# American Institute of Mathematical Sciences

• Previous Article
An application of a dynamical model with ecological predator-prey approach to extensive livestock farming in uruguay: Economical assessment on forage deficiency
• JDG Home
• This Issue
• Next Article
Economic evolution and uncertainty: Transitions and structural changes
April 2019, 6(2): 131-148. doi: 10.3934/jdg.2019010

## An application of minimal spanning trees and hierarchical trees to the study of Latin American exchange rates

 Institute for Latin American Studies, School of Business & Economics, Freie Universität Berlin, Rudesheimer Str. 54, 14197 Berlin, Germany

Received  December 2018 Revised  April 2019 Published  April 2019

Fund Project: Funding by the German Research Foundation (DFG) is gratefully acknowledged

This paper analyzes a group of nine Latin American currencies with the aim of identifying clusters of exchange rates with similar co-movements. In this work the study of currency relationships is formulated as a network problem, where each currency is represented as a node and the relationship between each pair of currencies as a link. The paper combines two methods, Symbolic Time Series Analysis (STSA) and a clustering method based on the Minimal Spanning Tree (MST), from which we obtain a Hierarchical Tree (HT). Symbolic Time Series Analysis consists in the transformation of a given time series into a symbolic sequence with the aim of identifying patterns in the set of data. The Minimal Spanning Tree condenses the core information on the global structure of the network and its main advantage is that it greatly simplifies comparisons by dramatically reducing the number of elements that must be compared. We identify two main clusters in the currency network, as well as specific currencies that function as transmission channels between clusters. Using data regarding the degree of financial liberalization, as well as the distinction between inflation targeting (IT) and non-IT countries, the analysis suggests that the obtained taxonomy is economically relevant.

Citation: Erick Limas. An application of minimal spanning trees and hierarchical trees to the study of Latin American exchange rates. Journal of Dynamics & Games, 2019, 6 (2) : 131-148. doi: 10.3934/jdg.2019010
##### References:
 [1] R. Almeida, Financial flows and exchange rates: Challenges faced by developing countries, IPC-IG, 97 (2012), 1-20. [2] R. Andrade and D. Prates, Exchange rate dynamics in a peripheral monetary economy, Journal of Post Keynesian Economics, 35 (2013), 399-416. doi: 10.2753/PKE0160-3477350304. [3] N. Boccara, Modeling Complex Systems, Springer-Verlag, New York, 2004. [4] J. Brida, D. Matesanz and W. Risso, Dynamical hierarchical tree in currency markets, Fundación de las Cajas de Ahorro (FUNCAS), 332 (2007). [5] J. Brida, D. Gómez and W. A. Risso, Symbolic hierarchical analysis in currency markets: An application to contagion in currency crises, Expert Systems with Applications, 36 (2009), 7721-7728. [6] J. Brida, D. Matesanz and W. Risso, Estructura jerárquica y dinámica en los mercados cambiarios latinoamericanos, Investigación Económica, 68 (2009), 115–146. [7] J. Brida, D. Matesanz and M. N. Seijas, Network analysis of returns and volume trading in stock markets: The Euro Stoxx case, Physica A, 444 (2016), 751-764. [8] J. Brida, D. Matesanz and M. N. Seijas, Debt and growth: A non-parametric approach, Physica A, 486 (2017), 883-894. doi: 10.1016/j.physa.2017.05.060. [9] J. Brida, S. London, M. Rojas, Una aplicación de los árboles de expansión mínima y árboles jerárquicos al estudio de la convergencia interregional en dinámica de regímenes, Revista de Métodos Cuantitativos para la Economía y la Empresa, 15 (2013), 3–29. [10] J. Brida and E. Limas, A post Keynesian framework of exchange