In 1944 Onsager published the formula for the partition function of the Ising model for the infinite square lattice. He was able to express the internal energy in terms of a special function, but he left the free energy as a definite integral. Seven
decades later, the partition function and free energy have yet to be written in closed form, even with the aid of special functions. Here we evaluate the definite integral explicitly, using hypergeometric series. Let $beta$ denote the reciprocal temperature, $J$ the coupling and $f$ the free energy per spin. We prove that $-beta f = ln(2 cosh 2K) - kappa^2, {}_4F_3 [1,1,tfrac{3}{2},tfrac{3}{2}; 2,2,2 ; 16 kappa^2 ] $, where $_p F_q$ is the generalized hypergeometric function, $K=beta J$, and $2kappa= {rm tanh} 2K {rm sech} 2K$.
We define the Ladyzhenskaya-Lions exponent $alpha_{rm {tiny sc l}} (n)=({2+n})/4$ for Navier-Stokes equations with dissipation $-(-Delta)^{alpha}$ in ${Bbb R}^n$, for all $ngeq 2$. We review the proof of strong global solvability when $alphageq alp
ha_{rm {tiny sc l}} (n)$, given smooth initial data. If the corresponding Euler equations for $n>2$ were to allow uncontrolled growth of the enstrophy ${1over 2} | abla u |^2_{L^2}$, then no globally controlled coercive quantity is currently known to exist that can regularize solutions of the Navier-Stokes equations for $alpha<alpha_{rm {tiny sc l}} (n)$. The energy is critical under scale transformations only for $alpha=alpha_{rm {tiny sc l}} (n)$.
We investigate the formation of singularities in the incompressible Navier-Stokes equations in $dgeq 2$ dimensions with a fractional Laplacian $| abla |^alpha$. We derive analytically a sufficient but not necessary condition for solutions to remain a
lways smooth and show that finite time singularities cannot form for $alphageq alpha_c= 1+d/2$. Moreover, initial singularities become unstable for $alpha>alpha_c$.
We propose a new linearizable model for the nonlinear photocurrent-voltage characteristics of nanocrystalline TiO$_2$ dye sensitized solar cells based on first principles and report predicted values for fill factors. Upon renormalization diverse expe
rimental photocurrent-voltage data collapse onto a single universal function. These advances allow the estimation of the complete current-voltage curve and the fill factor from any three experimental data points, e.g., the open circuit voltage, the short circuit current and one intermediate measurement. The theoretical underpinning provides insight into the physical mechanisms responsible for the remarkably large fill factors as well as their known dependence on the open circuit voltage.
We investigate a recently proposed non-Markovian random walk model characterized by loss of memories of the recent past and amnestically induced persistence. We report numerical and analytical results showing the complete phase diagram, consisting of
4 phases, for this system: (i) classical nonpersistence, (ii) classical persistence (iii) log-periodic nonpersistence and (iv) log-periodic persistence driven by negative feedback. The first two phases possess continuous scale invariance symmetry, however log-periodicity breaks this symmetry. Instead, log-periodic motion satisfies discrete scale invariance symmetry, with complex rather than real fractal dimensions. We find for log-periodic persistence evidence not only of statistical but also of geometric self-similarity.
