We prove that for a large class of functions P and Q, the discrete bilinear operator T P, Q (f, g)(n)= m Z{0} f (n P (m)) g (n Q (m)) 1 m is bounded from l 2 l 2 into l 1+ , for any (0, 1].

We prove that, under certain conditions on the function pair arphi_1 and arphi_1 , bilinear average arphi_1 along curve arphi_1 satisfies certain decay estimate. As a consequence, Roth type theorems hold in the setting of finite fields. In particular, if arphi_1 with arphi_1 are linearly independent polynomials, then for any arphi_1 with arphi_1 , there are arphi_1 triplets arphi_1 . This extends a recent result of Bourgain and Chang who initiated this type of problems, and strengthens the bound in a result of Peluse, who generalized Bourgain and Chang's work. The proof uses discrete Fourier analysis and algebraic geometry.

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We review several recent results concerning two-dimensional systems which exhibit a transport induced by a one-dimensional perturbation of a homogeneous magnetic field. The first concerns the local Iwatsuka model, where a charged particle interacts with a field which is homogeneous outside a finite strip and translationally invariant along it: we present two new sufficient conditions for absolute continuity of the spectrum and show that in most cases the number of open spectral gaps is finite. In the second model the perturbation is a periodic array of point obstacles. In this case the Landau levels remain to be infinitely degenerate eigenvalues, and between them the system has bands of absolutely continuous spectrum.

Graph consists of a set of vertices V={Xj: j I}, a set of finite edges L={Ljn:(Xj, Xn) IL I I} and a set of infinite edges L={Lj: Xj IC} attached to them. We regard it as a configuration space of a quantum system with the Hilbert space