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We report on the first experimental characterization of a laser-wakefield accelerator able to deliver, in a single pulse, doses in excess of \unit[1]{Gy} on timescales of the order of a hundred femtoseconds, reaching unprecedented average dose-rates up to \unit[10$^{13}$]{Gy/s}. The irradiator is demonstrated to deliver doses tuneable up to \unit[2.2]{Gy} in a cm$^2$ area and with a high degree of longitudinal and transverse uniformity in a single irradiation. In this regime, proof-of-principle irradiation of patient-derived glioblastoma stem-like cells and human skin fibroblast cells show indications of a differential cellular response, when compared to reference irradiations at conventional dose-rates. These include a statistically significant increase in relative biological effectiveness ($1.40\pm0.08$ at 50\% survival for both cell lines) and a significant reduction of the relative radioresistance of tumour cells. Data analysis provides preliminary indications that these effects might not be fully explained by induced oxygen depletion in the cells but may be instead linked to a higher complexity of the damages triggered by the ultra-high density of ionising tracks of femtosecond-scale radiation pulses. These results demonstrate an integrated platform for systematic radiobiological studies at unprecedented beam durations and dose-rates, a unique infrastructure for translational research in radiobiology at the femtosecond scale., Comment: 29 pages, 11 figures, Paper submission

We call a function $f: X\to Y$ $P$-preserving if, for every subspace $A \subset X$ with property $P$, its image $f(A)$ also has property $P$. Of course, all continuous maps are both compactness- and connectedness-preserving and the natural question about when the converse of this holds, i.e. under what conditions is such a map continuous, has a long history. Our main result is that any non-trivial product function, i.e. one having at least two non-constant factors, that has connected domain, $T_1$ range, and is connectedness-preserving must actually be continuous. The analogous statement badly fails if we replace in it the occurrences of "connected" by "compact". We also present, however, several interesting results and examples concerning maps that are compactness-preserving and/or continuum-preserving., Comment: 8 pages

We introduce the notion of compactifiable classes -- these are classes of metrizable compact spaces that can be up to homeomorphic copies ``disjointly combined'' into one metrizable compact space. This is witnessed by so-called compact composition of the class. Analogously, we consider Polishable classes and Polish compositions. The question of compactifiability or Polishability of a class is related to hyperspaces. Strongly compactifiable and strongly Polishable classes may be characterized by the existence of a corresponding family in the hyperspace of all metrizable compacta. We systematically study the introduced notions -- we give several characterizations, consider preservation under various constructions, and raise several questions., Comment: 32 pages, 1 figure, revised version

Given a Boolean algebra B and an embedding e:B -> P(N)/fin we consider the possibility of extending each or some automorphism of B to the whole P(N)/fin. Among other things, we show, assuming CH, that for a wide class of Boolean algebras there are embeddings for which no non-trivial automorphism can be extended.