rate determination: a dynamical approach, Dynamics of Continuous, Discrete and Impulsive Systems, 25 (2018), 409-426. [11] G. Calvo and C. Reinhart, Fear of floating, Quarterly Journal of Economics, 117 (2002), 379-408. [12] E. Carsamer, Exchange Rate Co-Movement and Volatility Spill Over in Africa, Ph.D thesis, School of Development Economics, National Institute of Development Administration, 2015. [13] M. Chinn and H. Ito, Capital Account Liberalization, Institutions and Financial Development: Cross Country Evidence, Working Paper No. 8967. Cambridge, MA: National Bureau of Economic Research, 2002. [14] M. Chinn and H. Ito, KAOPEN Index, 2017. Available from: http://web.pdx.edu/~ito/Chinn-Ito_website.htm. [15] L. F. de Paula, B. Fritz and D. Prates, Keynes at the periphery: Currency hierarchy and challenges for economic policy in emerging economies, Journal of Post Keynesian Economics, 40 (2017), 183-202. doi: 10.1080/01603477.2016.1252267. [16] C. Ebeke and A. Fouejieu, Inflation Targeting and Exchange Rate Regimes in Emerging Markets, IMF Working Paper, 228 (2015). [17] S. Edwards, The relationship between exchange rates and inflation targeting revisited, NBER Working Paper, 12163 (2006), 1-47. doi: 10.3386/w12163. [18] X. Feng and X. Wang, Evolutionary topology of a currency network in Asia, International Journal of Modern Physics C, 21 (2010), 471-480. doi: 10.1142/S0129183110015269. [19] D. Fenn, M. Porter, P. Mucha, M. McDonald, S. Williams, N. Johnson and N. Jones, Dynamical clustering of exchange rates, Quantitative Finance, 12 (2012), 1493-1520. doi: 10.1080/14697688.2012.668288. [20] J. Frankel and D. Xie, Estimation of de facto flexibility parameter and basket weights in evolving exchange rate regimes, NBER Working Paper, 15620 (2009), 1-22. doi: 10.3386/w15620. [21] J. Frankel and S. Wei, Yen bloc or dollar bloc? exchange rate policies of the east asian economies, in Macroeconomic Linkages (eds. I. Takatoshi and A. Krueger), Chicago: University of Chicago Press, (1994), 295–329. [22] J. Frankel and S. Wei, Assessing Chinas Exchange Rate Regime, Economic Policy, 51 (2007), 575-614. [23] J. Frankel and S. Wei, Estimation of de facto exchange rate regimes: Synthesis of the techniques for inferring flexibility and basket weights, IMF Staff Papers, 55 (2008), 384-416. doi: 10.3386/w14016. [24] A. Grski, S. Drozdz and J. Kwapien, Scale free effects in world currency exchange network, The European Physical Journal B, 66 (2008), 91-96. doi: 10.1140/epjb/e2008-00376-5. [25] A. Grski, J. Kwapien, P. Oswiecimka and S. Drozdz, Minimal spanning tree graphs and power like scaling in forex networks, Acta Physica Polonica A, 114 (2008), 531-538. doi: 10.12693/APhysPolA.114.531. [26] R. Henning, Choice and coercion in east asian exchange rate regimes, Working Paper Peterson Institute for International Economics, 12 (2012), 22pp. doi: 10.2139/ssrn.2151545. [27] R. Hill, International Comparisons Using Spanning Trees, in International and Interarea Comparisons of Income, Output, and Prices (eds.A. Heston and R. Lipsey), Chicago: University of Chicago Press, (1999), 109–120. [28] IMF, Annual Report on Exchange Arrangements and Exchange Restrictions, 2015. [29] A. Kaltenbrunner, Currency Internationalisation and Exchange Rate Dynamics in Emerging Markets: A Post Keynesian Analysis of Brazil, Ph.D thesis, University of London, England, 2011. [30] L. Kaufman and P. Rousseeu, Finding Groups in Data. An Introduction to Cluster Analysis, Wiley-Interscience, New York, 1990. doi: 10.1002/9780470316801. [31] B. Keddad, How do the renminbi and other east asian currencies co-move?, Journal of International Money and Finance, 91 (2019), 49-70. doi: 10.1016/j.jimonfin.2018.11.003. [32] M. Keskin, B. Deviren and Y. Kocakaplan, Topology of the correlation networks among