Jean Esterle
email: esterle@math.u-bordeaux.fr

Holomorphic Functional Calculus on Quotients of Fréchet Algebras and Michael's Problem

Keywords: Borel theorem, Picard theorem, entire function, Fréchet algebra, Michael's problem, prime ideal, Picard-Borel ideal, weak Picard-Borel ideal, holomorphic functional calculus MSC: 30D20, 30H50, 32A15, 46H10, 46J05 PACS:

The purpose of the paper is to give the state of the art on Michael's problem, the long-standing open question odf continuity of characters on commutative Fréchet algebras. We first quote two well-known consequences of the "abstract Mittag-Leffler theorem", the theorem of Arens, which shows that characters on finitely rationally generated Fréchet algebras are continuous, and the fact that the existence of a nonincreasing sequence (Ωn) n≥1 of Fatou-Bieberbach domains in C p such that ∩ n≥1 Ωn = ∅ would imply that all characters on commutative Fréchet algebras are continuous.

In the opposite direction the existence of a discontinuous character on some commutative unital Fréchet algebra is equivalent to the existence of a character on a quotient algebra of the form U/I where U is a 'test algebra' for Michael's problem and where I is a dense ideal of U which is a Picard-Borel ideal, which means that every family of pairwise linearly independent invertible elements of U/I is linearly independent. It was recently shown that all Picard-Borel ideals in commutative unital Fréchet algebras are prime, and Picard-Borel ideals of H(C) can be easily described. We raise a question concerning Picard-Borel ideals of H(C p ), p ≥ 2 which could lead to important general information about the quotient of commutative unital Fréchet algebras by Picard-Borel ideals.

The fact that entire functions of several variables oerate on quotients of Fréchet algebras by ideals which are not necessarily closed plays an essential role in the paper.

Introduction

Seventy years after the seminal memoir of Michael [START_REF] Michael | Locally multiplicatively convex topological algebras[END_REF] the continuity of characters on commutative Fréchet algebras is still an open question, known as Michael's problem. The fact that the action of entire functions of several complex variables on finite families of elements of commutative Fréchet algebras extends to quotients of Fréchet algebras by non necessarily closed ideals plays a crucial role in the investigations on this elusive problem.

In the first part of the paper we fill a gap in the litterature by giving a (folklore) extension of this functional calculus to holomorphic functions on C p × (C \ {0}) q acting on families u = (u 1 , . . . , u p+q ) of elements of a quotient A/I of a commutative unital Fréchet algebra A by an ideal I when u p+1 , . . . , u p+q are invertible in A/I. This necessitates some work since in general invertible elements u of A/I do not have the form u = a + I where a is invertible in A.

We quote two well-known consequences of the "abstract Mittag-Leffler theorem", the theorem of Arens, which shows that characters on finitely rationally generated Fréchet algebras are continuous, and the fact that the existence of a nonincreasing sequence (Ω n ) n≥1 of Fatou-Bieberbach domains in C p for some p ≥ 2 such that ∩ n≥1 Ω n = ∅ would imply that all characters on commutative Fréchet algebras are continuous.

In the opposite direction the existence of a discontinuous character on some commutative unital Fréchet algebra is equivalent to the existence of a character on a quotient algebra of the form U/I where U is a 'test algebra' for Michael's problem and where I is a dense ideal of U which is a Picard-Borel ideal, which means that every family of pairwise linearly independent invertible elements of U/I is linearly independent. It was recently shown that all Picard-Borel ideals in commutative unital Fréchet algebras are prime. The proof relies on a discussion of ideals of H(C 2 ) and on the description of Picard-Borel ideals of H(C) given in [START_REF] Esterle | Picard-Borel ideals in Fréchet algebras and Michael's problem[END_REF].

The class of Picard-Borel ideals admits maximal elements with respect to inclusion, and it would be interesting to see whether these maximal elements can have infinite codimension.

We raise at the end of the paper a question concerning Picard-Borel ideals of H(C p ), p ≥ 2 which could lead to important general information about the quotient of commutative unital Fréchet algebras by Picard-Borel ideals, which could open rthe gate for a construction of discontinous characters on U.

Section

Recall that a unital Fréchet algebra is a complex unital algebra A equipped with a nondecreasing sequence ( . n ) n≥1 of submultiplicative seminorms satisfying the following properties

• 1 n = 1, n ≥ 1, where 1 denotes the unit element of A , • ∩ n≥1 Ker( . n ) = {0},
• For every sequence (x p ) p≥1 in A satisfying the Cauchy condition with respect to each seminorm . n , there exists x ∈ A such that lim n→+∞ x -x p n = 0 for n ≥ 1.

In the sequel we will identify λ1 ∈ A with the complex number λ, and such elements of A will be called constants. We will denote Inv(A) the group of invertible elements of A.

If (A, ( . n ) n≥1 ) a unital Fréchet algebra, denote by the same symbol . n the norm induced by the seminorm . n on the quotient algebra A/ ker( . n ), denote by A n the completion of (A/ker( . ) n , . n ), denote by π n : A → A n the map x → x + ker( . n ) and for n ≥ m denote by π n,m : A n → A m the extension by continuity to A n of the map x + ker( . n ) → x + ker( . m ),.

The map π : . A deeper result due to Arens [START_REF] Arens | Dense inverse limit rings[END_REF] shows that σ A (a 1 , . . . , a k ) = {(χ(a 1 ), . . . , χ(a k )} χ∈Spec(A) for every finite family {a 1 , . . . , a k )} of elements of A.

x → (π n (x)) n≥1 is a Fréchet algebra isomorphism from A onto the inverse limit lim ← (A m , π n,m ) = {(x m ) m≥1 ∈ Π m≥1 A m | x m = π n,m (x n ), n ≥ m ≥ 1},
A commutative unital Fréchet algebra A is said to be polynomially finitely generated if there exists a finite family (a 1 , . . . , a k ) of elements of A such that elements of the form p(a 1 , . . . , a k ), where p ∈ C[X 1 , . . . , X k ] are dense in A. Similarly A is said to be rationally finitely generated if elements of the form p(a 1 , . . . , a k )q(a 1 , . . . , a k ) -1 , where p ∈ C[X 1 , . . . , X k ], q ∈ C[X 1 , . . . , X k ], q(a 1 , . . . , a k ) ∈ Inv(A), are dense in A.

The following consequence of Arens' result seems to be the most general positive partial answer to Michael's problem.

α = (α 1 , . . . , α k ) ∈ N p , a = (a 1 , . . . , a p ) ∈ A p , f ∈ H(U ), λ = (λ 1 , . . . , λ p ) ∈ U, set |α| = α 1 + • • • + α p , α! = α 1 ! . . . α p !, a α = a α1
1 . . . a αp p , and

∂ α f (λ) = ∂ |α| f ∂z α1 1 . . . ∂z αp p (λ 1 . . . λ p ), Set, for f ∈ H(C p ), a = (a 1 , . . . , a p ) ∈ A p , f (a) := α∈N p ∂ α f (0, . . . , 0) a α α! .
The map f → f (a) is a continuous algebra homomorphism from H(C p ) into A such that z j (a) = a j , wherez z j : (z 1 , . . . , z p ) → z j denotes the j th projection from C p onto C.

Denote by N p * the set of nonzero elements of N p . The above formula allows to compute ∂ α f (a) for f ∈ H(C p ), a ∈ A p , and we have,

for b = (b 1 , . . . , b p ) ∈ A p , f (b) = f (a) + α∈N p * ∂ α f (a) (b -a) α α! . (1)
So given f ∈ H(C p ), there exists g 1 , . . . , g p ∈ H(C 2p ) such that we have

f (b) = f (a) + p j=1 (b j -a j )g j (a, b). a = (a 1 , . . . , a p ) ∈ A p , b = (b 1 , . . . , b p ) ∈ A p . ( 2
)
This gives the following standard result.

Proposition 2.2. Let I be an ideal of a commutative Fréchet algfebra A, and let π I : A → A/I be the canonical surjection. Set, for f ∈ H(C p ), a = (a 1 , . . . , a p )

∈ A p , f (π I (a)) = f (π I (a 1 ), . . . , π I (a p )) = π I (f (a)).
Then the map f → f (π I (a)) is a well-defined algebra homomorphism from H(C p ) into A/I, and z j (π I (a)) = π I (a j ) for 1 ≤ j ≤ p.

We now give a folklore extension of this functional calculus by defining f (π

I (a)) for f ∈ H(C p × (C \ {0}) q ) when a = (a 1 , . . . , a p+q ) ∈ A p+q satisfies π I (a j ) ∈ Inv(A/I) for p + 1 ≤ j ≤ p + q. Proposition 2.3. (i) Set θ(z) = (z 1 , . . . , z p+q , z -1 p+1 , . . . , z -1 p+q ) for z = (z 1 , . . . , z p+q ) ∈ C p × (C \ {0}) q . Then the map θ : g → g • θ is a surjective continuous homomorphism from H(C p+2q ) onto H(C p × (C \ {.0}) q ).
(ii) Let I be an ideal of a commutative unital Fréchet algebra A, and let u :=

(u 1 , . . . , u p+q ) ∈ (A/I) p × Inv(A/I) q . Set, for f ∈ H(C p × (C \ {0}) q ), f (u) = g(a) + I, N. Surname, N. Surname: Short Title (pp. 1 -9)
where g ∈ H(C p+2q ) satisfies g•θ = f, and where a = (a 1 , . . . , a p+2q ) ∈ A p+2q satisfies π I (a j ) = u j for 1 ≤ j ≤ p + q and π I (a j ) = u -1 j-q for p + q + 1 ≤ j ≤ p + 2q.

Then the map f → f (u) is a well-defined homomorphism from H(C p × (C \ {0}) q ) into A/I such that z j (u) = u j for 1 ≤ j ≤ p + q. Proof: The map θ : g → g•θ is clearly a continuous homomorphism from H(C p+2q ) into H(C p × (C \ {0}) q ). Let f ∈ H(C p × (C \ {0}) q ).
We have

f (z 1 , . . . , z p+q ) = α∈N p ×Z q λ α z α ,
where λ α ∈ C, and where

z α = z α1 1 . . . z αp+q p+q for α = (α 1 , . . . , α p ) ∈ N p × Z q . Set (z 1 , . . . , z p+2q ) (α) = z|α1| 1 . . . z|αp+q| p+q ,
where zj = z j if α j ≥ 0, and where zj = z j+q if α j < 0.

