Guy Meunier
email: guy.meunier@ivry.inra.fr

Jean-Pierre Ponssard
email: jean-pierre.ponssard@polytechnique.edufrancisco.ruiz-aliseda@polytechnique.edu

Francisco Ruiz-Aliseda

Antitrust versus industrial policies, entry and welfare *

Keywords: JEL-Classification: L1, L4, L5 entry, industry dynamic, antitrust policy, divestiture

In industries with large sunk costs, the investment strategy of competing firms depends on the regulatory context. We consider ex-ante industrial policies in which the sunk cost may be either taxed or subsidized, and antitrust policies which could be either pro-competitive (leading to divestiture in case of high ex-post profitability) or lenient (allowing mergers in case of low ex-post profitability). Through a simple entry game we completely characterize the impact of these policies and examine their associated dynamic trade-offs between the timing of the investment, the ex-post benefits for the consumers, and the possible duplication of fixed costs. We find that merger policies are dominated by ex-ante industrial policies, whereas the latter are dominated by divestiture policies only under very special circumstances.

Introduction

In 2013, the UK government struck a deal with the French utility EDF to build the country's first new nuclear plant in a generation (Financial Times, October 21, 2013). The deal guaranteed a price in real terms for 35 years that roughly doubled the current price. In 2014, the UK Competition Commission required that, notwithstanding the late introduction of a new player in 2013 (Mittal), the cement market structure remained too concentrated, and that the dominant players in UK (Lafarge-Tarmac and Hanson) should divest plants to a newcomer (The Telegraph, October 9, 2014). The first example illustrates that industrial policies, and in particular those targeted to mitigate climate change, aim to promote investment possibly to the detriment of the consumers' future welfare due to limited future competition. The second example illustrates that antitrust polices aim to increase current consumer welfare, possibly to the detriment of future investment (and future welfare).

This paper explores the interaction between investment and regulatory policies. In particular, we aim to capture a fundamental ingredient in the underlying welfare analysis, the interdependencies between such regulatory policies that may often be shortsighted, as well as their possible feedback on the long term investment efforts of firms. To this end, we build upon a simple framework used by [START_REF] Cabral | Simultaneous Entry and Welfare[END_REF] that is known in the literature as the grab-the-dollar game (see Fudenberg and Tirole, 1991, p.127 for a textbook introduction). Grab-the-dollar games are the stationary variant of [START_REF] Fudenberg | Preemption and Rent Equalization in the Adoption of New Technology[END_REF] seminal work on preemptive strategic investment when the industry exhibits natural monopoly features. 1 In a grab-the-dollar game such as the one we analyze, a limited number of firms repeatedly decide whether or not to invest in a market. Investment is costly and irreversible. After some random time which may eventually be very long and is endogenously determined to some extent, no outsider further invests, and the market structure remains stationary. The limited size of the market implies that ex-post market structure may be either profitable or not.

This model provides a stylized and analytically tractable framework to discuss the welfare impact of entry dynamics. Indeed, [START_REF] Cabral | Simultaneous Entry and Welfare[END_REF] discusses whether some structural characteristics may lead to a situation of over-investment or under-investment. In case of over-investment (under-investment), the total discounted social welfare would increase (decrease) if investment were taxed (subsidized). We revisit this welfare analysis whenever the 1 As in [START_REF] Fudenberg | Preemption and Rent Equalization in the Adoption of New Technology[END_REF], most of the literature has analyzed games with complete information. In particular, natural extensions of their duopoly setting to one of oligopoly have been studied by [START_REF] Vettas | On Entry, Exit, and Coordination with Mixed Strategies[END_REF] and [START_REF] Shen | Strategic Entry before Demand Takes off[END_REF] (in stationary settings) as well as [START_REF] Argenziano | Clustering in N-Player Preemption Games[END_REF] (in nonstationary settings). It is worth noting though that [START_REF] Levin | To Grab for the Market or to Bide One's Time: A Dynamic Model of Entry[END_REF] have analyzed a stationary variant of [START_REF] Fudenberg | Preemption and Rent Equalization in the Adoption of New Technology[END_REF] when the sunk cost of entry borne by each firm constitutes private information.

social planner commits to some antitrust policy before firms choose their investment behavior. We consider three antitrust policies: no intervention, i.e., the base case for the model; a "lenient" policy in which mergers are allowed whenever the ex-post market structure generates a profitability deemed too low; a "pro-competitive" policy in which divestiture is forced whenever the ex-post market structure generates a profitability considered high. The impact of such an active antitrust policy will always be favorable ex-post. The model allows for exploring the feedback of investment on ex-ante welfare, which constitutes the focus of the analysis.

There is already a significant amount of literature on the interaction of certain competition policies and R&D incentives, pioneered by [START_REF] Segal | Antitrust in Innovative Industries[END_REF]. [START_REF] Shapiro | Competition and Innovation: Did Arrow Hit the Bull's Eye?. The Rate and Direction of Inventive Activity Revisited[END_REF] provides a useful and recent discussion of the interaction of merger policy and R&D, as opposed to the main focus of [START_REF] Segal | Antitrust in Innovative Industries[END_REF] on exclusionary behavior. [START_REF] Gans | Entrepreneurial Commercialization Choices and the Interaction between IPR and Competition Policy[END_REF] extend [START_REF] Segal | Antitrust in Innovative Industries[END_REF] framework to show that innovation policy and antitrust policy complement each other when firms can engage in cooperative forms of R&D such as licensing. See also [START_REF] Baker | Beyond Schumpeter vs. Arrow: How Antitrust Fosters Innovation[END_REF] for various economic arguments of when antitrust interventions are most likely to foster innovation. As for R&D, antitrust policy concerns the extent of market power that should be allowed ex-post but the public good nature of R&D implies that both the welfare analysis and the regulatory instruments differ from the ones we highlight.

In contrast with the R&D literature, there has been much less research on the interdependence of entry incentives and antitrust policy. To the best of our knowledge, [START_REF] Gowrisankaran | A dynamic model of endogenous horizontal mergers[END_REF] is the first one to point out this issue relative to merger policy. Typically, the welfare analysis of mergers is done in a static framework (see e.g. [START_REF] Salant | Losses from Horizontal Merger: The Effects of an Exogenous Change in Industry Structure on Cournot-Nash Equilibrium[END_REF][START_REF] Salant | Losses from Horizontal Merger: The Effects of an Exogenous Change in Industry Structure on Cournot-Nash Equilibrium[END_REF][START_REF] Farrell | Horizontal Mergers: An Equilibrium Analysis[END_REF][START_REF] Farrell | Horizontal Mergers: An Equilibrium Analysis[END_REF]), but [START_REF] Gowrisankaran | A dynamic model of endogenous horizontal mergers[END_REF] develops a dynamic computational model along the lines of [START_REF] Ericson | Markov-Perfect Industry Dynamics: A Framework for Empirical work[END_REF] in which firms endogenously decide whether or not to merge at each period. Allowing for mergers has an impact on the entry probability, the long term market structure, and eventually the associated welfare. This model is intended to illustrate that it could be feasible to carry out a welfare analysis in a concrete antitrust case. More recently, [START_REF] Mason | Merger Policy, Entry, and Entrepreneurship[END_REF] have examined the effect of merger policy on entry incentives when an incumbent is already active in the market. They analyze some determinants of the socially optimal merger policy, and then compare it with an entry subsidization policy. Their paper and ours differ in a number of respects: policies in our setting are set before the market structure forms, and our parameters of interest are the size of the sunk costs of entry and the intensity of competition, whereas they study changes in other parameters; we also consider a larger range of policies, such as forced divestiture. Despite these differences, it is worth noting that our results confirm their find-ing that merger policy is dominated by ex-ante industrial policy when firm subsidies involve negligible distortions at the economy-wide level. 2Our model allows for an extensive analysis of the impact of an antitrust policy with respect to three key aspects: ex-post welfare (consumer surplus and industry profits per period once the market structure becomes stationary), the delay to achieve the stationary market structure, and cost duplication. We shall not only consider allowing or not for mergers, but also analyze an antitrust policy which forces divestiture in case that a highly concentrated market arises. We obtain a complete characterization of the conditions under which the ex-ante discounted welfare increases when the feedback on investment is taken into account. Moreover, these antitrust policies will be compared to an ex-ante industrial policy which would either subsidize or tax investment.

We shall proceed as follows. In our framework, for analytical tractability, the number of competing firms is limited to two, as in [START_REF] Cabral | Simultaneous Entry and Welfare[END_REF] and [START_REF] Mason | Merger Policy, Entry, and Entrepreneurship[END_REF]. In a static context, the investment cost and the market size would be such that only a monopolist is profitable. We consider an infinite-horizon setting in which firms decide their investment actions simultaneously. In each period, the situation is one of the following three cases: (i) no firm invested in the past (in which case firms need to decide again whether or not to invest); (ii) there is a monopoly, and; (iii) there is a duopoly. The latter two cases can be seen as absorbing states of the game: the firm(s) active in the market remain in place forever, and the other firm never enters. Following [START_REF] Cabral | Simultaneous Entry and Welfare[END_REF], it is convenient to formalize the industry characteristics through two key parameters: the ratio of the investment cost to the monopoly profit, and the ratio of the duopoly profit to the monopoly profit. The higher the first parameter, the higher is the potential inefficiency in case of entry cost duplication. The lower the second parameter, the more intense is the ex-post competition in case of duopoly.

