Lionel De Boisde¤re
email: lionel.de.boisde¤re@wanadoo.fr

Dropping the Cass Trick and Extending Cass'Theorem to Asymmetric Information and Restricted Participation

Keywords: sequential equilibrium, perfect foresight, existence of equilibrium, rational expectations, incomplete markets, asymmetric information, arbitrage. JEL Classi…cation: D52

In a celebrated 1984 paper, David Cass provided an existence theorem for …nancial equilibria in incomplete markets with exogenous yields. The theorem showed that, when agents had symmetric information and ordered preferences, equilibria existed on purely …nancial markets, supported by any collection of state prices. This theorem built on the so-called "Cass trick", along which one agent had an Arrow-Debreu budget set, with one single constraint, while the other agents were constrained a la Radner (1972), that is, in every state of nature. The current paper extends Cass'theorem to asymmetric information, non-ordered preferences and restricted participation. It re…nes De Boisde¤re (2007), which characterized the existence of equilibria with asymmetric information by the no-arbitrage condition on purely …nancial markets. The paper de…nes no arbitrage prices with asymmetric information. It shows that any collection of state prices, in the agents' commonly expected states, supports an equilibrium. This result is proved without using the Cass trick, in the sense that budget sets are de…ned symmetrically across all agents. Thus, the paper suggests, in the symmetric information case, an alternative proof to Cass'.

Introduction

The current contribution needs be introduced with reference to standard concepts and properties of sequential equilibria, with symmetric and asymmetric information.

With symmetric information, the classical de…nition of sequential equilibrium relies on the assumption that agents know the map between future random states and the price to prevail on every spot market. Under this so-called 'perfect foresight'or 'rational expectation'hypothesis, the existence of equilibrium has been extensively studied since [START_REF] Radner | Existence of equilibrium plans, prices and price expectations in a sequence of markets[END_REF]. With nominal and numéraire assets, the span of pay-o¤s does not depend on prices. This insures the full existence of equilibrium in standard conditions, as shown by [START_REF] Cass | Competitive equilibrium with incomplete …nancial markets[END_REF][START_REF] Cass | Competitive equilibrium with incomplete …nancial markets[END_REF], Du¢ e (1987), [START_REF] Werner | Equilibrium in economies with incomplete …nancial markets[END_REF], for nominal assets, and [START_REF] Geanakoplos | Existence, regularity, and constrained suboptimality of competitive allocations when the asset market is incomplete[END_REF], for numéraire assets.

The literature shows an essential real indeterminacy of equilibrium with nominal assets [START_REF] Balasko | The structure of …nancial equilibrium with exogenous yields: the case of incomplete markets[END_REF]; Geanakoplos-Mas-Colell, 1989), whereas no-arbitrage prices coincide with equilibrium asset prices [START_REF] Cass | Competitive equilibrium with incomplete …nancial markets[END_REF]. Contrarily, with numéraire assets, [START_REF] Geanakoplos | Existence, regularity, and constrained suboptimality of competitive allocations when the asset market is incomplete[END_REF] proves the generic local uniqueness of equilibrium, but does not extend Cass' result. The extension of [START_REF] Cass | Competitive equilibrium with incomplete …nancial markets[END_REF] to numéraire assets and asymmetric information is poposed in De Boisde¤re (2021, a).

With other types of assets, the existence of equilibrium is not guaranteed, as shown by [START_REF] Hart | On the optimality of equilibrium when the market structure is incomplete[END_REF], when agents forecast prices perfectly. Hart's counterexample is based on the collapse of the span of assets' payo¤s, that occurs exceptionally at clearing-market prices. One response to that problem is to show, along Du¢ e-Shafer (1985), that the fall in rank of endogenous yields is, indeed, exceptional, and equilibrium therefore exists generically in payo¤s and endowments. Geanakoplos-1 Documents de travail du Centre d'Economie de la Sorbonne -2021.10 Shafer (1990), [START_REF] Bich | Fixed-point-like theorems on suspaces[END_REF], 2009), in the symmetric information case, and De Boisde¤re (2021, a), in the asymmetric information case, resume this argument.

Another approach is to drop the perfect foresight hypothesis, and let, instead, agents have an 'endogenous uncertainty' over future spot prices, akin to [START_REF] Kurz | On rational belief equilibria[END_REF]. Under a milder 'correct foresight'assumption, the full existence of sequential equilibrium may be restored for all types of assets and private information signals (see De Boisde¤re, 2021, b). For expositional purposes, the current paper does not retain this approach, which requires to deal with in…nite dimensional economies.

When agents have asymmetric or incomplete information, they seek to learn more information from markets. The traditional inference mechanism is described by the REE (rational expectations equilibrium) model by assuming, quoting Radner (1979), that "agents have a 'model' or 'expectations' of how equilibrium prices are determined". Under this assumption, agents know the maps between private information signals and equilibrium prices, along a so-called 'forecast function'. Since equilibrium prices are typically distinct across agents'joint information signals, that forecast function would theoretically enable consumers to infer all information detained by the other agents from observing such 'separating'prices. However, that function depends on all consumers'characteristics, which are private. Inferring and using it to reach a REE is seen as theoretical, if realistic. Along De Boisde¤re (2016), markets would not reveal information via a price model, but through arbitrage.

With this approach, in the simplest setting to study arbitrage, Cornet-De Bois-de¤re (2002) suggests an alternative model to [START_REF] Radner | Rational expectations equilibrium: generic existence and the information revealed by prices[END_REF], where asymmetric information is represented by private signals, informing each agent that tomorrow's true state will be in a subset of the state space. This speci…cation re ‡ects the fact than an information may always reduce a set of possibilities. The latter paper generalizes the de…nitions of equilibrium, no-arbitrage prices and no-arbitrage condition to asymmetric information. From De Boisde¤re (2007,2016), the latter no-arbitrage condition can be reached by agents observing available transfers, and characterizes the existence of equilibrium, on purely …nancial markets. Such results di¤er from [START_REF] Radner | Rational expectations equilibrium: generic existence and the information revealed by prices[END_REF] inferences and the ensuing generic existence of a REE.

If De Boisde¤re (2007) proves the full existence of equilibrium, it does not assess whether the two concepts of no-arbitrage price and equilibrium asset price coincide, as in the symmetric information economy studied by [START_REF] Cass | Competitive equilibrium with incomplete …nancial markets[END_REF][START_REF] Cass | Competitive equilibrium with incomplete …nancial markets[END_REF]. To prove this equivalence, the author relies on the so-called 'Cass trick', which lets one agent have an Arrow-Debreu budget set, with one single constraint, and the other agents be constrained a la [START_REF] Radner | Existence of equilibrium plans, prices and price expectations in a sequence of markets[END_REF], i.e., in each state of nature. Without using the Cass trick, the current paper extends Cass'theorem to asymmetric information, non-ordered preferences and restricted portfolio participation. It de-…nes no-arbitrage prices with reference to individual state prices. It shows that any collection of state prices, in the agents' commonly expected states, supports an equilibrium. In the particular setting of Cass, it thus suggests an alternative proof.

The current paper is not the …rst generalization of Cass'theorem which drops the Cass trick (see [START_REF] Cornet | Arbitrage and equilibrium with portfolio constraints[END_REF]. But it is, to our best knowledge, the …rst, after De Boisde¤re (2021, a), to extend the theorem to asymmetric information. The latter paper extends Cass' (1984) result to nominal and numéraire asset markets under asymmetric information. But it uses di¤erential topology arguments, which only yield interior equilibrium allocations. Moreover, the latter paper still applies the Cass trick, and provides no other insight on the price equivalence, stated above.

The current paper aims to …ll these gaps. It also addresses restricted portolio participation. This extension of the Cass model has been extensively studied, in the symmetric information case, by several papers culminating with [START_REF] Cornet | Arbitrage and equilibrium with portfolio constraints[END_REF]. Di¤erently from Cornet-Gopalan, we chose not to go beyond two periods, for expositional purposes. The extension of a purely …nancial economy from two periods to multiple periods poses no conceptual di¢ culty and is standard. But it implies introducing additional nodes and notations, which, in the present case, would have hampered the simplicity and focus of the model. This extension is all but necessary to introduce asymmetric information in the model and is deferred.

The paper is organized as follows: Section 2 presents the model. Section 3 states and proves the existence theorem in the general model, and illustrates the theorem when one agent has full information. An Appendix proves technical Lemmas.

The model

We consider a pure-exchange …nancial economy with two periods, t 2 f0; 1g, and an uncertainty, at t = 0, upon which state of nature will randomly prevail at t = 1. The economy is …nite in the sense that the sets, I, S, L and J, respectively, of consumers, states of nature, consumption goods and assets are all …nite. The observed state at t = 0 is denoted by s = 0 and we let 0 := f0g [ , whenever S.

Markets and information

Agents consume or exchange the consumption goods, l 2 L, on both periods' spot markets. At t = 0, each agent, i 2 I, receives privately the correct information that tomorrow's true state will be in a subset, S i , of S. We assume costlessly that S = [ i2I S i . Thus, the pooled information set, S := \ i2I S i , contains the true state, and the relation S = S characterizes symmetric information.

We let P := fp := (p s ) 2 R L S 0 : kp s k 6 1; 8s 2 S 0 g be the set of admissible commodity prices, which each agent is assumed to observe, or anticipate perfectly, a la Radner (1972). Moreover, each agent with an incomplete information forms her private forecasts in the unrealizable states she expects. Such forecasts, (s; p i s ), are pairs of a state, s 2 S i nS, and a price, p i s 2 R L ++ , that the generic i th agent believes to be the conditional spot price in state s. Non-fully informed agents may agree or disagree on forecasts, which never obtain and are henceforth given, along De Boisde¤re (2007).