Since the series

α∈N p ×Z q |λ α |R |α1|+•••+|αp+q| converges for every R > 0, the series α∈N p ×Z q λ α (z 1 , . . . , z p+2q ) (α) converges uniformly on compact subsets of C p+2q . Set g(z 1 , . . . , z p+2q ) = α∈N p ×Z q λ α (z 1 , . . . , z p+2q ) (α) .
Then g ∈ H(C p+2q ), and g

• θ = f, so θ is onto. Now let u = (u 1 , . . . , u p+q ) ∈ (A/I) p × Inv(A/I) q , let f ∈ H(C p × (C \ {0}) q
), and let g ∈ H(C p+2q ) such that g • θ = f . Let a = (a 1 , . . . a p+2q ) ∈ A p+2q and b = (b 1 , . . . b p+2q ) ∈ A p+2q such that π I (a j ) = π I (b j ) = a j for 1 ≤ j ≤ p + q and such that π I (a j ) = π I (b j ) = u -1 j-q for p + q + 1 ≤ j ≤ p + 2q. We have u j-q π I (a j ) = u j-q π I (b j ) = 1, hence u j-q (π I (a j ) -π I (b j )) = 0 for p + q + 1 ≤ j ≤ p + 2q. Since u j-q ∈ Inv(A/I), we have in fact π i (a j ) = π I (b j ) for 1 ≤ j ≤ p + 2q, and it follows from Proposition 2.2 that g(a) = g(b).

Clearly, (z

p+1 z p+q+1 -1)H(C p+2q ) + • • • + (z p+2q z p+2q-1)H(C p+2q ) ⊂ ker( θ). Now let g : (z 1 , . . . , z p+2q ) → α∈N p+2q λ α (z 1 , . . . , z p+2q ) α , where λ α = ∂ α (0,...,0) α! . Set αj = α j for 1 ≤ j ≤ p; for p + 1 ≤ j ≤ p + q, set αj = 0 if α j+q ≥ α j , αj = α j -α j+q otherwise; similarly for p + q + 1 ≤ j ≤ p + 2q, set αj = 0 if α j+q ≤ α j , αj = α j+q -α j otherwise.
Finally set αj = min(α j-q , α j-2q ) for p + 2q + 1 ≤ j ≤ p + 3q, and set, for

z = (z 1 , . . . , z p+3q ) ∈ C p+3q , g(z 1 , . . . , z p+3q ) = α∈N p+2q λ α (z 1 , . . . , z p+3q ) α.
Then g ∈ H(C p+3q ), and g(z 1 , . . . , z p+2q , z p+1 z p+q+1 , . . . z p+q z p+2q ) = g(z 1 , . . . , z p+2q ).

It follows from the definition of g that the Taylor series expansion at the origin of the function (z 1 , . . . , z p+2q ) → g(z 1 , . . . , z p+2q , 1 . . . , 1) has the form where ∆ = {β = (β 1 , . . . , β p+2q ) ∈ N p+2q | min(β j , β j+q ) = 0, p + 1 ≤ j ≤ p + q}. The map β → (z 1 , . . . , z p , z p+1 , . . . , z p+q , 1/z p+q+1 , . . . , 1/z p+2q ) β is a one to one map from ∆ into the set of Laurent monomials on C p × (C \ {0}) q .

So if g • θ = 0, then g(z 1 , . . . , z p+2q , 1, . . . , 1) = 0, and there exists h 1 , . . . , h q ∈ H(C p+3q ) such that we have g(z 1 , . . . , z p+2q ) = g(z 1 , . . . , z p+2q , z p+1 z p+q+1 . . . z p+q z p+2q )-g(z 1 , . . . , z p+2q , 1, . . . , 1)) = q j=1 (z p+j z p+q+j -1)h j (z 1 , . . . , z p+2q , z p+1 z p+q+1 , . . . z p+q z p+2q ) and so g ∈ q j=1

(z p+j z p+q+j -1)H(C p+2q ), and g(a 1 , . . . , a p+2q ) ∈ I if a p+j a p+q+j -1 ∈ I for 1 ≤ j ≤ q. So the definition of f (u 1 , . . . , u p+q ) does not depend on the choice of g, and the proposition follows.

The functional calculus described in Proposition 2.3 is easier to implement if Inv(A/I) = π I (Inv(A)), but this is not true in general.

Consider for example the Fréchet algebra A = H(C 2 ), and set I = (z 1 z 2 -1)A. Then A/I is isomorphic to H(C \ {0}, so A/I has an element without any square root while elements of π I (Inv(A)) = π I (exp(A)) have roots of all orders in A/I.

We now wish to describe the link between Michael's problem and iteration of holomorphic mappings pointed out by P.G. Dixon and the author in [START_REF] Dixon | Michael's problem and the Poincaré-Fatou-Bieberbach phenomenon[END_REF]. We will use the following notion. Definition 2.4. A (sequential) projective system is a sequence (En, φ n ) n≥1 where E n is a sequence of set and where φ n is an application from E n+1 into E n for n ≥ 1. The direct limit of the projective system is defined by the formula

lim ←- (E n , φ n )) := {(x n ) n≥1 ∈ Π n≥1 E n | x n = φ n (x n+1 ), n ≥ 1}.
We will use the notation π 1 (x) = x 1 for x = (x n ) n≥1 ∈ Π n≥1 E n . The proof of Theorem 2.1 [START_REF] Arens | Dense inverse limit rings[END_REF] uses the following general result.

Theorem 2.5. (abstract Mittag-Leffler theorem) Let (E n , φ n ) n≥1 be a projective system of complete metric spaces. Assume that φ n : E n+1 → E n is continuous and has dense range for n ≥ 1.

Then π 1 (lim ←-(E n , φ n )) is dense in E 1 .
Notice that since open subsets of complete metric spaces are homeomorphic to complete metric spaces, it follows from the abstract Mittag-Leffler theorem that the intersection of a countable family of dense open subsets of a complete metric space E is dense in E, but there is no way to deduce the abstract Mittag-Leffler theorem from the category theorem.

If be a nonincreasing sequence of Fatou-Bieberbach domains in C p and for n ≥ 1 let

f = C p → C is holomorphic,
F n ∈ H(C p ) such that Ω n = F n (C p ). Set G 1 = F 1 , and set G n = F -1 n-1 • F n for n ≥ 1. Then F n = G 1 • • • • • G n , and so lim ←- (G n , C p ) is isomorphic to ∩ n≥1 Ω n .
So the existence of a nondereasing sequence (Ω n ) n≥1 of Fatou-Bieberbach domains such that ∩ n≥1 Ω n = ∅ would imply that all characters on commutative unital Fréchet algebras are continuous. The research about Fatou-Bieberbach domains remains active [START_REF] Forstneric | Holomorphic families of Fatou-Bieberbach domains and applications to Oka manifolds[END_REF][START_REF] Forstneric | Fatou-Bieberbach domains in C n \ R k[END_REF][START_REF] Fornaess | A Fatou-Bieberbach domain intersecting the plane in the unit disc[END_REF][START_REF] Rosay | Holomorphic maps from C n → C n[END_REF][START_REF] Sabiini | Chaînes de contractions holomorphes et domaines de Fatou-Bieberbach[END_REF][START_REF] Vivas | Fatou-Bieberbach domains as basins of attaction of automorphisms tangent to the identity[END_REF] but the existence of such a sequence of Fatou-Bieberbach domains remains unknown.

We now outline a strategy to construct discontinuous characters on a 'big' Fréchet algebra. We will need the following notions, introduced under a different name by Roitman and Sternfeld in [START_REF] Roitman | When is a linear fuctional multiplicative?[END_REF].

Definition 2.8. (i) A weak Picard-Borel algebra is a commutative, unital algebra A such that card(C \ σ A (x)) ≤ 1 for every x ∈ A \ C.

(ii) A Picard-Borel algebra is a commutative, unital algebra A such that every family (u λ ) λ∈Λ of invertible elements of A which are pairwise linearly independent is linearly independent.

It follows from these definitions that every unital subalgebra of a (weak) Picard-Borel algebra is a (weak) Picard-Borel algebra.

A Picard-Borel (resp weak Picard-Borel) algebra A will be said to be trivial if Inv(A) = C. Notice that weak Picard-Borel algebras (resp. Picard-Borel algebras) are those commutative unital complex algebras which have the Picard property (resp. the Borel property) in the sense of [START_REF] Roitman | When is a linear fuctional multiplicative?[END_REF]. It is very easy to see that a Picard-Borel algebra is a weak Picard Borel algebra. More precisely the following result was proved in [START_REF] Roitman | When is a linear fuctional multiplicative?[END_REF].

Proposition 2.9. Let A be a commutative unital algebra. Then the following conditions imply each other (i) A is a weak Picard-Borel algebra.

(ii) Every family (u j ) 1≤j≤3 of three invertible elements of A which are pairwise linearly independent is linearly independent.

We refer to [START_REF] Roitman | When is a linear fuctional multiplicative?[END_REF] for a (very short) proof of this elementary result. Denote by A the subalgebra of C(X 1 , X 2 ) consisting of elements of the form f

= p Π m i=1 (X1+λiX2+λ 2 i )
, where (λ 1 , . . . , λ m ) is a finite family of complex numbers, and where p ∈ C[X 1 , X 2 ]. Roitman and Sternfeld showed in [START_REF] Roitman | When is a linear fuctional multiplicative?[END_REF] that A is a weak Picard-Borel algebra which is not a Picard-Borel algebra. This algebra has no character. We do not know any example of a Picard-Borel algebra without any character.

Let α, β ∈ C, and let λ ∈ C be such that λ 2 + βλ + α = 0. Then (X 1 -α) + λ(X 2β) = X 1 + λX 2 + λ 2 ∈ Inv(A), and so σ A (X 1 , X 2 ) = ∅, despite trhe fact that the spectum of every nonconstant element of A is very large.

Another example of a weak Picard-Borel algebra which is not a Picard-Borel algebra is given by the Fréchet algebra H(C 2 )/z 1 z 2 H(C 2 ), see below. Recall that the Jacobson radical Rad(A) of a commutative unital complex algebra A is the intersection of all maximal ideals of A, and that A is said to be semisimple if Rad(A) = {0}. The following easy result was proved in [START_REF] Esterle | Picard-Borel algebras[END_REF].

Proposition 2.10. Every weak Picard-Borel algebra is semisimple.

The algebra of entire functions on C is a Picard-Borel algebra, this follows from Borel's proof of the Picard theorem [START_REF] Borel | Sur les zéros des fonctions entières[END_REF], [START_REF] Nevanlinna | Le théorème de Picard-Borel[END_REF], and implies that H(C p ) is a Picard-Borel algebra for p ≥ 1.

An ideal I of a commutative unital algebra A is said to be a Picard-Borel ideal (resp. weak Picard-Borel ideal) if the quotient algebra A/I is a Picard-Borel algebra (resp. weak Picard-Borel algebra).

The author showed in [START_REF] Esterle | Picard-Borel algebras[END_REF] that Picard-Borel Fréchet algebras are integral domains, by using the fact that the space of continuous characters on a weak Picard-Borel Fréchet algebra is connected with respect to the Gelfand topology. This follows from the extension to Fréchet algebras of the Shilov idempotent theorem [START_REF] Dales | Banach algebras and automatic continuity[END_REF], [START_REF] Rosenfeld | Commutative F -algebras[END_REF]. This shows that closed Picard-Borel ideals in Fréchet algebras are prime. In particular the quotient algebra H(C 2 )/z 1 z 2 H(C 2 ) provides an example of a weak Picard-Borel Fréchet algebra which is not a Picard-Borel algebra. This result was extended to not necessarily closed ideals by the author in [START_REF] Esterle | Picard-Borel ideals in Fréchet algebras and Michael's problem[END_REF].

