Johan Kopra
email: jtjkop@utu.fi

Glider Automorphisms on Some Shifts of Finite Type and a Finitary Ryan's Theorem

Keywords: Mixing SFTs, Automorphisms, Cellular automata

For any mixing SFT X containing a fixed point we construct a reversible shift-commuting continuous map (automorphism) which breaks any given finite point of the subshift into a finite collection of gliders traveling into opposing directions. As an application we show that the automorphism group Aut(X) contains a two-element subset S whose centralizer consists only of shift maps.

Introduction

Let X ⊆ A Z be a one-dimensional subshift over a symbol set A. If X contains some constant sequence 0 Z (0 ∈ A), we may say that an element x ∈ X is finite if it differs from 0 Z only at finitely many coordinates. In this paper we consider the problem of constructing reversible shift-commuting continuous maps (automorphisms) on X which decompose all finite configurations into collections of gliders traveling into opposing directions. As a concrete example, consider the binary full shift X = {0, 1} Z and the map g = g 3 • g 2 • g 1 : X → X defined as follows. In any x ∈ X, g 1 replaces every occurrence of 0010 by 0110 and vice versa, g 2 replaces every occurrence of 0100 by 0110 and vice versa, and g 3 replace every occurrence of 00101 by 00111 and vice versa. In Figure 1 we have plotted the sequences x, g(x), g 2 (x), . . . on consecutive rows for some x ∈ X. It can be seen that the sequence x eventually diffuses into two different "fleets", the one consisting of 1s going to the left and the one consisting of 11s going to the right. It can be proved that this diffusion happens eventually no matter which finite initial point x ∈ X is chosen. 1 In Section 3 we construct on all mixing SFTs (that contain the point 0 Z ) a glider automorphism with the same diffusion property as the binary automorphism g above.

The existence of a glider automorphism g on a subshift X is interesting, because g can be used to convert an arbitrary finite x ∈ X into another sequence g t (x) (for some t ∈ N + ) with a simpler structure, which nevertheless contains all the information concerning the original point x because g is invertible. Such maps have been successfully applied to other problems e.g. in [START_REF] Salo | Transitive action on finite points of a full shift and a finitary Ryan's theorem[END_REF][START_REF] Salo | A One-Dimensional Physically Universal Cellular Automaton[END_REF].

We also consider a finitary version of Ryan's theorem. Let X be a mixing SFT and denote the set of its automorphisms by Aut(X), which we may consider as an abstract group. According to Ryan's theorem [START_REF] Ryan | The shift and commutativity[END_REF] the center of the group Aut(X) is generated by the shift map σ. There may also be subsets S ⊆ Aut(X) whose centralizers are generated by σ. Denote the minimal cardinality of such a finite set S by k(X). In [START_REF] Salo | Transitive action on finite points of a full shift and a finitary Ryan's theorem[END_REF] it was proved that k(X) ≤ 10 when X is the full shift over the four-letter alphabet. In the same paper it is noted that k(X) is an isomorphism invariant of Aut(X) and therefore computing it could theoretically separate Aut(X) and Aut(Y ) for some mixing SFTs X and Y . We use our glider automorphism construction to prove that k(X) = 2 for all mixing SFTs that contain the point 0 Z .

Preliminaries

A finite set A containing at least two elements (letters) is called an alphabet and the set A Z of bi-infinite sequences (configurations) over A is called a full shift. Formally any x ∈ A Z is a function Z → A and the value of x at i ∈ Z is denoted by x[i]. It contains finite and one-directionally infinite subsequences denoted by

x[i, j] = x[i]x[i+1] . . . x[j], x[i, ∞] = x[i]x[i+1] . . . and x[-∞, i] = . . . x[i-1]x[i].
A factor of x ∈ A Z is any finite sequence x[i, j] where i, j ∈ Z, and we interpret the sequence to be empty if j < i. Any finite sequence w = w[1]w [START_REF] Ryan | The shift and commutativity[END_REF] . . . w[n] (also the empty sequence, which is denoted by λ) where w[i] ∈ A is a word over A. The set of all words over A is denoted by A * , and the set of non-empty words is A + = A * \ {λ}. More generally, for any L ⊆ A * , let

L * = {w 1 . . . w n | n ≥ 0, w i ∈ L} ⊆ A * ,
i.e. L * is the set of all finite concatenations of elements of L. The set of words of length n is denoted by A n . For a word w ∈ A * , |w| denotes its length, i.e. |w| = n ⇐⇒ w ∈ A n . We say that the word

w ∈ A n occurs in x ∈ A Z at position i if w = x[i] . . . x[i + n -1]. We define the shift map σ A : A Z → A Z by σ A (x)[i] = x[i + 1] for x ∈ A Z , i ∈ Z. The subscript A in σ A is typically omitted.
The set A Z is endowed with the product topology (with respect to the discrete topology on A), under which σ is a homeomorphism on A Z . Any closed set X ⊆ A Z such that σ(X) = X is called a subshift, and the collection of words appearing as factors of elements of X is the language of X, denoted by L(X). The restriction of σ to X may be denoted by σ X , but typically the subscript X is omitted.

If X ⊆ A Z is a subshift and z ∈ X is such that σ(z) = z (i.e. z is a fixed point), then there exists a ∈ A such that z[i] = a for all i ∈ Z. For such subshifts we always fix one such point and denote a = 0, z = 0 Z . Then for x ∈ X we define its support supp(x) = {i ∈ Z | x[i] = 0} and say that x is finite if supp(x) is finite. Finite points x, y ∈ X with disjoint supports can be glued together;

if supp(x) ∩ supp(y) = ∅ we define x ⊗ y ∈ A Z by (x ⊗ y)[i] = x[i] when i ∈ supp(x) and (x ⊗ y)[i] = y[i] otherwise. Definition 1. A graph is a pair G = (V, E)
where V is a finite set of vertices (or nodes or states) and E is a finite set of edges. Each edge e ∈ E starts at an initial state denoted by ι(e) ∈ V and ends at a terminal state denoted by τ (e) ∈ V . We say that e ∈ A is an outgoing edge of ι(e) and an incoming edge of τ (e).

A sequence of edges e [START_REF] Lind | An introduction to symbolic dynamics and coding[END_REF] [START_REF] Lind | An introduction to symbolic dynamics and coding[END_REF]). We say that the path starts at e [START_REF] Lind | An introduction to symbolic dynamics and coding[END_REF] and ends at e [n]. A graph G is primitive if there is n ∈ N + such that for every v 1 , v 2 ∈ V there is a path of length n starting at v 1 and ending at v 2 . For any graph G = (V, E) we call the set

. . . e[n] in a graph G = (V, E) is a path (of length n) if τ (e[i]) = ι(e[i + 1]) for 1 ≤ i < n and it is a cycle if in addition τ (e[n]) = ι(e
{x ∈ E Z | τ (x[i]) = ι(x[i + 1])}
(i.e. the set of bi-infinite paths on G) the edge subshift of G. Definition 2. A subshift X ⊆ A Z is a mixing subshift of finite type (mixing SFT) if it is the edge subshift of a primitive graph G = (V, E) containing at least two edges (in particular E ⊆ A).

Example 3. Let A = {0, a, b}. The graph in Figure 2 defines a mixing SFT X also known as the golden mean shift. A typical point of X looks like . . . 000abab0ab00ab000 . . . i.e. the letter b cannot occur immediately after 0 or b and every occurrence of a is followed by b.

Definition 4. An automorphism of a subshift X ⊆ A Z is a continuous bijection f : X → X such that σ • f = f • σ. We say that f is a radius-r automorphism if f (x)[0] = f (y)[0]
for all x, y ∈ X such that x[-r, r] = y[-r, r] (such r always exists by continuity of f ). The set of all automorphisms of X is a group denoted by Aut(X). (In the case X = A Z automorphisms are also known as reversible cellular automata.) The centralizer of a set S ⊆ Aut(X) is

C(S) = {f ∈ Aut(X) | f • g = g • f for every g ∈ S}
and the subgroup generated by f ∈ Aut(X) is denoted by f . The following definition is from [START_REF] Salo | Transitive action on finite points of a full shift and a finitary Ryan's theorem[END_REF]: Definition 5. For a subshift X, let k(X) ∈ N ∪ {∞, ⊥} be the minimal cardinality of a set S ⊆ Aut(X) such that C(S) = σ if such a set S exists, and k(X) =⊥ otherwise.

The main result of [START_REF] Ryan | The shift and commutativity[END_REF] is that k(X) =⊥ whenever X is a mixing SFT. We say that subshifts

X ⊆ A Z and Y ⊆ B Z are conjugate if there is a continuous bijection ψ : X → Y such that ψ • σ X = σ Y • ψ. For conjugate subshifts X and Y it necessarily holds that k(X) = k(Y ).

Glider Automorphisms

In this section we define as a technical tool a subclass of mixing SFTs, and for any subshift X from this class we construct an automorphism which breaks every finite point of X into a collection of gliders traveling in opposite directions.

Note that if X is a mixing SFT with a fixed point 0 Z , then necessarily in its graph G = (V, E) it holds that τ (0) = ι(0). For such a graph we denote G = (V, E ) and E = E \ {0}, i.e. we get G from G by removing the 0-edge. Definition 6. A mixing SFT X with a fixed point 0 Z and defined by the graph G = (V, E) is called a 0-mixing SFT if the graph G is also primitive and contains at least two edges.

The golden mean shift given by the graph in Figure 2 is an example of a mixing SFT which strictly speaking isn't 0-mixing. Nevertheless, in the following lemma we show that the definition of a 0-mixing SFT is only technical and that it is not an actual restriction.

Lemma 7. Any mixing SFT with a fixed point is conjugate to a 0-mixing SFT.

Proof. Let X be a mixing SFT with a fixed point 0 Z defined by the graph G = (V, E) and let s = ι(0) = τ (0). Let 0, a 1 , . . . , a t be all the outgoing edges of s, let 0, b 1 , . . . , b u be all the incoming edges of s and construct a new graph

H = (V ∪ {s }, E ∪ {0 , b 1 , . . . , b u })
with the starting and ending nodes of e ∈ E the same as in G with the exception that ι(a i ) = s for 1 ≤ i ≤ t, and additionally ι(0 ) = s, τ (0 ) = s , ι(b j ) = ι(b j ) and τ (b j ) = s for 1 ≤ j ≤ u. 2 Let Y be the edge subshift of H; it is conjugate with X via the continuous shift-commuting map ψ : X → Y defined for x ∈ X, i ∈ Z as In the rest of the section we assume that X is a 0-mixing SFT defined by the graph G = (V, E). This means that the edge subshift of G is also a mixing SFT. Denote s = ι(0) = τ (0). Let v 1 be a cycle in G visiting s only at the beginning and ending, denote p = |v 1 | and let v 0 = 0 p . The words

ψ(x)[i] =    0 when x[i] = 0 and x[i + 1] ∈ {a 1 , . . . , a t }, b j when x[i] = b j for some 1 ≤ j ≤ u and x[i + 1] ∈ {a 1 , . . . , a t }, x[i] otherwise.
g = v 0 v 1 , g r = v 1 v 1
will be left-and rightbound gliders of the automorphism g defined later. The languages of left-and rightbound gliders are

L = (g 00 * ) * , L r = (0 * 0g r ) * .
We denote by ∞ 0 and 0 ∞ left-and right-infinite sequences of zeroes and define the glider fleet sets

GF = ∞ 0(g 00 * ) * 0 ∞ GF r = ∞ 0(0 * 0g r ) * 0 ∞ GF = GF ∪ GF r
(note that these consist of finite configurations). Denote u = v 1 v 1 v 1 and let n ∈ N + be a mixing constant of G (i.e. a number such that for every n ≥ n and s 1 , s 2 ∈ V there is a path of length n in G from s 1 to s 2 ) chosen such that n ≥ |u | = 3p. For every a ∈ E we may choose some path w a ∈ E 2n in G such that w a begins with u and 0w a a ∈ L(X). For every a ∈ E let W a = {w a,1 , . . . , w a,ka } ⊆ E 2n be the paths of length 2n in G such that w a,i does not have a prefix u and 0w a,i a ∈ L(X) for 1 ≤ i ≤ k a , and let W a = W a ∪ {w a }. Let U = {u 1 , . . . , u k } ⊆ (E ) + be the cycles from s to s (which may visit s several times) of length at most 2n -1 ≥ 5p which are different from v 1 and v 1 v 1 and do not have u as a prefix. Finally, these words are padded to constant length; u = 0 2n-1-|u | u and u i = 0 2n-1-|u i | u i . The words in W a and U are chosen so as to allow the following structural definition.

Definition 8. Assume that x / ∈ GF is a non-zero finite element of X. Then there is a maximal i ∈ Z such that x[-∞, i -1] ∈ ∞ 0L ,
and there is a unique word w ∈ {v

1 0} ∪ {v 1 v 1 0} ∪ {u } ∪ (U 0) ∪ ( a∈E W a a) such that w is a prefix of x[i, ∞]. If w = v 1 v 1 0 or w ∈ U 0, let j = i + |w| -1 and otherwise let j = i + |v 1 |.
We say that x is of left bound type (w, j) and that it has left bound j (note that j > i).

Similarly, if x / ∈ GF r is a non-zero finite element of X, then there is a minimal

j ∈ Z such that x[j + 1, ∞] ∈ L r 0 ∞
and we say that x has right bound j.