major currencies using hierarchical structure methods, Physica A: Statistical Mechanics and its Applications, 390 (2011), 719-730. doi: 10.1016/j.physa.2010.10.041. [33] J. Kruskal, On the shortest spanning subtree of a graph and the traveling salesman problem, Proceedings of the American Mathematical Society, 7 (1956), 48-50. doi: 10.1090/S0002-9939-1956-0078686-7. [34] J. Kwapien, S. Gworek, S. Drozdz and A. Grski, Analysis of a network structure of the foreign currency exchange market, Journal of Economic Interaction and Coordination, 4 (2009), 55-72. doi: 10.1007/s11403-009-0047-9. [35] J. Kwapien, S. Gworek and S. Drozdz, Structure and evolution of the foreign exchange networks, Acta Physica Polonica B, 40 (2009). [36] X. Li, How do exchange rates co-move? A study on the currencies of five inflation-targeting countries, Journal of Banking and Finance, 35 (2011), 418-429. [37] Y. Mai, H. Chen and S. Li, Currency co-movement and network correlation structure of foreign exchange market, Physica A: Statistical Mechanics and its Applications, 492 (2018), 65-74. doi: 10.1016/j.physa.2017.09.068. [38] R. Mantegna, Hierarchical structure in financial markets, Eur. Phys. J. B, 11 (1999), 193-197. doi: 10.1007/s100510050929. [39] R. Mantegna and H. Stanley, An Introduction to Econophysics, Cambridge University Press: Cambridge, UK, 2000. [40] D. Matesanz and G. Ortega, Network analysis of exchange data: Interdependence drives crisis contagion, Quality & Quantity, 48 (2014), 1835-1851. doi: 10.1007/s11135-013-9855-z. [41] M. McDonald, O. Suleman, S. Williams, S. Howison and N. Johnson, Detecting a currency dominance or dependence using foreign exchange network trees, Physical Review E, 72 (2005), 46-106. [42] T. Mizuno, H. Takayasu and M. Takayasu, Correlation networks among currencies, Physica A: Statistical Mechanics and its Applications, 364 (2006), 336-342. doi: 10.1016/j.physa.2005.08.079. [43] M. Naylor, L. Rose and B. Moyle, Topology of Foreign Exchange Markets using Hierarchical Structure Methods, 2006. [44] M. Reovsk, D. Horvth, V. Gazda and M. Sinikov, Minimum spanning tree application in the currency market, Ronk, 21 (2013), 21-23. [45] A. Subramanian and M. Kessler, The Renminbi Bloc is Here: Asia Down, Rest of the World to Go?, Working Paper 12-19, Peterson Institute for International Economics, 2013. [46] G. Wang, C. Xie, J. Chen and S. Chen, Statistical properties of the foreign exchange network at different time scales: evidence from detrended cross-correlation coefficient and minimum spanning tree, Entropy, 15 (2013), 1643-1662. doi: 10.3390/e15051643. [47] I. Yu, K. Fung and C. Tam, Assessing financial market integration in Asia equity markets, Journal of Banking and Finance, 34 (2010), 2874-2885.

show all references

##### References:
 [1] R. Almeida, Financial flows and exchange rates: Challenges faced by developing countries, IPC-IG, 97 (2012), 1-20. [2] R. Andrade and D. Prates, Exchange rate dynamics in a peripheral monetary economy, Journal of Post Keynesian Economics, 35 (2013), 399-416. doi: 10.2753/PKE0160-3477350304. [3] N. Boccara, Modeling Complex Systems, Springer-Verlag, New York, 2004. [4] J. Brida, D. Matesanz and W. Risso, Dynamical hierarchical tree in currency markets, Fundación de las Cajas de Ahorro (FUNCAS), 332 (2007). [5] J. Brida, D. Gómez and W. A. Risso, Symbolic hierarchical analysis in currency markets: An application to contagion in currency crises, Expert Systems with Applications, 36 (2009), 7721-7728. [6] J. Brida, D. Matesanz and W. Risso, Estructura jerárquica y dinámica en los mercados cambiarios latinoamericanos, Investigación Económica, 68 (2009), 115–146. [7] J. Brida, D. Matesanz and M. N. Seijas, Network analysis of returns and volume trading in stock markets: The Euro Stoxx case, Physica A, 444 (2016), 