Philippe Chartier
email: philippe.chartier@inria.fr

Ernst Hairer
email: ernst.hairer@unige.ch

Gilles Vilmart
email: gilles.vilmart@epfl.ch

Arieh P Iserles

Chartier

Algebraic Structures of B-series

Keywords: B-series, Rooted trees, Composition law, Substitution law, Butcher group, Hopf algebra of trees, Coproduct, Antipode, P-series, S-series Mathematics Subject Classification (2000) 65L06, 65P10, 37C10

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Introduction

Let us consider systems of differential equations

ẏ = f(y) (1)
with smooth vector field f(y). Since the work of Cayley [START_REF] Cayley | On the theory of the analytic forms called trees[END_REF] and Merson [START_REF] Merson | An operational method for the study of integration processes[END_REF]itis known that the expressions arising in the derivatives of its solution, ÿ = (f ′ f )(y), ... y = (f ′′ (f, f ))(y) + (f ′ f ′ f )(y), are in one-to-one correspondence with rooted trees. It is therefore natural to consider formal series of the form B(a,hf,y) = a(∅)y + ha( )f (y) + h 2 a( )(f ′ f )(y) + h 3 2 a( ) f ′′ (f, f ) (y)

+ h 3 a (f ′ f ′ f )(y) + h 4 a f ′′ (f, f ′ f) (y) +••• (2)
with scalar coefficients a(∅), a( ), a( ), etc. The exact solution of (1)i so ft h i s form with a(∅) = a( ) = 1, a( ) = 1/2, a( ) = 1/3, etc. In his fundamental work on order conditions, Butcher discovered in the 1960s (culminating in the seminal article [START_REF] Butcher | An algebraic theory of integration methods[END_REF]) that the numerical solution of a Runge-Kutta method is also a series of the form (2) with a(τ) depending only on the coefficients of the method. Hairer and Wanner [START_REF] Hairer | On the Butcher group and general multi-value methods[END_REF] considered series [START_REF] Ch | Runge-Kutta methods and renormalization[END_REF] with arbitrary coefficients and called them B-series. 1 They applied them to the elaboration of order conditions for general multivalue methods. B-series and extensions thereof are now exposed in various textbooks and articles, possibly with different normalizations; see e.g., [START_REF] Butcher | Numerical Methods for Ordinary Differential Equations[END_REF][START_REF] Hairer | Geometric Numerical Integration[END_REF]. B-series play an important role in the study and construction of numerical integrators. This is a consequence of the following two operations on B-series:

• Composition law [START_REF] Butcher | An algebraic theory of integration methods[END_REF][START_REF] Hairer | On the Butcher group and general multi-value methods[END_REF]. For b(∅) = 1, a B-series considered as a mapping y → B(b,hf,y) is O(h)-close to the identity. It is therefore possible to replace y in [START_REF] Ch | Runge-Kutta methods and renormalization[END_REF] with B(b,hf,y), and to expand all expressions around y. Interestingly, the result is again a B-series and we have B a,hf,B(b,hf,y) = B(b • a,hf,y).

• Substitution law [START_REF] Chartier | A substitution law for B-series vector fields[END_REF][START_REF] Chartier | Numerical integrators based on modified differential equations[END_REF]. For b(∅) = 0, the B-series B(b,hf,y) is a vector field that is a perturbation of hf ( y ) , multiplied by the scalar b( ). Therefore, we can substitute the vector field B(b,hf,•) for hf in [START_REF] Ch | Runge-Kutta methods and renormalization[END_REF]. Also in this case we obtain a B-series, which we denote

General formulae for both laws will be given in Sect. 3 below. The composition law is an important tool for the construction of various integration methods, such as Runge-Kutta methods, general linear methods, Rosenbrock methods, multi-derivative methods, etc. It allows the derivation of the order conditions for arbitrarily high orders in an elegant way avoiding tedious series expansions [START_REF] Hairer | Solving Ordinary Differential Equations I[END_REF][START_REF] Hairer | Solving Ordinary Differential Equations II. Stiff and Differential-Algebraic Problems[END_REF]. Another application is the composition of different numerical integrators yielding higher accuracy: effective order or pre-and post-processing of composition methods [START_REF] Blanes | Symplectic integrators with processing: a general study[END_REF][START_REF] Butcher | The effective order of Runge-Kutta methods[END_REF].

Applications of the substitution law are more recent and mainly in connection with structure-preserving algorithms (geometric numerical integration). This law gives much insight into the modified differential equation of backward error analysis [START_REF] Hairer | Geometric Numerical Integration[END_REF], and it is the main ingredient for the construction of modifying integrators [START_REF] Chartier | Numerical integrators based on modified differential equations[END_REF].

Group and Monoid Structures

Let T ={ , , ,...} be the set of rooted trees, and consider the set T 0 = T ∪{∅} including the empty tree. The set of mappings

G C = a : T 0 → R; a(∅) = 1 (7)
with the product (5) of the composition law is a group. The identity is the element that corresponds to the B-series B(a,hf,y) = y. Associativity follows from that of the composition of mappings and the existence of an inverse is obtained from the explicit formulae for the product. The group G C has been introduced in [START_REF] Butcher | An algebraic theory of integration methods[END_REF] and is called the Butcher group in [START_REF] Hairer | On the Butcher group and general multi-value methods[END_REF].

In a similar way, the substitution law (6) makes the set

G S = a : T 0 → R; a(∅) = 0 (8)
a monoid. It is a monoid of vector fields and has first been considered in [START_REF] Chartier | A substitution law for B-series vector fields[END_REF]. The identity element is the mapping that corresponds to the B-series B(a,hf,y) = hf ( y ) .

Invertible elements in G S are those with a( ) = 0 and yield the group

G * S = a : T 0 → R; a(∅) = 0,a( ) = 0 . (9)

Hopf Algebras of Trees

Independently of the theory of B-series, Connes and Moscovici [START_REF] Connes | Hopf algebras, cyclic cohomology and the transverse index theorem[END_REF] in the context of non-commutative geometry, and Connes and Kreimer [START_REF] Connes | Hopf algebras, renormalization and noncommutative geometry[END_REF][START_REF] Connes | Renormalization in quantum field theory and the Riemann-Hilbert problem. I. The Hopf algebra structure of graphs and the main theorem[END_REF] in the theory of renormalization consider a Hopf algebra of rooted trees whose co-product is for the first trees given by CK (∅) =∅⊗∅and

CK ( ) = ⊗∅+∅⊗ , CK ( ) = ⊗∅+ ⊗ +∅⊗ , CK ( ) = ⊗∅+ ⊗ + 2 ⊗ +∅⊗ , CK = ⊗∅+ ⊗ + ⊗ +∅⊗ . (10)
Brouder [START_REF] Ch | Runge-Kutta methods and renormalization[END_REF][START_REF] Ch | Trees, renormalization and differential equations[END_REF] (and also implicitly Dür [START_REF] Dür | Möbius Functions, Incidence Algebras and Power Series Representations[END_REF]) noticed the close connection between this co-product and the product (5) of the composition law. Indeed, it is obtained from ( 5) by writing the argument of the mapping a to the right of the ⊗ sign, and those of the mapping b to the left of it. To the last terms in [START_REF] Butcher | An algebraic theory of integration methods[END_REF], which do not contain any b(τ), one adds the trivial factor b(∅) = 1.

It is not surprising that a similar connection holds also for the substitution law. Inspired by the work [START_REF] Chartier | A substitution law for B-series vector fields[END_REF], Calaque et al. [START_REF] Calaque | Two interacting Hopf algebras of trees[END_REF] introduced a co-product which, for the first trees, is given by

CEM ( ) = ⊗ , CEM ( ) = ⊗ + 2 ⊗ , CEM ( ) = ⊗ + 2 ⊗ + 3 ⊗ , CEM = ⊗ + 2 ⊗ + 3 ⊗ . (11)
It gives rise to a new Hopf algebra of trees.

Outline of the Article

The aim of this paper, which can be seen as a mixture of survey and research article, is to discuss the composition and substitution laws, to explain their fundamental role in numerical analysis, and to explore their common algebraic structure and relationships. Section 2 rigorously introduces trees and B-series, and in particular also ordered subtrees and partitions of trees. The composition and substitution laws are discussed in Sect. 3, including explicit formulae for arbitrary trees and applications in numerical analysis. Various relations between the two laws are explored in Sect. 4 and a specific map related to the logarithm is considered. Section 5 gives more details of the two Hopf algebras of trees and their connection with the composition and substitution laws. Finally, Sect. 6 mentions an extension to P-series, which are of great use for partitioned or split systems of ordinary differential equations, and to S-series.

Trees, B-series, Ordered Subtrees, and Partitions

This section introduces trees, B-series, ordered subtrees and partitions of trees, concepts which are fundamental in this work. We closely follow the notation of [26, Chap. III].

Trees and B-series

Let T ={ , , ,...} be the set of rooted trees, and let ∅ be the empty tree. For τ 1 ,...,τ m ∈ T , we denote by τ =[τ 1 ,...,τ m ] the tree obtained by grafting the roots of τ 1 ,...,τ m to a new vertex which becomes the root of τ . The order |τ | of a tree τ is its number of vertices and its symmetry coefficient is defined recursively by

σ( ) = 1,σ ( τ ) = σ(τ 1 ) •••σ(τ m )μ 1 !μ 2 !••• , (12)
where the integers μ 1 ,μ 2 ,..., count equal trees among τ 1 ,...,τ m . The elementary differentials F f (τ ) are given by

F f ( )(y) = f(y), F f (τ )(y) = f (m) (y) F f (τ 1 )(y),...,F f (τ m )(y) .
For real coefficients a(∅) and a(τ),τ ∈ T , a B-series is a formal series of the form

B(a,hf,y) = a(∅)y + τ ∈T h |τ | σ(τ) a(τ)F f (τ )(y).
The factor h |τ | is included for historical reasons and motivated by the fact that originally the most important B-series were Taylor series expansions of the exact and numerical solution. One can assume h = 1 because, as the notation B(a,hf,y) suggests, the factor h is only a rescaling of the vector field. The normalization with the symmetry coefficient σ(τ) in the denominator has been proposed in [START_REF] Butcher | The number of conditions for a Runge-Kutta method to have effective order p[END_REF]t og i v et h e composition law a more elegant form.