The point of this definition is that if x is of left bound type (w, j), then the glider automorphism g defined later will create a new leftbound glider at position j and break it off from the rest of the configuration.

We define four maps g 1 , g 2 , g 3 , g 4 : X → X as follows. In any x ∈ X, g 1 replaces every occurrence of 0(v 0 v 1 )0 by 0(v 1 v 1 )0 and vice versa g 2 replaces every occurrence of 0(v 1 v 0 )0 by 0(v 1 v 1 )0 and vice versa g 3 replaces every occurrence of 0v 0 (v 1 v 0 v 1 ) by 0v 0 (v 1 v 1 v 1 ) and vice versa g 4 replaces every occurrence of 0w a a, 0w a,i a and 0w a,ka a by 0w a,1 a, 0w a,i+1 a and 0w a a respectively (for a ∈ E and 1 ≤ i < k a ) and every occurrence of 0u0, 0u i 0 and 0u k 0 by 0u 1 0, 0u i+1 0 and 0u0 respectively (for 1 ≤ i < k).

It is easy to see that these maps are well defined automorphisms of X. The glider automorphism g : X → X is defined as the composition

g 4 • g 3 • g 2 • g 1 .
The name is partially justified by the following lemma.

Lemma 9. If x ∈ GF (resp. x ∈ GF r ), then g(x) = σ p (x) (resp. g(x) = σ -p (x)).
Proof. Assume that x ∈ GF (the proof for x ∈ GF r is similar) and assume that i ∈ Z is some position in x where g occurs. Then

x[i -1, i + 2p] = 0g 0 = 0(v 0 v 1 )0 g 1 (x)[i -1, i + 2p] = 0(v 1 v 1 )0 g 2 (g 1 (x))[i -p -1, i + p] = 0v 0 (v 1 0) = 0g 0 g(x) = g 4 (g 3 (g 2 (g 1 (x)))) = g 2 (g 1 (x))),
so every glider has shifted by distance p to the left and g(x) = σ p (x).

In fact, the previous lemma would hold even if g were replaced by g 2 • g 1 . The role of the part g 4 • g 3 is, for a given finite point x ∈ X, to "erode" non-zero non-glider parts of x from the left and to turn the eroded parts into new gliders. This is the content of the following lemmas.

Lemma 10. Assume that x ∈ X has left bound j. Then there exists t ∈ N + such that the left bound of g t (x) is strictly greater than j.

Proof. Let x ∈ X be of left bound type (w, j) with w ∈ {v 1 0} ∪ {v 1 v 1 0} ∪ {u } ∪ (U 0) ∪ ( a∈E W a a). The gliders to the left of the occurrence of w near j move to the left at constant speed p under action of g without being affected by the remaining part of the configuration.

Case 1. Assume that w = v 1 v 1 0. Then g 1 (x)[j -(p + 1), j] = 0v 1 0. If g 1 (x)[j - (p + 1), j + p] = 0v 1 v 0 0, then g(x)[j -(p + 1), j + p] = g 2 (g 1 (x))[j -(p + 1), j + p] = 0v 1 v 1 0, g(x) is of left bound type (v 1 v 1 0, j + p) and we are done. Otherwise g 2 (g 1 (x))[j -(p + 1), j] = 0v 1 0. Denote y = g 3 (g 2 (g 1 (x))).
If y[j -(p + 1), j] = 0v 1 0, then g(x) = g 4 (y) is of left bound type (v 1 0, j) and we proceed as in Case 3. Otherwise y[j -(p + 1), j + (2p -1)] = 0u . If y[(j + 2p) -2n, j + 2p] = 0u0, then g(x)[(j + 2p) -2n, j + 2p] = 0u 1 0, g(x) is of left bound type (u 1 0, j + 2p) and we are done. On the other hand, if y[(j + 2p) -2n, j + 2p] = 0u0, then g(x) is of left bound type (w , j) for some w ∈ W a a ∪ {u } (a ∈ E) and we proceed as in Case 4 or Case 5. Case 2. Assume that w = u i 0 for 1 ≤ i ≤ k. There is a minimal t ∈ N such that g 3 (g 2 (g 1 (g t (x))))[j -2n, j] = 0u i 0. Because g t+k-i+1 (x)[j -2n, j] = 0u0, it follows that y = g t+k-i+1 (x) is of left bound type (u , j -2p). Then g(y)[j -5p, j] = g 3 (g 2 (g 1 (y)))[j -5p, j] = v 0 (v 0 v 1 )v 0 v 1 0 is of left bound type (v 1 0, j) and we proceed as in Case 3. Case 3. Assume that w = v 1 0. Then x[j -(2p + 1), j] = 0v 0 v 1 0 = 0g 0 because otherwise the left bound of x would already be greater than j, so

g 1 (x)[j - (p + 1), j] = 0v 1 0. If moreover g 1 (x)[j -(p + 1), j + p] = 0v 1 v 0 0, then g(x)[j - (p + 1), j + p] = g 2 (g 1 (x))[j -(p + 1), j + p] = 0(v 1 v 1 )0 so g(x)
is of left bound type (v 1 v 1 0, j + p) and we are done. Let us therefore assume that

g 1 (x)[j -(p + 1), j + p] = 0v 1 v 0 0, in which case g 2 (g 1 (x))[j -(2p + 1), j] = 0v 0 v 1 0. If g 2 (g 1 (x))[j -(2p + 1), j + 2p -1] = 0v 0 v 1 v 0 v 1 , then g(x)[j -(2p + 1), j] = g 3 (g 2 (g 1 (x)))[j -(2p + 1), j] = 0v 0 v 1 0.
The left bound of g(x) is now greater than j and we are done. Otherwise g 3 (g 2 (g 1 (x)))[j -(2p + 1), j + 2p -1] = 0v 0 u . If g 3 (g 2 (g 1 (x)))[(j + 2p) -2n, j + 2p] = 0u0, then g(x)[(j + 2p) -2n, j + 2p] = 0u 1 0 and the left bound of g(x) equals j + 2p. Finally, if g 3 (g 2 (g 1 (x)))[(j + 2p) -2n, j + 2p] = 0u0, then g(x) is of left bound type (w , j) for some w ∈ W a a ∪ {u } (a ∈ E) and we proceed as in Case 4 or Case 5. Case 4. Assume that w = w a,i a for a ∈ E and 1 ≤ i ≤ k a . Then g ka-i+1 (x)[jp, j + (2p -1)] = u and we proceed as in Case 5. Case 5. Assume that w = u . Then g 2 (g 1 (x))[j -(2p + 1), j + (2p -1)] = 0v 0 u , g 3 (g 2 (g 1 ))[j -(2p + 1), j + (2p -1)] = 0v 0 v 1 v 0 v 1 and the left bound of g(x) is at least j + 2p.

Lemma 11. Assume that x ∈ X has right bound j. Then there exists t ∈ N + such that the right bound of g t (x) is strictly less than j.

Proof. Let us assume to the contrary that the right bound of g t (x) is at least j for every t ∈ N + . Assume first that the right bound of g t (x) is equal to j for every t ∈ N + . By the previous lemma there is t ∈ N + such that the left bound of g t (x) is at least j + 3n, which means that g t (x) contains only g -gliders to the left of j + n and only g r -gliders to the right of j. This can happen only if

g t (x)[j +1, n-1] = 0 n-1 .
Then the right bound of g t+1 (x) is at least j -p, a contradiction.

Assume then that the right bound of g t (x) is strictly greater that j for some t ∈ N + and fix the minimal such t. This can happen only if g 1 (g t-1 (x))[j -p, j + p + 1] = 0v 1 v 0 0, in which case g t (x)[j -p, j + p + 1] = 0v 1 v 1 0 = 0g r 0. But then the right bound of g t (x) is less than j -p, a contradiction. Together these two lemmas yield the following theorem.

Theorem 12. If x ∈ X is a finite configuration, then for every N ∈ N there exists t ∈ N such that g

t (x)[-N, N ] = 0 2N +1 , g t (x)[∞, -(N + 1)] ∈ ∞ 0L and g t (x)[N + 1, ∞] ∈ L r 0 ∞ .

Finitary Ryan's Theorem

In this section we prove a finitary version of Ryan's theorem. The idea is that only very specific automorphisms commute with the glider map g : X → X defined in the previous section, so it will be relatively easy to choose another automorphism f on X such that only powers of the shift map commute with both g and f . We make a simple choice of such f . First we define maps f 1 , f 2 : X → X for a 0-mixing SFT X as follows. In any x ∈ X,

-f 1 replaces every occurrence of 0(v 1 v 1 )v 0 v 0 v 0 (v 1 )0 by 0(v 1 v 1 )v 0 v 0 (v 1 )v 0 0 and vice versa -f 2 replaces every occurrence of 0(v 1 v 1 )v 0 v 0 (v 1 )0 by 0v 0 (v 1 v 1 )v 0 (v 1 )0 and vice versa,
where v 0 and v 1 are as in the previous section. It is easy to see that these maps are well defined automorphisms of X. The automorphism f : X → X is then defined as the composition f 2 • f 1 . The map f has two important properties. First, it replaces any occurrence of 0(v

1 v 1 )v 0 v 0 v 0 (v 1 )0 by 0v 0 (v 1 v 1 )v 0 (v 1 )v 0 0. Second, if
x ∈ X is a configuration containing only gliders g and g r and every occurrence of g is sufficiently far from every occurrence of g r , then f (x) = x.

To prove our main result we need the following lemma.

Lemma 13 ([3], Lemma 7.5). If X is a mixing SFT containing a fixed point 0 Z and h : X → X is an automorphism which is not a power of σ, then there exists a finite configuration

x = 0 Z such that h(x) / ∈ O(x) {σ i (x) | i ∈ Z}.
Theorem 14. Let X ⊆ A Z be a 0-mixing SFT and g, f : X → X as above. The only automorphisms of X which commute with both g and f are powers of σ.

Proof. Assume to the contrary that h : X → X is a radius-r automorphism whose inverse is also a radius-r automorphism and which commutes with g and f but is not a power of σ. Let us first show that h(0 Z ) = 0 Z . Namely, if it were that h(0 Z ) = a Z , for some a ∈ A \ {0}, consider x = . . . 000g 000 . . . with the glider g at the origin and note that h(x)[i] = a for some -r ≤ i ≤ (2p -1) + r (recall:

g = v 0 v 1 , |v 0 | = |v 1 | = p) and h(x)[-∞, i -jp] = . . . aaa for some j ∈ N + . Then g t (h(x))[-∞, i -jp] = . . . aaa for every t ∈ Z but h(g j (x))[i -jp] = h(σ jp (x))[i -jp] = h(x)[i]
= a, contradicting the commutativity of h and g. Thus h maps finite configurations to finite configurations. We have h(GF ) ⊆ GF . To see this, assume to the contrary that there exists x ∈ GF such that h(x) / ∈ GF . Recall that g is reversible and g(GF ) = GF , so g t (h(x)) / ∈ GF for all t ∈ N. Combining this with Theorem 12 it follows that g t (h(x)) contains an occurrence of 0v 1 v 1 0 to the right of coordinate r for all sufficiently large t and therefore h -1 (g t (h(x)))[i] is non-zero for some i ≥ 0 which depends on t. This contradicts the fact that h -1 (g t (h(x)))[i] = g t (x)[i] = 0 for all i ≥ 0 given that t is sufficiently big. Similarly h(GF r ) ⊆ GF r .

For any finite x = 0 Z define its left and right offsets off (x) = min{supp(h(x))} -min{supp(x)}, off r (x) = max{supp(h(x))} -max{supp(x)}.

For all nonzero x ∈ GF and x r ∈ GF r we have off (x )-off r (x r ) = 0. If this did not hold, we could assume without loss of generality that off (x ) -off r (x r ) > 0 (by replacing h with h -1 if necessary) and that min{supp(x )} = (r + 2)p, max{supp(x r )} = -(r + 1)p -1 (by shifting x and x r suitably). Then consider x = x r ⊗x and note that from min{supp(x )} = (r +2)p > r, max{supp(x r )} = -(r+1)p-1 < -r it follows that h(x) = h(x r )⊗h(x ). Then g r (x)[-3p-1, 3p] = 0(v 1 v 1 )v 0 v 0 v 0 (v 1 )0 and g r (h(x)) contains no occurrence of the words mentioned in the definition of f 1 and f 2 by the assumption off (x ) -off r (x r ) > 0, so f (g r (x)) = g r (x) and f (g r (h(x))) = g r (h(x)). Now

x = g -r (f (g r (x))) =h -1 (g -r (f (g r (h(x)))))
=h -1 (g -r (g r (h(x)))) = x, a contradiction. It also follows that there is a fixed s ∈ Z such that off (x ) = off r (x r ) = s for all nonzero x ∈ GF , x r ∈ GF r . If x ∈ GF and x r ∈ GF r are configurations containing exactly one occurrence of g and g r respectively, then h(x ) = σ -s (x ) and h(x r ) = σ -s (x ). To see this, assume to the contrary (without loss of generality) that min{supp(x )} = (r+2)p (i.e. the occurrence of g in x is at (r+1)p), max{supp(x r )} = -(r+1)p-1 and h(x )[(r+1)p+s, (r+3)p-1+s] = h(x )[(r+1)p+s+k, (r+3)p-1+s+k] = g for some k > 2p (i.e. h(x ) contains at least two occurrences of g , the case in which h(x r ) contains at least two occurrences of g r being similar). Then consider x = x r ⊗ x and note that

g r (x)[-3p -1, 3p] = 0(v 1 v 1 )v 0 v 0 v 0 (v 1 )0 f (g r (x))[-3p -1, 3p] = 0v 0 (v 1 v 1 )v 0 (v 1 )v 0 0 g -1 (f (g r (x)))[-3p -1, 3p] = 0(v 1 v 1 )v 0 v 0 v 0 (v 1 )0 g -(r+1) (f (g r (x))) = x, therefore also h(x) = g -(r+1) (f (g r (h(x)))) y. On the other hand, f (g r (h(x)))[p + s + k, 3p -1 + s + k] =g r (h(x))[p + s + k, 3p -1 + s + k] = g g -(r+1) (f (g r (h(x))))[(r + 2)p + s + k, (r + 4)p -1 + s + k] = g , so in particular y[(r + 2)p + s + k, (r + 3)p -1 + s + k] = v 0 . Because we assumed that h(x)[(r + 2)p + s + k, (r + 3)p -1 + s + k] =h(x )[(r + 2)p + s + k, (r + 3)p -1 + s + k] = v 1 ,
it follows that h(x) = y, a contradiction.

