<?xml version="1.0" encoding="UTF-8"?>
<!DOCTYPE article PUBLIC "-//NLM//DTD Journal Publishing DTD v3.0 20080202//EN" "https://jats.nlm.nih.gov/nlm-dtd/publishing/3.0/journalpublishing3.dtd">
<article xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink" article-type="research-article" dtd-version="3.0" xml:lang="en">
<front>
<journal-meta>
<journal-id journal-id-type="publisher">NPG</journal-id>
<journal-title-group>
<journal-title>Nonlinear Processes in Geophysics</journal-title>
<abbrev-journal-title abbrev-type="publisher">NPG</abbrev-journal-title>
<abbrev-journal-title abbrev-type="nlm-ta">Nonlin. Processes Geophys.</abbrev-journal-title>
</journal-title-group>
<issn pub-type="epub">1607-7946</issn>
<publisher><publisher-name>Copernicus Publications</publisher-name>
<publisher-loc>Göttingen, Germany</publisher-loc>
</publisher>
</journal-meta>
<article-meta>
<article-id pub-id-type="doi">10.5194/npg-14-5-2007</article-id>
<title-group>
<article-title>An Ising model for earthquake dynamics</article-title>
</title-group>
<contrib-group><contrib contrib-type="author" xlink:type="simple"><name name-style="western"><surname>Jiménez</surname>
<given-names>A.</given-names>
</name>
<xref ref-type="aff" rid="aff1">
<sup>1</sup>
</xref>
</contrib>
<contrib contrib-type="author" xlink:type="simple"><name name-style="western"><surname>Tiampo</surname>
<given-names>K. F.</given-names>
</name>
<xref ref-type="aff" rid="aff1">
<sup>1</sup>
</xref>
</contrib>
<contrib contrib-type="author" xlink:type="simple"><name name-style="western"><surname>Posadas</surname>
<given-names>A. M.</given-names>
</name>
<xref ref-type="aff" rid="aff2">
<sup>2</sup>
</xref>
</contrib>
</contrib-group><aff id="aff1">
<label>1</label>
<addr-line>Department of Earth Sciences Biological and Geological Sciences, University of Western Ontario, London, Canada</addr-line>
</aff>
<aff id="aff2">
<label>2</label>
<addr-line>Department of Applied Physics, University of Almería, Spain</addr-line>
</aff>
<pub-date pub-type="epub">
<day>19</day>
<month>01</month>
<year>2007</year>
</pub-date>
<volume>14</volume>
<issue>1</issue>
<fpage>5</fpage>
<lpage>15</lpage>
<permissions>
<copyright-statement>Copyright: &#x000a9; 2007 A. Jiménez et al.</copyright-statement>
<copyright-year>2007</copyright-year>
<license license-type="open-access">
<license-p>This work is licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 2.5 Generic License. To view a copy of this licence, visit <ext-link ext-link-type="uri"  xlink:href="https://creativecommons.org/licenses/by-nc-sa/2.5/">https://creativecommons.org/licenses/by-nc-sa/2.5/</ext-link></license-p>
</license>
</permissions>
<self-uri xlink:href="https://npg.copernicus.org/articles/14/5/2007/npg-14-5-2007.html">This article is available from https://npg.copernicus.org/articles/14/5/2007/npg-14-5-2007.html</self-uri>
<self-uri xlink:href="https://npg.copernicus.org/articles/14/5/2007/npg-14-5-2007.pdf">The full text article is available as a PDF file from https://npg.copernicus.org/articles/14/5/2007/npg-14-5-2007.pdf</self-uri>
<abstract>
<p>This paper focuses on extracting the information contained in seismic
space-time patterns and their dynamics. The Greek catalog recorded from 1901
to 1999 is analyzed. An Ising Cellular Automata representation technique is
developed to reconstruct the history of these patterns. We find that there is
strong correlation in the region, and that small earthquakes are very
important to the stress transfers. Finally, it is demonstrated that this
approach is useful for seismic hazard assessment and intermediate-range
earthquake forecasting.</p>
</abstract>
<counts><page-count count="11"/></counts>
</article-meta>