Our main contribution is to shed light on the effects of various commonly used policies on the ex-ante incentives to invest in a socially optimal manner given that the industry may end up exhibiting substantial ex-post market power. Thus, we explicitly determine the unregulated probability of entry for all combinations of our two key parameters, and then study how such a probability is affected by the various policies examined (i.e., no regulation, ex-ante industrial policy and ex-post antitrust policies). This allows for an explicit ranking of all policies in terms of welfare. We show that allowing mergers is always dominated by an appropriate ex-ante industrial policy, even though it would almost achieve the same welfare in those cases in which the probability of entry is small. The industrial policy also dominates the divestiture policy, except whenever the probability of entry is close to one: in such cases, it would be beneficial to divest a monopoly into a duopoly. As soon as entry becomes less likely, however, we find that a divestiture policy would completely discourage entry. To the best of our knowledge, the current work provides the first welfare comparison of policies that are widely used by government agencies, so we can provide some guidance about their relative pros and cons.

The remainder of the paper is organized as follows. Section 2 introduces the model, solves it for the base case (unregulated equilibrium), and characterizes the optimal industrial policy. Section 3 analyzes the impact of the antitrust policies (whether to allow mergers or force divestiture). Section 4 discusses the policy relevance of the results using a numerical illustration. Section 5 concludes by summing up the results, and suggesting how this work could be extended.

The model

We consider a market for a homogeneous good and two potential entrants. A firm is active if it has already entered, or else it is inactive. Any inactive firm can enter the market and immediately become active by paying a fixed sunk cost K > 0. Time is discrete and infinite, and at each time t = 0, 1, ... inactive firms simultaneously decide whether to enter or not. We denote π n for the individual per-period profit if n ∈ {1, 2} firms are active in the market, and we assume it is collected at the end of the corresponding period (this is just a normalization). Firms discount future payoffs at rate r > 0, with δ ≡ (1 + r) -1 denoting the discount factor. We assume that 0 ≤ π 2 < rK < π 1 , (A1)

so the market has the potential to be profitable for at most one firm and exit never takes place (voluntarily).

The per-period consumer welfare is denoted by s n if n ∈ {1, 2} firms are active. Also, w n ≡ s n + nπ n denotes the per-period social welfare generated by n ∈ {1, 2} active firms. We use capital letters for discounted streams of surplus: Π n = +∞ t=1 δ t π n = π n /r for a firm's profit, S n = s n /r for consumers' surplus and W n = w n /r for the social welfare if n ∈ {1, 2} firms are active from the current date on. For convenience, define A n ≡ W n -nK for n ∈ {1, 2}.

Note that the product homogeneity assumption allows us to write consumer surplus and social welfare as a function of industry profit: w(nπ n ) and s(nπ n ) (n = 1, 2), respectively. Thus, w n ≡ w(nπ n ) = s(nπ n ) + nπ n . Welfare is decreasing with respect to industry profit because of the reduction in (allocative) efficiency, so consumer surplus is decreasing in such a profit with a slope smaller than -1. We shall assume that the per-period social welfare is weakly increasing and concave in the number of active firms:

w 2 ≥ w 1 and w 1 ≥ w 2 -w 1 . (A2)
The second condition in (A2) asserts that the entry of a single firm creates more gross welfare than the addition of a second firm; alternatively, a monopoly achieves more than half the gross welfare gains of a duopoly. A direct implication is that if a monopoly does not create net welfare (i.e., w 1 < rK), then neither will a duopoly. 3Our model can be seen as depending on two key parameters: on the one hand, π 2 /π 1 represents the intensity of competition, ranging from 0 (pure Bertrand competition) to 1/2 (collusion); on the other hand, rK/π 1 represents the profitability of the market for a monopoly: as rK/π 1 increases from π 2 /π 1 to 1, the profitability declines from the limit profitability needed to sustain a duopoly to the limit profitability needed to sustain a monopoly.

Unregulated equilibrium

Given our assumptions on profits, there are two stationary equilibria in pure strategies, with one firm entering in every period and the other staying out. These equilibria are asymmetric, so each firm should anticipate perfectly its role as an entrant or as a firm that remains inactive. From now on, we follow [START_REF] Cabral | Simultaneous Entry and Welfare[END_REF] and consider symmetric equilibria in stationary strategies. In subgames in which one firm is already active, the other firm chooses not to enter with probability one. The discounted payoff of an active firm when n ∈ {1, 2} firms have entered is then Π n , because no further entry will take place. And, when n ∈ {1, 2} firms have entered, the expected payoff of an inactive firm is zero.

In subgames with no active firm, symmetric behavior can be easily seen to require randomization. In particular, the two inactive firms must be indifferent between entering and waiting one more period to make the entry decision. Suppose that any one of the inactive firms chooses to enter with probability p 0 ∈ (0, 1). Denote the other one's expected payoff if it waits by V w , and let V e denote its payoff if it immediately enters, so that stationarity yields

V e = p 0 Π 2 + (1 -p 0 )Π 1 -K and V w = (1 -p 0 )δ max(V w , V e ).
Because V w = V e in a mixed-strategy equilibrium with stationary strategies, we have that V w = V e = 0, that is, the equilibrium mixing probability p * 0 solves:

p * 0 π 2 r + (1 -p * 0 ) π 1 r -K = 0. ( 1
)
It readily follows that the equilibrium probability as a function of π 1 , π 2 and rK is

p * 0 (π 1 , π 2 , rK) = (π 1 -rK)/(π 1 -π 2 ), (2)
a positive value not exceeding one because π 1 > rK > π 2 . The expected payoff of an inactive firm in these subgames is E * 0 = 0. That all profits from entry are dissipated in expectation because of aggressive entry behavior is clear given that there is room for only one firm in the market. The symmetric equilibrium probability of entry increases as either π 1 or π 2 grow and as the (per-period) cost of capital rK falls, all of them intuitive results as well.

Ex-ante industrial policy

Suppose now that a social planner could indirectly control the (nondiscriminatory) entry probability of firms by somehow taxing or subsidizing the sunk cost. Since a higher (lower) sunk cost decreases (increases) the entry probability, taxation (subsidization) could be referred as a situation of "excessive" ("insufficient") entry. This is indeed the approach followed by [START_REF] Cabral | Simultaneous Entry and Welfare[END_REF]: he studies the net marginal welfare gain relative to the sunk cost of entry. Even though he simply focuses on limit cases within the parameter space, we will focus on the entire parameter space, extending his pioneering analysis. To this end, we shall directly derive the optimal entry probability and then obtain the related tax or subsidy as well as the associated welfare.

Recall that W n denotes the discounted social welfare when n ∈ {1, 2} firms are active and that A n ≡ W n -nK for n ∈ {1, 2}. If the planner chooses an entry probability of p, then the equation of motion for expected welfare given that no firm is yet active is

W 0 (p) = 2p(1 -p)A 1 + p 2 A 2 + (1 -p) 2 δW 0 (p), (3)
which gives

W 0 (p) = p[2(1 -p)A 1 + pA 2 ] 1 -(1 -p) 2 δ . ( 4
)
For p ∈ (0, 1), an increase in the probability of entry on welfare has the following effect (by differentiation of (3)):

[1 -δ(1 -p) 2 ] ∂W 0 ∂p = 2p(A 2 -A 1 ) + 2(1 -p)(A 1 -δW 0 ). ( 5
)
An increase in the probability of entry makes it more likely that a duopoly results rather than a monopoly (first term on the right hand side), but also makes it more likely that a monopoly results as opposed to having no entry at all (second term). Which aspect dominates depends upon the probability of entry. The following proposition makes this precise.

Proposition 1 The discounted social welfare stream is (globally) maximized

• at p = 0 if w 1 ≤ rK; • at p = 1 if rK ≤ w 2 -w 1 ; • at p = p ≡ (2w 1 -w 2 ) 2 + 4(w 1 -rK)(w 1 + rK -w 2 )/r -(2w 1 -w 2 ) 2(w 1 + rK -w 2 )/r ∈ (0, 1)
otherwise.

Proof. See Appendix.

The social planner trades-off the social cost of capital (rK) against how much competition it would like to generate, keeping also in mind that entry delays may be socially costly. Given these aspects that the planner takes into account, the socially optimal probability of entry captures intuitive features. Indeed, the first two cases are clear: no entry is pursued when the per-period social cost of capital exceeds the per-period welfare generated by a monopoly; also, the immediate entry of the two firms is optimal if discounted welfare with a duopoly relative to a monopoly is higher than the per-period social cost of having one more firm active in the market. In the third case, that is, in circumstances in which a monopoly would be the social planner's preferred market structure if it could directly enforce it, the optimal probability of entry trades-off the probability of excessive entry with the risk of delay. In such a case, the optimal probability of entry can be shown to grow as the per-period cost of capital falls or as the intensity of competition increases, which constitutes our next result.

Proposition 2 Lowering either π 2 or K lead the social planner to pursue a larger extent of entry.

Proof. See Appendix.