Ordered Subtrees

The general formula for the composition law needs the following notions. An ordered subtree2 of τ ∈ T is a subset s of the set of all |τ | vertices which is (i) connected (by edges of the tree τ ) and (ii) contains the root of τ (if s is not empty). The set of all ordered subtrees of τ is denoted by S(τ ). It is given in Table 1 for a tree of order 5. Associated to an ordered subtree s ∈ S(τ ) are:

• τ \ s is the forest (collection of rooted trees) that remains when the vertices of the subtree s together with its adjacent edges are removed from the tree τ ; • s τ is the rooted tree given by the vertices of s with root and edges induced by that of the tree τ .

Notice that, due to the fact that we consider all vertices of τ as different, the second and third subtrees in Table 1 are different, even if s τ and τ \ s are identical for both of them. This is the reason why we use the notation "ordered" subtree.

Partitions of Trees

The substitution law needs the notation of partitions. A partition p of a tree τ is a subset of the edges of the tree. We denote by P(τ ) the set of all partitions p of τ (including the empty partition). Associated to such a partition are the following objects (see Table 2):

• τ \ p is the forest that remains when the edges of p are removed from the tree τ ;

• p τ , called skeleton [START_REF] Chartier | Reversible B-series[END_REF], is the tree obtained by contracting each tree of τ \ p to a single vertex and by re-establishing the edges of p.

Notice that a tree τ ∈ T has exactly 2 |τ |-1 partitions p ∈ P(τ ), and that different partitions may lead to the same skeleton p τ or the same forest τ \ p.

Composition and Substitution Laws

Using the notation introduced in Sect. 2 we present the general formulae for the composition and substitution laws. We also discuss their significance in the numerical treatment of differential equations.

The Composition Law

We extend3 maps b :

T 0 → R satisfying b(∅) = 1 to forests by putting b(τ 1 ...τ n ) := n i=1 b(τ i ). Theorem 3.1 Let a,b: T 0 → R be two mappings, with b(∅) = 1. Then, we have B a,hf,B(b,hf,y) = B(b • a,hf,y), where b • a : T 0 → R is defined by (b • a)(∅) = a(∅) and (b • a)(τ ) = s∈S(τ ) b(τ \ s)a(s τ ). (13)
For trees up to order 3 this formula corresponds to those of [START_REF] Butcher | An algebraic theory of integration methods[END_REF]. The proof for a general tree τ is by Taylor series expansion and can be read in [START_REF] Hairer | Geometric Numerical Integration[END_REF]Sect. III.1.4]. As mentioned in the Introduction, the set G C ={a : T 0 → R; a(∅) = 1} equipped with the product ( 13) is a group. Associativity can be verified by algebraic manipulations, but it is also a consequence of Theorem 3.1 and the associativity of mappings. The identity element in G C is given by

δ ∅ (∅) = 1 and δ ∅ (τ ) = 0f o r τ ∈ T, (14)
which corresponds to the B-series B(δ ∅ ,hf,y)= y. The existence of an inverse follows recursively from formula [START_REF] Chartier | An algebraic approach to invariant preserving integrators: the case of quadratic and Hamiltonian invariants[END_REF], because for a ∈ G C it has the form

(b • a)(τ ) = b(τ) + a(τ) +•••,
where dots indicate expressions involving trees with lower order than that of τ . It is essential for applications that Theorem 3.1 is valid without any restrictions on a(∅). The most important special case of this theorem is the formula

hf B(b,hf,y) = B(b ′ ,hf,y), (15)
where, as a consequence of the product (13), we have b ′ (∅) = 0 and

b ′ (τ ) = b(τ 1 ) •••••b(τ m ) for τ =[τ 1 ,...,τ m ]. (16)

Exact Solution of (1)

The solution y(t + h) of ( 1) with initial value y(t) = y is a B-series y(t + h) = B(e,hf,y). Differentiation with respect to h shows that h ẏ(t + h) is the B-series with coefficients |τ |•e(τ). Equating these coefficients with those of the B-series for hf (y(t

+ h)) = B(e ′ ,hf,y) yields the recurrence relation |τ |•e(τ) = e(τ 1 ) • ••• • e(τ m ).

Order Conditions for Runge-Kutta Methods

This is the origin of the consideration of B-series and of the Butcher group. A Runge-Kutta method is given by

g i = y n + h s j =1 a ij f(g j )
together with a similar formula for the numerical approximation y n+1 after one step with length h. Assuming g i = B(φ i ,hf,y n ), the Runge-Kutta equation can be written in terms of the coefficients of the B-series as φ i (∅) = 1 and, for τ =[τ 1 ,...,τ m ],

φ i (τ ) = s j =1 a ij φ ′ j (τ ) = s j =1 a ij φ j (τ 1 ) •••••φ j (τ m ).
These coefficients and those for the numerical approximation y n+1 = B(φ,hf,y n ) only depend on the parameters of the Runge-Kutta method. The order of the method, which is the largest integer p such that y(h)y 1 = O(h p+1 ) for all differential equations, is now expressed by the algebraic relations φ(τ) = e(τ) for |τ |≤p, where φ and e are the coefficients of the numerical and exact solution, respectively.

Effective Order

An early application of the group structure is the concept of effective order. The idea [START_REF] Butcher | The effective order of Runge-Kutta methods[END_REF] is to construct a numerical method y n+1 = Φ h (y n ) such that for a suitable mapping χ h the composition

Ψ h = χ -1 h • Φ h • χ h
is a method of higher order. In a constant step size implementation we have

Ψ n h = χ -1 h • Φ n h • χ h
, so that for the computational cost of the method Φ h we obtain a higher accuracy by slightly modifying the initial value and by correcting the output approximation. Method Φ h is called to be of effective order p,i fΨ h is of order p. Assuming Φ h (y) = B(a,hf,y) and χ h (y) = B(c,hf,y), the conditions for effective order p are (c

• a • c -1 )(τ ) = e(τ)
for |τ |≤p, where we have employed the product [START_REF] Chartier | An algebraic approach to invariant preserving integrators: the case of quadratic and Hamiltonian invariants[END_REF]. The mapping χ h is called a processor. Notice that this notion is different from preprocessed (modifying) vector field integrators described in the next Sect. 3.2.

Conjugate Methods

The idea of effective order has a wide applicability in the context of geometric numerical integration. The relation

Ψ h = χ -1 h • Φ h • χ h means that Φ h
and Ψ h are conjugate maps. If Φ h ,Ψ h ,χ h are B-series with coefficients a,b,c, respectively, then the conjugacy condition is best written in the form (c • a)(τ ) = (b • c)(τ ), so that the composition law (13) can be directly applied. When, for some reason, a method y n+1 = Φ h (y n ) cannot satisfy a desirable geometric property (like symplecticity) one can still ask whether it is conjugate to a method having this property. Extensive use of ( 13) has been made recently in proving conjugate-symplecticity up to order 2s + 2 for a class of energy-preserving B-series integrators of order 2s,see [START_REF] Hairer | Energy-preserving variant of collocation methods[END_REF].

The Substitution Law

The general formula for the substitution law is as follows: where b⋆a:

T 0 → R is defined by (b ⋆ a)(∅) = a(∅) and (b ⋆ a)(τ ) = p∈P(τ ) b(τ \ p)a(p τ ). (17)
For trees up to order 3 we recover the formulae [START_REF] Butcher | Numerical Methods for Ordinary Differential Equations[END_REF]. A detailed proof of the general case is given in [START_REF] Chartier | Numerical integrators based on modified differential equations[END_REF], see also [START_REF] Chartier | A substitution law for B-series vector fields[END_REF]. For the tree of Table 3,formula [START_REF] Chartier | Preserving first integrals and volume forms of additively split systems[END_REF] yields

(b ⋆ a) = a( )b + a( )b( )b( ) + 2a( )b( )b + a( )b( ) 2 b( ) + 2a b( ) 2 b( ) + a b( ) 4 .
The set G S ={a :

T 0 → R; a(∅) = 0},
The monoid and group properties are discussed further in Sect. 4. As we shall see in the following applications, it is important to note that Theorem 3.2 is valid without any restrictions on a(τ), e.g., also for a/ ∈ G S .

Backward Error Analysis

This is a fundamental tool for the study of the longtime behavior of geometric integrators (e.g., symplectic or reversible methods) [START_REF] Hairer | Geometric Numerical Integration[END_REF][START_REF] Leimkuhler | Simulating Hamiltonian Dynamics[END_REF]. The idea is to interpret the numerical solution of a method y n+1 = Φ h (y n ) applied to (1) as the exact solution of a modified differential equation ẏ = f h (y). For structured problems, such as Hamiltonian systems, the study of the flow of the modified differential equation gives much insight into the numerical solution.

For the case that the numerical integrator is represented by a B-series, Φ h (y) = B(a,hf,y), it turns out that also the modified differential equation is a B-series vector field hf h (y) = B(b,hf,y) [START_REF] Hairer | Backward analysis of numerical integrators and symplectic methods[END_REF]. It is defined by

Modifying Integrators

Modifying (or preprocessed) integrators [START_REF] Chartier | Numerical integrators based on modified differential equations[END_REF] permit to increase the order of accuracy of a basic integrator y n+1 = Φ h (y n ) without destroying its geometric properties. The idea is to find a modified vector field f h (y) (different from that of backward error analysis) such that the basic method applied to ẏ = f h (y) reproduces the exact solution of [START_REF] Blanes | Symplectic integrators with processing: a general study[END_REF]. Suitable truncation of the modified differential equation yields high order integrators. This idea has successfully been applied to the equations of motion for the rigid body [START_REF] Chartier | Numerical integrators based on modified differential equations[END_REF][START_REF] Hairer | Preprocessed discrete Moser-Veselov algorithm for the full dynamics of the rigid body[END_REF][START_REF] Mclachlan | The discrete Moser-Veselov algorithm for the free rigid body, revisited[END_REF].

In complete analogy to backward error analysis we have that for a B-series method Φ h (y) = B(a,hf,y) the modified differential equation is a B-series hf h (y) = B(b,hf,y). It is defined by Notice that the coefficients b ∈ G * S for backward error analysis and modifying integrators are inverse elements with respect to the substitution law.

Interactions Between the Groups

We study in this section the properties and connections between the composition law and the substitution law. The proofs provided here use the interpretation in terms of B-series, and have first been given in the unpublished report [START_REF] Chartier | A substitution law for B-series vector fields[END_REF].

A Monoid Action by Morphisms

We show that the substitution law can be seen as a monoid action of the monoid of vector fields on the Butcher group.

