Royal Society Open Science Mathematics
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Royal Society Open Science RSS feed -- recent Mathematics articles2054-5703Royal Society Open Science<![CDATA[Bistable fully developed mixed convection flow with viscous dissipation in a vertical channel]]>
http://rsos.royalsocietypublishing.org/cgi/content/short/5/7/171880?rss=1
It is shown that unstable dual solutions in fully developed mixed convection flow in a vertical channel disappear in the presence of relatively strong thermal radiation. In this case, we have a unique stable flow at each pressure gradient. When the effect of thermal radiation is weak another branch of stable solutions is created, resulting in bistable flows. In this case, the flow exhibits hysteresis with variation of the pressure gradient. Optically, a thin radiation model is used.
]]>2018-07-11T00:09:56-07:00info:doi/10.1098/rsos.171880hwp:master-id:royopensci;rsos.1718802018-07-11Mathematics57171880171880<![CDATA[Technology networks: the autocatalytic origins of innovation]]>
http://rsos.royalsocietypublishing.org/cgi/content/short/5/6/172445?rss=1
We analyse the autocatalytic structure of technological networks and evaluate its significance for the dynamics of innovation patenting. To this aim, we define a directed network of technological fields based on the International Patents Classification, in which a source node is connected to a receiver node via a link if patenting activity in the source field anticipates patents in the receiver field in the same region more frequently than we would expect at random. We show that the evolution of the technology network is compatible with the presence of a growing autocatalytic structure, i.e. a portion of the network in which technological fields mutually benefit from being connected to one another. We further show that technological fields in the core of the autocatalytic set display greater fitness, i.e. they tend to appear in a greater number of patents, thus suggesting the presence of positive spillovers as well as positive reinforcement. Finally, we observe that core shifts take place whereby different groups of technology fields alternate within the autocatalytic structure; this points to the importance of recombinant innovation taking place between close as well as distant fields of the hierarchical classification of technological fields.
]]>2018-06-27T06:04:49-07:00info:doi/10.1098/rsos.172445hwp:master-id:royopensci;rsos.1724452018-06-27Mathematics56172445172445<![CDATA[The physical and qualitative analysis of fluctuations in air and vapour concentrations in a porous medium]]>
http://rsos.royalsocietypublishing.org/cgi/content/short/5/5/171954?rss=1
This work presents the development and physical analysis of a sweat transport model that couples the fluctuations in air and vapour concentrations, and temperature, in a one-dimensional porous clothing assembly. The clothing is exposed to inherent time-varying conditions due to variations in the body temperature and ambient conditions. These fluctuations are governed by a coupled system of nonlinear relaxation–transport–diffusion PDEs of Petrovskii parabolic type. A condition for the well-posedness of the resulting system of equations is derived. It is shown that the energy of the diffusion part of the system is exponentially decreasing. The boundedness and stability of the system of equations is thus confirmed. The variational formulation of the system is derived, and the existence and uniqueness of a weak solution is demonstrated analytically. This system is shown to conserve positivity. The difficulty of obtaining an analytical solution due to the complexity of the problem, urges for a numerical approach. A comparison of three cases is made using the Crank–Nicolson finite difference method (FDM). Numerical experiments show the existence of singular coefficient matrices at the site of phase change. Furthermore, the steady-state profiles of temperature, air and vapour concentrations influence the attenuation of fluctuations. Numerical results verify the analytical findings of this work.
]]>2018-05-16T00:11:14-07:00info:doi/10.1098/rsos.171954hwp:master-id:royopensci;rsos.1719542018-05-16Mathematics55171954171954<![CDATA[Surrogate modelling for the prediction of spatial fields based on simultaneous dimensionality reduction of high-dimensional input/output spaces]]>
http://rsos.royalsocietypublishing.org/cgi/content/short/5/4/171933?rss=1
Time-consuming numerical simulators for solving groundwater flow and dissolution models of physico-chemical processes in deep aquifers normally require some of the model inputs to be defined in high-dimensional spaces in order to return realistic results. Sometimes, the outputs of interest are spatial fields leading to high-dimensional output spaces. Although Gaussian process emulation has been satisfactorily used for computing faithful and inexpensive approximations of complex simulators, these have been mostly applied to problems defined in low-dimensional input spaces. In this paper, we propose a method for simultaneously reducing the dimensionality of very high-dimensional input and output spaces in Gaussian process emulators for stochastic partial differential equation models while retaining the qualitative features of the original models. This allows us to build a surrogate model for the prediction of spatial fields in such time-consuming simulators. We apply the methodology to a model of convection and dissolution processes occurring during carbon capture and storage.
]]>2018-04-25T00:10:32-07:00info:doi/10.1098/rsos.171933hwp:master-id:royopensci;rsos.1719332018-04-25Mathematics54171933171933<![CDATA[Noise-induced transitions and shifts in a climate-vegetation feedback model]]>
http://rsos.royalsocietypublishing.org/cgi/content/short/5/4/171531?rss=1
Motivated by the extremely important role of the Earth’s vegetation dynamics in climate changes, we study the stochastic variability of a simple climate–vegetation system. In the case of deterministic dynamics, the system has one stable equilibrium and limit cycle or two stable equilibria corresponding to two opposite (cold and warm) climate–vegetation states. These states are divided by a separatrix going across a point of unstable equilibrium. Some possible stochastic scenarios caused by different externally induced natural and anthropogenic processes inherit properties of deterministic behaviour and drastically change the system dynamics. We demonstrate that the system transitions across its separatrix occur with increasing noise intensity. The climate–vegetation system therewith fluctuates, transits and localizes in the vicinity of its attractor. We show that this phenomenon occurs within some critical range of noise intensities. A noise-induced shift into the range of smaller global average temperatures corresponding to substantial oscillations of the Earth’s vegetation cover is revealed. Our analysis demonstrates that the climate–vegetation interactions essentially contribute to climate dynamics and should be taken into account in more precise and complex models of climate variability.
]]>2018-04-11T00:05:23-07:00info:doi/10.1098/rsos.171531hwp:master-id:royopensci;rsos.1715312018-04-11Mathematics54171531171531