Magnetoresistance of Highly Correlated Electron Liquid

The behavior in magnetic fields of a highly correlated electron liquid approaching the fermion condensation quantum phase transition from the disordered phase is considered. We show that at sufficiently high temperatures $T\geq T^*(x)$ the effective mass starts to depend on $T$, $M^*\propto T^{-1/2}$. This $T^{-1/2}$ dependence of the effective mass at elevated temperatures leads to the non-Fermi liquid behavior of the resistivity, $\rho(T)\propto T$ and at higher temperatures $\rho(T)\propto T^{3/2}$. The application of a magnetic field $B$ restores the common $T^2$ behavior of the resistivity. The effective mass depends on the magnetic field, $M^*(B)\propto B^{-2/3}$, being approximately independent of the temperature at $T\leq T^*(B)\propto B^{4/3}$. At $T\geq T^*(B)$, the $T^{-1/2}$ dependence of the effective mass is re-established. We demonstrate that this $B-T$ phase diagram has a strong impact on the magnetoresistance (MR) of the highly correlated electron liquid. The MR as a function of the temperature exhibits a transition from the negative values of MR at $T\to 0$ to the positive values at $T\propto B^{4/3}$. Thus, at $T\geq T^*(B)$, MR as a function of the temperature possesses a node at $T\propto B^{4/3}$.Comment: 7 pages, revtex, no figure

Composition of processes and related partial differential equations

In this paper different types of compositions involving independent fractional Brownian motions B^j_{H_j}(t), t>0, j=1,$ are examined. The partial differential equations governing the distributions of I_F(t)=B^1_{H_1}(|B^2_{H_2}(t)|), t>0 and J_F(t)=B^1_{H_1}(|B^2_{H_2}(t)|^{1/H_1}), t>0 are derived by different methods and compared with those existing in the literature and with those related to B^1(|B^2_{H_2}(t)|), t>0. The process of iterated Brownian motion I^n_F(t), t>0 is examined in detail and its moments are calculated. Furthermore for J^{n-1}_F(t)=B^1_{H}(|B^2_H(...|B^n_H(t)|^{1/H}...)|^{1/H}), t>0 the following factorization is proved J^{n-1}_F(t)=\prod_{j=1}^{n} B^j_{\frac{H}{n}}(t), t>0. A series of compositions involving Cauchy processes and fractional Brownian motions are also studied and the corresponding non-homogeneous wave equations are derived.Comment: 32 page

Isospectral deformations of the Dirac operator

We give more details about an integrable system in which the Dirac operator D=d+d^* on a finite simple graph G or Riemannian manifold M is deformed using a Hamiltonian system D'=[B,h(D)] with B=d-d^* + i b. The deformed operator D(t) = d(t) + b(t) + d(t)^* defines a new exterior derivative d(t) and a new Dirac operator C(t) = d(t) + d(t)^* and Laplacian M(t) = d(t) d(t)^* + d(t)* d(t) and so a new distance on G or a new metric on M.Comment: 32 pages, 8 figure