Measures of the performance of France in Mathematics

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STI 2018 Conference Proceedings

Proceedings of the 23rd International Conference on Science and Technology Indicators

All papers published in this conference proceedings have been peer reviewed through a peer review process administered by the proceedings Editors. Reviews were conducted by expert referees to the professional and scientific standards expected of a conference proceedings.

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Scientific Editors Rodrigo Costas Thomas Franssen Alfredo Yegros-Yegros

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The articles of this collection can be accessed at https://hdl.handle.net/1887/64521 ISBN: 978-90-9031204-0

This ARTICLE is licensed under a Creative Commons Atribution-NonCommercial-NonDetivates 4.0 International Licensed

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Wilfriedo Mesheba*, Frédérique Sachwald**

*wilfriedo.mescheba@hceres.fr

Observatoire des Sciences et Techniques, Hcéres, Rue Albert Einstein, Paris, 75013 (France)

**frederique.sachwald@hceres.fr

Observatoire des Sciences et Techniques, Hcéres, Rue Albert Einstein, Paris, 75013 (France)

Introduction

France is the third publishing country in mathematics, behind China and the United States. In 2015, its world share of publications in mathematics is 6.5%, well above its share for total publications (3.2%). France is indeed strongly specialized in mathematics. Besides, French researchers are internationally recognized through prestigious prizes. Between 1936 and 2014, French doctorate holders were awarded 14 such prizes (Fields medal, Abel, Gauss or Wolff prizes). France is second ex aequo with Russia on this measure, behind the United States (40).

Taking into account affiliation when the mathematicians receive these prizes, France would be second only to the United States (19 versus 59 laureates). Yet, bibliometric impact indicators measure a less outstanding performance of France in mathematics. For example the French share of the 1% most cited publications, at 5.5%, is lower than its share of publications in the discipline. This paper explores this uneven performance of France when different indicators are being used.

In order to do so, the paper compares different corpora. It first shows that French performance in mathematics is driven by its stronger results in Fundamental mathematics. It then compares results in two selective corpora: publications in the A* journals selected by the Australian Mathematical Society and publications by prize winning authors (Fields medal, Abel, Gauss and Wolff prizes). It finally discusses the profile and performance of French publications within the field of Statistics and Probability. The last section draws conclusions and suggests further research to better understand the roots of the performance of France in mathematics.

Data and corpora

Publication data is from OST’s version of the Web of Science, under license from Clarivate Analytics. The period under consideration is 2000-15 and the citation window for the impact indicators is 5 years.¹ Counts are fractional. France is compared with the countries publishing the most in mathematics, being the most specialized or having researchers with international prizes (Fields, Abel, Gauss, Wolff). Including France, 17 countries are compared (designated by their ISO codes).

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The discipline is considered as the sum of four WoS subject categories: Fundamental mathematics, Applied mathematics, Statistics & Probabilities (S&P) and Mathematics for interdisciplinary applications. The total number of publications in mathematics over the period is close to 600,000.

The A* corpus corresponds to the publications from the journals selected by the Australian Mathematical Society as being the best. This corpus comprises less than 8,000 publications over 2000-15.

The mathematical prizes corpus is that of the publications by the mathematicians who have won one of the four prizes listed above. There are 84 prize winning mathematicians with publications over 2000-15 with a little more than 1,000 publications. The country of affiliation is set for all the period depending on the country from which the mathematician holds its doctorate.

Interpretation of the results has been discussed with mathematicians during the course of this research (2017-18).

Findings from the comparison of corpora

Global corpus

France is specialized in all four fields of mathematics, which is the case of few countries (Figure 1). It is equally specialized at a high level in Fundamental mathematics and in S&P (1.8). Mathematics for interdisciplinary applications represents 7% of world publications in mathematics and will not be detailed in the rest of the paper.

Figure 1. Specialization index* by field in mathematics, 2010-12

* Share of Maths F in French publications / Share of Math F in world publications

Source: OST-Web of Science, OST treatment

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Figures 2a and 2b show first that the overall impact of France in mathematics is driven by its impact in Fundamental mathematics (1.13). The impact index is defined as the ratio between the number of citations and the number of publications. Impact of French publications is much lower in Applied mathematics and S&P.

Figure 2a. Impact index by field in mathematics, 2010-12

Source: OST-Web of Science, OST treatment Figure 2b. Field weighted impact in mathematics, 2000-12

Since Fundamental mathematics represents 49% of the total number of publications of France in mathematics, the field drives the overall performance. France is the 5^th country scoring best in Fundamental mathematics (figure 2a), but the 7^th overall in the discipline. It belongs to a second tier set of countries with an impact less than 10% above the world average number of

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citations per publication. The conclusion is similar when considering the top 1% most cited publications in mathematics. France is in 10^th position for the PPTop1% among the selected countries².