By Lemma 13 there exists a finite configuration x = 0 Z such that h(x) / ∈ O(x) and h(x) is finite. Use Theorem 12 to get t ∈ N such that g t (x)[-r, r] = 0 2r+1 and g t (x) = y ⊗ y r where y ∈ GF has max{supp(y )} < -r and y r ∈ GF r has min{supp(y r )} > r (it is possible that either y or y r is equal to 0 Z ). Then also h(g t (x)) = h(y ) ⊗ h(y r ) / ∈ O(g t (x)), and combining this with off (y ) = off r (y r ) it follows that h(y ) / ∈ O(y ) or h(y r ) / ∈ O(y r ). Without loss of generality assume that h(y ) / ∈ O(y ) (the case h(y r ) / ∈ O(y r ) is similar), that y contains a minimal number of occurrences of g (at least two by the previous paragraph) and that the distance from the leftmost g to the second-to-leftmost g in y is maximal (at most 2r + 2p since otherwise by dropping the leftmost g we would get a new configuration y such that h(y ) / ∈ O(y ), contradicting the minimal number of occurrences of g in y ). Let x r ∈ GF r contain exactly one occurrence of g r and assume that min{supp(y )} = (r + 2)p, max{supp(x r )} = -(r + 1)p -1.

Decompose y = x ⊗ x so that x contains only the leftmost g from y and x contains all the other occurrences of g from y . In a similar way as in the previous paragraph we see that g -(r+1) (f (g r ))(x r ⊗ y ) = x r ⊗ (x ⊗ (σ -p (x ))).

Denote h = (g -(r+1) •f •g r ) -1 . By the maximality of distance from the leftmost g to the second-to-leftmost g in y we know that h(x ⊗ (σ -p (x ))) ∈ O(x ⊗ (σ -p (x ))), and this is possible only if h(x ⊗ (σ -p (x ))) = σ -s (x ⊗ (σ -p (x ))), h(x r ⊗ x ⊗ (σ -p (x ))) = σ -s (x r ⊗ x ⊗ (σ -p (x ))), h(h (x r ⊗ x ⊗ (σ -p (x )))) = σ -s (h (x r ⊗ x ⊗ (σ -p (x )))), h(x r ⊗ y ) = σ -s (x r ⊗ y ), so in particular h(y ) ∈ O(y ), a contradiction.

Corollary 15 (Finitary Ryan's theorem). k(X) = 2 for every mixing SFT X with a fixed point.

Proof. The fact that k(X) ≥ 2 follows from the previous theorem and Lemma 7. To see that k(X) = 2, assume to the contrary that k(X) < 2. From k(X) = 0 it would follow that Aut(X) contains only powers of the shift, which is evidently false. Assume then that k(X) = 1 and that h is a single automorphism which commutes with h ∈ Aut(X) only if h is a power of the shift. Because h commutes with itself, it follows that h = σ i for some i ∈ Z. But all h ∈ Aut(X) commute with σ i and so Aut(X) contains again only powers of the shift, a contradiction.

Conclusions

We have constructed glider automorphisms g for mixing SFTs X which have a fixed point, and we have applied these glider maps to prove for such X that k(X) = 2. It seems that our construction of g should generalize to arbitrary mixing SFTs X which do not necessarily have any fixed points. In this case instead of a fixed point 0 Z we need to fix some periodic configuration p ∈ X (i.e. σ k (p) = p for some k ∈ N + ) and we consider points x ∈ X which are finite (in some sense) with respect to p instead of 0 Z . In light of this it is probable that k(X) = 2 for all mixing SFTs X.

Acknowledgments. The author thanks Ville Salo for helpful discussions concerning these topics.

Fig. 1 .

 1 Fig.1. The diffusion of x ∈ X under the map g : X → X. White and black squares correspond to digits 0 and 1 respectively.

a b 0 Fig. 2 .

 02 Fig. 2. The golden mean shift.

Fig. 3 .

 3 Fig. 3. Splitting the state s.

In other words we have performed an elementary state splitting of G at state s. State splitting is a well-known method to produce conjugate subshifts, see e.g. Chapter 2.4 of[START_REF] Lind | An introduction to symbolic dynamics and coding[END_REF].

The work was partially supported by the Academy of Finland grant 296018 and by the Vilho, Yrjö and Kalle Väisälä Foundation proof of Lemma 10 and Lemma 11.


L Khitri-Kazi-Tani

H Dib
email: hdib@mail.univ-tlemcen.dz

Hölderian convergence of fractional extended nabla operator to fractional derivative

Keywords: fractional power, fractional derivatives, Hölderian convergence, fractional differential equation, MSC (2010). 26A33, 47B38, 46E15, 39A70, 97N50, 47A60

In this paper, we construct the fractional extended nabla operator as fractional power of linear spline of backward difference operator. Then we prove the strong convergence of this operator to fractional derivative in a Hölder space setting. Finally numerical examples are presented.

Introduction

It is well-known that fractional calculus is a developing field both from the theoretical and applied point of view. The fractional differential equations turned out to be the best tool for modeling memory-dependent processes [START_REF] Das | Functional fractional calculus[END_REF]. We refer to the monograph [START_REF] Zhou | Basic Theory of Fractional Differential Equations[END_REF], which contains almost complete qualitative fractional differential equation theory, and to the monograph [START_REF] Diethelm | The Analysis of Fractional Differential Equations,An Application-Oriented Exposition Using Differential Operators of Caputo Type[END_REF] for an application oriented exposition.

Besides this rapid development, the notion of difference operators has been extended to fractional calculus in different ways [START_REF] Hilfer | Applications of Fractional Calculus in Physics[END_REF], [START_REF] Ortigueira | Fractional Calculus for Scientists and Engineers[END_REF], [START_REF] Samko | Fractional integrals and derivatives, Theory and Applications[END_REF]. The discrete calculus provides a natural setting to define such operators. However, in literature there is no single definition of fractional difference operators and this situation can be confusing (see for example [START_REF] Abdeljawad | On Riemann and Caputo fractional differences[END_REF], [START_REF] Abdeljawad | On the Definitions of Nabla Fractional Operators[END_REF] and [START_REF] Mozyrska | Overview of Fractional h-difference Operators[END_REF]).

Another way to define this operators is to consider the fractional power of positive discrete operators see [START_REF] Ashyralyev | A note on fractional derivatives and fractional powers of operators[END_REF].

Effectively, functional calculus is a consistent way to define operators of the form A α for a given linear operator A in a Banach space. The fundamental aspects of the theory of fractional powers of non-negative operators are given in [START_REF] Martinez | The Theory of Fractional Powers of Operators[END_REF]. Sectorial operators satisfy a resolvent condition that leads to define the fractional power of such operators. The functional calculus for sectorial operators has been developed by M. Haase in the book [START_REF] Haase | The Functional Calculus for Sectorial Operators, Operator Theory: Advances and Applications[END_REF].

Apart from [START_REF] Ashyralyev | A note on fractional derivatives and fractional powers of operators[END_REF], we do not know about any other work done on fractional difference derivative in terms of spectral operator theory.

In this paper, we define the fractional difference as fractional power of the nabla operator in a Hölder space. The Hölder spaces offer an interesting point of view in the analysis of fractional integrals and derivatives. This framework 1 was developed by Samko et al. for fractional operators in the sense of Marchaud [START_REF] Samko | Fractional integrals and derivatives, Theory and Applications[END_REF], [START_REF] Samko | Fractional type operators in weighted generalized Hölder spaces[END_REF]. In this functional framework we study the strong convergence of the extended backward differences to the derivative. We construct the fractional operators associated and the strong convergence result is proved. Lastly, some examples are provided to show the effectiveness of the approach.

This paper is organized as follows: The section 2 is devoted to preliminaries and some Hölderians tools. Then, the operators in this context are defined. In section 3, we give the basic definitions and results for the fractional power of sectorial operator and we construct the different fractional operators as fractional power of sectorial operator in Hölder spaces setting. In section 4, we discuss the strong convergence of the operators involved. Some examples are given in section 5.

Preliminaries on operators in Hölder spaces

Without loss of generality, we assume that the functions are defined on the interval [0, 1]. Let H β be the Banach space of Hölderian function on [0, 1] with exponent β, where 0 < β < 1 and such that f (0) = 0, endowed with the norm

f β = ω β (f, 1)
, where

ω β (f, δ) = sup s,t∈[0,1] 0<|s-t|≤δ |f (t) -f (s)| |t -s| β Let H β
0 be the subspace defined by

H β 0 = f ∈ H β , lim δ→0 ω β (f, δ) = 0 Remark 1 If f ∈ H β 0 then |f (t) -f (t -h)| = o(h β ) uniformly in t, for t = h we get |f (h)| = o(h β ). Remark 2 If f ∈ H β then f ∞ ≤ f β . Indeed, for every x ∈ ]0, 1] , |f (x)| = |f (x) -f (0)| x β x β ≤ f β . Remark 3 If f ∈ H β then ω β (f, h) tends to 0 as h tends to 0. Indeed, for every x , y ∈ [0, 1], x = y there exists some ξ ∈ ]x, y[ such that |f (x) -f (y)| |x -y| β = |x -y| 1-β |f (ξ)| ≤ |x -y| 1-β f β which leads to ω β (f, h) ≤ h 1-β f β (1)
The Hölder norm of the piecewise linear interpolation is given by the next lemma. As far as we know, this result was first proved by H. E. White, Jr, in a general setting see [23, 3.2 Corollary p 106] but we follow [START_REF] Račkauskas | Functional Laws of Large Numbers in Hölder Spaces, ALEA[END_REF] in the presentation. Lemma 2.1 (see lemma 3.1 [START_REF] Račkauskas | Functional Laws of Large Numbers in Hölder Spaces, ALEA[END_REF]) Let t 0 = 0 < t 1 < • • • < t n = 1 be a partition of [0, 1] and f be a real valued polygonal line function on [0, 1] with vertices at t i 's, i.e. f is continuous on [0, 1] and its restriction to each interval [t i , t i+1 ] is an affine function. Then for any 0

≤ β < 1, sup 0≤s<t≤1 |f (t) -f (s)| (t -s) β = max 0≤i<j≤1 |f (t j ) -f (t i )| (t j -t i ) β Definition 1 For 0 < h < 1 fixed, let ∆ h be the subdivision of [0, 1] in n subintervals with n = [1/h]
and t k = kh, for each k = 0, 1, . . . , n where [a] means the integer part of a. We denote by I h ∈ L(H β ) the piecewise linear interpolation operator defined by

(I h f ) (x) := n k=1 x -t k-1 h f (t k ) + t k -x h f (t k-1 ) 1 [t k-1 ,t k ] (x)
In the following lemma the remainder of piecewise linear interpolation is expressed in Hölder norm.

Lemma 2.2 Let (r h f )(x) = (I -I h ) f (x) then (r h f ) β ≤ 4ω β (f, h) Proof First let us suppose that x, y ∈ ]t k-1 , t k ] then (r h f )(x) -(r h f )(y) = f (x) -f (y) - x -y h (f (t k ) -f (t k-1
))

It follows, from |x -y| < h that

|(r h f )(x) -(r h f )(y)| |x -y| β ≤ |f (x) -f (y)| |x -y| β + |f (t k ) -f (t k-1 )| h β ≤ 2ω β (f, h)
Second, suppose that x ∈ ]t k-1 , t k ] and y ∈ ]t k , t k+1 ] then from the first case

|(r h f )(x) -(r h f )(y)| |x -y| β ≤ |(r h f )(x) -(r h f )(t k )| |x -t k | β + |(r h f )(t k ) -(r h f )(y)| |t k -y| β ≤ 4ω β (f, h) Third, suppose that x ∈ ]t k-1 , t k ] and y ∈ ]t m-1 , t m ] with |x -y| > h then |(r h f )(x) -(r h f )(y)| |x -y| β ≤ |(r h f )(x) -(r h f )(t k )| |x -y| β + |(r h f )(t k ) -(r h f )(t m-1 )| |x -y| β + |(r h f )(t m-1 ) -(r h f )(y)| |x -y| β Knowing that (r h f )(t k ) = (r h f )(t m-1 ) = 0 then |(r h f )(x) -(r h f )(y)| |x -y| β ≤ 4ω β (f, h)
We denote by A = d dx the differential operator acting on H β with domain,

D(A) = f ∈ H β , f ∈ H β ,
and for 0 < h < 1, let ∇ h ∈ L(H β ) the nabla operator defined by

(∇ h f ) (x) := f (x) -f (x -h) h , for x ∈ [h, 1] We set (∇ h f ) (x) = f (x)
h for all 0 < x < h. Lastly, we introduce the extended nabla operator as the polygonal line with vertices ∇f (t k ), k = 0, 1, . . . , n, in the following definition.

