Pavel Kalugin
email: kalugin@lps.u-psud.fr

André Katz

Constraints on pure point diffraction on aperiodic point patterns of finite local complexity

It is shown that the partial amplitudes of the pure point part of the diffraction spectrum of an aperiodic Delone point pattern of finite local complexity are linked by a set of linear constraints. These relations can be explicitly derived from the geometry of the prototile space of the underlying tiling.

Introduction

Delone sets of finite local complexity (FLC) are traditional objects of study in the diffraction theory of aperiodic solids. In this paper, instead of considering individual Delone sets which just happen to have the FLC property, we rather deal with families of such sets, having common allowed local configurations, and study the constraints on the pure point diffraction stemming from these local rules. The main motivation of this approach comes from the problem of the structural analysis in quasicrystals and is explained in details in [START_REF] Kalugin | Robust minimal matching rules for quasicrystals[END_REF]. However, in contrast with [START_REF] Kalugin | Robust minimal matching rules for quasicrystals[END_REF], this paper is not limited to the case of the quasiperiodic long-range order. Neither the restrictions on local environments need to have the strength of the matching rules, that is fix by themselves the long-range order of the structure. In particular, the results are also applicable to the pure point part of diffraction of random tiling models.

The results of this paper are formulated in terms of the partial diffraction amplitudes of the following distribution associated with an FLC Delone multiset (Λ 1 , . . . , Λ m ) in the d-dimensional Euclidean space E:

̺ = m p=1   y∈Λp w p δ y   , (1)
where the index p enumerates atomic sites distinguished by their local environment and the weights w p ∈ C represent their diffractive power. The partial amplitudes will be properly defined in Proposition 2, but it is reasonable to think of them informally as of the following quantities:

a k,p = lim r→∞ 1 vol(D r ) y∈Λp∩Dr w p exp(-2πik • y), (2)
where k ∈ E * is the wave vector corresponding to a Bragg peak and D r ⊂ E is the ball of radius r centered at the origin. The hypothesis that the purepoint part of the diffraction measure η of the distribution ( 1) is related to the amplitudes (2) by the formula

η({k}) = m p=1 w p a k,p 2 (3)
is commonly known as Bombieri-Taylor conjecture [START_REF] Bombieri | Which distributions of matter diffract? An initial investigation[END_REF]. We shall consider the sets Λ p in (1) as decorations of a tiling in E, and treat the corresponding prototile space as the embodiment of the local rules. In this setting, the informal expression (2) offers a glimpse of the nature of the constraints on the partial diffraction amplitudes a k,p , considered as functions on the prototile space. Indeed, as moving the p-th decorating point within a prototile shifts the entire set Λ p and thus modifies (2) by a phase factor, all these functions belong to a finite-dimensional linear space. Furthermore, since the decoration at a tile boundary can be assigned to either of the neighboring tiles, the partial amplitudes are subject to additional linear constraints akin to the Kirchhoff's current conservation law.

The paper is organized as follows. Section 2 sets up the model of Delone multisets of finite local complexity in terms of flat-branched semi-simplicial complexes and isometric windings. Section 3 is devoted to a formal definition of the partial diffraction amplitudes in terms of dynamical systems. The main results of the paper (Theorem 1 and Corollary 2) are exposed in Sections 4 and 5. Finally, Section 6 illustrates these results by several examples of aperiodic point patterns.

The model

A convenient way to impose local rules on the multiset (Λ 1 , . . . , Λ m ) in [START_REF] Kalugin | Robust minimal matching rules for quasicrystals[END_REF] consists in treating it as a decoration of a tiling of E. The range of the local rules is not limited by the tiles sizes, since the environment on a longer range can always be specified by discrete labels attached to the tiles. Without loss of generality, we can limit the consideration to the case of simplicial tilings. Note, that contrary to a common usage, we treat such a tiling as a partition of E by a countable set of interiors of affine simplices, in particular we accept simplicial tiles of any dimension from 0 to d (by a common convention, a vertex is its own interior). A class of tiles identically labeled and having the same shape (up to a translation) is called a prototile (this term may also refer to an arbitrary representative of the class). One can introduce an order on the vertices of each prototile by choosing a direction in E not orthogonal to any prototile edge, and ordering vertices by the values of their projections on this direction. Then the i-th face of a prototile is defined as its face not containing the i-th vertex. Let B n stand for the set of prototiles of dimension n. The matching constraints on the tiles are realized by defining the maps δ n,i : B n → B n-1 assigning to each s ∈ B n the prototile δ n,i s ∈ B n-1 glued to its i-th face. Since the ordering of vertices remains consistent across dimensions, the maps δ n,i satisfy the simplicial identity (54). We shall denote the resulting semi-simplicial set (see Appendix A) by B. The corresponding geometric realization |B| thus represents the prototile space of the tiling.

In addition to the combinatorial data encoded by B, the local rules must also specify the geometry of prototiles, which can be recovered from the directions and orientations of edges. The latter are constrained by the condition that the edges of each 2-dimensional face of a prototile form a triangle. This constraint can be conveniently formulated in terms of the chain complex C • (B, Z) (see Appendix A), leading to the following definition [START_REF] Kalugin | Robust minimal matching rules for quasicrystals[END_REF]:

Definition 1. A d-dimensional flat-branched semi-simplicial complex (FBS- complex) is a triple (B, E, ρ)
, where B is a finite semi-simplicial set of dimension d, E is a d-dimensional real Euclidean vector space and ρ is a homomorphism C 1 (B, Z) → E satisfying the following conditions:

• The homomorphism ρ vanishes on boundaries.

• For any s ∈ B d , the vectors ρ(e d,1 (s)), . . . , ρ(e d,d (s)) are linearly independent.

For any s ∈ B n let σ s ⊂ E be given by the formula

σ s := n i=1 c i ρ(e n,i (s)) c i ∈ R + and n i=1 c i < 1 (4)
The second condition of Definition 1 guarantees that σ s is a non-degenerate n-dimensional affine simplex. Identification of barycentric coordinates within |s| ⊂ |B| and σ s defines a homeomorphism

α s : |s| → σ s . (5)
The tilings obeying the matching rules encoded by an FBS-complex are in oneto-one correspondence with the isometric windings of the latter [START_REF] Kalugin | Robust minimal matching rules for quasicrystals[END_REF]:

Definition 2. A continuous map f : E → |B| is called isometric winding of an FBS-complex (B, E, ρ) if • For each s ∈ B the restriction f f -1 (|s|) is a covering map of |s|.
• The composition α s • f σ restricted to any connected component σ of f -1 (|s|) is a translation of σ by some vector of E.

The tiling T f corresponding to an isometric winding f : E → |B| is given by the partition of E by connected components of f -1 (|s|) for all s ∈ B. Fixing a point x ∈ |B| defines a decoration of this tiling by the set f -1 (x). We shall use this construction to define the sets Λ p in (1) by fixing m points x p ∈ |B| and setting Λ p := f -1 (x p ). This leads to the following formula for the distribution of the diffracting quantity:

̺ f := m p=1   y∈f -1 (xp) w p δ y   , (6)
The formula [START_REF] Baake | Spectral notions of aperiodic order[END_REF] describes the model of the distribution of matter which will be used throughout the rest of the paper.

Partial diffraction amplitudes

The main flaw of the informal expression ( 2) is the presence of the limit operation, which makes proving any result about a k,p a daunting task. The crucial step towards a closed expression for the diffraction amplitudes consists in using the theory of dynamical systems. The key element in this scheme is the hull of the diffracting distribution -a compact topological space representing arbitrarily large finite patches of an infinite system "all at once". The idea to use the hull in studying diffraction has been proposed by Dworkin in [START_REF] Dworkin | Spectral theory and x-ray diffraction[END_REF], and since then has lead to a significant progress in understanding the relation between the diffraction and dynamical spectra ( [START_REF] Deng | Dworkin's argument revisited: point processes, dynamics, diffraction, and correlations[END_REF][START_REF] Lenz | Stationary processes and pure point diffraction[END_REF][START_REF] Baake | Spectral notions of aperiodic order[END_REF]). In this section, we follow mostly the ideas of [START_REF] Deng | Dworkin's argument revisited: point processes, dynamics, diffraction, and correlations[END_REF] and [START_REF] Baake | Dynamical systems on translation bounded measures: Pure point dynamical and diffraction spectra[END_REF], but with the twist of using a matrix-valued diffraction measure.

In the theory of aperiodic order, hulls are built by adding limiting points to the orbit of an aperiodic structure under the action of translations. The actual construction may come in different guises, either as a closure of the orbit in an appropriate topological vector space (for the hulls of almost periodic functions or measures), or as a completion of the orbit in appropriate metric (in the case of point sets or tilings [START_REF] Lorenzo A Sadun | Topology of Tiling Spaces[END_REF]). Since we shall be mostly interested in the dependence of the pure point component of the diffraction measure on the parameters x p and w p in (6) (cf. the formula (20) below), the construction of the hull should depend only on the properties of the isometric winding f in [START_REF] Baake | Spectral notions of aperiodic order[END_REF]. The translation of the diffracting distribution by a vector t ∈ E corresponds to the following transformation of f :

T t f : y → f (y -t). (7)
For a given isometric winding f 0 : E → |B| we shall define its hull X(f 0 ) as the closure of its orbit in the compact-open topology of C(E, |B|):

X(f 0 ) := {T t f 0 | t ∈ E} (8)
Let us show that this construction is equivalent to that of the tiling space (a continuous hull) [START_REF] Lorenzo A Sadun | Topology of Tiling Spaces[END_REF] of T f0 . Consider the patch of a shifted tiling T f0 +t contained within a compact window K ⊂ E. Since T f0 has finite local complexity, there exists a compact K ′ ⊂ E such that all these patches are generated by shifts t ∈ K ′ . Therefore

(T t f 0 ) K | t ∈ E = (T t f 0 ) K | t ∈ K ′ (9)
Since the right-hand side of ( 9) is an image of the compact K ′ , the left-hand side is closed in the compact-open topology of C(K, |B|). Thus, for any f ∈ X(f 0 ) there exists t ∈ E such that f K = (T t f 0 ) K . Therefore all points of X(f 0 ) are isometric windings and thus correspond to tilings, while the convergence in X(f 0 ) is equivalent to that in the tiling space of T f0 .

The closed formula for the partial diffraction amplitudes requires one more ingredient -a translation invariant probability measure µ on X(f 0 ). Such a measure always exists since X(f 0 ) is compact, but it may be not unique. Although the results of this section remain valid for any translation invariant measure, it should be emphasized that the well-definedness of the limit in the formula (2) is guaranteed only in the case when there exists only one such measure, that is when the measure-preserving dynamical system (X(f 0 ), E, µ) is uniquely ergodic [START_REF] Lenz | Continuity of eigenfunctions of uniquely ergodic dynamical systems and intensity of Bragg peaks[END_REF]. On the other hand, in real physical applications the condition of unique ergodicity is not very stringent since it corresponds to an intuitive notion of macroscopic uniformity of the specimen. Bearing this in mind we shall treat the measure µ as a background parameter and omit to reference it unless necessary.