Selective corpora

Table 1 shows that countries’ shares vary substantially by corpus. The US and France have a much higher share of the two selective corpora. This is the contrary for China. Israel and Russia have a high share of the prizes corpus, but Russia has a low share of the A* corpus.

Similar positions for France and Russia were found by Dubois et al. (2013) on the basis of a selective corpus from the American Mathematical Society data base.³

A common feature of these two selective corpora is the high proportion of publications from Fundamental mathematics. In the case of France, the share of that field is close to 60% in the A* corpus and to 75% in the prizes corpus.

Table 1. World share of publications in mathematics, by corpus, 2000-15

Country

World share of Annual number of publications¹

in OST base OST base A* corpus International

prizes corpus

USA 20,1% 33,5% 34,5% 7,174

CHN 13,4% 6,1% - 5,401

FRA 6,5% 9,9% 12,8% 2,330

DEU 5,4% 6,9% 1,9% 1,949

ITA 4,2% 4,4% 1,0% 1,520

GBR 4,1% 6,0% 2,7% 1,465

JAP 4,0% 3,9% 1,0% 1444

RUS 3,5% 1,2% 9,9% 1,316

ESP 3,3% 3,0% - 1,122

CAN 3,0% 3,5% - 1102

IRN 1,4% 0,2% - 580

ISR 1,3% 1,8% 13,5% 463

ROU 1,0% 0,4% - 382

NLD 1,0% 1,1% - 354

BEL 0,9% 0,9% 10,4% 320

SWE 0,8% 0,9% 0,6% 284

AUT 0,7% 0,9% - 277

HUN * * 4,4% *

CHE * * 3,0% *

BRA * * 2,1% *

World 100% 100% 100% 37,255 1. Fractional count; * not in the benchmark

Figures 3a and 3b (in the appendix) show that the ranking of countries changes in a different way in terms of impact. The countries gaining most share in the A* corpus tend to loose in terms of impact (figure 3a). It is the case of both the US and France. Conversely, China and Russia have substantially higher impact indexes. A possible interpretation is that the few publications from China and Russia that have been published in the selective journals of the

2 This table is not shown; see OST (2018).

3 Actually, a similar corpus has also been tested and gives similar results as the A* corpus.

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A* corpus are among the best ones. This may not always be the case for the numerous publications from the US and France. Publications from prize winning mathematicians tend to have high impact indexes (figure 3b, appendix). The performance of those from the US is particularly high; the impact index of prize winning mathematicians from France is lower and similar to that of those from Belgium.

Figure 3a. Field weighted impact of publications in A* corpus, 2000-12

The high share of France in the two selective corpora is consistent with the common image of French mathematics. It is the second or third most publishing country in those corpora. The previous analysis of the total WoS corpus by field in mathematics suggests that it is because those selective corpora are concentrated on the field of fundamental mathematics. However, in those selective corpora, the performance of France in terms of impact is not particularly strong. It is clearly lower than that of the United States in the case of the A* corpus, and also lower than that of countries with a smaller share of publications such as the UK, China or Austria.

Findings from the focus on the field of S&P

France is as specialized in S&P as in Fundamental mathematics, but its performance is terms of impact is much lower. Interviews and discussions with mathematicians suggest that this does not reflect the actual strong contribution of French researchers in the fields of probabilities and mathematical statistics. The divergence between our statistical findings and their perception could be due to two non-exclusive factors. First, French mathematicians may try to publish their best probability papers in journals classified in Fundamental mathematics, or in both that field and S&P. Second, the citation rate in probability being lower than in statistics, in particular applied statistics, the classification could lower the measure of the French performance. This explanation is consistent with Smolinsky and Lercher (2012) analysis of different citation rates by subdisciplines in mathematics.

In order to test the second hypothesis, we have conducted a cluster analysis of journals classified in S&P. We have considered all S&P documents and built a Journal and Subject category matrix where each cell gives the number of documents of a journal that was co- assigned to another subject category. Thus each journal & subject category pair corresponds

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to the number of S&P documents published in that journal and also assigned to another subject category. Cosinus distance is used to measure the proximity between journals.

The method is detailed in Appendix 2 and Figures A2a to A2e show the map of S&P journals based on their distance for the world, France, the US, Belgium and the Netherlands.

Publications from France are more often in journals that are focused on the Core of S&P and not dealing with other subject or applied issues. Table 2 summarizes the results of this clustering. It shows that the UK and the Netherlands have a relatively high proportion of S&P publications dealing with Economics/Finance/Insurance and the US has the highest proportion of journals also classified in Biology. The US and Belgium have the highest proportions of journals also classified in Computer science.

