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Given a function f on (F_2_^n, we study the following problem. What is the largest affine subspace U such that when restricted to U, all the non-trivial Fourier coefficients of f are very small? For the natural class of bounded Fourier degree d functions f : (F_2)^n → [−1, 1], we show that there exists an affine subspace of dimension at least ˜Ω (n^(1/d!) k^(−2)), wherein all of f ’s nontrivial Fourier coefficients become smaller than 2^(−k) . To complement this result, we show the existence of degree d functions with coefficients larger than 2^(−d log n) when restricted to any affine subspace of dimension larger than Ω(dn^(1/(d−1))). In addition, we give explicit examples of functions with analogous but weaker properties. Along the way, we provide multiple characterizations of the Fourier coefficients of functions restricted to subspaces of (F_2)^n that may be useful in other contexts. Finally, we highlight applications and connections of our results to parity kill number and affine dispersers. 

This work studies the problem of finding a large affine subspace over the field F_2 such that a bounded function's nontrivial Fourier coefficients become small. We show that for any function f from F_2^n to [-1,1] with Fourier degree d, there exists an affine subspace of dimension at least roughly n^{1/d!}k^{-2}), wherein all of f's nontrivial Fourier coefficients become smaller than 2^{-k}. To complement this result, we show the existence of degree d functions with coefficients larger than 2^{-d log n} on any subspace of dimension larger than Omega(dn^{1/(d-1)}). In addition, we give explicit examples of functions with analogous but weaker properties. Along the way, we provide multiple characterizations of the Fourier coefficients of functions restricted to subspaces of $$\F_2^n$$ that may be useful in other contexts. Finally, we highlight applications and connections of our results to parity kill number and affine dispersers/extractors. 

Atmospheric oxidation of monoterpenes (C₁₀H₁₆) contributes to ambient particle number and mass concentrations due, in part, to the resulting peroxy radicals undergoing auto-oxidation to low-volatility highly oxygenated molecules (HOMs). 

Abstract Aircraft observations have revealed ubiquitous new particle formation in the tropical upper troposphere over the Amazon^1,2and the Atlantic and Pacific oceans^3,4. Although the vapours involved remain unknown, recent satellite observations have revealed surprisingly high night-time isoprene mixing ratios of up to 1 part per billion by volume (ppbv) in the tropical upper troposphere⁵. Here, in experiments performed with the CERN CLOUD (Cosmics Leaving Outdoor Droplets) chamber, we report new particle formation initiated by the reaction of hydroxyl radicals with isoprene at upper-tropospheric temperatures of −30 °C and −50 °C. We find that isoprene-oxygenated organic molecules (IP-OOM) nucleate at concentrations found in the upper troposphere, without requiring any more vapours. Moreover, the nucleation rates are enhanced 100-fold by extremely low concentrations of sulfuric acid or iodine oxoacids above 10⁵ cm⁻³, reaching rates around 30 cm⁻³ s⁻¹at acid concentrations of 10⁶ cm⁻³. Our measurements show that nucleation involves sequential addition of IP-OOM, together with zero or one acid molecule in the embryonic molecular clusters. IP-OOM also drive rapid particle growth at 3–60 nm h⁻¹. We find that rapid nucleation and growth rates persist in the presence of NO_xat upper-tropospheric concentrations from lightning. Our laboratory measurements show that isoprene emitted by rainforests may drive rapid new particle formation in extensive regions of the tropical upper troposphere^1,2, resulting in tens of thousands of particles per cubic centimetre. 

Abstract Chlorine radicals are strong atmospheric oxidants known to play an important role in the depletion of surface ozone and the degradation of methane in the Arctic troposphere. Initial oxidation processes of chlorine produce chlorine oxides, and it has been speculated that the final oxidation steps lead to the formation of chloric (HClO 3 ) and perchloric (HClO 4 ) acids, although these two species have not been detected in the atmosphere. Here, we present atmospheric observations of gas-phase HClO 3 and HClO 4 . Significant levels of HClO 3 were observed during springtime at Greenland (Villum Research Station), Ny-Ålesund research station and over the central Arctic Ocean, on-board research vessel Polarstern during the Multidisciplinary drifting Observatory for the Study of the Arctic Climate (MOSAiC) campaign, with estimated concentrations up to 7 × 10 6 molecule cm −3 . The increase in HClO 3 , concomitantly with that in HClO 4 , was linked to the increase in bromine levels. These observations indicated that bromine chemistry enhances the formation of OClO, which is subsequently oxidized into HClO 3 and HClO 4 by hydroxyl radicals. HClO 3 and HClO 4 are not photoactive and therefore their loss through heterogeneous uptake on aerosol and snow surfaces can function as a previously missing atmospheric sink for reactive chlorine, thereby reducing the chlorine-driven oxidation capacity in the Arctic boundary layer. Our study reveals additional chlorine species in the atmosphere, providing further insights into atmospheric chlorine cycling in the polar environment. 

Understanding the evolutionary stability and possible context dependence of biological containment techniques is critical as engineered microbes are increasingly under consideration for applications beyond biomanufacturing. While synthetic auxotrophy previously prevented Escherichia coli from exhibiting detectable escape from batch cultures, its long-term effectiveness is unknown. Here, we report automated continuous evolution of a synthetic auxotroph while supplying a decreasing concentration of essential biphenylalanine (BipA). After 100 days of evolution, triplicate populations exhibit no observable escape and exhibit normal growth rates at 10-fold lower BipA concentration than the ancestral synthetic auxotroph. Allelic reconstruction reveals the contribution of three genes to increased fitness at low BipA concentrations. Based on its evolutionary stability, we introduce the progenitor strain directly to mammalian cell culture and observe containment of bacteria without detrimental effects on HEK293T cells. Overall, our findings reveal that synthetic auxotrophy is effective on time scales and in contexts that enable diverse applications.