Let S(E) stand for the Schwartz space on E. For a given point x ∈ |B|, let Γ x be the linear map S(E) → L 2 (X(f 0 ), µ) defined by the formula

Γ x (ϕ) :=   f → y∈f -1 (x) ϕ(-y)   where ϕ ∈ S(E). ( 10
)
Since ϕ is rapidly decreasing and f -1 (x) is uniformly discrete, the sum in the above expression converges absolutely, therefore Γ x is continuous. Consider the following sesquilinear functional on S(E) (we use the convention that Dirac bracket is antilinear in the first argument):

(ϕ 1 , ϕ 2 ) → m p,q=1 w p w q Γ xp (ϕ 1 ), Γ xq (ϕ 2 ) .
This functional is translationally invariant, positive definite and continuous in each argument. Therefore (see [10, Chapter II.3, Theorem 6] and the discussion afterwards), there exists a positive tempered measure η on E * such that m p,q=1

w p w q Γ xp (ϕ 1 ), Γ xq (ϕ 2 ) = E * ϕ 1 (k) ϕ 2 (k) dη(k) (11)
We shall take [START_REF] Rosenberg | The square-integrability of matrix-valued functions with respect to a non-negative Hermitian measure[END_REF] as the definition of the diffraction measure η of the distribution (6) (see Appendix B for the proof that this definition is equivalent to more traditional ones).

The diffraction measure η in [START_REF] Rosenberg | The square-integrability of matrix-valued functions with respect to a non-negative Hermitian measure[END_REF] is sesquilinear in the weights w p :

η = m p,q=1 w p w q ζ pq , (12)
where ζ pq are complex-valued tempered measures on E * . It is convenient to consider them as elements of a matrix-valued measure on

E * ζ := (ζ pq ) where 1 ≤ p, q ≤ m.
The values of ζ on bounded Borel sets of E * are Hermitian positive semi-definite m × m matrices.

We shall now follow the ideas of [START_REF] Deng | Dworkin's argument revisited: point processes, dynamics, diffraction, and correlations[END_REF] and construct the isometric embedding of Hilbert spaces Θ :

L 2 (E * , ζ; C m ) → L 2 (X(f 0 ), µ), (13)
where L 2 (E * , ζ; C m ) stands for the space of (classes of) functions E * → C m square-integrable with respect to the matrix-valued diffraction measure ζ (see [START_REF] Rosenberg | The square-integrability of matrix-valued functions with respect to a non-negative Hermitian measure[END_REF] for the details). Let as start by defining the action of Θ on the Schwartz space

S(E * , C m ) ⊂ L 2 (E * , ζ; C m ) by the formula Θ( ϕ ⊗ e p ) := Γ xp (ϕ), (14)
where ϕ ∈ S(E) and (e p ) stands for the canonical basis in C m . As follows from ( 11) and ( 12), Θ intertwines the inner product of L 2 (E * , ζ; C m ) restricted to S(E * , C m ) with that of L 2 (X(f 0 ), µ). The isometric embedding ( 13) is then defined as the continuous extension the map ( 14) to L 2 (E * , ζ; C m ), which is unique by virtue of the following proposition:

Proposition 1. S(E * , C m ) is dense in L 2 (E * , ζ; C m )
Proof. Let us denote the scalar measure given by the trace of ζ by tr(ζ) and consider

L 2 (E * , tr(ζ)) ⊗ C m as a space of classes of C m -valued functions on E * . Since ζ is non-negative, the norm of L 2 (E * , tr(ζ)) ⊗ C m is stronger than that of L 2 (E * , ζ; C m ).
This defines a continuous linear map

L 2 (E * , tr(ζ)) ⊗ C m → L 2 (E * , ζ; C m ). ( 15
)
As follows from [START_REF] Rosenberg | The square-integrability of matrix-valued functions with respect to a non-negative Hermitian measure[END_REF]Theorem 3.11], simple functions are dense in

L 2 (E * , ζ; C m ).
By standard arguments so are also the simple functions with compact support. Since tr(ζ) is locally finite, the latter belong to the image of ( 15), which is there-

fore dense in L 2 (E * , ζ; C m ). Let C c (E * , C m
) stand for the space of continuous C m -valued functions with compact support. These functions are approximated by those of S(E * , C m ) uniformly, and thus also in the norm of

L 2 (E * , tr(ζ)) ⊗ C m . It remains to show that C c (E * , C m ) is dense in L 2 (E * , tr(ζ)) ⊗ C m , which follows immediately from [12, Theorem 3.14].
Let us show now that the isometric embedding ( 13) is equivariant with respect to translations of E. For any t ∈ E let S t stand for the unitary operator on L 2 (E * , ζ; C m ) defined by the formula

(S t g) (k) := exp(2πik • t)g(k), (16)
where g ∈ L 2 (E * , ζ; C m ) and k ∈ E * . The translation t also acts on L 2 (X(f 0 ), µ) by the unitary operator T t :

(T t h) (f ) := h(T -t f ), where h ∈ L 2 (X(f 0 ), µ), f ∈ X(f 0 )
and T -t f is given by the formula [START_REF] Baake | Dynamical systems on translation bounded measures: Pure point dynamical and diffraction spectra[END_REF]. In particular, for Γ x (ϕ) ∈ L 2 (X(f 0 ), µ) in [START_REF] Gel | Generalized Functions: Applications of Harmonic Analysis[END_REF] we have

(T t Γ x (ϕ)) (f ) = Γ x (ϕ)(T -t f ) = Γ x (T -t ϕ)(f ), (17)
where (T t ϕ)(y) := ϕ(y -t). Therefore, as follows from ( 14)

Θ(S t ( ϕ ⊗ e p )) = Γ xp (T -t ϕ) = T t Γ xp (ϕ) = T t (Θ( ϕ ⊗ e p ))
Then, by virtue of Proposition 1, the identity

ΘS t = T t Θ (18)
holds on the entire Hilbert space L 2 (E * , ζ; C m ). In other words, the isometric embedding [START_REF] Michael | Sheaves, cosheaves and applications, volume 1249 of Publicly Accessible Penn Dissertations[END_REF] intertwines the action of E on L 2 (E * , ζ; C m ) by S t with that on L 2 (X(f 0 ), µ) by T t . The notable consequence [START_REF] Deng | Dworkin's argument revisited: point processes, dynamics, diffraction, and correlations[END_REF][START_REF] Baake | Spectral notions of aperiodic order[END_REF][START_REF] Baake | Dynamical systems on translation bounded measures: Pure point dynamical and diffraction spectra[END_REF] of the identity [START_REF] Christopher R Iacovella | Selfassembly of soft-matter quasicrystals and their approximants[END_REF] is that the pure point part B of the diffraction measure ζ is a subset of the pure-point part E ⊂ E * of the dynamical spectrum of (X(f 0 ), E, µ), which is defined as the set of values k ∈ E * for which there exists a non-zero eigenfunction ψ k ∈ L 2 (X(f 0 ), µ):

T t ψ k = exp(2πik • t)ψ k for any t ∈ E
While the dependence of the diffraction measure η on the weights w p in ( 6) is captured by the matrix-valued measure ζ, the latter still depends on the positions of the atomic decorations {x 1 , . . . , x m } ⊂ |B|. The following result shows that the pure-point part of the diffraction measure can be expressed in terms of individual contributions of each atomic site x p , thus justifying the formula (3): Proposition 2. Let f 0 : E → |B| be an isometric winding and let µ be a translationally invariant probability measure on its hull X(f 0 ). Then for any eigenvalue k ∈ E and a corresponding normalized eigenfunction ψ k ∈ L 2 (X(f 0 ), µ) there exists a (not necessarily continuous) function a k : |B| → C such that for any set of atomic decorations {x 1 , . . . , x m } ⊂ |B| in (6) holds the identity

ψ k , Θ(1 {k} ⊗ e p ) = a k (x p ). ( 19
)
Moreover, if µ is ergodic, one also has

η({k}) = m p=1 w p a k (x p ) 2 . ( 20
)
We shall refer to the functions a k as the partial diffraction amplitudes of the dynamical system (X(f 0 ), E, µ).

Proof. Let us consider the distribution A k,x ∈ S ′ (E) defined by the formula

A k,x (ϕ) := ψ k , Γ x (ϕ) .
Using [START_REF] Pak | Dodecagonal order in a two-dimensional Lennard-Jones system[END_REF] we get

ψ k , Γ x (T t ϕ) = T t ψ k , Γ x (ϕ) ,
hence A k,x satisfies the following equation:

A k,x (T t ϕ) = exp(-2πik • t)A k,x (ϕ). ( 21
)
The solutions of this equation in S ′ (E) have the form

ϕ → a E ϕ(y) exp(-2πik • y) dy,
where a is an arbitrary complex constant. Therefore there exists a function

a k : |B| → C such that ψ k , Γ x (ϕ) = a k (x) ϕ(k) (22)
for any ϕ ∈ S(E) and any x ∈ |B|. Then as follows from ( 14)

ψ k , Θ( ϕ ⊗ e p ) = a k (x p ) ϕ(k) (23)
Since the multiplication by 1 {k} in L 2 (E * , ζ; C m ) is the projector on the eigenspace of S t (16), the embedding Θ intertwines it with the projector on the corresponding eigenspace of T t and we have

ψ k , Θ(1 {k} ϕ ⊗ e p ) = ψ k , Θ( ϕ ⊗ e p ) .
Combining this identity with ( 23) yields [START_REF] Gähler | The diffraction pattern of self-similar tilings[END_REF]. If µ is ergodic, all eigenvalues of the dynamical system (X(f 0 ), E, µ) are simple and

Θ(1 {k} ⊗ e p ) = ψ k , Θ(1 {k} ⊗ e p ) ψ k = a k (x p )ψ k .
Since Θ is an isometry, this yields

ζ pq ({k}) = 1 {k} ⊗ e p , 1 {k} ⊗ e q = a k (x p )a k (x q ),
and recalling [START_REF] Rudin | Real and complex analysis[END_REF] we finally get [START_REF]Nemo computer algebra package[END_REF].

Constraints on partial amplitudes of a single Bragg peak

We are now going to study the behavior of the partial amplitudes a k as functions on |B|. The results can be conveniently formulated in terms of semi-simplicial vector spaces (see Appendix A)

Y s (x) := exp(-2πik • α s (x)) if x ∈ |s| 0 if x / ∈ |s| (24)
Since all functions {Y s : s ∈ B • } are linearly independent, they form a basis of the direct sum

F (k),• := d n=0 F (k),n ,
which is naturally a subspace of the space of all complex-valued functions on |B|.

Proposition 3. Let a k be the partial diffraction amplitudes defined in Proposition 2. Then for any k ∈ E

a k ∈ F (k),• . (25)
Proof. Consider a simplex s ∈ B • and two points x 1 , x 2 ∈ |s| and let

t = α s (x 2 ) -α s (x 1 ).
Then f -1 (x 2 ) = f -1 (x 1 ) + t for any isometric winding f : E → |B| and

Γ x2 (ϕ) = Γ x1 (T t ϕ)
for any ϕ ∈ S(E). Therefore, as follows from [START_REF] Cornier-Quiquandon | Neutron scattering structural study of AlCuFe quasicrystals using double isotopic substitution[END_REF],

a k (x 2 ) = exp(-2πik • t)a k (x 1 ),
which proves [START_REF] Fritsch | Cellular structures in topology[END_REF].

