangible user interfaces, studies have shown that physicality offers cognitive help using, for instance, epistemic actions [START_REF] Fernaeus | Finding design qualities in a tangible programming space[END_REF][START_REF] Kirsh | On distinguishing epistemic from pragmatic action[END_REF]. We believe that these benefits can contribute to better supported collaborations.

CHALLENGES

The space of embedded personal physicalizations is ripe for exploration. Not only is it an almost untouched area of research, but it can also potentially impact and empower every person, promote the widespread use of visualization, and help to develop visualization literacy. Many challenges emerge when designing embedded personal physicalizations. In this section we describe the most salient of these challenges.

Adapting Infovis Priorities in the Home

The golden rules of visualization most likely do not apply in a home, which is a personal environment. Most information visualization guidelines are about faithfully and accurately presenting data, for fast and efficient perception of trends, values, and relations. In a personal context, we argue that other factors such as aesthetics, relatability, persistence, natural integration into the home, and curiosity-triggering, are likely to be more important.

Embedded personal physicalizations are very well suited to this context that does not necessarily require accurate representations of data. Embedded personal physicalizations are also particularly appropriate for monitoring data streams in real-time, acting as constant reminders, and becoming ambient visualizations, as we are used to living surrounded by physical, tangible objects.

In addition, information visualization research often assumes that people interacting with visualizations are fully dedicated to the act of decoding, analyzing, and interpreting data. This situation is less likely to be the case in personal environments. Under such conditions, where people split their attention and distribute their cognition in the physical spaces they live in, what makes a visualization effective is yet to be determined.

Fig. 3. The family uses a dynamic composite physicalization system [START_REF] Le Goc | Dynamic composite data physicalization using wheeled micro-robots[END_REF] to support collaboration and decision making.

In this particular context, we as a community need to question the efficiency and appropriateness of the visual encodings we employ to convey data. Some guidelines may not apply to embedded personal physicalizations. And perhaps some encodings that we quickly discard when designing visualizations are particularly relevant to embedded personal physicalizations. Rankings of encodings are based on perceptual studies that measure how fast and accurate people are at reading and comparing data values. But there is a growing discussion about how these rankings and guidelines can matter less, or even be meaningless, when factors like privacy, subjectivity, emotions and sharing of personal stories are more important than fast and accurate perception of data (e.g., [START_REF] Botros | Go and grow: Mapping personal data to a living plant[END_REF][START_REF] Huang | Personal visualization and personal visual analytics[END_REF][START_REF] Rodgers | Exploring ambient and artistic visualization for residential energy use feedback[END_REF][START_REF] Thudt | Subjectivity in personal storytelling with visualization[END_REF]).

Blending Streaming and Historical Physicalizations

Embedded personal physicalizations based on technologies such as swarm and robotic interfaces are particularly well-suited to providing real-time visualizations of streaming data, gathering insights, enhancing human awareness, and aiding collaborative decision-making activities. However, because they are more "things" than "stuff" [START_REF] Le Goc | Zooids: Building blocks for swarm user interfaces[END_REF], they are also ill-suited to visualizing large amounts or aggregations of data.

On one hand, highly dynamic visualizations that are well-suited to visualizing streaming data are likely to help people react immediately based on some data updates. On the other hand, visualizations that do not update much and are better-suited to comparisons and historical analysis, are likely to help people reflect and plan longer-term actions.

This tradeoff between highly dynamic, token-based visualizations and scalable, aggregated visualizations is not specific to physicalizations. For example, Visual Sedimentation [START_REF] Huron | Visual sedimentation[END_REF] explores the challenge of transitioning from digital data tokens to aggregates of data over time, using the metaphor of the natural process sedimentation process. However, the challenge appears to be bigger in the physical world, subject to more constraints than its digital counterpart. While strategies for smoothly transforming token-based visualizations into aggregated ones are relatively easy to design, explore, and study in the digital world, this poses a difficulty in the physical world.

One of the main challenges in enabling embedded personal physicalizations to support both awareness insights, as well as reflective and analytical insights, is to blend visualizations of streaming data with visualizations of aggregated data. While the former provides real-time feedback, the latter delivers historical information for comparison and analysis, helping self-reflection and behaviour changes.

Evaluating Physicalization in the Wild

As technologies become more and more available, we see more systems emerging as research prototypes or potential products. Yet, we need to understand their usability and benefits in personal usage contexts. Very much like the Tenison Road project [START_REF] Taylor | Data-in-place: Thinking through the relations between data and community[END_REF], longitudinal studies can help refine the new systems and most importantly, understand how they can inform and alter our behavior.