Table 2. Clusters of S&P journals based on their proximity with other subject categories, 2000-15 Share of publications

in cluster : FRA NLD USA BEL GBR World Core S&P 70.8% 61.2% 60.6% 53.8% 52.5% 59.3%

Economics 10.4% 23.1% 14.2% 19.2% 23.7% 14.6%

Computer science 6.8% 8.4% 10.7% 11.5% 9.0% 9.7%

Biology 2.3% 3.9% 9.8% 7.0% 7.1% 5.7%

Other 9.8% 3.3% 4.7% 8.6% 7.7% 10.7%

Source: OST-Web of Science, OST treatment

These results suggest that the profile of French publications in S&P could influence downward the impact measures of those publications. Table 3 (World) confirms that publications in the Core S&P cluster tend to be less cited than those from the Economics, Biology and Computer science clusters. However, it shows that the impact of French publications in each of the clusters is not high compared to the benchmark countries. Their impact is the lowest for the Core S&P and the Biology clusters. It is the second highest behind the USA for the Economics cluster.

Table 3. Citations by publication, normalized by the world total for S&P

Cluster FRA NLD USA BEL GBR World Core S&P 0.96 0.98 1.17 1.04 1.15 0.90 Economics 1.39 1.31 1.52 1.27 1.26 1.22 Computer science 2.64 2.46 2.75 2.18 2.94 2.10 Biology 1.59 1.63 1.60 1.65 1.68 1.52 Other 0.68 0.69 0.65 1.09 0.79 0.70

All S&P 1.00 1.38 1.27 1.53 1.51 1.00

Conclusion and further research

The high performance of France in mathematics is driven by its performance in fundamental mathematics. Its world share of publications is higher in corpora that are focused on fundamental mathematics, which is the case of the publications by international prize winners in particular.

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Performance within fundamental mathematics could be analysed in more detail in order to understand the contrast between the performance of France in terms of prize laureates or the proportion of its publications in the top 1% and its more modest performance in terms of average impact in selective but quite large corpora (corpus A*). It would be interesting for example to examine the distribution of impact measures of French publications in fundamental mathematics to see whether a small elite of mathematicians plays a particular role.

French publications have lower impacts in Applied mathematics, and even more so in S&P.

Interviews with mathematicians have suggested that the WoS classification, by aggregating statistics and probability fields could generate a bias against French publications. Our clustering of S&P journals on the basis of their simultaneous attribution to other WoS subject categories partially validates this hypothesis. It shows indeed, that French publications in S&P are concentrated in journals from the core of the subject category, the average impact of which is lower than those of the more applied journals. However, in most clusters, and in particular in the Core S&P one, the impact index of French publications tends to be lower than those of publications from the US, the UK, the Netherlands or Belgium. As a consequence, the modest performance of France in S&P cannot only be attributed to a classification issue. Further research could nevertheless try to examine this issue with other classifications, in particular to try to better distinguish statistics from probability. This could be interesting in view of the role of statistics in the areas of big data and data science.

References

Dubois, P., J-C. Rochet and J-M. Schlenker (2013), Productivity and mobility in academic research:

evidence from mathematicians, Scientometrics, 96

OST (2018), La position scientifique de la France dans le monde 2000-15, Hcéres, http://www.hceres.fr/PUBLICATIONS/Rapports/Rapport-sur-la-position-scientifique-de-la-France-dans-le- monde-2000-2015

Panaretos, J. and C. Malesios (2012), Influential Mathematicians: Birth, Education and Affiliation, MPRA Paper No. 68046, https://mpra.ub.uni-muenchen.de/68046/

Smolinsky, L. and A. Lercher (2012), Citation rates in mathematics: a study of variation by subdiscipline, Scientometrics (2012) 91:911–924 DOI 10.1007/s11192-012-0647-3

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Appendix 1. Prizes corpus

Figure 3b. Field weighted impact of publications in the prizes corpus*, 2000-12

* Countries with more than 50 publications in the corpus (2000-12 publications by mathematicians that were awarded prizes) . Source: OST-Web of Science, OST treatment

Appendix 2.

Maps of S&P journals according to their attribution to other subject categories

From S&P publications, we have created a profile for each journal based on the subject categories assigned to each publication. The set of profiles is then used to create journal clusters whose links to S&P are defined by the common subject categories.

Figure A2.a shows the global map of clusters. It contains 5 clusters for all S&P publications.

The Core S&P cluster corresponds to journals that are focused on the S&P and do not deal with other subject or applied issues.

In order to compare the selected countries, the next figures project the journal profiles of each country on the world map.

Figures A2.b to A2.e show that the share of France's publications in S&P related to the Economic cluster is less important than those of Belgium or the USA. In addition, the share of French publications in the Core S&P cluster is larger than that of other benchmark countries.

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Figure A2.a S&P publications by journal, World

Figure A2.b S&P publications by journal, France

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Figure A2.c S&P publications by journal, USA

Source: OST-Web of Science, OST treatment Figure A2.d S&P publications by journal, Belgium

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Figure A2.e S&P publications by journal, Netherlands