Agents may operate …nancial transfers across states in S 0 (actually in S 0 ) by exchanging, at t = 0, …nitely many nominal assets, j 2 J, which pay o¤, at t = 1, conditionally on the realization of the state. We assume that #J 6 #S, so that …nancial markets be typically incomplete. Assets'payo¤s de…ne a S J matrix, V , whose generic row in state s 2 S, denoted by V (s) 2 R J , does not depend on prices. Thus, at asset price, q 2 R J , agents may buy or sell portfolios of assets, z = (z j ) 2 R J , for q z units of account at t = 0, against the promise of delivery of a ‡ow, V (s) z, of conditional payo¤s across states, s 2 S. Moreover, the exchange of assets may be restricted, e.g., if agents are unaware or have no access to some available transfers. Participation to markets is then said to be restricted. For each i 2 I, we let Y i R J be the i th agent's set of admissible portfolios, henceforth set as given.

The consumer' s behaviour and concept of equilibrium

Each agent, i 2 I, receives an endowment, e i := (e is ), granting the commodity bundles, e i0 2 R L + at t = 0, and e is 2 R L + , in each expected state, s 2 S i , if it prevails. Given the market prices, p := (p s ) 2 P , for goods, q 2 R J , for assets, and her possible forecasts, the generic i th agent's consumption set is

X i := R L S 0 i +
, and budget set is:

B i (p; q) := f (x; z) 2 X i Y i : p 0 (
x 0 e i0 ) 6 q z and p s (x s e is ) 6 V (s) z; 8s 2 S and p i s (x s e is ) 6 V (s) z; 8s 2 S i nS g.

Each consumer, i 2 I, is endowed with a complete preordering,i , over her consumption set, representing her preferences. Her strict preferences, i , are represented, for each x 2 X i , by the set, P i (x) := f y 2 X i : x i y g, of consumptions which are strictly preferred to x. The above economy is denoted by E = f(I; S; L; J); V; (S i ); (p i s ); (e i ); ( i )g and yields the following concept of equilibrium:

De…nition 1 A collection of prices, p = (p s ) 2 P , q 2 R J , & decisions, (x i ; z i ) 2 B i (p; q),
for each i 2 I, is an equilibrium of the economy, E, if the following conditions hold:

(a) 8i 2 I; (x i ; z i ) 2 B i (p; q) and P i (x i ) R J \ B i (p; q) = ?; (b) P i2I (x is e is ) = 0; 8s 2 S 0 ; (c) P i2I z i = 0.
Endowments and preferences are called standard under the Assumptions:

A1 (strict monotonicity): 8(i; x; y) 2 I X i X i ; (x 6 y; x 6 = y) ) (x i y);

A2 (strong survival): 8i 2 I; e i 2 R L S 0 i ++ ;

A3 for every i 2 I, i is lower semi-continuous, convex-open-valued and such

that x i x + (y x), whenever (x; y; ) 2 X i P i (x) ]0; 1].
We now present algebraic notations and properties that will be used throughout.

The model' s notations and payo¤ structure

For every (i; s; z) 2 I S R J , we de…ne the following vector spaces and sums: We notice that the relation Z Z holds for any …nancial structure, (S i ), and that Z = f0g holds, if and only if Z = Z (in particular, if one agent is fully informed).

Z o i := fz 2 R J : V (s) z = 0; 8s 2 S i g & orthogonal complement, Z i := P s2Si RV (s); Z o := fz 2 R J : V (s) z = 0;
We denote j 1 := dim Z o , j := dim Z and j 2 := dim Z, which satisfy:

#J = j 1 + j + j 2 .
For convenience, we henceforth assume costlessly, by combining and reordering assets if needed, that portfolios selected amongst the …rst j 1 assets (if j 1 > 0) belong to Z o , those from the subsequent j assets (if j > 0) belong to Z , and those from the last j 2 assets (if any) belong to Z. For each s 2 S, the relation

V (s) = V (s) 2
follows from these assumptions, that is, the …rst j 1 +j components of V (s) are null.

Remark 1

The above structure of payo¤s may always be assumed and obtained by replacing (if needed) the columns of the matrix V by linear combinaisons of these columns, without changing the span of payo¤s and, hence, the …nancial structure.

Financial markets and participation are called standard under the Assumptions:

A4 for every i 2 I, Y i is closed, convex and contains zero;

A5 for every i 2 I, Z o + Y i Y i ; A6 (riskless asset) 9z + 2 \ i2I Y i : V (s) z + > 0; 8s 2 S.
The …nancial economy, E, is said to be standard if preferences, endowments, …nancial markets and participation are standard, along Assumptions A1 to A6. Assumption A6, which states that agents are always proposed to save a small amount of cash to insure risk in all realizable states, is met on all actual markets.

Under asymmetric information, Assumptions A5-A6 are made for technical purposes and serve to prove, in particular, Claim 3 below. Assumption A5 states that what would be a worthless portfolio to an informed agent may always be combined to an admissible portfolio. This possibility may be explained by the role of …nancial intermediaries, proposing to insure agents against their idiosyncratic risks, represented by states s 2 SnS. In the current model, the latter states never prevail.

We now present no-arbitrage prices, state prices, and some of their properties.

No-arbitrage prices and state prices with asymmetric information

We start with a de…nition:

De…nition 2 A no-arbitrage price is an asset price, q 2 R J , which meets one of the following equivalent conditions, and we let N A be their set:

(a) @(i; z) 2 I R J , q z > 0 and V (s) z > 0, 8s 2 S i , with one strict inequality;

(b) 8i 2 I, 9 i := ( is ) 2 R Si ++ , q = P s2Si is V (s).
Scalars, ( is ) 2 i2I R Si ++ , which meet the above condition (b), are said to support the no-arbitrage price, q 2 N A, and called (individual) state prices. For every := ( s ) 2 R S ++ , we denote N A( ) := fq 2 N A : We henceforth assume that (S i ) is arbitrage-free, i.e., admits a no-arbitrage price.

q 2 = P s2S s V (s)g and let N AC = [ 2R S ++ N A( ).

N.B. The equivalence between Conditions

Remark 3 The information structure, (S i ), is arbitrage-free when it is symmetric.

Otherwise, from Assumption A5, it is costless to assume that (S i ) is arbitrage-free. Indeed, it follows from Cornet-De Boisde¤re (2009) that agents may narrow down their information sets, if needed, from observing informative portfolios, which all belong to Z o . Then, agents infer these sets'coarsest arbitrage-free re…nements.

The following heuristic example and Claim 1 show that the sets N A and N AC di¤er, in general, but that any collection of state prices, := ( s ) 2 R S ++ , supports a no-arbitrage price, q 2 N A( ), and supports an equilibrium, from Theorem 1 below.

Example Consider an economy, E, with two agents, i 2 f1; 2g, three states, s 2 f1; 2; 3g, an information stucture, S 1 := f1; 2g and S 2 := f2; 3g, and one asset, whose price is q = 1. For a payo¤ matrix

V = 0 B B B B B B @ 1 0 1 1 C C C C C C A , the relations Z = R, Z = f0g, q 2 N AC hold. For V = 0 B B B B B B @ 1 1 1 1 C C C C C C A , the relations Z Z o = f0g and q 2 N A * N AC = R ++ hold.

Claim 1

The following Assertions hold:

(i) 8 2 R S ++ ; N A( ) 6 = ?; (ii) N AC N A, but N A * N AC, in general.
Proof Assertion (i) Let := ( s ) 2 R S ++ be given. As assumed above, N A 6 = ?.

Hence, we let q 2 N A and ( is ) 2 i2I R Si ++ be given, such that q = P s2Si is V (s), for each i 2 I. Since q is a no-arbitrage price, the relation

q 1 = 0 holds (q z = 0 if z 2 Z o ).
We refer to the notations, de…nitions and characteristics of the assets, stated in sub-Section 2.3. We recall that V (s) = V (s) 2 may be assumed for every s 2 S. For each i 2 I, we de…ne q i 2 Z by q i = P s2SinS is V (s) 2 , if S i 6 = S, and q i = 0 otherwise.

Since q i 2 Z, there exists a vector ( is ) 2 R S , such that q i = P s2S is V (s), for every i 2 I. For N 2 N large enough, the relations j is N j < s hold, for every pair and contains zero, for each i 2 I, and coincides with the orthogonal projection of Y i on Z i . The following sets are, therefore, well de…ned, for every (i; p; q) 2 I P Q:

B 1 i (p; q) := f (x; z) 2 X i (Z i \ Y i ) : p 0 (x 0 e i0 ) + q z + P s2S s p s (x s e is ) 6 1;
p s (x s e is ) 6 V (s) z; 8s 2 S; and p i s (x s e ) 6 V (s) z; 8s 2 S i nS g;

A(p; q) := f[(x i ; z i )] 2 i2I B i (p; q+q) : P i2I (x is e is ) s2S 0 = 0; (z i ) 2 i2I Z i ; P i2I z i 2 Z o g. Lemma 1 9r > 0 : 8(p; q) 2 P Q; 8 [(x i ; z i )] 2 A(p; q); P i2I (kx i k + kz i k) < r
Proof : See the Appendix.

Along Lemma 1, we let X i := fx 2 X i : kxk 6 rg and

Z i := fz 2 Z i \ Y i : kzk 6 rg,
and de…ne, for every (i; p := (p s ); q) 2 I P Q, the following convex compact sets:

B 0 i (p; q) := f (x; z) 2 X i Z i : p 0 (
x 0 e i0 ) + q z + P s2S s p s (x s e is ) 6 (p;q) ; p s (x s e is ) 6 V (s) z; 8s 2 S; and p i s (x s e is ) 6 V (s) z; 8s 2 S i nS g, where (p;q) := 1 min(1; kpk + kqk), so that B 0 i (p; q) B 1 i (p; q). Claim 2 For every i 2 I, B 0 i is upper semicontinuous.

