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We study BPS states of 5d N=1 SU(2) Yang-Mills theory on S^1×R^4. Geometric engineering relates these to enumerative invariants for the local Hirzebruch surface F_0. We illustrate computations of Vafa-Witten invariants via exponential networks, verifying fiber-base symmetry of the spectrum at certain points in moduli space, and matching with mirror descriptions based on quivers and exceptional collections. Albeit infinite, parts of the spectrum organize in families described by simple algebraic equations. Varying the radius of the M-theory circle interpolates smoothly with the spectrum of 4d N=2 Seiberg-Witten theory, recovering spectral networks in the limit.

We present an automatic reconstruction pipeline for large scale urban scenes from aerial images captured by a camera mounted on an unmanned aerial vehicle. Using state-of-the-art Structure from Motion and Multi-View Stereo algorithms, we first generate a dense point cloud from the aerial images. Based on the statistical analysis of the footprint grid of the buildings, the point cloud is classified into different categories (i.e., buildings, ground, trees, and others). Roof structures are extracted for each individual building using Markov random field optimization. Then, a contour refinement algorithm based on pivot point detection is utilized to refine the contour of patches. Finally, polygonal mesh models are extracted from the refined contours. Experiments on various scenes as well as comparisons with state-of-the-art reconstruction methods demonstrate the effectiveness and robustness of the proposed method.

We derive an asymptotic formula which counts the number of abelian extensions of prime degrees over rational function fields. Specifically, let l be a rational prime and K a rational function field F_q(t) with l\nmid q. Let Discf (F/K) denote the finite discriminant of F over K. Denote the number of abelian l-extensions F/K with deg (Discf (F/K)) = (l − 1)αn by a_l(n), where α = α(q, l) is the order of q in the multiplicative group (Z/lZ)^\times. We give a explicit asymptotic formula for a_l(n). In the case of cubic extensions with q ≡ 2 (mod 3), our formula gives an exact analogue of Cohn’s classical formula.