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Jury and Martin establish an analogue of the classical inner-outer factorization of Hardy space functions. They show that every function $f$in a Hilbert function space with a normalized complete Pick reproducing kernel has a factorization of the type $f=\mathrm{\phi <\#comment/>}g$, where $g$is cyclic, $\mathrm{\phi <\#comment/>}$is a contractive multiplier, and $\Vert <\#comment/>f\Vert <\#comment/>=\Vert <\#comment/>g\Vert <\#comment/>$. In this paper we show that if the cyclic factor is assumed to be what we call free outer, then the factors are essentially unique, and we give a characterization of the factors that is intrinsic to the space. That lets us compute examples. We also provide several applications of this factorization. 

Summary We develop a uniform test for detecting and dating the integrated or mildly explosive behaviour of a strictly stationary generalized autoregressive conditional heteroskedasticity (GARCH) process. Namely, we test the null hypothesis of a globally stable GARCH process with constant parameters against the alternative that there is an ‘abnormal’ period with changed parameter values. During this period, the parameter-value change may lead to an integrated or mildly explosive behaviour of the volatility process. It is assumed that both the magnitude and the timing of the breaks are unknown. We develop a double-supreme test for the existence of breaks, and then provide an algorithm to identify the periods of changes. Our theoretical results hold under mild moment assumptions on the innovations of the GARCH process. Technically, the existing properties for the quasi-maximum likelihood estimation in the GARCH model need to be reinvestigated to hold uniformly over all possible periods of change. The key results involve a uniform weak Bahadur representation for the estimated parameters, which leads to weak convergence of the test statistic to the supreme of a Gaussian process. Simulations in the Appendix show that the test has good size and power for reasonably long time series. We apply the test to the conventional early-warning indicators of both the financial market and a representative of the emerging Fintech market, i.e., the Bitcoin returns. 

Abstract The semiconductor tracker (SCT) is one of the tracking systems for charged particles in the ATLAS detector. It consists of 4088 silicon strip sensor modules.During Run 2 (2015–2018) the Large Hadron Collider delivered an integrated luminosity of 156 fb -1 to the ATLAS experiment at a centre-of-mass proton-proton collision energy of 13 TeV. The instantaneous luminosity and pile-up conditions were far in excess of those assumed in the original design of the SCT detector.Due to improvements to the data acquisition system, the SCT operated stably throughout Run 2.It was available for 99.9% of the integrated luminosity and achieved a data-quality efficiency of 99.85%.Detailed studies have been made of the leakage current in SCT modules and the evolution of the full depletion voltage, which are used to study the impact of radiation damage to the modules.