Theorem 2.11. Every Picard-Borel ideal of a commutative unital Fréchet algebra is prime.

We now explain the relevance of dense Picard-Borel ideals in Fréchet algebras to Michael's problem. Denote by C N [[X]] the algebra of all formal power series in infinititely many commuting variables (X n ) n≥1 , and denote by ∆ the space of sequences α = (α n ) n≥1 of nonnegative integers such that the set supp(α)

:= {n ≥ 1 | α n > 0} is finite or empty. For α ∈ ∆ set |α| = +∞ n=1 α n and X α = Π n∈supp(α) X αn n , with the obvious convention X α = 1 if α is the zero sequence. For n ≥ 1 set ∆ n := {α ∈ ∆ | supp(α) ⊂ {1, . . . , n}, |α| ≤ n}. For f ∈ C N [[X]], α ∈ ∆, denote by f (α) ∈ C the coefficient of X α in the power series expansion of f, and set p n (f ) = α∈∆n | f (α)| for n ≥ 1. The topological algebra (C N [[X]], (p n ) n≥1
) is a Fréchet algebra, and we have

f = α∈∆ f (α)X α (f ∈ C N [[X]]).
The following Fréchet algebra, discussed in [START_REF] Dixon | Michael's problem and the Poincaré-Fatou-Bieberbach phenomenon[END_REF], [START_REF] Esterle | Picard's theorem, Mittag-Leffler methods, and continuity of characters on Fréchet algebras[END_REF] is the second of the 'test algebras' for Michael's problem introduced by Clayton in [START_REF] Clayton | A reduction of the continuous homomorphism problem for Falgebras[END_REF]. Definition 2.12. The algebra U is a Fréchet algebra with respect to the sequence ( . n ) n≥1 of submultiplicative norms, and U is a 'test algebra' for Michael's problem, which means that the existence of a discontinuous character on any commutative Fréchet algebra would imply the existence of a discontinuous character on U. As usual c 00 denotes the space of complex sequences which vanish eventually, and 1 denotes the space of summable complex sequences. The following result was proved by the author in [START_REF] Esterle | Picard's theorem, Mittag-Leffler methods, and continuity of characters on Fréchet algebras[END_REF].

U := ∩ n≥1 {f ∈ C N [[x]] | f n := α∈∆ | f (α)|n |α| < +∞}, I p := X 1 U + • • • + X p U for p ≥ 1,
Theorem 2.13. Let λ = (λ n ) n≥1 ∈ 1 \ c 00 , and set I ∞,λ := I ∞ + +∞ n=1 λ n X n U.
Then I ∞,λ is a dense Picard-Borel ideal of U, and the following statements imply each other (i) There exists a discontinuous character on some commutative Fréchet algebra.

(ii) The Picard-Borel algebra U/I ∞,λ possesses a character.

Condition (ii) is indeed equivalent to the existence of a (necessarily discontinuous) character χ on U such that I ∞,λ ⊂ Ker(χ).

The union of a chain of (weak) Picard-Borel ideal is obviously a (weak) Picard Borel ideal, and so every (weak) Picard-Borel ideal is contained in a maximal (weak) Picard-Borel ideal.

Maximal weak Picard-Borel ideals of codimension > 1 cannot be maximal ideals, since σ

A (x) = ∅ for x ∈ A \ C if A is a division algebra of dimension > 1.
So any information on maximal weak Picard-Borel ideals or maximal Picard-Borel ideals would be relevant to Michael's problem. The fact that all Picard-Borel ideals in commutative unital Fréchet algebras are prime is already an important result.

We now give a (folklore) classification of singly rationally generated (weak) Picard-Borel Féchet algebras. on compact subsets of C, and u ∈ H(C), so that u = u(a). We thus see that the map u → u(a) is a continuous isomorphism from A onto H(C), and it follows from the closed graph theorem that this map is bicontinuous. Now assume that C \ σ A (a) = {α}. Replacing a by a -α, we may assume that σ A (a) = C \ {0}. Replacing C by C \ {0}, and replacing complex polynomials by complex Laurent polynomials in the proof above we see that A is isomorphic to H(C \ {0}).

The weak Picard-Borel ideals of H(C) (resp. H(C\{0}) have been characterized in [START_REF] Esterle | Picard-Borel ideals in Fréchet algebras and Michael's problem[END_REF]: these are the zero ideal and the kernels of point evaluations at elements of C (resp. C \ {0}). So these ideals are closed and are also Picard-Borel ideals. A consequence if this fact is that if I is a weak-Picard Borel ideal of a commutative unital Fréchet algebra and if f ∈ H(C) is not constant, then for λ ∈ C the only solutions in A/I of the equation f (x) = λ are the complex numbers µ such that f (µ) = λ.

In particular the solutions of the equation e x = 1 are the elements of 2iπZ, and e x and e y are linearly dependent if and only if x -y ∈ C. So if I is a Picard-Borel ideal of a commutative unital Fréchet algebra A, and if a family (x λ ) λ∈Λ of elements of A/I satisfies x λ -x µ / ∈ C for λ = µ, then the family (e x λ ) λ∈Λ is linearly independent.

The characterizatioin of Picard-Borel ideals of H(C) suggests the following question (we refer to [START_REF] Lelong | Entire functions of several complex variables[END_REF] for general information about H(C p ), p ≥ 2). A positive answer to this question would provide an interesting inductive structure on quotients of the form A/I, where A is a commutative unital Fréchet algebra and where I is a Picard-Borel ideal of A: if F = (u 1 , . . . , u k ) is a finite subset of A/I, set θ F (f ) = (f (u 1 ), . . . , f (u k )) for f ∈ H(C k ), and denote by U F the range of F. If the answer to question 2.15 were positive, then the Picard-Borel ideal ker(θ F ) would be closed, which would induce a Fréchet algebra structure on the Picard-Borel algebra

U F ≈ H(C k )/ker(θ F ). We have U F ⊂ U G if F ⊂ G. Denote i F,G : U F → U G the natural injection. It follows from theorem 2.1 that characters on U F and U G are continuous. If lim n→+∞ x n = a in U F and lim n→+∞ x n = b in U G , then χ(a) = χ(b)
for every character χ on U G and a = b since the Picard-Borel algebra U G is semisimple. It would follow then from the closed graph theorem that i F,G is continuous, and we would obtain A/I as the inductive limit of a family of finitely polynomially generated Fréchet algebras, which would suggest that A/I might possess characters.

Notice that if A is finitely rationally generated by a finite family (a 1 , . . . , a p ) and if I is a dense ideal of A then σ A/I (π I (a 1 ), . . . , π I (a p )) = ∅. To see this assume on the contrary that σ A/I (π I (a 1 ), . . . , π I (a p )) = ∅ and let (λ 1 , . . . , λ p ) ∈ σ A/I (π I (a 1 ), . . . , π I (a p )). Then (a 1 -λ 1 )A + • • • + (a p -λ p )A + uA A for every u ∈ I, and so there exists a continuous character χ u on A such that χ u (a 1 ) = λ 1 , . . . , χ u (a p ) = λ p , χ u (u) = 0. Since (a 1 , . . . , a p ) generates A, χ u does not depend on u, and I is not dense in A. But the example of the Roitman-Sternfeld algebra shows that the joint spectrum of two elements can be empty at least in weak Picard-Borel algebras.

N. Surname, N. Surname: Short Title (pp. 1 -9) Michael's problem, i.e. the question of continuity of characters on commutative unital Fréchet algebras is still unsolved. The fact that the answer is positive for finitely rationally generated Fréchet algebras, and the fact that the existence of a nonincreasing sequence of Fatou-Bieberbach domains in C p having an empty intersection for some p ≥ 2 would imply the continuity of characters on all commutative unital Fréchet algebras suggest that the answer to Michael's problem could be positive.

In the opposite direction the existence of a discontinuous character on some commutative unital Fréchet algebra is equivalent to the existence of a character on a quotient algebra of the form U/I where U is a 'test algebra' for Michael's problem and where I is a dense ideal of U which is a Picard-Borel ideal, which means that every family of pairwise linearly independent invertible elements of U/I is linearly independent. The class of such ideals admits maximal elements, and the study of these maximal elements could open the gate to a negative answer to Michael's problem. It was recently discovered that all Picard-Borel ideals in commutative unital Fréchet algebras are prime, and a characterization of Picard-Borel ideals of H(C p ), p ≥ 2, would provide general information on quotients of commutative unital Fréchet algebras by Picard-Borel ideals. .

  as was shown by Michael in his seminal memoir [14]. Denote by σ A (a) := {λ ∈ C | a -λ / ∈ Inv(A)} the spectrum of a ∈ A, and denote by σ A (a 1 , . . . , a k ) := {λ 1 , . . . , λ k ) ∈ C k | (a 1 -λ 1 )A + • • • + (a k -λ k )A A} the joint spectrum of a finite family (a 1 , . . . , a k ) of elements of A. When A is commutative, let Spec(A) be he space of continuous characters of A, equipped with the usual Gelfand topology, It follows from Michael's isomorphism theorem that σ(A)(a) = {χ(a)} χ∈Spec(A)

Theorem 2 . 1 .

 21 Characters on finitely rationally generated commutative unital Fréchet algebras are continuous.We now introduce the usual holomorphic functional calculus on quotients of a commutative unital Fréchet algebra A. Let H(U ) be the algebra of holomorphic functions on an open set U ⊂ C p , p ≥ 1.

For

  g(z 1 , . . . , z p+2q , 1 . . . , 1) = β∈∆ µ β (z 1 , . . . , z p+2q ) β , N. Surname, N. Surname: Short Title (pp. 1 -9)

  and if I is an ideal of a commutative unital Fréchet algebra A, Proposition 2.1 allows to define f (u) ∈ A/I for u ∈ (A/I) p by using the formula f (π I (a)) = π I (f (a)) (a ∈ A p ). N. Surname, N. Surname: Short Title (pp. 1 -9)

N

  . Surname, N. Surname: Short Title (pp.1 -9)

  and I ∞ := ∪ p≥1 I p . N. Surname, N. Surname: Short Title (pp. 1 -9)

Proposition 2 . 14 .

 214 Let A be a singly rationally generated Fréchet algebra which is a weak Picard-Borel algebra. Then there are three possibilities (i) Spec(A) is a singleton, and A ≈ C. (ii) Spec(A) is homeomorphic to C, and A ≈ H(C), (iii) Spec(A) is homeomorphic to C \ {0}, and A ≈ H(C \ {0}). Proof: Let a be a generator of A. The map χ → χ(a) is as well-known a homeomorphism from Spec A (a) onto σ A (a), and the only possibilities for σ A (a) are a singleton, C, or C \ {α} where α ∈ C. Since A is semisimple, the Gelfand transform G : u → u, where u(χ) = χ(u), is a one-to-one continuous homomorphism from A into C(Spec(A)). So A is isomorphic to C if σ A (a) is a singleton. Now assume that σ A (a) = C, and set u(λ) = χ λ (u) for u ∈ A, where χ λ ∈ Spec(A) denote the character of A satisfying χ λ (a) = λ. We can identify u with G(u), and so u ∈ C(C). If f ∈ H(C), λ ∈ C, we have f (a)(λ) = χ λ (f (a)) = f (χ λ (a)) = f (λ), so f = f. Let u ∈ A. There exists a sequence (p n ) n≥1 of complex polynomials such that lim n→+∞ p n (a) = u with respect to the topology of A. So lim n→+∞ p n (λ) = u(λ) uniformly N. Surname, N. Surname: Short Title (pp. 1 -9)

Question 2 . 15 :

 215 Are Picard-Borel ideals of H(C p ) closed for p ≥ 2?

N. Surname, N. Surname: Short Title (pp.1 -9)

Similarly if p ≥ 1, q ≥ 1, and if F = (f 1 , . . . f q ) : C p → C q ) is holomorphic we can define f (u) ∈ (A/I) q for u ∈ (A/I) p by using the formula F (π I (a)) = π I (F (a)) (a ∈ A p ).