Definition 2 We define the operator

A h for any f ∈ H β by A h f (x) = (I h ∇ h f ) (x)
Obviously A h is a linear bounded operator with A h β ≤ 2 h .

In the next proposition, It can be pointed out that the sequence (A h ) h has no uniform limit as h tends to 0.

Proposition 2.3 The sequence (A

h ) h of extended nabla operator is not a con- vergent sequence in L(H β ) as h tends to 0. Proof We need only to proof that (A h ) h is not a Cauchy sequence. Let ∆ h , ∆ h/2 be two subdivisions of [0, 1] and f (x) = x β . We then have for t = h/2 A h -A h/2 f (t) = h/2 h f (h) h - f (h/2) h/2 = h β-1 2 - h 2 β-1 ≥ 1 2 - 1 2 β-1 Since (A h -A 2h ) f β ≥ |(A h -A 2h ) f (h/2)| then A h -A 2h β f β ≥ (A h -A 2h ) f β ≥ 1 2 β-1 - 1 2
We have thus seen that (A h ) h is not a convergent sequence in L(H β ).

In the next proposition the strong convergence of extended nabla operator to the derivative operator is proved.

Proposition 2.4 For every f ∈ D(A), such that f ∈ H β 0 the sequence (A h ) h converges strongly to A as h tends to 0. Proof Note that (A -A h ) f (x) = (I -I h ) Af (x)+I h (A -∇ h ) f (x) = (r h f ) (x)+I h (A -∇ h ) f (x). Then |(A -A h ) f (x) -(A -A h ) f (y)| |x -y| β ≤ (r h f ) β + |I h (A -∇ h ) f (x) -I h (A -∇ h ) f (y)| |x -y| β From lemma 2.2 (r h f ) β ≤ 4ω β (f , h) For some i, j, ξ i ∈ ]t i-1 , t i [ and ξ j ∈ ]t j-1 , t j [ by lemma 2.1 we have |I h (A -∇ h ) f (x) -I h (A -∇ h ) f (y)| |x -y| β ≤ |f (t i ) -∇ h f (t i )| |t i -t j | β + |f (t j ) -∇ h f (t j )| |t i -t j | β ≤ |f (t i ) -f (ξ i )| |t i -t j | β + |f (t j ) -f (ξ j )| |t i -t j | β , ≤ 2ω β (f , h) Hence |(A -A h ) f (x) -(A -A h ) f (y)| |x -y| β ≤ 6ω β (f , h) (2)
Therefore

lim h→0 Af -A h f β = 0.
3 Fractional power of sectorial operator

B ∈ Sect(ω)) if: 1) σ(B) ⊂ S ω and 2) M (B, ω ) := sup λR(λ, B) , λ / ∈ S ω < ∞ for all ω < ω < π. A family of operators (B ι ) ι is uniformly sectorial of angle ω if B ι ∈ Sect(ω) for each ι and sup ι M (B ι , ω ) < ∞ for all ω < ω < π. Definition 4 A uniformly sectorial sequence (B n ) n∈N of angle ω is called a sectorial approximation on S ω for the operator B, if the sectorial convergence B n → B (S ω ) holds i.e., λ ∈ ρ(B) and R(λ, B n ) → R(λ, B) in L(Z) (3)
for some λ / ∈ S ω .

Note that if (3) is true for some λ / ∈ S ω then (3) is true for all λ / ∈ S ω . Moreover, B ∈ Sect(ω). We are now able to define the fractional power of a sectorial operator with help of the Balakrishnan representation (see [10, Proposition 3.1.12]) Proposition 3.1 Let B an operator with domain D(B), B ∈ Sect(ω), and let 0 < α < 1. Then for all f ∈ D(B)

B α f (x) = - sin απ π ∞ 0 λ α-1 R(-λ, B)Bf (x)dt (4)

Fractional power of the derivative

We define the fractional derivative as fractional power of sectorial operator in Hölder space. To do so we examine sectoriality of A.

Proposition 3.2 The operator A on H β is sectorial of angle π 2 .
Proof For all λ ∈ C,the resolvent of the operator A on H β is given by

R(λ, A)f (x) = - x 0 e λ(x-t) f (t)dt
Let x, h be such that 0 ≤ x -h < x ≤ 1. For some θ 1 ∈ (0, x -h) and θ 2 ∈ (x -h, x) we have, after performing a change of variable and integrating,

|R(λ, A)f (x) -R(λ, A)f (x -h)| ≤ sup t∈[0,x-h] |f (t) -f (t -h)| e λx -1 |λ| + sup t∈[0,h] |f (t)| e λx -e λ(x-h) |λ| ≤ ω β (f, h)h β e Re λx + 1 |λ| + e Re λx + e Re λ(x-h) |λ| If Re λ < 0, |R(λ, A)f (x) -R(λ, A)f (x -h)| ≤ 4h β |λ| ω β (f, h) which implies ω β (R(λ, A)f, h) ≤ 4 |λ| ω β (f, h) (5)
and

R(λ, A) β ≤ 4 |λ| f β (6)
Consequently, for every

λ ∈ C\S ω , π 2 < ω ≤ π, λR(λ, A) ≤ 4 Corollary 3.3 Let 0 < α < 1 and f ∈ D(A). Then A α f (x) = 1 Γ(1 -α) x 0 (x -t) -α f (t)dt
Proof Using the Balakrishnan representation of fractional power of sectorial operator (4), the previous representation follows.

Remark 5

The sectorial property of If A α was established by Ito [START_REF] Ito | On the Sectorial Property of the Caputo Derivative Operator[END_REF], in a slightly different way for 1 < α < 2.

In the next subsection the fractional power of operator ∇ h and A h are constructed.

Fractional nabla operators

Before studying the sectoriality of ∇ h and A h we begin by a surprising and useful result. In fact, elementary calculations show that the operator I h commutes with ∇ h . This property has an interesting consequence for the resolvent operator given in the next lemma and stated in general framework.

Lemma 3.4 Let X be a Banach space, B, T ∈ L(X ) such that T is idempotent and T commute with B then, for every λ ∈ ρ(B), λ = 0

R(λ, T B) = T R(λ, B) + 1 λ (I -T )
Proof To obtain the resolvent operator for T B we consider the equation, for f, g ∈ X , f = (λI -T B) g then by idempotence of the operator T and commutative property we get

T f = (λI -B) T g
combining the above two equations we have

f -T f = λ (I -T ) g using the fact that T g = (λI -B) -1 T f then g = R(λ, B)T f + 1 λ (f -T f ) Proposition 3.5 The family (∇ h ) h is uniformly sectorial of angle π 2 on H β .
Proof It can be easily proved using Laplace and inverse Laplace transforms that

R(λ, ∇ h )f (x) = -h n j=0 1 (1 -λh) j+1 f (x -t j ) First, we check the boundedness of R(λ, ∇ h ) in L H β . For every 0 ≤ x < y ≤ 1, |R(λ, ∇ h )f (x) -R(λ, ∇ h )f (y)| |x -y| β ≤ 1 |x -y| β    -h [x/h] j=0 1 (1 -λh) j+1 [f (x -t j ) -f (y -t j )] + -h [y/h] [x/h]+1 1 (1 -λh) j+1 f (x -t j )    ≤ f β n j=1 h |1 -λh| j
Using the sum of a geometric series we have

|R(λ, ∇ h )f (x) -R(λ, ∇ h )f (y)| |x -y| β ≤ h |1 -λh| -1 f β (7)
Now,observe that for any λ ∈ C\S ω ,

π 2 < ω < π we have |λh -1| > 1 and |λ| |R(λ, ∇ h )f (x) -R(λ, ∇ h )f (y)| |x -y| β ≤ |λ| h |1 -λh| -1 f β Knowing that |1 -λh| 2 = h 2 |λ| 2 + 1 -2 |λ| cos (arg λ) ≥ h 2 |λ| 2 + 1 -2h |λ| cos ω Then |λ| h |1 -λh| -1 ≤ |λ| h h 2 |λ| 2 + 1 -2h |λ| cos ω -1 Put ϕ(z) = z √ z 2 + 1 -2z cos ω -1 for z > 0.It is easy to see that ϕ(+∞) = 1, ϕ(0 + ) = -1
cos ω and the derivative satisfies

ϕ (z) = -1 + cos 2 ω z 2 √ z 2 + 1 -2z cos ω -1 2 √ z 2 + 1 -2z cos ω 1 -z cos ω + √ z 2 + 1 -2z cos ω < 0 Consequently, for every z ∈ ]0, +∞[ 1 < ϕ(z) ≤ -1 cos ω , which implies that |λ| |R(λ, ∇ h )f (x) -R(λ, ∇ h )f (y)| |x -y| β ≤ -1 cos ω f β (8)
We conclude that the family (∇ h ) h is uniformly sectorial of angle π 2 .

Consequently the extended nabla operator A h is also sectorial as shown in the next corollary.

Corollary 3.6 The family (A

h ) h is uniformly sectorial of angle π 2 on H β .
Proof From lemma 3.4 we have

R(λ, A h )f (x) = I h (R(λ, ∇ h )f ) (x) + 1 λ (f -I h f ) (x) Then |λ| |R(λ, A h )f (x) -R(λ, A h )f (y)| |x -y| β ≤ |λ| |I h R(λ, A h )f (x) -I h R(λ, A h )f (y)| |x -y| β + |(I -I h ) f (x) -(I -I h ) f (y)| |x -y| β from lemmas 2.1 and 2.2 , there exist 0 ≤ m, l ≤ n such that |λ| |R(λ, A h )f (x) -R(λ, A h )f (y)| |x -y| β ≤ |λ| |R(λ, A h )f (t m ) -R(λ, A h )f (t l )| |t m -t l | β +4ω β (f, h)
From proposition 3.5 we get the estimate

|λ| |R(λ, A h )f (x) -R(λ, A h )f (y)| |x -y| β ≤ -1 cos ω + 4 f β
As a result we are able to define the fractional power of ∇ h and A h . This is the purpose of the following theorem. Theorem 3.7 Let 0 < α < 1, then fractional nabla operator is

∇ α h f (x) = h 1-α Γ(1 -α) [x/h] i=1 Γ(j + 1 -α) Γ(j + 1) ∇ h f (x -t j )
and the fractional operator

A α h is A α h f (x) = n k=0 x -t k-1 h k i=1 Γ(j + 1 -α) Γ(j + 1) ∇ h f (t k -t j ) + t k -x h k-1 i=1 Γ(j + 1 -α) Γ(j + 1) ∇ h f (t k-1 -t j ) 1 [t k-1 ,t k ] (x)
We call A α h the fractional extended nabla operator.

Proof Using Balakrishnan representation of fractional power of sectorial operator (4) , we get when 0 < α < 1

∇ α h f (x) = sin απ π +∞ 0 λ α-1 (λ + ∇ h ) -1 ∇ h f (x)dλ = - sin απ π +∞ 0 λ α-1 R (-λ, ∇ h ) ∇ h f (x)dλ Then ∇ α h f (x) = sin απ π +∞ 0 λ α-1 h [x/h] j=0 1 (1 + λh) j+1 ∇ h f (x -t j )dλ = h [x/h] j=0 sin απ π +∞ 0 λ α-1 1 (1 + λh) j+1 dλ ∇ h f (x -t j )
Similar calculations to those in [START_REF] Ashyralyev | A note on fractional derivatives and fractional powers of operators[END_REF]Theorem 3.1] give

+∞ 0 λ α-1 1 (1 + λh) j+1 dλ = h -α Γ(j + 1 -α)Γ(α) Γ(j + 1) Therfore ∇ α h f (x) = h 1-α Γ(1 -α) [x/h] j=0 Γ(j + 1 -α) Γ(j + 1) ∇ h f (x -t j )
We now turn to the evaluation of A α h f . From lemma 3.4 we get

A α h f (x) = - sin απ π +∞ 0 λ α-1 R (-λ, A h ) A h f (x)dλ = - sin απ π +∞ 0 λ α-1 I h R (-λ, ∇ h ) I h ∇ h f (x)dλ = - sin απ π +∞ 0 λ α-1 I h R (-λ, ∇ h ) ∇ h f (x)dλ
the required evaluation of A α h f then follows.

Remark 6

The operator ∇ α h is nothing but the Grünwald-Letnikov operator

∇ α h f (x) = h -α [x/h] j=0 (-1) j α j f (x -jh)
Let us mention that this operator was defined in a formal way as a generalization of difference formulas of integer order by replacing the integer order by a real number.

It is worth to underscore that in [START_REF] Ashyralyev | A note on fractional derivatives and fractional powers of operators[END_REF] discrete Grünwald-Letnikov approximations was called the Riemann-Liouville fractional derivative of order α.