Requirements

The following are a few requirements that we deem important for an ideal embedded personal physicalization system from both implementation and design perspectives. Additional requirements may be identified with further development of implementation specifics.

Versatility. Embedded personal physicalizations should remain setup-agnostic, i.e., they should not need a specific or localized setup to work. In personal spaces, one may want to be able to re-arrange and move things around freely, bring objects to the living room or to a friend's house. The system should be flexible to be context independent.

Scalability. The size of the personal physicalizations should remain a design variable, to accommodate different spaces as well as different resolutions.

Dynamicity and Interactivity. Embedded personal physicalizations should be dynamic to react to real-time feeds of data. Furthermore, interactions should be supported to react to users' manipulations, allowing collocated input and output.

Trust. When entering private and intimate spaces, users need to create a trust relationship with the system. To this end, the personal physicalization system should provide transparency regarding the data collection and processing, and ensure ethics in the design.

Non-Intrusiveness and Privacy. Embedded personal physicalizations should be designed with reflection and relatability in mind. Personal physicalizations are not recommendation systems, and thus should not provide users with conclusions, but rather with all the element to allow them to make their own decisions. Consequently, embedded personal physicalizations should not disrupt or trouble everyday life. One should be able to hide the physicalizations, or obscure what they represent.

Feasibility

To our knowledge, only few currently available technologies allow to implement embedded personal physicalizations meeting the aforementioned requirements of versatility, scalability and interactivity. Zooids [START_REF] Le Goc | Zooids: Building blocks for swarm user interfaces[END_REF] are one of these technologies that support embedded personal physicalizations. As a result, the physicalizations presented in the scenarios in Sect. 2 are largely inspired by Zooids and what they are capable of, as well as potential future improvements. Yet, Zooids show clear limitations that prevent creating embedded personal physicalizations as presented in the scenarios. In particular, restrictions due to tracking range and cost restrain the extent to which these scenarios can be readily implemented. Other technological approaches such as ubiquitous robotics [START_REF] Kim | Ubiswarm: Ubiquitous robotic interfaces and investigation of abstract motion as a display[END_REF] as well as shape changing interfaces [START_REF] Siu | shapeshift: A mobile tabletop shape display for tangible and haptic interaction[END_REF] could allow for rich and diverse embedded personal physicalizations.

CONCLUSION

Embedded personal physicalizations have the potential to help people leverage the massive amounts of personal data that they collect in their everyday lives. They offer dynamic, actuated physical representations of data that can help families gain awareness insights, participate in collective decision-making activities, reflect on their behaviors, and change their habits. While embedded personal physicalizations offer some unique advantages, they also come with difficult challenges to tackle. We provide an initial discussion of these challenges as well as a first list of requirements for designing embedded personal physicalizations at home.

Fig. 1 .

 1 Fig. 1. An ambient physicalization representing the weather of the day.


Johannes Liem

Eirini Goudarouli

Steven Hirschorn

Jo Wood

Charles Perin

Charles Perin Ww100

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

WWI00 World War One Hundred

A hundred years ago, in November 1918, the First World War (WWI) came to an end. The National Archives (UK) preserved 1.5 million pages of hand-written diaries, documenting the story of the British Army and its units on the Western Front during the war. Since 2014, "citizen historians" annotate and tag scans of those diaries on the crowdsourcing platform Operation War Diary ([1], OWD). This process generates extensive time-series data about military units including their location, military activities, casualties, everyday army life, the names of unit members and weather information. However, this process also introduces uncertainty on many levels due to missing records, misspellings, unreadable parts, lost diaries, the reliability of workers, and even through post-processing of the gathered data (e.g., georeferencing place names).

Instead of giving the illusion that the data is complete and clean, we leverage this uncertainty in the visualization we designed and developed to produce a more "organic" view of a unit's movement over time, with the intent to convey the notion that a unit represents human beings.

Besides the spatiotemporal movement of UK's military units, the exhibit shows activities and day-to-day life on the Western Front during WWI. It aims to illustrate how soldiers spent their time by contrasting fighting and non-fighting activities.

The OWD data show that the balance between time spent in the trenches and time spent behind the trenches are not aligned with our common beliefs about war [2]. Revealing "life behind the trenches" to the general public might help to convey the more humane aspect of the war and clarify that a lot of time was not spent fighting in the trenches.

