By Casacuberta C. (ed.), Castellet M. (ed.)
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This ebook is the “Study booklet” of ICMI-Study no. 20, which was once run in cooperation with the overseas Congress on and utilized arithmetic (ICIAM). The editors have been the co-chairs of the research (Damlamian, Straesser) and the organiser of the learn convention (Rodrigues). The textual content features a finished record at the findings of the learn convention, unique plenary displays of the examine convention, reviews at the operating teams and chosen papers from far and wide global.
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For example, the bi-modular distribution was discovered from empirical data in option prices . One possible mechanism of polarized behavior is collective action studied in physics and social psychology . Sudden regime switch or phase transition may occur between uni-modular and bi-modular distribution when ﬁeld parameter changes across some threshold. Here, we discuss two possible models in statistical mechanics. One is the Ising model of ferromagnetism; and another is population model of social interaction.
Frisch shared the ﬁrst Nobel Prize in economics for this model in 1969. The noise-driven model is also behind the works by Lucas and RBC school. However, physicists already knew before Frisch that the demped harmonic cycles could not be kept alive by random shocks , since its amplitude would decay exponentially . We calibrate the Frisch model by the US data. We found that American business cycles would only last about 4–10 years . In fact, the NBER recorded US business cycles since 1854, more than 160 years ago.
B Horizontal flat distribution with b = a (denoted by long dashed line). Marginal case when individualistic orientation balances the social pressure. c Polarized distribution with b > a (denoted by solid line). It occurs when social pressure through mutual communication is stronger than independent judgment Fig. 5 Numerical autocorrelations from time series generated by random noise and harmonic wave. The solid line is white noise. 8 1 Linear Model of Harmonic Cycle and White Noise Linear harmonic cycles with unique frequency are introduced in business cycle theory .