The remaining problem is to study if the strong convergence (A h f ) h to Af can gives rise to the convergence of power operators; this is the aim of the next section.

Hölderian convergence of fractional extended nabla operator to fractional derivative

Before stating the convergence theorem, we define for all f in D(A), the function ϕ by

ϕ(x) := 1 Γ (1 -α) x 0 (x -t) -α A h f (t) dt
For the construction of the convergence result proof we need the following lemmas Lemma 4.1 For all 0 < β < 1 such that 1 -α -β > 0 we have

ω β (ϕ, h) ≤ 8 Γ (2 -α) f β h 1-α-β
Proof Remark that the function ϕ satisfies the following estimation

|ϕ(x) -ϕ(y)| |x -y| β ≤ 1 Γ (2 -α) A h f β (max(x, y)) 1-α ≤ 1 Γ (2 -α) A h f β then ϕ ∈ H β [0, 1] . Let us now estimate ω β (ϕ, h), we distinguish two cases, First t k-1 < x < y ≤ t k : Notice that Γ (1 -α) ϕ(x) = k-1 i=1 ti ti-1 (x -t) -α t -t i-1 h ∇ h f (t i ) + t i -t h ∇ h f (t i-1 ) dt+ x t k-1 (x -t) -α t -t k-1 h ∇ h f (t k ) + t k -t h ∇ h f (t k-1 ) dt Hence Γ (1 -α) (ϕ(x) -ϕ(y)) = k-1 i=1 ti ti-1 (x -t) -α -(y -t) -α t -t i-1 h ∇ h f (t i ) + t i -t h ∇ h f (t i-1 ) dt + x t k-1 (x -t) -α -(y -t) -α t -t k-1 h ∇ h f (t k ) + t k -t h ∇ h f (t k-1 ) dt+ y x (y -t) -α t -t k-1 h ∇ h f (t k ) + t k -t h ∇ h f (t k-1 ) dt and Γ (1 -α) |ϕ(x) -ϕ(y)| ≤ 2 f β k-1 i=1 ti ti-1 (x -t) -α -(y -t) -α dt + x t k-1 (x -t) -α -(y -t) -α dt + y x (y -t) -α which leads to |ϕ(x) -ϕ(y)| ≤ 2 Γ (2 -α) f β k-1 i=1 (x -t i-1 ) 1-α -(y -t i-1 ) 1-α -(x -t i ) 1-α + (y -t i ) 1-α + 2 Γ (2 -α) f β (x -t k-1 ) 1-α -(y -t k-1 ) 1-α + 2(y -x) 1-α finally |ϕ(x) -ϕ(y)| |x -y| β ≤ 4 Γ (2 -α) f β |y -x| 1-α-β ≤ 4 Γ (2 -α) f β h 1-α-β Second t k-1 < x ≤ t k < y ≤ t k+1 : |ϕ(x) -ϕ(y)| |x -y| β ≤ |ϕ(x) -ϕ(t k )| |x -t k | β + |ϕ(t k ) -ϕ(y)| |t k -y| β ≤ 8 Γ (2 -α) f β h 1-α-β
Therefore

ω β (ϕ, h) ≤ 8 Γ (2 -α) f β h 1-α-β
Lemma 4.2 There exists C > 0 such that for every 0

≤ k ≤ n |ϕ(t k ) -∇ α h f (t k )| h β ≤ 1 + C Γ (1 -α) f β h 1-α-β + 2 β Γ (2 -α) ω β (f , 2h)
Proof Obviously if k = 0 the lemma holds for every C > 0. Assume now that k > 0, then

ϕ(t k ) = 1 Γ (1 -α) k i=1 tj tj-1 (t k -t) -α t -t j-1 h ∇ h f (t j ) + t j -t h ∇ h f (t j-1 ) dt A simple integration leads to ϕ(t k ) = 1 Γ (2 -α) k i=1 (t k -t j-1 ) 1-α ∇ h f (t j-1 ) -(t k -t j ) 1-α ∇ h f (t j ) + 1 Γ (2 -α) k i=1 (t k -t j-1 ) 2-α -(t k -t j ) 2-α (2 -α)h [∇ h f (t j ) -∇ h f (t j-1 )]
which can be arranged as follows

ϕ(t k ) = 1 Γ (2 -α) k-1 i=1 (t k -t j ) 1-α -(t k -t j+1 ) 1-α ∇ h f (t j ) + 1 Γ (2 -α) k i=1 (t k -t j-1 ) 2-α -(t k -t j ) 2-α (2 -α)h -(t k -t j ) 1-α (∇ h f (t j ) -∇ h f (t j-1 ))
Then

ϕ(t k ) -∇ α h f (t k ) = S 1 + S 2 -h 1-α ∇ h f (t k ) where S 1 = h 1-α Γ (1 -α) k-1 i=1 j 1-α -(j -1) 1-α 1 -α - Γ(j + 1 -α) Γ(j + 1) ∇ h f (t k -t j )
and

S 2 = 1 Γ (2 -α) k i=1 (t k -t j-1 ) 2-α -(t k -t j ) 2-α (2 -α)h -(t k -t j ) 1-α (∇ h f (t j ) -∇ h f (t j-1 ))
By using the fact that

0 ≤ (t k -t j-1 ) 2-α -(t k -t j ) 2-α (2 -α)h -(t k -t j ) 1-α = 1 h tj tj-1 (t k -t) 1-α -(t k -t j ) 1-α dt ≤ (t k -t j-1 ) 1-α -(t k -t j ) 1-α and |∇ h f (t j ) -∇ h f (t j-1 )| ≤ (2h) β ω β (f , 2h) |S 2 | can be estimated by |S 2 | ≤ (2h) β ω β (f , 2h) Γ (2 -α)
It remain to estimate |S 1 |. To doing so, we use the following expansion (see [14, lemma 2.4] ),

Γ(j + 1 -α) Γ(j + 1) = j -α + 1 -α Γ(α) Φ (j) with |Φ (j)| ≤ Γ(1 + α) 2 j -α-1
Therefore

j 1-α -(j -1) 1-α 1 -α - Γ(j + 1 -α) Γ(j + 1) = tj tj-1 s -α -j -α ds - 1 -α Γ(α) Φ (j)
and for every j ≥ 2

j 1-α -(j -1) 1-α 1 -α - Γ(j + 1 -α) Γ(j + 1) ≤ (j -1) -α -j -α + 1 -α Γ(α) |Φ (j)| This leads to k-1 j=1 j 1-α -(j -1) 1-α 1 -α - Γ(j + 1 -α) Γ(j + 1) ≤ 1 1 -α -Γ (2 -α) + 1 -(k -1) -α + α (1 -α) 2 k-1 j=1 j -α-1 ≤ C with C = 1 1 -α -Γ (2 -α) + 1 + α (1 -α) 2 ζ(1 + α) > 0
where ζ(•) is the Riemann zeta function. Finally

|S 1 | ≤ h 1-α Γ (1 -α) C f β
Now we can put the pieces together to get

|ϕ(t k ) -∇ α h f (t k )| h β ≤ 1 + C Γ (1 -α) f β h 1-α-β + 2 β Γ (2 -α) ω β (f , 2h)
The following theorem shows that the sequence (A α h ) h converges strongly to A α . Theorem 4.3 Let X β be the space X β = f ∈ H β such that f ∈ H β 0 . Then for all β such that 1 -α -β > 0 the sequence (A α h ) h converges strongly to the fractional derivative A α on X β as h tends to 0.

Proof For every 0 ≤ x < y ≤ 1,

(A α -A α h ) (f )(x) -(A α -A α h ) (f )(y) = - sin πα π +∞ 0 λ α-1 (R (-λ, A) (Af ) -R (-λ, A h ) (A h f )) (x) dλ + sin πα π +∞ 0 λ α-1 (R (-λ, A) (Af ) -R (-λ, A h ) (A h f )) (y) dλ
by introducing a mixed term we get

(A α -A α h ) (f )(x) -(A α -A α h ) (f )(y) = - sin πα π +∞ 0 λ α-1 R (-λ, A) ((Af -A h f ) (x) -(Af -A h f ) (y)) dλ + sin πα π +∞ 0 λ α-1 (R (-λ, A) -R (-λ, A h )) (A h f (x) -(A h f ) (y)) dλ
Denote by I 1 (x, y) and I 2 (x, y) respectively the first and the second integral in the equality above.

We begin by the first integral

|I 1 (x, y)| ≤ +∞ 0 λ α-1 |R (-λ, A) ((Af -A h f ) (x) -(Af -A h f ) (y))| dλ ≤ +∞ 0 λ α-1 x 0 e -λt |(Af -A h f ) (y -t) -(Af -A h f ) (x -t)| dtdλ + +∞ 0 λ α-1 y x e -λt |(Af -A h f ) (y -t)| dtdλ
which leads to

I 1 (x, y) |x -y| β ≤ Af -A h f β +∞ 0 λ α-1 y 0 e -λt dt dλ
The estimate

I 1 (x, y) |x -y| β ≤ 6Γ(α) 1 -α ω β (f , h)
follows from Fubini's theorem and inequality 2 . Consider now the second integral First notice that

sin πα π +∞ 0 λ α-1 R (-λ, A) A h f (x)dλ = sin πα π +∞ 0 λ α-1 x 0 e -λ(x-t) A h f (t) dt dλ = sin πα π x 0 +∞ 0 λ α-1 e -λ(x-t) dλ A h f (t) dt = 1 Γ(1 -α) x 0 (x -t) -α A h f (t) dt Then sin πα π +∞ 0 λ α-1 (R (-λ, A) -R (-λ, A h )) A h f (x) dλ = 1 Γ(1 -α) x 0 (x -t) -α A h f (t) dt -I h ∇ α h f (x) = (r h ϕ)(x) + I h (ϕ(x) -∇ α h f (x))
From lemmas 2.2 and 2.1 we have for some k and m sin πα π

|I 2 (x, y)| |x -y| β ≤ 4ω β (ϕ, h) + |ϕ(t k ) -∇ α h f (t k ) -ϕ(t m ) + ∇ α h f (t m )| |t k -t m | β
From lemmas 4.1 and 4.2 we deduce sin πα π

|I 2 (x, y)| |x -y| β ≤ 2 16 Γ (2 -α) + C Γ (1 -α) + 1 f β h 1-α-β + 2 β+1 Γ (2 -α) ω β (f , 2h) Hence (A α -∇ α h ) (f ) β ≤ 2 16 Γ (2 -α) + C Γ (1 -α) + 1 f β h 1-α-β + 2 β+1 + 6 Γ (2 -α) ω β (f , 2h)
and the conclusion of the theorem holds.

Numerical examples

In this section two examples are discussed.

Exemple 1

Consider the fractional derivative of f (x) = x µ ln x. The analytical expression of the fractional derivative of f is

A α f (x) = Γ(µ + 1) Γ(µ + 1 -α) x µ-α [ln x + ψ(µ + 1) -ψ(µ + 1 -α)]
Where ψ(•) denote the digamma function see [START_REF] Valério | Fractional calculus: A survey of useful formulas[END_REF]Formula (103)]. In the next tables error at the step size h is the Hölderian error defined by

max 0≤i<j≤1/h |(A α f -∇ α h f ) (t i ) -(A α f -∇ α h f ) (t j )| |t i -t j | β (9)
According to our theoretical consideration the convergence is ensured by h 1-α-β and ω β (f , 2h). Let us establish an estimation of ω β (f , h) For every 0 ≤ x < y ≤ 1

f (y) -f (x) = µ(y µ-1 ln y -x µ-1 ln x) + y µ-1 -x µ-1
Using the fact that y µ-1 ln y-x µ-1 ln x = Then for every 1 + β < β < µ we have

Exemple 2

For the second example we consider the fractional differential equation presented in [START_REF] Diethelm | Detailed error analysis for fractional Adam method[END_REF] ,for t ∈ [0, 1]. D α y(t) = 40320 Γ(9 -α) t (10) The initial condition is y(0) = 0. The exact solution of this problem is y(t) = t 8 -3t 4+ α 2 + 9 4 t α .

For α = 0.5, we display the results in table 2 for β = 0.1 and β = 0.01 respectively.

Apparently, we need to use small values for β to increase the accuracy.

h Errors for β = 0.1 Errors for β = 0.01 2 -7 0.0347581 0.0224598 2 -8 0.0269360 0.0163528 2 -9 0.0206910 0.0118018 2 -10 0.0158085 0.0084716 2 -11 0.0120388 0.0060613 2 -12 0.0091502 0.0043283 2 -13 0.0069465 0.0030872

Table 2: Hölderian errors for problem [START_REF] Haase | The Functional Calculus for Sectorial Operators, Operator Theory: Advances and Applications[END_REF] with α = 0.5

Conclusion

In this work, we defined a fractional operator as a fractional power of a piecewise linear interpolation of a backward difference on a Hölder space. We proved the strong convergence of this operator to fractional derivative, and we supported our results with examples. We think that we have now a kind of process to define Euler-like formulas which contribute to solve numerically fractional differential equations in Hölder spaces. However, several questions can be the subject of further works, in particular, the analysis of the order of approximation, and the results that can be expected if one replace the linear spline I h by a spline of degree n > 1, or if one replaces the operator ∇ h by another more accurate approximation of the derivative.

