Many Sparse Cuts via Higher Eigenvalues

Cheeger's fundamental inequality states that any edge-weighted graph has a vertex subset $S$ such that its expansion (a.k.a. conductance) is bounded as follows: \phi(S) \defeq \frac{w(S,\bar{S})}{\min \set{w(S), w(\bar{S})}} \leq 2\sqrt{\lambda_2} where $w$ is the total edge weight of a subset or a cut and $\lambda_2$ is the second smallest eigenvalue of the normalized Laplacian of the graph. Here we prove the following natural generalization: for any integer $k \in [n]$, there exist $ck$ disjoint subsets $S_1, ..., S_{ck}$, such that $$\max_i \phi(S_i) \leq C \sqrt{\lambda_{k} \log k}$$ where $\lambda_i$ is the $i^{th}$ smallest eigenvalue of the normalized Laplacian and $c0$ are suitable absolute constants. Our proof is via a polynomial-time algorithm to find such subsets, consisting of a spectral projection and a randomized rounding. As a consequence, we get the same upper bound for the small set expansion problem, namely for any $k$, there is a subset $S$ whose weight is at most a \bigO(1/k) fraction of the total weight and $\phi(S) \le C \sqrt{\lambda_k \log k}$. Both results are the best possible up to constant factors. The underlying algorithmic problem, namely finding $k$ subsets such that the maximum expansion is minimized, besides extending sparse cuts to more than one subset, appears to be a natural clustering problem in its own right

W and Z Production in pp Collisions at 7TeV with ATLAS

Measurements of W and Z cross-sections in pp collisions at ECM = 7 TeV at the Large Hadron Collider are reported from the ATLAS experiment. From an observation of 118 leptonic W candidates, the inclusive W cross-section times leptonic branching fraction is measured as [9.3 \pm 0.9(stat) \pm 0.6(syst) \pm 1.0(lumi)] nb. The result for the Z boson is [0.83 \pm 0.07(stat) \pm 0.06(syst) \pm 0.09(lumi)] nb. These results agree with theoretical expectations from NNLO QCD.Comment: 3 pages, 6 figure

Infrared line parameters at low temperatures relevant to planetary atmospheres

Employing the techniques that were described in several publications for measuring infrared lineshifts, linewidths and line intensities with a tunable diode laser, these parameters were measures for lines in the important infrared bands of several molecules of interest to the planetary astronomer at low temperatures that are relevant to planetary atmospheres using He, Ne, Ar, H2, N2, O2, and air as the perturbers. In addition to obtaining the many original data on the temperature dependence of the intensities and linewidths, it was also the first measurement of the same for the collision-induced lineshift of an infrared line and it showed that it was markedly different from that of the corresponding collision-broadened linewidth