Proof Let i 2 I be given. The correspondences B 0 i is, as standard, upper semicontinuous, for having a closed graph in a compact set.

The …xed-point-like argument

Budget sets were modi…ed in sub-section 3.1, so that their interiors be non-empty. This was required to prove the lower semi-continuity of the reaction correspondences of Lemma 2, below. For every (p; q) 2 P Q, these interior budget sets are as follows:

B 00 i (p; q) := f (x; z) 2 X i Z i : p 0 (x 0 e i0 ) + q z + P s2S s p s (x s e is ) < (p;q) and p s (x s e is ) < V (s) z; 8s 2 S; and p i s (x s e is ) < V (s) z; 8s 2 S i nS g, for each i 2 I.

11 Documents de travail du Centre d'Economie de la Sorbonne -2021.10

Claim 3 The following Assertions hold, for each i 2 I:

(i) 8(p; q) 2 P Q, B 00 i (p; q) 6 = ?;

(ii) the correspondence B 00 i is lower semicontinuous.

Proof Let i 2 I, z + 2 Q ? , along Assumptions A6, and (p; q) 2 P Q be given.

Assertion (i) If (p;q) = 1, from Assumptions A4-A6, the relation (0; t:z + ) 2 B 00 i (p; q)
holds, for t > 0 small enough. If p 6 = 0, from Assumptions A2-A4-A6 and the relation

z + 2 Q ? , there exist (x; t) 2 X i R ++ , such that (x; t:z + ) 2 B 00 i (p; q). If q 6 = 0, from Assumptions A4-A5-A6, there exist (z; t) 2 Z R ++ , such that (0; t:z + + z) 2 B 00 i (p; q). Assertion (ii) Let V be an open subset of X i Z i , such that V \ B 00
i (p; q) 6 = ? and let (x; z) 2 V \ B 00 i (p; q). From the de…nition, there exists a neigbourhood, U , of (p; q), such that (x; z) 2 V \ B 00 i (p 0 ; q 0 ), if (p 0 ; q 0 ) 2 U , i.e., B 00 i is lower semicontinuous at (p; q).

We now introduce an agent representing markets (i = 0), a convex compact set,

:= P Q ( i2I X i Z i )
, and a lower semicontinous reaction correspondence on the set , for each agent. Thus, for every i 2 I and every := (p; q; [(x i ; z i )]) 2 , we let: 0 ( ) := f (p 0 ; q 0 ) 2 P Q : (q 0 q)

P i2I z i + P s2S 0 (p 0 s p s ) P i2I (x is e is ) > 0 g; i ( ) := 8 > > < > > : B 0 i (p; q) if (x i ; z i ) = 2 B 0 i (p; q) B 00 i (p; q) \ P i (x i ) Z i if (x i ; z i ) 2 B 0 i (p; q)
Lemma 2 For each i 2 I [ f0g, i is lower semicontinuous.

Proof See the Appendix.

The latter correspondences admit a …xed point, , along the following Claim:

Claim 4 There exists := (p ; q ; [(x i ; z i )]) 2 , such that:

(i) 8(p; q) 2 P Q; (q q) P i2I z i + P s2S 0 (p s p s )

P i2I (x is e is ) > 0;
(ii) 8i 2 I; (x i ; z i ) 2 B 0 i (p ; q ) and B 00 i (p ; q ) \ P i (x i ) Z i = ?.

Proof Quoting [START_REF] Gale | An equilibrium existence theorem for a general model without ordered preferences[END_REF][START_REF] Radner | Rational expectations equilibrium: generic existence and the information revealed by prices[END_REF]: "Given X = m i=1 Xi, where X i is a non-empty compact convex subset of R n , let ' i : X ! X i be m convex (possibly empty) valued correspondences, which are lower semicontinuous. Then, there exists x := (x i ) in X such that for each i either x i 2 ' i (x) or ' i (x) = ?". The correspondences i , for each i 2 I [ f0g, meet all conditions of the above theorem and yield Claim 4.

An equilibrium of the economy E

The above …xed point, , meets the following properties, proving Theorem 1: Claim 5 Given := (p ; q ; [(x i ; z i )] ) 2 , along Claim 4, the following holds:

(i) P i2I z i = 0, where, for each i 2 I, z i is the normal projection of z i on Z ;

(ii) P i2I (x is e is ) = 0, for every s 2 S 0 ;

(iii) for every i 2 I; (x i ; z i ) 2 B 0 i (p ; q ) and B 0 i (p ; q ) \ P i (x i ) Z i = ?;

(iv) p s 2 R L S 0 ++ and (p ;q ) = 0;

(v) P i2I z i 2 Z o , and we set as given (z o i ) 2 i2I Z o i , such that

P i2I z i = P i2I z o i ;
(vi) given (z i ) := (z i z o i ), q := P s2S s V (s) and q := (q + q) 2 N A( ), the collection of prices and strategies, (p ; q; [(x i ; z i )]), de…nes an equilibrium of the economy E.

Proof Assertion (i) Assume, by contraposition, that P i2I z i 6 = 0. Then, from Claim 4-(i), the relations q P i2I z i = q P i2I z i > 0 and (p ;q ) = 0 hold. Moreover, from Claim 4-(i), the relations 0 6 P i2I p s (x is e is ) hold, for every s 2 S 0 . From Claim 4-(ii), the relations p 0 (x i0 e i0 ) + q z i + P s2S s p s (x is e is ) 6 0 hold, for every i 2 I. Summing them up (for i 2 I) yields, from above: This contradiction proves that

P i2I z i = 0.
Assertion (ii) From Claim 4-(i), p s P i2I (x is e is ) > 0 holds, for every s 2 S 0 , and

p 0 P i2I (x i0 e i0 )+ P s2S s P i2I p s (x is e is ) > 0 holds whenever ( P i2I (x is e is )) s2S 0 6 = 0.
Assume, by contraposition, that P i2I (x is e is ) 6 = 0, for some s 2 S 0 . Then, from Claim 4, the relation (p ;q ) = 0 and the following budget constraints hold, for each i 2 I: p 0 (x i0 e i0 ) + q z i + P s2S s p s (x is e is ) 6 0.

Summing them up yields, from Assertion (i) and above:

0 < p 0 P i2I (x i0 e i0 ) + P s2S s P i2I p s (x is e is ) 6 0.
This contradiction proves that P i2I (x is e is ) = 0, for every s 2 S 0 . Assertion (iii) Let i 2 I be given. From Claim 4, it su¢ ces to show the relation:

B 0 i (p ; q ) \ P i (x i ) Z i = ?.
By contraposition, we assume that there exists a strategy, (x i ; z i ) 2 B 0 i (p ; q ) \ P i (x i ) Z i , and we set as given (x 0 i ; z 0 i ) 2 B 00 i (p ; q ) B 0 i (p ; q ), along Claim 3. From Assumptions A3-A4, the relations

(x n i ; z n i ) := [ 1 n (x 0 i ; z 0 i )+(1 1 n )(x i ; z i )] 2
B 00 i (p ; q ) hold, for every n 2 N, and

(x N i ; z N i ) 2 P i (x i ) Z i holds, for N 2 N big enough. The relation (x N i ; z N i ) 2 B 00 i (p ; q )\P i (x i ) Z i follows and contradicts Claim 4.
Assertion (iv) From Assertion (ii), we may assume that kx i k 6 P i2I ke i k < r holds for each i 2 I in Lemma 1. Then, the relation p 2 R LS 0 ++ is standard from Assumption A1 and Assertion (iii). It follows from Assumption A1 and Assertion (iii) that the budget constraints of (x i ; z i ) 2 B 0 i (p ; q ) are all binding. Summing them up, for i 2 I, at the …rst period yields, from Assertions (i)-(ii): 0 = P i2I p 0 (x i0 e i0 ) + q P i2I z i + P s2S s P i2I p s (x is e is ) = #I: (p ;q ) . Assertion (v) From Assertion (iii)-(iv) and Assumption A1, the budget constraints of (x i ; z i ) 2 B 0 i (p ; q ) (for i 2 I) are all binding. Then, the relations,

P i2I (x is e is ) = 0,
of Assertion (ii), yield: 0 = P i2I p s (x is e is ) = P i2I V (s) z i = V (s) P i2I z i , for each s 2 S. The latter are written, with the notations of sub-Section 2.3, ( P i2I z i ) 2 Z o . Moreover, P i2I z i = 0 holds from Assertion (i). It follows that

P i2I z i 2 Z o .
Assertion (vi) Let C := (p ; q; j(x i ; z i )]) be de…ned along Claim 5. That collection meets Conditions (b) and (c) of De…nition 1, from Assertions (ii) and (v) above, and, moreover, q 2 N A( ) holds from the de…nition.

For each i 2 I, the relations (x i ; z i ) 2 B i (p ; q) and (x i ; z i ) 2 B i (p ; q) hold from the de…nitions of z i and q, and from Assertions (iii)-(iv)-(v) (which imply that all constraints of (x i ; z i ) 2 B 0 i (p ; q) are binding). The relation

P i2I (kx i k + kz i k) < r
follows, from Lemma 1 and Assertions (ii)-(v), that is, (x i ; z i ) is interior to X i Z i .

Let i 2 I be given and assume, by contraposition, that B i (p ; q) \ P i (x i ) Y i 6 = ?.

