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In this paper, the differential transformation method (DTM) is applied to solve singular initial problems represented by certain classes of Lane-Emden type equations. Some new differential transformation formulas for certain exponential and logarithmic nonlinearities are derived. The approximate and exact solutions of these equations are calculated in the form of series with easily computable terms. The results obtained with the proposed methods are in good agreement with those obtained by other methods. The advantages of this technique are shown as well.

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A network of Timepix (TPX) devices installed in the ATLAS cavern measures the Large Hadron Collider (LHC) luminosity as a stand-alone system. The data were recorded from 13-TeV proton-proton collisions in 2016. Using two TPX devices, the number of hits created by particles passing the pixel matrices was counted. Absolute luminosity is determined with the van der Meer scan technique by separating the LHC proton beams and measuring the widths of the beams in low-intensity LHC proton-proton collisions. The exact determination of the activation background contributes to the overall precision of the TPX luminosity measurements. The activation background varies in time due to induced radioactivity at the different positions of the TPX devices in the ATLAS cavern. The activation at a given time depends on the history of the LHC operation. A detailed study of induced radioactivity has been performed to reduce the uncertainty on both the relative and absolute luminosity measurements.

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