Jean-Bernard Chatelain
email: jean-bernard.chatelain@u-paris10.fr

Andrea Generale
email: generale.andrea@insedia.interbusiness.it

Ignacio Hernando
email: hernando@bde.es

Ulf Von Kalckreuth
email: ulf.von-kalckreuth@bundesbank.de

Philip Vermeulen
email: philip.vermeulen@ecb.int.

Christopher Allsopp

Michael Artis

Paul Mizen

New Findings on Firm Investment and Monetary Policy Transmission in the Euro Area

Keywords: JEL Classification numbers: E22, E50 Keywords: investment, monetary transmission, user cost of capital

In this paper we present comparable results on the determinants of firms' investment and their link to monetary policy. The results have been obtained by the Eurosystem Monetary Transmission Network. This network has produced a series of papers in which the use of micro data permits estimating and quantifying the relevance of two channels of monetary policy transmission: the interest rate and the broad credit channel. The research findings provide evidence of an operative interest rate channel in all countries examined. Moreover, the results indicate that variables which proxy firms' financial conditions play a role. Firms characterised by weaker balance sheets show higher liquidity sensitivity.

INTRODUCTION

Business investment decisions are thought to be influenced by monetary policy through two important channels. First, changes in market interest rates imply changes in the cost of capital, which in turn affect investment decisions. This is known as the interest rate channel. Second, changes in market interest rates might affect firms differently. Given imperfect capital markets, firms with less collateral to offer might see the terms they face in the debt market worsen more than others when market interest rates rise. This is generally referred to as the broad credit channel.

This article provides an overview of new findings on the existence of these two channels based on results from a series of studies on firm investment in the euro area.

As in its companion paper (Ehrmann et al., this issue), it reports on the main findings obtained by the Eurosystem Monetary Transmission Network. This network has produced a set of comparable papers investigating the determinants of investment. A special feature of these papers is their use of firm panel datasets to identify the relative importance of the interest rate and broad credit channels. In addition, they use similar methodologies. Especially in the study by Chatelain et al. (2001), the results of harmonised regressions are compared among the four largest countries in the euro area. Although harmonised regressions make comparison between countries more straightforward, they need to be interpreted with caution. By default, individual country idiosyncrasies are not taken into account. Individual country results in which country-specific features are taken into account are available for Germany [START_REF] Von Kalckreuth | Monetary Transmission in Germany: New Perspectives on Financial Constraints and Investment Spending[END_REF], France [START_REF] Chatelain | Investment, the Cost of Capital and Monetary Policy in the Nineties in France: A Panel Data Investigation[END_REF], Italy [START_REF] Gaiotti | Does Monetary Policy Have Asymmetric Effects? A Look at the Investment Decisions of Italian Firms[END_REF], Belgium [START_REF] Butzen | The Interest Rate and Credit Channel in Belgium: an Investigation with Micro-Level Firm Data[END_REF], Austria [START_REF] Valderrama | Credit Channel and Investment Behavior in Austria: A Micro-econometric approach[END_REF] and Luxembourg [START_REF] Lünnemann | Monetary Transmission: Empirical Evidence from Luxembourg Firm Level Data[END_REF].

The rest of the article is structured as follows. In section II we describe the motivation behind the analysis. In section III we describe the theoretical framework. In section IV we present data and regression results. In section V we provide an assessment of the magnitude of the interest rate channel. In section VI we report on evidence of whether a broad credit channel is operative in the euro area. Finally, conclusions are provided in the last section of the article.

II. MOTIVATION

The interest of central banks in the causes of investment fluctuations is natural.

Investment fluctuations probably represent the most important part of cyclical changes in the business cycle in the euro area. [START_REF] Vermeulen | Business fixed investment: evidence of a financial accelerator in Europe[END_REF] reports that the decline in gross fixed capital formation was the major cause of the decline in total domestic expenditure during the downturn in the early 1990s in Europe. In addition, investment fluctuations are generally believed to be closely linked to monetary policy.

Macroeconomic models at central banks also assign a large role to investment fluctuations after monetary policy shocks. [START_REF] Van Els | Monetary Policy transmission in the Euro area: What do aggregate and national structural models tell us?[END_REF] report on simulations carried out using central bank aggregate models. They find that after an increase in the policy-controlled interest rate in the euro area, the decrease in investment accounts for between 30 to 50 percent of the drop in GDP during the first year. This share rises to as much as 80 percent of the fall in GDP after three years.

Policy-makers at central banks are naturally interested in the channels through which interest rate changes affect investment. They need to know whether only the interest rate channel is at work or whether the condition of a firm's balance sheet also matters after monetary policy shocks. If balance sheet conditions are important, financial condition indicators might provide useful additional information beyond traditional interest rate information. Macro-models usually do not include balance sheet information on firms and are therefore ill-suited to testing for the presence of broad credit channel effects.

Microanalysis of firm investment is generally better suited to identifying the different channels through which investment is affected. In studies using aggregate investment data, it has often been hard to identify user cost effects on investment and therefore the existence of an interest rate channel. Micro data on firms have a major advantage in providing user cost variation in both time and cross-section dimensions. In addition, the broad credit channel emphasises the effect of balance sheet strength on investment response. Firm datasets allow separating firms with weak balance sheets from firms with strong ones. This makes it possible to identify broad credit channel effects. Below we describe the theoretical framework used by the Eurosystem researchers in identifying both channels.

III. THE THEORETICAL FRAMEWORK

The models in the investment literature can be divided into two broad categories depending on whether dynamic elements are treated implicitly or explicitly (Chirinko, 1993a(Chirinko, , 1993b)). Models are included in the latter category if dynamic elements appear explicitly in the optimisation problem and if the estimated coefficients are linked explicitly to the underlying technology and expectation parameters. The implicit category contains those investment models that do not meet these criteria. Explicit models have the notable advantages of being based on a choice-theoretic framework and having coefficients in the econometric equation that can be identified with technology and expectation parameters. However, their empirical performance has been disappointing, their parameter estimates appear fragile in many applications, and they do not provide a framework for assessing either the interest rate or broad credit channels for investment. Consequently, the Eurosystem researchers have opted for an implicit model. The model platform corresponds to that used recently by, among others, [START_REF] Bond | Investment, R&D and Financial Constraints in Britain and Germany[END_REF][START_REF] Bond | Financial Factors and Investment in Belgium, France, Germany, and the United Kingdom: A Comparison Using Company Panel Data[END_REF], [START_REF] Chirinko | How Responsive is Business Capital Formation to its User Cost? An Exploration with Micro Data[END_REF], [START_REF] Hall | Firm-Level Investment in France and the United States: an Exploration of What we Have Learned in Twenty Years[END_REF], 2001[START_REF] Von Kalckreuth | Exploring the Role of Uncertainty for Corporate Investment Decisions in Germany[END_REF], [START_REF] Harhoff | Investment and Taxation in Germany: Evidence from Firm-Level Panel Data[END_REF], as well as [START_REF] Bloom | The dynamics of investment under uncertainty[END_REF]. The point of departure is the static neo-classical equation for capital demand. Using a generalised CES production function, [START_REF] Eisner | Investment Behavior and the Neo-Classical Theory[END_REF] derive the following linear equation from the first-order conditions of profit maximisation:

t t t t h UC S K log log log log + σ - θ = , (1)
with In the following, it is assumed that the production possibilities are given by the capital stock at the beginning of the current period. Taking account of installation costs and the short-run dynamics in the formation of expectations, one can generalise the static capital-demand equation by using distributed lags. For firm i, this results in:

      ν σ - + σ = θ 1 and ( ) σ ν - σ να ⋅ = 1 t t A h , (2)
t i t i t i t i t i UC L C S L B K L A , , , , log ) ( log ) ( log ) ( ε + λ + φ + ∆ σ + ∆ θ = ∆ , ( 4
) ) ( ), ( ), ( L C L B L A
being polynomials in the lag operator, not necessarily of the same degree. The latent term is composed of a firm-specific constant i φ that reflects depreciation and a possible trend in the capital-demand equation ( 1), a time-specific shock t λ equal for all firms, and finally, an idiosyncratic transitory shock t i, ε . In this quite general specification, the data are allowed to determine the adaptation dynamics.

Imposing the additional constraints

1 ) 1 ( ) 1 ( = A B and 1 ) 1 ( ) 1 ( = A C
, the long-run effects of changes in the level of sales or user costs are the same as in the static model (1). In the regression, the growth rates

t i K , log ∆
and lags are substituted by current and lagged values of investment outlays per unit of capital,

1 , , - t i t i K I .
In addition, one can also estimate an extension of equation ( 4). Contemporaneous and lagged real cash-flow per unit of capital,

1 , , - t i t i K CF
are added as further regressors to capture financial constraints and possible dynamics of expectation formation.1 This yields:

t i t i t i t i t i t i t i K CF L D UC L C S L B K L A , 1 , , , , , ) ( log ) ( log ) ( log ) ( ε + λ + φ + + ∆ σ + ∆ θ = ∆ - (5)
Following [START_REF] Auerbach | Taxation, corporate financial policy and the cost of capital[END_REF] and [START_REF] Hayashi | The Cost of Capital, Q, and the Theory of Investment[END_REF], one can obtain a weighted-average definition of the user cost of capital where the cost of debt and equity are weighted by their respective share of the total liabilities of the firm. Chatelain et al. (2001) use the accounting proportions of debt and equity which matters for taxation:

] ) 1 ( ) )( ( ) 1 )( ( [ ) 1 ( ) 1 ( 1 s st I I st s it it it t t it it it it t s t t st I st it P P E D E LD E D D AI z itc p p UC δ + ∆ δ - - + + τ - + τ - τ - - = + (6)
where s is the sector-specific index, st p the price of final goods, I st p the price of capital goods of sector s; t τ the corporate income tax rate, against which interest payments and depreciation are assumed to be deductible, z the present value of depreciation allowances, and itc the investment tax credit. AI is the apparent interest rate, measured as interest payment over gross debt, LD the long-term debt rate used as a proxy for the opportunity cost of equity, E the book value of equity, D the book value of debt, and s δ the industry-specific rate of economic depreciation.