Then, there exists (x 0 i ; z 0 i ) 2 B i (p ; q) \ (P i (x i ) \ X i ) Z i , from the interiority of (x i ; z i ), Assumptions A3-A4-A5 and the de…nition of Z i . Moreover, from Assumption A1 and Assertion (iv), we may assume that all budget constraints of (x 0 i ; z 0 i ) in B i (p ; q) are binding. Then, the relation (x 0 i ; z 0 i ) 2 B 0 i (p ; q ) holds from Assertion (iv) and the de…nition of B 0 i (p ; q ). It follows that (x 0 i ; z 0 i ) 2 B 0 i (p ; q )\P i (x i ) Z i , which contradicts Assertion (iii). This contradiction ends the proof of Assertion (vi) and Theorem 1.

The model' s outcomes when one agent is fully informed

We now assume that one agent is fully informed and illustrate Theorem 1 simply.

Resuming the notations of sub-Section 2.3, the relation Z = f0g (or Z o = Z o ) holds. The de…nition of no-arbitrage prices and the proof simplify, with Q = f0g:

8s 2 6

 26 Sg and orthogonal complement, Z := P s2S RV (s);Z o := P s2Si Z o i , its orthogonal complement, Z := \ i2I Z i , and Z := Z \ Z o ; z = z 1 + z + z 2 , the orthogonal decomposition of z on Z o Z Z.Documents de travail du Centre d'Economie de la Sorbonne -2021.10

Remark 2

 2 Under symmetric information, the relation Z o = f0g holds, from the elimination of redundant assets, and Assumption A5 vanishes. Assumption A4 is made throughout Cornet-Gopalan (2010) and is a minimal requirement in most models of portfolio choice with …nancial constraints (Elsinger and Summer, 2001).

  (a) and (b) of De…nition 2 is standard and proved in Cornet-De Boisde¤re (2002, Lemma 1, p. 398).

13

  Documents de travail du Centre d'Economie de la Sorbonne -2021.10

	0 < p 0	P	i2I (x i0 e i0 ) + q	P i2I z i +	P s2S	s	P i2I p

s (x is e is ) 6 0.

Documents de travail du Centre d'Economie de la Sorbonne -2021.10

Documents de travail du Centre d'Economie de la Sorbonne -2021.10

Documents de travail du Centre d'Economie de la Sorbonne -2021.10

Documents de travail du Centre d'Economie de la Sorbonne -2021.10

(i; s) 2 I S. Then, we let i := ( is ) 2 R Si ++ be de…ned, for each i 2 I, by is = is N , for every s 2 S i nS, and is = s is N , for every s 2 S. By construction, the state prices, i := ( is ) 2 R Si ++ , de…ned for all i 2 I, support some no-arbitrage price, q 2 N A( ).

Assertion (ii) results from the de…nitions and the heuristic example above. [START_REF] Bich | Existence of pseudo-equilibria in a …nancial economy[END_REF] The existence theorem A standard economy, E, and vectors, := ( s ) 2 R S ++ and q = P s2S s V (s) are henceforth given and we prove that the state prices, , support an equilibrium:

Theorem 1 Let 2 R S ++ be given. A standard economy, E, admits an equilibrium,

along De…nitions 1 and 2 above.

The proof's main argument is the Gale-Mas-Colell (1975,1979) …xed-point-like theorem. We apply this theorem to lower semi-continuous reaction correspondences, which are de…ned on convex compact sets and formally represent agents'behaviours.

In particular, in the symmetric information case, the proof is an alternative to Cass'.

Sub-Section 3.1 derives from the economy E an auxiliary compact economy. Sub-Section 3.2 de…nes the reaction correspondences in the compact economy, to apply the GMC theorem. A so-called (with slight abuse) "…xed point" obtains. Sub-Section 3.3 derives from this …xed point an equilibrium of the economy E. Sub-Section 3.4 illustrates the result of Theorem 1 when one agent is fully informed.

An auxiliary compact economy with modi…ed budget sets

Using the notations and assumptions of sub-Section 2.3, we de…ne the price set,

Assumption A5 is no longer required and may be smoothen to Z o i + Y i Y i , for each i 2 I, whose interpretation and relevance are clear. In De…nition 2, the relations

s)g hold, for every := ( s ) 2 R S ++ , and the identity N AC = N A obtains. A no-arbitrage price, q 2 N A, is characterized by the state prices of the informed agent, say := ( s ) 2 R S ++ , and written as standard,

With one fully informed agent, the two concepts of no-arbitrage price and equilibrium asset price coincide in a clear standard way from Theorem 1. This theorem extends Cass'to asymmetric information, restricted participation and non-ordered preferences. However, its proof drops the Cass trick, that is, de…nes budget sets

Appendix

Lemma 1 9r > 0 : 8(p; q) 2 P Q; 8 [(x i ; z i )] 2 A(p; q);

Proof Let = P i2I ke i k, (p; q) 2 P Q and [(x i ; z i )] 2 A(p; q) be given. The relations

x is 2 [0; ] L hold, for every pair (i; s) 2 I S 0 , from the market clearance conditions of A(p; q). From the fact that there exists > 0, such that p i s 2 [ ; +1[ L , for every (i; s) 2 I S i nS, it su¢ ces to prove the following Assertion:

Assume, by contraposition, that, for every k 2 N, there exist

For every k 2 N, we let

The bounded sequence, fz 0k g, may be assumed to converge in a closed set, say to

hold, for every k 2 N, and imply, for every (i; s; k) 2 I S i N:

, hence, V (s) z 0k i > =k and, in the limit, V (s) z i > 0, for each s 2 S i ;

i , be given. Then, the relations,

, for each (i; s) 2 I S i , from above and from Assumption A5, whereas

The latter relation contradicts the former, kzk = 1. This contradiction proves Lemma 1.

Lemma 2 For each i 2 I [ f0g, i is lower semicontinuous.

Proof The correspondence 0 is lower semicontinuous for having an open graph. We now set as given i 2 I and := (p; q; [(x i ; z i )]) 2 := P Q ( i2I X i Z i ).

Assume that (x i ; z i ) = 2 B 0 i (p; q). Then, i ( ) = B 0 i (p; q). Let V be an open set in X i Z i , such that V \ B 0 i (p; q) 6 = ?. It follows from the convexity of B 0 i (p; q) and the non-emptyness of the open set B 00 i (p; q) B 0 i (p; q) that V \ B 00 i (p; q) 6 = ?. From Claim 3, there exists a neighborhood U of (p; q), such that

and W j := X j Z j , for each j 2 Infig, be a neighbourhood of in . Then, i ( 0 ) = B 0 i (p 0 ; q 0 ), and V \ i ( 0 ) 6 = ? hold, for every 0 := (p 0 ; q 0 ; [(x 0 i ; z 0 i )]) 2 W . That is, i is lower semicontinuous at .

Assume that (x i ; z i ) 2 B 0 i (p; q). Then, i ( ) = B 00 i (p; q) \ P i (x) Z i .

Lower semicontinuity results from the de…nition if i ( ) = ?. Assume, now, that i ( ) 6 = ?. We notice that P i is lower semicontinuous with open values, from Assumption A3, and that B 00 i has an open graph in P Q X i Z i . As a corollary, the correspondence (p 0 ; q 0 [(x 0 i ; z 0 i )]) 2 7 ! B 00 i (p 0 ; q 0 ) \ P i (x 0 i ) Z i B 0 i (p 0 ; q 0 ) is lower semicontinuous at , and i is also, from the latter inclusions.


Josselin Josselin

Josselin Massot

Nicolas Nicolas

Nicolas Crouseilles

Anaïs Anaïs

Anaïs Crestetto

Crestetto Crestetto

Exponential Runge-Kutta methods for Vlasov-Poisson equation Exponential Runge-Kutta methods for Vlasov-Poisson equation Exponential Runge-Kutta methods for Vlasov-Poisson equation Exponential Runge-Kutta methods for Vlasov-Poisson equation Exponential Runge-Kutta methods for Vlasov-Poisson equation

High order methods are needed in phase space to study lamentation.

▸ Simplified model

We study the stability problem on a simplify model:

We discretize in space with WENO method [START_REF] Jiang | E cient Implementation of Weighted ENO Schemes[END_REF], and we test various Runge-Kutta methods for the time integrator. We apply von Neumann analysis on space discretization to study ampli cation factor as a curve in complex plane .

Von Neumann analysis usually used on linear method: rst studies on stability of WENO method [START_REF] Wang | Linear instability of the fth-order WENO method[END_REF], and CFL estimate [START_REF] Motamed | On the Linear Stability of the Fifth-Order WENO Discretization[END_REF] have been done on linearized WENO.

We can try to apply von Neumann analysis to the full non-linearized WENO method. WENO method is weakly non-linear then we can compute ampli cation factor.

Ampli cation factor of WENO method (linearized or not)

We de ne stability domain of an explicit Runge-Kutta method RK( , ) as: with the number of stages, the order and the stability function of RK( , ).

▸ Automatic CFL estimate

It is possible to interpret CFL number between time integrator and space method as the biggest homothety ratio that wedges all the ampli cation factor curve into the stability domain of considered Runge-Kutta methods. We estimate numerically the CFL (respectively ) for various couples RK( , ) with WENO (respectively linearized WENO). All CFL have been approved with a long time simulation of ( * ).

Geometric CFL interpretation

▸ The chosen one

We are interested in the numerical cost of RK( , ). To compare each time integrator, we compute total energy in Vlasov-Poisson system: which is preserved in time. We propose to select the best method by considering:

Tested strategy:

Spectral scheme in , WENO method in , Lawson method of the underlying RK( , ) in time.

as a function of numerical cost in log scale

▸ Conclusion

Promising approach for Vlasov-Poisson simulation, less stages than splitting strategy.