</front>
<body/>
<back>
<ref-list>
<title>References</title>
<ref id="ref1">
<label>1</label><mixed-citation publication-type="other" xlink:type="simple"> Aki, K.: Correlogram analyses of seismograms by means of a simple automatic computer, J. Phys. Earth, 4, 71&amp;ndash;79, 1956. </mixed-citation>
</ref>
<ref id="ref2">
<label>2</label><mixed-citation publication-type="other" xlink:type="simple"> Allègre, C J. and Le Mouel, J L.: Introduction of Scaling Technique in Brittle Failure of Rocks, Phys. Earth Planet Inter., 87, 85&amp;ndash;93, 1994. </mixed-citation>
</ref>
<ref id="ref3">
<label>3</label><mixed-citation publication-type="other" xlink:type="simple"> Allègre, C J., Le Mouel, J L., and Provost, A.: Scaling Rules in Rock Fracture and Possible Implications for Earthquake Predictions, Nature, 297, 47&amp;ndash;49, 1982. </mixed-citation>
</ref>
<ref id="ref4">
<label>4</label><mixed-citation publication-type="other" xlink:type="simple"> Anghel, M., Klein, W., Rundle, J B., and Martins, J S S.: Scaling in a cellular automaton model of earthquake faults, ArXiv Condensed Matter e-prints, 2000. </mixed-citation>
</ref>
<ref id="ref5">
<label>5</label><mixed-citation publication-type="other" xlink:type="simple"> Bak, P. and Tang, C.: Earthquakes as a Self-Organized Critical Phenomenon, J. Goephys. Res., 94, 15 635&amp;ndash;15 637, 1989. </mixed-citation>
</ref>
<ref id="ref6">
<label>6</label><mixed-citation publication-type="other" xlink:type="simple"> Bak, P., Tang, C., and Wiesenfeld, K.: Are Earthquakes, Fractals, and 1/f Noise Self-Organized Critical Phenomena, Cooperative Dynamics in Complex Systems, edited by: Takayama, H., Springer-Verlag, Tokyo, 1988. </mixed-citation>
</ref>
<ref id="ref7">
<label>7</label><mixed-citation publication-type="other" xlink:type="simple"> Barriere, B. and Turcotte, D L.: A scale-invariant cellular automaton model for distributed seismicity, Geophys. Res. Lett., 18, 2011&amp;ndash;2014, 1991. </mixed-citation>
</ref>
<ref id="ref8">
<label>8</label><mixed-citation publication-type="other" xlink:type="simple"> Boettcher, S. and Hartmann, A K.: Reduction of Two-Dimensional Dilute Ising Spin Glasses, Phys. Rev. B, 72, 014 429, prefixhttp://arxiv.org/abs/cond-mat/0503486, 2005. </mixed-citation>
</ref>
<ref id="ref9">
<label>9</label><mixed-citation publication-type="other" xlink:type="simple"> Bowman, D D. and King, G. C P.: Accelerating seismicity and stress accumulation before large earthquakes, Geophys. Res. Lett., 28, 4039&amp;ndash;4042, 2001. </mixed-citation>
</ref>
<ref id="ref10">
<label>10</label><mixed-citation publication-type="other" xlink:type="simple"> Bufe, C G. and Varnes, D J.: Predictive Modeling of the Seismic Cycle of the Greater San Francisco Bay Regiony, J. Geophys. Res., 98, 9871&amp;ndash;9883, 1993. </mixed-citation>
</ref>
<ref id="ref11">
<label>11</label><mixed-citation publication-type="other" xlink:type="simple"> Burridge, R. and Knopoff, L.: Model and theoretical seismicity, Bull. Seism. Soc. Am., 57, 341&amp;ndash;371, 1967. </mixed-citation>
</ref>
<ref id="ref12">
<label>12</label><mixed-citation publication-type="other" xlink:type="simple"> Burton, P W.: Dicing with earthquakes, Geophys. Res. Lett., 23, 1379&amp;ndash;1382, 1996. </mixed-citation>
</ref>
<ref id="ref13">
<label>13</label><mixed-citation publication-type="other" xlink:type="simple"> Burton, P W., Xu, Y., Tselentis, G., Sokos, E., and Aspinall, W.: Strong ground acceleration seismic hazard in Greece and neighboring regions, Soil Dyn. Earthqu. Eng., 23, 159&amp;ndash;181, 2003. </mixed-citation>
</ref>
<ref id="ref14">
