Abstract:
We introduce and study the basic notion of polarized Poisson manifolds generalizing the classical case of Poisson manifolds and extend this last notion for the ${k-}$% symplectic stuctures. And also, we show that for any polarized Hamiltonian map, the associated Nambu's dynamical system and polarized Hamiltonian system are connected by relations characterizing the mechanical aspect of the $k-$symplectic geometry.

Abstract:
The eThe existence of common zero of a family of polynomials has led to study of inertial forms whose homogeneous part of degree 0 constitutes the ideal resultant. The Koszul anc Cech cohomology groups play a fundamental role in this study. An analogueous of Hurwitz theorem is given and also we finds a MCoy theorem in a particular case of this study.

Abstract:
The Fermi surface calculated within the rotating antiferromagnetism theory undergoes a topological change when doping changes from p-type to n-type, in qualitative agreement with experimental data for n-type cuprate Nd2？xCexCuO4 and p-type La2？xSrxCuO4. Also, the reconstruction of the Fermi surface, observed experimentally close to optimal doping in p-type cuprates, and slightly higher than optimal doping in the overdoped regime for this n-type high-TC cuprate, is well accounted for in this theory. This reconstruction is a consequence of the quantum criticality caused by the disappearance of rotating antiferromagnetism. The present results are in qualitative agreement with recently observed quantum oscillations in some high-TC cuprates. This paper presents new results about the application of the rotating antiferromagnetism theory to the study of the electronic structure for n-type materials.

Abstract:
The phase of the rotating order parameter in rotating antiferromagnetism is calculated using a combination of mean-field theory and Heisenberg equation. This phase shows a linear time dependence, which allows us to interpret rotating antiferromagnetism as a synchronized Larmor-like precession of all the spins in the system or as an unusual ${\bf q}=(\pi,\pi)$ spin-wave around a zero local magnetization. We discuss implications for the pseudogap state of high-$T_C$ superconducting materials. Rotating antiferromagnetism has been proposed to model the pseudogap state in these materials.

Abstract:
The paramagnetic phase of the one-dimensional Kondo lattice model is investigated for electron densities below half-filling using a new mean-field approach. The physical parameters that govern this phase are identified to be the spin-flip processes of both the localized and itinerant spins. A nonmagnetic quantum state, where the local magnetization is a rotating vector with a nonzero average length, is proposed in order to describe this phase. This state does not break SU(2) symmetry in agreement with Mermin-Wagner theorem. The line boundary between this phase and the ferromagnetic phase is calculated in the coupling-density phase diagram. Also, expressions are calculated for the velocities of the conduction electrons excitations, and heat capacity and entropy versus temperature are analyzed. Good agreement with many of the available numerical data is achieved.

Abstract:
The magnetic response expected from a state characterized by rotating antiferromagnetism in a neutron-scattering experiment is calculated. We predict the occurrence of a peak at the frequency of the rotation of the rotating antiferromagnetic order parameter. The doping dependence of this frequency is very similar to that of the frequency of the magnetic resonance observed in the neutron-scattering experiments for the hole-doped high-$T_C$ cuprates. This leads us to propose the rotating antiferromagnetism as a possible mechanism for this magnetic resonance. We conclude that while the magnitude of the rotating antiferromagnetic order parameter was previously proposed to be responsible for the pseudogap and the unusual thermodynamic and transport properties, the phase of the rotating order parameter is proposed here to be responsible for the unusual magnetic properties of the high-$T_C$ copper-oxide superconductors.

Abstract:
The Fermi surface calculated within the rotating antiferromagentism theory undergoes a topological change when doping changes from p-type to n-type, in qualitative agreement with experimental data for n-type cuprate Nd$_{2-x}$Ce$_x$CuO$_4$ and p-type La$_{2-x}$Sr$_x$CuO$_4$. Also, the reconstruction of the Fermi surface observed experimentally close to optimal doing in p-type cuprates, and slightly higher than optimal doping in the overdoped regime for this n-type high-$T_C$ cuprate is well accounted for in this theory, and is a consequence of quantum criticality caused by the disappearance of rotating antiferromagnetism. The present results are in qualitative agreement with the recently observed quantum oscillations in some high-$T_C$ cuprates regarding the change in the size of the Fermi surface as doping evolves and the location of its reconstruction. This paper presents new results about the application of the rotating antiferromagnetism theory to the study of electronic structure for n-type materials.

Abstract:
Algebraic relations that characterize quantum statistics (Bose-Einstein statistic, Fermi-Dirac statistic, supersymmetry, parastatistic, anyonic statistic, ...) are reformulated herein in terms of a new algebraic structure, which we call para-algebra.

Abstract:
The motion of holes on the triangular lattice is studied using the t-J model. Within the Born self-consistent approximation and the exact Lanczos diagonalization, the single hole physics is first analyzed. Then the spiral theory of Shraiman and Siggia is used to investigate the case of a finite density of holes.

Abstract:
A mean-field theory of the spin Peierls systems based on the two dimensional dimerized Heisenberg model is proposed by introducing an alternating bond order parameter. Improvements with respect to previous mean-field results are found in the one-dimensional limit for the ground state and the gap energies. In two dimensions, the analysis of the competition between antiferromagnetic long range order and the spin-Peierls ordering is given as a function of the coupling constants. We show that the lowest energy gap to be observed does not have a singlet-triplet character in agreement with the low temperature thermodynamic properties of CuGeO3.