IV. THE EMPIRICAL ANALYSIS

The analysis of monetary transmission through firm investment within the Eurosystem's Monetary Transmission Network has been mainly based on the use of firm balance sheet data collected by the NCBs of the euro area. The richness of these databases (in particular, in their cross-sectional dimension)2 makes them especially suitable for identifying the different transmission channels. Although these databases differ in many respects, in general, their coverage of the population of small and unlisted companies is higher than that which is standard in comparable empirical studies. And, in spite of being generally biased towards larger firms, these databases are reasonably representative of the corporate sector of these countries.

In order to better address the challenge of making cross-country comparisons on the impact of monetary policy on firm investment, Chatelain et al. ( 2001) make a further harmonising effort. In that paper, which focuses on the four largest economies in the euro area, additional attempts are made to standardise, to the extent feasible, the definition of the variables and to homogenise the composition of the samples. In particular, the analysis is restricted to the manufacturing sector and the period covered by the samples used in the estimations is 1985-1999, with the exception of Germany for which the time period available for estimation is 1988-1997. The total number of observations is over 200,000 and the coverage of the samples ranges from 19 percent for Spain to 42 percent for Germany3 .

Chatelain et al. ( 2001) estimate a common specification, using the same estimation method, for the four largest countries of the euro area, France, Germany, Italy and Spain. Thus, the comparison of results obtained across countries provides a first indication of the existence of asymmetries in the transmission mechanism through firm investment4 . Furthermore, companion research undertaken within the Monetary Transmission Network of the Eurosystem [START_REF] Valderrama | Credit Channel and Investment Behavior in Austria: A Micro-econometric approach[END_REF][START_REF] Butzen | The Interest Rate and Credit Channel in Belgium: an Investigation with Micro-Level Firm Data[END_REF][START_REF] Lünnemann | Monetary Transmission: Empirical Evidence from Luxembourg Firm Level Data[END_REF] performs a similar analysis for other countries in the euro area. These papers, although adopting a country-specific approach in order to take into account national particularities, share with Chatelain et al. the main elements of the analytical framework and, therefore, offer a set of comparable results for other countries in the euro area5 . An important difference from Chatelain et al. (2001) is that the firms in the samples used in these papers are not exclusively from the manufacturing sector.

Table 1 provides summary statistics of the studies by Chatelain et al. (2001), [START_REF] Valderrama | Credit Channel and Investment Behavior in Austria: A Micro-econometric approach[END_REF], [START_REF] Butzen | The Interest Rate and Credit Channel in Belgium: an Investigation with Micro-Level Firm Data[END_REF] and Lünneman and Mathä (2001). The table

shows the investment/capital ratio, real sales growth, real user cost growth and the cash-flow/capital ratio. The summary statistics for the four largest euro area countries are broadly similar, revealing that manufacturing firms in those countries are comparable. The average investment rate for French and Italian firms is somewhat lower than German and Spanish firms. Spanish firms also have the highest sales growth rate. The better performance of Spanish firms in terms of both investment and sales probably reflects the overall growth performance of Spain in the 1990s. Italian firms have lower cash-flows. Comparisons of national averages have to be interpreted with caution however, because, as is usually the case with panel data, there is a wide dispersion of the variables in all the countries studied. The summary statistics of the Austrian, Belgian and Luxembourg samples are not strictly comparable. For instance, 40% of the firms in the Luxembourg sample are outside the manufacturing sector. This might explain the relatively low average investment-to-capital ratio. Equations ( 4) and ( 5) have been estimated in a harmonised fashion in Chatelain et al. (2001) for France, Germany, Italy and Spain. The results are summarised in panel A of table 2. A similar analysis is conducted in the country-specific studies of [START_REF] Valderrama | Credit Channel and Investment Behavior in Austria: A Micro-econometric approach[END_REF], [START_REF] Butzen | The Interest Rate and Credit Channel in Belgium: an Investigation with Micro-Level Firm Data[END_REF] and [START_REF] Lünnemann | Monetary Transmission: Empirical Evidence from Luxembourg Firm Level Data[END_REF] for Austria, Belgium and Luxembourg, respectively. In these papers, comparable results to those presented in Chatelain et al. can be found and are reported in panel B of table 2. In both panels, we limit ourselves to reporting only the estimated long-run coefficients that represent the long-run elasticity of the capital stock with respect to the regressors6 . User costs have a significant long-run effect on capital stock in Germany, Italy, Belgium and Luxembourg, and for France when cash-flow is dropped from the regression. Moreover, Chatelain et al (2001) note that it is also significant for Spain when non-significant lags are excluded. The point estimates of the long-run elasticity of capital stock with respect to user costs lay in a range of -0.03 to -0.66. In a recent study using similar methodology, [START_REF] Chirinko | How Responsive is Business Capital Formation to its User Cost? An Exploration with Micro Data[END_REF] find an elasticity of -0.25 for US manufacturing firms. However, they provide other estimates in the neighbourhood of -0.40 using another methodology [START_REF] Chirinko | That elusive elasticity of capital: A long-panel approach to estimating the price sensitivity of business capital[END_REF]. [START_REF] Chatelain | Investment, the Cost of Capital and Monetary Policy in the Nineties in France: A Panel Data Investigation[END_REF] suggest a reason for the loss in significance of long-run user cost elasticity in France when the cash-flow/capital ratio enters the regression. They argue that this is due to an omitted variables bias. When not only cash-flow but also cash-flow multiplied by dummy variables related to the risk of bankruptcy of firms (e.g. using Fisher discriminant analysis with several financial ratios) are included in the specification, long-run user cost elasticity becomes significant again with a value similar to that in the specification without cash-flow. They then find that about 80% of the firms in the French sample exhibit an investment cash-flow sensitivity significant but close to zero (similar to firms in the German sample), with a user cost elasticity significant but at a much lower level (below -0.26) than in Germany. By contrast, risky firms (about 20% of the French sample) exhibit investment cash-flow sensitivity and user cost elasticity rather similar to that in the Italian and Spanish samples. This French result is particularly interesting since it demonstrates that even within a relatively homogenous sample of manufacturing firms, sub-samples of firms can behave quite differently from the average firm in the sample.

It is interesting to see that Germany has the lowest cash-flow sensitivity among the large European countries, whereas its user cost elasticity is highest. Furthermore, von [START_REF] Von Kalckreuth | Monetary Transmission in Germany: New Perspectives on Financial Constraints and Investment Spending[END_REF] does not find significant differences between the investment behaviour of large and small firms in Germany. He argues that it is possible to interpret this result as being related to a special feature of the German financial system, the importance of so-called house banks7 . The term "house bank" is used to denote a credit institution which enters into a long-term business relationship with a certain company based on an intensive exchange of information and implied insurance against liquidity shortfalls and sharp fluctuations in refinancing costs. It is possible that relationship banking has helped firms, especially small firms, to overcome informational asymmetries that otherwise would have distorted their investment decisions.

Overall, the significance of the user cost variable in the regressions suggests that monetary policy has a significant influence on corporate investment in the euro area through the interest rate channel. However, the exact amount of user cost elasticity remains sensitive to the inclusion of cash-flow in the regression (and to the sample of firms).

Sales growth, as a proxy for expected demand, has a substantial impact on the investment rate and, with the exceptions of Austria and Spain, this effect is significant even in models including cash-flow. Consistent with most of the previous empirical literature, cash-flow is significant in each country, with the exception of Luxembourg8 . The potential relevance of a broad credit channel is addressed in section VI.

V. THE DYNAMICS OF INVESTMENT

In this section we present the short-run profile of the investment/capital ratio in the different euro area countries after shocks to user costs and cash-flow. The profile of the investment/capital ratio after shocks to user costs is especially interesting for policy-makers at central banks. In principal, policy-makers can influence user costs by changing the market interest rate. Changes in market interest rates should lead to changes in user costs since they are an element of user costs. Clearly, user costs also contain other elements such as tax parameters, price variables and depreciation.

Changes in market interest rates therefore do not lead to one-to-one changes in user costs. Chatelain et al. (2001) provide estimates of the elasticity of user costs to changes in the market interest rate ranging from 0.32 in Germany to 0.70 in Italy. (a) For France the coefficients are from the regression without cash-flow . Sources: Chatelain et al. (2001); The figures for Austria, Belgium and Luxembourg have been obtained from the estimates reported in [START_REF] Valderrama | Credit Channel and Investment Behavior in Austria: A Micro-econometric approach[END_REF], [START_REF] Butzen | The Interest Rate and Credit Channel in Belgium: an Investigation with Micro-Level Firm Data[END_REF] and [START_REF] Lünnemann | Monetary Transmission: Empirical Evidence from Luxembourg Firm Level Data[END_REF], respectively.

Table 3 shows the effect of a one percentage point change in the growth rate of user costs on the investment/capital ratio. Although the magnitude of the effects differs quite a lot across countries (as was evident from table 2), the dynamic pattern is quite similar across countries. The largest effect occurs in the first year. Most of the effect disappears after two years.

Table 4 shows the effect of a one percentage point change in the cash flowcapital ratio. In general, the contemporaneous impact is significant and most of the effect is also here disappeared after the second year. On the whole these effects, although relatively small in most countries, are by no means negligible9 . Chatelain et al. (2001); The figures for Austria, Belgium and Luxembourg have been obtained from the estimates reported in [START_REF] Valderrama | Credit Channel and Investment Behavior in Austria: A Micro-econometric approach[END_REF], [START_REF] Butzen | The Interest Rate and Credit Channel in Belgium: an Investigation with Micro-Level Firm Data[END_REF] and [START_REF] Lünnemann | Monetary Transmission: Empirical Evidence from Luxembourg Firm Level Data[END_REF], respectively.

VI.

THE BROAD CREDIT CHANNEL.

In this section we report on the results of tests of whether different categories of firms show different investment reactions depending on their liquidity.