<label>14</label><mixed-citation publication-type="other" xlink:type="simple"> Burton, P W., Xu, Y., Qin, C., Tselentis, G., and Sokos, E.: A catalogue of seismicity in Greece and the adjacent areas for the twentieth century, Tectonophysics, 390, 117&amp;ndash;127, 2004a. </mixed-citation>
</ref>
<ref id="ref15">
<label>15</label><mixed-citation publication-type="other" xlink:type="simple"> Burton, P W., Qin, C., Tselentis, G., and Sokos, E.: Extreme earthquake and earthquake perceptibility study in Greece and its surrounding area, Nat. Hazards, 32, 277&amp;ndash;312, 2004b. </mixed-citation>
</ref>
<ref id="ref16">
<label>16</label><mixed-citation publication-type="other" xlink:type="simple"> Carlson, J M. and Langer, J S.: A mechanical model of an earthquake fault, Phys. Rev. A, 40, 6470&amp;ndash;6484, 1989a. </mixed-citation>
</ref>
<ref id="ref17">
<label>17</label><mixed-citation publication-type="other" xlink:type="simple"> Carlson, J M. and Langer, J S.: Properties of earthquake generated by fault dynamics, Phys. Rev. Lett., 62, 2632&amp;ndash;2635, 1989b. </mixed-citation>
</ref>
<ref id="ref18">
<label>18</label><mixed-citation publication-type="other" xlink:type="simple"> Carpinteri, A. and Pugno, N.: Are scaling laws on strength of solids related to mechanics or to geometry?, Nature Materials, 4, 421&amp;ndash;423, 2005. </mixed-citation>
</ref>
<ref id="ref19">
<label>19</label><mixed-citation publication-type="other" xlink:type="simple"> Christensen, K. and Olami, Z.: Variation of the Gutemberg-Richter b values and nontrivial temporal correlations in a spring-block model of earthquakes, J. Geophys. Res., 97, 8729&amp;ndash;8735, 1992. </mixed-citation>
</ref>
<ref id="ref20">
<label>20</label><mixed-citation publication-type="other" xlink:type="simple"> Colella, V S., Klopfer, E., and Resnick, M.: Adventures in Modeling: Exploring Complex Dynamic Systems with Starlogo, Teachers College Press, 2001. </mixed-citation>
</ref>
<ref id="ref21">
<label>21</label><mixed-citation publication-type="other" xlink:type="simple"> Cover, T. and Thomas, J.: Elements of information theory, Wiley and Sons, 1991. </mixed-citation>
</ref>
<ref id="ref22">
<label>22</label><mixed-citation publication-type="other" xlink:type="simple"> Daley, D J. and Vere-Jones, D.: Scoring probability forecasts for point processes: the entropy score and information gain, J. Appl. Probab., 41A, 297&amp;ndash;312, 2004. </mixed-citation>
</ref>
<ref id="ref23">
<label>23</label><mixed-citation publication-type="other" xlink:type="simple"> Geophys. Lab. A.U.TH. and ITSAK: Collection and Procession of Seismological data and computation of seismic hazard map of Greece according to the current Greek Seismic Code and Eurocode, Tech. rep., Geophysical Laboratory of Aristotle University of Thessaloniki and Institute of Engineering Seismology and Earthquake Engineering (ITSAK), 2002. </mixed-citation>
</ref>
<ref id="ref24">
<label>24</label><mixed-citation publication-type="other" xlink:type="simple"> Gould, H. and Tobochnik, J.: Thermal and Statistical Physics, available at http://stp.clarku.edu/notes/, 2004. </mixed-citation>
</ref>
<ref id="ref25">
<label>25</label><mixed-citation publication-type="other" xlink:type="simple"> Gutenberg, B. and Richter, C.: Earthquake magnitude, intensity, energy, and acceleration, Bull. Seism. Soc. Am., 46, 105&amp;ndash;145, 1956. </mixed-citation>
</ref>
<ref id="ref26">
<label>26</label><mixed-citation publication-type="other" xlink:type="simple"> Hirata, T.: A correlation between the b value and the fractal dimension of earthquakes, J. Geophys. Res., 94, 7507&amp;ndash;7514, 1989. </mixed-citation>
</ref>
<ref id="ref27">
