We study the Fröhlich Hamiltonian with cut-off. We show that this Hamiltonian is self-adjoint in the Fock space and semibounded when the number of electrons is kept constant. The lower bound we give generalizes the bound obtained for one electron by Lee and Pines to the case of several electrons. We also give a domain where the perturbation series for the resolvent of the Hamiltonian does converge in the operator norm. Finally, we study the influence of the Coulomb interaction.
Solutions x(t) of the Lipschitz equation x = f(x) with an arbitrary vector norm are considered. It is proved that the sharp lower bound for the distances between successive extremums of xk(t) equals π/L where L is the Lipschitz constant. For non-constant periodic solutions, the lower bound for the periods is 2π/L. These estimates are achieved for norms that are invariant with respect to permutation of the indices.
The best lower bound known on the crossing number of the complete bipartite graph is : $$cr(K_{m,n}) \geq (1/5)(m)(m-1)\lfloor n/2 \rfloor \lfloor(n-1)/2\rfloor$$ In this paper we prove that: $$cr(K_{m,n}) \geq (1/5)m(m-1)\lfloor n/2 \rfloor \lfloor (n-1)/2 \rfloor + 9.9 \times 10^{-6} m^2n^2$$ for sufficiently large $m$ and $n$.