751-764. [8] J. Brida, D. Matesanz and M. N. Seijas, Debt and growth: A non-parametric approach, Physica A, 486 (2017), 883-894. doi: 10.1016/j.physa.2017.05.060. [9] J. Brida, S. London, M. Rojas, Una aplicación de los árboles de expansión mínima y árboles jerárquicos al estudio de la convergencia interregional en dinámica de regímenes, Revista de Métodos Cuantitativos para la Economía y la Empresa, 15 (2013), 3–29. [10] J. Brida and E. Limas, A post Keynesian framework of exchange rate determination: a dynamical approach, Dynamics of Continuous, Discrete and Impulsive Systems, 25 (2018), 409-426. [11] G. Calvo and C. Reinhart, Fear of floating, Quarterly Journal of Economics, 117 (2002), 379-408. [12] E. Carsamer, Exchange Rate Co-Movement and Volatility Spill Over in Africa, Ph.D thesis, School of Development Economics, National Institute of Development Administration, 2015. [13] M. Chinn and H. Ito, Capital Account Liberalization, Institutions and Financial Development: Cross Country Evidence, Working Paper No. 8967. Cambridge, MA: National Bureau of Economic Research, 2002. [14] M. Chinn and H. Ito, KAOPEN Index, 2017. Available from: http://web.pdx.edu/~ito/Chinn-Ito_website.htm. [15] L. F. de Paula, B. Fritz and D. Prates, Keynes at the periphery: Currency hierarchy and challenges for economic policy in emerging economies, Journal of Post Keynesian Economics, 40 (2017), 183-202. doi: 10.1080/01603477.2016.1252267. [16] C. Ebeke and A. Fouejieu, Inflation Targeting and Exchange Rate Regimes in Emerging Markets, IMF Working Paper, 228 (2015). [17] S. Edwards, The relationship between exchange rates and inflation targeting revisited, NBER Working Paper, 12163 (2006), 1-47. doi: 10.3386/w12163. [18] X. Feng and X. Wang, Evolutionary topology of a currency network in Asia, International Journal of Modern Physics C, 21 (2010), 471-480. doi: 10.1142/S0129183110015269. [19] D. Fenn, M. Porter, P. Mucha, M. McDonald, S. Williams, N. Johnson and N. Jones, Dynamical clustering of exchange rates, Quantitative Finance, 12 (2012), 1493-1520. doi: 10.1080/14697688.2012.668288. [20] J. Frankel and D. Xie, Estimation of de facto flexibility parameter and basket weights in evolving exchange rate regimes, NBER Working Paper, 15620 (2009), 1-22. doi: 10.3386/w15620. [21] J. Frankel and S. Wei, Yen bloc or dollar bloc? exchange rate policies of the east asian economies, in Macroeconomic Linkages (eds. I. Takatoshi and A. Krueger), Chicago: University of Chicago Press, (1994), 295–329. [22] J. Frankel and S. Wei, Assessing Chinas Exchange Rate Regime, Economic Policy, 51 (2007), 575-614. [23] J. Frankel and S. Wei, Estimation of de facto exchange rate regimes: Synthesis of the techniques for inferring flexibility and basket weights, IMF Staff Papers, 55 (2008), 384-416. doi: 10.3386/w14016. [24] A. Grski, S. Drozdz and J. Kwapien, Scale free effects in world currency exchange network, The European Physical Journal B, 66 (2008), 91-96. doi: 10.1140/epjb/e2008-00376-5. [25] A. Grski, J. Kwapien, P. Oswiecimka and S. Drozdz, Minimal spanning tree graphs and power like scaling in forex networks, Acta Physica Polonica A, 114 (2008), 531-538. doi: 10.12693/APhysPolA.114.531. [26] R. Henning, Choice and coercion in east asian exchange rate regimes, Working Paper Peterson Institute for International Economics, 12 (2012), 22pp. doi: 10.2139/ssrn.2151545. [27] R. Hill, International Comparisons Using Spanning Trees, in International and Interarea Comparisons of Income, Output, and Prices (eds.A. Heston and R. Lipsey), Chicago: University of Chicago Press, (1999), 109–120. [28] IMF, Annual Report on Exchange Arrangements and Exchange Restrictions, 2015. [29] A. Kaltenbrunner, Currency Internationalisation and Exchange Rate Dynamics in Emerging Markets: A Post Keynesian Analysis