G S × G C → G C , (b, a) → b⋆a.
In particular, we have the compatibility relations

(b 1 ⋆b 2 )⋆a= b 1 ⋆(b 2 ⋆a), (19) b⋆
(a 1 • a 2 ) = (b ⋆ a 1 ) • (b ⋆ a 2 ), (20)
(b ⋆ a) -1 = b⋆ a -1 (21) for all b, b 1 ,b 2 ∈ G S ,a,a 1 ,a 2 ∈ G C . Here, a -1 denotes the inverse in G C .
Proof The connection of the products with B-series (Theorem 3.1 and Theorem 3.2) permits us to give simple proofs:

• (19) is a consequence of the associativity of the composition of functions, here of the form hf → B(c,hf,y) for different mappings c. • (20) means that considering the composition of the flows of two B-series B(a 1 ,hf,y) and B(a 2 ,hf,y), it is equivalent to substitute the vector field hf by another B-series B(b,hf,•) before or after the composition of the flows. • (21) is an immediate consequence of (20), putting a 1 = a and a 2 = a -1 .

A purely algebraic proof of these properties has recently been given in [START_REF] Calaque | Two interacting Hopf algebras of trees[END_REF].

As already explained in Sect. 3.1, the coefficients a -1 (τ ) can be computed straightforwardly by induction on |τ | from the relation a • a -1 = δ ∅ , and for the first trees we obtain

a -1 ( ) =-a( ), a -1 ( ) =-a( ) + a( ) 2 , a -1 ( ) =-a( ) + 2a( )a( ) -a( ) 3 , a -1 =-a + 2a( )a( ) -a( ) 3 . (22)

The Exponential and Logarithmic Maps

For a B-series method Φ h (y) = B(a,hf,y), the coefficients b for backward error analysis are given by b⋆e= a (see Sect.

a ∈ G C ; a(u • v) + a(v • u) = a(u)a(v) for u, v ∈ T
corresponds to the submonoid of Hamiltonian vector fields characterized by [START_REF] Hairer | Backward analysis of numerical integrators and symplectic methods[END_REF] b

∈ G S ; b(u • v) + b(v • u) = 0foru, v ∈ T , where u • v := [u 1 ,...,u n ,v] for u =[u 1 ,...,u n ],v ∈ T
y n+1 = y n + hf ( y n ) (23)
which is the simplest B-series integrator, with coefficients δ ∅ + δ , see definitions ( 14) and [START_REF] Connes | Hopf algebras, renormalization and noncommutative geometry[END_REF]. We denote by

ω := log(δ ∅ + δ ) (24)
the coefficients of its modified vector field for backward error analysis.

Theorem 4.2

The inverse of a in the Butcher group G C is given explicitly by

a -1 = (a -δ ∅ )⋆(δ ∅ + δ ) -1 (25)
and coefficients for backward error analysis are obtained explicitly in terms of ω by

log(a) = (a -δ ∅ )⋆ω. ( 26
)
Proof Any B-series integrator y n+1 = B(a,hf,y n ) can be interpreted as the explicit Euler method [START_REF] Hairer | Order conditions for numerical methods for partitioned ordinary differential equations[END_REF], where the vector field hf is replaced by the B-series vector field with coefficients aδ ∅ , which yields a = (aδ ∅ )⋆(δ ∅ + δ ). Then, application of property (21) yields relation [START_REF] Hairer | Energy-preserving variant of collocation methods[END_REF].

Coefficients ω are defined by ω⋆e= δ ∅ + δ . Subtracting δ ∅ on both sides we obtain ω⋆(e-δ ∅ ) = δ , which means that ω = (eδ ∅ ) ⋆-1 is the inverse for the substitution law of the exact solution. Now, log(a) is defined by log(a) ⋆ e = a. Subtracting δ ∅ on both sides, and multiplying by ω from the right side yields the statement [START_REF] Hairer | Geometric Numerical Integration[END_REF].

Explicit Formula for the Inverse in the Butcher Group

Since for a = δ ∅ + δ only two terms are non-zero in [START_REF] Chartier | An algebraic approach to invariant preserving integrators: the case of quadratic and Hamiltonian invariants[END_REF], we get by induction on |τ | that (δ ∅ + δ ) -1 (τ ) = (-1) |τ | . Using the general formula [START_REF] Chartier | Preserving first integrals and volume forms of additively split systems[END_REF] for the substitution law, formula [START_REF] Hairer | Energy-preserving variant of collocation methods[END_REF] allows to recover the following formula for the inverse of an element a in G C , a result first discovered in [START_REF] Connes | Hopf algebras, renormalization and noncommutative geometry[END_REF] in the context of Hopf algebras of trees:

a -1 (τ ) = p∈P(τ ) (-1) |p τ | a(τ \ p). ( 27
)
For trees up to order 3 this formula yields again [START_REF] Ebrahimi-Fard | A Magnus and Fer-type formula in dendriform algebras[END_REF].

Explicit Formula for the Logarithmic Map

Once the coefficients ω(τ) are tabulated (see Table 4 for trees up to order 5), [START_REF] Hairer | Geometric Numerical Integration[END_REF] together with [START_REF] Chartier | Preserving first integrals and volume forms of additively split systems[END_REF] give an explicit formula for the logarithm log(a) of backward error analysis. In the context of combinatorial Hopf algebras the coefficients ω can be traced back to [START_REF] Chapoton | Rooted trees and an exponential-like series[END_REF] under the name log * and are studied also in [START_REF] Calaque | Two interacting Hopf algebras of trees[END_REF][START_REF] Chapoton | A rooted-trees q-series lifting a one-parameter family of Lie idempotents[END_REF][START_REF] Ebrahimi-Fard | A Magnus and Fer-type formula in dendriform algebras[END_REF][START_REF] Murua | The Hopf algebra of rooted trees, free Lie algebras, and Lie series[END_REF].

The coefficients ω(τ) may be computed by induction from the relation ω⋆e= δ ∅ + δ using formula [START_REF] Chartier | Preserving first integrals and volume forms of additively split systems[END_REF]. The coefficients ω(τ) for the bushy trees , , ,... are the Bernoulli numbers B i . They correspond to quadrature problems ẏ = f(t),see Proposition 4. [START_REF] Ch | Trees, renormalization and differential equations[END_REF] The coefficients ω satisfy the following relation for all u, v ∈ T ,

[26, Example IX.7.1], ω( ) = B 1 =-1/2,ω ( ) = B 2 = 1/6,
ω(u • v) + ω(v • u) + ω(u × v) = 0. ( 28
)
This generalizes to three trees (and more),

ω u • (vw) + ω v • (wu) + ω w • (uv) + ω (u × v) • w + ω (v × w) • u + ω (w × u) • v + ω(u × v × w) = 0. ( 29
)
Moreover, ω is the unique mapping of G S satisfying both ω( ) = 1 and (28)-( 29).

Here, • and × denote respectively the Butcher product and the merging product,

u • (vw...)=[u 1 ,...u n ,v,w...],u × v ו••=[u 1 ,...u n ,v 1 ...v m ,...],
for all trees u =[u 1 ,...u n ],v=[v 1 ,...v m ], w,.... A direct proof of ( 28)-( 29)i s obtained using the substitution law formula in [START_REF] Chartier | A substitution law for B-series vector fields[END_REF]Prop. 4.4]. An algebraic proof using quasi-shuffle products is given in [8, Sect. 9]. The uniqueness of ω is a consequence of results in [START_REF] Chartier | An algebraic approach to invariant preserving integrators: the case of quadratic and Hamiltonian invariants[END_REF], where it is shown that among B-series methods, only the (time-scaled) exact flow conserves both quadratic and cubic invariants, i.e., satisfies

e(u • v) + e(v • u) = e(u)e(v), e u • (vw) + e v • (wu) + e w • (uv) = e (u)e(v)e(w)
.

Two Hopf Algebras of Trees

There is a close connection between the results of the previous sections and Hopf algebras of trees. The composition law is related to a Hopf algebra introduced by Connes, Kreimer and Moscovici [START_REF] Connes | Hopf algebras, renormalization and noncommutative geometry[END_REF][START_REF] Connes | Renormalization in quantum field theory and the Riemann-Hilbert problem. I. The Hopf algebra structure of graphs and the main theorem[END_REF][START_REF] Connes | Hopf algebras, cyclic cohomology and the transverse index theorem[END_REF] (see the review [START_REF] Manchon | Hopf algebras and renormalisation[END_REF]), whereas the substitution law is related to a Hopf algebra introduced by Calaque et al. [START_REF] Calaque | Two interacting Hopf algebras of trees[END_REF]. We consider the set H of linear combinations of forests of rooted trees. As for standard polynomials we consider scalar multiplication, addition and multiplication which makes this set a commutative algebra. For example, the product of two polynomials is

( + 2 ) + 5 = 2 + 5 + 2 + 10 ,
and the identity element for the multiplication is the empty tree e =∅. Such an algebra equipped with a coproduct : H → H ⊗ H, which is coassociative,

(Id ⊗ ) • = ( ⊗ Id) • , (30)
and compatible with the algebra laws is called a bialgebra. It is a Hopf algebra if in addition it is equipped with an antipode S : H → H which is an algebra map 4satisfying

μ • (S ⊗ Id) • = μ • (Id ⊗ S) • = eδ e (31)
where μ : H ⊗ H → H is the multiplication of elements in H.

The Hopf Algebra of Connes & Kreimer

Let us consider the commutative R-algebra of polynomials on trees T 0 (including the empty tree). We recall that elements a in the Butcher group G C satisfy a(∅) = 1 and can be extended to unital algebra maps by a(τ 1 ...τ n ) = a(τ 1 ) •••(τ n ) and by linearity.