Table 5 provides an overview of the effect of liquidity on investment in six euro area countries including results from [START_REF] Valderrama | Credit Channel and Investment Behavior in Austria: A Micro-econometric approach[END_REF], [START_REF] Butzen | The Interest Rate and Credit Channel in Belgium: an Investigation with Micro-Level Firm Data[END_REF], Chatelain andTiomo (2001), von Kalckreuth (2001), [START_REF] Lünnemann | Monetary Transmission: Empirical Evidence from Luxembourg Firm Level Data[END_REF] and [START_REF] Gaiotti | Does Monetary Policy Have Asymmetric Effects? A Look at the Investment Decisions of Italian Firms[END_REF]. These studies tested the differences in the coefficient estimates of the cash-flow/capital ratio10 . By testing whether the long-run effect of the cashflow/capital ratio is significantly different for different groups of firms, they were able to compare the behaviour of firms that tend to have weaker balance sheets with that of other firms. This test aims at identifying possible broad credit channel effects. No single criterion was found to be relevant in all countries by which firms are more sensitive to liquidity. For example, size in itself does not seem to be sufficient to capture the presence of differences in external finance premia11 . However, all criteria that are relevant in individual countries point towards firms with less potential to provide adequate collateral. We take these differences in liquidity sensitivity to be indicative of potential broad credit channel effects. For instance, in both France and Germany, firms with poor credit ratings show higher cash-flow sensitivity. In Italy and Belgium, small firms are more sensitive to cash-flow. Also, particular industries could be more vulnerable to credit channel effects. Small services firms in Belgium and equipment manufacturers in France are found to be more liquidity sensitive. For Austria, firms without a close relationship to a house bank show higher liquidity sensitivity. In the case of Spain, although according to Chatelain et al. (2001) the investment behaviour of small firms does not display a significantly higher cash-flow sensitivity than that of large firms, the large magnitude of the cash-flow effect, its robustness to changes in the specification and the evidence obtained for the Spanish economy using alternative specifications, datasets, sample periods and split criteria seem to support the relevance of informational asymmetries to explain investment behaviour12 .

For Germany, it was possible to obtain evidence of the effect of financial constraints that does not rely on the traditional method of comparing cash-flow sensitivities between supposedly constrained and unconstrained firms. Von [START_REF] Von Kalckreuth | Monetary Transmission in Germany: New Perspectives on Financial Constraints and Investment Spending[END_REF] views the rating score (overall rating ratio, ORR) of the Deutsche Bundesbank -being the result of a discriminant analysis -as a direct measure of the perceived default risk as seen from the perspective of the lender. Therefore, it ought to vary with the external finance premium, if there is one. Technically, the costs of finance are modelled as the risk free user costs of capital, augmented by a mark up that is a function of the rating score, ORR. An autoregressive distributed lag equation was estimated with both the change in ORR and the growth rate of a risk-free user cost variable that was constructed according to the principles laid down by King and Fullerton (1984). Since user costs enter the model with a negative effect, we would anticipate that ∆ORR has a positive coefficient, indicating that investment increases with the credit rating. This effect could actually be verified. Using a GMM procedure, the rating variable was found to be significant on the 1% level. The size of the coefficient indicates that rating is of importance for capital accumulation, although not overwhelmingly so: a permanent increase in rating by one standard deviation would increase capital demand by 2.4% in the long run.

Overall, considering the above results, one can be reasonably confident that internal funds are a crucial determinant of firms' investment, with an effect that is stronger for firms that are more likely to face financial constraints, so that the credit channel in the euro area is also operative.

VII. CONCLUSION

This article has presented a comparable set of results on the monetary transmission channel on investment in the euro area countries. These results have been obtained by the Eurosystem Monetary Transmission Network in a series of papers using firm panel datasets. Evidence for both interest rate channel effects and broad credit channel effects is found. There is clear evidence that user costs matter for investment. Most of the effect of user cost changes on investment can be found during the first two years.

Investment by various firms responds differently depending on cash-flow and liquidity variables. Investment by firms with weaker balance sheets especially displays higher liquidity sensitivity indicating the presence of broad credit channel effects.

  Notes: (a) For Austria and Luxembourg, liquid assets instead of cash-flow. (b) The reported statistics are those corresponding to the sample of large firms. (c) For Belgium, value-added growth instead of sales growth. Sources: Panel A: Chatelain et al. (2001); Panel B : Austria: Valderrama (2001); Belgium: Butzen et al. (2001); Luxembourg: Lünneman and Mathä (2001).

Belgium

  Notes: */** denotes significance at the 5%/1% levels. Numbers in brackets are standard errors. Estimation method: two step GMM of equation (1) and (2) in first differences. For details on the estimation technique and the instrumentation seeChatelain et al (2001). (a) Model (2) for Austria and Luxembourg includes liquid assets instead of cash-flow. (b) The results for Belgium are those corresponding to the sample of manufacturing firms. (c) For Belgium, value-added growth instead of sales growth. (d) In the case of Luxembourg the estimation method is OLS. Sources: Panel A: Chatelain et al. (2001); Panel B : Austria: Valderrama (2001); Belgium: Butzen et al. (2001); Luxembourg: Lünneman and Mathä (2001).

Table 1 : Summary Statistics

 1

		I/K	∆log S (c)	∆log UC	CF/K (a)
		Mean Std.d.	Mean Std.d.	Mean Std.d. Mean Std.d.
	Panel A:
	France	0.122 0.141	0.029 0.153	-0.009 0.140	0.330 0.330
	Germany	0.181 0.219	0.021 0.158	0.025 0.110	0.276 0.464
	Italy	0.124 0.155	0.034 0.196	-0.012 0.263	0.196 0.220
	Spain	0.186 0.217	0.043 0.171	0.006 0.150	0.370 0.469
	Panel B:
	Austria (a)	0.099 0.091	0.032 0.145	n.a.	n.a.	0.484 0.330
	Belgium (b)(c)	0.169 0.233	-0.013 0.323	-0.009 0.559	0.346 0.657
	Luxembourg (a)	0.052 0.227	0.003 0.180	-0.052 0.347	0.822 3.313

Table 2

 2

	: Long-run coefficients
		Equation (4)		Equation (5)
		Without CF/K in the regression		With CF/K in the regression
		Sales (c)	User cost	Sales (c)	User cost	Cash-flow (a)
	Panel A: Estimation results with an harmonised approach
	France	0.35**	-0.11**	0.12**	-0.03	0.20**
		(0.04)	(0.05)	(0.05)	(0.04)	(0.04)
	Germany	0.45**	-0.66**	0.39**	-0.52**	0.08*
		(0.07)	(0.17)	(0.08)	(0.15)	(0.03)
	Italy	0.29**	-0.11**	0.14**	-0.20**	0.30**
		(0.05)	(0.04)	(0.05)	(0.06)	(0.03)
	Spain	0.14*	-0.26	0.02	-0.28	0.15**
		(0.09)	(0.20)	(0.09)	(0.20)	(0.04)
	Panel B: Estimation results of firm studies for single countries
	Austria (a)	0.27**	-0.14	0.02	-0.04	0.28**
		(0.02)	(0.39)	(0.84)	(0.72)	(0.00)

Table 3 . Effect of a one-time 1 percentage point change in the user cost growth on I/K (*)

 3

		Germany	France(a)	Italy	Spain
	t	-0.22	-0.06	-0.08	-0.28
	t+1	-0.18	-0.05	-0.07	-0.05
	t+2	-0.08	-0.01	-0.03	0.03
	t+3	-0.03	0.00	-0.01	0.02
		Austria	Belgium	Luxembourg
	t	-0.02	-0.01	-0.15
	t+1	-0.02	-0.01	-0.04
	t+2	0.00	-0.01	-0.01
	t+3	0.00	-0.01	0.00
	Notes: (*) Figures represent the deviation of I/K after a 1% increase in the user cost
	growth rate.

Table 4 . Effect on I/K of a one-time 1 percentage point change in CF/K (*)

 4

		Germany	France	Italy	Spain
	t	0.04	0.06	0.26	0.12
	t+1	0.02	0.10	0.02	0.05
	t+2	0.01	0.03	0.02	-0.01
	t+3	0.01	0.01	0.01	-0.01
		Austria	Belgium	Luxembourg
	t	0.20	0.04	0.00
	t+1	0.08	0.10	0.00
	t+2	0.00	0.01	0.00
	t+3	0.00	0.04	0.00
	Note: (*) Figures represent the deviation of I/K after a 1 percentage point increase in
	CF/K.
	Sources:

Table 5 : Country findings for broad credit channel effects

 5

		Model specification	Results: relevant firm
				characteristics
		Sample period	No. of obs	Highest Liquidity sensitivity
	Austria	1994-1999	2652	Firms with no close relationship to
				a house bank
	Belgium	1986-1998	157547	Small services, manufacturing
	France	1986-1999	61237	Equipment manufacturing, firms
				with bad credit rating
	Germany	1988-1997	44345	Firms with bad credit rating
	Italy	1984-1999	43912	Small firms, firms with high share
				of intangible assets
	Luxembourg	1992-1998	517	Young firms
	Sources: Austria: Valderrama (2001); Belgium: Butzen et al. (2001); France: Chatelain and
	Tiomo (2001); Germany: von Kalckreuth (2001); Italy: Gaiotti and Generale (2001);
	Luxembourg (2001) Lünnemann and Mathä (2001).

[START_REF] Chirinko | That elusive elasticity of capital: A long-panel approach to estimating the price sensitivity of business capital[END_REF] develop this equation by introducing delivery lags and adaptive expectations. However, as in all implicit models, the parameters belonging to the expectation formation mechanism are not separately identified.

In this respect, the annual accounts database of the National Bank of Belgium used in[START_REF] Butzen | The Interest Rate and Credit Channel in Belgium: an Investigation with Micro-Level Firm Data[END_REF], which covers almost the entire population of Belgian firms, is especially remarkable.

See Chatelain et al. (2001) for additional details on the datasets.

Note, however, that since cross-country comparisons are not performed on a pooled dataset, a formal test on the significance of the differences is precluded.

See also[START_REF] Chatelain | Investment, the Cost of Capital and Monetary Policy in the Nineties in France: A Panel Data Investigation[END_REF],[START_REF] Gaiotti | Does Monetary Policy Have Asymmetric Effects? A Look at the Investment Decisions of Italian Firms[END_REF] Generale (2001) and[START_REF] Von Kalckreuth | Monetary Transmission in Germany: New Perspectives on Financial Constraints and Investment Spending[END_REF] for country-specific studies on the link between monetary policy and firm investment in France, Italy and Germany, respectively.

The long-run coefficients are calculated as the sum of the coefficients of the various lags of the indicated variable, divided by one minus the sum of the coefficients on the lagged endogenous variable.

See von Kalckreuth (2001) and Deutsche[START_REF] Bundesbank | Monetary Policy and Investment Behaviour, an Empirical Study[END_REF]. For the house banking principle, see[START_REF] Elsas | Is Relationship Lending Special? Evidence from Credit File Data in Germany[END_REF].

Even in this case, a significant (although small) coefficient for cash-flow is obtained, when using a Within estimator.