Based on this proposition and on the fact that the equilibrium probability grows with π 2 , it is clear that, for any K ∈ (0, π 1 /r), there exists a value π C 2 (K) ∈ (0, π 1 ] such that there is excess entry if π 2 > π C 2 (K) and insufficient entry otherwise. 4 There is a clear tension related to the intensity of competition: if competition is more intense, entry is more desirable but less likely in equilibrium. From a social welfare standpoint, the unregulated equilibrium results in insufficient entry for low π 2 , and in excessive entry for high π 2 . As for the entry cost, an increase in K reduces the equilibrium probability of entry as well as the socially optimal one. The monotonicity of the comparison is ambiguous. Still, if the fixed cost is large (close to Π 1 ) the probability of entry is close to zero, and (by Proposition 1, since w 1 < rK in such cases) the unregulated equilibrium probability of entry leads to insufficient entry because of the cost of delaying an investment that would create some consumer surplus. In that case, a monopoly is much more likely than a duopoly and the intensity of competition has a negligible effect on the comparison. In Figure 1, we plot on the (rK/π 1 , π 2 /π 1 )-plane the areas in which there is socially insufficient entry and socially excessive entry. The shapes of the areas illustrates the preceding discussion. In figure 1, we use a linear demand and a constant marginal production cost and r = 0.1 (Cabral, 2004, Fig. 2 depicts r = 0.01). More on this numerical illustration in section 4. It is worth stressing that, in our framework, maximizing expected total welfare is equivalent to maximizing consumer surplus net of the expected subsidy (provided there is no extra cost of public funds, an aspect that is easy to incorporate into the analysis). The point is that all the subsidies received by firms are dissipated due to the more intense competition for the market (in equilibrium, expected net profits are null). More formally, for a subsidy σ ∈ given upon entry, the equilibrium probability of investment becomes p * (π 1 , π 2 , r(K -σ)), written p * as a shorthand in the remainder of this subsection. Social welfare in expression (4) could be rewritten by isolating expected consumers surplus and firm profits. Then with the aid of expression (1) (having replaced K with K -σ), we can eliminate the profits so that the following holds:

W 0 (σ) = p * 1 -(1 -p * ) 2 δ [2(1 -p * )(S 1 -σ) + p * (S 2 -2σ) +2(1 -p * )(Π 1 -(K -σ)) + p * (2Π 2 -2(K -σ))] = p * 1 -(1 -p * ) 2 δ [2(1 -p * )(S 1 -σ) + p * (S 2 -2σ)] = p * 1 -(1 -p * ) 2 δ [2(1 -p * )S 1 + p * S 2 ] -σ, where σ ≡ 2p * σ/[1 -(1 -p * ) 2 δ]
denotes the discounted expected public cost of an ex-ante policy that implements a probability of entry equal to p * through a subsidy of size σ. Treating σ as a function of p * helps to clarify the trade-off generated by the ex-ante industrial policy:5 the benefit for consumers owing to a greater probability of entry should be compared with the larger expected budget of the policy.

Ex-post antitrust policies

Two antitrust interventions are considered in this section: allowing duopolists to merge or forcing a monopolist to divest. Intuitively, allowing mergers would be appropriate in case of insufficient entry, whereas divestiture would be desirable in case of excessive entry. However, each antitrust intervention modifies the probability of entry through a modification of the ex-post profit of firms. Our analysis sheds light on the relationship.

Decentralized entry with merger in duopoly

We first consider a situation where the social planner allows firms to merge if the market structure happens to be a duopoly. When firms are allowed to merge, there is a welfare loss in the duopoly case relative to the unregulated equilibrium. The possible gain arises from the greater incentive to enter the market, though. If there is already excessive entry, allowing merger would both increase the inefficiencies associated to cost duplication and those associated to market power. Therefore, it can only be welfare enhancing to allow mergers in situations in which entry is insufficient. This leads to the following proposition.

Proposition 3 Allowing mergers can only be welfare enhancing in a situation of insufficient entry.

Further clarification of the relationship between this antitrust policy and insufficient entry can be obtained on the basis of Proposition 2 and Figure 1. Entry might be insufficient because of intense competition (low π 2 ) or a large fixed cost. If entry is insufficient because competition is too intense, softening competition by allowing merger cannot increase welfare. However, if entry is insufficient because of a large fixed cost that is hardly covered by monopoly profit, it could be welfare enhancing to promote entry by allowing mergers. In such a case, a monopoly is much more likely than a duopoly and still better than nothing, so sacrificing welfare in the (unlikely) duopoly outcome is worthwhile. This reasoning explains why, in Figure 1, merger dominates on the right (large fixed cost) of the insufficient entry area. Even though the linear case is further described in Section 4, the following proposition formally states this result in the general case.

Proposition 4 For all π 2 there is a threshold entry cost K(π 2 ) below π 1 /r such that mergers should be allowed for K > K(π 2 ).

Proof. See Appendix.

Decentralized entry with asset divestiture in monopoly

Asset divestitures only concern monopoly situations, and they are modeled in a simple way: if only one firm happens to enter the industry, the regulator (instantaneously) forces it to resell half of its assets. Two cases need be distinguished: resale to an outsider firm or resale to the firm that remains inactive. The latter case is more complicated since a waiting motive may arise, so we first discuss the former case, and then extend the results to the latter one.

We assume that purchasing divested assets allows a firm to avoid having to pay K. In these conditions, the divestiture results in a duopoly without duplication of the fixed costs, so s 1 is equal to s 2 and A 1 is equal to A 2 + K. Even though the regulator cannot directly control the entry probability, it can indirectly do so via the value of the assets divested. More precisely, we assume that the regulator can choose the share α ∈ [0, 1] of the value of divested assets that the divested monopoly obtains, so it holds that π 1 = (1 + α)π 2 , and α can be interpreted as a parameter that captures the incumbent's bargaining power. If the planner set institutional rules that induced intense competition for the divested assets, α would be close to 1; in a situation close to asset expropriation, α would be close to 0, however. Even if the regulator might not have a large control over α in some occasions, providing the planner with this lever is useful from an abstract point of view, since it enjoys an additional instrument to implement its desired outcome. As we shall see, the social planner cannot improve upon ex-ante industrial policies in many situations even in the best case scenario in which it has this extra lever at its disposal. And even if it can, this requires a very precise knowledge of industry features, since small mistakes in their assessment may have dramatic consequences. As we argue in the next section, this lack of robustness, coupled with the possibility that it may be hard for the regulator to commit ex-ante to a specific value of α, suggests that it is hard to defend divestitures as an approach preferred over ex-ante industrial policy. 6Turning to the formal analysis, when a firm foresees that a monopolist will be (immediately) forced to divest, the equilibrium entry probability becomes

p * 0 = (1 + α)π 2 -rK απ 2 , ( 6
)
which is an increasing function of α, as one would expect. By choosing α, the regulator can modify the probability of entry without affecting the ex-post efficiency. However, the regulator is constrained by the total value of the assets. Recalling that π 2 represents the extent of competition when in duopoly, one can then prove the following result.

Proposition 5 Suppose that the planner can fully control the price of divested assets. Then there exists π 2 ∈ (rK/2, rK) such that

• if π 2 > π 2 , then the planner's optimal choice is α * < 1.

• if π 2 < π 2 , then the planner's optimal choice is α * = 1.

• if π 2 < rK/2, entry never occurs and the expected welfare is null.

Proof. See Appendix.

The proposition highlights the fact that if competition is anticipated to be mild, entry is not highly desirable, and in addition the economic value of divested assets is high. Consequently, the regulator can implement the optimal extent of entry given the ex-post duopoly structure by giving only a share of the value of divested assets to the divested monopoly. If competition is intense, entry is highly desirable but not very profitable. Taking into account that the regulator is constrained by the low value of divested assets, the firm that initially entered as monopolist gets the full economic value of its divested assets, even though this will still induce insufficient entry from a welfare perspective, given the allocative efficiency of a duopoly. In the extreme case in which competition is very fierce and the fixed cost of entry is relatively high, asset divestiture completely eliminates the incentive to enter and the expected welfare is null. In such a case, a duopoly does not generate enough profit to cover the fixed cost.

Since the planner can implement the optimal extent of entry for π 2 > π 2 , it should come as no surprise using a continuity argument that asset divestiture is socially useful even when π 2 is below π 2 . In particular, the following holds.

Proposition 6 There is a threshold π2 ∈ (rK/2, π 2 ) such that expected welfare is strictly higher with asset divestiture than without it if and only if π 2 > π2 .

Proof. See Appendix. This proposition can be compared to Proposition 3. When competition is mild (i.e., π 2 has a high value), if there is excessive entry in an unregulated equilibrium, divestiture will be welfare enhancing as long as the regulator controls the resale value of the assets. In that case, divestiture achieves both a short-term gain by enhancing competition in the product market and a long-term one by reducing the extent of entry. However, as will be shown shortly in our numerical illustration in the next section, it may well hold that divestiture is welfare enhancing even if there is insufficient entry. This situation arises if the gain in the post-entry welfare offsets the lower probability of entry. Our illustration will also show that this gain rapidly turns into a loss as duopoly competition becomes more intense. With fierce competition, the full value of a duopoly is so low that firms are reluctant to enter and welfare is reduced. At the extreme, asset divestiture fully discourages entry. To sum up, while we could expect that divestiture would be an appropriate policy whenever there is excessive entry, a detailed analysis shows that the situation is not clear-cut.

Suppose now that the assets up for resale are (immediately) bought by the other potential entrant. If only one firm enters, the other firm obtains a profit (1 -α)π 2 from the divested assets. This profit should be taken into consideration by a firm contemplating the profitability of entry. Suppose that a firm enters with probability p 0 . Then, if the other firm enters with probability p 0 as well, the firm must expect to gain

V e = p 0 Π 2 + (1 -p 0 )(1 + α)Π 2 -K. Staying out yields V w = p 0 (1 -α)Π 2 + (1 -p 0 )δ max(V e , V w ).
In equilibrium, V w = V e must hold, so the equilibrium entry probability p * 0 solves

p * 0 Π 2 + (1 -p * 0 )(1 + α)Π 2 -K = p * 0 (1 -α)Π 2 r + 1 r + p * 0 .
The left-hand side is the value of entering, which is decreasing with respect to the probability that the rival enters, and the right-hand side is the value of waiting and possibly obtaining the divested assets, which increases with the probability of the rival's entry. From this equation, one can conclude that there is a unique equilibrium probability of entry, it is positive if (1 + α)Π 2 > K and null otherwise.

In that case, the equilibrium probability of entry is still increasing with respect to α: for α = 1, it is equal to the probability obtained with a third party given by equation ( 6); otherwise, it is strictly lower if it is positive (i.e., when α > (K -Π 2 )/Π 2 ) or obviously it is null. Similar results as those in Propositions 5 and 6 could be obtained, but the value of the divested assets that a monopoly obtains should be larger in that case, that is, the regulator should anticipate that the perspective of obtaining the divested assets will soften the incentive to enter.