According to [START_REF] Connes | Hopf algebras, renormalization and noncommutative geometry[END_REF][START_REF] Connes | Renormalization in quantum field theory and the Riemann-Hilbert problem. I. The Hopf algebra structure of graphs and the main theorem[END_REF][START_REF] Connes | Hopf algebras, cyclic cohomology and the transverse index theorem[END_REF] we define the coproduct CK :

H → H ⊗ H on trees, CK (τ ) = s∈S(τ ) (τ \ s) ⊗ s τ ,
and we extend CK to an unital algebra map. There is a striking similarity [START_REF] Ch | Runge-Kutta methods and renormalization[END_REF][START_REF] Ch | Trees, renormalization and differential equations[END_REF] with formula (13) for the composition law of the Butcher group. Indeed, if we apply b on the left side of the tensor product and a on the right side, we obtain (b • a)(τ ).This connection can be expressed by the formula

(b • a)(τ ) = μ • (b ⊗ a) • CK (τ ), (32)
where μ : R ⊗ R → R denotes the multiplication. In the context of combinatorial Hopf algebra, this is called a convolution product. We say that the Butcher group corresponds to the group of characters of the Hopf tree algebra of Connes & Kreimer. The validity of (30) can be understood as follows: by definition of the coproduct, we have

(Id ⊗ CK ) • CK (τ ) = s∈S(τ ) s ′ ∈S(s τ ) (τ \ s) ⊗ (s τ \ s ′ ) ⊗ s ′ τ
and for the composition law we have

a • (b • c) (τ ) = s∈S(τ ) s ′ ∈S(s τ ) a(τ \ s) b(s τ \ s ′ )c(s ′ τ ).
Similar formulae for the right side of (30) together with the associativity a

• (b • c) = (a • b) • c
S CK (τ ) = p∈P(τ ) (-1) |p τ | (τ \ p),
which yields for the first trees,

S CK (∅) =∅, S CK ( ) =-, S CK ( ) =-+ 2 , S CK ( ) =-+ 2 -3 , S CK =-+ 2 -3 .
Again, there is a strong similarity with the Butcher group, compare with formulae ( 22) and ( 27): the inverse in the Butcher group is related to the antipode S CK by

a -1 (τ ) = (a • S CK )(τ ).
This can be seen from ( 31) and ( 32) which yield a • (a

• S CK ) = (a • S CK ) • a = δ ∅ .

A Bialgebra Based on the Substitution Law

In the recent article [START_REF] Calaque | Two interacting Hopf algebras of trees[END_REF] a Hopf algebra of trees has been constructed, with a new coproduct that is closely related to the substitution law [START_REF] Chartier | Preserving first integrals and volume forms of additively split systems[END_REF]. For the same algebra as in Sect. 5.1, we consider here the coproduct (see [START_REF] Chapoton | Rooted trees and an exponential-like series[END_REF])

CEM (τ ) = p∈P(τ ) (τ \ p) ⊗ p τ . (33)
which is slightly different from that of [START_REF] Calaque | Two interacting Hopf algebras of trees[END_REF], cf. Sect. 5.3.

In analogy to the relation between the coproduct of Connes & Kreimer and the composition law, the convolution product for the coproduct [START_REF] Lundervold | Hopf algebras of formal diffeomorphisms and numerical integration on manifolds[END_REF] is exactly the substitution law of [START_REF] Chartier | Preserving first integrals and volume forms of additively split systems[END_REF]. For a,b ∈ G S ,wehave

(b ⋆ a)(τ ) = μ • (b ⊗ a) • CEM (τ ).
The coassociativity of CEM follows as in Sect. 5.1. We thus get a bialgebra. 5 Since G S is a monoid but not a group, it is not surprising that no antipode associated to this coproduct exists.

The Hopf Algebra of Calaque, Ebrahimi-Fard & Manchon

In the article [START_REF] Calaque | Two interacting Hopf algebras of trees[END_REF] it is shown how the algebra H has to be modified so that the coproduct CEM of [START_REF] Lundervold | Hopf algebras of formal diffeomorphisms and numerical integration on manifolds[END_REF] gives rise to a Hopf algebra structure.

We let H CEM be the commutative R-algebra of polynomials on trees T (excluding the empty tree), with e = as the identity element (in contrast to the algebra of Sects. 5.1 and 5.2). We equip this algebra with the coproduct CEM of [START_REF] Lundervold | Hopf algebras of formal diffeomorphisms and numerical integration on manifolds[END_REF]. Since is the identity element, it can be removed when multiplied with other trees. For example, we have CEM = ⊗ + 2 ⊗ + ⊗ which should be compared to [START_REF] Chapoton | Rooted trees and an exponential-like series[END_REF]. In the context of B-series, the choice e = for the identity element corresponds to the subgroup of G S G 1 S = a : T → R; a( ) = 1 , which corresponds to B-series vector fields of the form hf ( y ) + higher order terms. The inverse a ⋆-1 (τ ) of a ∈ G 1 S (with respect to the substitution law) can be computed recursively from [START_REF] Chartier | Preserving first integrals and volume forms of additively split systems[END_REF]. For trees up to order 3 we obtain

a ⋆-1 ( ) = a( ) = 1, a ⋆-1 ( ) =-a( ), a ⋆-1 ( ) =-a( ) + 2a( ) 2 , a ⋆-1 =-a + 2a( ) 2 , S CEM ( ) = , S CEM ( ) =-, S CEM ( ) =-+ 2 2 , S CEM =-+ 2 2 . ( 34
)
This readily permits us to define an antipode by the formulas to the right of [START_REF] Manchon | Hopf algebras and renormalisation[END_REF], which makes the bialgebra a Hopf algebra. A general formula for this antipode is given in [START_REF] Calaque | Two interacting Hopf algebras of trees[END_REF]. Since it requires additional notations, we do not reproduce it here. Similarly to the relation between G C and H CK (Sect. 5.1), the group G 1 S corresponds to the group of characters of the Hopf algebra H CEM .

Extensions

There exist many different extensions of B-series in the numerical analysis literature, e.g., P-series [START_REF] Hairer | Order conditions for numerical methods for partitioned ordinary differential equations[END_REF] for partitioned differential equations, DAE-series for differentialalgebraic equations, Lie-Butcher series [START_REF] Munthe-Kaas | Lie Butcher theory for Runge-Kutta methods[END_REF], S-series [START_REF] Murua | Formal series and numerical integrators. I. Systems of ODEs and symplectic integrators[END_REF] and LS-series [START_REF] Munthe-Kaas | On the Hopf algebraic structure of Lie group integrators[END_REF]f o r differential operators. We briefly present the ideas of P-series and S-series.

P-series

Partitioned systems of differential equations ṗ = f [1] (p, q), q = f [2] (p, q)

arise in many situations. Second-order differential equations when written in firstorder form and Hamiltonian systems are interesting special cases. There are important numerical integrators that treat the variables p and q in a different manner, e.g., symplectic methods based on the Störmer-Verlet integrator [START_REF] Hairer | Geometric Numerical Integration[END_REF]. P-series are an extension of B-series adapted to partitioned systems [START_REF] Hairer | Solving Ordinary Differential Equations I[END_REF]Sect. II.15]. Let TP 0 ={∅ p , ∅ q , , , , , ,...} denote the set of bi-colored trees, and ∅ p , ∅ q denote empty trees. For a mapping a : TP 0 → R aP-seriesisoftheform P(a,hf,y)= a(∅ p )p a(∅ q )q + h a( )f [1] a( )f [2] + h 2 a( )(f [1] p f [1] ) + a( )(f [1] q f [2] ) a( )(f [2] p f [1] ) + a( )(f [2] q f [2] )

+••• ,
with functions evaluated at y = (p, q). The upper component contains terms corresponding to trees with black root, and the lower component to trees with white root. Results for B-series of previous sections can be extended straight-forwardly to P-series. We again have a composition law P a,hf,P(b,hf,y) = P(b• a,f,y) and a substitution law P a,P (b,hf,•), y = P(b⋆a,f,y).

The formulas of Theorem 3.1 and Theorem 3.2 are still valid, where trees, subtrees, and skeletons are now in TP 0 . The only ambiguity that could arise is in the definition of the skeleton p τ : the color of a vertex of p τ is that of the root of the tree which is replaced by the vertex.

For instance, for the partition p = we have p τ = . An example for the substitution law is

(b ⋆ a)( ) = a( )b( ) + a( )b( )b( ) + a( )b( )b( ) + a( )b( ) 2 b( ).
6.2 S-series S-series have been introduced by Murua [START_REF] Murua | Formal series and numerical integrators. I. Systems of ODEs and symplectic integrators[END_REF] for the purpose of analyzing the preservation of invariants, either in the context of differential-algebraic equations or in the context of Hamiltonian dynamics. If g is a first integral of (1), i.e., g ′ (y)f (y) ≡ 0, it is natural to investigate whether the numerical approximation y n+1 = B(a,hf,y n ) provided by a B-series method satisfies g(y n+1 ) = g(y n ). This leads to the study of g B(a,hf,y) and motivates the following definitions.

Let g(y) be a scalar-or vector-valued smooth function and f(y) the vector field of the system (1). We let

F g,f ( )(y) = g(y), F g,f (τ )(y) = g (m) (y) F f (τ 1 )(y),...,F f (τ m )(y) , (35)
where τ =[τ 1 ,...,τ m ] and F f (τ )(y) are the elementary differentials of Sect. 2.1.

For real coefficients a(τ), τ ∈ T , an S-series is a formal series of the form

S(a,hg,hf,y) = τ ∈T h |τ | σ(τ) a(t)F g,f (τ )(y). (36)
For g = f we recover B-series vector fields. For g = Id, we obtain S(a,hId,hf,y)= hB( a,hf,y)

with a(τ) = a([τ ]) for τ ∈ T 0 , and we see that all B-series can be interpreted as S-series.

Such a series [START_REF] Merson | An operational method for the study of integration processes[END_REF] can be seen as a formal linear differential operator acting on g (notice that g → F g,f (y) is a linear differential operator). In the original6 literature [START_REF] Murua | Formal series and numerical integrators. I. Systems of ODEs and symplectic integrators[END_REF][START_REF] Murua | The Hopf algebra of rooted trees, free Lie algebras, and Lie series[END_REF] on S-series and also in the context of Lie theory [START_REF] Lundervold | Hopf algebras of formal diffeomorphisms and numerical integration on manifolds[END_REF][START_REF] Munthe-Kaas | On the Hopf algebraic structure of Lie group integrators[END_REF] (where S-series correspond to "pullback series"), this interpretation as differential operator is very important. Indeed, for a given flow ϕ t (y) of a vector field f(y) on a differentiable manifold M, the "Vertauschungssatz" [26, p. 88] in the theory of Lie-series allows to interpret g(ϕ t (y)) as a differential operator,

g • ϕ t = exp(tF )[g] (37)
where F(y)= n j =1 f j (y) ∂ ∂y j (here for the special case M = R n ) is the differential operator (Lie derivative) associated to the vector field f(y).