See Chatelain et al. (2001) for an estimation of the effect of an increase in interest rate on investment, induced through a reduction in cash-flow.

The Austrian and Luxembourg study did not use cash-flow but rather cash stock as their liquidity measure.

[START_REF] Hernando | Financial constraints and investment in France and Spain: a comparison using firm level data[END_REF], estimating the investment Euler equation model of[START_REF] Bond | Dynamic investment models and the firm's financial policy[END_REF] for two samples of firms (French and Spanish) very similar to the ones used in this study, find a significantly higher cash-flow sensitivity for those firms paying zero dividends.[START_REF] Chatelain | Comparing several specifications of financial constraints and adjustment costs in investment Euler equations[END_REF] estimating the investment Euler equation model where the financial constraint is designed according to[START_REF] Estrada | Investment and financial structure in Spanish manufacturing firms[END_REF] for a French sample, find similar results for low dividend payout firms and for debt increasing firms, using tests of misspecification of cash-flow for various groups of firms. Thus, it can be argued that size might not always be the best indicator of informational asymmetries.

See, for instance, Alonso-Borrego (1994),[START_REF] Estrada | Investment and financial structure in Spanish manufacturing firms[END_REF] and[START_REF] Hernando | Financial constraints and investment in France and Spain: a comparison using firm level data[END_REF].


Bernard Girau
email: bernard.girau@loria.fr

Amine Boumaza
email: amine.boumaza@loria.fr

EMBEDDED HARMONIC CONTROL FOR DYNAMIC TRAJECTORY PLANNING ON FPGA

This paper presents a parallel hardware implementation of a well-known navigation control method on reconfigurable digital circuits. Trajectories are estimated after an iterated computation of the harmonic functions, given the goal and obstacle positions of the navigation problem. The proposed massively distributed implementation locally computes the direction to choose to get to the goal position at any point of the environment. Changes in this environment may be immediately taken into account, for example when obstacles are discovered during an on-line exploration. The implementation results show that the proposed architecture simultaneously improves speed, power consumption, precision, and environment size.

Introduction

Trajectory planning consists in finding a way to get from a starting position to a goal position while avoiding obstacles within a given environment or navigation space.

Harmonic functions have been proposed as potential fields for trajectory planning in [START_REF] Connolly | Path planning using laplace's equation[END_REF]. This "harmonic control" approach was motivated by the fact that harmonic functions do not have local extrema (unlike other potential based methods as in [START_REF] Khatib | Real-time obstacle avoidance for manipulators and mobile robots[END_REF]). In this approach, obstacles correspond to maxima of the potential, while goals correspond to minima. Control algorithms then reduce to locally descend the potential until they reach a global minimum.

Harmonic control has had some impact on the robotics community [START_REF] Masoud | Motion planning in the presence of directional and obstacle avoidance constraints using nonlinear anisotropic, harmonic potential fields: A physical metaphor[END_REF][START_REF] Zelek | Complete real-time path planning during sensor-based discovery[END_REF][START_REF] Alvarez | Online motion planning using laplace potential fields[END_REF][START_REF] Hjs Feder | Real-time path planning using harmonic potentials in dynamic environments[END_REF][START_REF] Huber | A control basis for multilegged walking[END_REF][START_REF] Kazemi | An incremental harmonic function-based probabilistic roadmap approach to robot path planning[END_REF][START_REF] Sweeney | Scalability and schedulability in large, coordinated, distributed robot systems[END_REF][START_REF] Wang | A new potential field method for robot path planning[END_REF]. Nevertheless, very few hardware implementations have been proposed. They are usually analog, therefore they suffer from a very long and complex design process, and a lack of flexibility (environment size, precision). This paper proposes an embeddable digital implementation on reconfigurable circuits, so as to reduce design time and to improve flexibility.

The proposed implementation maps the massively distributed structure of the space-discretized estimation of the harmonic function onto the circuit. An area-saving serial arithmetic is used, within a global scheme that simultaneously ensures pipelining and parallelism of the iterative computations. The decision process is also taken into account, through local estimations of the direction to take from any point.

Though the speed performance reported in this paper outperform software implementations, the main advantages of the described architecture is to provide low-power and scalable solutions that are able to handle the very large precisions required by harmonic control within large navigation environments.

Section 2 describes the principles of harmonic functions and of their use for trajectory planning. Section 3 summarizes the advantages of hardware parallel implementations on FPGAs for embedded navigation in autonomous systems, and it justifies the choice of serial arithmetics, especially with respect to the precision requirements of harmonic control. The proposed hardware architecture and its implementation results are described in section 4.

Harmonic control

In this part, we begin by a brief reminder of harmonic functions and some of their properties after which we will discuss their application to the navigation problem.

Harmonic functions

Let Ω be an open subset of IR n , ∂Ω its boundary and Ω its closure such that Ω = Ω ∪ ∂Ω.

Definition Let u : Ω → IR be a real function, twice continuously differentiable, and

Ω ⊆ IR n with n > 1. The function u is harmonic iff ∆u = n i=1 ∂ 2 u ∂x 2 i = 0.
This equation is known as Laplace's equation. Harmonic functions satisfy interesting properties:

• The maximum principle states that: if u is a nonconstant continuous function on Ω that is harmonic on Ω, then u attains its maximum and minimum values over Ω on ∂Ω.

• Applying the divergence theorem on harmonic functions, the following equation holds:

s ∇u • n ds = 0
where s is the boundary of a closed region strictly in Ω and n is a normal vector of s. This equation expresses that the normal flux of the gradient vector field through the region bounded by s is zero. It follows that there can be no local minimum or maximum of the potential inside a bounded region of Ω.

Figure 1. Trajectories generated by harmonic control (100 × 100 grid, equally spread starting points, two goals).

Application to navigation

To solve the navigation problem using harmonic functions, we consider the problem as a Dirichlet problem: its solution is to find the function u that is harmonic on Ω (the navigation space) and that satisfies boundary conditions on ∂Ω (obstacles and navigation goal):

∀x ∈ Ω, ∆u (x) = 0 ∀x ∈ ∂Ω, u (x) = g (x)
where the function g : ∂Ω → IR represents boundary conditions on ∂Ω. These conditions define the values of the navigation function on obstacles and goals. Without loss of generality we choose g(x) = 1 for obstacles and g(x) = 0 for goals. Solving the Dirichlet problem consists in finding the function u that is harmonic on Ω and that has value 1 on obstacle positions and value 0 on goal positions.

The navigation problem is then solved as follows: a simple descent along the gradient of u provides a trajectory towards a given goal from any starting position. The properties of harmonic functions ensure that such a path exists and it is free of local optima.

Numerical method to solve Laplace's equation

In this part we consider the case where Ω = IR 2 . Traditionally, Laplace's equation is solved using methods from finite differences on a regular grid (discrete sampling of Ω). Using a Taylor approximation of the second derivatives we obtain the following discrete form of u (x 1 , x 2 ):

∆ δ u (x 1 , x 2 ) = 1 δ 2 [u(x 1 + δ, x 2 ) + u(x 1 -δ, y 1 ) +u(x 1 , x 2 + δ) + u(x 1 , x 2 -δ) -4u (x 1 , x 2 )]
where δ is the sampling of the grid that represents Ω. In this form, the equation can be solved using relaxation methods such as Jacobi or Gauss-Seidel whose principle is to iteratively replace each grid point value with the simple average of its four neighbors. Figure 1 shows different trajectories generated by simulations using this numerical scheme.

Properties of harmonic navigation functions

Harmonic navigation functions have many interesting properties which motivated their use in numerous applications especially in robotics [START_REF] Tarassenko | Analogue computation of collision-free paths[END_REF][START_REF] Connolly | On the applications of harmonic functions to robotics[END_REF][START_REF] Huber | A control basis for multilegged walking[END_REF][START_REF] Hjs Feder | Real-time path planning using harmonic potentials in dynamic environments[END_REF][START_REF] Zelek | Complete real-time path planning during sensor-based discovery[END_REF][START_REF] Wang | A new potential field method for robot path planning[END_REF][START_REF] Masoud | Motion planning in the presence of directional and obstacle avoidance constraints using nonlinear anisotropic, harmonic potential fields: A physical metaphor[END_REF][START_REF] Sweeney | Scalability and schedulability in large, coordinated, distributed robot systems[END_REF][START_REF] Alvarez | Online motion planning using laplace potential fields[END_REF][START_REF] Kazemi | An incremental harmonic function-based probabilistic roadmap approach to robot path planning[END_REF]:

Global navigation: Complete trajectories may be generated towards a goal position from anywhere in the environment, since there are no local minima.

Dynamic trajectory planning:

Unexpected updates of the environment may be taken into account, since harmonic functions are computed by iterative relaxation methods. Therefore newly detected obstacles may be integrated in the model as new boundary conditions during computation, so that harmonic control is available in dynamic environments or in environments explored on-line [START_REF] Zelek | Complete real-time path planning during sensor-based discovery[END_REF][START_REF] Boumaza | Mobile robot sensor fusion using flies[END_REF].

Parallel computation: An interesting property of the computations described above is their massively parallel distribution. Computing grid point values only requires local information of the neighboring cells. Fine-grain parallel implementations appear as an opportunity, as discussed in the next section.

Towards an embedded implementation

The results shown in figure 1 were obtained from software simulations carried out on a PC. The aim of our work is to design an embedded system for robot navigation. Computation speed is not the only criterion (trajectory decision must be performed in real time). Power consumption is also essential for autonomous systems. Moreover, computation precision and scalability appear as critical issues for harmonic control, as discussed below. These combined aspects motivate the design of a parallel hardware implementation. In such a work, the number of inputs/outputs and above all the level of parallelism have a direct influence on the obtained implementation consumption and speed. A massively parallel implementation is a real challenge, taking into account constraints such as precision, grid size, dynamic updates, etc.

Implementation environment

Since the appearance of programmable circuits, such as field programmable gate arrays (FPGAs), algorithms may be implemented on fast integrated circuits with softwarelike design principles. Usual VLSI designs lead to very high performances. But the time needed to realize an ASIC (application specific integrated circuit) is too long, especially when different configurations must be tested. The chip production time is usually very long (up to 6 months).