Let us now provide the graded vector space F (k),• with the face operators δ n,i :

F (k),n → F (k),n-1 , defined via the face maps δ n,i : B n → B n-1 as: δ n,i (Y s ) = exp(-2πik • t s,i )Y δn,is (26)
where n = dim(s) and t s,i ∈ E is given by

t s,i := 0 if i = 0 ρ(e n,1 (s)) if i = 0 ( 27
)
Note that with the expression (27) for t s,i , the affine simplex α δn,is (|δ n,i s|)+t s,i is the i-th face of the affine simplex α s (|s|) for any 0 ≤ i ≤ d (see Figure 1). Consequently, the linear operator δ n,i in (26) acts on the basis function Y s by its continuation to the i-th face of |s|. Therefore, the operators δ n,i satisfy the simplicial identity (54) and provide F (k),• with the structure of a semi-simplicial vector space which we shall denote by F (k) .

To formulate the main result of this section, we shall need an orientation of d-simplices S :

B d → {-1, 1},
N s := {(s ′ , i) ∈ B d × {0, . . . , d} | δ d,i s ′ = s}
As follows from ( 24) and ( 26), for any (s ′ , i) ∈ N s and for any points x ∈ |s| and x ′ ∈ |s ′ | one has

(δ n,i (Y ′ s )) (x) = exp (-2πik • (α s (x) -α s ′ (x ′ ) + t s ′ ,i )) Y s ′ (x ′ ) (30)
The identity (30) allows to express the value of a k at a point x ∈ |s| via the values of a k at arbitrarily chosen points x s ′ ,i ∈ |s ′ | (one for each (s ′ , i) ∈ N s ):

(∂ d (sa k )) (x) = (s ′ ,i)∈Ns (-1) i S(s ′ )a k (x s ′ ,i ) exp (-2πik • (α s (x) -α s ′ (x s ′ ,i ) + t s ′ ,i )) (31)
To finish the proof, we shall use an appropriate choice of the points x s ′ ,i to show that that the right-hand side of (31) is zero. For any (s

′ , i) ∈ N s , the affine simplex α s ′ (|s ′ |) -t s ′ ,i has α s (|s|) as its i-th face. Let us denote the opposite vertex of α s ′ (|s ′ |)-t s ′ ,i by v s ′ ,i . The hyperplane of E containing α s (|s|) divides the set of points {v s ′ ,i |(s ′ , i) ∈ N s } in two parts, following the sign of the expression Ω •   v s ′ ,i ∧   d-1 j=1 ρ(e d-1,j (s))     (32)
The set N s is thus naturally split as N s = N + s ⊔ N - s , according to the sign of (32). As follows from [START_REF] Baake | Aperiodic Order: Volume 1, A Mathematical Invitation[END_REF], the position of v s ′ ,i is given by the formula

v s ′ ,i = -ρ(e d,1 (s ′ )) if i = 0 ρ(e d,i (s ′ )) if i = 0
To compute the sign of (32) one can use the identity

v s ′ ,i ∧   d-1 j=1 ρ(e d-1,j (s))   = (-1) i+1 d j=1
ρ(e d,j (s ′ ))

together with (28), which gives rise to

N ± s = (s ′ , i) ∈ N s (-1) i+1 S(s ′ ) = ±1
Let Σ + s and Σ - s stand for the intersections of the affine simplices α s ′ (|s ′ |) -t s ′ ,i for (s ′ , i) belonging to N + s and N - s respectively:

Σ ± s = (s ′ ,i)∈N ± s (α s ′ (|s ′ |) -t s ′ ,i ) (33)
(see Figure 2). Both Σ + s and Σ - s are non-empty open polyhedra having α s (|s|) as a face. Therefore, one can choose a vector t ∈ E such that α s (x) ± t ∈ Σ ± s and fix the points x s ′ ,i in (31) in such a way such that

α s ′ (x s ′ ,i ) -t s ′ ,i = α s (x) ± t for (s ′ , i) ∈ N ± s ( 34
)
The formula (31) then yields

(∂ d (sa k )) (x) = exp (-2πik • t) x ′ ∈X - a k (x ′ ) -exp (2πik • t) x ′ ∈X + a k (x ′ ) (35)
where

X ± := x s ′ ,i (s ′ , i) ∈ N ± s
Note now that as follows from (34), f (y) = x if and only if f (y ± t) ∈ X ± for any isometric winding f : E → |B|. In other words,

f -1 (X ± ) = f -1 (x) ± t
and for any ϕ ∈ S(E)

x ′ ∈X ± Γ x ′ (ϕ) = Γ x (T ±t ϕ).
Then, as follows from ( 21)

exp (±2πik • t) x ′ ∈X ± a k (x ′ ) = a k (x) (36)
and the terms at the right-hand side of (35) cancel each other, which proves (29).

As follows from the equations ( 25) and ( 29), the functions a k restricted to the d-dimensional simplices of |B| belong to a linear space of dimension not exceeding rank(H d (F (k),• , ∂ • )). On the other hand, as can be seen from ( 36), the values of a k on the simplices of dimension d -1 depends linearly on those on the neighboring d-dimensional simplices. The reasoning leading to (36) can be easily generalized to simplices of lower dimensions. Therefore, if the number of atomic positions m in the model ( 6) is big enough, the contributions of these positions in the pure point diffraction are subject to linear constraints: Corollary 1. There exist at least m -rank(H d (F (k),• , ∂ • )) linear constraints on the partial amplitudes a k (x p ).

Bragg peaks densely filling a subspace

Let V ⊂ E * be a non-zero linear subspace of E * . We are going now to establish a connection between constraints on partial amplitudes of different Bragg peaks in the case where B is dense in an open subset of V , and this is the second main result of this paper. Let us start by showing that the semi-simplicial vector spaces F (k) for any k ∈ V can be obtained by an appropriate extension of scalars from only one semi-simplicial module.

Let us denote by L V ⊂ V * the image of the group homomorphism ν : H 1 (B, Z) → V * given by the formula

ν(c) := (k → k • ρ * (c)) for c ∈ H 1 (B, Z) (37)
For k ∈ V , let φ k,V stand for the ring homomorphism Z[L V ] → C extending the following character of L V :

l → exp(-2πik • l) for l ∈ L V (38)
and let φ k,V ! stand for the functor of extension of scalars by φ k,V .

Proposition 4. There exists a semi-simplicial

Z[L V ]-module G V such that for any k ∈ V the semi-simplicial vector space φ k,V ! G V is isomorphic to F (k) .
Proof. We shall define G V by providing an ordinary free graded

Z[L V ]-module Z[L V ] B•
∂c s = δ 1,1 (e n,1 (s)) -s 0 ( 39
)
(this is always possible since B is connected). The face homomorphisms of G V are defined via the face maps δ n,i : B n → B n-1 as follows:

δ n,i (ǫ s ) = l s,i ǫ δn,i(s) (40)
where dim(s) = n and l s,i ∈ L V is given by

l s,i := ν(c s -c δn,is ) if i = 0 ν(c s -c δn,is + e n,1 (s)) if i = 0
(this expression is well-defined since in both cases the arguments of ν are cycles).

To check that G V is indeed a semi-simplicial Z[L V ]-module, it remains to verify that the homomorphisms (40) satisfy the simplicial identity (54). This result stems from the following identity in L V :

l δn,js,i + l s,j = l δn,is,j-1 + l s,i if i < j (41)
(note that we use the multiplicative notation for the action of l s,i as an element of the group ring Z[L V ] in (40) and the additive notation for the group operation in L V in (41)). Let (ǫ ′ s ) s∈B• be the basis in φ k,V ! G V corresponding to (ǫ s ) s∈B• . The functorial image of the face homomorphisms is then given by the following formula:

φ k,V ! δ n,i (ǫ ′ s ) = φ k,V (l s,i )ǫ ′ δn,i(s) = exp(-2πik • l s,i )ǫ ′ δn,i(s) .
Consider now the bijective linear map ω k : φ k,V ! G V → F (k) defined by its action on the basis (ǫ ′ s ) s∈B• :

ω k (ǫ ′ s ) = Φ k,s Y s , (42)
where Y s is defined in [START_REF] Beucher | The morphological approach to segmentation: the watershed transformation[END_REF] and the unitary factors Φ k,s are given by

Φ k,s := exp(-2πik • ρ(c s )) (43)
As follows from the identity

Φ k,s exp(-2πik • t s,i ) = Φ k,δn,is exp(-2πik • l s,i )
ω k commutes with the face operators and is therefore an isomorphism of semisimplicial vector spaces φ k,V ! G V and F (k) .

Proposition 5. Let J be a generating set of d-cycles of G V (which can always be chosen finite since Z[L V ] is a Noetherian ring) and let r j,s ∈ Z[L V ] be the coefficients of the cycle j ∈ J in the basis of G V,d :

j = s∈B d r j,s ǫ s
Then for almost all k ∈ V with the possible exception of a nowhere dense subset of V , the space of d-cycles of F (k) is spanned by the set of vectors

s∈B• φ k,V (r j,s )Φ k,s Y s j∈J , (44)
where the phase factors Φ k,s are given by ( 43)).

Corollary 2. For any linear subspace V ⊂ E * , there exists a finite set of smooth functions a j : V → C m indexed by a generating set of the d-cycles of G V , such that for almost all k ∈ B ∩ V (with the possible exception of a subset nowhere dense in V ), the vector (a k (x 1 ), . . . , a k (x m )) ∈ C m of partial diffraction amplitudes defined in Proposition 2 belongs to a subspace spanned by a j (k).

As follows from ( 24), ( 43) and (44), the components of a j are finite exponential sums of the form

a j,p (k) = l∈Xj,p C lj,p exp (-2πik • (l + y p )) , (45)
where X j,p is a finite subset of L V , C lj,p ∈ Z and y p ∈ V * . It is remarkable that while the dependence of the diffraction amplitudes on k ∈ B ∩ V is usually utterly irregular, the constraints (45) for the partial amplitudes are smooth functions of k. Note, however, that these constraints are effective only if the Bragg peaks fill densely an open subset of V and when the number m of distinct atomic sites in ( 6) is large enough (e.g. if m exceeds the number of generators of the d-cycles of G V ).

To prove Proposition 5 we shall need the following technical result:

Proposition 6. If ϑ is a homomorphism of free Z[L V ]-modules, then for all k ∈ V rank (φ k,V ! ϑ) ≤ rank(ϑ).
Moreover, the subset V ϑ ⊂ V for which this inequality is strict

V ϑ := {k ∈ V : rank (φ k,V ! ϑ) < rank(ϑ)} is nowhere dense in V .
Proof. Let rank(ϑ) = r. Since Z[L V ] is a commutative domain, the rank of a homomorphism of free modules coincides with that of its matrix over the quotient field of Z[L V ] [14, Ex. 5.23A]. Hence, all minors of order r + 1 of the matrix of ϑ (if any) are zero, and there exists a non-zero minor of order r. All these minors are elements of Z[L V ] and the corresponding minors of φ k,V ! ϑ are obtained from them by the ring homomorphism φ k,V . Therefore, rank (φ k,V ! ϑ) ≤ r for any k ∈ V . Let 0 = P ∈ Z[L V ] be a non-vanishing minor of ϑ of order r. Then the expression

k → φ k,V (P )
defines an entire analytic function on V ⊗ R C vanishing on V ϑ and thus also on the closure of V ϑ in V . If this closure contains an open subset of V , this function is zero. Since the set of functions

V → C {k → exp(-2πik • l) | l ∈ L V }
is linearly independent, P = 0. This contradiction proves the Proposition.