There are now three possibilities of defining composition or substitution laws. One can substitute in (36) a B-series with b(∅) = 1fory, a B-series with b(∅) = 0forhf , and finally an arbitrary S-series for hg. All of these operations lead to S-series which we denote as follows:

S(b • a,hg,hf,y) = S a,hg,hf,B(b,hf,y) ,

S(b ⋆ a,hg,hf,y) = S a,hg,B(b,hf,•), y ,

S(b * a,hg,hf,y) = S a,S(b,hg,hf,•), hf, y .

The proof of Theorem 3.

where S 0 (τ ) is the set of non-empty, ordered subtrees of τ . Notice that the empty tree is not involved in the S-series [START_REF] Merson | An operational method for the study of integration processes[END_REF].

The product (b ⋆ a)(τ ) in ( 39) is closely related to the substitution law of Theorem 3.2. The difference is that in the S-series only the function f is substituted with a B-series and the function g (corresponding to the root) is not touched. We therefore obtain the formula

with b ′ (τ ) given by ( 16). This relation is at the origin of considering S-series as mentioned in the beginning of this section. It is related to [START_REF] Munthe-Kaas | Lie Butcher theory for Runge-Kutta methods[END_REF] and expresses in terms of S-series the exponential of the Lie derivative corresponding to the vector field B(log b, hf, y). Furthermore, replacing hg in (43) by an S-series with general coefficients a(τ) we obtain the relation

b • a = a * b ′
which links the products (41) and (42).

Similarly to the case of B-series, the substitution law for S-series (42) can be turned into a coproduct on the algebra of Sect. 5.3 (excluding the empty tree)

(τ ) = s∈S 0 (τ ) s τ ⊗[τ \ s]
which yields again a Hopf algebra of trees.

  B a,B(b,hf,•), y = B(b ⋆ a,hf,y).(4)A straightforward computation yields for the composition law (b • a)(∅) = a(∅) and(b • a)( ) = a(∅)b( ) + a( ), (b • a)( ) = a(∅)b( ) + a( )b( ) + a( ), (b • a)( ) = a(∅)b( ) + a( )b( ) 2 + 2a( )b( ) + a( ), (b • a) = a(∅)b + a( )b( ) + a( )b( ) + a .(5) Similarly, for the substitution law we obtain (b ⋆ a)(∅) = a(∅) and (b ⋆ a)( ) = a( )b( ), (b ⋆ a)( ) = a( )b( ) + a( )b( ) 2 , (b ⋆ a)( ) = a( )b( ) + 2a( )b( )b( ) + a( )b( ) 3 , (b ⋆ a) = a( )b + 2a( )b( )b( ) + a b( ) 3 .

Theorem 3 . 2

 32 Let a,b: T 0 → R be two mappings, with b(∅) = 0. Then we have B a,B(b,hf,•), y = B(b ⋆ a,hf,y),

  considered in the Introduction, together with the product (17) forms a monoid. Invertible elements are those with a( ) = 0 and form the group G * S . The identity element is δ defined by δ (∅) = 0,δ ( ) = 1 and δ (τ ) = 0f o r |τ |≥2.

  B e, B(b, hf, •), y = B(a,hf,y), where the coefficients e(τ) are those of the exact solution, given in Sect. 3.1.T h e coefficients of the B-series for the modified differential equation are recursively given by (b ⋆ e)(τ ) = a(τ), because (b ⋆ e)(τ ) = b(τ) + lower order terms.

  B a,B(b,hf,•), y = B(e,hf,y), which leads to the condition (b ⋆ a)(τ ) = e(τ) for the coefficients of the arising Bseries. Again, b(τ) can be computed recursively from this relation.

Theorem 4 . 1

 41 The monoid of vector fields (G S ,⋆) acts by morphisms on the Butcher group (G C , •) via the substitution law:

3 . 2 )

 32 . This relation means that the B-series B(a,hf,y) is the exact flow of the differential equation with vector field B(b,hf,y). Motivated by the standard notation of flows we use exponential and logarithm in place of b⋆e= a (see[START_REF] Murua | The Hopf algebra of rooted trees, free Lie algebras, and Lie series[END_REF]), a = exp(b) and b = log(a). The exponential and logarithmic maps exp : G S → G C log : G C → G S allow us to interpret important results in geometric numerical integration in terms of one-to-one correspondences between subgroups of the Butcher group G C and submonoids of the monoid of vector fields G S : • the subgroup of methods of order at least p {a ∈ G C ; a(τ) = e(τ) for 1 ≤|τ |≤p} corresponds to the submonoid {b ∈ G S ; b( ) = 1, b(τ) = 0for2≤|τ |≤p}; • the subgroup of symmetric B-series methods {a ∈ G C ; B(a -1 ,hf,y) = B(a,-hf , y ) } is in correspondence with the submonoid {b ∈ G S ; b(τ) = 0f o r |τ | even};• the subgroup of symplectic B-series methods characterized by[START_REF] Calvo | Canonical B-series[END_REF]

Table 4

 4 3 = 0,ω () = B 4 ,.... Coefficients for tall trees , , ,... correspond to linear problems ẏ = λy , and are simply those of the series of log(1 + x): ω( ) = 1,ω ( ) =-1/2,ω = 1/3, ω =-1/4,ω = 1/5,.... Coefficients ω(τ) for trees of order ≤ 5

  1 shows that the product (b • a)(τ ) in (38) is given by(b • a)(τ ) = s∈S 0 (τ ) b(τ \ s)a(s τ ),

  (b ⋆ a)(τ ) = p∈P 0 (τ ) b {τ \ p} 0 a(p τ ),where the sum is only over those partitions of τ which contain all edges leaving the root of τ (the set of such partitions is denoted by P 0 (τ )), and where {τ \ p} 0 is the forest obtained by removing the edges of p and also the root of τ .The product (b * a)(τ ) in (40) is given by(b * a)(τ ) = s∈S 0 (τ ) b(s τ )a [τ \ s] ,(42)where [τ \ s] is the tree obtained from τ by contracting the subtree s τ to a single vertex, which becomes the root of [τ \ s]. All proofs are very similar to those for B-series. Putting a = δ in[START_REF] Munthe-Kaas | On the Hopf algebraic structure of Lie group integrators[END_REF], we get in analogy to (15) that hg B(b,hf,y) = S(b ′ ,hg,hf,y)

Table 1

 1 All ordered subtrees of a tree with associated functions

	s
	τ \ s	∅
	s τ

Table 3

 3 All 8 partitions of a tree with skeletons and tree forests

	p ∈ P
	p τ
	τ \ p

  denotes the Butcher product, see[START_REF] Hairer | Geometric Numerical Integration[END_REF] Def. III.3.7];• the subgroup of energy-preserving B-series methods in G C is in correspondence with the submonoid of vector fields in G S having the energy as first integral.• the subgroup of volume-preserving methods in G C is in correspondence with the submonoid of divergence-free vector fields in G S . Theses classes of B-series integrators and vector fields were studied recently in[START_REF] Chartier | Preserving first integrals and volume forms of additively split systems[END_REF][START_REF] Iserles | B-series methods cannot be volume-preserving[END_REF].Proofs of these statements are given in Theorems IX.1.2, IX.2.2, IX.3.1, and Corollaries IX.5.4, IX.9.13 of [26]. It is interesting to note that geometric properties are non-linear conditions in G C and become linear in G S .

	4.3 The Special Role of the Explicit Euler Method
	Consider the explicit Euler method

  in the Butcher group prove the coassociativity property[START_REF] Hairer | Solving Ordinary Differential Equations II. Stiff and Differential-Algebraic Problems[END_REF]for CK . The antipode S CK : H → H is given by

Originally called a Butcher series.

Ordered subtrees are called "admissible cuts" in[START_REF] Connes | Hopf algebras, renormalization and noncommutative geometry[END_REF].

This extension from trees to forests can be interpreted in the context of Hopf algebras: the extended map on the Hopf algebra Connes & Kreimer is a character, i.e. a unital algebra map (see Sect. 5 below).

In general, the antipode is an algebra antimorphism: S(u)S(v) = S(vu). If the Hopf algebra is commutative, it reduces to an algebra map. This is the case here, but not in[START_REF] Lundervold | Hopf algebras of formal diffeomorphisms and numerical integration on manifolds[END_REF][START_REF] Munthe-Kaas | On the Hopf algebraic structure of Lie group integrators[END_REF] in the context of Lie group integrators.

In a private communication, Dominique Manchon pointed out that H equipped with the coproduct CEM is a bialgebra which is graded but not connected. In fact, a graded bialgebra which is connected is automatically a Hopf algebra, i.e. an antipode exists.

In this article, we have changed the notation because it is more convenient in our context to consider g and f on the same level.

Acknowledgements

The authors are grateful to Damien Calaque and Dominique Manchon for helpful discussions and comments. This work was partially supported by the Fonds National Suisse, project No. 200020-109158.


PhD Archana Singh-Manoux

MD, PhD Michael G Marmot

SD Maria Glymour

PhD Séverine Sabia

PhD Mika Kivimäki

MSc Aline Dugravot

Archana Singh

Does cognitive reserve shape cognitive decline?

Objectives: Cognitive reserve is associated with a lower risk of dementia but the extent to which it shapes cognitive aging trajectories remains unclear. Our objective is to examine the impact of three markers of reserve from different points in the lifecourse on cognitive function and decline in late adulthood.

Methods: Data are from 5234 men and 2220 women, mean age 56 years (standard deviation=6) at baseline, from the Whitehall II cohort study. Memory, reasoning, vocabulary, phonemic and semantic fluency were assessed three times over 10 years. Linear mixed models were used to assess the association between markers of reserve (height, education, and occupation) and cognitive decline, using the 5 cognitive tests and a global cognitive score composed of these tests.

Results: All three reserve measures were associated with baseline cognitive function, with strongest associations with occupation and the weakest with height. All cognitive functions except vocabulary declined over the 10 year follow-up period. On the global cognitive test, there was greater decline in the high occupation group (-0.27; 95% confidence interval (CI): -0.28, -0.26) compared to the intermediate (-0.23; 95% CI: -0.25, -0.22) and low groups (-0.21; 95% CI: -0.24, -0.19); p=0.001. The decline in reserve groups defined by education (p=0.82) and height (p=0.55) was similar.

Interpretation: Cognitive performance over the adult lifecourse was remarkably higher in the high reserve groups. However, rate of cognitive decline did not differ between reserve groups except occupation where there was some evidence of greater decline in the high occupation group.