<label>27</label><mixed-citation publication-type="other" xlink:type="simple"> Hirata, T. and Imoto, M.: Multifractal analysis of spatial distribution of microearthquakes in the Kanto region, Geophys. J. Int., 107, 155&amp;ndash;162, 1991. </mixed-citation>
</ref>
<ref id="ref28">
<label>28</label><mixed-citation publication-type="other" xlink:type="simple"> Hirata, T. and Imoto, M.: A probabilistic cellular automaton approach for a spatiotemporal seismic activity pattern, Zisin, 2(49), 441&amp;ndash;449, 1997. </mixed-citation>
</ref>
<ref id="ref29">
<label>29</label><mixed-citation publication-type="other" xlink:type="simple"> Jiménez, A. and Posadas, A M.: A Moore&apos;s cellular automaton model to get probabilistic seismic hazard maps for different magnitude releases: A case study for Greece, Tectonophysics, 423, 35&amp;ndash;42, 2006. </mixed-citation>
</ref>
<ref id="ref30">
<label>30</label><mixed-citation publication-type="other" xlink:type="simple"> Jiménez, A., Posadas, A M., and Marfil, J M.: A Probabilistic Seismic Hazard Model based on Cellular Automata and Information Theory, Nonlin. Processes Geophys., 12, 381&amp;ndash;396, 2005. </mixed-citation>
</ref>
<ref id="ref31">
<label>31</label><mixed-citation publication-type="other" xlink:type="simple"> Jiménez, A., Tiampo, K F., Levin, S., and Posadas, A M.: Testing the persistence in earthquake catalogs: The Iberian Peninsula, Europhys. Lett., 73 (2), 171&amp;ndash;177, 2006. </mixed-citation>
</ref>
<ref id="ref32">
<label>32</label><mixed-citation publication-type="other" xlink:type="simple"> Klein, W., Rundle, J B., and Ferguson, C D.: Scaling and Nucleation in Models of Earthquake Faults, Phys. Rev. Lett., 78, 3793&amp;ndash;3796, 1997. </mixed-citation>
</ref>
<ref id="ref33">
<label>33</label><mixed-citation publication-type="other" xlink:type="simple"> Klein, W., Gould, H., Tobochnik, J., Alexander, F J., Anghel, M., and Johnson, G.: Clusters and Fluctuations at Mean-Field Critical Points and Spinodals, Phys. Rev. Lett., 85, 1270&amp;ndash;1273, 2000. </mixed-citation>
</ref>
<ref id="ref34">
<label>34</label><mixed-citation publication-type="other" xlink:type="simple"> Kroll, D M. and Gompper, G.: Wetting in fcc Ising antiferromagnets and binary alloys, Phys. Rev. B, 36, 7078&amp;ndash;7090, 1987. </mixed-citation>
</ref>
<ref id="ref35">
<label>35</label><mixed-citation publication-type="other" xlink:type="simple"> Mai, P M. and Beroza, G C.: Source Scaling Properties from Finite-Fault-Rupture Models, Bull. Seism. Soc. Am., 90, 604&amp;ndash;615, 2000. </mixed-citation>
</ref>
<ref id="ref36">
<label>36</label><mixed-citation publication-type="other" xlink:type="simple"> Makropoulos, K C. and Burton, P W.: A catalogue of seismicity in Greece and adjacent areas, Geophys. J. R. Astron. Soc., 65, 741&amp;ndash;762, 1985. </mixed-citation>
</ref>
<ref id="ref37">
<label>37</label><mixed-citation publication-type="other" xlink:type="simple"> Makropoulos, K C., Drakopoulos, J K., and Latousakis, J B.: A revised and extended earthquake catalogue for Greece since 1900, Geophys. J. Int. Res. Note, 98, 391&amp;ndash;394, 1989. </mixed-citation>
</ref>
<ref id="ref38">
<label>38</label><mixed-citation publication-type="other" xlink:type="simple"> Nakanishi, H.: Cellular-automaton model of earthquakes with deterministic dynamics, Phys. Rev. A, 41, 7068&amp;ndash;7089, 1990. </mixed-citation>
</ref>
<ref id="ref39">
<label>39</label><mixed-citation publication-type="other" xlink:type="simple"> Nakanishi, H.: Statistical properties of the cellular-automaton model for earthquakes, Phys. Rev. A, 43, 6613&amp;ndash;6621, 1991. </mixed-citation>
</ref>
<ref id="ref40">