Proof of Proposition 5. Consider the homomorphism κ :

Z[L V ] (J) → G V,d map-
ping each basis element of the free Z[L V ]-module over the set J to the corresponding element of J ⊂ G V,d . Since J is the generating set of the submodule of d-cycles, the following sequence of free Z[L V ]-modules

Z[L V ] (J) κ / / G V,d ∂ d / / G V,d-1 (46)
is exact. Applying the extension of scalars φ k,V ! to (46) yields the upper row of the following diagram (which is commutative by Proposition 4):

C (J) φ k,V ! κ / / φ k,V ! G V,d ω k φ k,V ! ∂ d / / φ k,V ! G V,d-1 ω k C (J) ω k •(φ k,V ! κ) / / F (k),d ∂ d / / F (k),d-1 (47)
Since the sequence (46) remains exact when extended to the vector spaces over the quotient field, one has

rank(κ) + rank(∂ d ) = #B d .

Then by Proposition 6, for all

k ∈ V \ (V κ ∪ V ∂ d ) holds the equality rank (φ k,V ! κ) + rank (φ k,V ! ∂ d ) = #B d .
and thus the rows of (47) are also exact. As follows from (42), the vectors of the set (44) are images of the the basis vectors of C (J) in the bottom row of (47). Therefore, for all k ∈ V \ (V κ ∪ V ∂ d ) the subspace of d-cycles in F (k) is spanned by (44). Since V κ ∪ V ∂ d is nowhere dense in V , this proves the Proposition.

In the physically relevant case where V = E * (and therefore L V can be considered as a subgroup of E), is possible to give a geometric meaning to formula (45) in the following way. Since L V is a quotient of H 1 (B, Z) by the kernel of (37), there exists a normal semi-simplicial covering B V → B for which L V is the group of deck transformations. The corresponding action of L V on Z ( BV ) commutes with the face homomorphisms and therefore defines on Z ( BV ) the structure of a semi-simplicial Z[L V ]-module. It is straightforward to check using formula (40) that this module is isomorphic to G V . Hence, the d-chains of Z ( BV ) can be seen as formal finite integer linear combinations of copies of prototiles translated by vectors of L V . The d-cycles of Z ( BV ) (and thus also those of G V ) then correspond to the combinations with boundaries canceling out. The advantage of this approach is that it can be directly applied to tilings with tiles of arbitrary shapes, without preliminary triangulation, as illustrated by Figure 3. This representation can be used directly to calculate the sum at the right hand side of (45). Assuming that Figure 3 depicts the cycle j, and the decoration corresponds to the point x p , the vectors l + y p ∈ E in (45) are given by the positions of the copies of the decorating point, and the coefficients C lj,p are the weights of the corresponding tiles (+1 and -1 for the case shown on Figure 3).

Examples

Binary patterns in one dimension

One-dimensional binary patterns are decorated tilings of the real line by two types of intervals. Let v 1 and v 2 stand for the length of the intervals and let u 1 and u 2 be the positions of the decorating points (relative to the left end of the respective intervals). The diffracting distribution (1) of a binary pattern is defined by assigning complex weights w 1 and w 2 to the respective decorations (see Figure 4). Let us now describe the FBS-complex (B, E, ρ) encoding the Figure 4: One-dimensional binary pattern. binary patterns. The graded set B • contains three elements:

B 0 = {s 0 } B 1 = {s 1 , s 2 }
with the face maps given by

δ 1,0 s i = δ 1,1 s i = s 0 for i ∈ {1, 2}.
Therefore, the geometric realization |B| of the semi-simplicial set B is a bouquet of two circles (see Figure 5). The group of 1-cycles of C • (B, Z) is generated by the one-dimensional simplices s 1 and s 2 . Finally, the homomorphism ρ :

H 1 (B, Z) → E is defined by the formula ρ(s i ) = v i for i ∈ {1, 2}
(recall that in this case E is a real line).

Figure 5: The geometric realization of the FBS-complex of a binary pattern and its decorations.

Let us now assume that the set of Bragg peaks of the considered binary pattern is dense in E * . With the notation used in of Section 5, this amounts to the assumption that V = E * with V * naturally isomorphic to E. Therefore, the group L V (see (37)) is generated by v 1 and v 2 .

Let us first consider the case when v 1 and v 2 are commensurate, that is there

exists v ∈ E such that v i = n i v
where n 1 , n 2 ∈ Z are coprime. In this case L V is a free abelian group of rank 1 generated by v. It is convenient to write the group operation in multiplicative notation, and treat the group ring Z[L V ] as a ring of Laurent polynomials:

Z[L V ] = Z[ξ, ξ -1 ]
where the multiplication by the indeterminate ξ corresponds to the action of v ∈ L V . Since B contains only one vertex, one can set in (39) c s = 0 for all s ∈ B • . Then the face homomorphisms (40) of G V take the following form

δ 1,0 (ǫ si ) = ξ ni ǫ s0 δ 1,1 (ǫ si ) = ǫ s0 for i ∈ {1, 2}
The generating set J of Proposition 5 contains only one cycle with the coefficients

r s1 = ξ n2 -1 r s2 = 1 -ξ n1
Therefore, the partial amplitudes a k (x 1 ) and a k (x 2 ) for all Bragg peaks with the exception of a nowhere dense set obey the following constraint:

a k (x 1 ) a k (x 2 ) = exp (2πik(u 2 -u 1 )) exp(-2πikvn 2 ) -1 1 -exp(-2πikvn 1 ) , valid for k / ∈ v -1 Z.
In the case n 1 = n 2 = 1 and u 1 = u 2 this amounts to a k (x 1 ) = -a k (x 2 ), which reflects the trivial fact that for w 1 = w 2 in this case one recovers the periodic Dirac comb.

When v 1 and v 2 incommensurate, L V is a free abelian group of rank 2 generated by v 1 and v 2 . Again, we shall use the multiplicative notations

Z[L V ] = Z[ξ 1 , ξ -1 1 , ξ 2 , ξ -1 2 ]
with the multiplication by the indeterminate ξ i corresponding to the action of v i . The face homomorphisms (40) are then given by

δ 1,0 (ǫ si ) = ξ i ǫ s0 δ 1,1 (ǫ si ) = ǫ s0 for i ∈ {1, 2}
and again, J contains a single cycle with the coefficients

r s1 = ξ 2 -1 r s2 = 1 -ξ 1
leading to the following constraint

a k (x 1 ) a k (x 2 ) = exp (2πik(u 2 -u 1 )) exp(-2πikv 2 ) -1 1 -exp(-2πikv 1 ) ,
valid for k = 0.

Decorated canonical tilings

Let us fix n vectors v 1 , . . . , v n ∈ E, such that any d of them are linearly independent over R. The prototiles of an (n, d)-canonical tiling [START_REF] Bodini | Crystallization by stochastic flips[END_REF] of E are parallelotopes with edges {v i1 , . . . , v i d } (one prototile for every subset {i 1 , . . . , i d } ⊂ {1, . . . , n}). For the sake of simplicity we shall limit the consideration to the case when v 1 , . . . , v n are linearly independent over Q and shall also assume that the Bragg peaks are dense in E * . The results of this paper are not directly applicable to canonical tilings for d 1 since the prototiles are not simplices. However, as a d-dimensional parallelotope can be straightforwardly triangulated by d! simplices, we shall tacitly assume such triangulation applied to every tile. As follows from [START_REF] Fritsch | Cellular structures in topology[END_REF], partial diffraction amplitudes at the points belonging to the same simplex are related by a trivial phase factor. Since the triangulation of a prototile is purely formal, the same applies to the points belonging to the same prototile of the canonical tiling. To keep focus on the non-trivial constraints only, we shall therefore consider only the case when each prototile is decorated by a single point at its center (see Figure 6). Similarly to the previous example, we assume V = E * . The group L V ⊂ E ∼ = V * is then freely generated by v 1 , . . . , v n and we shall use the multiplicative notation for the ring Z[L V ]:

Z[L V ] ∼ = Z[ξ 1 , ξ -1 1 , . . . , ξ n , ξ -1 n ]
The space |B| of the FBS-complex of an (n, d)-canonical tiling is a (triangulated) d-skeleton of the standard CW-decomposition of an n-dimensional quasiperiodic order observed in molecular dynamics simulation [START_REF] Pak | Dodecagonal order in a two-dimensional Lennard-Jones system[END_REF], and has also attracted attention recently in soft mater physics [START_REF] Christopher R Iacovella | Selfassembly of soft-matter quasicrystals and their approximants[END_REF]. Square-triangle tilings are commonly decorated at vertices, but we shall consider decorations positioned at the center of tiles instead (see Figure 7). We shall impose no additional constraint on the tiling beyond the assumption that the pure-point part of its diffraction spectrum is dense in E * (a lot of examples of such tilings can be obtained by some inflation procedure, see e.g. [START_REF] Gähler | The diffraction pattern of self-similar tilings[END_REF]). The local order in the tiling is thus described by an FBS-complex obtained by gluing together the (triangulated) prototiles of Figure 7. The group L V is generated by the edges of the tiling and is a free abelian subgroup of E of rank 4. The computation performed with the computer algebra system Nemo [START_REF]Nemo computer algebra package[END_REF] shows that the null space of the boundary operator of G V in degree 2 has rank 2 (over the quotient field of Z[L V ]). Fortunately, it is possible to visualize the generating 2-cycles using the approach given at the end of Section 5 (see Figure 8). Figure 8: A metaphorical representation of two 2-cycles of Z ( BV ) for the squaretriangle tiling (see also Figure 3). The solid circle represents the decoration of the tile s 1 . Top: the first cycle is a formal difference of two dodecagonal patches of tiling. Bottom: the tiles entering with the same sign in the second cycle are overlapping, but their effective boundary has 12-fold symmetry. The tiles of type s 1 are outlined.

To give explicit formulas for the constraints on the partial diffraction amplitudes, we shall use the following notation for the twelve vectors corresponding to the edges of the tiling (assumed to be of unit length):

l i = cos(π(i -1)/6) sin(π(i -1)/6) for i ∈ Z/12Z
We shall denote the points on |B| corresponding to the decoration of the square s p and the triangle t q by x sp and x tq respectively. To obtain the constraints on the partial amplitude at a given decorating point, one has to find all copies of the corresponding tiles on Figure 8 and take into account their weights and the positions of decorations, as explained at the end of Section 5. This yields the following formulas:

a k (x sp ) = A 1 (k) 3 i=0 (-1) i exp iπ( √ 3 + 1)k • l 3i+p + A 2 (k) 3 i=0 (-1) i+1 exp iπ( √ 3 -1)k • l 3i+p (52)
and

a k (x tq ) = A 1 (k) 2 i=0 exp -2πi √ 3 3 k • l 4i+q -exp -2πi 1 + √ 3 3 k • l 4i+q + A 2 (k) 2 i=0 exp -2πi √ 3 3 k • l 4i+q -exp -2πi √ 3 -1 3 k • l 4i+q , (53)
where the complex-valued functions A 1 (k) and A 2 (k) depend on the considered tiling. Therefore, for almost all Bragg peaks of a square-triangle tiling, with the possible exception of a set nowhere dense, the seven partial amplitudes (52) and (53) depend on only two unknown quantities!