Selected Runge-Kutta method is Dormand-Prince RK(6,5), this is an embedded Runge-Kutta method (adaptative stepsize, relaxation of the CFL).

Extension to multi-dimensional Vlasov-Poisson is easier than splitting strategy.

Simulation of the bump on tail test case :

Bump on tail simulation ( on the left, on the right)

(x, v) f = f (t, x, v) E = E(t, x) (VP) { ∂ f + v∂ f + E∂ f = 0 t x v ∂ E = f dv -1 x ∫ R x ∈ Ω ⊂ R, v ∈ R (x, v) ( * ) ∂ u + t a∂ u = x 0. C s n D = (s,n) z ∈ C : |p (z)| ≤ 1 { (s,n) } s n p (s,n) s n σ σ ℓ s n Δt ≤ Δx a σ σ ℓ σ x ∂ + t f ^iκv + f ^= (E∂ f ) v 0 ∂ u + t Lu + N (u) = 0. ∂ (e ) + t iκvt f ^e = iκvt (E∂ f ) v 0 ∂ (e u) +

  of the form Next we use an exponential integrator (Duhamel formula): of the form With this exponential form we can use a classical Runge-Kutta method, in this context we call them Lawson methods or Integrating factor Runge-Kutta methods [4]. Stability function of Lawson schemes can be expressed in terms of the underlying Runge-Kutta method: with so stability domain is the same. Our numerical CFL study, on simpli ed model, still works on Duhamel formulation of Vlasov-Poisson.

1 ▸

 1 Vlasov-Poisson equationVlasov equation plays an important role in the modeling of plasma physics, it is a

	non-linear transport in	of a density distribution	; coupled
	with Poisson equation for estimate electric eld	:
		Example of lamentation process in a solution


Nathan Cogné
email: nathan.cogne@univ-rennes1.fr

Kerry Gallagher

Some comments on the effect of uranium zonation on fission track dating by LA-ICP-MS

Keywords: Fission track dating, LA-ICP-MS, single grain age dispersion, Uranium zonation 1-Introduction

The use of LA-ICP-MS for uranium determination in the fission track dating technique is becoming increasingly popular because of several advantages over the classical external detector method and a variety of analytical and statistical protocols have been developed. However, two important issues remain unresolved in the context of the LA-ICP-MS approach (i) how to best deal with low track density (rs) samples, and (ii) does a correlation between age and uranium content (or eU) reflect an annealing dependence or not? To assess the impact of the analytical methodology on these issues, we compare the multi-spot and more classical single spot methods on samples of known ages, variably zoned and / or with low track densities. To make the comparison we use an approach, implemented in a Python script, that randomly samples our multi-spot ICP-MS data to choose a single U measurement per grain, simulating the single spot approach. We then calculate the central age, p(c 2 ) and dispersion of the simulated single spot analysis and repeat this 2000 times. Our results show that the multi-spot approach is robust for low rs and zoned samples, yielding both accurate and precise results without over-dispersion.

Additionally, our random sampling approach shows that a single spot measurement can induce an overdispersion coupled to a relationship between single grain age and U content. This is at least partly attributable to zonation that creates a mismatch between the U in the counted area and the spot-measured U. Therefore, we recommend that if over-dispersion is observed for basement samples, when one typically expects a single age population, then multiple spot analysis should be carried out to assess if the excess dispersion is linked to undetected zoning and / or laser spot misplacement rather than to U dependent annealing behaviour.

calibration factor, z (see [START_REF] Hurford | The zeta age calibration of fission-track dating[END_REF] for more details). This factor is estimated by analyzing samples of known ages and recasting the age equation in terms of z as an unknown. Despite the advantages of this coupled approach, the EDM suffers from some setbacks including (i) the need for three different counts of tracks: in the mineral, the detector and the dosimeter and (ii) the manipulation of hazardous hydrofluoric acid to etch the muscovite detector (iii) increasingly difficult access to low energy neutron irradiation to induce fission tracks in the detector.

Therefore, during the early 2000's researchers explored the use of LA-ICP-MS to determine U concentration of the mineral to be dated. [START_REF] Hasebe | Apatite fissiontrack chronometry using laser ablation ICP-MS[END_REF] demonstrated the feasibility of this technique. During the past decade several protocols have been developed for FT dating, mainly on the calcium-phosphate mineral, apatite. Building on [START_REF] Donelick | Apatite Fission-Track Analysis[END_REF][START_REF] Chew | Combined apatite fission track and U-Pb dating by LA-ICP-MS and its application in apatite provenance analysis[END_REF][START_REF] Cogné | LA-ICP-MS apatite fission track dating: a practical zeta-based approach[END_REF] proposed a z-based protocol, while other authors have suggested using an absolute approach (e.g. De [START_REF] De Grave | Late Palaeozoic and Meso-Cenozoic tectonic evolution of the southern Kyrgyz Tien Shan: Constraints from multi-method thermochronology in the Trans-Alai, Turkestan-Alai segment and the southeastern Ferghana Basin[END_REF][START_REF] Soares | Novel calibration for LA-ICP-MS-based fission-track thermochronology[END_REF][START_REF] Gleadow | The Fish Canyon Tuff: a new look at an old low-temperature thermochronology standard[END_REF]. Importantly, statistical methods for processing the LA-ICP-MS data have been developed recently [START_REF] Vermeesch | Statistics for LA-ICP-MS based fission track dating[END_REF](Vermeesch, , 2019) ) and we incorporate these into the analysis we present later.

It has also been shown that laser spot ablation can also produce accurate U-Pb and trace element data simultaneously during the analysis required for AFT data (e.g. [START_REF] Chew | Combined apatite fission track and U-Pb dating by LA-ICP-MS and its application in apatite provenance analysis[END_REF], Chew et al., 2016[START_REF] Cogné | LA-ICP-MS apatite fission track dating: a practical zeta-based approach[END_REF]. This is a big advantage of the LA-ICP-MS method as the additional data can help to better understand the AFT age data in both the analytical and geological contexts. In contrast to the EDM method, application of the LA-ICP-MS approach can be problematic in case of samples zoned in U. The EDM method produces an approximate map of U concentration and typically both spontaneous and induced tracks are counted on the same area, implicitly allowing for zoning to some degree.

For LA-ICP-MS only spontaneous tracks are counted and typically only a limited part of the counted area is analyzed for U concentration by the ablation spot. Although zonation in the z axis direction can be dealt with (Chew andDonelick, 2012, Cogné et al., 2020), in-plane or horizontal zonation is less readily resolved by the routine (i.e. single ablation spot) methods used in AFT dating. For samples with high spontaneous track densities (rs) the zonation is often visible in the spatial distribution of spontaneous tracks. In that case, an analyst can select a counting area that is unzoned and/or close to the shape and size of the laser spot. However, dealing with potential zoning becomes more problematic for samples with low rs. If an analyst selects a small area that mimics the laser spot then the number of counted tracks is likely to be low and the uncertainty on a single grain age (SGA) increases. Two methods have been proposed to overcome the zoning/low rs issue, both requiring counting tracks on a large area. [START_REF] Vermeesch | Statistics for LA-ICP-MS based fission track dating[END_REF] suggested using multiple laser spots (multi-spot) to improve understanding of the U distribution spatially. More recently, [START_REF] Cogné | LA-ICP-MS apatite fission track dating: a practical zeta-based approach[END_REF] proposed mapping the entire grain with a raster. Both approaches, however, are time consuming and more expensive compared to single spot analysis. Ideally, they then should be used only when necessary for example if a large SGA dispersion is detected.

A second issue with the use of LA-ICP-MS is the commonly larger dispersion of SGA compared to those from EDM. For example, [START_REF] Ketcham | Reproducibility of thermal history reconstruction from apatite fission-track and (U-Th)/He data[END_REF] reported an interlaboratory experiment which consisted of blind dating two samples S1 and S2. Of thirteen analysts, five used LA-ICP-MS and eight used EDM. The mean dispersion for EDM was 2.5% for S1 and 2.1% for S2 while for LA-ICP-MS the mean dispersion was 17.3% for S1 and 10.4% for S2. Regarding the canonical c 2 test, that is used to assess the possibility of single versus multiple or over-dispersed age populations in a given sample, none of the EDM data sets failed the test (p(c 2 ) < 0.05) while three out of five of LA-ICP-MS dataset failed for S1 and two out of five for S2. Such over-dispersion can be problematic because it could be used to infer multiple age populations that do not really exist. In this context some authors have suggested that the dispersion in SGA can be explained by the U content, or eU content (eU = [U] + 0.235 [Th]) (e.g. [START_REF] Fernie | Thermochronological insights into reactivation of a continental shear zone in response to Equatorial Atlantic rifting (northern Ghana)[END_REF]McDannell et al., 2019).

These authors demonstrated that the SGA from LA-ICP-MS are often older for lower U grains and McDannell et al. (2019) argue for a radiation damage control on annealing. Previously, Hendricks and Redfield (2005) proposed a similar a-radiationenhanced-annealing (REA) for EDM SGA data from the Fennoscandian shield.

However, [START_REF] Kohn | A reappraisal of low-temperature thermochronology of the eastern Fennoscandia Shield and radiation-enhanced apatite fission-track annealing[END_REF] reconsidered the question using a different set of data from various cratonic settings and concluded that there was no link between eU and AFT age. This question is perhaps still open for the EDM, but has also become topical for the LA-ICP-MS approach in the light of new datasets.

In this contribution we aim to test the multiple spot approach of Vermeesch (2017) on samples of known age and also zoned and/or low rs samples. We aim to assess if this approach leads to decreased SGA dispersion in case of undetected zonation. The second objective is to investigate if an inferred relationship between U content and SGAs could be explained in part by such zonation.