The following result [START_REF] Dixon | Michael's problem and the Poincaré-Fatou-Bieberbach phenomenon[END_REF][START_REF] Esterle | Picard's theorem, Mittag-Leffler methods, and continuity of characters on Fréchet algebras[END_REF] is a conqsequence of the abstract Mittag-Leffler theorem.

Theorem 2.6. Let (C pn , F n ) n≥1 be a projective system, where F n : C pn+1 → C pn is holomorphic for n ≥ 1, let A be a commutative unital Fréchet algebra and let I be an ideal of A.

If I is dense in A, then lim ←-

Proof: We give a proof for the convenience of the reader. Set E 1 = A p1 , E n = A pn × I p1+•••+pn-1 for n ≥ 2, and equip A with the given Fréchet topology, equip I with the discrete topology and equip E n with the associated product topology. The discrete topology on I is the topology associated to the distance d defined by the formulae d(x, y) = 1 if x = y, d(x, y) = 0 if x = y, and so E n is homeomorphic to a complete metric set.

For u = (v, w p1 , . . . , w pn ) ∈ E pn+1 , set

We have u n = (v n , w (n) ), where v n ∈ A n and w (n) ∈ I p1+•••+pn-1 , and it follows from the definition of

Let χ be a character on a commutative unital Fréchet algebra A. It follows from formula (2) that χ(f (a)) = F (χ p (a)) for a = (a 1 , . . . , a p ) ∈ A p , f ∈ H(C p ), where χ p (a) = (χ(a 1 ), . . . , χ(a p )).

Corollary 2.7. [START_REF] Dixon | Michael's problem and the Poincaré-Fatou-Bieberbach phenomenon[END_REF] Assume that there exists a discontinuous character on some commutative unital Fréchet algebra A. Then lim

Proof: Assume that a character χ on some commutative unital Fréchet algebra A is discontinuous, and let I = Ker(χ). It follows from Theorem 2.6 that there exists a sequence (a (n) ) n≥1 ∈ Π n≥1 A pn such that a (n) -F n (a (n+1) ) ∈ I pn for n ≥ 1. We have

and so (χ pn (a

Classical results of Bieberbach and Fatou, see for example [START_REF] Rosay | Holomorphic maps from C n → C n[END_REF], show that there exist for p ≥ 2 some one-to-one holomorphic maps from C p into itself having nondense range, and the ranges of such maps are called Fatou-Bieberbach domains. Let (Ω n ) n≥1 N. Surname, N. Surname: Short Title (pp. 1 -9)


Westerners consider hugging to be both highly meaningful and intrinsically beneficial.

The oft-touted (but seemingly apocryphal) quote from Family Therapy founder Virginia Satir says it all: 'We need four hugs a day for survival. We need eight hugs a day for maintenance. We need twelve hugs a day for growth.' Unsurprisingly, prolonged, intense hugging-between people, but also with trees, animals or visualised beings-is one of the hallmarks of ritualised practices associated with what have been called contemporary spiritualities: New Age-inspired movements, Contemporary Paganisms, Personal Development initiatives, etc. This chapter explores the interactive workings of such pointedly meaningful embracing which I will refer to as 'immersive' hugging. Drawing on ethnographic fieldwork and other illustrative material, I attempt to identify some of the recurrent features of immersive hugs in order to speculate about what is at stake in their performance.

HUGS GALORE

Hugging as a component of everyday interaction is both partially idiosyncratic and, like most everything else, governed by cultural codes. This has been made clear by the profusion of YouTube videos on hugging etiquette that has accompanied the recent upswing of hugging in the United States. Excluding the many videos on animal or tree hugging, as well as those on how to hug in a variety of online games, more than 150 such clips have been uploaded since 2007, the most popular of which, dating from May 2018, totals almost 15 million views. Consider, for example, the videos providing instructions on how to 'man hug' or 'bro hug'. Some with wacky humour, others in dead seriousness, advise men on how to avoid imbuing this samesex embrace with homosexual undertones: keep your feet toes-to-toes, place one arm over and the other under your friend's arms, keep it short, don't talk while hugging, and no snuggling or stroking. Other videos specify the proprieties and pitfalls of heterosexual hugging (interestingly, there are no videos on hugging between women, which seems to be deemed unproblematic). In principle, such ordinary, mundane hugs, regardless of the conventions that guide them, are presumed to reflect the everyday relations between the huggers concerned. All else being equal, the more intimate people are, the more intense their hugging is likely to be.

Hugging may also be a canonical ceremonial practice. Examples include the solemn accolade that confers military and other honours in France, the consoling, darshanbestowing embrace of the Hindu guru Mātā Amritānandamayī (Amma, the 'hugging saint'), the three-fold hug that ushers in the Muslim festival of Eid al-fitr, and the liturgical greeting known as the 'kiss of peace' or 'holy kiss' in many Eastern and Western Christian traditions. While such hugs may be inflected by interpersonal ties, they are above all indicative of a more impersonal, ritual relationship the hugging parties are held to enact: allegiance to a higher authority, the communication of blessedness to a disciple, reciprocal well-wishing between members of a devotional community.

There are also hugs that stand midway between commonplace and ceremonial hugging, such as the warm, public handshake-hug sometimes performed by male politicians and celebrities. This one-armed embrace Wikipedia describes as a 'pound hug' (also dude hug, cootie hug, homie hug, bro hug and thug hug) is often announced in a visible yet understated I-know-you-know-I-know fashion by leaning slightly back with arms extended downward at a 30-degree angle, and consists in a tight handshake inserted between the huggers' bodies, accompanied by several firm, open-handed taps on the back.

In all these cases, hugging assumes a form that, as a rule, is consonant with the relationship between those who hug. The formal qualities of the hug itself-how long it lasts, the emotions it exhibits, the positions adopted, the body parts involved, the degree of pressure exerted, the symmetries or asymmetries brought into play, etc.articulate the nature of this relationship. Immersive hugging, however, seems to proceed from somewhat different premises.

BEING INTIMATE WITH CASUAL ACQUAINTANCES

Immersive hugs, sometimes called 'spiritual', 'therapeutic', 'heart-to-heart' or simply 'real', are embraces of exceptional intensity and duration presumed to have salutary effects in and of themselves. Known in some circles as the 'Findhorn full-frontal' in reference to the renowned British New Age 'colony of seekers' (Sutcliffe 2003: 164), immersive hugging is an inevitable feature of Pagan festivals, well-being workshops, spirituality training sessions and personal development practices of all sorts. In the wake of a mutually accepted invitation (a smile, an eyebrow flash, an opening of arms), people wrap their arms around each other and enter into a tight, face-to-face embrace. Eyes closed or unfocused, bodies still or gently rocking, participants appear to be fully absorbed in a blissful, self-contained whole. As their hug lengthens, they rearrange their grasp several times, repositioning their arms, moving their heads closer or further apart, pulling slightly back so as to fit their bodies together once again, as though exploring new ways of expressing their caring attentiveness and of savouring the sensations of safety and warmth their hugging provides. After a long while, they gently draw apart, wordlessly communicating feelings of gratitude and affection with a knowing smile, a kiss, a caress, a slow blink or a quick final hug.

What is remarkable about this way of hugging is that it acts out a heightened interpersonal familiarity between people who usually know each other only casually, if at all. Indeed, one of characteristic features of immersive hugging is the striking hiatus between the intense personal intimacy it displays and the lack of interpersonal commitment this display might ordinarily be expected to imply. In this regard, it is significant that immersive hugging typically occurs in the context of semi-public, structured events whose specialness is readily acknowledged. Speaking for example of personal growth-oriented dance practices like Biodanza or 5 Rhythms in which immersive hugs abound (Houseman and Mazzella di Bosco 2020), one participant explained that at such venues 'you can connect with [people] on another, higher, […] but then, outside, it's as if we suddenly don't speak the same language anymore'.

Another said she feels 'moments of profound connection with other dancers, […] but after the dance session, if I try to speak and socialize with them, it's as though everything has flattened out, the magic is lost'. The expectations that hold sway in such contexts diverge significantly from those of everyday interaction. I learnt this to my cost when, on the train platform going home from my first Pagan festival, my open-armed invitation to hug was violently rebuffed as an unwarranted aggression by someone I (wrongly) assumed had been at the festival as well. Inversely, as one person put it: 'If I go to a Pagan festival and encounter a few people I know well, I'll also hug them with a similar intensity to those strangers I hug at the same event.' Thus, partaking in an immersive hug with someone neither presupposes a close relationship with them, nor does it anticipate that such a relationship will follow.

Immersive hugging tends to take place between those Karen [START_REF] Fingerman | Consequential Strangers and Peripheral Partners: The Importance of Unimportant Relationships[END_REF] has called 'consequential strangers': people one may be familiar with, but with whom one feels free not to entertain a close relationship.

In light of this, one might be tempted to see the immersive hug as a ritual act. I will be arguing precisely along these lines. Specifically, in keeping with Véronique Altglas's analysis of contemporary religious exoticism (2014: chapter 7), I see immersive hugging as a ritualised enactment of certain Euro-American middle-class aspirations and sensibilities closely linked with the emergent sociological category Pierre Bourdieu, writing at the end of the 1970s, identified as the 'new petty bourgeoisie ' (1979: 409-431). However, because the 'spontaneous', 'heartfelt qualities of immersive hugs are such a far cry from the distanced conventionality typical of ceremonial hugging generally, the idea that immersive hugging might be a ritual practice needs to be spelled out in detail. This will then allow me to envisage immersive hugging as a possible model for thinking about much contemporary Western ritual.

FEELING ONESELF WITH OTHERS

To get some idea of what immersive hugging as a ritual practice might be about, let us look at some of its formal features. Immersive hugs are voluntary and emphatic.

Whether they are entered into gently or with force, they are enthusiastic and intense.

Huggers hold each other tightly, and often tenderly caress each other's bodies (firm back-tapping almost never occurs). However, such hugs are also markedly undifferentiating. An immersive hug is a full-body, frontal embrace that emphasises participants' personal presence while being largely indifferent to individual-specific attributes like gender, age, size and physical condition. As was made clear to me during one Pagan festival, although huggers increasingly seek consent, it is because personal peculiarities are so earnestly disregarded in this way that some attendees make it known beforehand that they do not wish to be hugged.