Conclusions and discussion

We have considered the pure-point part of the diffraction spectrum of the families of Delone point patterns in the Euclidean space E, obeying local rules in a wide sense of the term (in particular, including disordered systems such as models of decorated random tilings). The partial diffraction amplitudes of such patterns are constrained by linear equations explicitly derivable from the local rules. More specifically, these equations depend on the properties of the corresponding FBS-complex -a geometric object encoding the local order of the pattern (see Definition 1). Whenever Bragg peaks fill densely a linear subspace V ⊂ E, for almost all of them, with the possible exception of a subset nowhere dense in V , the coefficients of these equations depend smoothly on the wave vector k ∈ V . For a given FBS-complex, these coefficients can be calculated explicitly in terms of finite trigonometric sums. It has been argued in [START_REF] Kalugin | Robust minimal matching rules for quasicrystals[END_REF] that the goal of the structure analysis of aperiodic solids should be the determination of the local environments responsible for the formation of the long range order, rather than finding the position of each and every atom in the structure. The local environments are naturally described by decorated FBS-complexes, and the constraints on partial diffraction amplitudes could be used to evaluate the validity of such structure models. This brings up a question: are partial amplitudes a k experimentally measurable? Since formally a k can be derived (up to a common phase factor) through the dependence of the Bragg peak intensities (20) on the weights w p , one can think of using the method of isotopic substitution in neutron diffraction experiments [START_REF] Cornier-Quiquandon | Neutron scattering structural study of AlCuFe quasicrystals using double isotopic substitution[END_REF]. However, this approach does not allow to distinguish the contributions of the same chemical element in different local environments. A alternative way to access the partial amplitudes is made possible by the recent progress in the development of phasing algorithms [START_REF] Palatinus | The charge-flipping algorithm in crystallography[END_REF]. Namely, one could separate the contribution of different atomic sites to the diffraction via the segmentation of the reconstructed electronic density (see e.g. Figure 2 of [START_REF] Takakura | Atomic structure of the binary icosahedral Yb-Cd quasicrystal[END_REF]). Technically, such a segmentation can be performed by means of a watershed algorithm [START_REF] Beucher | The morphological approach to segmentation: the watershed transformation[END_REF].

Corollary 2 provides a way to prove for a given set of local rules that the pure point part of the diffraction measure of any decorated tiling respecting these rules cannot be dense everywhere. The case of local rules enforcing periodic tilings is a trivial example of such a situation. An interesting open question is whether there are less trivial tilings for which the absence of everywhere dense pure point diffraction can be proven in this way.

Appendix A

Let ∆ inj stand for the small category of finite ordered sets [n] := {0 < 1 < • • • < n} and order preserving injective maps, and ∆ op inj be the corresponding opposite category. Definition 3. A semi-simplicial2 object in a category C is a functor S : ∆ op inj → C (or equivalently a contravariant functor ∆ inj → C).

The functor S is entirely characterized by its value on objects and morphisms of ∆ op inj . Thus, the first part of the data defining a semi-simplicial object S is a sequence {S n , n ∈ N} of objects of C, where we use the notation S n for the functorial image S [n]. Let δ n,i stand for the injective map from [n-1] to [n] missing the element i ∈ [n]. The entire set of morphisms of ∆ inj is a transitive closure of δ n,i (called elementary coface maps). Let δ n,i := S δ n,i stand for the functorial image of δ n,i . Therefore, the second part of the data defining S is a set of the elementary face morphisms δ n,i : S n → S n-1 . These morphisms must satisfy the so-called simplicial identity:

δ n-1,i δ n,j = δ n-1,j-1 δ n,i whenever i < j, (54)
which follows directly from the identity δ n,j δ n-1,i = δ n,i δ n-1,j-1 in the category ∆ inj .

In the case where C is the category of sets and maps, semi-simplicial objects are called semi-simplicial sets. If B is a semi-simplicial set, the disjoint union of the sets B n has naturally the structure of an N-graded set, which we will denote by B • . The semi-simplicial set B is called d-dimensional if B d = ∅ and B n = ∅ for all n > d. A finite-dimensional semi-simplicial set is finite if all sets B n are finite. For a semi-simplicial set B we shall denote by |B| its geometric realization (that is the topological cellular complex with the combinatorial structure given by B, see e.g. [START_REF] Milnor | The geometric realization of a semi-simplicial complex[END_REF] and [START_REF] Fritsch | Cellular structures in topology[END_REF]Chapter 4.2]). Similarly, for s ∈ B n , the notation |s| refers to the corresponding n-dimensional simplicial cell |s| ⊂ |B|.

Let e n,i : [1] → [n] stand for the morphism in ∆ inj given by e n,i (0) = 0 e n,i (1) = i.

We define the reference edge maps e n,i : B n → B 1 as the functorial image of e n,i : e n,i := Be n,i

Another important case of semi-simplicial objects corresponds to the situation when C is the category of modules over a commutative ring R. Such semi-simplicial objects are called semi-simplicial R-modules (or semi-simplicial vector spaces if R is a field). If M is a semi-simplicial R-module, the face operators δ n,i are R-module homomorphisms:

δ n,i : M n → M n-1 .
As follows from (54), the operators ∂ n : M n → M n-1 defined as

∂ n = n i=0
(-1) i δ n,i satisfy the equation ∂ n-1 ∂ n = 0 and thus make the N-graded module

M • = n∈N M n into a chain complex (M • , ∂ • ).
Given a commutative ring R and a semi-simplicial set B, one can construct the free semi-simplicial R-module R (B) by postcomposing B with the free R-module functor R (-) . It is noteworthy that the chain complex (Z (B) • , ∂ • ) is tautologically isomorphic to the complex of cellular chains of |B| considered as a CW-complex. For this reason we shall use for (Z (B) • , ∂ • ) the more traditional notation C • (B, Z).

Appendix B

Traditionally, the diffraction measure is defined as a Fourier transform of the autocorrelation (or Patterson) measure [START_REF] Baake | Dynamical systems on translation bounded measures: Pure point dynamical and diffraction spectra[END_REF][START_REF] Baake | Aperiodic Order: Volume 1, A Mathematical Invitation[END_REF] of the diffracting quantity. In this Appendix, we shall show that the distribution η defined by the formula [START_REF] Rosenberg | The square-integrability of matrix-valued functions with respect to a non-negative Hermitian measure[END_REF] equals the Fourier transform of the autocorrelation measure of [START_REF] Baake | Spectral notions of aperiodic order[END_REF].

Let us start by constructing the autocorrelation measure γ of the weighted Dirac comb ̺ f in the sense of the dynamical system (X(f 0 ), E, µ). For a given f ∈ X(f 0 ), let us consider the product measure ̺ f × ̺ f on E 2 . Averaging this measure over the hull X(f 0 ) yields a positive translation bounded measure on E 2 X(f0)

̺ f × ̺ f dµ(f ),
which is invariant with respect to translations of the form (y 1 , y 2 ) → (y 1 +t, y 2 + t). Therefore, there exists a positive translation bounded measure γ on E such that for any ψ ∈ S(E 2 ) holds the following It can be shown following the Dworkin's argument [START_REF] Dworkin | Spectral theory and x-ray diffraction[END_REF] (see also [START_REF] Baake | Spectral notions of aperiodic order[END_REF] for a detailed account) that if the dynamical system (X(f 0 ), E, µ) is uniquely ergodic, then the naïve autocorrelation measure of ̺ f exists and is equal to γ. The former is defined (see for instance [START_REF] Baake | Aperiodic Order: Volume 1, A Mathematical Invitation[END_REF]) as the Eberlein convolution ̺ f ⊛ ̺ f , where the symbol stands for complex conjugation and changing the sign of the function argument, i.e. ̺ f (y) = ̺ f (-y).

Let us now express the left hand side of [START_REF] Rosenberg | The square-integrability of matrix-valued functions with respect to a non-negative Hermitian measure[END_REF] in terms of γ. The right-hand side of ( 10) is a square-integrable function with well-defined values everywhere on X(f 0 ) (and not just µ-almost everywhere). Therefore, taking into account [START_REF] Baake | Spectral notions of aperiodic order[END_REF], for any f ∈ X(f 0 ) and any ϕ 1 , ϕ 2 ∈ S(E) one has m p,q=1 w p w q Γ xp (ϕ 1 )Γ xq (ϕ 2 ) (f ) = Therefore, since the functions of the form ϕ 1 ϕ 2 are dense in S(E * ), we have the identity η = γ.

Figure 1 :

 1 Figure1: The affine simplex α s (|s|) ⊂ E and its i-th face (bold solid line). The cross-hair represents the origin of E, numbers are the indices of the vertices of |s|. Top: in the case i = 0, the zeroth vertex of |δ n,i s| coincides with the zeroth vertex of |s|, thus α δn,is (|δ n,i s|) is the i-th face of α s (|s|). Bottom: in the case i = 0, the zeroth vertex of |δ n,i s| coincides with the first vertex of |s|, and α δn,0s (|δ n,0 s|) (dashed line) has to be shifted by ρ(e n,1 (s)) to obtain the zeroth face of α s (|s|).

Theorem 1 .

 1 which can be introduced by fixing a non-zero constant d-form Ω ∈ d E * : S(s) := sgn Ω • d i=1 ρ(e d,i (s)) (28) Let s : |B| → {-1, 0, 1} be defined as s(x) := S(s) if x ∈ |s| where s ∈ B d 0 otherwise Let a k be the partial diffraction amplitudes defined in Proposition 2. Then for any k ∈ E, the function sa k is a d-cycle of the chain complex (F (k),• , ∂ • ): ∂ d (sa k ) = 0. (29) Proof. Note first that since a k ∈ F (k),• by Proposition 3 and s is constant on all d-simplices of |B| and zero elsewhere, we have sa k ∈ F (k),d . To prove the cycle condition (29) it suffices to show that ∂ d (sa k ) vanishes on any (d -1)-simplex |s| ⊂ |B|. As follows from (26), the values of ∂ d (sa k ) on |s| depend only on the values of sa k on the neighboring d-simplices of |B|. Let us denote the set of those neighbors (considered together with the index of the face corresponding to s) by N s :

Figure 2 :

 2 Figure 2: The superposition of tiles (triangles outlined by thin solid lines) which may have as a face the affine simplex α s (|s|) ⊂ E of dimension d -1 (bold solid line). The cross-hair represents the origin of E. Shaded areas correspond to the open polyhedra Σ +s and Σ - s defined in (33). Given a point x ∈ |s|, one can always choose a vector t ∈ E in (34) in such a way that α s (x) ± t ∈ Σ ± s .

  over the formal basis (ǫ s ) s∈B• with the face homomorphisms. Let us start by fixing an element s 0 ∈ B 0 and associating with each s ∈ B • an arbitrarily chosen 1-chain c s ∈ C 1 (B, Z) satisfying the condition

Figure 3 :

 3 Figure 3: A metaphorical representation of an integer linear combination of translated prototiles corresponding to a d-cycle of Z ( BV ) , for the case of the tiling of plane by squares and 45 degrees rhombi. The solid circles represent the decoration of the square tile.

