Nonlinear Estimates in Anisotropic Gevrey Spaces

2002 Mathematics Subject Classification: 35G20, 47H30

Systems of coupled clamped beams equations with full nonlinear terms: Existence and location results

This work gives sufficient conditions for the solvability of the fourth order coupled system┊ u⁽⁴⁾(t)=f(t,u(t),u′(t),u′′(t),u′′′(t),v(t),v′(t),v′′(t),v′′′(t)) v⁽⁴⁾(t)=h(t,u(t),u′(t),u′′(t),u′′′(t),v(t),v′(t),v′′(t),v′′′(t)) with f,h: [0,1]×ℝ⁸→ℝ some L¹- Carathéodory functions, and the boundary conditions {┊ u(0)=u′(0)=u′′(0)=u′′(1)=0 v(0)=v′(0)=v′′(0)=v′′(1)=0. To the best of our knowledge, it is the first time in the literature where two beam equations are considered with full nonlinearities, that is, with dependence on all derivatives of u and v. An application to the study of the bending of two elastic coupled campled beams is considered.

Machine learning approach to the extended Hubbard model

The 1d extended Hubbard model with soft-shoulder potential has proved itself to be very difficult to study due its non solvability and to competition between terms of the Hamiltonian. Given this, we tried to investigate its phase diagram for filling n=2/5 and range of soft-shoulder potential r=2 by using Machine Learning techniques. That led to a rich phase diagram; calling U, V the parameters associated to the Hubbard potential and the soft-shoulder potential respectively, we found that for V<5 and U>3 the system is always in Tomonaga Luttinger Liquid phase, then becomes a Cluster Luttinger Liquid for 57, with a quasi-perfect crystal in the U<3V/2 and U>5 region. Finally we found that for U<5 and V>2-3 the system shall maintain the Cluster Luttinger Liquid structure, with a residual in-block single particle mobility.