INTRODUCTION

There is considerable inter-individual variability in cognitive aging, some individuals or groups of individuals experience slower rates of cognitive decline than others. [START_REF] Holland | The Course and causes of Cognitive Change with Advancing age[END_REF][START_REF] Brayne | The elephant in the room -healthy brains in later life, epidemiology and public health[END_REF] This variability is also evident in the inconsistent relationship between the clinical and the pathological severity of dementia. [START_REF] Nelson | Neuropathology and cognitive impairment in Alzheimer disease: a complex but coherent relationship[END_REF] Hypothesized explanations for these discrepancies include functional reserve capacity, including neuroplasticity, biological variation between individuals and incomplete understanding of disease mechanisms. [START_REF] Nelson | Neuropathology and cognitive impairment in Alzheimer disease: a complex but coherent relationship[END_REF] Autopsies of individuals with "normal" brain aging, where an older person"s cognition remains intact, reveal almost as many neurofibrillary tangles and amyloid plaques as seen in patients with Alzheimer"s disease. [START_REF] Crystal | Clinico-pathologic studies in dementia: nondemented subjects with pathologically confirmed Alzheimer's disease[END_REF] These findings have led to the elaboration of the concept of cognitive reserve, defined as the capacity that creates a delay in time between pathology and clinical expression of dementia. [START_REF] Stern | What is cognitive reserve? Theory and research application of the reserve concept[END_REF] Anatomical features of the brain are likely to influence reserve, [START_REF] Stern | What is cognitive reserve? Theory and research application of the reserve concept[END_REF][START_REF] Christensen | Education and decline in cognitive performance: compensatory but not protective[END_REF] but complex or enriched environments may also be important as demonstrated in animal studies. [START_REF] Markham | Experience-driven brain plasticity: beyond the synapse[END_REF] Education and other markers of socioeconomic circumstances have been shown to be associated with a lower risk of dementia, [START_REF] Stern | Influence of education and occupation on the incidence of Alzheimer's disease[END_REF][START_REF] Qiu | The influence of education on clinically diagnosed dementia incidence and mortality data from the Kungsholmen Project[END_REF][START_REF] Evans | Education and other measures of socioeconomic status and risk of incident Alzheimer disease in a defined population of older persons[END_REF][START_REF] Launer | Rates and risk factors for dementia and Alzheimer's disease: results from EURODEM pooled analyses. EURODEM Incidence Research Group and Work Groups[END_REF][START_REF] Ngandu | Education and dementia: what lies behind the association?[END_REF][START_REF] Gatz | Accounting for the relationship between low education and dementia: a twin study[END_REF][START_REF] Mortimer | Head circumference, education and risk of dementia: findings from the Nun Study[END_REF] and reduced clinical manifestation of neuropathological changes. [START_REF] Bennett | Education modifies the relation of AD pathology to level of cognitive function in older persons[END_REF] Although most of the research on cognitive reserve focuses on dementia or cognitive decline immediately pre-or post-diagnosis, the concept of cognitive reserve has wider implications for cognitive aging. [START_REF] Stern | What is cognitive reserve? Theory and research application of the reserve concept[END_REF] It is possible that greater cognitive reserve is associated with both high function and lower rates of cognitive decline in adulthood. Our objective is to examine the extent to which cognitive reserve shapes adult cognitive aging trajectories, starting in midlife.

We use three markers of reserve: height, an anthropometric measure of development and seen to be a measure of cognitive reserve, [START_REF] Brickman | White matter hyperintensities and cognition: Testing the reserve hypothesis[END_REF] alongside education and occupation.

MATERIALS AND METHODS

Study Population

The target population for the Whitehall II study was all London-based office staff, aged 35-55 years, working in 20 civil service departments. [START_REF] Marmot | Cohort Profile: the Whitehall II study[END_REF]

Cognitive function

The cognitive test battery, administered at three clinical examinations over 10 years, consists of five standard tasks, described below, chosen to provide a comprehensive assessment of cognitive function.

The Alice Heim 4-I (AH4-I) is composed of a series of 65 verbal and mathematical reasoning items of increasing difficulty. [START_REF] Heim | AH 4 group test of general Intelligence[END_REF] It tests inductive reasoning, measuring the ability to identify patterns and infer principles and rules. Participants had 10 minutes to do this section.

Short-term verbal memory

was assessed with a 20-word free recall test. Participants were presented a list of 20 one or two syllable words at two second intervals and were then asked to recall in writing as many of the words in any order and had two minutes to do so.

We used two measures of verbal fluency: phonemic and semantic [START_REF] Borkowski | Word fluency and brain damage[END_REF] . Phonemic fluency was assessed via "S" words and semantic fluency via "animal" words. Subjects were asked to recall in writing as many words beginning with "S" and as many animal names as they could.

One minute was allowed for each test.

Vocabulary was assessed using the Mill Hill Vocabulary test, [START_REF] Raven | Guide to using the Mill Hill vocabulary test with progressive matrices[END_REF] used in its multiplechoice format, consisting of a list of 33 stimulus words ordered by increasing difficulty and six response choices.

Global cognitive score was created using all five tests described above by first standardizing the raw scores on each test to z-scores (mean=0; standard deviation (SD)=1) using the baseline mean and standard deviation value in the entire cohort for each test. Then the z scores were averaged to yield the global cognitive score. Previous research on cognitive ageing has used global scores of this description in order to minimize problems due to measurement error. [START_REF] Wilson | Neurodegenerative basis of age-related cognitive decline[END_REF][START_REF] Arvanitakis | Relation of NSAIDs to incident AD, change in cognitive function, and AD pathology[END_REF]

Cognitive reserve

We used three markers: occupational position, education and height. Occupational position was assessed by the British Civil Service grade of employment at the baseline of this study (Phase 5 of the Whitehall II study), concurrent to the first assessment of cognition. It is a three level variable representing high (administrative grades), intermediate (professional or executive grades) and low (clerical or support grades) position. This measure is a comprehensive marker of socioeconomic circumstances and is related to salary, social status and level of responsibility at work. [START_REF] Marmot | Cohort Profile: the Whitehall II study[END_REF] Education was measured as the highest qualification on leaving full-time education and categorised as lower secondary school or less, higher secondary school, university or higher degree. Height was measured using a stadiometer with the participant standing completely erect with the head in the Frankfort plane. The mean height in men was 177 cm leading us to use 175-179 cm as the middle category and height above or below this range to constitute the other two categories. In women, the mean height was 163 cm, making the 160-164 cm group the middle category.

Statistical Analysis

All three markers of reserve, occupation, education and height, comprised three categories: high, intermediate and low. We first examined the mean cognitive scores at baseline as a function of reserve, using analysis of variance. These analyses were stratified by sex as the interaction term between sex and markers of cognitive reserve (all p<0.03 except that between occupation and sex for memory test, p=0.74, and phonemic fluency, p=0.17) suggested gender differences in the association between reserve markers and cognitive function.

Linear mixed models [START_REF] Laird | Random-effects models for longitudinal data[END_REF] were then used to estimate the association between the markers of reserve and decline in the 5 cognitive tests. Mixed models use all available data over the follow-up, take into account the fact that repeated measures on the same individual are correlated with each other, and can handle missing data. In these analyses both the intercept and the slope were fitted as random effects, allowing them to vary between individuals. The mixed models were used to estimate 10 year decline and the associated 95% confidence interval (CI) in each measure of cognitive function using three cognitive assessments over ten years. We first examined gender differences in cognitive decline in models that included terms for time (here age centred at age 60 and divided by 10 for the coefficients to yield effects of decline over 10 years), age at baseline to adjust for cohort effects, gender, an interaction term between gender and age at baseline, an interaction term between gender, time and the marker of cognitive reserve plus all lower order interaction terms contained within this triple interaction. The triple interaction term suggested that the association between cognitive reserve and cognitive decline was similar in men and women (p values between 0.15 and 0.86 except for memory test with height, p<0.001 and semantic fluency with education, p<0.02), leading us to combine men and women in the analysis for cognitive decline.

We then examined the association between each marker of reserve and cognitive decline using 3 assessments of cognitive function over the 10 year follow-up. The mixed models for these analyses included terms for time, age at baseline, one marker of reserve at a time (occupation, education or height) specified as a categorical variable, sex, an interaction term between sex and the marker of reserve, an interaction term between cognitive reserve and age at baseline, and finally an interaction term between the marker of reserve and time. This last interaction term is the key objective of our paper and provides a test for the null hypothesis that there were no differences in cognitive decline between the low, intermediate and high reserve groups. It also allows calculation of the estimate of 10-year decline in the high, intermediate and low reserve groups. All the analyses were carried out using Proc Mixed procedure using the SAS software version 9.1 (SAS Institute Inc., Cary, NC, USA).

RESULTS

The baseline (1997-1999) of this study was Phase 5 of the Whitehall II study that started in 1985 on 10,308 individuals. Our analysis was based on 7,454 individuals (5,234 men and 2,220 women) with data on at least one of the three repeat measurements of cognition; 62.0% of these individuals had complete data at all three waves and 22.8% on two waves. Restricting the analyses to those with complete data or data at a minimum of two phases (N between 6310 and 6335, depending on the cognitive test) did not change the conclusions of the paper leading us to present analyses on 7,454 individuals. Compared to the baseline population of the Whitehall II study, N=10,308, the analysis reported in this paper was based on individuals who were younger (55.3 years vs.54.1 years at the start of the cognitive data collection, p<0.001), more educated (29.5% vs. 20.5% with a university degree, p<0.001) and more likely to be from the higher occupational position (42.1% vs. 23.6% at the start of the cognitive data collection, p<0.001). The three measures of cognitive reserve were modestly correlated with each other: the Spearman" rank correlation coefficient of height with education and occupation was 0.06 (p<0.0001) and 0.12 (p<0.001) respectively and that between education with occupation was 0.45 (p<0.001).

Of the 7,454 individuals in the analysis, 98 died between the first two waves of cognitive data collection and a further 213 died between the second and the third wave. Those who died did not have lower values on the markers of reserve (p=0.19 for occupational position, p=0.21 for education and p=0.09 for height) but had poorer cognitive scores (p<0.01 on all tests). Excluding these individuals from the analysis did not much modify the observed associations leading us to retain data before death from these individuals in the analysis.