Our piece is based on what we call

GeoBlob: an abstract representation of moving entities on a map with uncertain positions. Instead of showing a unit at a given point in time, GeoBlobs convey an unordered estimation of the possible locations over a temporal window using enclosed shapes, or blobs. To this end, we Johannes Liem 1 , Eirini Goudarouli 2 , Steven Hirschorn 2 , Jo Wood 1 , Charles Perin 3,1 1 City, University of London, London, UK 2

The National Archives, London, UK 3 University of Victoria, Canada apply heuristics to weigh each location within the temporal window. Then, at the presentation stage, it is possible to specify dynamically which sets of locations form the GeoBlobs. GeoBlobs leverage the uncertainty in the data to produce an "organic" view of a moving unit on a map over time, in contrast to the visual certainty conveyed by crisp line/dot visualizations. GeoBlobs also suffer less from scalability issues, while line/dot visualizations become cluttered when the number of data points is large.

We submitted a five-minute talk about this topic to the Information+ conference 2018, which is co-located with IEEE VIS 2018. With Berlin as the location in mind, we believe the exhibition and talk will be of interest to the overlapping audiences, but also the more general, local public.

Data and Identities

Experts have collected historical information in analog forms throughout history. Today, modern technology, such as applications, devices or sensors, change the ways of collecting massive amounts of historical data entirely. Modern techniques now allow us to transform such analog information into digital data (as described above), making more and more data available in the Digital Humanities. In the case of OWD data, the digitized and qualified hand-written diaries (OWD data) provide an interface to otherwise buried information. This data is an unprecedented source of information that makes it possible to look at WWI through the lens of well-structured and rather exhaustive war diaries, thus provide new historical perspectives we can perhaps better relate to on a personal level.

History, or past events in general, influences our cultural identity in manifold ways. Both World Wars greatly influenced how we live together in Europe, shaping our collective as well as our personal identities. This exhibit is not intended to communicate battles, victories or defeats of WWI only. Instead, it aims at communicating a sense of the day-to-day life in the war zone. Communicating the OWD data as a whole, including fighting and

Snapshots of two units moving from Belgium into Germany after the war in early 1919

non-fighting activities, we aim at bringing to light aspects of war that can contribute to the better understanding of Europe's collective past. In the light of the current political climate, we think that it is important to highlight our collective memory, rather than a silo national mentality, that may help young generations to understand current developments.

Technical Details

The browser-based visualization uses the mapping framework leaflet [3] with custom D3 SVG overlays [START_REF]D3SvgOverlay[END_REF] to visualize the OWD data (spatial distribution and activities). To calculate the GeoBlobs, we make use of the Bubble Sets algorithm [5] The custom background map was designed with Map Box Studio to convey a historical and clean aesthetics.

Currently, we are experimenting with different narrative designs [START_REF] Henry-Riche | Data-Driven Storytelling[END_REF] for enriching the visualization to provide an intriguing and more emotional experience for visitors. Amongst others, this includes text, icon, and media-based annotations of unit events and activities. We also investigate how motion can be applied to convey information (e.g., a shaking map or unit GeoBlob on fighting days).

For the exhibit setup, the visualization will play a looping animation that shows the movements and activities of several units from the 3rd division from 1914 to 1920. The selection of a single division that consists of a number of units, based on discussions with historians, is well-suited to depicting the picture of coherent military activities as a whole, where each unit is part of a bigger picture.

For a better-controlled exhibition environment, we limit the interaction to a set of sliders, which allow the visitor to explore and vary the visualization design space to a certain degree. It includes interactive adjustment of the temporal window (how many days are visualized by a unit GeoBlob), blob properties (e.g., size), the distance from a blobs centroid to remove outlier locations forming the GeoBlob, playback speed, or duration and fade out time of events. We do not allow any map interaction or advanced temporal navigation.

Visitors interact with a tablet-sized touch screen that communicates with the visualization projected onto a flat surface through local WebSocket.

Installation Requirements

For the interactive WWI00 installation, we require a projector, a flat projection surface (wall or screen) and a touch interface to allow visitors to explore the visualization (e.g., tablet or a laptop, running a local host). We can provide a touch-enabled laptop. Regarding space, we would require approximately an area of 3x3 meters. Optionally, the projector can be


Charles Perin

The Symmetry of My Life II

INTRODUCTION AND CONCLUSION

Last year tat the IEEE VIS conference, I presented a poster entitled "The Symmetry of My Life" that showed personal and work-related activities for the past 18 months of my life. A year has passed, and it is time to update the visualization. I recognize there is not much to say research-wise. Instead of self-plagiarizing I will point to the poster that was exhibited last year [START_REF] Perin | The Symmetry of My Life: An Autobiographical Visualization[END_REF] and that is also shown in Figure 1.