of Brazil, Ph.D thesis, University of London, England, 2011. [30] L. Kaufman and P. Rousseeu, Finding Groups in Data. An Introduction to Cluster Analysis, Wiley-Interscience, New York, 1990. doi: 10.1002/9780470316801. [31] B. Keddad, How do the renminbi and other east asian currencies co-move?, Journal of International Money and Finance, 91 (2019), 49-70. doi: 10.1016/j.jimonfin.2018.11.003. [32] M. Keskin, B. Deviren and Y. Kocakaplan, Topology of the correlation networks among major currencies using hierarchical structure methods, Physica A: Statistical Mechanics and its Applications, 390 (2011), 719-730. doi: 10.1016/j.physa.2010.10.041. [33] J. Kruskal, On the shortest spanning subtree of a graph and the traveling salesman problem, Proceedings of the American Mathematical Society, 7 (1956), 48-50. doi: 10.1090/S0002-9939-1956-0078686-7. [34] J. Kwapien, S. Gworek, S. Drozdz and A. Grski, Analysis of a network structure of the foreign currency exchange market, Journal of Economic Interaction and Coordination, 4 (2009), 55-72. doi: 10.1007/s11403-009-0047-9. [35] J. Kwapien, S. Gworek and S. Drozdz, Structure and evolution of the foreign exchange networks, Acta Physica Polonica B, 40 (2009). [36] X. Li, How do exchange rates co-move? A study on the currencies of five inflation-targeting countries, Journal of Banking and Finance, 35 (2011), 418-429. [37] Y. Mai, H. Chen and S. Li, Currency co-movement and network correlation structure of foreign exchange market, Physica A: Statistical Mechanics and its Applications, 492 (2018), 65-74. doi: 10.1016/j.physa.2017.09.068. [38] R. Mantegna, Hierarchical structure in financial markets, Eur. Phys. J. B, 11 (1999), 193-197. doi: 10.1007/s100510050929. [39] R. Mantegna and H. Stanley, An Introduction to Econophysics, Cambridge University Press: Cambridge, UK, 2000. [40] D. Matesanz and G. Ortega, Network analysis of exchange data: Interdependence drives crisis contagion, Quality & Quantity, 48 (2014), 1835-1851. doi: 10.1007/s11135-013-9855-z. [41] M. McDonald, O. Suleman, S. Williams, S. Howison and N. Johnson, Detecting a currency dominance or dependence using foreign exchange network trees, Physical Review E, 72 (2005), 46-106. [42] T. Mizuno, H. Takayasu and M. Takayasu, Correlation networks among currencies, Physica A: Statistical Mechanics and its Applications, 364 (2006), 336-342. doi: 10.1016/j.physa.2005.08.079. [43] M. Naylor, L. Rose and B. Moyle, Topology of Foreign Exchange Markets using Hierarchical Structure Methods, 2006. [44] M. Reovsk, D. Horvth, V. Gazda and M. Sinikov, Minimum spanning tree application in the currency market, Ronk, 21 (2013), 21-23. [45] A. Subramanian and M. Kessler, The Renminbi Bloc is Here: Asia Down, Rest of the World to Go?, Working Paper 12-19, Peterson Institute for International Economics, 2013. [46] G. Wang, C. Xie, J. Chen and S. Chen, Statistical properties of the foreign exchange network at different time scales: evidence from detrended cross-correlation coefficient and minimum spanning tree, Entropy, 15 (2013), 1643-1662. doi: 10.3390/e15051643. [47] I. Yu, K. Fung and C. Tam, Assessing financial market integration in Asia equity markets, Journal of Banking and Finance, 34 (2010), 2874-2885.
Illustration of symbolic encoding: Variation of the exchange rate of the Mexican peso against the U.S. dollar. The horizontal line represents the symbol partition; data below the partition are represented by 0, and data above the partition are represented by 1. In this case, the frontier level is the trend of the series, $\mu = 0.0025$. Then, the original data time series is represented by the symbol sequence S = 100100101
Minimal Spanning Tree (left) and Hierarchical Tree (right). Period: 2007
Minimal Spanning Tree (left) and Hierarchical Tree (right). Period: 2008-2010.