FPGAs, such as Xilinx FGPA ( [START_REF]The Programmable Logic Data Book[END_REF]), are based on a matrix of configurable logic blocks (CLBs). Each CLB is able to implement small logical functions (4 or 5 inputs) with a few elementary memory devices (flip-flops or latches) and some multiplexors. Each CLB is independently programmable. Similarly, the routing structure that connect the CLBs can be configured. A FPGA approach simply adapts to the handled application, whereas a usual VLSI implementation requires costly rebuildings of the whole circuit when changing some characteristics. A design on FPGAs requires the description of several operating blocks. Then the control and the communication schemes are added to the description, and an automatic "compiling" tool maps the described circuit onto the chip.

Technological choices

The main issues when a massively distributed model is mapped onto a FPGA are the huge number of operators, and the routing problems due to the dense interconnections. A first standard technological choice to solve these problems is to use serial arithmetics: smaller operators may be implemented, and they require less connection wires. Another essential technological choice is to estimate the minimum precision required to keep satisfactory results with as small as possible operators and memory resources.

Serial arithmetics

Serial arithmetics correspond to computation architectures where digits are provided digit after digit. Serial arithmetics lead to operators that need small implementation areas and less inputs/outputs, and that easily handle different precisions, without an excessive increase of the implementation area. Serial systems are characterised by their delay, i.e, the number δ such that p digits of the result are deduced from p + δ digits of the input values.

Two main kinds of serial arithmetics are available: LSBF or MSBF (least/most significant bit first). Standard serial operators work in a LSBF mode. Though our model requires the computation of a minimum value (gradient descent), that may only be computed in a MSBF mode, we use standard serial operators to optimize the required area1 .

Computation precision

Software simulation are usually performed to study the precision that is required by an application before its hardware implementation. Precision issues appear as a critical problem for harmonic control, that has already been mentioned as a major limitation for analog implementations [START_REF] Tarassenko | Analogue computation of collision-free paths[END_REF]. Computing a harmonic potential over a large grid may result in gradients that are too small to use because the allowable precision is easily reached. Connolly [START_REF] Connolly | On the applications of harmonic functions to robotics[END_REF] presents a relationship between the precision required for floating point representation on a computer and the number of nodes on the grid. He argues that the precision should at least represent 1 N , where N is the total number of grid points, to circumvent precision problems. This further motivates the use of an embeded implementation in which we can allow very high precisions (see 4.3).

We have carried out experimental simulations with different navigation spaces. They have shown that for some 50 × 50 grids, a 22-bit precision is required at least to ensure significant differences between neighboring grid point values so as to generate correct trajectories. Other simulations have shown that very large grids may require more than 64-bit precision numbers. It should be pointed out that in case of insufficient precision, large areas of the grid may share the same value, which results in wrong trajectories.

Implementations based on serial arithmetics may be more easily extended to larger precisions than implementations based on parallel arithmetics. Since the size of serial adders and comparators does not depend on precision (unlike multipliers and elementary functions), our implementation may handle large precisions by means of rather simple changes in the control modules.

Hardware implementation of harmonic control

Though harmonic control has been widely used in robotics, few hardware implementations have been proposed. Their technological choices are mostly motivated by the fact that analog resistive grids may easily compute the harmonic function as in 2.2.1. For example in [START_REF] Stan | Analog vlsi for robot path planning[END_REF] an analog implementation of a 16 × 16 grid is proposed. The main limitation of this work is the precision (as for most analog implementations). To our knowledge, digital FPGA-based implementations have not yet been proposed. In this work, the discretized computation of the harmonic function is performed to make navigation decisions for a robot in an environment that may be explored on-line. Navigation orders are also discretized: at each position, the robot is ordered to move north, east, west or south. This simplification implies that the optimal trajectory along the gradient is only roughly estimated by elementary movements along the axis'. It still results in global trajectories towards the goal position from any starting point.

General architecture

The fine grain pipelined internal structure of the proposed digital architecture was implemented using fixed point arithmetic. Since grid point values range from 0 to 1, an unsigned representation with 25 bits is chosen, with a 24bit fractional part. As mentioned above, larger wordlengths may easily be handled.

Figure 2 illustrates the general architecture of the implementation of harmonic control for a n × n grid. It con- The role of each component is the following: • Each node computes its corresponding grid point value, as well as the direction (north, east, west or south) to follow to get to the goal position. All nodes use loop computations within an internal pipeline scheme. This scheme is particularly efficient for serial implementations of iterated computations within massively distributed models ( [START_REF] Girau | FPGA implementation of an integrate-and-fire legion model for image segmentation[END_REF]). The control of these computations are synchronized in the whole grid so that nodes may serially communicate their grid point values to their neighbors. In order to simplify the block-diagram of figure 2, only few buses and wires are shown: the signals that carry the grid point values from and to any neighboring node.

• The nodes are split in groups of 4×4 nodes that share common storage resources: a single dual port SRAM block stores the values of the 16 grid points, and its R/W accesses are controlled by a single set of counters (see 4.2).

• The border nodes are very simple. They serially generate the grid point value of obstacles.

• The interaction with the robot has not yet been implemented. It strongly depends on the exact configuration of the robot and of the FPGA board. It includes a position modules, which role is mainly to compute the coordinates (X,Y) of the closest grid point around the real coordinates (x,y) of the robot in its environment.

• The control module generates the enable signals that are sent to all nodes to control their individual behaviour when a non-synchronized event occurs:

• convergence of the computation of the harmonic function (it depends on the local convergence signals computed by all nodes, see 4.2)

• detection of an unknown obstacle (at node (X,Y))

• The decision modules collects the navigation directions locally indicated by each node, and it forwards the selected direction of node (X,Y) to the robot.

In the following subsections, the hardware architecture for the node model and its main components will be described in some detail.

Node implementation

The architecture for the node model is shown in the simplified block diagram on figure 3.

It uses 1bit inputs and outputs to exchange data among nodes and with the global modules. The proposed hardware node model is constituted by five main modules: the iterative computation of the harmonic function is performed by the Update module, the local convergence of this computation is detected by the Cvg module, the local navigation decision is performed by the Direction module, the node receives orders to behave as an obstacle or a goal through the Saturation module, and communication with the dual port SRAM block that stores the grid point value is controled by the Mem module. Figure 3 shows this architecture, as well as its interaction with shared resources (the local Counters and RAM modules are shared by a group of 4 × 4 nodes, and the Enable module is part of the global control module). The functionality of the main modules and their implementation details are described below.

Update: This module performs the iterated computation of the harmonic function value h i,j where (i, j) are the coordinates of the node in the grid. As described in 2.2.1, each iteration computes: hi,j(t + 1) = hi-1,j (t) + hi,j-1(t) + hi+1,j (t) + hi,j+1(t) Direction: This module operates after convergence of the iterations. It computes the minimum grid point value among the 4 neighbors. Its output is the 2-bit code of the direction to choose. This code is computed by the NEWS module that receives the results of the three comparators. This computation is performed in a MSBF mode, thanks to the reverse address counter r_cnt.

Counters: This module is shared by 16 nodes. It generates the read (resp. write) addresses for the dual port RAM by means of counters cntR (resp. cntW). The read address is multiplexed with the reverse counter r_cnt to change from LSBF to MSBF mode. When handling d-bit precision data, these counters are reset each d + 2 cycles (the RAM is written with a 2-clock cycles delay). Value 1 (for obstacles) is computed as a logical function of cntR.

Mem and Saturation:

The local grid point value is not directly the output of the Update module. It may also be a constant 0 or 1 (goal or obstacle). A multiplexer selects the correct value with respect to a control given by the Saturation module that memorizes the sat value to be the constant value of the grid point when the node is selected by the global control module (signal sel).

Implementation results

This work uses a PCI bus board equiped with a Virtex XC2V6000-4FF1517 FPGA from Xilinx, with up to 6,000,000 system gates. Such a FPGA contains 67,584 logic cells, to be compared with the 200,448 ones of the current largest Virtex-4. The design was synthesized, placed and routed automatically in Xilinx Foundation ISE 7.1i. Each node requires 20 logic cells, so that each block of 16 nodes requires 360 logic cells (counters included). On a XC2V6000, 144 dual port SRAM blocks are available, that may be configured as 1K ×18 RAMs to be shared by blocks of 16 nodes. The whole architecture implements a 48 × 48 grid on less than 77 % of the XC2V6000 logic cells. Larger grids (up to 72 × 72) may be implemented on the current largest FPGAs with this approach (the available SRAM blocks being the critical resource).

Software implementations of the harmonic function computation on a microprocessor based computer, Pentium 4, 2 GHz, require around 100 µs per iteration with a 50×50 grid (100 to 1000 iterations are required to converge, depending on the environment). In the proposed hardware implementation, 27 clock cycles are required per iteration, with an estimated clock frequency of 140 MHz. Thus, the architecture provides a speed factor up to 500×, that would even increase with the number of nodes in the grid (sequential vs parallel implementation). But the implementation speed is not the main advantage of our implementation: real-time computation is easily reached by software implementations for such precisions and size grids. Power consumption is a key factor for embedded implementations, and above all very large precisions may be easily handled by the proposed serial implementation (up to 1K bits when fully using the SRAM blocks).

Conclusion and future work

We have presented an embedded architecture to solve the navigation problem in robotics, that computes trajectories along a harmonic potential, using a FPGA implementation. This architecture includes the iterated estimation of the harmonic functions. The goals and obstacles of the navigation problem may be changed during computation. The trajectory decision is also performed on-chip, by means of local computations of the preferred direction at each point of the discretized environment. The proposed architecture uses a massively distributed grid of identical nodes that interact with each other within mutually dependant serial streams of data to perform pipelined iterative updates of the local harmonic function values until global convergence.

The proposed architecture enables us to handle very large precisions using SRAM blocks, which is a benefit over computer resolution of harmonic funtions. Furthemore, a greatly improved speed and a low power consumption are other non-negligible advantages, since the goal is to embed this implementation on mobile robots. The current hardware model is already able to handle significantly large discretized environments.

Future architecture improvements are already considered to extend the capacities of the model to navigation within larger and more complex environments. A studied extention efficiently uses the available SRAM to handle very large grid, by means of an iterated computation mode that is both globally asynchronous and block-synchronous. Current efforts are also made to extend our implementation to optimal control, a more generic (and tunable) trajectory planning method.

Figure 2 .