2-Samples and Methods

Samples

We selected two apatite age standards: Fish Canyon Tuff (FCT -28.1 ± 0.1 Ma, Boenhnke and[START_REF] Boehnke | A meta-analysis of geochronologically relevant half-lives: what's the best decay constant[END_REF]Mount Dromedary (MtDrom -98.5 ± 0.5 Ma, McDougall and[START_REF] Mcdougall | Calibration of GA1550 biotite standard for K/Ar and 40 Ar/ 39 Ar dating[END_REF] because both are well characterized in terms of their AFT ages, often being used as age standards. We selected eight additional samples, all previously dated with AFT (EDM and/or LA-ICP-MS) from crystalline basement from various tectonic settings (high and low exhumation rate) with known (because of high track densities) or possible (unidentifiable because of low track densities) U zoning, see Table 1 for details. Because all samples were previously dated, we can test the accuracy of the multi-spot approach. The sample M1 was chosen because of its visible zonation due to the high track density. However, this sample is treated as if the zonation was not visible (i.e. spots for the multi-spot approach were randomly positioned on the counted area). The motivation for doing this was to simulate what could happen (increasing dispersion, SGA-U content correlation) when one has not detected the zonation. The other samples were chosen because of the possible zonation and / or low track density to assess how an unknown zonation can affect the results of single versus multi-spot analysis.

Methods

All analyses were performed at the GeOHeLiS analytical platform (Univ.

Rennes, France) and follow the protocol of [START_REF] Cogné | LA-ICP-MS apatite fission track dating: a practical zeta-based approach[END_REF]. All samples were mounted in a 1.5 mm thick epoxy disc. The apatites were etched using a 5.5 M HNO3 at 21°C for 20s. The grain mounting and etching are similar to the protocol described by [START_REF] Donelick | Apatite Fission-Track Analysis[END_REF]. Spontaneous fission track counting was carried out using a Zeiss AxioImage M1 equipped with an automated stage system using TrackWorks software at a magnification of 1000´. The U/Ca ratios, determined using an ESI NWR193UC Excimer laser coupled to an Agilent 7700x Q-ICP-MS, were used to calculate the ages (see [START_REF] Cogné | LA-ICP-MS apatite fission track dating: a practical zeta-based approach[END_REF] for details on the protocol). The laser spot size was 30 µm with a repetition rate of 5 Hz and a fluence of 4 J/cm 2 . All instrumental conditions are summarized in the supplementary table S1.

For each grain the spontaneous track counting area was chosen to be large enough to accommodate three to four ablation spots when possible. For the sample M1 a smaller area was also delimited and counted to compare a "guided" single spot approach with the multi-spot approach on the same grains. Guided here means that the counted area is close in size and shape to the ablation spot and selected from a region considered not too zoned, the low zonation being assessed on the basis of the reasonably high and uniform track density (see Figure 1). For sample HIM 622/244 the whole grain area was counted irrespective of the size of the grain. We did this because the track densities in this sample are so low that not taking into account the whole area of the grains would tend to increase the estimated density and lead to older SGA. The multi-spot approach of Vermeesch (2017) consists of calculating the mean and variance U/Ca ratio from individual spots. We used IsoplotR [START_REF] Vermeesch | IsoplotR : A free and open toolbox for geochronology[END_REF], in which the approach is implemented, to calculate central age, dispersion and p(c2) for each sample. Pooled ages and associated uncertainties are calculated using the spreadsheet discussed in [START_REF] Cogné | LA-ICP-MS apatite fission track dating: a practical zeta-based approach[END_REF] with single grain mean U/Ca ratios and associated uncertainties calculated according to [START_REF] Vermeesch | Statistics for LA-ICP-MS based fission track dating[END_REF].

Routinely, the U analysis for LA-ICP-MS FT dating is performed using only one spot (e.g. [START_REF] Cogné | The thermal history of the western Irish onshore[END_REF][START_REF] Cogné | Measuring plumerelated exhumation of the British Isles in Early Cenozoic times[END_REF]. As previously explained, application of this protocol to low rs samples is problematic as it is not always obvious where to best put the small ablation spot in the larger counted area. To simulate an effectively random choice of ablation position, we used random sampling of multi-spot U/Ca measurements to select one value for each grain and then calculate the equivalent central age, dispersion and p(c2) of the sample. We repeated this sampling 2000 times, the goal being to compare what a standard single spot ablation approach would have yield to the multi-spot approach in terms of accuracy and precision of the central age and the dispersion of the SGA.

3-Results

The results are summarized in Table 2 and single grain data are provided as supplementary data (Table S2). All multi-spot samples yielded similar pooled and central ages and have p(c 2 )>0.05. A large part of the samples actually exhibits high p-value (>0.8) which might indicate an overestimation of the SGA uncertainty. The low dispersion of SGA indicates a high probability of single age population, as it is expected for bedrock sample. Guided single spot results for sample M1 also show p(c 2 )>0.05, although the dispersion of the SGA is larger than for multi-spot results (Figure 1b and1c). All ages are indistinguishable at the 2s level of the published ages except for HIM622/244 (Table 1, Figure 2). For that sample, the EDM reference age is 0.03±0.04 Ma [START_REF] Treloar | Geochronological constraints on the evolution of the Nanga Parbat syntaxis, Pakistan Himalaya[END_REF], somewhat younger than the age in the present study. However, our age is similar to that given by [START_REF] Cogné | LA-ICP-MS apatite fission track dating: a practical zeta-based approach[END_REF] also with LA-ICP-MS. We will discuss this difference in more detail below. The relative central age uncertainties (presented as 2s throughout the text) range from 5-7% for 'normal' to high track density samples (MtDrom, FC1) to 25-33% for very low track density samples (SG19, HIM 622/244), which is in the precision range of fission track dating. The radial plots for the multi-spot data are shown on Figure 3, except for sample M1 that is presented in detail on Figure 1. Single grain dots are color-coded according to their mean U/Ca content.

To assess the degree of grain U/Ca zonation (Z) for a given sample we calculate the relative uncertainty of the mean U/Ca ratio for each grain as calculated by [START_REF] Vermeesch | Statistics for LA-ICP-MS based fission track dating[END_REF], and then we take the average and standard deviation of that ratio over all grains (equation 1).

𝑍 = ! " ∑ % # ! "# $ % $ %& & ' (((((((( & " )*! (1)
where N is the number of grains of the sample. If this average is low, then the intra grain zonation is rather limited, even if the U concentrations in each grain may be very different. Conversely if this value is high, this indicates a high proportion of zoned grains in the sample.

In figure 4 Figure 5 shows the radial plots for a single random draw for each sample, except sample M1 that is already presented in detail on Figure 1. Single grain ages are color-coded according to the U/Ca content. This figure shows that, even for less zoned samples (e.g. FCT, SG10, SG19, FC1), the dispersion of random draw single spot SGA increases relative to the multi-spot data. The increased dispersion will potentially lead to failure of the c 2 test. This dispersion is also associated with a generally more pronounced apparent inverse correlation of the SGA with U content (see supplementary material (Fig. S3) for plots of eU v age for the multi-spot and single spot data).

4-Discussion

Age accuracy and precision with multi-spot approach

The central and pooled ages estimated using the multi-spot approach reproduce the published literature ages well (Table 1 and Fig. 2), except the EDM age of HIM622/244 [START_REF] Treloar | Geochronological constraints on the evolution of the Nanga Parbat syntaxis, Pakistan Himalaya[END_REF] which is discussed below. The precision of SGAs using the multi-spot approach tends to be lower than when using a single spot (see for example guided single spot vs multi-spot data for M1, Fig. 1b and1c). This effect seems to be larger for more highly zoned samples (compare Fig. 3 and5), such that the precision on SGA with multi-spot can half that when using a single spot.

For the single spot approach, the uncertainty on the U/Ca ratio comes directly from a single ICP-MS measurement, while for the multi-spot approach it is calculated from the different measurements of U/Ca on the same grain. Consequently, the more zoned a sample is, the more variability of the individual U/Ca will increase, as will the overall variance. Indeed, where the uncertainty (2s) of single U/Ca measurement was usually ca. 7% during this study, the uncertainty of mean U/Ca for a grain rises to ca 60-70% for zoned samples or even 130% for HIM622/244. Therefore, while the multi-spot approach gives a better understanding of the U/Ca distribution, the resulting precision decreases strongly if the zonation is important. The decreasing precision on the U/Ca ratio may induce an under-dispersion of the age data, leading in turn to a relatively high p-value. This effect is potentially due to an over estimation of U variability because we only use few measurements per grains (i.e. 2 to 4 spots).

In any case the under dispersion remains minor and we thus suggest that this does not preclude the use of the multi-spot approach.

In a low rs sample it is not possible to infer the zonation before ablation.

Therefore, reliably determining the central age of a sample using a single spot approach would require using a counting area that mimics exactly the ablation spot.

For example, a sample with rs ca. 1e5 tr/cm 2 , a precision of U/Ca measurement of 7% and a precision of zeta of 1.5% (typical values in this study), 30 grains with a spot of 30 µm diameter (typical number of counted grains and spot size) would result in a central age uncertainty of ca. 45%. With a rs of 1e4 tr/cm 2 the uncertainty rises to ca.

140%. Our samples with comparable rs have central age uncertainties of ca. 15 to 25% using the multi-spot approach. To achieve similar precision with a single spot approach, one would need to ablate ca. 100 to 1000 grains, for rs of 1e5 and 1e4 tr/cm 2 respectively, which is time-consuming, and in turn could increase the risk of over-dispersion by increasing the probability of using outliers grains.