Minimal Spanning Tree (left) and Hierarchical Tree (right). Period: 2011-2014
Minimal Spanning Tree (left) and Hierarchical Tree (right). Period: 2015-2017
Financial openness: the Chinn-Ito index (2007-2016 average). Source: [14]
Countries, currencies and three-letter codes
Monetary Policy Framework. Argentina maintains a de facto exchange rate anchor to the U.S. dollar. Uruguay has an inflation target regime with monetary aggregates control. Source: [28]
 [1] Lluís Alsedà, David Juher, Pere Mumbrú. Minimal dynamics for tree maps. Discrete & Continuous Dynamical Systems - A, 2008, 20 (3) : 511-541. doi: 10.3934/dcds.2008.20.511 [2] Maria Cameron. Computing the asymptotic spectrum for networks representing energy landscapes using the minimum spanning tree. Networks & Heterogeneous Media, 2014, 9 (3) : 383-416. doi: 10.3934/nhm.2014.9.383 [3] Yaru Xie, Genqi Xu. The exponential decay rate of generic tree of 1-d wave equations with boundary feedback controls. Networks & Heterogeneous Media, 2016, 11 (3) : 527-543. doi: 10.3934/nhm.2016008 [4] Alba Málaga Sabogal, Serge Troubetzkoy. Minimality of the Ehrenfest wind-tree model. Journal of Modern Dynamics, 2016, 10: 209-228. doi: 10.3934/jmd.2016.10.209 [5] Vincent Delecroix. Divergent trajectories in the periodic wind-tree model. Journal of Modern Dynamics, 2013, 7 (1) : 1-29. doi: 10.3934/jmd.2013.7.1 [6] Miaohua Jiang, Qiang Zhang. A coupled map lattice model of tree dispersion. Discrete & Continuous Dynamical Systems - B, 2008, 9 (1) : 83-101. doi: 10.3934/dcdsb.2008.9.83 [7] Frédéric Bernicot, Bertrand Maury, Delphine Salort. A 2-adic approach of the human respiratory tree. Networks & Heterogeneous Media, 2010, 5 (3) : 405-422. doi: 10.3934/nhm.2010.5.405 [8] Sergei Avdonin, Jonathan Bell. Determining a distributed conductance parameter for a neuronal cable model defined on a tree graph. Inverse Problems & Imaging, 2015, 9 (3) : 645-659. doi: 10.3934/ipi.2015.9.645 [9] Rostyslav Kravchenko. The action of finite-state tree automorphisms on Bernoulli measures. Journal of Modern Dynamics, 2010, 4 (3) : 443-451. doi: 10.3934/jmd.2010.4.443 [10] Alberto Bressan, Michele Palladino. Well-posedness of a model for the growth of tree stems and vines. Discrete & Continuous Dynamical Systems - A, 2018, 38 (4) : 2047-2064. doi: 10.3934/dcds.2018083 [11] Reuven Cohen, Mira Gonen, Avishai Wool. Bounding the bias of tree-like sampling in IP topologies. Networks & Heterogeneous Media, 2008, 3 (2) : 323-332. doi: 10.3934/nhm.2008.3.323 [12] Shigeki Akiyama, Edmund Harriss. Pentagonal domain exchange. Discrete & Continuous Dynamical Systems - A, 2013, 33 (10) : 4375-4400. doi: 10.3934/dcds.2013.33.4375 [13] Giacomo Micheli. Cryptanalysis of a noncommutative key exchange protocol. Advances in Mathematics of Communications, 2015, 9 (2) : 247-253. doi: 10.3934/amc.2015.9.247 [14] Carlos Gutierrez, Simon Lloyd, Vladislav Medvedev, Benito Pires, Evgeny Zhuzhoma. Transitive circle exchange transformations with flips. Discrete & Continuous Dynamical Systems - A, 2010, 26 (1) : 251-263. doi: 10.3934/dcds.2010.26.251 [15] Ivan Dynnikov, Alexandra Skripchenko. Minimality of interval exchange transformations with restrictions. Journal of Modern Dynamics, 2017, 11: 219-248. doi: 10.3934/jmd.2017010 [16] Min Ye, Alexander Barg. Polar codes for distributed hierarchical source coding. Advances in Mathematics of Communications, 2015, 9 (1) : 87-103. doi: 10.3934/amc.2015.9.87 [17] Getachew K. Befekadu, Eduardo L. Pasiliao. On the hierarchical optimal control of a chain of distributed systems. Journal of Dynamics & Games, 2015, 2 (2) : 187-199. doi: 10.3934/jdg.2015.2.187 [18] Ellina Grigorieva, Evgenii Khailov. Hierarchical differential games between manufacturer and retailer. Conference Publications, 2009, 2009 (Special) : 300-314. doi: 10.3934/proc.2009.2009.300 [19] Cesare Bracco, Annalisa Buffa, Carlotta Giannelli, Rafael Vázquez. Adaptive isogeometric methods with hierarchical splines: An overview. Discrete & Continuous Dynamical Systems - A, 2019, 39 (1) : 241-261. doi: 10.3934/dcds.2019010 [20] Christopher F. Novak. Discontinuity-growth of interval-exchange maps. Journal of Modern Dynamics, 2009, 3 (3) : 379-405. doi: 10.3934/jmd.2009.3.379

Impact Factor: