321 research outputs found
Eigenvalue distribution of bipartite large weighted random graphs. Resolvent approach
We study eigenvalue distribution of the adjacency matrix
of weighted random bipartite graphs . We assume that the
graphs have vertices, the ratio of parts is and
the average number of edges attached to one vertex is or
. To each edge of the graph we assign a weight
given by a random variable with the finite second moment.
We consider the resolvents of and
study the functions and in the limit
. We derive closed system of equations that uniquely determine the
limiting functions and . This system of equations
allow us to prove the existence of the limiting measure
. The weak convergence in probability of normalized
eigenvalue counting measures is proved.Comment: 12 pages. arXiv admin note: text overlap with arXiv:0911.5684 by
other author
Scattering through a straight quantum waveguide with combined boundary conditions
Scattering through a straight two-dimensional quantum waveguide Rx(0,d) with
Dirichlet boundary conditions on (-\infty,0)x{y=0} \cup (0,\infty)x{y=d} and
Neumann boundary condition on (-infty,0)x{y=d} \cup (0,\infty)x{y=0} is
considered using stationary scattering theory. The existence of a matching
conditions solution at x=0 is proved. The use of stationary scattering theory
is justified showing its relation to the wave packets motion. As an
illustration, the matching conditions are also solved numerically and the
transition probabilities are shown.Comment: 26 pages, 3 figure
Singular-hyperbolic attractors are chaotic
We prove that a singular-hyperbolic attractor of a 3-dimensional flow is
chaotic, in two strong different senses. Firstly, the flow is expansive: if two
points remain close for all times, possibly with time reparametrization, then
their orbits coincide. Secondly, there exists a physical (or
Sinai-Ruelle-Bowen) measure supported on the attractor whose ergodic basin
covers a full Lebesgue (volume) measure subset of the topological basin of
attraction. Moreover this measure has absolutely continuous conditional
measures along the center-unstable direction, is a -Gibbs state and an
equilibrium state for the logarithm of the Jacobian of the time one map of the
flow along the strong-unstable direction. This extends to the class of
singular-hyperbolic attractors the main elements of the ergodic theory of
uniformly hyperbolic (or Axiom A) attractors for flows.Comment: 55 pages, extra figures (now a total of 16), major rearrangement of
sections and corrected proofs, improved introductio
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