It is thus preferable to use the multi-spot approach than the single spot approach for low rs samples. In addition to delivering a better understanding of U distribution in each grain, it is more efficient in that we obtain a more reliable and precise age. In high rs samples, however, one might prefer to use a guided single spot approach instead of using multi-spot as it is more precise, even with a lower number of tracks counted, and faster, as demonstrated in the comparison for our M1 discrepant age, we reiterate that the central ages determined in this study using the multi-spot approach agree well with the vast majority of published ages.

The effects of U zonation on LA-ICP-MS FT dating

In the examples of our random draws (Fig. 5), we see that high-U grains (red colours on Fig. 5) tend to have younger ages while the older ages are from low-U grains (yellow colours). Some correlation of U and SGA is expected based on statistical arguments [START_REF] Pearson | Mathematical contributions to the theory of evolution.-On a form of spurious correlation which may arise when indices are used in the measurement of organs[END_REF]. The fission track age equation can be approximated to a good precision by a linear equation in which the age (t) is proportional to the ratio of rs to U concentration following (eq.5 of [START_REF] Vermeesch | Statistics for LA-ICP-MS based fission track dating[END_REF]:

𝑡 = ! + ' 𝑙𝑛 *1 + + ' + ( ," ) [ $ *+, ]/ ) 01 -≈ ,2 ) + ( [ $ *+, ]01 ∝ 2 ) [ $ *+, ] (2)
Given this we expect t to be negatively correlated with [U] to some extent (we can imagine plotting

2 )

[$] 𝑣 [U] ). Thus, even on multi-spot data a correlation may exist but it is clearly more important when looking at the single spot data (Fig. S3), and it becomes visible on radial plots.

Moreover, when using single U/Ca measurement the dispersion increases, and p(c 2 ) decreases, compared to the multi-spot results (Fig. 4). Therefore, an inverse correlation between U/Ca content and SGA appears along with overdispersion. This is illustrated at the scale of a grain in Figure 6a. This radial plot

shows SGA for one grain of sample M1. The multi-spot age includes the total number of tracks counted on the larger blue area illustrated on Fig. 1a. As the number of spontaneous tracks appropriately reflects the mean U/Ca measured on the 3 spots, the resultant multi-spot single grain age is in line with the central age of the sample.

The "guided" single count is similar to the multi-spot age, but more precise as explained above. Finally spot 1, spot 2 and spot 3 ages use the large area for the spontaneous track count and only one U/Ca measurement. The calculated ages are more precise but more dispersed compare to the central age as the single U/Ca measured is not representative of the whole counted area. Several similarly unrepresentative measurements would lead to over-dispersion. Furthermore, the ages are inversely related to the U content. To summarise, both the over-dispersion and inverse correlation are attributable in this example to single spot U/Ca measurements not being representative of the mean in the region where the spontaneous tracks were counted for the given grain.

At the scale of the whole M1 sample we also plotted all the possible SGAs depending on which spot is used (i.e. 1 to 4 ages per grain, using the same count and area but the different U/Ca measurements, Fig. 6b). This clearly highlights the tendency for higher U/Ca to be associated with younger and generally more precise ages (the bottom right part of the radial plot). When considering the ensemble of single spots on figure 6b an apparent inverse correlation between U content and age is obvious. However this is an artefact of the spot position, as the multi-spot and "guided" single spot data (figure 1 b,c) do not show this behaviour. While the zonation is visible on this sample, the same effect would appear on sample on which the zonation would have remained undetected.

Random draw data for Mt Dromedary or FC1 show that, even on relatively unzoned and 'normal' track density samples, misplacing a spot on only few slightly zoned grains can results in an over-dispersion from time to time. On MtDrom especially, about 60% of our random draws failed the c 2 test despite a low degree of U zoning. As increasing precision of single grain ages increases the possibility of inferring over-dispersion statistically, this is likely due to the relatively high track densities in this sample. As detailed above, over-dispersion of SGA in a sample can also arise from an unrepresentative determination of the U content of the counted area. Thus, with a larger number of tracks the SGA becomes more precise and smaller variations of U/Ca can increase the dispersion.

Therefore, over-dispersion of SGA can be the result of older or younger ages relative to the true age. However, as summarised in Table 2, sample central ages calculated from random draws for single spots are undistinguishable from the central ages determined using the multi-spot data at the 2s level (except for 5 draws out of 2000 on M1 sample). Even at the 1s level the vast majority of the central ages (almost 100% of the random draws for most samples, and >90% for the others) are undistinguishable from the multi-spot central ages. This is likely the result of compensation between too old grains and too young grains for the single spot data.

This result, based on the experiments presented here, suggests that although the dispersion increases slightly the uncertainty on the single spot central age, it has minimal impact on its accuracy. In case of extreme core rim zonation, the problem could be enhanced but such configuration is probably rare. This is particularly important, because it implies that LA-ICP-MS ages for basement samples that show a high degree of dispersion likely remain accurate, the dispersion potentially coming from undetected zoning and / or spot misplacement during laser ablation.

However, the U-SGA relationship reported in some publications (e.g. [START_REF] Fernie | Thermochronological insights into reactivation of a continental shear zone in response to Equatorial Atlantic rifting (northern Ghana)[END_REF], McDannell et al., 2019) could still reflect other controls. As the factors that determine track annealing kinetics are crucial to understand and interpret FT data, it is clearly of primary importance to know if U content is an influential factor or not. In assessing this, we suggest here that when a relationship between U content and SGA is apparent in data collected using one ablation spot per grain, a second LA-ICP-MS session should be performed to see if this is really linked to U content or may due to initially undetected zoning.

Recommendations when dealing with low rs samples

When the spontaneous track density is too low to visualise possible zonation, three different approaches are possible, (i) counting an area that exactly mimics the ablation spot, (ii) using the mapping approach of Ansberque et al. ( 2020) and (iii) using the multi-spot approach of [START_REF] Vermeesch | Statistics for LA-ICP-MS based fission track dating[END_REF]. The first solution is the fastest but is not precise as it limits the size of the counting area and consequently reduces the number of countable tracks. The two other approaches are more time consuming, limiting output to roughly two basement samples for a day of ICP-MS analysis. Comparing the samples that were presented in both Ansberque et al.

(2020) and our study (RM 13, HIM622/244 and FCT), it appears that the mapping approach yields more precise SGA data, and so a more precise central age. This is mostly due to the U/Ca measurement protocol and counting statistics. The mapping approach relies on averaging U/Ca for all pixels of the grain map. Given a pixel is about 3.5 µm wide, one ends up with a few hundred pixels for a grain and thus a few hundred U/Ca ratios. With the multi-spot approach, we typically use 3 to 4 spots per grain. It is obvious that the standard error of the mean U/Ca concentration will be larger with four counts than with a few hundred and so leads to a less precise age estimate. However, the mapping approach necessitates equipment and materials that are not currently available in all LA-ICP-MS laboratories. Additionally, once one has decided to use the mapping approach, it has to be done for the whole sample and this choice implies half a day of ICP-MS time. In contrast the multi-spot approach is more practical and flexible for basement samples as one could decide to do only one spot during a first session. If the single grain dispersion is low then there is no obvious need to perform more analysis. Alternatively, if the dispersion is high, one should do more spots, at least on the grains that are responsible for the overdispersion, to assess if the dispersion reflects a zoning issue or could be a sign of differential annealing behaviour. For detrital samples, the preferred procedure is harder to define as we may anticipate multiple age populations and different annealing effects, typically requiring analysis of a large number of grains (c. 120 grains per sample) to characterise adequately the distribution of SGA. Using the multi-spot approach on for each grain would be time consuming but in detrital samples, one cannot generally rely on the SGA dispersion to indicate analytical issues. In such cases it is possible to try and initially characterise different populations using other data acquired simultaneously, such as U-Pb ages or trace element concentrations (e.g. [START_REF] O'sullivan | The trace element composition of apatite and its application to detrital provenance studies[END_REF], Westerweel et al., 2020). Then, at least in principle, different populations or components can be isolated and the analyst can decide if more spots are needed on some grains.

5-Conclusion

In this study we assessed the multi-spot approach proposed by Vermeesch (2017) in terms of its ability to decrease SGA dispersion in low rs and / or zoned samples. It also increases the precision on the central age by allowing the counting larger areas. Our results show that this method is efficient and should be used when SGA over-dispersion is apparent for a basement sample. Moreover, we show that an inverse correlation between SGA and U content coupled to over dispersion can be induced due to zoning and single spot U contents being unrepresentative of the whole grain. This means that U zoning could give a potentially false impression that high/low U content leads to faster/slower annealing. Therefore, it is important to assess if multi-spot datasets continue to give such relationships in slowly cooled basement samples to better understand and better interpret FT data.

6-Acknowledgements

NC thanks the following people for sharing their samples: Marc Jolivet for SG9, SG10 and SG19, Claire Ansberque for RM13 and HIM 622/244, Simon Nachtergaele for FC1, Stéphanie Brichau for M1 and Raymond Donelick for Mt Drom and FCT. We thank S. Glorie for his comments that improved the manuscript. We also thank P. Vermeesch for his helpful review and especially for his comments (including eq. 2 and the reference to Pearson 1986) about the expected U -SGA correlation. For all samples the SGA dispersion increases compared to the multi-spot data (Fig. 3). In some samples the SGA seem correlated to U/Ca content (color-coded for each grain) -younger and often more precise ages with higher U/Ca.