An illustration 4.1 A linear specification

To illustrate our findings, let us assume that the inverse demand is P (q) = a -bq for a total production q and that the marginal cost production is constant, in which case we can set it equal to zero without loss of generality. The unregulated monopoly profit is obtained for a quantity of q m = a/(2b) and it is equal to π m = bq 2 m = a 2 /(4b). Welfare can be written as a function of the total production q: w(q) = (a -bq/2)q = s(q) + π * (q), where s(q) = bq 2 /2 stands for consumer surplus and π * (q) for industry profit. Note that Assumptions (A1) and (A2) are satisfied. Both consumer surplus and welfare could be written as a function of the industry profit π * as follows: the profit obtained with a production q is π * (q) = (a -bq)q = π m -b(q -q m ) 2 , so that the quantity q(π * ) > q m corresponding to profit π * is q(π * ) = q m + [(π m -π * )/b] 1/2 , and consumer surplus and total welfare are s(q(π * )) and w(q(π * )), respectively.

The results can be displayed in a two-dimensional diagram. The horizontal axis represents rK/π 1 varying from 0 to 1, whereas the vertical axis represents π 2 /π 1 varying from 0 to .5 (that is, as q goes from a/b to q m ). In the area in which π 2 /π 1 < rK/π 1 , Assumption (A1) is satisfied, i.e. the equilibrium probability of entry is smaller than one. Otherwise, it is equal to one and the market structure is a duopoly. In the former area, the entry probability declines as rK/π 1 increases and becomes null exactly when rK/π 1 = 1. In the latter area, our model should be extended to allow for more competitors.

Without loss of generality, we take a = 1 and b = 1/4 so that π 1 = 1; the discount rate is r = 0.1, so Π 1 = 10. The gross welfare associated with a monopoly is W 1 = 15, whereas the maximum gross welfare obtained with a Bertrand duopoly is W 2 = 20.

Comparison of the policies

Given this specification, we first derive the areas in which there is insufficient entry or excessive entry, respectively (see Figure 2(a)). In case of insufficient entry, the darker area depicts a configuration in which the optimal subsidy leads to an entry probability of one, extending the duopoly area to the right. Conversely for the case of excessive entry, the duopoly area is restricted on the left.

In Figure 2(b), we depict the optimal antitrust policy. As expected, merger policy dominates only in the area in which there is insufficient entry (Proposition 3). Divestiture may be optimal whenever these is excessive entry, but this is neither necessary nor sufficient. We can gain further insights from these results by drawing the welfare as a function of π 2 /π 1 for two given values of rK/π 1 , namely 0.6 (high monopoly profit) and 0.9 (low monopoly profit). This gives Figure 3 andFigure 4. In these figures, the vertical axis units are measured as a percentage of the welfare obtained with a regulated monopoly, which we denote by W RM and note that it is independent from the horizontal axis (W RM is obtained by setting the price of the good so that the per-period profit of the firm equals rK).7 This shows that the policies at hand induce a large welfare loss, in particular in the case of low monopoly profit (Figure 4). We shall return to this aspect shortly, but consider first the comparison between ex-ante and ex-post policies. Observe, on the one hand, that an ex-ante industrial policy always dominates merger, although the difference gets smaller as rK/π 1 becomes close to 1 The dotted line corresponds to π 2 = rK above it the equilibrium probability of entry is 1. In the brown area the optimal probability of entry is 1.

(Figure 3). On the other hand, divestiture may be better than an ex-ante industrial policy, but it may also be much worse; in case of intense duopolistic competition (low π 2 /π 1 ), a divestiture policy might totally discourage entry.

We return now to the welfare loss. It is conceivable to design a policy that combines both ex-ante and ex-post features though its implementation is unlikely since it would require coordination between currently independent regulatory agencies. Such a "hybrid" policy would encourage investment ex-ante through subsidies but discourage market power ex-post through divestiture of monopoly. It is a simple matter to prove that this hybrid policy dominates both a pure ex-ante industrial policy and a pure divestiture policy,8 and it significantly reduces the welfare loss relative to the case of the regulated monopoly (see Figures 3 and4).

It is also important to stress that the policies under consideration differ in terms of commitment. An industrial policy is a once-and-for-all commitment. For antitrust policies they are typically implemented after the fact, that is, ex-post and rarely explicit ex-ante. In our framework, we assume that the authorities can commit ex-ante. To emphasize the role of this assumption, consider for instance a situation in which the authorities are supposed to implement an industrial policy and no antitrust policy, and then, after the fact, "hold up" a resulting monopolist by forcing it to divest. This may be more realistic than our hybrid policy.9 It may reduce the welfare loss as long as the firms are myopic. We leave the analysis of such out-of-equilibrium scenarios for further research.

How each policy balances the trade-offs

Each policy trades-off three aspects: (i) the ex-post market structure (a duopoly being more favorable to the consumers; more generally, the probability of getting a monopoly versus that of getting a duopoly); (ii) the delay in obtaining such ex-post market structure (the earlier the better, so, everything else equal, a higher probability of entry would be better), and; (iii) the expected discounted investment costs (a duopoly duplicates fixed costs). Figure 5 and Figure 6 explicit the trade-offs associated to each policy.

Take the no-merger policy as the benchmark (section 2.1). In a configuration such as π 2 /π 1 = 0.4 and rK/π 1 = 0.6, we have excessive entry (see Figure 2(a)). An ex-ante policy would tax investment, increase the delay, make a monopoly situation more likely and reduce the overall expected investment cost. A divesture policy would increase the delay even further, make sure that a duopoly prevails, and reduce the expected fixed cost even more. As can be seen in Figure 3, divestiture dominates the ex-ante policy; alas, as soon as π 2 /π 1 < 0.3, divestiture leads to no entry. For rK/π 1 = 0.9, the range on which divestiture dominates shrinks to approximately [0.46, 0.5] (see Figure 4).

As can be seen in Figures 5 and6, a merger policy arbitrates in favor of reducing the delay, at the cost of guaranteeing that a monopoly prevails and at the expense of increasing the entry cost. It is only appropriate in case of insufficient entry due to a very high investment cost relative to the size of the market (i.e. rK/π 1 = 0.9). Then its sensitivity relative to π 2 /π 1 is low. The interested reader will note in Figure 7 that differences among policies persist in the neighborhood of π 2 /π 1 = 0.5 for rK/π 1 = 0.9.

Why a dynamic structure matters

The linear specification can also help to understand the effect of the dynamic structure of our framework. To do so, it is worth comparing our dynamic game to a "static game" in which firms first decide whether to enter and then produce to earn the stream Π i for i = 1, 2. This is the typical framework of a two-stage entry game, and [START_REF] Mason | Merger Policy, Entry, and Entrepreneurship[END_REF] analyze merger control in such a framework. The difference with the dynamic game is that entry, in the static game, occurs only once and if a firm does not enter it will never do so. From the firms' point of view, the value of entering and staying out are the same in the dynamic and the static game, so that the equilibrium probabilities of entry are identical in the two games. However, from a welfare perspective, entry is more valuable in the static game than in a dynamic game because entry cannot be postponed to the next period in the former case. The implications are that insufficient entry is more frequent and hence the ex-ante industrial policy consists of a larger entry subsidy in the static game than in the dynamic one. Any antitrust intervention that decreases the probability of entry has a larger long-term welfare cost in the static game. Consequently, the dynamic framework is more favorable to no merger and divestiture than the static one. To illustrate and quantify the difference, the optimal antitrust policy in the static game for our linear specification is depicted in Figure 8.

Conclusion

We have proposed a model in which industrial and antitrust policies can be compared taking into account the dynamic feedback that these policies generate on the firms' investment strategies. In our model, two firms repeatedly compete (through incurring or not a sunk fixed cost) for a natural monopoly situation: in case only one firm invests, it will secure the market; in case both firms invest, the discounted stream of duopoly profit will not compensate the sunk cost of entry. An ex-ante industrial policy takes the form either of a tax or a subsidy for the investment that allows the regulator to decrease or increase the extent of entry. An ex-post antitrust policy may either (if pro-competitive) divest a monopoly into a duopoly or (if lenient) allow a duopoly to merge into monopoly. We compare these policies in terms of their discounted welfare, which captures consumers' surplus, entry delays and the possible duplication of fixed costs. Usually, the industrial policy dominates the antitrust policies. A lenient policy is relatively good whenever the sunk cost is so large that the entry probability is very small: in such cases, the loss in consumers' surplus of allowing a duopoly and the duplication of fixed cost are more than compensated by the reduction in the entry probability; still, an ex-ante industrial policy would achieve a somewhat higher welfare. A pro-competitive policy is better than an industrial policy in specific circumstances, that is, whenever the probability of entry is close to one, but as this probability decreases such a policy would quickly destroy all investment incentives. Moreover, in order to be really better than the ex-ante policy, the bargaining power of the monopolist when reselling its assets may have to be closely monitored by the regulator.

We now discuss both limitations and possible extensions. Our model is limited to two firms, which is a strong limitation, so it would be useful to extend our results to the case of an oligopolistic structure involving n > 2 firms. However, this will make the model more complicated. For instance, assume that the market size is such that it can profitably accommodate 4 firms and say that n = 7. Without any policy, the long term market structure may be 4, 5, 6, 7 (assuming that market structures with 0, 1, 2 or 3 firms would be transient). The impact of an industrial policy is easily extended, but modeling the impact of antitrust policies is more elaborate. The targeted market structures for the authorities need to be defined; this depends on the capacity of the regulator to fully appreciate the structural characteristics of the industry. Also, the merger process needs be precisely designed, eventually giving up to the convenient assumption of symmetry among firms. Finally, the nature and the degree of differentiation on the market would need to be more specific, cost functions and capacity constraints may need to be introduced, etc.

The boundary between an analytical model and a simulation model such as the one in [START_REF] Gowrisankaran | A dynamic model of endogenous horizontal mergers[END_REF] is easily crossed at the expense of tractability. Despite its simplicity, we believe that the current model already delivers some qualitative policy recommendations. Then, on a case by case approach, a more refined model should be made to precisely discuss the benefits or pitfalls associated with implementations of policies in a real word situation. We can think of two sectors in which this approach would seem worthwhile. The first one would require extending the model so that investment leads to a new (green) technology, whereas firms that do not invest would remain stuck to an old (dirty) technology. Delay would then be associated not so much with a loss in consumer surplus but rather with the environmental damage due to the persistence of the old technology. Such an extension looks both feasible and interesting even in the case of a two-firm model. The competition between various powertrains (electric, hydrogen, hybrids) for clean transportation could provide motivation for this extension, and it could deliver some insights for the analysis of the relative efficiency of various public policies intended to promote their deployment (see for instance [START_REF] Rösler | Electricity versus hydrogen for passenger cars under stringent climate change control[END_REF].

A second context in which this approach could be both interesting and relevant concerns the cement sector. Indeed, this seems quite feasible since [START_REF] Ryan | The Costs of Environmental Regulation in a Concentrated Industry[END_REF] precisely provides a numerical dynamic model of this industry for the US using the approach of [START_REF] Ericson | Markov-Perfect Industry Dynamics: A Framework for Empirical work[END_REF]. He shows that the 1990 Clean Air Act Amendment increased the setup cost for new entrants by 20%, leading to higher market concentration, lower profit and loss of consumer surplus. Building on the same approach, Perez-Saiz (2014) and [START_REF] Fowlie | Market-Based Emissions Regulation and Industry Dynamics[END_REF] investigate how various regulatory policies may indirectly affect the market structure. However, in these applied studies antitrust policies are not explicitly considered in spite of their empirical relevance for cement.

.

Expected firms profits are null at equilibrium, so expected welfare is equal to expected consumer surplus. Consumer surplus only depends indirectly, via the entry probability, on the fixed cost K, property that will be exploited in what follows.

In particular, we show that the difference between the expected consumer surplus with merger and the expected consumer surplus without merger is positive for large K. First, note that the difference in expected consumers surplus is continuous and null for K = π 1 /r. Second, the effect of K on expected consumer surplus is (by analogy with equation ( 7)):

dS 0 dK = ∂S 0 ∂p p * 0 ∂K = - 2{p 0 [1 -(1 -p 0 )δ](s 2 -s 1 ) + (1 -p 0 )(1 -δ)s 1 } (π 1 -π 2 )[1 -δ(1 -p 0 ) 2 ] 2 .
When K goes to π 1 /r from below, the equilibrium probability goes to 0, and the derivative of consumer surplus goes to -2(1 -δ)s 1 /(π 1 -π 2 ). Given that π 2 = π 1 /2 with merger, while π 2 < π 1 /2 without merger, the derivative of the difference of consumers surplus is negative for large K. Finally, the difference of expected consumers surplus with and without merger is decreasing for large K and null at K = π 1 /r, so there must be a range of K such that the difference is positive. Proof of Proposition 5. Welfare is quasi-concave with respect to the entry probability (c.f. proof of Proposition 1). The equilibrium probability p * 0 ((1 + α)π 2 , π 2 , rK) is increasing with respect to α. Therefore, welfare is first increasing then decreasing w.r.t. α, and it is maximized either at an interior solution such that p * 0 = p, or at α = 1. The former arises if and only if p * (2π 2 , π 2 , rK) > p.

The equilibrium probability p * 0 (2π 2 , π 2 , rK) is increasing w.r.t. π 2 . The optimal probability p is increasing with respect to the duopoly welfare w 2 (from the first order condition ( 7)), so it is decreasing w.r.t. to the duopoly profit. Furthermore, for π 2 ≥ rK/2, p * (2π 2 , π 2 , rK) = 0 while p > 0. Thus, there is π 2 > 0 such that α * = 1 if and only if π 2 ≤ π 2 .

Proof of Proposition 6. Let us respectively denote W D 0 (π 2 ) and W 0 (π 2 ) for the expected welfare with divestiture (and optimal α) and the expected welfare without divestiture as functions of π 2 .

If π 2 ≥ π 2 , then asset divestiture induces the optimal probability of entry given that there is a post-entry duopoly. Without asset divestiture, the ex-post welfare is lower if one firm enters, so the expected welfare is lower whatever the entry probability is. Therefore, the expected welfare without divestiture is lower than the optimal expected welfare with divestiture: W 0 (π 2 ) < W D 0 (π 2 ). However, if π 2 < π 2 the expected welfare with divestiture is sub-optimal, in the sense that the probability of entry is sub-optimal. The probability of entry is null for π 2 = rK/2, and W D 0 is null for lower π 2 while W 0 (π 2 ) > 0 in such cases. By continuity, there must exist π2 ∈ (rK, π 2 ) such that W D 0 (π 2 ) > W 0 (π 2 ) if and only if π 2 > π2 .

Figure 1 :

 1 Figure1: The ex-ante industrial policy as a function of the fixed cost (rK/π 1 ) and the intensity of competition (π 2 /π 1 ) with a linear demand and r = 0.1. The dotted line corresponds to π 2 = rK.

Figure 2 :

 2 Figure2: Optimal policy as a function of the intensity of competition and sunk cost. The dotted line corresponds to π 2 = rK above it the equilibrium probability of entry is 1. In the brown area the optimal probability of entry is 1.

Figure 3 :

 3 Figure 3: Welfare relative to welfare with a regulated monopoly (W RM = 19.3$) as a function of the degree of competition for K = 0.6π 1 /r.

Figure 4 :

 4 Figure 4: Welfare relative to welfare with a regulated monopoly (W RM = 17.6$) as a function of the degree of competition for K = 0.9π 1 /r

Figure 5 :Figure 6 :Figure 7 :Figure 8 :

 5678 Figure 5: Expected delay before entry as a function of the degree of competition.

This result follows from their fourth proposition assuming no extra cost of public funds (R = 1), a welfare function that puts equal weight on consumer surplus and industry profit (λ = 1), and the lack of "ineffective entry" (ρ = 0).

Assumption (A2) holds for all π 2 ∈ [0, π 1 /2] if and only if it holds for π 2 = 0 (perfect competition), that is, a monopoly should achieve more than 50% of the maximal welfare. It always holds for the market of a homogeneous product with p(q) = a -bq α in which α > -1 and bα > 0. In that case, the percentage welfare loss associated with a monopoly is 1 -(2 + α)/(1 + α) (α+1)/α , which is smaller than 50% (for an extensive discussion of welfare losses in Cournot oligopoly, see[START_REF] Corchón | Welfare Losses under Cournot Competition[END_REF]. Furthermore, if the demand function is concave,[START_REF] Johari | Efficiency Loss in Cournot Games[END_REF] show that the percentage welfare loss is at most 33% for a monopoly.

Note that[START_REF] Cabral | Simultaneous Entry and Welfare[END_REF] simply gives sufficient conditions for excessive entry based on π 2 , whereas our conditions are both necessary and sufficient.

Note that the subsidy could be written as a function of the equilibrium probability of entry:σ = s(p * ) ≡ (π 1 /r -π 2 /r)p * -(π 1 /r -K),which is strictly increasing. The expected subsidy σ then becomes a monotone function of p * (differentiation shows that it is strictly increasing), so it is equivalent for the planner to choose the probability directly or to work with σ.

Note also that divestiture has no ex-post social costs (e.g., no share of K needs to be paid by the firm that gets the divested assets), so the scenario we consider should be highly favorable towards divestiture.

We consider a first-best regulation (having the fixed investment cost be directly paid by consumers while selling the product at marginal cost) to be irrelevant for our comparison. For an extensive presentation of the theory of the regulation of a natural monopoly, see[START_REF] Train | Optimal regulation: the economic theory of natural monopoly[END_REF].

With this hybrid policy, the regulator subsidizes entry and divests a monopoly into a duopoly. Writing W 0 (w 1 , w 2 , r, K, p) in expression (4), it holds that expected welfare with the hybrid policy is then max p W 0 (w 2 , w 2 , r, K, p), which is higher than welfare with the ex-ante industrial policy (since W 0 (w 2 , w 2 , r, K, p) ≥ W 0 (w 1 , w 2 , r, K, p) for all p ∈ [0, 1]), and higher than welfare with divestiture (it is equal if α < 1 and strictly higher if α = 1).

In some sectors such as electricity, we have rather observed a sequence of divestiture of monopolies followed by investment subsidies for capacity investments. See for instance[START_REF] Busnello | Evolution of the Italian power sector after its liberalisation[END_REF] for the Italian case.

The authors gratefully acknowledge the financial support from the Europlace Institute of Finance's Business Sustainability Initiative and from the ANR/Investissements d'avenir (ANR -11-IDEX-0003-02).

APPENDIX

Proof of Proposition 1. Note that A 1 = w 1 /r -K ≤ 0 implies that A 2 ≤ 2A 1 ≤ 0 because we have assumed that w 2 -w 1 ≤ w 1 . So A 1 ≤ 0 implies the expression in ( 4) is nonpositive and clearly maximized at p = 0.

Turn now to the case in which A 1 > 0. Using expression (4) after multiplying through by δ yields that

so expression ( 5) can be rewritten as 7) the derivative is positive for p = 0 and negative for p = 1, so that (the numerator being a second order polynomial) both first-order and second-order conditions are satisfied at (and only at)

Hence, W 0 (p) is maximized at p = p ∈ (0, 1). Proof of Proposition 2. Since the welfare under duopoly, w 2 , is a decreasing function of π 2 (recall that w 2 ≡ w(2π 2 )), it follows (from (7)) that

The strict concavity of W 0 (p) whenever w 1 ≥ rK > w 2 -w 1 yields the desired result.

Proof of Proposition 4. In order to analyze the situation, it is worth considering the expected consumers surplus: S 0 (s 1 , s 2 , r, p) = 2p(1 -p)s 1 /r + p 2 s 2 /r 1 -δ(1 -p) 2 .


