Limits of Boolean Functions on F_p^n

We study sequences of functions of the form F_p^n -> {0,1} for varying n, and define a notion of convergence based on the induced distributions from restricting the functions to a random affine subspace. Using a decomposition theorem and a recently proven equi-distribution theorem from higher order Fourier analysis, we prove that the limits of such convergent sequences can be represented by certain measurable functions. We are also able to show that every such limit object arises as the limit of some sequence of functions. These results are in the spirit of similar results which have been developed for limits of graph sequences. A more general, albeit substantially more sophisticated, limit object was recently constructed by Szegedy in [Sze10].Comment: 12 page

Δ+300\Delta+300 is a Bound on the Adjacent Vertex Distinguishing Edge Chromatic Number

An adjacent vertex distinguishing edge-coloring or an \avd-coloring of a simple graph $G$ is a proper edge-coloring of $G$ such that no pair of adjacent vertices meets the same set of colors. We prove that every graph with maximum degree $\Delta$ and with no isolated edges has an \avd-coloring with at most $\Delta+300$ colors, provided that $\Delta >10^{20}$

Electron-electron interaction induced spin thermalization in quasi-low-dimensional spin valves

We study the spin thermalization, i.e., the inter-spin energy relaxation mediated by electron-electron scattering in small spin valves. When one or two of the dimensions of the spin valve spacer are smaller than the thermal coherence length, the direct spin energy exchange rate diverges and needs to be regularized by the sample dimensions. Here we consider two model systems: a long quasi-1D wire and a thin quasi-2D sheet.Comment: Contribution to a Special Issue on Spin Caloritronic