Summary. Equilibrium equations and stability conditions for the simple deformable elastic body are derived by means of considering a minimum of the static energy principle. The energy is supposed to be sum of the volume (elastic) and the surface terms. The ability to change relative positions of different material particles is taken into account, and appropriate natural definitions of the first and second variations of the energy are introduced and calculated explicitly. Considering the case of negligible magnitude of the surface tension, we establish that an equilibrium state of a nonhydrostatically stressed simple elastic body (of any physically reasonable elastic energy potential and of any symmetry) possessing any small smooth part of free surface is always unstable with respect to relative transfer of the material particles along the surface. Surface tension suppresses the mentioned instability with respect to sufficiently short disturbances of the boundary surface and thus can probably provide local smoothness of the equilibrium shape of the crystal. We derive explicit formulas for critical wavelength for the simplest models of the internal and surface energies and for the simplest equilibrium configurations. We also formulate the simplest problem of mathematical physics, revealing peculiarities and difficulties of the problem of equilibrium shape of elastic crystals, and discuss possible manifestations of the abovementioned instability in the problems of crystal growth, materials science, fracture, physical chemistry, and low-temperature physics.
In this paper we use arguments based on Picard-Fuchs equations and transversality of intersections of level curves to obtain an exact count of the number of stationary solutions of the one-dimensional Cahn-Hilliard equation with a cubic nonlinearity.
We introduce a class of agent-based market models founded upon simple descriptions of investor psychology. Agents are subject to various psychological tensions induced by market conditions, and endowed with a minimal 'personality'. This personality consists of a threshold level for each of the tensions being modeled, and the agent reacts whenever a tension threshold is reached. This paper considers an elementary model including just two such tensions. The first is 'cowardice', which is the stress caused by remaining in a minority position with respect to overall market sentiment, and leads to herding-type behaviour. The second is 'inaction', which is the increasing desire to act or re-evaluate one's investment position. There is no inductive learning by agents, and they are only coupled via the global market price and overall market sentiment. Even incorporating just these two psychological tensions, important stylized facts of real market data, including fat-tails, excess kurtosis, uncorrelated price returns and clustered volatility over the timescale of a few days, are reproduced. By then introducing an additional parameter that amplifies the effect of externally generated market noise during times of extreme market sentiment, long-time volatility correlations can also be recovered.
The distribution of capture zones formed during the nucleation and growth of point islands on a one-dimensional substrate during monomer deposition is considered for general critical island size i. A fragmentation theory approach yields the small and (for i = 0) large size asymptotics for the capture zone distribution (CZD) under the assumption of no neighbourneighbour gap size correlation. These CZD asymptotic forms are different to those of the Generalised Wigner Surmise which has recently been proposed for island nucleation and growth models, and we discuss the reasons for the discrepancies.
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