Immersive hugs are deeply resonant with feeling and yet carelessly generic. They are performed to be highly expressive emotionally and intentionally, while being at the same time conspicuously depersonalised. Now, the depersonalised, generic nature of immersive hugging is fully congruent with the idea that the relationship it enacts is a conventional, ceremonial one. But what, then, are we to make of the intense intimacy and exacerbated affectivity its performance also displays? I suggest that the latter bears witness to the fact that unlike ceremonial hugging generally, immersive hugging is centred less on the acting out of a special (ritual) relationship, than on the acting out of special (ritual) selves. In other words, the value and meaning of an immersive hug pertain less to participants' shared experience of an extra-ordinary connection with one another, than to their respective experiences of becoming exceptional individuals capable of making such a connection.

To see how this might be so, consider another 'obvious aspect' [START_REF] Rappaport | Ecology, Meaning and Religion[END_REF] of immersive hugging: it is inordinately long. In contrast to the three (plus or minus two) seconds of both everyday and ceremonial hugs (cf. Nagy 2011), twenty seconds is often presented as the minimum duration for a 'real' hug. According to popular renderings of some experimental work done between 2005 and 2015 (see [START_REF] Field | Social Touch, CT Touch and Massage Therapy: A Narrative Review[END_REF] for a review), this is the time needed for huggers' oxytocin (dubbed the 'love hormone') to be released. Now, several laboratory studies have demonstrated the comforting effects of touch before and during stressful situations (Field 2019: 375).

However, it remains that a close embrace between adults, if prolonged, tends to make people anxious, as borne out by popular 'prank' videos documenting overlylong hugs between strangers (e.g. YouTube posts by Angrypicnic, BigDawsVlogs, and The Daily Dropout). This is especially true in the absence of an intimate relationship that provides extended body contact with shared meaning. As a result, immersive huggers inevitably experience what marital and couple therapist David Schnarch calls 'the jolt ' (1997: 166). This is the sudden dawning of a heightened selfawareness that occurs when a hug which normally would have broken off (after three to four seconds) keeps going. It gives rise to noticeable shifts in muscle tension, balance and breathing. Indeed, immersive hugs always have two phases: a short coming together part during which huggers grasp each other in a tight, fusional clinch, and a lengthy staying together part during which they self-consciously explore the sensations and feelings that accompany each other's bodily readjustments and gestured initiatives (changes of grip, caresses, eye-contact, etc.). When initiating an immersive hug, participants surrender to rushes of compassionate intention and to the diffuse pleasures of bodily touch. In doing so, they jointly aspire to the deeper connection their intense hugging is presumed to exhibit and/or bring about. However, because their hug is so long-lasting, their efforts are inescapably modulated by various complications-a flash of erotic excitement, a sense of disquiet, moments of physical discomfort, an unexpected smell, an awareness of possible observers, etc.-the handling of which prompts them to attend to their own personal sentiments and attitudes. Continually jolted in this manner, hugging partners are repeatedly thrown back upon themselves, moved to harken to and acknowledge their respective thoughts and feelings. In the case of prolonged hugging between intimate partners, Schnarch explains, making such 'a solid connection with yourself' is 'the key to [the] interdependence' that life as a couple entails (1997: 167). However, immersive hugging typically takes place between non-intimates who do not aspire to this type of open-ended interdependency. Instead, becoming aware of one's own feelings during sustained close contact with another takes on meaning and value in its own right as an intimation of one's capacity to successfully 'connect'.

Immersive hugging allows participants to demonstrate-to themselves, to each other, to onlooking parties-their respective capacities for 'really' hugging and for being 'really' hugged. It lets them experience themselves, and be experienced by others, as individuals endowed with certain exemplary if difficult-to-define qualities of thought and feeling deemed essential to the fulfilment of contemporary (middle-class) Western personhood: 'spontaneity', 'authenticity', 'deep feeling', 'self-awareness', 'openness to others', etc. In other words, those who partake in immersive hugs do not so much enter into intense, intimate relationships with others as they are afforded salient experiences of what current Euro-American understandings of personal identity invite them to construe as their 'potential' to do so. In this respect, purposeful, fervent hugging can be seen as a ritualised process of self-construction in which participants willingly act as resources for each other's development.

This reciprocally enabled, self-directed quality of immersive hugging comes out clearly, for example, in the 'Hugging meditation' video posted on YouTube in 2010 by journalist and writer Todd Aaron Jensen, author of On Gratitude (2010). This meditation, Jensen earnestly explains, is a 'simple' practice that 'works with everybody' to help us 'be […] our higher selves'. It requires a partner, who 'can be a child, a parent, a friend, a lover, a complete stranger', and is made up of three successive hugs. The 'meditative thought' for the first embrace is 'I may not be here long, but I am here now', for the second 'This person may not be here long, but there're here now', and for the third 'there is a mutual acknowledgement: we may not be here long, but we're here now'. Jensen concludes: 'If we can practice our love with that level of mindfulness, and acceptance, and surrender, and gratitude, we are truly free, and our heart truly open to give and to receive.' There exist a number of other versions of this practice, such as that promoted as a mindfulness exercise by the Buddhist monk Thich Nhat Hanh and others. It entails bowing ('offering a lotus') to each other and then taking three deep, long breaths together while hugging 'heart-toheart' (left sides touching). During the first breath participants think 'I am alive', during the second 'This person is alive', and during third 'I feel happy' or 'I feel gratitude', before finishing with a closing bow. In 'absorptive' [START_REF] Luhrmann | The Absorption Hypothesis: Learning to Hear God in Evangelical Christianity[END_REF] practices such as these, as in immersive hugging generally, hugging partners' personal identities are plainly subordinate to the role each plays in affording the other with an intensified experience of whom they feel they can become.

Immersive hugging takes full advantage of the potential two-facedness of hugging as a close interaction in which participants do not meet each other's gaze. In the absence of eye-contact, sensations aroused by the prolonged pressing of another person's body against one's own easily take on a life of their own, with the other person being temporarily abstracted from the picture. In this light, it is tempting to take tree hugging, not to prevent logging but to promote well-being, as an exemplary model. As many Internet sites attest, this pleasurable display of embellished intimacy with an unfamiliar other involves consciously imbuing one's behaviour towards one's (other-than-human) partner with heart-felt, well-meaning intentions, all the while paying careful attention to one's own perceptions and feelings. And, in the course of tree hugging as it is usually construed (but not always, cf. [START_REF] Pike | For the Wild: Ritual and Commitment in Radical Eco-Activism[END_REF], what personal agency this partner might have becomes sublimated into a source of 'energy' or 'connection' that allows the human hugger to knowingly raise their vibratory level, commune with Nature, recycle their worries, and reduce their stress. There are virtual adaptations of this in which the tree is materially absent, such as the Qigong meditation exercise of 'hugging the tree' (also called 'standing like a tree'). Some Personal Development advocates recommend a still more radically self-serving form of immersive hugging in which one (physically or mentally) hugs oneself (see wikiHow staff 2019 for a simple illustrated example). Eye Movement Desensitization and Reprocessing (EMDR) therapy, for instance, promotes the self-administered 'butterfly hug' to alleviate the distress associated with traumatic memories [START_REF] Shapiro | Getting Past Your Past: Take Control of Your Life With Self-Help Techniques from EMDR Therapy[END_REF][START_REF] Artigas | The Butterfly Hug Method for Bilateral Stimulation[END_REF]. During the 2020 Covid-19 lockdown, British massage therapist Sally Morris provided a mindfulness-inspired 'self-hug meditation' YouTube video as a self-care support for those 'who are missing that physical contact with other people', and Becky Rosenthal posted a self-hugging meditation 'to soothe and take care of ourselves' on the UNION SA YouTube channel ('We empower positive transformation'). Finally, somewhere between another being such as a tree and one's own body are a variety of special vests and other electromechanical devices that generate heat and/or pressure to simulate hugging.

PERFORMING A TRANSCENDENT SOCIALITY

It would be a mistake, however, to understand immersive hugging as directed solely towards individual self-fulfilment. An immersive hug not only requires the presence of a consenting partner, whose reactions (and my reactions to their reactions) I am expected to be aware of, but like many spiritual practices, typically takes place in the company of still further attending others who are often similarly engaged. In short, another formal feature of immersive hugging is that it is exhibited. Now, to the extent that immersive hugs are presumed to be witnessed, their enactment becomes a performance. Participants' embodied representation of the ideal dispositions to which they aspire becomes addressed, and the participants themselves become both the wellsprings of desire that inspire hugging to happen, and the distanced, monitoring observers of their own feelings and behaviour. This assumption of an outside point of view underscores the conventional aspects of immersive hugging. It also forestalls possible abusive misconduct and acts as a hedge against the interpersonal alienation that prolonged bodily contact can easily bring about. The performative dimension of their embrace allows hugging parties to interact in a highly intimate, affectively engaged fashion while minimising the risks of the emotional commitment and interpersonal 'interdependency' [START_REF] Schnarch | Passionate Marriage: Sex, Love and Intimacy in Emotionally Committed Relationships[END_REF] or 'contamination' [START_REF] Moisseeff | Dealing with Death in Contemporary Western Culture: A View from Afar[END_REF]) such behaviour would naturally imply if they were on their own. To the degree to which an immersive hug is freed in this way from the sticky, endlessly negotiated entanglements of close interpersonal ties, it provides an exceptional-ritual-space in which participants' individual autonomy is preserved, allowing them to more fully explore, physically and emotionally, whom they feel they might really be. In this regard, the profuse picture-taking and video-recording that accompanies Free Hugs campaigns and other 'social experiments' involving pointedly meaningful hugging is a telling example of the attendant third-party perspective immersive hugging implies (see for instance ikizlerTV's 2014 'I Trust You, Do You Trust Me? Hug Me' video on YouTube). Moreover, as previously mentioned, immersive hugging usually takes place as an expected feature of certain special venues or practices: festivals, self-development workshops, etc. It may occur in an impromptu fashion, or may be purposely initiated by others, such the Hug Patrols 'who wander around [Rainbow] Gatherings hugging people' (Niman 1997: 289). It is often provoked by those leading or facilitating these special activities or events. In this respect, immersive hugs, whether they are tacitly anticipated or explicitly encouraged, generally happen in response to incitements that are less directives than verbal or non-verbal 'invitations' whose authority-depreciating quality is characteristic of power relations in the context of alternative spiritualities (cf. [START_REF] Wood | Possession, Power and the New Age: Ambiguities of Authority in Neoliberal Societies[END_REF]. Archetypal in this respect is the paradoxical injunction that ended the closing ritual of a four-day English Druid Camp I took part in in 2015: 'Let the spontaneous hugging begin!' Finally, a purely egocentric appreciation of immersive hugging is belied by another of its constitutive features which are the expressions of mutual recognition and gratitude it invariably occasions. Immersive hugs always end with participants looking deeply into each other's eyes and with some kind of concluding sign: a knowing look, a shared smile, a bow, a slow blink. By ratifying their common involvement in this way, participants acknowledge the part each willingly plays in allowing the other to experience the more accomplished self-spontaneous, authentic, emotionally expressive, open to others, etc.-they have momentarily become. More than the act of hugging itself, it is this shared awareness of oneself-with-another as the enabling environment of one's own self-discovery that underlies the sentiments of glad companionship and fortuitous like-mindedness that participants generally take away from a rewarding immersive hug. When immersive hugging occurs with multiple partners within the context of regular or extended activities-weekly dance sessions, weekend workshops, week or month-long festivals, etc.-such feelings of reciprocal recognition and demonstrative gratefulness easily reverberate outward to include the group as a whole, prompting invocations of 'family' or 'home ' (cf. Pike 2001: 13, 29-37) as well as drawn-out group hugs ('cuddle puddles' in Rainbow parlance [Niman 1997: 285]). Inversely, as several long-time Pagans have pointed out to me, immersive hugging, as the embodiment of a shared world-view and set of attitudes, can also lose a measure of its ritual character to become a customary expression of community membership (cf. Prince and Riches 2000: 116). In such cases, the final mutual acknowledgement phase of immersive hugging tends to prevail over the warm but abbreviated embrace itself, the latter becoming an index of the antecedent hugs that are presumed to have taken place between the parties concerned.