Observed data over the three repeat assessments on all five tests are shown in Figure 1A to E. These data are stratified by occupational position and participants have been grouped in 5-year age groups using age at baseline. In the analysis there was some evidence of accelerated decline at older ages for all tests except vocabulary (results not shown but available from authors). However, in order to simplify the presentation of the results of the main analysis we combined all age-groups and adjusted for age at baseline. Table 1 presents mean cognitive scores (and those at the 25 th and 75 th percentile) at baseline as a function of the reserve measures in men and women. The interaction term between sex and the markers of cognitive reserve suggested gender differences in the association with cognitive tests; R 2 estimates confirmed the stronger association between reserve and cognitive function in women. There was a finely graded association between all three markers of reserve and the five measures of cognitive function in both men and women (all p<0.01). The scores at the 25 th and 75 th percentile in each reserve group show little evidence of ceiling/floor effects in the tests. The estimates of variance explained (R 2 ) suggest that of the three markers of reserve, occupation was most, and height the least strongly associated with cognitive function. The association between markers of reserve and cognition was weakest for the measure of memory (R 2 between 5 and 10%) and strongest for the measure of reasoning (R 2 between 5 and 43%).

The mean age and sex adjusted decline in reasoning (AH4-I) was estimated at -3.49 (95% CI: -3.64, -3.35), in memory at -0.65 (95% CI: -0.72, -0.59), at -1.57 (95% CI: -1.66, -1.49) in phonemic fluency and at -1.32 (95% CI: -1.40, -1.24) in semantic fluency. There was some evidence of an improvement in vocabulary scores (0.09, 95% CI: 0.04, 0.15). Table 2 presents estimates of cognitive decline over 10 years as a function of occupational position. Decline in all functions except vocabulary was evident in all three occupational groups. There was greater decline in reasoning (AH4-I) in the lower occupational group (p=0.04). However, on tests of memory (p=0.003), phonemic (p=0.04) and semantic fluency (p<0.001) the decline was faster in the higher occupation group. On the global cognitive score, compared to the high occupational group (-0.27 (95% CI: -0.28, -0.26), there was less decline in the intermediate (-0.23 (95% CI: -0.25, -0.22) and low (-0.21 (95% CI: -0.24, -0.19) occupational groups, both p<0.001.

The analysis using education as the marker of cognitive reserve (Table 3) showed greater decline in reasoning (AH4-I) in the lower education group (p=0.002) but similar decline in all education groups for memory (p=0.30), phonemic fluency (p=0. 19),semantic fluency (p=0.08) and global cognitive score (p=0.82). Vocabulary improved over the testing period more in the higher than the lower education group (p=0.009). Table 4 shows results for cognitive decline in analyses stratified by height. The results were similar to that for education in that the high reserve group (taller than average) had slower decline in reasoning (AH4-I), p=0.01, and vocabulary (p=0.05) although the global cognitive score suggested similar declines in all height groups (p=0.55).

DISCUSSION

Our results showed all three markers of cognitive reserve used in this study to be strongly associated with cognitive function scores but there was little evidence of slower rates of decline over a 10-year period in the higher reserve group. At the baseline cognitive assessment, occupation was most strongly associated with cognition, explaining between 13 and 26% of the variance in test scores in men and between 10 and 43% in women. The impact of occupation was such that men aged 75-79 years in the high occupational group had a higher mean reasoning score (45.1, SD=9.2) than men aged 45-49 years (33.3, SD=10.9) from the low occupational group. Of the three measures, height had the weakest association with the measures of cognitive function at baseline. Clinical studies, often on small number of subjects, assume that cognitive function declines only in the period immediately prior to the diagnosis of dementia. [START_REF] Scarmeas | Education and rates of cognitive decline in incident Alzheimer's disease[END_REF][START_REF] Hall | Education delays accelerated decline on a memory test in persons who develop dementia[END_REF] However, there is now enough evidence from large scale studies to suggest a much longer period of cognitive decline. The Framingham data show lower cognitive scores in those with dementia up to 10 years before the diagnosis of dementia. [START_REF] Elias | The preclinical phase of alzheimer disease: A 22-year prospective study of the Framingham Cohort[END_REF] The French PAQUID study, based on adults aged 65 or older at baseline, shows the trajectory of cognitive decline in those who develop dementia to change course 12 years before diagnosis. [START_REF] Amieva | Prodromal Alzheimer's disease: Successive emergence of the clinical symptoms[END_REF] Data from the 1921 Scottish Birth Cohort have shown childhood cognitive scores to also predict late-life dementia. [START_REF] Whalley | Childhood mental ability and dementia[END_REF] The case for continuities in aging, in that prior changes shape later life changes, makes it meaningful to examine age-related cognitive decline earlier than that in studies on dementia.

The extent to which cognitive reserve influences cognitive decline has been the subject of much research. [START_REF] Lee | The relation of education and income to cognitive function among professional women[END_REF][START_REF] Koster | Socioeconomic differences in cognitive decline and the role of biomedical factors[END_REF][START_REF] Richards | Cognitive ability in childhood and cognitive decline in mid-life: longitudinal birth cohort study[END_REF][START_REF] Anstey | Education, activity, health, blood pressure and apolipoprotein E as predictors of cognitive change in old age: a review[END_REF] However, the conclusion drawn from previous two-wave studies has been shown to be problematic, [START_REF] Glymour | When is baseline adjustment useful in analyses of change? An example with education and cognitive change[END_REF] as adjustment for the baseline measure of cognitive function in regression models of change biases results. Longitudinal analyses, using education and income in the AHEAD study [START_REF] Karlamangla | Trajectories of cognitive function in late life in the United States: demographic and socioeconomic predictors[END_REF] (mean age of participants 77 years) and education alone in the Cambridge City over 75s Cohort Study [START_REF] Muniz-Terrera | Education and trajectories of cognitive decline over 9 years in very old people: methods and risk analysis[END_REF] suggest no consistent impact of reserve on the rate of decline in cognitive function. This is in agreement with studies on younger age-groups; an investigation based on adults aged 60-64 years at baseline showed education and measures of brain volume not to influence cognitive decline. [START_REF] Christensen | Education, atrophy, and cognitive change in an epidemiological sample in early old age[END_REF] A similar null finding was reported in a study of adults aged 49-81 years at baseline where education had no effect on the rate of cognitive decline. [START_REF] Van Dijk | No protective effects of education during normal cognitive aging: results from the 6-year follow-up of the Maastricht Aging Study[END_REF] In our 10-year follow-up of individuals, whose mean age at baseline was 56 years (SD = 6 years), age-related decline was evident for all cognitive functions except vocabulary. The results for vocabulary were expected as some cognitive abilities, verbal knowledge for example, are known to be preserved into old age. [START_REF] Holland | The Course and causes of Cognitive Change with Advancing age[END_REF][START_REF] Hedden | Insights into the ageing mind: a view from cognitive neuroscience[END_REF] Analyses based on the global cognitive score, constructed as in previous studies, [START_REF] Wilson | Neurodegenerative basis of age-related cognitive decline[END_REF][START_REF] Arvanitakis | Relation of NSAIDs to incident AD, change in cognitive function, and AD pathology[END_REF] may be less affected by measurement error than those based on individual cognitive tests. In our study, the results using this summary measure suggest faster decline in the high reserve group defined by occupation but similar decline in reserve groups defined by education and height.

The results for occupation, showing greater decline in the high reserve group, mirror findings from the AHEAD study [START_REF] Karlamangla | Trajectories of cognitive function in late life in the United States: demographic and socioeconomic predictors[END_REF] where greater decline in memory and verbal fluency was observed in the high reserve group, defined by income but not education. In our data, of the three measures of reserve the association with cognitive tests was strongest with occupation.

It is possible that faster decline in the high occupation group is the result of statistical phenomena such as regression to the mean. [START_REF] Barnett | Regression to the mean: what it is and how to deal with it[END_REF] However, the persistence of this effect with the global cognitive score in the present study suggests that regression to the mean might not completely explain the finding. An alternative explanation is that cognitive gains associated with work content in different occupations are transient, explaining the faster decline in the higher occupational group. However, it is important not to over interpret this finding as for all three measures of reserve, the decline in reasoning (AH4-I) was fastest in the lowest reserve group. The overall conclusion from our results is that cognitive reserve does not much alter the rate of cognitive decline.

There are two possible ways in which cognitive reserve could impact the risk of cognitive impairments in old age: by their impact on peak adult cognitive function and by influencing the trajectory of cognitive decline itself (the slope of cognitive change with age).

Either of these effects would lead to poor cognitive status, and perhaps greater risk of dementia, in old age among those with low cognitive reserve. Much research on cognitive reserve focuses on its role in the interface between brain pathology and function among those with dementia. [START_REF] Stern | What is cognitive reserve? Theory and research application of the reserve concept[END_REF][START_REF] Bennett | Education modifies the relation of AD pathology to level of cognitive function in older persons[END_REF][START_REF] Roe | Education and Alzheimer disease without dementia: support for the cognitive reserve hypothesis[END_REF] A simple formulation of the reserve hypothesis for aging is that greater cognitive reserve allows individuals to cope better or more successfully with aging. A distinction has been made between the passive (threshold) and the active model of reserve. [START_REF] Stern | What is cognitive reserve? Theory and research application of the reserve concept[END_REF][START_REF] Brickman | White matter hyperintensities and cognition: Testing the reserve hypothesis[END_REF] The passive model assumes that functioning below a certain threshold will lead to the diagnosis of dementia, greater reserve is protective simply by creating distance from the cutoff threshold of functioning for dementia. [START_REF] Katzman | Education and the prevalence of dementia and Alzheimer's disease[END_REF] The active model promotes the view that greater efficiency and compensatory mechanisms in those with great reserve will lead to better aging outcomes. [START_REF] Stern | What is cognitive reserve? Theory and research application of the reserve concept[END_REF] Our results support the passive models of reserve: individuals with high reserve must experience substantial cognitive deterioration before they will reach any threshold of impairment. Although we do not find evidence that reserve slows rate of cognitive change, the difference in cognitive function at baseline are very large. For example, at baseline, men with high-occupational position score 6.7 points above men with intermediate occupational level on the reasoning test; this is equivalent to 19 years of decline at the average rate experienced by the high-occupational position men. The baseline reasoning score difference between high and low occupational position men is 55 times the annual average rate of decline for high occupational position men.