<label>40</label><mixed-citation publication-type="other" xlink:type="simple"> Olami, Z., Feder, H. J S., and Christensen, K.: Self-organized criticality in a continuous, nonconservative cellular automaton modeling earthquakes, Phys. Rev. Lett., 68, 1244&amp;ndash;1247, 1992. </mixed-citation>
</ref>
<ref id="ref41">
<label>41</label><mixed-citation publication-type="other" xlink:type="simple"> Omori, F.: On the aftershocks of earthquakes, J. Coll. Sci., 7, 111&amp;ndash;200, 1895. </mixed-citation>
</ref>
<ref id="ref42">
<label>42</label><mixed-citation publication-type="other" xlink:type="simple"> Otsuka, M.: A simulation of earthquake occurrence, Phys. Earth Planet. Inter., 6, 311&amp;ndash;315, 1972. </mixed-citation>
</ref>
<ref id="ref43">
<label>43</label><mixed-citation publication-type="other" xlink:type="simple"> Papaioannou, C A. and Papazachos, B C.: Time-independent and time-dependent seismic hazard in Greece based on seismogenic sources, Bull. Seismol. Soc. Am., 90, 22&amp;ndash;33, 2000. </mixed-citation>
</ref>
<ref id="ref44">
<label>44</label><mixed-citation publication-type="other" xlink:type="simple"> Papazachos, B C. and Comninakis, P E.: Geophysical and tectonic features of the Aegean arc, J. Geophys. Res., 76, 8517&amp;ndash;8533, 1971. </mixed-citation>
</ref>
<ref id="ref45">
<label>45</label><mixed-citation publication-type="other" xlink:type="simple"> Papazachos, B C. and Papazachou, C.: The Earthquakes of Greece, Editions Ziti, Thessaloniki, 1997. </mixed-citation>
</ref>
<ref id="ref46">
<label>46</label><mixed-citation publication-type="other" xlink:type="simple"> Papazachos, B C., Karakaisis, G F., and Hatzidimitriou, P M.: Further information on the transform fault of the Ionian sea, XXIV Gen. Ass., Europ. Seism. Com., Athens, 19&amp;ndash;24 September, 1, 377&amp;ndash;384, 1994. </mixed-citation>
</ref>
<ref id="ref47">
<label>47</label><mixed-citation publication-type="other" xlink:type="simple"> Pe&amp;ntilde;a, J., Vidal, F., Posadas, A M., Morales, J., Alguacil, G., De Miguel, F., Ibá&amp;ntilde;ez, J., Romacho, M., and López-Linares, A.: Space clustering properties of the Betic-Alboran earthquakes in the period 1962&amp;ndash;1989, Tectonophysics, 221, 125&amp;ndash;134, 1993. </mixed-citation>
</ref>
<ref id="ref48">
<label>48</label><mixed-citation publication-type="other" xlink:type="simple"> Polimenakos, L C.: Shallow seismicity in the area of Greece: its character as seen by means of a stochastic model, Nonlin. Processes Geophys., 2, 136&amp;ndash;146, 1995. </mixed-citation>
</ref>
<ref id="ref49">
<label>49</label><mixed-citation publication-type="other" xlink:type="simple"> Posadas, A M., Hirata, T., Vidal, F., and Correig, A.: Spatiotemporal seismicity patterns using mutual information application to southern Iberian peninsula (Spain) earthquakes, Phys. Earth Planet. Inter., 122, 269&amp;ndash;276, 2000.   </mixed-citation>
</ref>
<ref id="ref50">
<label>50</label><mixed-citation publication-type="other" xlink:type="simple"> Rundle, J B.: A physical model for earthquakes, 2. Application to Southern California, J. Geophys. Res., 93, 6255&amp;ndash;6274, 1988. </mixed-citation>
</ref>
<ref id="ref51">
<label>51</label><mixed-citation publication-type="other" xlink:type="simple"> Rundle, J B.: Magnitude-frequency relations for earthquakes using a statistical mechanical approach, J. Geophys. Res., 98, 21 943&amp;ndash;21 949, 1993. </mixed-citation>
</ref>
<ref id="ref52">
<label>52</label><mixed-citation publication-type="other" xlink:type="simple"> Rundle, J B., Turcotte, D L., Shcherbakov, R., Klein, W., and Sammis, C.: Statistical physics approach to understanding the multiscale dynamics of earthquake fault systems, Rev. Geophys., 41, 1019&amp;ndash;1049, 2003. </mixed-citation>
</ref>
<ref id="ref53">