There are four paramount reasons why I send this updated version.

1. Reason #1: the poster triggered a lot of interest and discussions from the VIS attendees. I expect this second version to generate the same interest. Not that my life is particularly fascinating, but students and academics can easily reflect on these events and work-life patterns.

Reason #2:

This poster aligns well with the VISAP theme this year on data and identities, where I will also (hopefully) present an annotated portfolio on the project (http://visap.net/). I believe both submissions will complement each other well. A few minor things changed compared to last year's design. For example, I added "travel" as a new category that exists for both personal and work-related activities. In the previous version "travel" was associated to "vacation", which is definitely less accurate.

Perhaps the most interesting fact in this version is how little I have had a personal life since the beginning of September 2017. Except the period around Christmas during which I went on vacation, my life has been largely dominated by my work. The reason is quite simple: in September I started to work on my job applications, working first on my personal website, before working on my application package then my job talk, until the interviews took place in February and March. An that was on top of my regular job, with a lot of teaching happening during the winter and the inevitable March 31st deadline. It shows quite well that finding a job in academia is a long and demanding process, but also a unique opportunity that is quite enjoyable.

* e-mail: cperin@uvic.ca

3.

  Reason #3: It's a nice poster to hang on a wall. 4. Reason #4: Last but not least, I spent a lot of time and energy collecting my data for 12 more months, and I am not sure I can keep doing that for long without endangering my mental health.

Figure 1 :

 1 Figure 1: The poster presented last year at the IEEE VIS conference.

Figure 2 :

 2 Figure 2: The updated visualization. Streamgraphs show work and personal daily activities on the two sides of a spiral timeline. Coloured squares shows daily steps, floors, emails and SVN commits. Annotations provide contextual information, such as travels, deadlines, and important events.

ACKNOWLEDGMENTS

I want to thank the reviewers of this poster for being so enthusiastic about this work; I also want to thank Toggl (https://toggl. com/), for having built a simple and efficient logging tool; and of course my family for not complaining too much about this singular obsession.


Aidan Slingsby

Andy Paterson

Mark Rigby

Katherine Grace

Phong Nguyen

Charles Perin

Cagatay Turkay

Dalal Aljasem

Characterising Farms by the Movement of Animals through Them

Keywords:

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

INTRODUCTION

The Animal and Plant Health Agency (APHA) is the UK Government agency responsible for monitoring plant and animal diseases and taking appropriate action to help reduce the impacts on agriculture, the environment and the general public.

Cattle is an important sector of UK agriculture. For the last twenty years, individual cattle movements between 'holdings' (farms and agriculture-related sites including abattoirs, markets and shows) have been captured, forming the "Cattle Tracing System (CTS)" database (as in many other EU countries). The data are used in various operational settings, e.g. helping designing surveillance protocols, where the movements of animals need to be identified to help control the spread of diseases when they are found. However, these data have not really been used beyond these operational uses than respond to particular incidents.

The movement of animals and humans is the most important factor affecting disease or pest outbreak. For APHA epidemiologists and other modellers, knowledge and understanding of the movement of animals is key to the decisions and interventions that happen in response. Identifying holdings at risk involve understanding the sources, destinations and timings of animal movement between them. Interventions may include forbidding animal movements to and from certain areas within certain timescales, or even forbidding movement altogether. Although relatively easy to investigate for specific incidents (because there are more constraints on what to consider), a more general understanding of how cattle are moved around is more difficult as movement behaviour can change by season and over time. For non-expert senior decision-makers, an understanding of how policy decisions are informed by the analysis is important. For organisations representing the needs of farmers, advice and policy need to be justified in terms of the science.

To assist, we are characterising holdings by the movement of cattle through them, by designing metrics that capture aspects of movement that might have implications for disease spread. Through visual summaries in an interactive visualisation interface, we are helping establish the effectiveness of these metrics for characterising holdings. This work is helping AHPA better understand the significance of their operations in terms of surveillance. * e-mail: a.slingsby@city.ac.uk

PILOT STUDY

We investigate how we might usefully characterise the movement of cattle in and out of holdings, eventually using these as bases to cluster holdings. There are obvious simple cases where this may relate to function -animals do not leave abattoirs alive; agricultural show animals travel from a permanent base to different shows and then back to their permanent base; markets take animals from a wide geographical range and redistribute them -but there may be more subtle examples. These clusters may also relate to risk of disease spread -holdings whose animals regularly move in and out within a short space of time may have a higher risk of unwittingly spreading disease further, especially if sources from high-risk areas and move to geographically diverse destinations. Our work takes inspiration from existing work that infers the semantics of places by temporal characteristics of the visit of people (e.g. [START_REF] Andrienko | Scalable and privacy-respectful interactive discovery of place semantics from human mobility traces[END_REF]). Our pilot study is structured thus:

• Talk to APHA staff as to what are important issues and why.