 2 Figure 2. General architecture

  Inputs are mainly used to receive the neighboring grid point values (signals h_N,h_E,h_W,h_S) and global control signals (standard signals clk,reset,enable, signals sel,sat to indicate obstacle/goal changes, and SRAM controls EN,R,W). Outputs are twofold: the local value h of the harmonic function is sent to all 4 neighbors (signals to_N,to_E,to_W,to_S) and signals NE and WS code the 4 possible directions to which a robot located around this grid point should move.

1 Figure 3 .

 13 Figure 3. Architecture of a node

The only existing radix-2 MSBF serial arithmetics is called on-line arithmetics. It uses a redundant number representations system, which induces less area-saving operators.


Mihail Vladkov
email: mihail.vladkov@lpmcn.univ-lyon1.fr

Jean-Louis Barrat
email: jean-louis.barrat@lpmcn.univ-lyon1.fr

Modelling thermal conductivity and collective effects in a simple nanofluid

Molecular dynamics simulations are used to model the thermal properties of a fluid containing solid nanoparticles (nanofluid). The flexibility of molecular simulation allows us to consider the effects of particle mass, particle-particle and particle-fluid interaction and that of the spatial distribution of the particles on the thermal conductivity. We show that the heat conductivity of a well dispersed nanofluid is well described by the classical Maxwell Garnett equation model. In the case of particle clustering and strong inter particle interactions the conductivity can be again described by effective medium calculation taking into account the aspect ratio of the cluster.

Heat transfer is increased when particles are aligned in the direction of the temperature gradient. This kind of collective effects could be a first step to understand the substantial increase in the conductivity observed in some experiments.

Many experimental studies have suggested that the thermal conductivity of colloidal suspensions referred to as "nanofluids" is unusually high 1,2 . Predictions of effective medium theories are accurate in some cases 3 but generally fail to account for the large enhancement in conductivity. In spite of a large number of -sometimes conflicting or controversial -suggestions and experimental findings 4 , the microscopic mechanisms for such an increase remain unclear. One of the possibilities that was suggested was the effect of Brownian motion 5 , and appeared to be an attractive and generic explanation.

The essential idea is that the Brownian velocity of the suspended particle induces a fluctuating hydrodynamic flow 6,7 , which on average influences (increases) thermal transport. This mechanism is different from transport of heat through center of mass diffusion, the contribution of which was previously shown to be negligible 8 . However, some recent experimental high precision studies reported a normal conductivity in nanoparticle suspensions at very small volume fractions below 1% 9 , questioning the validity of this assumption.

Recent simulations also showed that normal conductivity is expected for low volume fractions (around 3.3%) 11 as well as for volume fractions up to 13% 10 establishing that the physical parameter determining thermal properties should be the particle interfacial thermal resistance.

We also recently established by simulations that Brownian motion and Brownian velocity field have no effect on the heat transfer of a single nanoscopic particle with the surrounding fluid 10 . Thus the explanation of the increase in conductivity is currently to be looked for mostly in some collective effects between particles -a field that has not been studied through microscopic simulation.

In this work we use non equilibrium molecular dynamics "experiments" to explore further the transfer of heat in a model fluid containing nanoparticles. We make use of the flexibility allowed by molecular simulations to explore extreme cases in terms e.g. particle/fluid mass density mismatch. We concentrate on model systems that are expected to be representative of generic properties. We explore a large range of parameters and make a quantitative comparison with effective medium calculations. By studying the thermal conductivity in a system with two particles and by precisely controlling their positions we are able to observe the influence of collective effects consisting in different particle-particle interactions and displacement with respect to the temperature gradient.

We start by describing our simulation methodology and presenting results about the conductivity of a fluid containing a single nanoparticle in a temperature gradient. We then study the conductivity of a system containing two particles varying their positions and the intensity of their interactions.

I. SIMULATION METHOD

The model fluid used in this study is a simple Lennard-Jones liquid. The nanoparticles (solid phase) are obtained by a spherical cut of a bulk FCC crystal. All atoms in our system interact through Lennard-Jones interactions

U lj (r) = 4ε((σ/r) 12 -c(σ/r) 6 ), r ≤ r c 0, r > r c (1)
where r c = 2.5σ. The coefficient c is equal to 1 for atoms belonging to the same phase, but can be adjusted to modify the wetting properties of the liquid on the solid particle 12,13 .

A coefficient of c = 1 defines a wetting interaction while the non wetting case is modelled by c = 0.5. Within the solid particles, atoms are linked with their neighbors through a FENE (Finite extension non-linear elastic) bonding potential:

U F EN E (r) = k 2 R 0 ln(1 -( r R 0 ) 2 ), r < R 0 (2)
where R 0 = 1.5σ and k = 30.0ε/σ 2 . This potential, combined with the Lennard-Jones interaction results in a narrow distribution of the distance between linked atoms around 0.97σ. A solid particle in the fluid was prepared as follows: starting from a FCC bulk arrangement of atoms at zero temperature, the atoms within a sphere were linked to their first neighbors by the FENE bond. Then the system was equilibrated in a constant NVE ensemble with energy value corresponding to a temperature T = 1. A particle contains 555 atoms, surrounded by the atoms of the liquid. The number density in the system is ρ = 0.85σ -3 . As the simulated particles are not exactly spherical, but present some FCC facets, their radius was estimated from the radius of gyration:

R 2 g = 1 N N 1 (r i -r CM ) 2 = 3 5 R 2 p (3)
where R 2 g is the measured radius of gyration of the particle atoms, and the second equality applies to an ideal sphere. Taking σ = 0.3nm this corresponds to a particle radius of order R part ∼ 1.5nm. The solid particles obtained by this procedure behave very closely as ideal harmonic solids. Non linear effects associated with the non linearity of the bonding potential are absent in a wide temperature range. This assumption was verified through equilibrium simulations at different temperatures, monitoring the energy per particle. The observed relation is linear and indicates a particle heat capacity very close to 3k B T N, as for an harmonic ideal solid, in a temperature range from T = 1 to T = 3.5. We study systems of wetting ("hydrophilic", c = 1) and non wetting ("hydrophobic" c = 0.5) particles, different values of the density mismatch between the solid and liquid phase (m p /m l = 1, 50, 100 where m l is the mass of a fluid atom and m p is the mass of a particle atom) and different particle volume fractions.

A. Simulating Heat Flow and Measuring Conductivity

The most simple and direct method of measuring thermal conductivities in a simulation is undoubtedly non equilibrium molecular dynamics. With this set up a temperature gradient is established through the sample and thermal properties are calculated from the measurement of energy fluxes. This method involves locally injecting and evacuating heat in the system. In our study this is achieved by applying two thermostats with different temperatures in the two ends of a fluid slab. The periodicity of the box is maintained in the directions perpendicular to the temperature gradient and the system is non periodic in the direction of the heat flux. If one allows the gradient to change sign in the simulation box a stationary heat flow can be achieved in a fully periodic box 11 . However with this setup special attention should be payed in the regions of abrupt change in the temperature gradient, and the simulated system is less realistic.

The systems we study are periodic in the x and y directions. In the transverse z direction, the liquid is confined by a repulsive potential (ideal flat wall). The thermostats are applied to a fluid slice in the vicinity of the walls of width of around 3 atom diameters.

The thermostat consists in a rescaling of the velocities of the particles currently present in the slice at a given time interval. The temperature measured locally in slices parallel to the heat flow direction shows that a linear profile is established in the fluid slab. The value of the temperature gradient depends on the temperature of the thermostats and their rescaling time constant (see fig. 1). A smaller rescaling interval increases slightly the energy transfer between the thermostat and the fluid thus maintaining the thermostatted layer at an average temperature closer to the temperature of the thermostat. It is thus important to have the same time constant for the thermostat in all the systems in order to compare the values of the energy flow and detect its variations in the different setups. We set this value to dt = 1.5τ in the simulations. Using this heat transfer setup, the local thermal conductivity is given by:

λ(z) = J(z) ∇T (z) (4)
where J(z) is the heat flow and T (z) the temperature. This formula is easily generalized to the whole slab by taking the heat flow at the thermostats and a mean slope of the temperature profile. The problem is that such an estimation of the mean gradient is prone to error. The mean temperature profile for a pure liquid has a well defined slope, even if the fluctuations and the fitting procedure cause error. The situation is worse in the presence of nanoparticles. The temperature profile averaged over liquid and solid atoms has noticeable fluctuations around the particles as the gradient is different in the two phases. That is why measuring an effective value of the conductivity that does not involve fitting and assuming linear temperature profile is preferable. If the thermostat relaxation time is constant for all systems, a well defined values are the two thermostats temperatures. Thus we define the slab conductivity as:

λ ef f = J (T 1 -T 2 )/L ( 5
)
where J is the mean stationary heat flux measured by the thermostats, T i is the temperature of the thermostat i and L is the length of the simulation box in the z dimension.

The conductivity defined in this way is sensitive only to variations in the energy flux and is a precise method for capturing variations in conductivities between systems.

To avoid any effect of thermophoresis or coupling of the thermostat to the particle, the particles are constrained to stay away from the thermostatted regions by tethering weakly their center of mass to a fixed point by harmonic springs of stiffness k = 30.

Controlling the particles position also allows us to study different configurations and explore the influence of the spatial distribution of the particles on the thermal properties.

The temperatures of the two thermostats were T 1 = 2 and T 2 = 1. In order to compare the conductivity results for the different systems they were first equilibrated to the same pressure at a temperature T = 1.5, then a non equilibrium run was performed for about 1500-2000 τ LJ to make sure the pressure stays the same for the systems of different nature and finally a production run of about 15000 τ LJ during which thermostats energy, particle positions and temperature profiles are monitored. In nanoscale systems it has been observed that interfacial effects are very important 10 . In order to compare quantitatively the conductivity variations to the predictions of effective medium calculations it is necessary to know the value of the interfacial (Kapitza) thermal resistance. The values used in our study were determined for different wetting and particle masses by a transient adsorption simulation as explained in ref. 10 . In real units, a value R K = 1 corresponds typically to an interfacial conductance G = 1/R K , of the order of 100MW/Km 2 1 . We study systems 1 The conversion to physical units is made by taking a Lennard-Jones time unit τ LJ = 10 -12 s, and a length unit σ = 0.3nm. The unit for G is energy/temperature/(length) 2 /time. As the energy/temperature ratio is given by the Boltzmann constant k B , we end up with a unit for G equal to with particle volume fraction of 13% or 12%. The volume fraction is defined as the volume of the particle divided by the volume of the fluid outside the thermostats.