<label>53</label><mixed-citation publication-type="other" xlink:type="simple"> Rundle, P B., Rundle, J B., Klein, W., Martins, J. S S., Tiampo, K F., Donnellan, A., and Kellogg, L H.: GEM plate boundary simulations for the plate boundary observatory: Understanding the physics of earthquakes on complex fault systems, Pure. Appl. Geophys., 159, 2357&amp;ndash;2381, 2002. </mixed-citation>
</ref>
<ref id="ref54">
<label>54</label><mixed-citation publication-type="other" xlink:type="simple"> Ryan, M J. and Frater, M R.: Communications and information systems, Argos Press, 2002. </mixed-citation>
</ref>
<ref id="ref55">
<label>55</label><mixed-citation publication-type="other" xlink:type="simple"> Sammis, C G. and Smith, S W.: Seismic Cycles and the Evolution of Stress Correlation in Cellular Automaton Models of Finite Fault Networks, Pure Appl. Geophys., 155, 307&amp;ndash;334, 1999. </mixed-citation>
</ref>
<ref id="ref56">
<label>56</label><mixed-citation publication-type="other" xlink:type="simple"> Schaap, H G.: Ising models and neural networks, Ph.D. thesis, University of Groningen, 2005. </mixed-citation>
</ref>
<ref id="ref57">
<label>57</label><mixed-citation publication-type="other" xlink:type="simple"> Smalley, R F., Chatelain, J L., Turcotte, D L., and Prevot, R.: A fractal approach to the clustering of earthquakes: applications to the seismicity of the New Hebrides, Bull. Seism. Soc. Amer., 77, 1368&amp;ndash;1381, 1987. </mixed-citation>
</ref>
<ref id="ref58">
<label>58</label><mixed-citation publication-type="other" xlink:type="simple"> Sornette, A. and Sornette, D.: Self-organized criticality and earthquakes, Europhys. Lett., 9, 197&amp;ndash;202, 1989. </mixed-citation>
</ref>
<ref id="ref59">
<label>59</label><mixed-citation publication-type="other" xlink:type="simple"> Sornette, D. and Sammis, C G.: Complex Critical Exponent from Renormalization Group Theory of Earthquakes: Implications for Earthquake Predictions, J. Phys. I France, 5, 607&amp;ndash;619, 1995. </mixed-citation>
</ref>
<ref id="ref60">
<label>60</label><mixed-citation publication-type="other" xlink:type="simple"> Tiampo, K F., Rundle, J B., McGinnis, S., Gross, S J., and Klein, W.: Mean-field threshold systems and phase dynamics: an application to earthquake fault systems, Europhys. Lett., 60, 481&amp;ndash;488, 2002a. </mixed-citation>
</ref>
<ref id="ref61">
<label>61</label><mixed-citation publication-type="other" xlink:type="simple"> Tiampo, K F., Rundle, J B., McGinnis, S A., and Klein, W.: Pattern Dynamics and Forecast Methods in Seismically Active Regions, Pure Appl. Geophys., 159, 2429&amp;ndash;2467, 2002b. </mixed-citation>
</ref>
<ref id="ref62">
<label>62</label><mixed-citation publication-type="other" xlink:type="simple"> Toussaint, R. and Pride, S R.: Interacting damage models mapped onto Ising and percolation models, ArXiv Condensed Matter e-prints, 2004. </mixed-citation>
</ref>
<ref id="ref63">
<label>63</label><mixed-citation publication-type="other" xlink:type="simple"> Vicsek, T.: Fractal growth phenomena, World Scientific, 1992. </mixed-citation>
</ref>
<ref id="ref64">
<label>64</label><mixed-citation publication-type="other" xlink:type="simple"> Wells, D L. and Coppersmith, K J.: New empirical relationships among magnitude, rupture length, rupture width, rupture area, and surface displacement, Bull. Seism. Soc. Am., 84, 974&amp;ndash;1002, 1994. </mixed-citation>
</ref>
<ref id="ref65">
<label>65</label><mixed-citation publication-type="other" xlink:type="simple"> Wong, S K.: A cursory study of the thermodynamic and mechanical properties of Monte-Carlo simulations of the Ising model, Ph.D. thesis, University of Notre Dame, Indiana, 2005. </mixed-citation>
</ref>
</ref-list>
</back>
</article>