  (a) Solid circles represent the decorations, gray lines are the tile boundaries. (b) The decorations are positioned at the centers of tiles.

Figure 6 :

 6 Figure 6: A decorated (3, 2)-canonical tiling.

Figure 7 :

 7 Figure 7: Seven prototiles of the square-triangle tiling. Solid circles represent decorations placed at the centers of prototiles.

E 2 X

 2 (f0) ̺ f (y 1 )̺ f (y 2 ) dµ(f ) ψ(y 1 , y 2 ) dy 1 dy 2 = E E ψ(t, y + t) dt dγ(y) (55)

E 2 ̺

 2 f (y 1 )̺ f (y 2 )ϕ 1 (-y 1 )ϕ 2 (-y 2 ) dy 1 dy 2 By integrating this identity over the hull X(f 0 ) and taking into account (55), we get m p,q=1w p w q Γ xp (ϕ 1 ), Γ xq (ϕ 2 ) = E ( ϕ 1 * ϕ 2 )(-y) dγ(y)Using[START_REF] Rosenberg | The square-integrability of matrix-valued functions with respect to a non-negative Hermitian measure[END_REF] for the left-hand side and expressing the right-hand side through the Fourier transform of γ yieldsE * ϕ 1 (k) ϕ 2 (k)dη(k) ≡ η( ϕ 1 ϕ 2 ) = γ( ϕ 1 ϕ 2 )

  1 . Let us start by constructing a family of such spaces parameterized by a vector k ∈ E * .Let F (k),n be the space of functions |B| → C spanned by {Y s : s ∈ B n }, where

In the case when the semi-simplicial set B describes a regular cellular complex the results of Sections 4 and 5 can also be formulated in terms of homology of cellular cosheaves[START_REF] Michael | Sheaves, cosheaves and applications, volume 1249 of Publicly Accessible Penn Dissertations[END_REF] on |B|. While this approach might be more natural for the case of no-simplicial tilings, it is not immediately applicable to some relevant examples when the cellular complex associated with B is not regular.

This construction is also called a presimplicial object[START_REF] Fritsch | Cellular structures in topology[END_REF] Chapter 4.1]. We follow here the terminology of[START_REF] Charles | An introduction to homological algebra[END_REF] Chapter 8.1].

Acknowledgments P.K. thanks Marat Rovinski for fruitful discussions.

torus T n . We can use this fact to calculate the homology of the chain complex (G V,• , ∂ • ) in the following way. For 1 ≤ i ≤ n, let (M i,• , ∂ • ) stand for the chain complex of free Z[ξ i , ξ -1 i ]-modules of rank 1:

The complex (M i,• , ∂ • ) is isomorphic to the complex of submodules of G V corresponding to the the edge v i and the unique vertex of |B|. The CWdecomposition of the entire torus then corresponds to the tensor product of (M i,• , ∂ • ) over Z for i = 1 . . . n:

Note that (49) is naturally a complex of modules over the ring

The d-skeleton of the torus is described by the truncated chain complex

The triangulation of the prototiles of the canonical tiling thus yields a chain quasi-isomorphism (see e.g. [16, Chapter 1.1]) of ( 51) to (G V , ∂ • ). Since the chain complex (48) is acyclic, so is (M • , ∂ • ) and the d-cycles of the truncated complex (51) are precisely the boundaries of the (d + 1)-chains of (M • , ∂ • ). Therefore, the module of d-cycles of G V is a free Z[L V ]-module of rank n d+1 . Taking into account that an (n, d)-canonical tiling has n d prototiles, the constraints of Corollary 2 are effective in the case n ≤ 2d.

It is instructive to consider in details the case n = d + 1. In this situation, the generating set of d-cycles of G V contains only one element. Therefore, the partial amplitudes a k at d+1 decorating points must belong to a one-dimensional subspace of C d+1 , depending smoothly on k. Let us denote by x p the decorating point belonging to the prototile not having v p as its edge (see Figure 6b). A straightforward computation (using the approach presented at the end of Section 5) then leads to the following formula for the partial amplitudes:

where the coefficient A(k) depends on the tiling under consideration and is clearly not a regular function of k.

Decorated square-triangle tiling

The square-triangle tiling is a popular model for the structure of planar aperiodic systems with twelve-fold symmetry. It describes remarkably well the


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the OPTiMiSE cohort study

Keywords: Inflammation, pro-inflammatory cytokines, remission, psychosis, longitudinal study

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Baseline levels of C-reactive protein and proinflammatory cytokines are not associated with early response to amisulpride in patients with First Episode Psychosis

A c c e p t e d M a n u s c r i p t

Introduction

Early treatment response is an excellent predictor of long-term symptomatic and functional outcome in psychosis. Unfortunately, and despite extensive research, reliable biomarkers of early treatment response and clinical outcome in FEP patients remain to be identified [START_REF] Emsley | Remission in early psychosis: Rates, predictors, and clinical and functional outcome correlates[END_REF] . In a typical study aimed at identifying biomarkers of treatment response, FEP patients are sampled at baseline for blood or neuroimaging biomarkers and assessed at inclusion as well as several weeks or months later for clinical outcomes. In most cases, response to treatment is defined as a reduction in symptom severity to the levels required by the remission criteria of the Schizophrenia Working Group Consensus [START_REF] Andreasen | Remission in schizophrenia: proposed criteria and rationale for consensus[END_REF] . This consensus established a set of criteria that provide an absolute threshold in severity of symptoms that should be reached for clinical improvement. Alternatively, some authors have used a continuous measure of treatment response which is defined as change in PANSS total scores from baseline to follow-up, considering baseline PANSS total score and subtracting a score of 30, as even individuals without any mental health problem could score 30 in the PANSS [START_REF] Leucht | Defining 'response' in antipsychotic drug trials: recommendations for the use of scale-derived cutoffs[END_REF] . In this latter c se the following formul w s used: (b seline PAN tot l score -30) -(follow-up PAN tot l score -30)/(b seline PAN tot l score -30) × 100. Other clinic l outcomes such as changes in the Clinical Global Impression Scale (CGI), the Brief Psychiatric Rating Scale (BPRS), the Calgary Depression Scale for Schizophrenia (CDSS), and occurrence of Serious Adverse Events (SAEs) may also be assessed.

While several studies have demonstrated the feasibility of using neuroimaging biomarkers for predicting clinical outcome in FEP patients [START_REF] Dazzan | Magnetic resonance imaging and the prediction of outcome in first-episode schizophrenia: a review of current evidence and directions for future research[END_REF] , diagnostics tools based on biological fluids may be more cost and time efficient. Indeed studies, several authors have aimed at identifying soluble biomarkers in blood or saliva which, when measured at baseline in FEP patients, were associated with clinical outcomes several weeks, months or years later. A wide range of biomarkers have been tested including the acute phase protein CRP 5- 7 , many interleukins [START_REF] Mondelli | Cortisol and Inflammatory Biomarkers Predict Poor Treatment Response in First Episode Psychosis[END_REF][START_REF] Martinuzzi | Stratification and prediction of remission in first-episode psychosis patients: the OPTiMiSE cohort study[END_REF] and chemokines [START_REF] Martinuzzi | Stratification and prediction of remission in first-episode psychosis patients: the OPTiMiSE cohort study[END_REF] , TNF-α [START_REF] Mondelli | Cortisol and Inflammatory Biomarkers Predict Poor Treatment Response in First Episode Psychosis[END_REF][START_REF] Martinuzzi | Stratification and prediction of remission in first-episode psychosis patients: the OPTiMiSE cohort study[END_REF] , interferon-γ (IFN-γ) [START_REF] Mondelli | Cortisol and Inflammatory Biomarkers Predict Poor Treatment Response in First Episode Psychosis[END_REF][START_REF] Martinuzzi | Stratification and prediction of remission in first-episode psychosis patients: the OPTiMiSE cohort study[END_REF] , vascular endothelial growth factor (VEGF) [START_REF] Mondelli | Cortisol and Inflammatory Biomarkers Predict Poor Treatment Response in First Episode Psychosis[END_REF][START_REF] Martinuzzi | Stratification and prediction of remission in first-episode psychosis patients: the OPTiMiSE cohort study[END_REF] , complement component 4 (C4) [START_REF] Mondelli | Baseline high levels of complement component 4 predict worse clinical outcome at 1-year follow-up in first-episode psychosis[END_REF] , lipids and glucometabolic biomarkers (glycated hemoglobin (HbA1c), triglycerides (TG), total cholesterol, Low Density Lipoprotein (LDL) cholesterol, fasting glucose) [START_REF] Nettis | Metabolic-inflammatory status as predictor of clinical outcome at 1-year follow-up in patients with first episode psychosis[END_REF] , redox biomarkers (glutathione (GSH), GSH peroxidase and reductase activities (GPx, GR), thioredoxin (Trx), and GSH-related metabolites), amino acids [START_REF] Fournier | Topology predicts long-term functional outcome in early psychosis[END_REF] and stress biomarkers (cortisol, sulfated dehydroepiandrosterone (DHEA-S)) [START_REF] Babinkostova | Association between Serum Cortisol and DHEA-S Levels and Response to Antipsychotic Treatment in Schizophrenia[END_REF] . In a 12 weeks follow-up study on 68 FEP patients, salivary levels of cortisol measured shortly after awakening as well as blood levels of IFN-γ and IL-6 were lower in responders compared to non-responders in the early phases of psychosis [START_REF] Mondelli | Cortisol and Inflammatory Biomarkers Predict Poor Treatment Response in First Episode Psychosis[END_REF] . In a longitudinal study on 42 FEP patients, the authors used a principal component analysis (PCA) to reduce the dimensionality of the dataset accounting for both inflammation and metabolic status 7 . Among the three identified PCA factors, factor 1 that accounted for hsCRP and Body Mass Index (BMI), was associated with treatment response at one year. Lastly, in a naturalistic longitudinal study on 25 FEP patients, serum levels of complement component 4 (C4) were associated with treatment response at one year after adjustment for baseline severity of symptoms and CRP levels [START_REF] Mondelli | Baseline high levels of complement component 4 predict worse clinical outcome at 1-year follow-up in first-episode psychosis[END_REF] .

While several studies have allowed for the identification of candidate neuroimaging and soluble biomarkers that could be used for predicting clinical outcome in FEP patients, many of them have suffered from several shortcomings including small sample size and treatment heterogeneity. As an attempt to overcome these shortcomings, we used clinical data and biological samples from the multinational, multi-center, randomized, double-blind OPTiMiSE study in which 481 FEP patients were treated with the second-generation antipsychotic amisulpride [START_REF] Kahn | Amisulpride and olanzapine followed by open-label treatment with clozapine in first-episode schizophrenia and schizophreniform disorder (OPTiMiSE): a three-phase switching study[END_REF] . We assessed serum samples collected at baseline for wellstudied biomarkers of inflammation, i.e. IL-6, IL-8, TNF-α nd C P. We then used multivariable regression to study the association between these biomarkers and PANSS score and sub-scores, CDSS, remission and SAEs.