3. 1 Definition 3

 13 Sectorial propertyWe first, recall the Haase concept of sectorial operators [10, Section 2.1 ,p19]. In the following R(λ, B) = (λI -B) -1 , ρ(B) and σ(B) = C\ρ(B) denote respectively the resolvent, the resolvent set and the spectrum of a linear operator B on a Banach space Z. Let S ω denote the open sector {z ∈ C, z = 0 and |arg z| < ω} , 0 < ω ≤ π. The operator B is sectorial of angle ω < π (in short:

Remark 4

 4 Sectorial operator in Haase definition don't have to be densely defined see [12, Definition 3.8, p 97].

  -1 ln t)dt = (µ -1) y x t µ-2 ln tdt+ 1 µ -1 y µ-1 -x µ-1

y β - 1 -x β - 1

 11 Therefore(a) A α f in the continuous line, A α h f in dotted line for h = 2 -4 . (b) Holderian error with respect to step size h.

Figure 1 :

 1 Figure 1: Comparison between A α f and A α h f when f (x) = x µ ln x for µ = 3/2, α = 0.3 and β = 0.1.

It follows that

The results concerning errors are presented in 9) when f (x) = x µ ln x for µ = 3/2, α = 0.3 and β = 0.1.

In figure 1, on the left the graphs of A α f and A α h f are shown. On the right we give the Hölderian errors. e-mail: l khitri@mail.univ-tlemcen.dz H. Dib Laboratoire de Statistiques et modélisation stochastique Department of Mathematics, Abou Bekr Belkaid University, Tlemcen, Algeria e-mail: h dib@mail.univ-tlemcen.dz


Rakotoarisoa Anjara

Lalaina Jocelyn

Corruption and economic growth in Madagascar

Keywords: Economic growth, Corruption, Madagascar, Econometric approach Economic growth, Corruption, Madagascar, Economic Approach

published or not. The documents may come

Introduction

Corruption is a universal phenomenon. Nothing is more common, yet nothing is more difficult to grasp than corruption. First, its contours are very variable. "Wherever you have massive corruption, development is lagging behind. Said Akinwimi Adesina, President of the African Development Bank (ADB).

A report issued by the General Accounting Office (GAO) in Washington, in 2016, stated that corruption seriously undermines Africa's long-term economic growth. In addition to this, research by the International Monetary Fund (IMF) has shown that investments made in corrupt countries are 5% lower than those made in relatively uncorrupted countries.

Corruption has become a concern. This is one reason why a diagnosis of the relationship between corruption and economic growth is needed. According to the World Bank, corruption can imply a decline in the rate of economic growth of 0.5 to 1 percentage point per year. A study conducted by Alberto Ades and Rafael Di Tella in 1995 also concluded that if per capita income rises by $ 4400, the ranking of a country concerned has its ranking, in terms of corruption perception index (CPI), improved by two points. Recall that the CPI is a standard index between 0 and 100. A zero CPI indicates that the country is totally corrupt and a CPI equal to 100 reflects that the country is entirely non-corrupt.

Madagascar is one of the most corrupt countries in particular. Its CPI score is 24/100, ranking it 155th out of 180 countries according to Transparency International. The generalization of corruption in Madagascar seems to penalize its economy. The situation discourages investment and public development aid.

It is to this end that we ask ourselves the question: how is the link between corruption and growth in Madagascar? And to provide an answer to this question, we used an econometric approach. This is the purpose of this document.

Analytical framework

Definitions and concepts of "corruption" Alesina and Weder (2002) define corruption as the misuse of state property by an official for personal gain. This definition is close to that of Langseth (1997) defines corruption as an abuse of power given to obtain a personal benefit.

For Celentani and Ganuza (2002) corruption is in the form of an exchange, a favor or a facility in the public service. Corruption is thus linked to the administrative activities of the state. It is "An act [...] results from an abuse of public power to obtain a private benefit. As [START_REF] Tanzi | Corruption around the world[END_REF] indicates.

Empirical studies of the link between corruption and growth

Several studies have been done on the link between corruption and growth. Corruption affects growth either directly or indirectly, either on human capital, on public spending, or on investment.

Lui (1974), Klitgaard (1988) and Rose-Ackerman (1989) reported the early work on this topic. But the first empirical evaluation was maintained by [START_REF] Mauro | Corruption and Growth[END_REF]. The harmful nature of corruption on investment and growth has been highlighted by these studies. Tanzi and Davvodi (2000) did a study and these authors found that a negative relationship also exists. This idea follows the order of study of Mookherjee (1997) who considers that corruption is beneficial to the economy since it is the basis of incentive mechanism to work.

Corruption-growth relationship in Madagascar

Contexts of corruption and growth in Madagascar

Despite statements about the fight against corruption, Madagascar has declined in perception of corruption. If in 2016 its CPI was 26/100, the year 2017 recorded a CPI of 24/100, a decrease of two points. The most affected sectors are the Decentralized Territorial Communities, the Gendarmerie as well as the National Police and Justice, the Land Service. The causes are attributed to poverty, abuse of power and thirst for money.

On the growth side, Madagascar achieves, on average, an economic growth rate of 2 to 3%. Extractive industry and tourism are the most promising sectors. Agriculture remains underdeveloped despite the fact that almost 85% of the population works in this sector.

Corruption versus growth adjustment in Madagascar

Our study uses the least squares method for diagnosis. It is an adjustment between Madagascar's CPI and its economic growth rate by drawing an equation of form:

The The positive correlation between the CPI and the growth rate reflects a positive relationship between these variables. The lower the level of corruption, the better the growth. Despite the fact that this relationship is statistically not significant enough, the influence of the perception of corruption on growth is not negligible.

Conclusion

Corruption is generally perceived as a bad thing. This is obvious according to the result of our study. This reflects a parallel link between the perception of corruption and economic growth in the case of Madagascar. Despite the fact that a relationship may exist as Mookherjee (1997) suggests, Mauro's perception is more valid in the situation in Madagascar. Given this state of affairs, the authorities should intensify the fight against corruption to allow economic growth to flourish.

  data provided by Transparency International and the Ministry of Economy and Planning are the basis for estimating parameters A and B of the equation, which results in the following


Amelia Carolina Sparavigna

Digital restoration of a marble head of Julius Caesar from Noviomagus (Nijmegen)

Keywords: Face recognition, 3D rendering, History of art, Caesar's portraitures, Tusculum bust

A marble head, today at the National Museum of Antiquities (Rijksmuseum van Oudheden) in Leiden, was probably found in the fort of Hunerberg in Noviomagus, the Roman Nijmegen. This marble head, which is considered as a portraiture of Julius Caesar, is heavily damaged. Here we propose a digital restoration of the face of this head. For

Turin [7,8]. "D'Hollosy took 3D scans of both busts and combined the details into the Leiden 3D scan. She then removed the top layer of the bust and used clay and silicone to achieve a more lifelike appearance". In [START_REF] Sparavigna | On Maja d'Hollosy reconstruction of Caesar's head[END_REF][START_REF] Sparavigna | On d'Hollosy reconstruction of Caesar -continued[END_REF], we compared the two busts and the reconstruction, giving some measurements. Actually, the d'Hollosy reconstruction is not the average of the two marble busts.

The Rijksmuseum van Oudheden of Leiden possesses two marble busts which can be considered as portraitures of Julius Caesar (Figure 1). We discussed one of them in [START_REF] Sparavigna | On a possible reconstruction of the face of Julius Caesar using a Leiden marble head[END_REF] (that on the right of the Figure 1). Here we consider the bust that had been used by d'Hollosy, and make on it a digital reconstruction using the face of Tusculum bust (Figure 1, on the left). So, let us consider the head in the Figure 1, on the left, and restore it digitally using the Tusculum head. First, let us adjust it with Paint to add nose and lips. The result is given in the Figure 2. After, let us consider the Tusculum and the Leiden heads and make some measurements to see if the Tusculum face could fit into the Leiden head. Let us use two rectangles (red and purple as in the Figure 3). The result, obtained by means of GIMP, the GNU image processing software, is given in the Figure 4. The result is quite interesting because it is telling us that the two heads are congruent in some manner.

For the discussion of other portraitures of Julius Caesar see please [START_REF] Sparavigna | Portraits of Julius Caesar: a proposal for 3D analysis[END_REF][START_REF] Corazzi | The Rhone Caesar[END_REF][START_REF] Corazzi | Il Cesare di Arles[END_REF][START_REF] Sparavigna | Facial transformations of ancient portraits: the face of Caesar[END_REF][START_REF] Griffin | A Companion to Julius Caesar[END_REF][START_REF] Long | Secrets du Rhône[END_REF][START_REF] Long | le Rhône pour mémoire[END_REF][START_REF] Johansen | Antichi ritratti di Caio Giulio Cesare nella scultura[END_REF][START_REF] Johansen | The Portraits in Marble of Gaius Julius Caesar, in Ancient Portraits inthe P[END_REF][START_REF] Carotta | Il Cesare incognito. Sulla postura del ritratto tuscolano di Giulio Cesare[END_REF]. The Figure 5 is showing four of them.

  Written in Turin, 27 June 2018. DOI: 10.5281/zenodo.1299230 On 22 June 2018, an article published by the National Museum of Antiquities (Rijksmuseum van Oudheden) of Leiden [1], disclosed a 3D reconstruction of Julius Caesar's head based on a marble bust of the museum. Of the result of this reconstruction and of its consequences, we discussed in [2-5]; therefore, let us just tell the following. The reconstruction was made by Maja d'Hollosy. As told by [6], d'Hollosy used a head of Caesar from the National Museum of Antiquities of Leiden and the Tusculum bust from

  .

Figure 1 :

 1 Figure 1: The two Caesar's heads in Leiden, as we can see in [9] (Courtesy: National Museum of Antiquities, Leiden). Let me stress that these images are here used only for scientific and cultural purposes

Figure 2 :

 2 Figure 2: Using Paint, let us try to add nose and lips to the Leiden head.

Figure 3 :

 3 Figure 3: Rectangles of the Tusculum bust (left) and of the Leiden head. Let me stress that the images are here used for scientific and cultural purposes. The sizes are in pixels.

Figure 4 :

 4 Figure 4: Digital restoration of the face of the Leiden bust

Figure 5 :

 5 Figure 5: Heads of Caesar. From the left: Tusculum, Arles, the head from a private collection and proposed by F. Johansen as a replica of the Arles bust., and Pantelleria.


Amelia Carolina Sparavigna

Keywords: Face recognition, 3D rendering, History of art, Caesar's portraitures, Tusculum bust, Arles bust

come

The 3D reconstruction of the bust was revealed at an event during which a new book on Caesar written by archaeologist Tom Buijtendorp had been promoted [START_REF] Buijtendorp | Caesar in de Lage Landen: De Gallische oorlog langs Rijn en Maas[END_REF]. As told in [2], "Buijtendorp came up with the idea of recreating Caesar's head based on a marble bust of the ancient Roman leader that's held at the Rijksmuseum van Oudheden". However, on the cover of his book, the author, or the Editor, used the Tusculum portrait of Caesar [4].

Today, the Tusculum bust is exhibited at the Museo Archeologico of Torino [5].

Actually, in 2012 [START_REF] Sparavigna | Portraits of Julius Caesar: a proposal for 3D analysis[END_REF], I suggested the use of a 3D scanning and rendering to create some virtual copies of ancient objects to study and compare them. In particular, I told that this approach could be interesting for some Roman marble busts, two of which are well-known portraits of Julius Caesar (Tusculum and Farnese [7]), and the third is a realistic portrait of a man, probably Caesar too, found at Arles, France (see Figure 1).

It was probably between 49 and 46 BC, when Caesar had close relationships with Arles, that, according to the French archaeologist Luc Long who found it in 2007, the marble bust of Arles had been carved. This beautiful portrait, discovered in the depths of the right bank of the river near Arles, has been undoubtedly attributed by Long to Julius Caesar [START_REF] Long | Secrets du Rhône[END_REF][START_REF] Long | le Rhône pour mémoire[END_REF].

Besides comparing to the Tusculum bust, in [START_REF] Corazzi | The Rhone Caesar[END_REF][START_REF] Corazzi | Il Cesare di Arles[END_REF], we compared the Arles bust with Pantelleria bust and a bust of a private collection, proposed by F. Johansen as a replica of the Arles bust [START_REF] Johansen | Antichi ritratti di Caio Giulio Cesare nella scultura[END_REF][START_REF] Johansen | The Portraits in Marble of Gaius Julius Caesar, in Ancient Portraits in the P[END_REF]. The reader can find this head in the Figure 2 and more detail at [START_REF] Corazzi | The Rhone Caesar[END_REF]. Let us note that the Tusculum bust is considered the more ancient and faithful portrait of Caesar. After its discovery, also the Arles bust was proposed as a portrait of Caesar.

Because of the period of time it was sculpted, the Arles bust could be a faithful portrait too. Therefore, in [START_REF] Corazzi | The Rhone Caesar[END_REF], we considered the following question. "What is then, between the portraits of Arles and Tusculum, the most likely portrait of Caesar?" To find a proper answer, as we told in [START_REF] Corazzi | The Rhone Caesar[END_REF], we need a systematic comparison with other busts. Moreover, the discovery of other portraits, such as the one recently found in an ancient cistern of Pantelleria, could help in the analysis of Caesar's portraitures.