Recall, however, that immersive hugging does not imply ongoing interpersonal intimacy. Mentions of 'home' and 'family' notwithstanding, those taking part in groups and/or networks in which immersive hugs regularly occur are rarely embroiled in the ambivalent co-dependencies and continually renegotiated accountabilities so characteristic of lasting emotional ties. Their interactions, undertaken in a spirit of affable complicity implying limited interpersonal liability, are upheld by a more ephemeral, freer sociality in which each is expected to pursue their personal fulfilment while acting as a willing resource for the personal fulfilment of others. Thus, while the social dimensions of immersive hugging are assuredly present, they pertain less to everyday relations than to this special, ritually circumscribed mode of relatedness in which the culturally constructed opposition between individual autonomy and collective solidarity is happily resolved. Indeed, in many respects immersive hugging is emblematic of this eminently Western utopic social model that one of [START_REF] Tavory | A Collective of Individuals": Between Self and Solidarity in a Rainbow Gathering[END_REF] informants, in reference to Rainbow gatherings, dubbed a 'collective of individuals' (cf. Houseman 2020 on 'collective individualism'). This is a situation, explain these two authors, in which 'both solidarity and individuality are constructed simultaneously-as an ongoing set of pragmatic achievements, mutually shaping each other in the field ' (2009: 262-263; see also [START_REF] Pike | Magical Selves, Earthly Bodies: Contemporary Pagans and the Search for Community[END_REF] on Pagan festivals generally, and Batchelor 2014 on New Age yoga and Sufi dance practices in England). In this perspective, it is perhaps no accident that intense, meaningful bodily contact between 'consequential strangers' so often occurs in contemporary spiritual practices that configure individual realisation and community fellowship as conjunct ideals. It may well be that such close dyadic interactions that accentuate feelings of personal presence while downplaying interpersonal commitment, notably when deployed in a round-robin pattern, are especially instrumental in mediating the connection between these two allegedly antagonistic poles (cf. [START_REF] Moisseeff | L'orchestration rituelle du partage des émotions et ses ressorts interactionnels[END_REF]. Exemplary in this regard, as far as new religious movements are concerned, is the YouTube video in which 'People from all over the world meet at Osho Humaniversity in the Netherlands celebrating the international Hugging Day 2009'. As several dozen people tenderly embrace one another, a deeply resonant voice, accompanied by soothing music, slowly explains: I know we can make Osho's dream into a reality, just by hugging you right now, exchanging warm vibrations of love and allowing our hearts to say yes. … We are one, and God is happy. … Hugging is only a [unintelligible] of love, of want, of gaiety. … I know we can make Osho's dream into a reality. You see, together we can create a beautiful world to live in, filled with friends and lovers.

A POSSIBLE MODEL?

Immersive hugging, then, is an (1) overly emphatic, (2) markedly undiscriminating (3) protracted embrace (4) between non intimates that is (5) exhibited, (7) enjoined and (8) gratefully acknowledged. I have suggested that it is best understood as a ritual act. As such, it affords participants with memorable, conventional if somewhat hard to pin down experiences in which supposed opposites-close intimacy and interpersonal remove, absorptive bodily engagement and heightened selfconsciousness, individual autonomy and mutual recognition in the service of collective solidarity-are made to go hand in hand. In the manner of rituals generally, to the extent that these special experiences remain difficult to account for in terms of everyday attitudes and patterns of behaviour, they are ill-suited to provide participants with viable models for their daily lives. What they do provide them with, however, are largely unassailable, lived-through touchstones in light of which their everyday interactions can be customarily appraised. Close contact with non-intimate others during immersive hugging can, for example, act as a privileged yardstick for appreciating what intimate contact with close others might and should be.

Concomitantly, on a collective level, immersive hugging is one of the myriad ways whereby simply through the performance of certain acts, the axiomatic idea-values [START_REF] Dumont | The Anthropological Community and Ideology[END_REF]) of a particular cultural tradition-'spontaneity', 'authenticity', 'deep feeling', 'bodily pleasure', 'self-awareness', 'openness to others', etc.-are enshrined, embodied and passed on.

I would like to conclude by using the analysis offered here to think about ritualised practices linked with contemporary spiritualities in general. Immersive hugging displays on an easily accessible scale many of these practices' characteristic properties, the foremost of which is a kind of built-in reflexivity. As we have seen, because immersive hugs are so long-lasting, hugging parties are repeatedly jolted into becoming attentive to their own thoughts and feelings. This type of heightened self-consciousness is not a necessary component of ritual per se. Indeed, in many rituals in which certain ordained actions must be undertaken correctly, it is important that participants pay attention to what they are doing (Smith 1987:103), but it is not necessary that they be knowingly aware of their own frame of mind while doing so. This is because the distinctive complexity of such rituals that gives their performance special potency pertains above all to the somewhat mysterious actions participants are expected to undertake. Take, for example, the case of a priest declaring 'This is my body' during a Catholic mass while displaying a wafer of bread. This construes what he is holding to be at once bread and the body of Christ, regardless of what his personal dispositions might be. He may be heedful of his own thoughts and feelings, but his vigilance in this regard is not constitutive of the ritual act he performs, which, as the reiteration of behaviour presumed to have been undertaken previously by more authoritative others, is deemed to have value and efficacy in and of itself.

In contrast to this, mindful reflexivity is an integral aspect of many ritual practices associated with contemporary spiritualities. This is often thought to be because these practices have yet to become part of established traditions. However, new rituals crop up all the time in many cultural contexts without self-attentiveness being emphasised. In light of this, it is tempting to link such an emphasis to Western modernity. Tanya Luhrmann (2012), for example, has argued that the 'self-conscious epistemological complexity' found not only in Contemporary Paganism and Neo-Shamanism, but also in evangelical Christianity, is strongly stimulated by participants' awareness of how much their practices and beliefs deviate from mainstream secular thinking. At the same time, from a historical perspective, the self-conscious epistemology she refers to can be understood as but the latest religious expression of a cultural tradition-what Colin [START_REF] Campbell | The Romantic Ethic and the Spirit of Modern Consumerism[END_REF] has called 'the romantic ethic'deriving from Pietistic Protestantism and dating back to the eighteenth century. All told, it would seem that the exacerbated self-awareness characteristic of many present-day spiritual practices, while perhaps not exclusive to Euro-American culture, is particularly well-developed in the contemporary West.

I understand this type of enhanced reflexivity to be an essential component of a specific mode of ritualisation that has gone largely unrecognised, at least by anthropologists. As I have suggested elsewhere (2007,2016,2020), the distinctive complexity of ceremonial activities organised according to this mode relates less to exceptional items of behaviour participants are supposed to undertake, than to special ways of thinking and feeling their behaviour is alleged to embody. For example, many ritual practices, ranging from Contemporary Pagan ceremonies to New Age inspired guided meditations begin with acts of 'centring' or 'grounding'.

Participants may be invited to form a circle holding hands, quiet their minds, take a deep breath, become fully present and feel their connection with the earth. While the actions involved-standing in a circle, holding hands, taking a deep breath-may be unusual, they are readily intelligible in and of themselves in a way that calling a piece of bread the body of Christ, for instance, is not. More puzzling, on the other hand, are the special dispositions these actions are taken to express and/or bring about: the ability to 'quiet one's mind', 'become fully present' and 'feel a connection with the earth'. Now, what bridges the gap between the participants' unremarkable actions and the remarkable things their performance of these actions allows them to accomplish are the participants themselves, construed as at once ordinary and extraordinary beings. Indeed, the mysterious, highly salient focus of rituals such as these-equivalent to the priest holding something that is both bread and the body of Christ-is the two-fold experience of self they engender and display.

Participants are induced both to experience certain special ways of thinking and feeling, and to consciously experience themselves as doing so. Immersive hugging incites huggers to appreciate themselves and each other as exceptional individuals able to 'connect' with others in a singular fashion, and also as regular men and women thankfully moved by their (more or less successful) attempts to do so.

Similarly, by taking part in rituals structured in the manner of our centring/grounding example, participants are encouraged to feel themselves both as endowed with certain extraordinary aptitudes, and as ordinary persons affected by the performances in which they aspire to put these aptitudes into effect. In other words, rituals such as these give rise to refracted, dilated identities in which participants experience themselves as becoming more than what they seem (e.g. magical beings, higher selves, shamans, etc.) without ceasing to be whom they recognise themselves to be (e.g. members of a Western or Westernised middle-class). This happy uncertainty as to who exactly participants become within the context of the rituals they perform is upheld, for themselves and others, by a built-in reflexivity that these performances prompt them to entertain. Under these conditions, what Caroline Humphrey and James Laidlaw (1994) have called the 'stipulated' quality of ritual takes on a specific form. Thus, one of the hallmarks of much contemporary ceremonial is a tendency to replace appeals to prescriptive traditions (that require participants to do X, Y and Z) by appeals to regulatory auto-injunctions (that incite participants to ask themselves how they can best do X, Y and Z). What we might call a canonical 'action-centred' mode of ritualisation, posits, for example, that participants must light a white candle with their eyes closed to call upon the Goddess; if a candle of another colour is used, or if one opens one's eyes, the Goddess has not been properly called upon. On the other hand, in the 'actor-centred' ritual mode typical of many contemporary spiritual practices, having one's eyes open or closed can be left up to each of the participants who, having been informed that lighting a white candle is usually thought to be best, are invited to ponder what they feel the candle's colour should be in order for them to call upon the Goddess as properly as possible. This does not mean that anything goes. Nor is it simply a matter of acting according to what one thinks and feels.

Rather, it is a question of acting according to what one thinks and feels one should be thinking and feeling. There is a correct, optimally effective way for each of us to call upon the Goddess and we need to draw on both others' knowhow and our own personal sensibilities to identify what it must be.