Bruno Bon
email: bruno.bon@irht.cnrs.fr

Krzysztof Nowak
email: krzysztofn@ijp-pan.krakow.pl

Wiki Lexicographica. Linking Medieval Latin Dictionaries with Semantic MediaWiki

Keywords: medieval Latin, wiki-interface, multilingual dictionaries linking, dictionary alignment

The paper demonstrates results of the survey of which the main aim was to scrutinize consequences of adopting wiki model in alignment of medieval Latin dictionaries. In the first section, the objectives of the project as well as its methodology are presented.

As a framework Semantic MediaWiki (SMW) has been chosen. For the purpose of the research several entries from four dictionaries were selected. In the following chapters authors scrutinize presentation, search, and collaboration features provided by SMW.

Authors demonstrate how intrinsic wiki concepts, such as namespaces, templates, property-value pairs etc., may be employed in macro-and microstructure display. Next, alternative modes of accessing lexicographical data are presented such as maps, timelines, charts etc. After that search capabilities are analysed, among which the most important appear to be semantic properties search and faceted browsing. Lastly, the paper focuses on different ways SMW can encourage researchers to collaborate and enrich dictionary content.

Introduction

It is in 1913 that the idea of Pan-European dictionary of medieval Latin had been clearly expressed by research community, but it is not until early 1920s that the work began on preparing Novum Glossarium Mediae Latinitatis which would cover four centuries (IX-XII) of Latin language use (Langlois, 1924), replacing older and at that time already obsolete Glossarium of Charles du Fresne, sieur du Cange (Du Cange, 1883). From the very beginning it was also clear that, due to various periodization of Middle Ages, national dictionaries should be compiled as well. This is the reason why there now exist a dozen of dictionaries which vary not only in their chronological (500-1600 AD) and regional (from Spain to Poland; from Sweden to Italy) coverage, but also in their advancement (three of them completed, but the majority of projects still in progress).

Yet, it is with the advent of the e-lexicography that the founding idea of the common dictionary of European Latin should be reconsidered. A first step was made during the congress of medieval Latin lexicography held in Barcelona in 2004, where several elements of microstructure were proposed as a basis for dictionaries' alignment, among them headword, etymology and sense definitions (Heid, 2004). This proposal, however, was put forward without major consideration of such "technical" issues as software framework, encoding schema or data structure. During the next years community witnessed emergence of several elexicography and e-corpora projects, among which one should mention:

(1) electronic editions of Du Cange's Glossarium 1 , Novum Glossarium Mediae Latinitatis 2 , dictionaries of medieval Latin from Polish 3 and Catalan 4 sources;

(2) corpora of medieval Latin in Catalonia 5 , Galicia

Objectives and procedures

As a framework for our survey we choose MediaWiki only text formatting, but also semantic annotation of data.

For instance, when a content author types {{headword| mandragora}}, a template "headword" is called with first argument set to "mandragora". Once triggered, the template:

(1) sets property "headword" to "mandragora" and displays text string "mandragora" on entry page 16 ;

(2) sets property "headword_canonical" to "mandragora" without displaying word itself on the entry page 17 .

Annotation Main namespace has been reserved for so-called "superentries", i.e. entries of unified dictionary which serve as an index for all headwords. Super-entry page for headword depost, then, will provide the list of dictionaries the word is attested in (with appropriate links), as well as other information about the word in question that can be retrieved by means of the embedded queries. This information is presented in form of timelines and maps of the attested word occurrences which have been extracted from respective dictionary entries:

Figure 1: Super-entry page Spatio-temporal information retrieval which is in fact a heart of the WikiLexicographica was possible, because each source quotation is stored as so-called "semantic internal object" (SIO) 20 , a complex data structure which permits encapsulation of multiple property-value pairs.

SIOs include in particular:

(1) reference to the entry they belong to;

(2) source reference abbreviation;

(3) bibliographical data (page, verse etc.);

(4) proper citation;

(5) date of text composition;

(6) geographical provenance of the text. (1) sort by frequency in dictionaries;

(2) browse them on a timeline;

(3) access them on a google map.

In our survey, the last form of lexicographical data

Search and Browse Capabilities

Search and browse capabilities of the presented infrastructure are partly known from Wikipedia and its derivatives. It comes of no surprise that entries may be retrieved by means of simple full text search. As in Wikipedia, when typing word beginning in ajax-based search form, user is given suggestions. Naturally, it is also possible in advanced mode to limit search results to specific namespaces, i.e. dictionaries. 23 Framework which is object of the present study seems to reach its full potential, however, thanks to semantic layer provided by SMW. Semantic properties embedded in each entry can be browsed, for instance, thanks to the factbox displayed on the bottom of the entry page (Figure 7). 'feminine') of property "gender", one is redirected to the page where all the feminine substantives from all the dictionaries included in the wiki are listed. Similar results can be obtained from the "Special Page" on which user can process a simple semantic search, by directly specifying in two-fields form the property and its value s/he is looking for (Figure 8). Users who are familiar with wiki syntax can be assigned edit rights and contribute to entries or source pages without any difficulty. However, even non-technical oriented users may contribute to the wiki, if they are given chance to use simple edit forms. In the framework demonstrated in the present paper, this is the case of the source pages which can be modified in traditional way, by typing wiki syntax code, or by filling in forms provided by wiki developers (Figure 11). 25 One can expect that the wiki will be fed at first with data from ongoing lexicographic projects, and next enriched by the users themselves. Batch import of lexicographical information from existing dictionaries may be carried out, e.g., by means of RDF import plugin. In spite of the fact that the dictionaries under analysis may seem to differ essentially, at least as far as their micro-and macrostructure are concerned, their electronic versions are considered to be TEI compliant, and follow rules indicated in the chapter of TEI Guidelines devoted to encoding of machine-readable dictionaries (TEI Consortium, 2013). 26 Since WikiLexicographica has to serve as a common interface of data retrieval, shared information schema should be conceived as well. Extent of data extraction could be then decided according to time 25This is possible thanks to the plugin called Semantic Forms (http://www.mediawiki.org/wiki/Extension:Semantic_Forms). 26So far no attempt has been made in order to standardize medieval Latin dictionaries schema according to, for instance, Lexical Markup Framework.

or financial limitations, however burden of mapping between particular schemas and the common one need to be shifted to each lexicographic team.

The second main contributor of WikiLexicographica is expected to be research community, namely philologists, linguists, historians, palaeographers, briefly, all those who work with medieval Latin texts. Apart from simple form or meaning corrections and additions, users may be encouraged, e.g., to propose adding new words found in their sources or deleting existing ones if manuscripts deny lexicographer's reading; to supply entries with world knowledge facet which in turn can greatly support text comprehension; to create links between words by making their relations explicit, and so on.

Conclusions

MediaWiki, the software underlying Wikipedia, enhanced with semantic data annotation capabilities offered by Semantic MediaWiki extension appears to be a tool mature enough to serve as an interface for lexicographical data retrieval. It provides presentation and collaboration features an average Wikipedia user can already be familiar with. Interface popularity itself is likely to encourage contributions even from those less technical-oriented researchers. It is, however, retrieval of semantic properties that should attract a major interest of researchers since charts, timelines and maps, as well as embedded queries, offer a fresh and inventive look at the lexicographical data.

(

  MW) 9 which is best known as an application running in the background of Wikipedia. Once installed, it was subsequently supplemented with a bunch of plugins of which the essential one was Semantic MediaWiki (SMW) 10 , an extension that enhances MediaWiki with semantic dimension, enabling advanced data annotation and as a consequence their finer retrieval. It allows explicit declaration of exact meaning of the data contributed to the wiki page by annotating it with property name of which value are the data: [[Property_name::Property_value]] eg. [[Headword::Mandragora]].

  task has been primarily conducted by authors of this paper with the help of Renaud Alexandre (IRHT CNRS). Next, several members of other lexicographical teams have been familiarized with wiki editing interface (especially wiki syntax) and asked to correct or edit entries from scratch. 18 Macrostructure Although the main goal of the presented database is enabling unified retrieval of dispersed dictionaries, the provenance of lexicographical data should be always easily traceable. Firstly, it permits to acknowledge the institutional and research teams' effort of which machinereadable dictionaries are the result. Secondly, it provides users with the possibility of limiting their search results. In our prototype separation of dictionary entries has been assured by resorting to the mechanism of wiki namespaces 19 , each entry being preceded by 2-letter prefix indicating the dictionary it originates from: namespace:entry_headword, eg. for Latin word decipula 'a snare, trap', a full page title is PL:Decipula which results in following entry link: .../ i ndex.php? title=PL:Decipula . This separation allows user to browse each dictionary in traditional way by looking at the entry list (... /index.php?title= PL ).

Figure 2 :

 2 Figure 2: SIO structure

Figure 3 :

 3 Figure 3: Source description page

  analysis has been enriched (Figure4) thanks to the map layers provided by the project Digital Atlas of Roman and Medieval Civilization 21 . One is now able to view source citations in the context of administrative boundaries of the medieval world and in the light of regional variation of medieval intellectual culture.22

Figure 4 :

 4 Figure 4: Map layers

Figure 5 :

 5 Figure 5: Dictionary entry (basic view) In the next tab, one can consult full entry with all the idiosyncrasies each dictionary editorial system is affected with. The relative variety of typographic conventions, as well as different levels of data explicitness render preserving original entry display for each dictionary a very difficult, if not impossible task. This is one of the reasons why, in our opinion, wiki interface should not be considered a means of text-oriented digitization of lexicographical work. The other major problem stems from the relatively flat textual data representation in Semantic Mediawiki: it is not easy (if possible) toproperly reflect nested tree structure of the dictionary entry by means of wiki syntax only. It can seem a serious limitation, considering that medieval Latin lexicographers tend to make a heavy use of sense nesting in order to account for semantic change. Thus, more appropriate approach seems to be data-oriented recompilation of source data into desired output format, even if some of the original data (in particular, formatting) are lost. Burden of preserving original work in its typographic, sequential etc. order may, then, be arguably shifted towards each

Figure 6 :

 6 Figure 6: Dictionary entry ('Other resources' tab)

Figure 7 :

 7 Figure 7: Entry page factbox After clicking "magnifying glass" icon near the value of each property, user is taken to the page where all the entries with the same value set for selected property will be listed. So, for example, in the case of mandragora, if one clicks the zoom icon near the value f. (femininum, lat.

