On traces of operators associated with the actions of compact Lie groups

Let M be a compact smooth closed manifold, and G a discrete group. A G-operator on M is an operator of finite sum defined by D=sumlimits_{gin G} D_gPhi_g:H^s(M)to H^{s-m}(M), where the D_g are (pseudo)differential operators with orders m and gto Phi_g is a representation of the group G by operators which operate on functions on M. par The main aim of this paper is to study a new class of elliptic G-operators, associated with a representation of the group G by quantum canonical transformations Phi_g. par The following theorem is the main theorem of this paper: par Theorem 2. Let the G-operator 1+D:L^2(M)to L^2(M) be elliptic. Then 1+D is Fredholm

The Numerical Simulation of Shallow Water: Estimation of the Roughness Coefficient on the Flood Stage

The results of the numerical simulations of the dynamics of shallow waters for Volga-Akhtuba Floodplain are discussed. The mathematical model is based on the system of Saint-Venant equations. Numerical solution applies a combined Lagrangian-Eulerian (cSPH-TVD) algorithm. We have investigated the features of the spring flood in 2011 and found the inapplicability of the hydrodynamical model with the constant roughness coefficient n M . We have found a good agreement between the results of numerical simulations and the observational data at gauging stations which allows us to estimate n M in low water n M min = 0.02 and the maximum water level in the river Volga n M max = 0.06–0.07

The Biogenesis of the Golgi Ribbon: The Roles of Membrane Input from the ER and of GM130

The Golgi complex in mammalian cells forms a continuous ribbon of interconnected stacks of flat cisternae. We show here that this distinctive architecture reflects and requires the continuous input of membranes from the endoplasmic reticulum (ER), in the form of pleiomorphic ER-to-Golgi carriers (EGCs). An important step in the biogenesis of the Golgi ribbon is the complete incorporation of the EGCs into the stacks. This requires the Golgi-matrix protein GM130, which continuously cycles between the cis-Golgi compartments and the EGCs. On acquiring GM130, the EGCs undergo homotypic tethering and fusion, maturing into larger and more homogeneous membrane units that appear primed for incorporation into the Golgi stacks. In the absence of GM130, this process is impaired and the EGCs remain as distinct entities. This induces the accumulation of tubulovesicular membranes, the shortening of the cisternae, and the breakdown of the Golgi ribbon. Under these conditions, however, secretory cargo can still be delivered to the Golgi complex, although this occurs less efficiently, and apparently through transient and/or limited continuities between the EGCs and the Golgi cisternae