Abstract:
In this work, we show how to obtain for non-compact manifolds the results that have already been done for Monge Transport Problem for costs coming from Tonelli Lagrangians on compact manifolds. In particular, the already known results for a cost of the type $d^r,r>1$, where $d$ is the Riemannian distance of a complete Riemannian manifold, hold without any curvature restriction.

Abstract:
We give a proof of Ilmanen's lemma, which asserts that between a locally semi-convex and a locally semi-concave function it is possible to find a C$^{1,1}$ function.

Abstract:
In this paper, we consider a time independent $C^2$ Hamiltonian, sa\-tisfying the usual hypothesis of the classical Calculus of Variations, on a non-compact connected manifold. Using the Lax-Oleinik semigroup, we give a proof of the existence of weak KAM solutions, or viscosity solutions, for the associated Hamilton-Jacobi Equation. This proof works also in presence of symmetries. We also study the role of the amenability of the group of symmetries to understand when the several critical values that can be associated with the Hamiltonian coincide.

Abstract:
Under appropriate assumptions on the dimension of the ambient manifold and the regularity of the Hamiltonian, we show that the Mather quotient is small in term of Hausdorff dimension. Then, we present applications in dynamics.

Abstract:
Given a smooth compact Riemannian manifold $M$ and a Hamiltonian $H$ on the cotangent space $T^*M$, strictly convex and superlinear in the momentum variables, we prove uniqueness of certain ergodic invariant Lagrangian graphs within a given homology or cohomology class. In particular, in the context of quasi-integrable Hamiltonian systems, our result implies global uniqueness of Lagrangian KAM tori with rotation vector $\rho$. This result extends generically to the $C^0$-closure of KAM tori.

Abstract:
We consider a continuous coercive Hamiltonian $H$ on the cotangent bundle of the compact connected manifold $M$ which is convex in the momentum. If $u_\lambda:M\to\mathbb R$ is the viscosity solution of the discounted equation $$ \lambda u_\lambda(x)+H(x,d_x u_\lambda)=c(H), $$ where $c(H)$ is the critical value, we prove that $u_\lambda$ converges uniformly, as $\lambda\to 0$, to a specific solution $u_0:M\to\mathbb R$ of the critical equation $$ H(x,d_x u)=c(H). $$ We characterize $u_0$ in terms of Peierls barrier and projected Mather measures.

The mixed spin-2 and spin-5/2 Ising ferrimagnetic system with different anisotropies (D_{A}/z_{}｜J｜) for the spin-2 and (D_{B}/z｜J｜) for the spin-5/2 is studied by the use of the mean-field theory based on the Bogoliubov inequality for the free energy. First, the ground state phase diagram of the system at zero temperature is obtained on the (D_{A}/z｜J｜,D_{B}/z｜J｜) plane. Topologically, different kinds of phase diagrams are achieved by changing the temperature and the values of the single ion anisotropies D_{A}/z｜J｜ and D_{B}/z｜J｜. Besides second-order transition lines, first order phase transition lines terminating at tricritical points, are found. The existence and dependence of a compensation temperature on single-ion anisotropies is also investigated.

The mixed spin-3/2 and spin-2 Ising ferrimagnetic system with
different single-ion anisotropies in the absence of an external magnetic field
is studied within the mean-field theory based on Bogoliubov inequality for the
Gibbs free energy. Second-order critical lines are obtained in the
temperature-anisotropy plane. Tricritical line separating second-order
and first-order lines is found. Finally, the existence and dependence of a
compensation points on single-ion anisotropies is also investigated for the
system. As a result, this mixed-spin model exhibits one, two or three compensation
temperature depending on the values of the anisotropies.

Hydrometallurgical technology offers a unique possibility for developing countries to exploit their mineral resources locally instead of shipping them as concentrates. Production plants may start on a small scale with small capital investment then increase productivity later when the economy permits without financial penalty. This is in contract to smelting operations which necessitates large scale production from the start with high capital investment that may not be available locally.

The magnetic properties of a mixed Ising ferrimagnetic system consisting of spin-3/2 and spin-2 with different single ion anisotropies and under the effect of an applied longitudinal magnetic field are investigated within the mean-field theory based on Bogoliubov inequality for the Gibbs free energy. The ground-state phase diagram is constructed. The thermal behaviours of magnetizations and magnetic susceptibilities are examined in detail. Finally, we find some interesting phenomena in these quantities, due to applied longitudinal magnetic field.