Figure 8 :

 8 Figure 8: Direct search for semantic properties' values More advanced semantic queries can be formulated from within two other search interfaces available in SMWbased wiki which are accessible from "Special Pages":Special:Ask and Special:BrowseData. The first (Figure9) requires from the users a basic knowledge of SMW syntax, but it also provides them with numerous output formats they can choose from, e.g. different types of charts, timelines, maps, tables, slides etc.24

Figure 9 :

 9 Figure 9: Semantic search (Special:Ask page)

Figure 10 :

 10 Figure 10: Semantic faceted browsing (Special:BrowseData page)

1Accomplished, available at http://ducange.enc.sorbonne.fr/. 2In progress, due to be finished in 2013, more information on http://glossaria.eu.

Acknowledgment

This work was supported by following grants: ANR Omnia "Outils et Méthodes Numériques pour l'Interrogation et l'Analyse des textes médiolatins"; NCN 3736/B/H03/2011/40 "eLexicon Mediae et Infimae Latinitatis Polonorum". Collaboration on this paper was made possible by support of COST Action 1005 "Medioevo Europeo" (www.medioevoeuropeo.org). Content of the WikiLexicographica was partially typed by members of respective dictionaries teams: Renaud Alexandre (Novum Glossarium), Susanna Allés Torrent (Glossarium Mediae Latinitatis Cataloniae), Pavel Nývlt (Lexicon Bohemorum), Marta Segarrés Gisbert (GLMC).


Guntars Kitenbergs
email: guntars.kitenbergs@lu.lv

Kaspars Ērglis

Régine Perzynski

Andrejs Cēbers

Magnetic particle mixing with magnetic micro-convection for microfluidics

Keywords: Microfluidics, Mixing, Magnetic Fluid, Ferrofluid, Magnetic micro-convection, Diffusion

In this paper we discuss the magnetic micro-convection phenomenon as a tool for mixing enhancement in microfluidics systems in cases when one of the miscible fluids is a magnetic particle colloid. A system of a water-based magnetic fluid and water is investigated experimentally under homogeneous magnetic field in a Hele-Shaw cell. Subsequent image analysis both qualitatively and quantitatively reveals the high enhancement of mixing efficiency provided by this method. The mixing efficiency dependence on the magnetic field and the physical limits is discussed. A suitable model for a continuous-flow microfluidics setup for mixing with magnetic micro-convection is also proposed and justified with an experiment. In addition, possible applications in improving the speed of ferrohydrodynamic sorting and magnetic label or selected tracer mixing in lab on a chip systems are noted.

Introduction

Over the last 15 years the field of microfluidics has continuously advanced, providing an interesting framework for various applications and scientific studies. As these systems typically have a small Reynolds number, a lot of effort has been devoted to enhance mixing, which is otherwise limited by diffusion speed [START_REF] Suh | A Review on Mixing in Microfluidics[END_REF], causing long channel lengths. Mixers in microfluidics can be divided into passive and active, where the latter need an external energy supply [START_REF] Hessel | Micromixers -a review on passive and active mixing principles[END_REF]. A convenient energy source for active mixing systems is an external magnetic field, as the energy can be transmitted to the microfluidics chip or cell without direct connectors. It is particularly interesting for systems with magnetic particles. Magnetic fluid, being a colloidal disperion of magnetic particles, can be used as a model system for future applications. Several examples of magnetic and non-magnetic fluid mixing have already been demonstrated [START_REF] Wen | Rapid magnetic microfluidic mixer utilizing AC electromagnetic field[END_REF][START_REF] Zhu | Rapid magnetofluidic mixing in a uniform magnetic field[END_REF]. Here we evaluate the possible use of the magnetic micro-convection phenomenon for mixing applications, applying the knowledge obtained in previous theoretical and experimental studies [START_REF] Erglis | Magnetic field driven micro-convection in the Hele-Shaw cell[END_REF][START_REF] Tatulcenkovs | Magnetic field driven micro-convection in the Hele-Shaw cell: Brinkman model and its comparison with experiment[END_REF].

Theory, Materials and Methods

Magnetic micro-convection, first described in 1980s [START_REF] Maiorov | Magnetic microconvection on the diffusion front of ferroparticles[END_REF], is caused by a ponderomotive force of the non-homogeneous self magnetic field of the magnetic fluid. Above a certain magnetic field threshold, an instability forms on the magnetic/non-magnetic fluid interface. A characteristic fingering pattern results from the magnetic particle flow induced by the acting force, pushing magnetic particles into the non-magnetic liquid. This force depends on particle concentration gradient, which decreases with mixing time as a result of the ongoing particle diffusion. Here we will focus on the practical application of the magnetic micro-convection, whereas more information on the fundamental aspects can be found in our previous studies [START_REF] Erglis | Magnetic field driven micro-convection in the Hele-Shaw cell[END_REF][START_REF] Tatulcenkovs | Magnetic field driven micro-convection in the Hele-Shaw cell: Brinkman model and its comparison with experiment[END_REF].

Experimentally magnetic micro-convection is tested in a Hele-Shaw cell as a microfluidics model system. It is placed on a stage with a coil system in an inverted microscope (Leica DMI3000B) with a 10x magnification (See Fig. 1). The coil system (Fig. 1 (1)) consists of two identical coils that are 19 mm high and have inner and outer diameters of d in = 45 mm and d out = 57 mm respectively. Each coil has 200 turns of copper wire with a diameter d = 0.7 mm. Coils are fixed on both sides of a microscope stage (1 mm thick, Fig. 1 (2)) so that their axes and the optical axis of the microscope coincide. A current (up to 3 A) flowing through the coils that are connected in series (R tot = 2.9Ω) creates a ≈ 25 mm 2 large area with a homogeneous magnetic field in z direction (up to B = 20 mT) where the center of the cell is placed. The cell (Fig. 1 (3)) is made of two Parafilm M R spacers, cut in 'U' shapes, that are put opposite each other between two glass slides to form a rectangular cell leaving two outlets for air on the sides. Two metal tubes, glued in drilled holes in the upper glass slide, provide tubing connections (Fig. 1 (4)) to a syringe pump. After assembly, glass slides with spacers are welded together on a hot plate (75 • C, 5 min), creating a cell with a 5 × 20 × 0.12 mm 3 size. The process is filmed in the bright-field mode with a fast camera (Mikrotron MC1363, maximum resolution 1280×1024 px). It is recording true color images at f = 50 Hz and its white balance is calibrated for the tungsten halogen lamp used for illumination. Color images are later converted to 8-bit grayscale images (I GRAY = 0.2989 × I R + 0.5870 × I G + 0.1140 × I B ) for subsequent analysis. The miscible fluids are water based magnetic fluid (Fig. 1 (5)) and distilled water (Fig. 1 (6)). The magnetic fluid is made by Massart's coprecipitation method [START_REF] Massart | Preparation of aqueous magnetic liquids in alkaline and acidic media[END_REF] and results in maghemite particles that are stabilized with citrate ions and have a mean diameter d = 7.0 nm, saturation magnetization M sat = 8.4 kA/m at B sat = 1 T, susceptibility χ 0 = 0.2 (SI units) in the range up to B = 20 mT and volume fraction φ = 2.9 % (from magnetization measurements).

Both fluids are slowly brought into contact inside the cell through the tubing connections by a syringe pump, while the magnetic field is already present. When droplets touch and an interface is formed along y axis, the pump is turned off, stopping the flow of fluids. The magnetic micro-convection is recorded for further analysis in 0.5x0.5 mm 2 area (Fig. 1 (7)) with the microscope camera, forming images I t (i, j), where t is time, and i and j denote spatial indices with the total length N = 360 px for both x and y axes (see Fig. 2 (a)).

To characterize a mixing system quantitatively, information on concentration distribution is necessary. As we use bright field microscopy and magnetic fluid

c t (i, j) = lg I t (i, j) -lg I H2O lg I FF -lg I H2O , (1)
where I H2O and I FF are the intensities of initial water and magnetic fluid concentrations, found in images at t = 0. A spatially averaged concentration ct (i) is then calculated from concentration data, to characterize concentration in the mixing direction along x axis (see Fig. 2 (c)):

ct (i) = 1 N N j=1 c t (i, j). (2)
To quantify mixing dynamics, we define mixing efficiency M eff (t) as follows:

M eff (t) = 1 - 1 N N i=1 (c t (i) -c ∞ (i)) 2 1 N N i=1 (c 0 (i) -c ∞ (i)) 2 , (3)
where c 0 and c ∞ are theoretical concentration distributions before mixing starts (t = 0) and when mixing has finished (t → ∞) (see Fig. 2 (c)). Definition of M eff is a slight variation of other measurements from literature, e.g. mixing ratio [START_REF] Wen | Rapid magnetic microfluidic mixer utilizing AC electromagnetic field[END_REF] and percentage mixed [START_REF] Johnson | Rapid microfluidic mixing[END_REF], adjusted for better representation of the experimental data.