II. RESULTS AND DISCUSSION

A. Single Particle in a Heat Flux

First we investigate the effect of the presence of a single nanoparticle on the thermal conductivity of the fluid. In this system the center of mass of the particle is held at equal distance from the two thermostats (fig. 2).

FIG. 2: Snapshot of the system used to evaluate the thermal conductivity with a particle of 13% volume fraction.

We performed simulations for a small volume fraction (Φ ∼ 2%), where we were not able to detect any change in thermal conductivity compared to the bulk fluid. At a higher volume fraction (Φ ∼ 13%), on the other hand, we observe a clear decrease in the heat conductivity associated with the presence of the nanoparticle (fig. 3). The Kapitza resistance R K for the considered particles ranges from 1 to 7 so that the associated

k B /σ 2 /τ LJ ≃= 10 8 W/m 2 /s
characteristic Kapitza length is in all cases of the order of the particles diameter. This means that the decrease must be interpreted in terms of interfacial effects. To quantify these effects, we use the Maxwell-Garnett approximation for spherical particles, modified to account account for the Kapitza resistance at the boundary between the two media.

The resulting expression for the effective conductivity 14 (see appendix) is

λ ef f λ l = λp λ l (1 + 2α) + 2 + 2Φ λp λ l (1 -α) -1 λp λ l (1 + 2α) + 2 -Φ λp λ l (1 -α) -1 (6)
where λ l and λ p are the liquid and particle conductivities, Φ is the particle volume fraction and α = R K λ l Rp is the ratio between the Kapitza length (equivalent thermal thickness of the interface) and the particle radius. This model predicts an increase in the effective conductivity for α > 1 and a decrease for α < 1, regardless of the value of the conductivity of the particles or of the volume fraction. The prediction depends very weakly on the ratio λ p /λ l , less than 1% for 10 < λ p /λ l < 100. The minimum value of λ ef f /λ l , obtained when α → ∞, is 1-Φ 1+Φ/2 while the maximum possible enhancement (for λ p → ∞ and R K → 0) is 1+2Φ 1-Φ . The Kapitza resistance can be modified by tuning either the liquid solid interaction coefficient c, or the mass density of the solid, or a combination of these two parameters.

Figure 3 illustrates the variation of the measured effective conductivity for several values of the Kapitza resistance, determined independently for various values of these parameters.

It is seen that the observed variation (decrease in our case) in the effective conductivity is very well described by the Maxwell-Garnett expression. This expression also allows us to understand why the heat conductivity does not vary in a perceptible manner for small volume fractions (∼ 2%), for which the predicted change would be less than 2%, within our statistical accuracy.

B. Study of Collective Effects

Next we study a system of volume fraction 12% containing two nanoparticles in various configurations, in order to investigate the influence of collective effects. In order to model the influence of microscopic particle clustering we study two particles tethered by soft springs and forced to stay in "contact", directly interacting with each other (fig. 4. We modify the position of the centers of mass of the particles with respect to the temperature gradient -the particles are either aligned parallel or perpendicular to the gradient. In these two situations the solid phase has an aspect ratio of either a = 2 (parallel) or a = 1/2 (perpendicular). We also varied the Lennard-Jones interaction intensity between the two particles (ε pp ) in order to increase the rate of particle-particle energy transfer. The particles-liquid thermal resistance is also modified by choosing different wetting properties or different ratios of the masses of the atoms belonging to the two phases. Finally, to explore a broader range of configurations, we also modified the distance between the centers of mass of the particles. In most configurations the particles are directly interacting with each other and in two setups they are separated along the z direction by a layer of atoms from the liquid phase, with a thickness of the order of a particle diameter . An overview of the systems investigated is shown in table II B.

First we study systems with aspect ratio a = 2 and strong particle-particle interaction, as a function of the interfacial resistance with th efluid(fig. 4). The strong interaction is essentially equivalent to a chemical bonding between the particles. Te two particles move as a single rigid body and the heat transfer between them is considerably enhanced.

We can therefore reasonably compare the obtained conductivities with the values an effective medium calculation would predict for an ellipsoid with the same aspect ratio.

A calculation including interfacial thermal resistance for ellipsoidal particles of the same size, aligned with the thermal gradient, relevant for this situation can be derived from a general expression in ref. 14 . The ratio of the effective conductivity to the conductivity of the pure liquid is given by:

λ ef f λ l = 1 + Φβ(1 -L zz ) 1 -ΦβL zz ( 7
)
where Φ is the volume fraction, L zz is given by

L zz = 1 -2 a 2 2(a 2 -1) - a 2 2(a 2 -1) 3/2 cosh -1 a (8)
with aspect ratio a > 1. The parameter β is given by

β = λ c -λ l λ l + L zz (λ c -λ l ) (9)
where

λ c = λ p 1 + λp λ l (2 + 1/a) R K λ l Rp L zz ( 10
)
FIG. 4: Snapshot of the system containing two particles aligned with the temperature gradient having strong particle-particle interaction.

In the above λ p and λ l are the conductivities of the particles and the liquid, R p is the particles radius and R K -the Kapitza resistance of the particle-liquid interface. The relations hold for an aspect ratio a > 1.

As can be seen in figure 5 the measured conductivities are in reasonable agreement with the calculation. For the smallest interfacial resistance there is an enhancement of the composite system conductivity by 5 to 10%. For larger values of R K the small radius of the particles results in a decrease in the conductivity, due to interfacial effects.

Next, we study the thermal behavior of the systems with particles aligned with the thermal gradient but much weaker interactions. The particle-particle interaction is taken to be neutral, ε pp = 1 (configurations a2ε1m50c1, a2ε1m1c1 in table II B). For this weaker interaction, we find that the conductivity is typically 4% below the one obtained with the strong attractive interaction. This difference can be understood from the temperature profiles shown in figure 6. The weaker interaction results in a higher resistivity in the "neck" region, at the boundary between the two particles. As the strongly attractive interaction decreases the particle-particle thermal resistance, the particles temperatures are closer for ε pp = 10. As the particles are much more conductive than the liquid phase their temperature varies very little within their dimension. We observed that the temperature difference between the particles decreases by a factor of two when the interaction intensity is increased by a factor of ten: ∆T = 0.22 for ε pp = 1 and ∆T = 0.12 for ε pp = 10 in the systems of m = 1. In the m = 50 case ∆T (ε pp = 1) = 0.44 and ∆T (ε pp = 10) = 0.2.

Because of the layers of fluid (of thickness around 5σ) remaining in both cases between the particles and the thermostats the global conductivity of the slab is still dominated by interfacial effects and its increase is small. When the particle-particle distance is increased, so that the particles do not interact directly with each other, the conductivity of the sample stays nearly identical (difference ∼ 1% in favor for the system where the particles are closer) to the case where they are in contact and with neutral interaction. Knowing that the contact area is very small when the particles are close and also given that the interaction is not strong, being equal to the interaction with the liquid atoms ε pp = ε = 1, the inter particle distance in this case does not play an important role in the value of the conductivity. As can be seen in fig. 6, whenever ε pp = ε = 1 the average temperature of the particle is close to the average temperature of the liquid at the z coordinate of its center of mass, the particle thermalizes with the surrounding fluid. In contrast, when ε pp = 10 the inter particle heat flux becomes important and the two particles behave more like a single body.

We now turn to the conductivity of the systems where the particles are in the plane situated in the middle of the box and orthogonal to the temperature gradient. The heat fluxes measured for such systems of aspect ratio 1/2, with strong and neutral particleparticle interaction, were identical within our statistical accuracy (difference less than one percent). The conductivities in this case are slightly lower, with about 4.5% (2% for m = 1 and 7% for m = 50) than in the case where the particles are aligned with the temperature gradient and has neutral interactions (ε pp = 1). The difference is further increased to ∼ 8% (7% for m = 1 and 10% for m = 50) when the a = 0.5 systems are compared with the a = 2 and ε pp = 10 systems.

In summary, we showed that the mutual positions of the particles in suspension in the fluid has an influence on the thermal conductivity of the system. If the particles interact so that clustering occurs in the suspension, the global conductivity of the nanofluid can increase to values higher than the one of a pure fluid, even if a well dispersed suspension at the same volume fraction has a conductivity lower than the pure system. The alignment with the temperature gradient enhances the conductivity and its effect on a microscopic level can be predicted by effective medium calculation taking into account the aspect ratio of the particle cluster. According to the effective medium prediction the ratio of the slab conductivity to the conductivity of the pure liquid grows essentially linearly with the number of aligned particles (or the aspect ratio) (see fig. 7) when this number is smaller than the ratio of the conductivities of the two phases (λ p /λ l ). Hence even a moderate clustering (e.g. in strings of 3 to 4 particles) could be sufficient to interpret increases in the thermal conductivity compared to standard effective medium predictions. The enhancement tends to (1 -Φ) + Φλ p /λ l when the aspect ratio tends to infinity.

Thus, following this calculation that seems to give reasonable results at least on the scale of two particles and given that the ratio λ p /λ l is usually high in real nanofluids, a large enhancement can be expected if aggregates of particles are formed in the suspension. Our calculation concerns string like aggregates, but it is likely that similar effects would be observed with other types of aggregates, e.g. fractal ones with a low enough dimension.

III. CONCLUSION

We have explored some aspects of the thermal properties of "nanofluids", at the level of model system, individual solid particles and collective effects involving two particles on a microscopic scale.

By varying interaction parameters or mass density, we are able to vary the interfacial resistance between the particle and the fluid in a large range. This allowed us to estimate, over a large range of parameters, the effective heat conductivity of a model nanofluid in which the particles would be perfectly dispersed. The results for this setup can be simply explained in terms of the classical Maxwell-Garnett model, provided the interfacial resistance is taken into account. The essential parameter that influences the effective conductivity turns out to be the ratio between the Kapitza length and the particle radius, and for very small particles a decrease in conductivity compared to bulk fluids is found.

We also examined the effect of particle clustering and alignment with respect to the temperature gradient. We found that alignment improves the conductivity of the nanofluid in accordance with calculation with effective medium approach. Increased inter particle interaction further enhances conductivity. The results for small clusters of particles can again be described by effective medium theory, taking into account anisotropy and interface effects. Clustering of particles into string like objects is therefore suggested as a possible mechanism for obtaining larger conductivities, compared to the case of well dispersed suspensions.