While inflammation is an intricate process involving dozens of molecules, we have chosen to focus on IL-6, IL-8, TNF-α nd C P for the following re sons. First ntips choticnaive FEP and acute psychotic relapse are associated with increased serum concentrations of IL-6 [START_REF] Miller | Meta-analysis of cytokine alterations in schizophrenia: clinical status and antipsychotic effects[END_REF] , TNF-α 12 and CRP [START_REF] Fernandes | C-reactive protein is increased in schizophrenia but is not altered by antipsychotics: meta-analysis and implications[END_REF] . Second, population-based longitudinal studies have shown an association between higher levels of IL-6 [START_REF] Khandaker | Association of serum interleukin 6 and C-reactive protein in childhood with depression and psychosis in young adult life: a population-based longitudinal study[END_REF] and CRP [START_REF] Metcalf | Serum C-reactive protein in adolescence and risk of schizophrenia in adulthood: A prospective birth cohort study[END_REF] and risk for psychotic experience and a diagnosis of schizophrenia subsequently in adulthood. Third, IL-6, IL-8, and TNF-α re three of the main proinflammatory cytokines that are produced by macrophages when activated with Pathogen-Associated Molecular Patterns (PAMPs) [START_REF] Duque | Macrophage cytokines: involvement in immunity and infectious diseases[END_REF] . Four, IL-8 is a potent chemotactic factor that cause neutrophils and other granulocytes that promote inflammation to migrate toward the site of infection [START_REF] Moore | Attracting Attention: Discovery of IL-8/CXCL8 and the Birth of the Chemokine Field[END_REF] .

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Clinical assessment

A screening visit was conducted during which eligibility was assessed. Baseline data were obtained regarding sex, age, diagnoses, body mass index (BMI), waist circumference, recreational drug use, alcohol consumption, caffeine consumption, smoking and current treatments. Patients were clinically assessed at baseline for positive, negative and general psychopathology symptoms using the PANSS scale [START_REF] Kay | The positive and negative syndrome scale (PANSS) for schizophrenia[END_REF] , depression using the CDSS [START_REF] Addington | A depression rating scale for schizophrenics[END_REF] , social functioning using the Personal and Social Performance Scale (PSP) [START_REF] Morosini | Development, reliability and acceptability of a new version of the DSM-IV Social and Occupational Functioning Assessment Scale (SOFAS) to assess routine social functioning[END_REF]

Blood samples

Venous blood was obtained from fasting subjects between 7:00 am and 9:00 am. Five millimiters of blood were drawn into serum Vacutainer tubes and allowed to clot for 1 h before centrifugation (1500 x g, 10 min). Sera were stored in 0.5 ml aliquots at -80°C for 3-4 years. Samples were thawed once and immediately assessed for cytokine and CRP levels by immunoassay.

Immunoassay

Serum levels of IL-6, IL-8, TNF-α nd C P were me sured using the Pro-inflammatory Panel 1 and Vascular Injury Panel 2 v-PLEX® kits (MSD) at a single site at the Centre National de la Recherche Scientifique (CNRS). All assays were performed according to the m nuf cturer's instructions. D t were cquired on the V-PLEX® Sector Imager 2400 plate reader and analyzed using the Discovery Workbench 3.0 software (MSD). Standard curves for each cytokine were generated using standards provided in the kits. Serial 4-fold dilutions of the standards were run to generate a 7-standard concentration set, and the diluent alone was used as a blank. Cytokine concentrations were determined from the standard curve using a 4-parameter logistic curve fit to transform the mean luminescence intensities into concentrations. As the lower limit of detection (LLOD), we used the median LLOD compiled over multiple plates and corresponding to the concentration calculated based on the signal recorded for the blank plus 2.5 standard deviations. For each cytokine and the CRP, samples that fell below the LLOD were given a substituted value equal to half the LLOD. Supplementary Table 1 shows, for each biomarker and each cohort the LLOD, the number and percentage of samples that fall below the LLOD, the lower (min), upper (max), median and mean values, as well as the standard deviation (SD).

Data Analysis

Data preprocessing. Our dataset did not contain any missing data. For each cytokine and the CRP, samples that fell below the LLOD were given a substituted value equal to half the LLOD. As cytokines and CRP are believed to be stable when stored at -80°C and since the samples were thawed only once, biomarker levels were not adjusted for storage duration.

Unsupervised clustering analysis. We stratified patients based on serum cytokine levels at baseline using the k-spectral clustering algorithm [START_REF] Von Luxburg | A tutorial on spectral clustering[END_REF] , with the aim to identify two homogeneous clusters within our study sample. We first normalized each feature (serum levels of each cytokine) independently so that the norm of each feature becomes equal to one. The k-spectral clustering method uses the spectrum (eigenvalues) of the similarity matrix of the data to perform dimensionality reduction before clustering in fewer dimensions.

The similarity matrix is provided as an input and consists of a quantitative assessment of the relative similarity of each pair of points in the dataset. Silhouette values were computed as a measure of clustering appropriateness. We used the t-Distributed Stochastic Neighbor Embedding (t-SNE) method for graphical representation of the clusters. Codes will be available under request.

Statistical analyses.

Comparison analyses between clusters were performed using the twotailed non-parametric Mann & Whitney U-test or the Chi-square test (as indicated).

Comparison analyses of cytokine and CRP levels between cluster 1 and 2 patients, and healthy individuals were performed using the Kruskal-Wallis two-tailed non-parametric test followed by post-hoc analysis.

Results

A total of 481 FEP patients were included in the OPTiMiSE clinical trial [START_REF] Kahn | Amisulpride and olanzapine followed by open-label treatment with clozapine in first-episode schizophrenia and schizophreniform disorder (OPTiMiSE): a three-phase switching study[END_REF] . Out of the 446 patients in the intention-to-treat (ITT) sample, 348 had serum samples collected at baseline before the study treatment was initiated. From these patients, 289 were included in this study and assessed clinically at baseline and after 4-5 weeks of treatment with the antipsychotic amisulpride (Table 1). Serum samples at baseline were assessed for the pro-inflammatory cytokines IL-6, IL-8 and TNF-α nd for the cute ph se protein C P (Supplementary Table 1). Low to medium positive correlations were observed between IL-6, IL-8 and TNF-α but not CRP, suggestive of coordinated but not similar variations in the levels of these biomarkers in FEP patients (Supplementary Figure 1).

We first investigated the association between IL-6, IL-8, TNF-α nd C P considered s continuous variables and PANSS total score and subscores, CDSS, remission and SAEs.

After adjusting for age, gender, BMI, waist circumference, recreative drugs use, consumption of coffee and alcohol, and tobacco smoking, no association was found between any of these biomarkers and total PANSS score (Table 2), CDSS (Table 3), remission (Table 4), SAEs (Table 5), positive PANSS (PPANSS) (Supplementary Table 2), negative PANSS (NPANSS) (Supplementary Table 3) or general psychopathology PANSS (GPANSS) (Supplementary Table 4) subscores.

Because inflammatory biomarkers may be associated with a clinical outcome only when above a specific threshold, we dichotomized each biomarker at the 15 th /85 th upper/lower percentile before studying its association and clinical outcomes. None of the four tested inflammatory biomarkers was associated with total PANSS score (Supplementary Table 5), PPANSS (Supplementary Table 6), NPANSS (Supplementary Table 7), GPPANSS (Supplementary Table 8) subscores, CDSS (Supplementary Table 9), remission (Supplementary Table 10) or SAEs (Supplementary Table 11).

Bec use n individu l's inflammatory status may be better captured by a combination of pro-inflammatory cytokines than by any individual one, we used an unsupervised cl ssific tion ppro ch to str tif p tients into n "infl med" nd "non-infl med" cluster based on serum levels of IL-6, IL-8 and TNF-α. These two clusters ccounted for % nd 34% of the total sample respectively (Table 1). In agreement with the relatively high mean silhouette value of the clustering solution, cluster exhibited a lower BMI compared to those of cluster 2, they were similar with respect to sex ratio, age, waist circumference, used of recreative drugs, and consumption of coffee and alcohol (Table 1). Cluster 2 patients had higher serum levels of IL-6, IL-8 and TNF-α t baseline when compared to cluster 1 patients or healthy controls (Supplementary Figure 3).

In contrast, serum levels of IL-8 and TNF-α were simil r in both cluster 1 p tients nd he lth controls, while IL-6 more more abundant in cluster 1 patients. Of note, CRP levels were similar in cluster 1 patients, cluster 2 patients and healthy controls. We next studied the association between the patient cluster and clinical outcomes. After adjustment for covariates and confounders, we did not find any association between patient cluster and total PANSS score (Table 2), CDSS (Table 3), remission (Table 4), SAEs (Table 5), PPANSS (Supplementary Table 2), NPANSS (Supplementary Table 3) or GPPANSS (Supplementary Table 4) subscores.

Discussion

Main findings

Here, we have investigated the association between baseline levels of four inflammatory biomarkers and clinical outcomes in 289 FEP patients 4-5 weeks after treatment with amisulpride was initiated. We first analyzed each biomarker independently and assessed their association with PANSS total score and subscores, CDSS, remission and SEAs. We performed two types of analyses: one in which we considered each biomarker as a continuous variable, and another in which we dichotomized them at the 15 th upper/85 th lower percentile. After adjustment for age, sex, BMI, waist circumference, use of recreative drug, coffee and alcohol consumption and tobacco smoking, neither IL-6 nor IL-8, TNF-α nd CRP was associated with any of the tested clinical outcomes.

We next used an unsupervised clustering method to stratify FEP patients into two clusters based on serum levels of IL-6, IL-8 and TNF-α. While these c tokines were present at normal levels in cluster 1 patients, their levels were abnormally elevated in cluster 2 patients, indicative of an inflamed profile. The patient cluster was neither associated with total PANSS score at baseline, nor with PPANSS, NPANSS or GPPANSS subscores, CDSS, remission or SAEs. To summarize, we could not identify any association between the patient's infl mm tor profile nd n of the clinic l outcome th t we h ve ssessed.