In a recent paper, Stern and colleagues propose height, alongside other anthropometric measures of development, to represent a measure of passive reserve and measures of education and occupation are seen to represent reserve from the perspective of the active model of reserve. [START_REF] Brickman | White matter hyperintensities and cognition: Testing the reserve hypothesis[END_REF] In our data, height, education and occupation were all associated with the baseline scores of the five cognitive tests. Occupation was the most strongly associated with the measures of cognition. It is possible that the greater effect of occupation is attributable to the impact of environments encountered after the completion of education that induce practice and reinforcement of cognitive abilities. These cognitive activities induced by work may lead to maintenance of higher levels of function. In this sense cognitive reserve is not fixed but continues to be shaped by life experiences. [START_REF] Stern | Cognitive reserve[END_REF] This reflects findings from animal studies where enriched environments have been shown to improve brain structure and function. [START_REF] Markham | Experience-driven brain plasticity: beyond the synapse[END_REF][START_REF] Kozorovitskiy | Experience induces structural and biochemical changes in the adult primate brain[END_REF] However, greater decline in the higher occupation groups might imply that some of these gains may be lost with ageing.

The primary caveat to the results reported here is that the Whitehall II study is based on a white collar cohort with stable jobs and thus is not representative of the general population.

However, the study includes a wide occupational spectrum with salary difference of over 10fold between the top and bottom of the hierarchy. A further limitation is the lack of neuropathological data in our study. Although there is evidence to suggest that all cognitive decline has a neuropathological basis, [START_REF] Wilson | Neurodegenerative basis of age-related cognitive decline[END_REF][START_REF] Johnson | Longitudinal study of the transition from healthy aging to Alzheimer disease[END_REF] it is possible that the effect of cognitive reserve is mostly apparent in groups of older individuals with significant neuropathology.

Advantages of this cohort is the large size, multiple and accurate measures of cognitive reserve, low attrition and a large battery of cognitive tests. Cognitive decline is a central feature of the aging process. [START_REF] Brayne | The elephant in the room -healthy brains in later life, epidemiology and public health[END_REF][START_REF] Nelson | Neuropathology and cognitive impairment in Alzheimer disease: a complex but coherent relationship[END_REF][START_REF] Hedden | Insights into the ageing mind: a view from cognitive neuroscience[END_REF][START_REF] Finch | The neurobiology of middle-age has arrived[END_REF][START_REF] Filippini | Distinct patterns of brain activity in young carriers of the APOE-epsilon4 allele[END_REF] An important goal of understanding cognitive aging trajectories over the adult lifecourse is to identify factors that are neuroprotective. The association of education, and other markers of reserve, with dementia, [START_REF] Stern | Influence of education and occupation on the incidence of Alzheimer's disease[END_REF][START_REF] Qiu | The influence of education on clinically diagnosed dementia incidence and mortality data from the Kungsholmen Project[END_REF][START_REF] Evans | Education and other measures of socioeconomic status and risk of incident Alzheimer disease in a defined population of older persons[END_REF][START_REF] Launer | Rates and risk factors for dementia and Alzheimer's disease: results from EURODEM pooled analyses. EURODEM Incidence Research Group and Work Groups[END_REF][START_REF] Ngandu | Education and dementia: what lies behind the association?[END_REF][START_REF] Gatz | Accounting for the relationship between low education and dementia: a twin study[END_REF][START_REF] Mortimer | Head circumference, education and risk of dementia: findings from the Nun Study[END_REF] suggests that it ≠ The interaction term assesses whether the decline was different in the three occupational groups. Further tests compared the high reserve group to the other two groups and here *p<0.05, **p<0.01, ***p<0.001.

± Score calculated by converting raw scores on each test to z-scores using the baseline mean and standard deviation and then averaged across the five tests. ≠ The interaction term assesses whether the decline was different in the three occupational groups. Further tests compared the high reserve group to the other two groups and here *p<0.05, **p<0.01, ***p<0.001.

± Score calculated by converting raw scores on each test to z-scores using the baseline mean and standard deviation and then averaged across the five tests. ≠ The interaction term assesses whether the decline was different in the three occupational groups. Further tests compared the high reserve group to the other two groups and here *p<0.05, **p<0.01, ***p<0.001.

± Score calculated by converting raw scores on each test to z-scores using the baseline mean and standard deviation and then averaged across the five tests.

  At study inception (Phase 1), 1985-1988, 10,308 participants (67% men) underwent a clinical examination and completed a selfadministered questionnaire. Cognitive testing was introduced to the full cohort at Phase 5 (1997-1999) of the study and repeated at Phases 7 (2002-2004) and 9 (2007-2009). All participants provided written consent and the University College London ethics committee (UCLH Committee Alpha, #85/0938) approved this study.

Table 1 .

 1 Mean(25 th , 75 th percentile) scores at baseline in men and women as a function of cognitive reserve markers.

Table 4 .

 4 Cognitive decline as a function of height, estimates derived from linear mixed models using 3 assessments over 10 years.‡ 22 (-3.48, -2.97) -3.48 (-3.75, -3.22) -3.74 (-3.99, -3.50)* 0.01 Memory (20-word list) -0.71 (-0.82, -0.60) -0.63 (-0.74, -0.52) -0.62 (-0.73, -0.52) 0.46 Phonemic fluency ("s" words) -1.59 (-1.74, -1.44) -1.52 (-1.68, -1.36) -1.61 (-1.75, -1.47) 0.70 Semantic fluency (animal names) -1.29 (-1.43, -1‡ Height in the tall group in men (women) was ≥ 180cm (165cm), in the average group was 175-179cm (160-164cm) and in the shorter that average group it was ≤ 175cm (160cm). † Range of cognitive tests: Memory (0-20), Reasoning (0-65), Phonemic and Semantic Fluency (0.35) and Vocabulary (0-33).

Figure 1 .

 1 Figure 1. Observed data on cognitive tests (Reasoning (A), Memory (B), Phonemic Fluency (C), Semantic Fluency (D) and E (Vocabulary) at three repeat assessments in the high, intermediate and low occupation groups.

Table 2 .

 2 Cognitive decline as a function of occupational position, estimates derived from linear mixed models using 3 assessments over 10 years.

			OCCUPATION POSITION
		High	Intermediate	Low
		(N=3140)	(N=3289)	(N=1025)
		14% women	31% women	73% women
		10-year decline	10-year decline	10-year decline	≠ P for
	Cognitive tests †	(95% CI)	(95% CI)	(95% CI)	interaction
	Reasoning (AH4-I)	-3.47 (-3.69, -3.26)	-3.43 (-3.64, -3.21)	-4.04 (-4.47, -3.60)*	0.04
	Memory (20-word list)	-0.77 (-0.87, -0.68)	-0.61 (-0.70, -0.51)*	-0.46 (-0.64, -0.28)**	0.003
	Phonemic fluency ("s" words)	-1.72 (-1.85, -1.59)	-1.48 (-1.61, -1.35)*	-1.54 (-1.79, -1.29)	0.04
	Semantic fluency (animal names)	-1.53 (-1.65, -1.41)	-1.26 (-1.38, -1.14)**	-0.89 (-1.12, -0.65)***	<0.001
	Vocabulary (Mill-Hill)	0.11 (0.03, 0.19)	0.08 (0.00, 0.16)	0.06 (-0.10, 0.22)	0.83
	Global cognitive score

± -0.27 (-0.28, -0.26) -0.23 (-0.25, -0.22)*** -0.21 (-0.24, -0.19)*** 0.001 † Range of cognitive tests: Memory (0-20), Reasoning (0-65), Phonemic and Semantic Fluency (0.35) and Vocabulary (0-33).

Table 3 .

 3 Cognitive decline as a function of education, estimates derived from linear mixed models using 3 assessments over 10 years. ‡

				EDUCATION
			High	Intermediate	Low
			(N=2092)	(N=1848)	(N=3157)
			21% women	25% women	37% women
			10-year decline	10-year decline	10-year decline	≠ P for
	Cognitive tests †
			(95% CI)	(95% CI)	(95% CI)	interaction
	Reasoning (AH4-I)		-3.17 (-3.43, -2.90)	-3.41 (-3.70, -3.12)	-3.78 (-4.00, -3.55)***	0.002
	Memory (20-word list)	-0.73 (-0.85, -0.62)	-0.61 (-0.73, -0.48)	-0.65 (-0.74, -0.55)	0.30
	Phonemic fluency ("s" words)	-1.57 (-1.73, -1.42)	-1.70 (-1.87, -1.53)	-1.50 (-1.63, -1.37)	0.19
	Semantic fluency (animal names) -1.45 (-1.60, -1.31)	-1.30 (-1.45, -1.14)	-1.23 (-1.36, -1.11)*	0.08
	Vocabulary (Mill-Hill)	0.20 (0.10, 0.30)	0.11 (0.00, 0.21)	0.00 (-0.08, 0.09)**	0.009
	Global cognitive score	±	-0.25 (-0.27, -0.23)	-0.24 (-0.26, -0.23)	-0.26 (-0.26, -0.23)	0.82
	‡ Note that data on education were missing for 357 individuals; these analyses are based on 5018 men and 2079
	women.
	† Range of cognitive tests: Memory (0-20), Reasoning (0-65), Phonemic and Semantic Fluency (0.35) and
	Vocabulary (0-33).

ACKNOWLEDGEMENTS

ASM is supported by a "European Young Investigator Award" from the European Science Foundation and the National Institute on Aging, NIH, USA (R01AG013196; R01AG034454).

MK by the Academy of Finland, the BUPA Foundation, the National Heart, Lung, and Blood Institute, USA (R01HL036310) and the National Institute on Aging, NIH, USA (R01AG034454) . The Whitehall II study has been supported by grants from the Medical

1 Research Council; British Heart Foundation; Health and Safety Executive; Department of Health; and the National Institutes of We thank all of the participating civil service departments and their welfare, personnel, and establishment officers; the British Occupational Health and Safety Agency; the British Council of Civil Service Unions; all participating civil servants in the Whitehall II study; and all members of the Whitehall II study team. The Whitehall II Study team comprises research scientists, statisticians, study coordinators, nurses, data managers, administrative assistants and data entry staff, who make the study possible.

could be one such factor. It is important to identify neuroprotective factors as even in the absence of dementia, poor cognitive status at older ages is taxing for the individual and society at large. Our data suggests that cognitive reserve is associated with cognitive performance but not rate of decline in cognitive function. One of the key tasks of future research is to identify mechanisms that lead to the large differences in cognitive performance over the adult lifecourse.