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Royal Society Open Science RSS feed -- recent Mathematics articles2054-5703Royal Society Open Science<![CDATA[Pedestrian motion modelled by Fokker-Planck Nash games]]>
http://rsos.royalsocietypublishing.org/cgi/content/short/4/9/170648?rss=1
A new approach to modelling pedestrians' avoidance dynamics based on a Fokker–Planck (FP) Nash game framework is presented. In this framework, two interacting pedestrians are considered, whose motion variability is modelled through the corresponding probability density functions (PDFs) governed by FP equations. Based on these equations, a Nash differential game is formulated where the game strategies represent controls aiming at avoidance by minimizing appropriate collision cost functionals. The existence of Nash equilibria solutions is proved and characterized as a solution to an optimal control problem that is solved numerically. Results of numerical experiments are presented that successfully compare the computed Nash equilibria to the output of real experiments (conducted with humans) for four test cases.
]]>2017-09-13T00:09:20-07:00info:doi/10.1098/rsos.170648hwp:master-id:royopensci;rsos.1706482017-09-13Mathematics49170648170648<![CDATA[Huygens clocks revisited]]>
http://rsos.royalsocietypublishing.org/cgi/content/short/4/9/170777?rss=1
In 1665, Huygens observed that two identical pendulum clocks, weakly coupled through a heavy beam, soon synchronized with the same period and amplitude but with the two pendula swinging in opposite directions. This behaviour is now called anti-phase synchronization. This paper presents an analysis of the behaviour of a large class of coupled identical oscillators, including Huygens' clocks, using methods of equivariant bifurcation theory. The equivariant normal form for such systems is developed and the possible solutions are characterized. The transformation of the physical system parameters to the normal form parameters is given explicitly and applied to the physical values appropriate for Huygens' clocks, and to those of more recent studies. It is shown that Huygens' physical system could only exhibit anti-phase motion, explaining why Huygens observed exclusively this. By contrast, some more recent researchers have observed in-phase or other more complicated motion in their own experimental systems. Here, it is explained which physical characteristics of these systems allow for the existence of these other types of stable solutions. The present analysis not only accounts for these previously observed solutions in a unified framework, but also introduces behaviour not classified by other authors, such as a synchronized toroidal breather and a chaotic toroidal breather.
]]>2017-09-06T01:49:28-07:00info:doi/10.1098/rsos.170777hwp:master-id:royopensci;rsos.1707772017-09-06Mathematics49170777170777<![CDATA[Modelling radicalization: how small violent fringe sects develop into large indoctrinated societies]]>
http://rsos.royalsocietypublishing.org/cgi/content/short/4/8/170678?rss=1
We model radicalization in a society consisting of two competing religious, ethnic or political groups. Each of the ‘sects’ is divided into moderate and radical factions, with intra-group transitions occurring either spontaneously or through indoctrination. We also include the possibility of one group violently attacking the other. The intra-group transition rates of one group are modelled to explicitly depend on the actions and characteristics of the other, including violent episodes, effectively coupling the dynamics of the two sects. We use a game theoretic framework and assume that radical factions may tune ‘strategic’ parameters to optimize given utility functions aimed at maximizing their ranks while minimizing the damage inflicted by their rivals. Constraints include limited overall resources that must be optimally allocated between indoctrination and external attacks on the other group. Various scenarios are considered, from symmetric sects whose behaviours mirror each other, to totally asymmetric ones where one sect may have a larger population or a superior resource availability. We discuss under what conditions sects preferentially employ indoctrination or violence, and how allowing sects to readjust their strategies allows for small, violent sects to grow into large, indoctrinated communities.
]]>2017-08-30T00:09:44-07:00info:doi/10.1098/rsos.170678hwp:master-id:royopensci;rsos.1706782017-08-30Mathematics48170678170678<![CDATA[The impact of weather and storm water management ponds on the transmission of West Nile virus]]>
http://rsos.royalsocietypublishing.org/cgi/content/short/4/8/170017?rss=1
West Nile virus (WNV) is the most widely distributed arbovirus in the world and the spread is influenced by complex factors including weather conditions and urban environmental settings like storm water management ponds (SWMP). The purpose of this work was to develop an ordinary differential equation model to explore the impacts of SWMP, temperature and precipitation on WNV vector abundance and the transmission of WNV between mosquito and bird populations. The model was used to analyse how weather conditions and SWMP can influence the basic reproduction number. The results found that an excess of precipitation and fiercer intraspecific competition will reduce vector population and the peak value of infectious vectors and birds. This information can be used to identify measures that would be useful to control larval abundance in SWMP and the transmission of WNV.
]]>2017-08-16T00:08:56-07:00info:doi/10.1098/rsos.170017hwp:master-id:royopensci;rsos.1700172017-08-16Mathematics48170017170017<![CDATA[Stochastic phase segregation on surfaces]]>
http://rsos.royalsocietypublishing.org/cgi/content/short/4/8/170472?rss=1
Phase separation and coarsening is a phenomenon commonly seen in binary physical and chemical systems that occur in nature. Often, thermal fluctuations, modelled as stochastic noise, are present in the system and the phase segregation process occurs on a surface. In this work, the segregation process is modelled via the Cahn–Hilliard–Cook model, which is a fourth-order parabolic stochastic system. Coarsening is analysed on two sample surfaces: a unit sphere and a dumbbell. On both surfaces, a statistical analysis of the growth rate is performed, and the influence of noise level and mobility is also investigated. For the spherical interface, it is also shown that a lognormal distribution fits the growth rate well.
]]>2017-08-16T00:08:56-07:00info:doi/10.1098/rsos.170472hwp:master-id:royopensci;rsos.1704722017-08-16Mathematics48170472170472<![CDATA[Integrating sentiment and social structure to determine preference alignments: the Irish Marriage Referendum]]>
http://rsos.royalsocietypublishing.org/cgi/content/short/4/7/170154?rss=1
We examine the relationship between social structure and sentiment through the analysis of a large collection of tweets about the Irish Marriage Referendum of 2015. We obtain the sentiment of every tweet with the hashtags #marref and #marriageref that was posted in the days leading to the referendum, and construct networks to aggregate sentiment and use it to study the interactions among users. Our analysis shows that the sentiment of outgoing mention tweets is correlated with the sentiment of incoming mentions, and there are significantly more connections between users with similar sentiment scores than among users with opposite scores in the mention and follower networks. We combine the community structure of the follower and mention networks with the activity level of the users and sentiment scores to find groups that support voting ‘yes’ or ‘no’ in the referendum. There were numerous conversations between users on opposing sides of the debate in the absence of follower connections, which suggests that there were efforts by some users to establish dialogue and debate across ideological divisions. Our analysis shows that social structure can be integrated successfully with sentiment to analyse and understand the disposition of social media users around controversial or polarizing issues. These results have potential applications in the integration of data and metadata to study opinion dynamics, public opinion modelling and polling.
]]>2017-07-12T00:08:29-07:00info:doi/10.1098/rsos.170154hwp:master-id:royopensci;rsos.1701542017-07-12Mathematics47170154170154<![CDATA[Population patterns in Worlds administrative units]]>
http://rsos.royalsocietypublishing.org/cgi/content/short/4/7/170281?rss=1
Whereas there has been an extended discussion concerning city population distribution, little has been said about that of administrative divisions. In this work, we investigate the population distribution of second-level administrative units of 150 countries and territories and propose the discrete generalized beta distribution (DGBD) rank-size function to describe the data. After testing the balance between the goodness of fit and number of parameters of this function compared with a power law, which is the most common model for city population, the DGBD is a good statistical model for 96% of our datasets and preferred over a power law in almost every case. Moreover, the DGBD is preferred over a power law for fitting country population data, which can be seen as the zeroth-level administrative unit. We present a computational toy model to simulate the formation of administrative divisions in one dimension and give numerical evidence that the DGBD arises from a particular case of this model. This model, along with the fitting of the DGBD, proves adequate in reproducing and describing local unit evolution and its effect on the population distribution.
]]>2017-07-05T00:08:13-07:00info:doi/10.1098/rsos.170281hwp:master-id:royopensci;rsos.1702812017-07-05Mathematics47170281170281<![CDATA[A quantum Samaritans dilemma cellular automaton]]>
http://rsos.royalsocietypublishing.org/cgi/content/short/4/6/160669?rss=1
The dynamics of a spatial quantum formulation of the iterated Samaritan’s dilemma game with variable entangling is studied in this work. The game is played in the cellular automata manner, i.e. with local and synchronous interaction. The game is assessed in fair and unfair contests, in noiseless scenarios and with disrupting quantum noise.
]]>2017-06-14T00:39:50-07:00info:doi/10.1098/rsos.160669hwp:master-id:royopensci;rsos.1606692017-06-14Mathematics46160669160669