A c c e p t e d M a n u s c r i p t

Page 14

Other findings

An unexpected finding of this study is that cluster 1 and cluster 2 patients had comparable blood levels of CRP, and that these levels were comparable to those found in healthy controls. This is an important result as previous studies attempting to use CRP to identify people at risk of later schizophrenia have been mixed with some, but not others, finding that patients with increased levels of CRP were at increased risk for later schizophrenia 26 . CRP is produced mainly in the liver in response to inflammation 27 and is commonly used to assess the presence and severity of inflammation in a wide range of diseases. However, CRP is an ambiguous pro-inflammatory marker 28 as it is also constitutively produced in the absence of inflammation in a pentameric form (pCRP) with anti-inflammatory properties 29 . In contrast, the monomeric CRP (mCRP) is produced locally by dissociation of the pCRP, in response to inflammatory cues in inflamed or damaged tissues and has potent pro-inflammatory effects 28 . Because pro-inflammatory mCRP is mostly retained in tissues, the CRP measured in serum by immunoassays exclusively corresponds to the anti-inflammatory pCRP form, in particular in the case of low-grade inflammation. It is only in the case of acute inflammation that the overload of CRP in tissues is reflected in the serum. Based on these data, Del Giudice and Gangestad have pointed out that measuring serum CRP levels is insufficient to establish the existence of a low-grade inflammatory state, and have proposed that other pro-inflammatory cytokines such as TNF-α and IL-1ß could be more reliable as biomarkers of inflammation 28 . Our results not only support this view but also suggest that combinations of pro-inflammatory cytokines, rather than individual ones, should be used to identif "infl med" p tients. This m h ve implications for the design of prospective placebo-controlled clinical trials aimed at comp ring dise se outcome in "infl med" nd "non-infl med" p tients or for selecting "infl med" p tients for dd-on anti-inflammatory treatment.

While the primary objective of this study was to investigate association between inflammatory biomarkers and clinical outcome after treatment with amisulpride, our results provided insight into the association between clinical variables at baseline and clinical outcome. As for PANSS total score and subscores after treatment, they were negatively associated with age in all models, andas expectedpositively associated with their respective score and subscores at baseline. Remission was negatively and positively associated with age and NPANSS at baseline respectively. As for SAEs, it was associated with sex, waist circumference and GPPANSS at baseline in models in which IL- TNF-⍺ were used as covariates. Also, lower levels of CRP decreased the odds of SAEs but the effect size was extremely low.

Discrepancies with other studies

An association between inflammatory biomarkers at baseline and response treatment was first reported by Mondelli et al in a highly cited paper [START_REF] Mondelli | Cortisol and Inflammatory Biomarkers Predict Poor Treatment Response in First Episode Psychosis[END_REF] . In this study, 68 FEP patientsamong whom 33 were sampled at baselinewere treated as usual (TAU) and assessed 12 weeks later to identify those who met remission criteria [START_REF] Mondelli | Cortisol and Inflammatory Biomarkers Predict Poor Treatment Response in First Episode Psychosis[END_REF] . In these 33 patients among whom 18 and 21 were responders and non-responders respectively, blood levels of IL-6 and IFN-γ were lower in responders compared to non-responders [START_REF] Mondelli | Cortisol and Inflammatory Biomarkers Predict Poor Treatment Response in First Episode Psychosis[END_REF] . These authors concluded that "Infl mm tor biom rkers predict poor response in FEP". The discrep nc between our results and this previous finding may be explained by several reasons.

Limitations

Our results should be considered and interpreted in light of the following limitations. First, we did not study the association between IL-1α nd IL-1ß because the immunoassay platform that we used was not sensitive enough to detect these cytokines in the majority of serum samples. Second, our results were adjusted to age, gender, BMI, consumption of alcohol and tobacco and use of recreational drugs, but not to other variables that may be associated with either cytokine levels or clinical outcome such as socioeconomical status, physical exercise, diet. Third, patients with an autoimmune or an allergic disease, with an infectious disease and/or who have been treated anti-inflammatory drugs, corticosteroids or antibiotics within the prior months were not excluded. Third, we only assessed the association between inflammation at baseline and clinical outcome in FEP patients after 4-5 weeks of treatment with amisulpride. It remains to be established whether similar results would be obtained when patients are treated with another antipsychotic and for a longer period. Fourth, the serum samples from healthy individuals that we have used in this study were not collected as part as the OPTiMiSE clinical trial but as part of another clinical study. However, because the Standard Operating Procedures (SOP) for blood collection and serum preparation were the same in the two studies, and all serum samples were assessed in the same laboratory using the same procedure, we believe that biomarker levels could readily be compared across samples.

Implications and Future Directions

The present study does not support an association between baseline levels of biomarkers of inflammation and early clinical outcomes in FEP patients. However, our data do not rule out the possibility that inflammatory biomarkers could predict long-term clinical outcomes such as relapse in chronic schizophrenia. Testing this hypothesis in a prospective study is warranted. While the results presented here could be considered as disappointed, we believe that publication of these negative findings is important for the field of immuno-psychiatry to move forward.

A c c e p t e d M a n u s c r i p t Page 9

 9 , wellbeing using the Subjective Wellbeing under Neuroleptics (SWN) and overall severity of symptoms with the CGI scale 22 . Patients were treated with up to 800 mg/day amisulpride in an open-label design and clinically assessed 4-5 weeks later for psychopathology using the PANSS scale, symptomatic remission according to the criteria of Andreasen et al. 2 ( score ≤3 simultaneously on 8 PANSS items: P1, P2, P3, N1, N4, N6, G5, and G9), percentage of weight gain and the occurrence of SAE. Downloaded from https://academic.oup.com/schizbullopen/advance-article/doi/10.1093/schizbullopen/sgab017/6286940 by guest on 14 July 2021 Healthy individuals Healthy individuals were recruited as part of a study registered in ClinicalTrials.gov (ID: NCT02209142). An extensive clinical and psychological examination, using both Structured Clinical Interview for DSM Disorders 23 and a self-rating questionnaire, as well as clinical and biological examinations (including body temperature, blood cell count) were used to determine the absence of any medical conditions or any psychiatric disorders. None of the subjects had received any vaccinations within a month before inclusion. All subjects gave informed written consent. The study was conducted in accordance with the latest version of the Declaration of Helsinki and was approved by the appropriate French ethical and medical authorities.

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  To study associations at baseline clusters based on proinflammatory cytokines at baseline and clinical scores, we implemented a multivariable linear Downloaded from https://academic.oup.com/schizbullopen/advance-article/doi/10.1093/schizbullopen/sgab017/6286940 by guest on 14 July 2021Page 11 variables. To study associations between remission after treatment and cluster at baseline, we implemented a multivariable logistic regression model using the outcome (remission/nonremission) as the dependent variable and cluster, PANSS sub-scores at baseline, age, sex, BMI, waist circumference, use of recreational drugs, alcohol consumption and current smoking as independent variables. To study associations between PANSS sub-scores after treatment and cluster at baseline, we implemented a multivariable linear regression model using PANSS sub-scores after treatment as the dependent variables and cluster, PANSS sub-scores at baseline, age, sex, BMI, waist circumference, use of recreational drugs and alcohol consumption and current smoking as independent variables. Statistical significance was set according to a p-value (p) < 0.05. Statistical analyses were performed using R Stats package25 and graphs were generated using GraphPad Prism version 6.00 for iOS (GraphPad Software, USA).

	A c c e p t e d M a n u s c r i p t A c c e p t e d M a n u s c r i p t	Downloaded from https://academic.oup.com/schizbullopen/advance-article/doi/10.1093/schizbullopen/sgab017/6286940 by guest on 14 July 2021
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regression model using the cluster as the dependent variable and baseline PANSS total score and sub-scores at baseline, age (years), sex (Female/Male), BMI (kg/m 2 ), waist circumference (cm), use of recreational drugs (Yes/No), coffee consumption (number of cups/day), alcohol consumption (yes/no), and tobacco smoking (yes/no) as independent

  6, IL-8 and Downloaded from https://academic.oup.com/schizbullopen/advance-article/doi/10.1093/schizbullopen/sgab017/6286940 by guest on 14 July 2021

  First, while Mondelli et al. studied response to clinician-led antipsychotic treatment after 12 weeks, we searched for biomarkers that were associated with response to a single antipsychotic, i.e. amisulpride, 4-5 weeks after the treatment was initiated. Second, our study sample consisted of 289 FEP p tients comp red to 33 in Mondelli's p per. Third our results were adjusted for age, gender, BMI, waist circumference, use of recreative drugs, consumption of coffee and lcohol tob cco smoking nd clinic l s mptoms' severit t b seline while Mondelli et al.StrengthsOne important strength of this study is that it takes advantage of clinical and biological samples collected within the frame of one of the largest longitudinal clinical studies in FEP patients. Patients were not only medication naïve (or almost medication naïve) at baseline, but also treated with the same antipsychotic and assessed 4-5 weeks after the treatment was initiated, therefore precluding biases due to different medications or assessment at different time points. Second, the longitudinal design of the OPTiMiSE clinical trial allowed for assessing the association of inflammatory biomarkers collected at baseline and clinical outcomes 4-5 weeks later. Third, and with at least two noticeable exceptions 30, 31 , we are not aware of any other study in which pro-inflammatory cytokines, and more generally inflammatory biomarkers, were considered altogetherrather than individuallyto determine the "infl med" or "non-infl med" st tus of n individu l.

	used univariate statistical methods to analyze their data. In another longitudinal study on 42 FEP patients, the authors aimed at identifying biomarkers of metabolism and inflammation that were associated with clinical outcomes 7 . To this aim, they measured high sensitivity C-reactive protein (hsCRP) as a biomarker of inflammation, and BMI, lipid profile and gluco-metabolic parameters as metabolic variables. A principal component analysis (PCA) was then used to reduce the dimensionality of the dataset accounting for both inflammation and metabolic status. Results showed that a PCA factor that accounted for hsCRP, BMI and triglycerides (TG) measured at baseline was associated with total PANSS score, PPANSS and NPANSS subscores, and treatment response rate at 1-year follow-up. While these latter results did show an effect of abnormal metabolic and inflammatory status on the brain, the use of PCA did not allow for assessing the relative contribution of metabolic and inflammatory biomarkers to the outcome. A c c e p t e d M a n u s c r i p t	Downloaded from https://academic.oup.com/schizbullopen/advance-article/doi/10.1093/schizbullopen/sgab017/6286940 by guest on 14 July 2021
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	M a n u s c r i p t
	A c c e p t e d
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Acknowledgments

We thank all the patients who participated in the OPTiMiSE study. The OPTiMiSE study was funded by the European Commission within the 7th Program (HEALTH-F2-2010-242114). This work was also supported by the Investissements d'Avenir program managed by the Agence Nationale pour la Recherche (ANR) under reference ANR-11-IDEX-0004-02 (Labex BioPsy) (to WBHA). Additional financial support was obtained from the Fondation de France (NG) and the Fondation FondaMental (NG), the LABEX SIGNALIFE (#ANR-11-LABX-0028-01) and the FHU OncoAge. ML's te m is ffili ted with the P ris chool of Neuroscience (ENP) and the Bio-Psy Laboratory of Excellence. Downloaded from https://academic.oup.com/schizbullopen/advance-article/doi/10.1093/schizbullopen/sgab017/6286940 by guest on 14 July 2021

Conflict of interest

RSK declares personal fees for consultancy from Alkermes, Minerva Neuroscience, Gedeon Richter, and Otsuka; and personal (speaker) fees from Otsuka/Lundbeck. SL declares fees for consulting from LB Pharma, Lundbeck, Otsuka, Teva, and Gedeon Richter; and fees for lectures from Janssen, Eli Lilly, Lundbeck, Otsuka, Sanofi, and Servier. SWL declares personal fees from Otsuka. CA declares grants, personal fees, and other fees from Janssen-Cilag, Lundbeck, Otsuka, Acadia, Abbott, Amgen, AstraZeneca