As stressed in [START_REF] Corazzi | The Rhone Caesar[END_REF], the use of biometric software for face detection and morphing can help us for rendering these marble portraits in more realistic faces, turning them out in some sort of pictures of real people. An example of the results is given in the Figure 3, which is showing that proposed in [START_REF] Corazzi | The Rhone Caesar[END_REF], obtained by a software that in 2013 was freely available at the site In20Years.com. The face of Arles and the portrait of the private collection, proposed by F. Johansen, appear greatly resembling. Other "faces" of Caesar are given in the Figure 4 (see the discussion in Ref. [START_REF] Sparavigna | Facial transformations of ancient portraits: the face of Caesar[END_REF]. In [START_REF] Sparavigna | A software for aging faces applied to ancient marble busts[END_REF][START_REF] Sparavigna | The Appearance of Roman Emperors Rendered by a Face Detection Software[END_REF] we proposed the same rendering for Augustus and other Roman emperors. In particular, in [START_REF] Sparavigna | A software for aging faces applied to ancient marble busts[END_REF], we used In20Years.com for ageing the face of young Augustus, to compare with his portraits as emperor. For what concerns the Leiden bust, as told in [2], "Archaeologist and anthropologist Maja d'Hollosy brought the bust to life, using a 3D scan of the marble piece before drawing on other existing portraits of Caesar to fill in gaps where the marble was damaged. Caesar was then given a realistic-looking face using silicon and clay." In [1], we can see d'Hollosy's rendering. We gain also the information that the other bust used is the Tusculum bust.

In the Figure 5 we can see the Tusculum head in profile and frontal view. A remarkable work by Francesco Carotta, [START_REF] Carotta | Il Cesare incognito. Sulla postura del ritratto tuscolano di Giulio Cesare[END_REF], is discussing this head and the best perspective we have to use to see it. Let us concentrate on the face. Since we have not a 3D scan of Tusculum bust, let us apply a different approach, which we have used for articles [START_REF] Sparavigna | 3D Faces Reconstruction Applied to Some Paintings of Leonardo Da Vinci[END_REF][START_REF] Sparavigna | 3D Faces from 2D Pictures[END_REF] (in [START_REF] Sparavigna | 3D Faces from 2D Pictures[END_REF] I investigated the Arles bust).

In the research field of face recognition, a very difficult problem to solve is that of having a 3D facial reconstruction from a single 2D image. A remarkable software which is solving this task has been recently developed by Jackson, Bulat, Argyriou and Tzimiropoulos at the University of Nottingham and Kingston University. The researchers have obtained their result by training a Convolutional Neural Network (CNN) on an appropriate dataset consisting of 2D images and 3D facial models [START_REF] Jackson | Large pose 3D face reconstruction from a single image via direct volumetric CNN regression[END_REF]. The researchers are also providing a site for experiments, at the address http://www.cs.nott.ac.uk/~psxasj/3dme/index.php . Let us call here this software as "3DME". For experiments, a frontal image is required. After the image is uploaded, the site brings us to the corresponding 3D model, which is easy to move by means of the mouse. An OBJ file of the 3D model is also given. So, in [START_REF] Sparavigna | 3D Faces from 2D Pictures[END_REF], we used 3MDE for the Arles bust. And here, let us use 3MDE for the Tusculum bust. The frontal image used is that of Wikipedia page [21] (Courtesy images Users Soerfm and Gautier Poupeau). In the Figures 6 and7 we can see some results that we can obtain from a picture of the Tusculum bust. Let me stress once more that the 3D rendering of the 3DME software [START_REF] Jackson | Large pose 3D face reconstruction from a single image via direct volumetric CNN regression[END_REF] is obtained from one image only. I consider the software remarkable and of fundamental importance for several applications. One application is in the field of the history of arts and, in particular, in the analysis of the marble portraitures.

For what concerns the Caesar's bust of Leiden [1], it would be interesting to have more detail on the technique and the data used for the 3D rendering.

Ref.2 is telling that this 3D reconstruction is "including the bizarre proportions of his [Caesar's] cranium." However, Suetonius, in De vita Caesarum [START_REF] Suetonius | Available at www[END_REF], is not mentioning any bizarre proportion. And, to the author's knowledge, no witty remark exists on Caesar's head, besides his baldness of course.

Let us conclude reporting the Suetonius' works. "It is said that he was tall, of a fair complexion, round limbed, rather full faced, with eyes black and piercing; and that he enjoyed excellent health, except towards the close of his life, when he was subject to sudden fainting-fits, and disturbance in his sleep. … He was so nice in the care of his person, that he not only kept the hair of his head closely cut and had his face smoothly shaved, but even caused the hair on other parts of the body to be plucked out by the roots, a practice for which some persons rallied him. His baldness gave him much uneasiness, having often found himself on that account exposed to the jibes of his enemies. He therefore used to bring forward the hair from the crown of his head; and of all the honours conferred upon him by the senate and people, there was none which he either accepted or used with greater pleasure, than the right of wearing constantly a laurel crown" [START_REF] Suetonius | Available at www[END_REF].

  Written in Turin, 24 June 2018. DOI: 10.5281/zenodo.1297051 On 22 June 2018, an article has been published by the web site of the National Museum of Antiquities (Rijksmuseum van Oudheden), in Leiden [1]. It is showing a new 3D reconstruction of Julius Caesar's head based on a bust of the museum. Ref.2 is telling that this 3D reconstruction is "including the bizarre proportions of his cranium."

Figure 1 :

 1 Figure 1: Reproduction of Figs.1 and 2 from [6]. Upper part. Arles bust on the left and Tusculum bust on the right (Image courtesy: Wikipedia). In the middle, the two images superposed. Lower part: Arles bust on the left and Farnese bust (Courtesy, [7]) on the right.

Figure 2 :

 2 Figure 2: Arles and the head from a private collection, proposed by F. Johansen as a replica of the Arles bust. The man on the right seems the old Arles one.

Figure 3 :

 3 Figure 3: Caesar from Arles, Tusculum, Pantelleria and from the private collection, processed by means of a morphing from site In20Years.com, which makes a face looking older of about twenty years. Note the similarity between the Arles portrait and that of the private collection, disclosed by F. Johansen.

Figure 4 :

 4 Figure 4: Portraits of Caesar (more detail at Ref. 14).

Figure 5 .

 5 Figure 5. The Tusculum head.

Figure 6 :

 6 Figure 6: Results of the 3DME software [20], applied to the top left image, which is that given by Wikipedia [21].

Figure 7 :

 7 Figure 7: Results of the 3DME software (middle and right), applied to the image given by Wikipedia [21], and compared to the profile of the statue (left).


Amelia Carolina Sparavigna

Luca Vinci

Johannes Pacioli

Fra Kepler

Domenico Giovanni Da Verona

Contrefait Giovanni Ambrogio Remps

Egidius Milanese

Lorenz Lobenigk

Grollier Zick

Chinese and European Ivory Puzzle Balls

Keywords: Ivory Carving, Chinese Art, Feng Shui, Renaissance Art, Geometry, Platonic Solids, Dodecahedron, Cabinet of Curiosities, Wunderkammer, Leonardo da

published or not. The documents may come

In the item of Wikipedia about the Ivory Carving [1], it is told that, by the 18th century, China started to have a considerable market of artistic objects made for the export to Europe. In the later 19th century, thanks to the African ivory, these objects became as large as the material would allow. A speciality was a round ball of openwork that contained a series of smaller concentric balls, which could rotate freely inside each other (Figure 1). Because of the vast expenditure of time and effort required for making them, these "devil's work balls", as they are also known [START_REF] Gabriel | Beyond the Cube: The Architecture of Space Frames and Polyhedra[END_REF], became some of the preferred artistic "playthings" of the rich people [3]. We can find more information on the Chinese puzzle balls in an article discussing the collection at the Sam Noble Oklahoma Museum of Natural History [START_REF] Cronin | [END_REF]. For what concerns the dates for the ivory ball carvings, Ref. 4 tells that they are usually no earlier than the Qing Dynasty , and that some occurrences of similar products appeared in Canton as early as the 14th century [START_REF] Cronin | [END_REF][START_REF] Laufer | Ivory in China[END_REF].

But in Ref. [START_REF]Antikitera[END_REF] we find that the first puzzle balls appeared during the Song Dynasty, around 1000 AD (this dynasty began in 960 and continued until 1279). Actually, the first reference to a puzzle ball seems to be that given by the Chinese writer Cao Zhao, in his Essential Criteria of Antiquities, first published in Nanjing in 1388. He tells the following: "I have seen a hollow-centred ivory ball, which had two concentric balls inside it, which can both revolve. It is called 'witch ball'. I was told that it was made for the Palaces of the Song dynasty" [START_REF] Tarnai | Turned Ivory Polyhedra[END_REF].

The process for making a puzzle ball is very difficult. The preferred material for them is the ivory, but other materials (jade, wood, or soapstone) were used too [START_REF] Cronin | [END_REF]. "In ancient times the balls were carved completely by hand. In more recent times, a lathe was used to shape the balls". First, coneshaped holes are drilled towards the centre at varying depths. Once the holes are made, an "L" shaped carving tool is used to separate the layers which are creating the balls within the balls. As explained in [START_REF] Cronin | [END_REF], the artist starts with the smallest core ball and then carves out spherical layers slightly larger than the previous one. Then the balls are decorated with micro-tools to pierce their surfaces and to decorate the outermost ball.

The puzzle balls are considered as symbols of good luck and are decorated with a variety of feng shui elements. In [3], we find mentioned that the outermost layer has usually the features of the phoenix and of the dragon, which are symbols of yin and yang. "The phoenix represents the wife while the dragon is the husband and emperor, and balls decorated with these symbols are thought to bring good luck and happiness to a marriage" [3]. So we have that almost all of the symbols that we can find decorating the puzzle balls are associated with ensuring a long and happy marriage.

"Some balls even have different symbols on different layers, though the most common is a highly decorative outer ball and 'latticed' balls inside (with geometric patterns of holes)." [3] An example of inner latticed spheres is given in the Figure 2. In the Figure 2 we see a detail of an example of Canton ivory carving. In the item of Wikipedia on the Lingnan culture, or Cantonese culture [8], we read that the ivory woodcarving is a well-known product from Lingnan. "With a history of 2000 years", it traditionally uses ivory as raw material to make sculptures, with a style renowned for being delicate and detailed without being brittle. Also Ref. 9 is giving further information about the Chinese puzzle balls. "These detailed works of art are usually made up of at least 3 to 7 layers, but the world's largest puzzle ball is actually made of 42 concentric balls all enclosed one within the other". And also: "Although the inner balls can be manipulated to align all the holes, Chinese puzzle balls got their name from people who, through the ages, pondered the mystery of making such objects" [9].

If we consider just the Wikipedia item about the Chinese ivory balls [1], we could conclude that Europe knew such concentric spheres only through the Chinese products of the 19th century. But this is not true. Puzzle balls were produced in Europe during 16th and 17th centuries too. Here we can see an example of European production given in the Figure 3. The balls shown in the images of the Figure 3 are remarkable because they look like Roman Dodecahedra. Renato Bucci, who served as honorary inspector to the Soprintendenza Archeologica of Tuscany, was so kind to send me the pictures because of the similarity to the archaeological object [START_REF] Sparavigna | Roman Dodecahedron as dioptron: analysis of freely available data[END_REF][START_REF] Sparavigna | An Ancient Rangefinder for Teaching Surveying Methods[END_REF]. The objects depicted in the Figure 3 were probably some of the items of a Wunderkammer, that is, of a Room of Curiosities and Wonder (in Italiano, camera delle meraviglie o gabinetto delle curiosità o delle meraviglie), a depository (or simply a cabinet) for the items of a collection. The Wunderkammer had its origin from the rooms used to contain the encyclopedic collections of remarkable and exotic objects, which were assembled under the patronage of the Renaissance courts. In [12], it is told that a Wunderkammer or (Kunstkammer) was like a "microcosm or theatre of the world", which was conveying symbolically the "patron's control of the world through its indoor, microscopic reproduction" [12].

In a modern web Kunstkammer [13], we find told that it was the Chinese work of the puzzle balls, which fascinated the European travellers, and that consequently stimulated the German craftsmen to create their own ivory puzzle balls. This hypothesis is also mentioned in [START_REF] Gabriel | Beyond the Cube: The Architecture of Space Frames and Polyhedra[END_REF]. However, after comparing Chinese and European works, we have to note the following facts and differences.

For the Chinese balls, the aesthetic appearance is fundamental and the geometry just necessary to create them [START_REF] Tarnai | Ivory shells and polyhedra[END_REF]. Their surfaces are full of decorations and symbols, whereas the European balls have a smooth surface and a polyhedron-like shape. As a consequence, the beauty of the European works comes explicitly from their perfect geometry [START_REF] Tarnai | Ivory shells and polyhedra[END_REF]. Another feature of some of the European balls is the presence of radial spikes, which are passing through the holes, having origin from the innermost core ball [START_REF] Tarnai | Ivory shells and polyhedra[END_REF] (see Figure 4). These spikes are missing from the Chinese balls. The perfect geometry of the European balls is not surprising. The artists who made them were the heirs of those Renaissance artists that were deeply attracted to the geometry of solid shapes and, in particular, to the Platonic Solids. Also Leonardo da Vinci considered these Solids and drew them for a book of the mathematician Luca Pacioli [START_REF] Frank | Leonardo da Vinci's Geometric Sketches[END_REF].