However, all central ages are undistinguishable from those estimated with the multispot data. CA = Central age Figure 6: (a) Radial plot of a single grain age from M1 grain 30 data calculated using multi-spot, "guided" single count (tracks counted in the white area in Fig. 1a), and spot 1, spot 2, spot 3, that use tracks counted in the blue area in Fig. 1a and the measured U/Ca ratio from spot 1, 2 or 3 respectively. Spot 1 samples more or less the average U/Ca ratio. We see that when the ablation spot hits a higher than average U area, the age is younger (spot 3) while when it hits a lower U area the age is older (spot 2). (b) Radial plots of all the possible SGA, implying an apparent inverse relation between U/Ca content and SGA. See text for further discussion.

  we summarise the results from the 2000 random draws of one of the multi-spot U/Ca ratios per grain. For highly zoned samples (M1, SG9, HIM622/244), most of the draws lead to a large SGA dispersion (averaging >40%), involving failure of the c 2 test 80-100% of the time. In contrast, samples with little zoning (FCT, SG10, SG19, FC1) have low dispersion, rarely failing the c 2 test (0-10% of the time). Two moderately zoned samples (Pb1 and RM13) show intermediate behaviour (c 2 test 25-60% failure rate). An exception to this trend is the relatively uniform MtDrom sample with draws that often (~60%) fail the c 2 test. This case will be further discussed below.

Figure 1 :

 1 Figure 1: Example of grain area determination and data for M1 sample (a) Example

Figure 2 :Figure 3 :

 23 Figure 2: Plot of the central ages calculated in this study versus literature central

Figure 5 :

 5 Figure 5: An example of the random sampling to simulate single spot data for the

  Figure 1

  Figure3

  partly explain the older ages obtained using LA-ICP-MS. Firstly, the zero track grains represent half of the grains counted in this sample.[START_REF] Cogné | LA-ICP-MS apatite fission track dating: a practical zeta-based approach[END_REF] protocol, following[START_REF] Vermeesch | Statistics for LA-ICP-MS based fission track dating[END_REF], does not attribute a zero Ma age to zero spontaneous track grains, while this was most likely the case for[START_REF] Treloar | Geochronological constraints on the evolution of the Nanga Parbat syntaxis, Pakistan Himalaya[END_REF]. If we assign a zero age to these zero track grains then the central age falls to 0.86 ± 0.76 Ma; closer, but still different, to the EDM age, and closer to our pooled age of 0.45±0.21 Ma (Table1). Secondly it is possible that we selected more grains with tracks than[START_REF] Treloar | Geochronological constraints on the evolution of the Nanga Parbat syntaxis, Pakistan Himalaya[END_REF], which can explain the older age we obtain or equivalently[START_REF] Treloar | Geochronological constraints on the evolution of the Nanga Parbat syntaxis, Pakistan Himalaya[END_REF] selected more zero track grains. The actual number of zero track grains clearly impacts the final central or pooled ages. A random draw of 20 grains from the 50 actually measured in this study, again repeated 2000 times,

	sample.
	HIM622/244 is the only sample where we do not adequately reproduce the
	EDM central age. However, our age is similar to that determined by Ansberque et al.

(2020), also obtained with LA-ICP-MS. These authors stated that the reasons for discrepancy remain unclear. While we agree with that statement, we suggest two possible reasons to and using a zero Ma age for the zero track grains reproduced the central age of

[START_REF] Treloar | Geochronological constraints on the evolution of the Nanga Parbat syntaxis, Pakistan Himalaya[END_REF]

ca. 13% of the time. Therefore, irrespective of this single

Table 1 :

 1 Vermeesch, P., 2019. Statistics for fission-track thermochronology, in: Fission-track thermochronology and its application to geology, M. Malusà and P. Fitzgerald Eds., Springer, Berlin. 393pp. Samples used in this study.

	Westerweel, J., Licht, A., Cogné, N., Roperch, P., Dupont-Nivet, G., Kay Thi, M.,
	Swe, H. H., Huang, H., Win, Z., Aung, D.W., 2020. Burma Terrane collision and
	northward indentation in the Eastern Himalayas recorded in the Eocene-Miocene
	Chindwin Basin (Myanmar). Tectonics, 39, e2020TC006413.
	Woodhead, J., Hellstrom, J., Hergt, J., Greig, A. and Maas, R., 2007. Isotopic and
	elemental imaging of geological materials by laser ablation Inductively Coupled
	Plasma mass spectrometry. Journal of Geostandards and Geoanalytical
	Research 31, 331-343.
	Figure caption

Table 2 :

 2 Results for fission track analysis and random draws. Detailed grain data are provided in supplementary material.

  Defined as the average ± standard deviation of the relative standard deviation of U/Ca ratio of each grain (b) Percentage of random draw that yielded the same central age than multiple spots within 2σ /within 1σ (c) Percentage of random draw that have p(χ2) >0.05 (d) Average and 95% asymetric interval of dispersion of single grain age of random draw

	Sample Name and Location		lithology	Reported age	Characteristics	central age±2σ	Reference pooled age±2σ		Uranium	Random	Random	Random
	Fish Canyon tuff (USA) nb grains Ns		Tuff Area (cm2) ρs (track/cm2) 28.1 ± 0.1 U/Ca	Age standard p(χ2) Dispersion (%)		Boenhnke and Harrison, 2014 (Ma) (Ma) reference age±2σ (Ma)	variation (a)	Ages (%) (b)	p(χ2) (%) (c)	dispersion (d)
	M1 (Single spot) Mt Dromedary (NSW, 30 754 M1 (Multi spots) 30 1642 Australia) RM13 40 244		6.33E-04 Granite 1.63E-03 2.29E-03	1.19E+06 98.5 ± 0.5 1.03E-01 1.00E+06 7.25E-02 1.06E+05 6.94E-03	0.06 Age standard 13.2 1 0 0.87 0		10.3 ± 0.9 McDougall and Wellman, 2011 10.0 ± 0.8 12.5 ± 2.2* 10.0 ± 1.5 9.6 ± 1.8 12.5 ± 2.2* 11.2 ± 1.7 10.3 ± 1.6 9.9±0.6°0.22 ± 0.02 -0.37 ± 0.05	99.9 / 92.9 100 / 99.9	0 37.1	0.53 [0.36-0.66] 0.26 [0.00-0.44]
	FC1 (Duluth Complex, USA) SG9 30 145 SG10 29 70 SG19 30 62 Pb1 28 228	1.40E-03 Anorthositic serie 1.79E-03 1.82E-03 9.95E-04	proposed reference 0.97 0 material; low exhumation 1.50E-02 861 ± 29 1.03E+05 °* 3.91E+04 6.45E-03 0.93 0 3.40E+04 4.75E-03 1 0 rate 2.29E+05 7.33E-04 0.94 0 214.8 ± 34.0 5.0 ± 1.1 5.2 ± 1.3 Iwano et al., 2019 4.7 ± 1.0 4.4 ± 1.1 5.5 ± 1.4 4.9 ± 1.3 203.3 ± 34.6	5.7 ± 0.7* 4.5 ± 0.6* 4.7 ± 0.7* 220.8 ± 52.0°0.19 ± 0.06 0.31 ± 0.06 0.14 ± 0.02 0.12 ± 0.02	100 / 98.5 100 / 100 100 / 100 100 / 99.6	19.9 100 100 74.8	0.44 [0.00-0.74] 0.00 [0.00-0.12] 0.00 [0.00-0.00] 0.19 [0.00-0.52]
	M1 (Mykonos, Greece) Monzogranite 12.5 ± 2.2* FC1 20 1862 7.60E-04 2.45E+06 1.83E-03 High track density; visible 0.92 0 882.4 ± 44.4 zonation; high HIM622/244 50 30 8.21E-03 3.65E+03 5.56E-03 0.2 37.5 1.2 ± 0.5 Brichau et al., 2008 879.7 ± 46.6 0.45 ± 0.21 0.03 ± 0.04*; 0.8 ± 0.3°0.64 ± 0.04 861 ± 29 °* 0.04 ± 0.01 Fish Canyon Tuff 28 236 1.23E-03 1.92E+05 4.81E-03 0.84 0 29.0 ± 4.0 27.7 ± 3.8 28.1 ± 0.1 0.09 ± 0.01	100 / 99.8 100 / 100 100 / 100	91.2 0 100	0.04 [0.00-0.11] 0.62 [0.55-0.74] 0.01 [0.00-0.14]
	MtDromedary	20	945		7.38E-04	1.28E+06	9.13E-03	exhumation rate 0.63 0		96.5 ± 7.0	95.8 ± 7.1	98.5 ± 0.5	0.06 ± 0.01	100 / 99.5	40.9	0.11 [0.00-0.19]
	RM13 (Paros, Greece) ° Reference Age determined with LA-ICP-MS * Reference Age determined withEDM (a)	Paragneiss	9.9±0.6°l	ow track density; known zonation; high exhumation rate	Henrichs et al., 2018; Ansberque et al., 2020
								ow track density;
	Pb1 (Wales)			Granite	220.8 ± 52.0°l possible zonation; low		Cogné et al., 2016
								exhumation rate
	HIM 622/244 ( Nanga Parbat Massif, Pakistan)		Gneiss	0.03 ± 0.04*; 0.8 ± 0.3°v ery low track density; possible zonation, high exhumation rate	Treloar et al., 2000; Ansberque et al., 2020
	SG9 (Danba Dome, Eastern Tibet)		Gneiss	5.7 ± 0.7*	low track density; exhumation rate possible zonation, high	Jolivet et al., 2015
	SG10 (Danba Dome, Eastern Tibet)		Leucocratic dyke	4.5 ± 0.6*	low track density; exhumation rate possible zonation, high	Jolivet et al., 2015
	SG19 (Danba Dome, Eastern Tibet)		Granite	4.7 ± 0.7*	low track density; exhumation rate possible zonation, high	Jolivet et al., 2015
	° Reference Age determined with LA-ICP-MS
	* Reference Age determined withEDM