• Calculate different measures at a holding level, including those that characterise temporal and geographical differences in movement. Get APHA feedback on these. • Visually summarise these measures.

• Facilitate geographical and temporal filtering to explore the stability of these measures. • Experiment with clustering holdings based on these features.

INITIAL RESULTS

With 185 million individual movements, we selected geographicallyand temporally-constrained subset of animals that passed through Gloucestershire in 2010, reducing the set to 400,000, whilst retaining the complete movement history of every animal in our the set. Zooming and panning the map serves as a geographical filter in Fig. 1) for the holdings and their "movement signatures", listed on the right and sorted by the number of movements, with the red/blue portion of the bar indicating in/out movement (some of the holdings have in-but not out-movement). The mouse pointer is over an abattoir with only in-movement, identified by a green dot on the map. The is too cluttered to interpret, but serves as the (geographical) selection mechanism for "movement signatures" in the right. These capture aspects of the movement of cattle through the holdings. The orange histograms show (from left to right) frequency distributions of all cattle movements for each holding:

• By year. Although the observed patterns are generally expected -given the temporal filtering -some holdings appear to take older cattle than others. • By month of year: A bimodal spring/autumn distribution can be observed.The question is does this apply to all holdings? • By duration spent before moving on: As expected, the majority of holdings move cattle on within half a year (the leftmost bin). However, some keep cattle on the premises much longer. APHA are particularly interested in those identified as regularly moving cattle on within a short time. We can begin to characterise and quantify how this could help inform risk-based interventions. • By distance moved out: Holdings whose animals come from or to distant locations may represent a greater risk in terms of contributing to longer-range spread of disease, though the riskiness of the area may be more important. • By frequency moved by individuals: holdings that receive animals that regularly move may be at higher risk of disease as they are exposed to more places. • By day of week: This indicates routine behaviours that correspond to markets, abattoirs, etc. whose activities are on particular days of the week.

The green charts show proportions of:

• Unique holdings: This tells us whether the cattle come from a small or large pool of holdings; the latter of might be of concern.

• Unique animals: This tells us the extent animals revisit the holding. Some holdings receive the same animal multiple times, for example bullocks being hired out or animals entering shows. • Dairy vs beef cattle: Indicates the particular type of holding.

The characteristics of the movement of cattle through holdings can also inform other risk-based approaches including welfare concerns and general compliance with other responsibilities, particularly with regards to "traceability" requirements.

Fig. 2 is a DO maps (OD map [START_REF] Wood | Visualisation of origins, destinations and flows with od maps[END_REF] with origin and destination reversed), showing that source of the animals tends to be from south of the destination -but not in all cases -and with different sizes of geographical spread.

REFLECTION AND FURTHER WORK

This preliminary work so far shown that these movement signatures are a good basis with which to explore movement characteristics of holdings with APHA. It is helping establish the characteristics that may help APHA usefully characterise holdings. The characteristics we are investigating are based on conversations with APHA and we plan other workshop to consider these and others. Once we've established a good working set of characterisations, we will be experimenting with clustering holdings based on these and considering changes in movement characterisation over time. We are working closely with APHA to develop effective visualisation techniques and characterisations of movement.

Figure 1 :

 1 Figure 1: Red dots indicate holdings. Black lines are the movements of cattle between them (see Fig. 2 for clearer representation). Movement signatures for each holding are on the right, where each row is a farm (names blurred) within the map view, sorted by the number of cattle movements. Orange histograms show (from left to right) frequency distributions of cattle movements and the green charts show proportions as indicated in section 3. Data source: DEFRA.

Figure 2 :

 2 Figure2: DO map (OD map[START_REF] Wood | Visualisation of origins, destinations and flows with od maps[END_REF] with origin and destination reversed) is a better representation then in Fig.1(left). Indicates geographical origins of animals in the UK maps for the destinations indicated by the rectangles. Destinations tend to have animals that originate south of them (yellow dots).

ACKNOWLEDGMENTS

The authors wish to thank City, University of London for funding this pilot study and the other DEFRA staff who were involved.