In a nested form, the Platonic Solids became the famous representation of the macrocosm in the Mysterium Cosmographicum written by Johannes Kepler (1596) (see the Figure 5). This is the book which explains Kepler's cosmological theory based on the Copernican system. Kepler started from the five Pythagorean regular polygons as dictating the structure of the universe, according to a God's plan achieved through geometry. After doing further calculations he realized that he could not use the polygons, but that he had to use the five Platonic solids, nested in the form figured as in Fig. 5, to represent the solar system [16]. Besides attracting the studies of philosophers and artists, the Platonic Solids and the related geometrical forms attracted the interest of collectors too. Therefore, we find them depicted in the images of the period reproducing the Cabinets of Curiosities (see Figures 6 and7).To the images of solids shown in these Figures, let us also add four of those proposed by Wenzel Jamnitzer (1507/08 -1585) in his book on the Perspectiva Corporum Regularium (Figure 8). Probably, these images were used to create some real models for the cabinets of artists and collectors. Let us continue the discussion on the puzzle balls. After the analysis of their geometry, we can tell that many of the European puzzle balls were inspired by the Platonic solids. We find many balls having the geometry of nested dodecahedra, such as those created by Egidius Lobenigk (1581 -1584), or of spheres and other solids as shown in [START_REF] Tarnai | Ivory shells and polyhedra[END_REF] (some Lobenigk dodecahedra, made of ivory, are exhibited at the Staatliche Kunstsammlungen in Dresden [17]). As we can see from the "The great masters of turning and their pupils, who were often sovereign rulers, employed the most advanced machine of their day: the lathe. ... Lathe technology originated at least as early as the eighth century BC, and simple lathes were used through the middle ages in Europe for the production of wooden cabinet elements and small decorative items. It was not until the 16th century, however, that technical and artistic advancements brought the practice into a realm far above the ranks of craftsmen. Turners such as Giovanni Ambrogio Maggiore, Georg Wecker, and Philip Senger began to exploit the more responsive medium of ivory". These artists were able of "inventing fantastic forms" according to the spirit of the scientific and philosophical exploration of their day. These remarkable objects, which are exceedingly rare today, are showing both the wealth and intellect of their past creators and owners.

"Perhaps the highest accomplishment of the art of turning ivory -continues the catalogue -are the hollow spheres containing internally-turned capsules and nesting spheres known as contrefait.

These were among the most difficult forms in the turner's repertoire, with almost eggshell-thin ivory walls and complex forms turned within through a small aperture. The attempt to understand the technique by which these spheres are created is baffling to the viewer, and in this astonishment lies their intellectual appeal." That is, these spheres were the analogue of the Chinese puzzle spheres that we have discussed at the beginning of this article.

And here, it is strongly required to note what Ref.7 is telling about the Chinese nested spheres. "To our knowledge [of the author of Ref.7, Tibor Tarnai], there is no trace of "Devil's work ball" in the 15th, 16th and 17th centuries. In the 18th and 19th centuries, these balls were made in the Canton region for the European trade, after the first European lathe was imported in 1722. [START_REF] Ching-Fei | Concentric Ivory Spheres and the Exchange of Craft Techniques: Canton, the Ch'ing Court and the Holy Roman Empire[END_REF]." (see also the articles [START_REF] Ching-Fei | Concentric Ivory Spheres and the Exchange of Craft Techniques: Canton, the Ch'ing Court and the Holy Roman Empire[END_REF] and [START_REF] Shih | Unknown transcultural objects: Turned ivory works by the European Rose Engine Lathe in the Eighteenth-Century Qing Court[END_REF]).

As Shih Ching-fei is telling in [START_REF] Ching-Fei | Concentric Ivory Spheres and the Exchange of Craft Techniques: Canton, the Ch'ing Court and the Holy Roman Empire[END_REF], the Chinese puzzle balls, so representatives of Chinese civilization, "are rarely the subject of academic inquiry". The study in [START_REF] Ching-Fei | Concentric Ivory Spheres and the Exchange of Craft Techniques: Canton, the Ch'ing Court and the Holy Roman Empire[END_REF] begins with an important question: "Do the concentric spheres represent nothing more than an ingenious style of ivory carving?" This question is proposed by the author, because of the existence of the concentric balls made in Europe during the 16th and 17th centuries. As a consequence, to Shih Ching-fei the concentric ivory spheres are not simply "objects of wonder or emblems of China for Western imaginations", but are objects that need to be considered in "a full-fledged re-examination of both the general relationship between Europe and the Ch'ing court".

It seems therefore that it was the import from Europe of the "rose engine lathe" that stimulated the production in Canton of the puzzle balls. To have specific information about lathes and techniques, the reader can use the Manuel du tourneur by L.-E. Bergeron [START_REF] Manuel | précédé de notions élémentaires sur la connoissance des bois, la menuiserie, la forge, la trempe, la fonte[END_REF]. For what concerns the history of the European ivory spheres, the Sotheby's catalogue [21] tells that "The earliest known example [of contrefait] is a sphere turned by Giovanni Ambrogio Maggiore of Milan in 1582 now preserved in the Museo degli Argenti in Florence. Egidius Lobenigk of Dresden was another innovator of the form, and in the Grünes Gewölbe there are four signed spheres of the early 17th century by Georg Friedel". It seems therefore that Lobenigk was the artist who introduced the dodecahedron in the nested form of the contrefait. About Giovanni Ambrogio Maggiore, we find some information in [START_REF]The Grove Encyclopedia of Northern Renaissance Art in Oxford Reference[END_REF]. Maggiore, (Milan 1550 -Rome 1598), was an Italian turner of ivory and wood, active in Bavaria. He or his brother, Dionigi Maggiore, "succeeded in making a rotary lathe that was capable of producing not only the usual round turnings but also oblique forms. Little is known of the early stages of this invention; possibly the brothers based it on a project (now untraced) by Leonardo da Vinci" [START_REF]The Grove Encyclopedia of Northern Renaissance Art in Oxford Reference[END_REF]. In fact, in [START_REF] O'keeffe | The Art of Ornamental Turning[END_REF], we find that the earliest illustration of an ornamental turning lathe is dated 1578. This machine was described by Jacques Besson. And Besson was, like Leonardo da Vinci, an engineer to the French court.

In 1573, it happened that a Milanese art dealer, Prospero Visconti, brought Giovanni Ambrogio He was a cousin of Jean Grolier de Servières, Treasurer of France and famed bibliophile. Nicolas was born in Lyon. In his youth, he followed a military career that took him to Flanders, Germany, Italy and Constantinople. As a military engineer, he specialized in movable bridges. After he retired at his home in Lyon, he started creating ornamental objects based on a series of fantastic models, which he displayed in a cabinet that he opened to the public once a week. This cabinet of the In this article we have discussed the Chinese ivory puzzle balls and the European ivory nested solids. Of course, the cultural backgrounds of these objects are different but it seems that, in both cases, it was the use of a modern lathe that determined their success. For what concerns them as philosophical objects, it seems that they were aiming to represent the macrocosm in a microcosm of nested solids. Today, this artistic turning is also appreciated by some European artists. We have objects turned in ivory, as those of Pierre Meyer, or created, as in the case of Andrea Pacciani, by means of the new generation of 3D technologies. As Pacciani is telling, using these technologies we can bring back in vogue a production of objects having a great visual impact [30]; that is, we can create again the amazing objects of Renaissance and Baroque arts, at least virtually, for our modern Wunderkammer.

Figure 1 :

 1 Figure 1: Chinese puzzle ball, with openwork and a series of twelve smaller balls, ivory, 19th century. British Museum. Original photograph by Ged Carroll for Wikipedia.

Figure 2 :

 2 Figure 2: Detail of an ivory ball on show in the Rautenstrauch-Joest Museum, Köln. It has 16 layers, which can spin. Courtesy :Till Niermann, Wikipedia.

Figure 3 :

 3 Figure 3: European puzzle ball and Rosary, 16th-17th century (Image Courtesy: Maureen and Renato Bucci, Italy, 2014). These objects were exhibited in Stuttgart.

Figure 4 :

 4 Figure 4: Ivory balls from German workshops, Bode-Museum, Berlin, Germany. 17th century. Lorenz Zick, c. 1650, Courtesy image: Daderot for Wikipedia.

Figure 5 :

 5 Figure 5: The Kepler solar system.

Figure 6 :

 6 Figure 6: Fra Giovanni da Verona. Detail of a tarsia showing a cabinet with polyhedra. Courtesy Laurom di Wikipedia in italiano. Fra Giovanni da Verona (1457 circa -1525) was an Italian artist who worked from the end of the 15th Century and the beginning of the 16th Century.

Figure 7 :

 7 Figure 7: Domenico Remps, (1620-1699), Cabinet of the Curiosities, Painting, Detail, Date 1690s, Current location: Opificio delle pietre dure, Firenze. Note the spikes inside the spheres, a real puzzle.

Figure 8 :

 8 Figure 8: Dodecahedra and Icosahedra from Wentzel Jamnitzer's Perspectiva Corporum Regularium.

Figure 4 ,

 4 Figure 4, which is showing two balls from works by Lorenz Zick (1594-1666), the representation of nested dodecahedra was a favourite subject during 17th century too.Lorenz was a member of the Zick family[18,[START_REF] Campbell | The Grove Encyclopedia of Decorative Arts: Two-volume Set[END_REF], the son of the turner Peter Zick and grandson of the turner Martin Zick[START_REF] Campbell | The Grove Encyclopedia of Decorative Arts: Two-volume Set[END_REF]. Lorenz Zick was called to Vienna in 1642-4 to instruct Emperor Ferdinand III and was appointed Kammerdrechsler of the imperial court. Peter Zick was the

Figure 9 :

 9 Figure 9: Ivory cups in the Grünes Gewölbe, Dresden. Courtesy: Heidelberg Historische Bestände. From [20] at the following link: http://digi.ub.uni-heidelberg.de/diglit/sponsel1932bd4/0063

Figure 10 :

 10 Figure 10: Detail of the spires of three ivory vases in the Grünes Gewölbe, Dresden. Courtesy: Heidelberg Historische Bestände. Page from [20] at the following link: http://digi.ub.uniheidelberg.de/diglit/sponsel1932bd4/0067/image

Figure 11 :

 11 Figure 11: Ivory objects from German workshop, XVII century. Lorenz Zick, c. 1650. Bode-Museum, Berlin, Germany. Courtesy image: Szilas for Wikipedia. The puzzle spheres are those in the Figure 4.

  Maggiore to the attention of Crown Prince Wilhelm, who reigned from 1579 to 1598 as Wilhelm V, Duke of Bavaria. "The Prince, who was interested in the discovery, brought Maggiore to Bavaria to work and to teach the new technique. In Munich, "Maggiore passed on the technique to his pupils, and as early as 1576 Georg Wecker, son of the court turner Hans Wecker, took a new lathe and the new techniques to Saxony". This had the consequence of the development of the Dresden school of ivory turning. Due to the Prince Wilhelm's unstable financial position, Maggiore received only temporary appointments at the residence at Landshut and later in Munich. However, we know that In 1582 Wilhelm sent to the Medici in Florence a Kontrafetkugel, that is a contrefait made by Maggiore, which was set up in the Tribuna of the Uffizi [27, 28]. It is made of ebony and ivory. As told in [28], this sphere is the first of the contrefait spheres. In the Maggiore's contrefait, the spheres "can be moved independently of each other by pulling on the attached strings. Inside are painted miniature portraits of Duke Wilhelm of Bavaria and his family" [27]. Maggiore's sphere was the model for the spheres with strings that we see in the Figure 11. Little is known of the later life of Maggiore. From Milan, he thanked the Prince for a gift in 1593. In the book La nobiltà di Milano of 1595, historian Paolo Morigia praised him as a successful Milanese artist. In 1597 Grand Duke Ferdinand I de' Medici recommended Maggiore to Rome, In 1598, Maggiore was still working for the Medici court in Florence. Today, some European artists are turning ivory again and are making contrefait balls too. We have, for instance, the contemporary puzzle ivory dodecahedra, created by the artists Pierre Meyer, as we can see at the web page [29]. A new production, in polymeric materials, of the ornamental turning ivory of the 17th century is also evidenced by the works of Andrea Pacciani, architect in Parma. As given at the web page [30], a piece was inspired by the masterpieces of the collection of Rosenborg Castle in Denmark [31]. Another Pacciani's work is inspired to the drawings of Grollier de Serviere, (1596-1689), French inventor and ornamental turner.

Figure 12 :

 12 Figure 12: A drawing from the book on the works of Grollier de Servière.

  wonder became famous enough to attract many visitors [32]. Even the king Louis XIV paid a visit to Grollier de Servière's cabinet. After his death in 1689, his son Gaspard (1646-1716) and then his grandson Gaspard II (1676-1745) continued to display the contents of the cabinet [32]. Gaspard published a book cataloguing the curiosities: Recueil d'Ouvrages Curieux de Mathématique et de Mécanique, ou Description du Cabinet de Monsieur Grollier de Servière. An image of the book is shown in the Figure 12: note the spikes in the balls, which look like those depicted in the Figure 7. Probably, Domenico Remps was inspired by the works of Grollier.