The reflexive dynamic built into immersive hugging and other 'actor-centred' rituals underlies the distinctive efficacy of these practices. The inherently unstable, two-fold sense of self it engenders allows participants to experience their potential to become extraordinary beings without deceiving themselves. Ritual experiences, however, are notoriously difficult to put into words, and the lived-through quality of this in-built reflexivity is no exception. One might be tempted to describe it as an inner dialogue or as an act of reasoned introspection. But such discourse-inspired or vision-based 'conceptual metaphors' [START_REF] Lakoff | Metaphors We Live By[END_REF], because they proceed from a clear distinction between interlocutors or between an observer and the observed, seem basically misleading. As is applying a relational vocabulary-to speak, for example, of 'a relationship with oneself'-to an experience that clearly lacks the latent uncertainties, shifting accountabilities and ongoing negotiations that any interaction with a genuine Other necessarily entails. The reflexivity involved here seems to be more a matter of enfolding oneself, of becoming one's own customarily sanctioned 'holding environment' [START_REF] Winnicott | The Theory of the Parent-Infant Relationship[END_REF]. In this light, conceptualisations based on hearing, touch or even proprioceptive sensation might be more appropriate to convey the process of intimate self-encompassment whereby those taking part in these practices can become newly present to themselves. It is more like listening to one's own breathing, or feeling one's weight-or giving oneself a hug.

  psychology and well-being literature attests, contemporary

ACKNOWLEDGEMENTS I am very grateful to Laurent Gabail, Ron Grimes, Graham Harvey, Arnaud Halloy, Marie Mazzella di Bosco, Marika Moisseeff, Sarah Pike and Steven Sutcliffe for their helpful comments on earlier drafts. BIBLIOGRAPHY Altglas, Véronique. 2014. From Yoga to Kabbalah : Religious Exoticism and the Logics of Bricolage. Oxford and New York: Oxford University Press.


S Mennou
email: saidmennou@yahoo.fr

A Chillali

A Kacha
email: ali.kacha@yahoo.fr

A Chillali -Sidi

Mohamed Ben Abdellah

Matrix continued fractions and Expansions of the Error Function

Keywords: MSC 2020 40A15, 15A60, 47A63 Matrix continued fractions, Convergence criteria, Error function

In this paper we recall some results and some criteria on the convergence of matrix continued fractions. The aim of this paper is to give some properties and results of continued fractions with matrix arguments. Then we give continued fraction expansions of the error function erf(A) where A is a matrix. At the end, some numerical examples illustrating the theoretical results are discussed.

Introduction and motivation

The theory of continued fractions has been a topic of great interest over the last two hundred years. The basic idea of this theory over real numbers is to give an approximation of various real numbers by the rationals. A continued fraction is an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of its integer part and another reciprocal, and so on. One of the main reasons why continued fractions are so useful in computations is that they often provide representations for transcendental functions that are much more generally valid than the classical representation by, say, the power series. Further, in the convergent case, continued fraction expansions have the advantage that they converge more rapidly than other numerical algorithms.

Recently, the extension of continued fraction theory from real numbers to the matrix case has seen several developments and interesting applications (see [START_REF] Cuyt | Handbook of continued fractions for special functions[END_REF], [START_REF] Golub | Matrix computations[END_REF], [START_REF] Lorentzen | Continued fractions with applications[END_REF]). Since calculations involving matrix valued functions with matrix arguments are feasible with large computers, it will be an interesting attempt to develop such a matrix theory. In this direction, and generally in a Banach space, few convergence results on non-commutative continued fraction are known.

Two theorems are stated in [START_REF] Wynn | One some recent developments in the theory of continued fractions[END_REF], where Wynn reviews many aspects of the theory of continued fractions, whose elements do not commute under a multiplication law. In Banach space, extensions of Worpitsky's have been proven by Haydan [START_REF] Hayden | Continued fractions in Banach spaces[END_REF] and Negoescu [START_REF] Negoescu | Convergence theorems on non-commutative continued fractions[END_REF].

In [START_REF] Raissouli | Convergence of matrix continued fractions[END_REF], the authors give several convergence criteria on non-commutative continued fractions whose arguments are m × m matrices of the form

K(B n /A n ).
The error function erf is a special function that is important since it appears in the solutions of diffusion problems in heat, mass and momentum transfer, probability theory, theory of errors and various branches of mathematical physics. The closely related Fresnel integrals, which are fundamental in the theory of optics, can be derived directly from the error function.

Preliminaries and notations

Throughout this paper, we denote by M m the set of m × m real (or complex) matrices endowed with the subordinate matrix infinity norm defined by,

∀A = (a i,j ), A ∈ M m , A = max 1≤i≤m m j=1 |a i,j |.

This norm satisfies the inequality

AB ≤ A B . Let A ∈ M m , A is said to be positive semi-definite (resp. positive definite) if A is symmetric and ∀x ∈ R m , (Ax, x) ≥ 0 (resp. ∀x ∈ R m , x = 0, (Ax, x) > 0)
where (•, •) denotes the standard scalar product of R m defined by

∀x = (x 1 , . . . , x m ) ∈ R m , ∀y = (y 1 , . . . , y m ) ∈ R m : (x, y) = m i=1 x i y i .
For any A, B ∈ M m with B invertible, we write A/B := B -1 A, in particular, if A = I, where I is the m th order identity matrix, then we write I/B = B -1 . It is clear that for any invertible matrix C, we have

CA CB = A B . Definition 2.1. Let (A n ) n≥0 , (B n ) n≥1 be two nonzero sequences of M m .
The continued fraction of (A n ) and (B n ), denoted by K(B n /A n ), is the quantity

A 0 + B 1 A 1 + B 2 A 2+... = A 0 ; B 1 A 1 , B 2 A 2 , . . . .

Sometimes, we use the notation

A 0 ; B k A k +∞ k=1
or K(B n /A n ), where

A 0 ; B k A k n k=1 = A 0 ; B 1 A 1 , B 2 A 2 , . . . , B n A n .
The fractions Bn An and Pn Qn :

= A 0 ; B k A k n k=1
are called, respectively, the n th partial quotient and the n th convergent of the continued fraction K(B n /A n ). When B n = I for all n ≥ 1, then K(I/A n ) is called an ordinary continued fraction. The following proposition gives an adequate method to calculate K(B n /A n ).

Proposition 2.2. The elements (P n ) n≥-1 and (Q n ) n≥-1 of the n th convergent of K(B n /A n ) are given by the relationships

P -1 = I, P 0 = A 0 Q -1 = 0, Q 0 = I and P n = A n P n-1 + B n P n-2 Q n = A n Q n-1 + B n Q n-2 , n ≥ 1 .
Proof. This can be done by induction.

The proof of the next Proposition is elementary and we leave it to the reader.

Proposition 2.3. For any two matrices C and D with C invertible, we have

C A 0 ; B k A k n k=1 D = CA 0 D; B 1 D A 1 C -1 , B 2 C -1 A 2 , B k A k n k=3 . (1)
The continued fraction

K(B n /A n ) converges in M m if the sequence (F n ) = P n Qn = (Q -1 n P n )
converges in M m in the sense that there exists a matrix F ∈ M m such that

lim n→+∞ ||F n -F || = 0.
In the opposite case, we say that K(B n /A n ) is divergent. It is clear that

P n Q n = A 0 + n i=1 P i Q i - P i-1 Q i-1 , (2)
and from (2), we see that the continued fraction K(B n /A n ) converges in M m if and only if the series

+∞ n=1 ( Pn Qn -P n-1 Q n-1 ) converges in M m . Definition 2.4. Let (A n ), (B n ), (C n )
and (D n ) be four sequences of matrices. We say that the continued fractions

K(B n /A n ) and K(D n /C n ) are equivalent if we have F n = G n for all n ≥ 1, where F n and G n are the n th convergent of K(B n /A n ) and K(D n /C n ) respectively.
The following lemma characterizes equivalence of continued fractions.

Lemma 2.5 ([4]). Let (r n ) be a non-zero sequence of real numbers. The continued fractions

a 0 ; r 1 b 1 r 1 a 1 , r 2 r 1 b 2 r 2 a 2 , . . . , r n r n-1 b n r n a n , . . . and a 0 ; b 1 a 1 , b 2 a 2 , . . . , b n a n , . . . are equivalent.
We also recall the following Lemma. From the expansion of a function given by its Taylor series, we give the expansion in continued fractions of the series that was established by Euler.

Lemma 2.6 ([5]). Let f be a function with Taylor series expansion

f (x) = +∞ n=0 c n x n in D ⊂ R. Then, the expansion in continued fraction of f (x) is f (x) = c 0 ; c 1 x 1 , -c 2 x c 1 + c 2 x , -c 1 c 3 x c 2 + c 3 x , . . . -c n-2 c n x c n-1 + c n x , . . . = c 0 1 , -c 1 x c 0 + c 1 x , -c 0 c 2 x c 1 + c 2 x , -c 1 c 3 x c 2 + c 3 x , . . . , -c n-2 c n x c n-1 + c n x , . . . .
Remark 2.7. Let (A n ) and (B n ) be two sequences of M m . Then we notice that we can write the first convergents of the continued fraction K(B n /A n ) by:

F 1 = A 0 + A -1 1 B 1 = A 0 + (B -1 1 A 1 ) -1 . F 2 = A 0 + (A 1 + A -1 2 B 2 ) -1 B 1 = A 0 + (B -1 1 A 1 + (B -1 2 A 2 B 1 ) -1 ) -1 . If we put, A * 1 = B -1 1 A 1 and A * 2 = B -1 2 A 2 B 1 , we have F 1 = A 0 + I A * 1 , F 2 = A 0 + I A * 1 + I A * 2 .
Generally, we prove by a recurrence that if we put for all k ≥ 1,

A * 2k = (B 2k . . . B 2 ) -1 A 2k B 2k-1 . . . B 1 and A * 2k+1 = (B 2k+1 . . . B 1 ) -1 A 2k+1 B 2k . . . B 2 , then the continued fractions A 0 + K(B n /A n ) and A 0 + K(I/A * n ) are equivalent.
So, the convergence of one of these continued fractions implies the convergence of the other continued fraction.

Theorem 2.8 ( [START_REF] Zhao | Matrix-valued continued fractions[END_REF]). Let all the elements of A n (n = 1, 2, . . .) be positive, namely, A n are positive matrices for all n, then the matrix continued fraction K(I/A n ) converges if and

only if +∞ n=0 ||A n || = ∞.
We will use the following Theorem to prove our main result.

Theorem 2.9 ([9]). Let (A n ), (B n ) be two sequences of M m . If ||(B 2k-2 . . . B 2 ) -1 A -1 2k-1 B 2k-1 . . . B 1 || ≤ α and ||(B 2k-1 . . . B 1 ) -1 A -1 2k B 2k . . . B 2 )|| ≤ β for all k ≥ 1, where 0 < α < 1, 0 < β < 1 and αβ ≤ 1/4, then the continued fraction K(Bn/An) converges in M m .
We need to present the following Proposition:

Proposition 2.10 ([7]). Let C ∈ M m such that ||C|| < 1, then the matrix I -C is invertible and we have ||(I -C) -1 || ≤ 1 1 -||C|| . (3)
To end this section, we give the following Theorem.

