In this talk, we are concerned with the analytic aspects of the mean field equation for equilibrium turbulence with arbitrarily-signed vortices. Especially we develop the blow-up analysis, that is, the classification of the limits of the non-compact solution sequences to the equation and establish...
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the variations of the refractive index, the gradient of the refractive index, the divergence of the rays, the frequency of modulation, and the widths of light pulses, which are related to the mean refractive index, the absorption coefficient, and the reduced scattering coefficient of the medium....
are simple yet very powerful tools for describing phenomena such as motility-induced phase separation. No comparable theory has been derived yet for the underdamped case. In this work, we introduce active model I+, an extension of active model B+ to particles with inertia. The governing equatio...
[0,+\infty \right)and is the time-like coordinate on which the system evolves, while\tau \in \left[-{T}_{{\rm{M}}}/2,+{T}_{{\rm{M}}}/2\right]and is the space-like coordinate on whichFandgverify the periodic boundary conditions (16) (\overline{G}is a mean field which ...
disparate mass binary mixtures from a consistent asymptotic analysis of the underlying kinetic equations, taking into account the smallness of the mass ratio between the particles simultaneously with letting the Knudsen number (ratio between the collision mean free-path to the macroscopic scale) to ...
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For the hydrogen atom, the radial equation can be written in the form d2Rdr2+2rdRdr+2E+2Zr−ℓ(ℓ+1)r2R=0, where e is the energy of the hydrogen atom and Z is a constant.6 When E < 0 and we let p=−2E, a solution of the radial equation for the hydrogen atom has ...
In contrast to (6.5) below, (6.2) is satisfied for all times s∈R, and we use the word “ancient” to emphasize this fact. That is, the solution is a stationary process, its distribution is constant in time. The law of the sequence of mean-zero, Gaussian random variables (gn)n∈Z...
mean plant species richness per sampling point analyzed with a General Linear Mixed Model based on a normal distribution with the nlme package (Pinheiro et al., 2015); (2) carabid species richness per sampling point analyzed with a General Linear Mixed Model based on a normal distribution with...