In order to draw more precise conclusions concerning the conductivity dependence on the complex physics of collective effects larger systems containing more complex aggregates should be examined. The present study provides guidelines and outlines general tendencies that are to be expected in a larger and more complex system.

IV. APPENDIX

We present here a brief derivation of the Maxwell Garnett equation for the effective thermal conductivity in a two phase media (matrix with spherical inclusions) taking into account the interfacial thermal resistance. We consider a macroscopically homogeneous material with thermal conductivity λ ef f in a temperature gradient following some axis:

T ef f (r) = -g • r. We concentrate on a spherical inclusion of radius r 0 and conductivity ∆T (r, θ) = 0 (11) where θ is the angle between r and the external gradient g. The solutions of equation 11 in the three different regions are given by: T 1 (r, θ) = Ar cos θ, 0 < r ≤ r 0

T 2 (r, θ) = (Br + E/r 2 ) cos θ, r 0 < r ≤ r 1

T ef f (r, θ) = -gr cos θ, r 1 < r

The unknown constants in the equations above are to be determined from the appropriate boundary conditions. These are obtained by expressing the continuity of the heat flow at the domain boundaries and the value of the temperature field:

T 1 (r 0 , θ) -T 2 (r 0 , θ) = -λ 1 ∂T 1 ∂r (r 0 )R K (15)

λ 1 ∂T 1 ∂r (r 0 ) = λ 2 ∂T 2 ∂r (r 0 ) (16) T 2 (r 1 , θ) = T ef f (r 1 , θ) (17)
λ 2 ∂T 2 ∂r (r 1 ) = λ ef f ∂T ef f ∂r (r 1 ) (18)
where R K is the interfacial thermal resistance responsible for a temperature jump at the matrix-inclusion interface. Substituting the solutions for the temperature fields in equations 16-18, we end up with the following relations:

Ar 3 0 -Br 3 0 -E + λ 1 R K Ar 2 0 = 0 (19)
λ ef f λ 2 = λ 1 λ 2 (1 + 2α) + 2 + 2Φ λ 1 λ 2 (1 -α) -1 λ 1 λ 2 (1 + 2α) + 2 -Φ λ 1 λ 2 (1 -α) -1 (23)
where α = R K λ 2 /r 0 .

FIG. 1 :

 1 FIG.1: Dependence of the temperature profile on the thermostat time constant for a pure liquid. A smaller rescaling interval increases the efficiency of the energy transfer between the thermostat and the liquid inducing a smaller temperature jump.

FIG. 3 :

 3 FIG.3: Comparison between the ratio of the effective conductivity to the conductivity of the pure liquid of the simulated systems and the values obtained from the Maxwell Garnett equation.

5 FIG. 5 :

 55 FIG.5: Comparison of the measured variation in conductivity in the ε = 10, a = 2 systems to the prediction of the effective medium calculation with aspect ratio 2 and volume fraction 0.12 (equation 7).

FIG. 6 :

 6 FIG. 6: Temperature profiles in the particles in the m = 1 systems in stationary heat flow. The temperature profile within the liquid is also shown for one of the systems. For the other systems the liquid temperature profile is very similar. Stronger interaction decreases the temperature difference between the particles by approximately 50% compared to particles separated by the same distance with neutral interaction.

9 FIG. 7 :

 97 FIG.7: Prediction of the effective medium calculation (equation 7) for the ratio of the system conductivity to the conductivity of the liquid for an interfacial resistance R K = 1, a volume fraction Φ = 0.1, and a ratio of 10 between the thermal conductivities of the solid and of the liquid. .

λ 1

 1 FIG. 8: Schematic presentation of the matrix with dispersed inclusions

λ 1 3 0 -λ 2 λ 2 3 1 -

 13221 Ar Br Br 2λ 2 E + λ ef f gr 3 1 = 0 (22) Using this set of equations, one can obtain the Maxwell Garnett equation for the effective conductivity with interfacial thermal resistance:

TABLE I :

 I System name Aspect ratio a ε pp m p /m l c R K Particle-particle ∆r Description of the systems containing two particles . The last column indicates the distance between the centers of mass of the two particles.

	a2ε1m1c1	2.0	1.0	1	1 0.8	2R p
	a2ε10m1c1	2.0	10.0	1	1 0.8	2R p
	a2ε1m50c1	2.0	1.0	50	1 4.0	2R p
	a2ε10m50c1	2.0	10.0 50	1 4.0	2R p
	a2ε10m1c0.5	2.0	1.0	1	0.5 3.2	2R p
	a0.5ε1m1c1	0.5	1.0	1	1 0.8	2R p
	a0.5ε10m1c1	0.5	10.0	1	1 0.8	2R p
	a0.5ε1m50c1	0.5	1.0	50	1 4.0	2R p
	a0.5ε10m50c1	0.5	10.0 50	1 4.0	2R p
	ε1m1c1	×	1.0	1	1 0.8	4R p
	ε1m50c1	×	1.0	50	1 4.0	4R p


Arthur Capelier-Mourguy

Philippe Blache

Christian Retoré

Laurent Prevot

Laurent Prevot Quantifier

Capelier-Mourguy Arthur

Blache Philippe

Retoré Christian

Prévot Laurent

Quantifier scope: a formal and experimental study

This work is part of a general research program, whose aim is to confront the predictions of linguistic formalisation to actual human processing mainly for lexical, syntactic and above all semantic issues, excluding as much as possible the interference with pragmatics. Examples of questions to be address are the following: Are there preferred readings? Is a sentence complex to interpret? Which linguistic ingredients are responsible for its complexity?

We present here a semantic claim about quantifier scope taking, together with an experiment (described below) that we designed to test this claim, and which actually justifies our claim.

On the formal side, we view quantification with Hilbert's epsilon terms as in the type-theoretical computational semantic framework that we introduced recently [START_REF] Retoré | Typed hilbert epsilon operators and the semantics of determiner phrases[END_REF]. Epsilon terms, that can be compared with choice functions and opposed to generalised quantifiers, denote generic individuals: they provide a logical structure with in situ quantification that follow the syntactic structure and thus naturally express underspecification. This is quite adequate to our study which involves variants and reformulations of sentences with two quantifiers.

As opposed to apparently related work like [START_REF] Szymanik | Comprehension of simple quantifiers: Empirical evaluation of a computational model[END_REF] we clearly focus on the interpretation of the utterance: the choice of the lexical items, the syntactic structure, and the semantic interpretation itself. As such, we tried to minimise the influence of reasoning tasks, computational issues and world knowledgealthough the observation of the semantic interpretation requires some of these.

The precise task to which we applied our view in this paper is quantifier scope disambiguation in universal-existential double-quantified sentences. These are sentences such as "An identification number is given to every student". It has two readings, and common knowledge, syntax and the choice of the quantifying determiners influence its disambiguation. Here we focus on the determiners expressing universal quantification chaque and tous les, and we observed they do not perfectly match English quantifiers: tous les resembles all and every, while chaque rather corresponds to each (although chacun corresponds to everyone). For the existential quantifier, its simplest expression un(e) -the indefinite article (a)is homonymous with the numeral un(e) (one). Even though work on such sentences has been done in English like [START_REF] Tunstall | The interpretation of quantifiers: semantics & processing[END_REF], the equivalent constructions remain understudied in French and furthermore our view on the issue is slightly different.

Our aim is to look at the comprehension of sentences in terms of visual situations. To prevent any effect of pragmatic inference on sentence comprehension, items are made as neutral as possible : squares and circles. The computational aspects are also minimized by giving a very little number of items (3 of each), and asking a response time as short as possible. We also tried to prevent any natural left-to-right lecture of images by positioning the items horizontally. We observed the reaction time, and recorded eye gaze to have a look at internal comprehension processes. Our material thus resulted in figure combinations as the one presented in Fig. 1 paired with auditory-presented sentences of the form "[U/E] carré est relié à [E/U] rond " ([U/E] square is connected to [U/E] circle), existential (E) quantifier always being un and universal (U) quantifier being either chaque (U chaque ) or tous les (U tous les ). Each of the four figures shown on the stimulus represents a lecture (here, from left to right and top to bottom, U tous les E, E U, a lure item, and U chaque E) 1 . Then we collect interpretation (figure choice), reaction time, and eye gaze. All this give us information on the computed meaning of the sentence, its complexity, and possible distractors from target.

Our results (see Fig. 2) confirm some theoretical assumptions about general phenomena influencing quantifier scope disambiguation, such as linear order in sentence or lexical Error bars on plots represents 95% confidence intervals, thus statistical significance is achieved whenever error bars do not overlap (more precise statistical data available on demand). A Pearson's χ 2 analysis showed a significant effect on the distribution of choices across sentences. realization of quantifiers, confirming part of the scale of [START_REF] Ioup | Some universals for quantifier scope[END_REF]. Indeed, chaque has been shown to be a stronger quantifier than tous les, whith respect to taking scope -this is probably related to their different meanings see e.g. [START_REF] Tunstall | The interpretation of quantifiers: semantics & processing[END_REF].

Furthermore, our results suggest that un should take wide scope on tous les but not on chaque, as it is suggested by the end of [START_REF] Ioup | Some universals for quantifier scope[END_REF]. More precisely, un interferes more with chaque than with tous les. It thus seems to be the case that un is closer to chaque with respect to taking scope over different quantifiers than to tous les, giving a greater precision to binary relations between quantifiers.

These results are concordant with Default theory see e.g. [5] and Grice's Maxims of Quantity, and opposed to Relevance theory [START_REF] Sperber | Relevance: communication and cognition[END_REF]. Indeed, while Relevance theory expect a logical reading, thus no difference between the logically equivalment universal terms (chaque and tous les), they interact significantly differently depending on the context. This suggest that the use of one rather than the other is informative for the addressee.

Finally, our study presents a good start in this domain, providing precise knowledge about quantifier scoping. It also paves the way to further investigations to test the relation between formal semantics and actual human processing, in particular as regards reading preferences and the complexity of ambiguous sentences.

Figure 1 .

 1 Figure 1. Exemple of visual stimulus

Figure 2 .

 2 Figure 2. Main results, figures of fixation times across sentences and reaction times across sentences are not shown here but lead to same conclusions.Error bars on plots represents 95% confidence intervals, thus statistical significance is achieved whenever error bars do not overlap (more precise statistical data available on demand). A Pearson's χ 2 analysis showed a significant effect on the distribution of choices across sentences.

We use here the convention symbol as corresponding to "takes wide scope on"