The experiment involves an interface formation, which creates a slightly mixed state that differs from one time to another, due to the experimental limitations instead of a theoretical step like concentration distribution c 0 . We remove this influence by introducing a relative mixing efficiency M r (t) for t > t 0 , which subtracts the mixing efficiency that has been made due to the interface formation. This value M eff (t 0 ) is taken at a manually chosen time t 0 , when it can be seen that the interface formation is finished (typically t 0 = 0.04 s):

M r (t) = M eff (t) -M eff (t 0 ). ( 4
)
3 Results and Discussion The experiments are performed for various magnetic field values. Snapshots of magnetic micro-convection development at several time moments can be seen in Fig. 3 (a). Larger field provokes a faster evolution of the instability, enhancing mixing. It becomes more apparent, when one observes the spatially averaged concentration ct dynamics, shown in Fig. 3 (b) contour plots. In the case of diffusion (B = 0), more than 2 seconds are needed for mixing to change the initial concentrations c 0 near the edges of the x-axis 0.5 mm field of view, whereas for the largest field B = 13.3 mT it happens in less than 0.5 s. Clearly, an increase in the field strength increases the mixing development, although, interface formation influence makes it less notable. A more quantitative result of the magnetic micro-convection influence can be seen in Fig. 4 (a), where relative mixing efficiency M r (t) is showed for the magnetic field values. Achieved M r values might seem small, but it is important to remember its definition (Eq.3&4) and the experimental cell, which is much larger than the considered field of view (0.5 × 0.5 mm 2 ) and accordingly has two large basins of the original concentrations, making it very long to reach completely mixed state. Overall, small fields (B < 7 mT) seem to enhance mixing over diffusion only slightly, while further increase of magnetic field boosts mixing. This can be explained by Fig. 4 (b), where relative mixing efficiency is plotted as a function of magnetic field squared for several time values. Data points in graph agree well with the fitted lines, implying mixing efficiency to be square dependent on the field. This result is consistent with previous findings of the field dependence of magnetic micro-convection characteristics [START_REF] Tatulcenkovs | Magnetic field driven micro-convection in the Hele-Shaw cell: Brinkman model and its comparison with experiment[END_REF].

Magnetic field influence on mixing can be analyzed by observing the change in slopes of the fitted lines in Fig. 4 (b). The slopes are 0.7, 2.0, 2.0, 1.7 and 1.5 ×10 -3 mT -2 for 0.1, 0.3, 0.5, 1.0 and 2.0 s time values respectively. The variation in slopes for different times arises from the phenomenon and diffusion interplay. At small, but finite times (t = 0.1 s, t = 0.3 s), the instability forms and enhances mixing, increasing the slope until a maximum. Afterwards (t > 0.5 s), a slow decrease of the slope can be seen, as the concentrations tend to equate. In addition, these fits offer an estimate of the relative mixing efficiency after a selected time as a function of magnetic field. For example, after t = 0.5 s (linear fit for data marked with asterisk in Fig. 4 (b)):

M r (t = 0.5 s) = 0.002 × (B[mT]) 2 + 0.092, (5)
which is valid for this particular system with B < 20 mT.

A direct comparison between the magnetic micro-convection and other active mixing methods for magnetic/non-magnetic fluid systems is difficult due to the different geometries and fluids used. Wen et. al [START_REF] Wen | Rapid magnetic microfluidic mixer utilizing AC electromagnetic field[END_REF], where mixing in a microchannel is achieved with an AC magnetic field, has mentioned that the characteristic fingers reach both channel walls, which are ∆x = 0.15 mm apart, after t = 0.5 s at B = 14.6 mT. In comparison, after the same time and a similar field strength, we have observed that fingers created by micro-convection are ∆x = 0.5 mm long, which is 3 times more, although the particle concentration and the initial susceptibility for our magnetic fluid is smaller. Partly this comes from the differences in thicknesses. In Zhu et al. [START_REF] Zhu | Rapid magnetofluidic mixing in a uniform magnetic field[END_REF] mixing with an in-plane DC field is achieved in a reservoir with a 0.5 mm radius. The mixing efficiency in this paper is insufficiently explained, allowing us only to surmise that magnetic micro-convection creates instabilities in similar length and time scales for alike magnetic fields.

(2) (

x y z The results showed above clearly indicate the advantages of using magnetic micro-convection for mixing. It is very simple to realize it in an already existing microfluidics setup, by adding small coils in the desired place of mixing, as proposed in Fig. 5.

To test this concept, we have made a preliminary experiment in a simple continuous flow microfluidics setup as similar as possible to the proposed concept. The flow cell is made of a 'Y' shape cut in a single Parafilm M R layer (20 × 50 mm 2 ) with two inlet channels (≈ 5 × 20 mm 2 ) with a ≈ 15 • angle between them that join in a common outlet channel (≈ 10 × 20 mm 2 ). Two glass slides are welded together with this layer as described previously in the section 2, making a flow cell with a 0.12 mm thickness. The upper glass slide has glued metal tubing connections for inlets (magnetic fluid and water) and outlet. Syringe pump produces a v = 0.15 mm/s flow in x direction over the field of view, as determined with a micro Particle Image Velocimetry (Dantec) setup.

Magnetic micro-convection in the flow cell test (see Fig. 6) creates a similar pattern as seen above, most likely inducing a comparable mixing enhancement. A more detailed study in a specially designed microfluidics channels should be performed to characterize the applicability of the phenomenon in specific systems, but the concept and its benefit has been showed.

In these experiments the magnetic field is created with a coil system, because it can be easily installed on a microscope stage, without blocking the observation ability of the microscope. Besides, the field strength is adjustable with a power supply and does not involve any moving parts on the microscope stage. However, a homogeneous magnetic field needed for evoking magnetic microconvection can also be obtained differently. For example, permanent magnets can be advantageous in applications where a connection to a power supply is impossible and a flexible adjustment of the used field is not needed. In addition, they can provide high fields while having a small size. Another apparent method is micro-fabrication, where micro-coil or permanent magnet systems can be implemented in the microfluidics chip during its production. In fact, the magnetic field source should be selected after considering the requirements of the desired application and the critical field [START_REF] Tatulcenkovs | Magnetic field driven micro-convection in the Hele-Shaw cell: Brinkman model and its comparison with experiment[END_REF] needed for the magnetic micro-convection.

A limitation of boosting mixing by increasing the magnetic field strength comes from the saturation of magnetic fluid particle magnetization. It is efficient to increase the field only during the linear regime, which can be found from the magnetic fluid magnetization curve and is B sat ≈ 20 mT for this particular magnetic fluid, as mentioned before (magnetization curve is not showed).

Magnetic nanoparticles are well known in many scientific and clinical applications [START_REF] Ito | Medical application of functionalized magnetic nanoparticles[END_REF]. As magnetic fluid is a colloid that consists of magnetic nanoparticles, it should be possible to realize magnetic micro-convection with any magnetic particle ensemble, if colloidal, magnetic and setup properties are properly combined. For example, we can consider a colloidal dispersion of magnetic particles that is functionalized with antibodies. Mixing it with an analyte in a microfluidics flow cell by using the magnetic micro-convection should decrease the time needed for cell or biomolecule magnetic labeling. Together with a consecutive magnetic sorting and subsequent analysis, an accelerated immunoassay test in a lab on a chip system can be achieved. Other possible applications are in speeding up sorting mechanisms based on ferrohydrodynamics [START_REF] Zhu | Continuous-flow ferrohydrodynamic sorting of particles and cells in microfluidic devices[END_REF] or increasing the mixing speed of selected tracers that are suspended in the magnetic fluid.

Conclusions

We have shown that the magnetic micro-convection is a convenient method to enhance mixing in microfluidics systems that use magnetic particle colloids. Comparing to other methods proposed in literature, it is better or of similar efficiency. In addition, it is easy to implement due to the simple requirements of a homogeneous magnetic field in the place wanted. The experiments show the capability to achieve M r ≈ 45% over a large 0.5 mm distance in 0.4 seconds with a magnetic field B < 15 mT, which is 4 times more and 10 times faster than with diffusion. Mixing efficiency quickly grows with an increasing magnetic field, following a square law, but it is limited by the magnetization saturation. A simple microfluidics test measurement proves the concept. But further measurements in a specially designed microfluidics setup are necessary for full characterization of the magnetic microconvection usage capabilities, including determining the influence of channel size, flow-rate and colloidal properties and reducing the interface formation effect.

Figure 1 :

 1 Figure 1: Experimental setup consists of a coil system (1) fitted on a microscope stage (2). A Hele-Shaw cell (3) with tubing connections for fluid introduction (4) is placed in the center. A closer view of the cell displays how magnetic fluid (5) and water (6) droplets are brought to a contact in the center of the Hele-Shaw cell. Camera is recording only the central part of the cell (7) where droplets merge.

Figure 2 :

 2 Figure 2: Example of image analysis sequence for finding concentration distribution. (a) Original image I. (b) Concentration plot c, found via Beer-Lambert law (Eq.1). (c) Spatially averaged concentration c calculated with Eq.2 is marked with dots, while theoretical initial state c 0 and final mixed state c ∞ are marked with solid and dashed lines..

Figure 3 :

 3 Figure 3: (a) Snapshots of the magnetic micro-convection development at several times for various magnetic fields. The field of view is 0.5x0.5 mm 2 . (b) Spatially averaged concentration time dependence for the same magnetic fields as in (a) qualitatively revealing mixing enhancement and dynamics.

Figure 4 :

 4 Figure 4: (a) Relative mixing efficiency as a function of time for various magnetic fields. A clear mixing enhancement is visible. (b) Relative mixing efficiency as a function of magnetic field squared for various time moments.

Figure 5 :

 5 Figure 5: A proposed test setup with microfluidics channel (1), inlets (2), outlet (3) and small coils (4).

Flow

Figure 6 :

 6 Figure 6: A preliminary test of mixing with magnetic micro-convection in a simple microfluidics flow cell. Scale bar is 0.5 mm.

Acknowledgements

This research is supported by a European Social Fund project Nr.2013/0028/1DP/ 1.1.1.2.0/13/APIA/VIAA/054