Theorem 2.11 ([11]). If the function f (x) can be expanded in a power series in the circle |x -x 0 | < r, namely

f (x) = +∞ p=0 α p (x -x 0 ) p , (4)
then this expansion remains valid when the scalar argument x is replaced by a matrix A whose characteristic values lie within the circle of convergence.

3 Main results

The real case

In mathematics, the error function, also called the Gauss error function, is a special function of sigmoid shape that occurs in probability, statistics and partial differential equations describing diffusion. It is defined as

erf(x) = 2 √ π x 0 e -t 2 dt, |x| < ∞
where the coefficient in front of the integral normalizes erf(+∞) = 1. A plot of erf(x) over the range -3 ≤ x ≤ 3 is shown as follows.

Figure 1: plot of function error.

The power series expansion for the error function is given by erf

(x) = 2 √ π +∞ n=0 (-1) n (2n + 1)n! x 2n+1 . (5)
Accordingly, we have

erf(x) = 2 √ π +∞ n=0 (-1) n x n+1 (2n + 1)n! x n . (6)
Lemma 3.1. Let x be a real number. Then the continued fraction expansion of the error function is

erf(x) = 0; (2/ √ π)x 1 , x 2 3 -x 2 , -(n -1)(2n -1) 2 x 2 (-1) n-1 (n(2n + 1) -(2n -1)x 2 ) +∞ n=2 . (7)
Proof. We use Lemma 2.6 for the function

g(x) = +∞ n=0 (-1) n x n+1 (2n + 1)n! x n , c n = (-1) n x n+1 (2n + 1)n! . So, we have c 0 1 = x 1 , -c 1 x c 0 + c 1 x = x 3 3 3x-x 3 3 , -c 0 c 2 x c 1 + c 2 x = -x 5 10 -10x 2 +3x 4 30
.

For n ≥ 3, we get

c n-2 c n x = (-1) n-2 x n-2+1 (2(n -2) + 1)(n -2)! • (-1) n x n+1 (2n + 1)n! x = x 2n+1 n(n -1)(2n + 1)(2n -3)((n -2)!) 2 .
Furthermore, we have

c n-1 + c n x = (-1) n-1 x n-1+1 (2(n -1) + 1)(n -1)! + (-1) n x n+1 (2n + 1)n! x = (-1) n-1 x n (n -1)! ( 1 2n -1 + -x 2 n(2n + 1) ) = (-1) n-1 (n(2n + 1) -(2n -1)x 2 ) (2n + 1)(2n -1)n! x n .
Then, we obtain

-c n-2 .c n x c n-1 + c n x = -x 2n+1 n(n-1)(2n+1)(2n-3)((n-2)!) 2 (-1) n-1 (n(2n+1)-(2n-1)x 2 ) (2n+1)(2n-1)n! x n .
Therefore, the continued fraction expansion of erf

(x) = 2 √ π g(x) is erf(x) = 0; b n a n +∞ n=1 = 0; (2/ √ π)x 1 , x 3 3 3x-x 3 3 , -x 5 10 -10x 2 +3x 4 30 , -x 2n+1 n(n-1)(2n+1)(2n-3)((n-2)!) 2 (-1) n-1 (n(2n+1)-(2n-1)x 2 ) (2n+1)(2n-1)n! x n +∞ n=3 .
Let us define the sequence (r n ) n≥1 by r 1 = 1 r n = (2n-1)(2n-3)((n-1)!)

x n-1

, for n ≥ 2.

Then, we have

               r 1 b 1 r 1 a 1 = (2/ √ π)x 1 , r 1 r 2 b 2 r 2 a 2 = x 2 3 -x 2 , r n r n+1 b n+1 r n+1 a n+1 = -(n -1)(2n -1) 2 x 2 (1-) n-1 (n(2n + 1) -(2n -1)x 2 )
for n ≥ 2.

By applying the result of Lemma 2.5 to the sequence (r n ) n≥1 , we obtain

erf(x) = 0; (2/ √ π)x 1 , x 2 3 -x 2 , -9x 2 -10 + 3x 2 , -(n -1)(2n -1) 2 x 2 (-1) n-1 (n(2n + 1) -(2n -1)x 2 ) +∞ n=3 = 0; (2/ √ π)x 1 , x 2 3 -x 2 , -(n -1)(2n -1) 2 x 2 (-1) n-1 (n(2n + 1) -(2n -1)x 2 ) +∞ n=2
and the proof is complete.

The matrix case

According to Theorem 2.11, we have Definition 3.2. Let A be a matrix in M m . Then we define the error function by the expression

erf(A) = 2 √ π +∞ n=0 (-1) n (2n + 1)n! A 2n+1 . (8)
Now, we treat the matrix case, Theorem 3.3. Let A be a matrix in M m , such that ||A|| = α, where 0 < α < 1 2 . The continued fraction 0;

(

2/ √ π)A I , A 2 3I -A 2 , -(n -1)(2n -1) 2 A 2 (-1) n-1 (n(2n + 1)I -(2n -1)A 2 ) +∞ n=2 converges in M m . Furthermore, this continued fraction represents erf(A). So erf(A) = 0; (2/ √ π)A I , A 2 3I -A 2 , -(n -1)(2n -1) 2 A 2 (-1) n-1 (n(2n + 1)I -(2n -1)A 2 ) +∞ n=2 .
Proof. We study the convergence of the continued fraction K(B k /A k ) with

A 1 = I, A 2 = 3I -A 2 , B 1 = (2/ √ π)A, B 2 = A 2 ,
and for k ≥ 3, we have:

A k = (-1) k-2 ((k -1)(2k -1)I -(2k -3)A 2 ), B k = -(k -2)(2k -3) 2 A 2 ,
we check that the conditions of Theorem 2.9 are satisfied:

B 2k-2 . . . B 2 = ±((2k -2) -2)(2(2k -2) -3) 2 . . . (4 -2)(2 • 4 -3) 2 A 2(k-1) = ±(2k -4)(4k -7) 2 . . . 50A 2(k-1) , A -1 2k-1 = -((2k -2)(4k -3)I -(4k -5)A 2 ) -1
and

B 2k-1 B 2k-3 . . . B 1 = ±(2k -1 -2)(2(2k -1) -3) 2 . . . (3 -2)(2 • 3 -3) 2 • (2/ √ π)A 2k-1 = ±(2/ √ π)(2k -3)(4k -5) 2 • . . . • 9A 2k-1 .
Then, we have:

(B 2k-2 . . . B 2 ) -1 A -1 2k-1 B 2k-1 B 2k-3 . . . B 1 = 1 (2k -4)(4k -7) 2 . . . 50 A -2(k-1) ((2k -2)(4k -3)I -(4k -5)A 2 ) -1 (2/ √ π)(2k -3)(4k -5) 2 . . . 9A 2(k-1)+1 ≤ (2/ √ π)(2k -3)(4k -5) 2 . . . 9 (2k -4)(4k -7) 2 . . . 50 A -2(k-1) ((2k -2)(4k -3)I -(4k -5)A 2 ) -1 A 2(k-1)+1 .
Now, the matrices ((2k -2)(4k -3)I -(4k -5)A 2 ) -1 and A -2(k-1) commute, so the above inequality becomes

(B 2k-2 . . . B 2 ) -1 A -1 2k-1 B 2k-1 B 2k-3 . . . B 1 ≤ (2/ √ π)(2k -3)(4k -5) 2 . . . 9 (2k -4)(4k -7) 2 . . . 50 (I - (4k -5) (2k -2)(4k -3) A 2 ) -1 A .
By Proposition 2.10 and the fact that ||A|| < 1/2, we obtain

(I - (4k -3) (2k -1)(4k -1) A 2 ) -1 ≤ 1 1 - (4k-5) (2k-2)(4k-3) A 2 < 1
It implies that for all sufficiently large k, we get

||B 2k-2 . . . B 2 ) -1 A -1 2k-1 B 2k-1 B 2k-3 . . . B 1 || ≤ ||A|| = α < 1/2.
To prove the second inequality of Theorem 2.9, we have

(B 2k-1 B 2k-3 . . . B 1 ) -1 A -1 2k B 2k . . . B 2 = (2k -2)(4k -3) 2 . . . 50 (2/ √ π)(2k -3)(4k -5) 2 . . . 9 A 2(k-1)+2 A 2(k-1)+1 I - 4k -3 (2k -1)(4k -1) A 2 -1
Again using the fact that the matrices I -

4k-3 (2k-1)(4k-1) A 2 -1
and A 2k-1 commute, the Proposition 2.10 and passing to the norm, we get

||(B 2k-1 B 2k-3 . . . B 1 ) -1 A -1 2k B 2k . . . B 2 || ≤ ||A|| = α < 1/2
which completes the proof.

Numerical applications

This section will provide some numerical data to illustrate the preceding results. The focus will be on two cases:

Real case:

• The following table clarifies the differences between erf(x) and its first convergents when applying Lemma 3.1.

x (erf -F 1 )(x) (erf -F 2 )(x) (erf -F 3 )(x) (erf -F 4 )(x) (erf -F 5 )(x)/ erf(x) 0.005
-0.47015e-7 0.1e-11 0.1e-11 0.1e-11 0.1e-11 0.05 -0.00004698055 0.3525e-7 -0.2e-10 0.1e-10 0 0.075 -0.15841090e-3 0.26742e-6 -0.35e-9 -0.1e-10 -0.1e-10 0.1 -0.3750007e-3 0.11257e-5 -0.27e-8 0 0 0.15 -0.12609036e-2 0.85229e-5 -0.457e-7 0.3e-9 0 0.2 -0.29732442e-2 0.357669e-4 -0.3412e-6 0.27e-8 0 0.25 -0.57684016e-2 0.1085733e-3 -0.16201e-5 0.196e-7 -0.4e-9 0.3 -0.98869906e-2 0.2684219e-3 -0.57742e-5 0.1014e-6 -0.12e-8 0.35 -0.155506548e-1 0.5757641e-3 -0.168817e-4 0.4037e-6 -0.81e-8 0.4 -0.229593118e-1 0.11127771e-2 -0.426832e-4 0.13344e-5 -0.351e-7 0.45 -0.322889054e-1 0.19856118e-2 -0.965652e-4 0.38255e-5 -0.1275e-6

We can clearly see that F 5 is approximately the exact value of erf(x).

• The following graphics illustrate the approximations of erf(x) in terms of continued fractions.

After the first iteration, the convergence of (F n (x)) to erf(x) is very rapid. It is hard to distinguish between the curve of the convergents and that of erf(x), for 0 ≤ x ≤ 0.5.

              .
The value of erf(A) is given by erf

(A) =               0

Figure 2 :

 2 Figure 2: Iteration 1.

Figure 3 :

 3 Figure 3: Iteration 2.

Figure 4 :

 4 Figure 4: Iteration 3.

4. 2 .

 2 Matrix case: Example 4.1. Let A be a matrix such that Using the expansion of Theorem 3.3, we can obtain the following convergents of erf(A) :We calculate erf(A) by using the expansion given in Theorem 3.3. The first convergents are