Certain Integrals Arising from Ramanujan's Notebooks

In his third notebook, Ramanujan claims that $$\int_0^\infty \frac{\cos(nx)}{x^2+1} \log x \,\mathrm{d} x + \frac{\pi}{2} \int_0^\infty \frac{\sin(nx)}{x^2+1} \mathrm{d} x = 0.$$ In a following cryptic line, which only became visible in a recent reproduction of Ramanujan's notebooks, Ramanujan indicates that a similar relation exists if $\log x$ were replaced by $\log^2x$ in the first integral and $\log x$ were inserted in the integrand of the second integral. One of the goals of the present paper is to prove this claim by contour integration. We further establish general theorems similarly relating large classes of infinite integrals and illustrate these by several examples

Accretion Product Formation from Self- and Cross-Reactions of RO2 Radicals in the Atmosphere

Hydrocarbons are emitted into the Earth's atmosphere in very large quantities by human and biogenic activities. Their atmospheric oxidation processes almost exclusively yield RO2 radicals as reactive intermediates whose atmospheric fate is not yet fully unraveled. Herein, we show that gas-phase reactions of two RO2 radicals produce accretion products composed of the carbon backbone of both reactants. The rates for accretion product formation are very high for RO2 radicals bearing functional groups, competing with those of the corresponding reactions with NO and HO2. This pathway, which has not yet been considered in the modelling of atmospheric processes, can be important, or even dominant, for the fate of RO2 radicals in all areas of the atmosphere. Moreover, the vapor pressure of the formed accretion products can be remarkably low, characterizing them as an effective source for the secondary organic aerosol.Peer reviewe

Accretion Product Formation from Ozonolysis and OH Radical Reaction of alpha-Pinene : Mechanistic Insight and the Influence of Isoprene and Ethylene

alpha-Pinene (C10H16) represents one of the most important biogenic emissions in the atmosphere. Its oxidation products can significantly contribute to the secondary organic aerosol (SOA) formation. Here, we report on the formation mechanism of C-19 and C-20 accretion products from alpha-pinene oxidation, which are believed to be efficient SOA precursors. Measurements have been performed in a free-jet flow system. Detection of RO2 radicals and accretion products was carried out by recent mass spectrometric techniques using different ionization schemes. Observed C-10-RO2 radicals from alpha-pinene ozonolysis were O,O-C10H15(O-2)(x)O-2 with x = 0, 1, 2, 3 and from the OH radical reaction HO-C10H16(O-2)(alpha)O-2 with alpha = 0, 1, 2. All detected C 20 accretion products can be explained via the accretion reaction RO2 + R'O-2 -> ROOR' + O-2 starting from the measured C-10-RO2 radicals. We speculate that C-19 accretion products are formed in an analogous way assuming CH2O elimination. Addition of isoprene (C5H8), producing C-5-RO2 radicals, leads to C-15 accretion products formed via cross-reactions with C-10-RO2 radicals. This process is competing with the formation of C-19/C-20 products from the pure alpha-pinene oxidation. A similar behavior has been observed for ethylene additives that form C-12 accretion products. In the atmosphere, a complex accretion product spectrum from self- and cross-reactions of available RO2 radicals can be expected. Modeling atmospheric conditions revealed that C-19/C-20 product formation is only reduced by a factor of 1.2 or 3.6 in isoprene-dominated environments assuming a 2- or 15-fold isoprene concentration over alpha-pinene, respectively, as present in different forested areas.Peer reviewe

First oxidation products from the reaction of hydroxyl radicals with isoprene for pristine environmental conditions

Isoprene, C5H8, inserts about half of the non-methane carbon flux of biogenic origin into the atmosphere. Its degradation is primarily initiated by the reaction with hydroxyl radicals. Here we show experimentally the formation of reactive intermediates and corresponding closedshell products from the reaction of hydroxyl radicals with isoprene for low nitric oxide and low hydroperoxy radical conditions. Detailed product analysis is achieved by mass spectrometric techniques. Quantum chemical calculations support the usefulness of applied ionization schemes. Observed peroxy radicals are the isomeric HO-C5H8O2 radicals and their isomerization products HO-C5H8(O-2)O-2, bearing most likely an additional hydroperoxy group, and in traces HO-C5H8(O-2)(2)O-2 with two hydroperoxy groups. Main closed-shell products from unimolecular peroxy radical reactions are hydroperoxy aldehydes, C5H8O3, and smaller yield products with the composition C5H8O4 and C4H8O5. Detected signals of C10H18O4, C10H18O6, and C5H10O2 stand for products arising from peroxy radical self- and cross-reactions.Peer reviewe

Cryogel-supported stem cell factory for customized sustained release of bispecific antibodies for cancer immunotherapy

Combining stem cells with biomaterial scaffolds provides a promising strategy for the development of drug delivery systems. Here we propose an innovative immunotherapeutic organoid by housing human mesenchymal stromal cells (MSCs), gene-modified for the secretion of an anti-CD33-anti-CD3 bispecific antibody (bsAb), in a small biocompatible star-shaped poly(ethylene glycol)-heparin cryogel scaffold as a transplantable and low invasive therapeutic machinery for the treatment of acute myeloid leukemia (AML). The macroporous biohybrid cryogel platform displays effectiveness in supporting proliferation and survival of bsAb-releasing-MSCs overtime in vitro and in vivo, avoiding cell loss and ensuring a constant release of sustained and detectable levels of bsAb capable of triggering T-cell-mediated anti-tumor responses and a rapid regression of CD33 + AML blasts. This therapeutic device results as a promising and safe alternative to the continuous administration of short-lived immunoagents and paves the way for effective bsAb-based therapeutic strategies for future tumor treatments

The lipoxygenase-dependent oxygenation of lipid body membranes is promoted by a patatin-type phospholipase in cucumber cotyledons

Oilseed germination is characterized by the mobilization of storage lipids as a carbon and energy source for embryonic growth. In addition to storage lipid degradation in germinating oilseeds via the direct action of a triacylglycerol lipase (TGL) on the storage lipids, a second degradation pathway that is dependent on a specific lipid body trilinoleate 13-lipoxygenase (13-LOX) has been proposed in several plant species. The activity of this specific 13-LOX leads first to the formation of ester lipid hydroperoxides. These hydroperoxy fatty acids are then preferentially cleaved off by a TGL and serve as a substrate for glyoxysomal β-oxidation. As a prerequisite for triacylglycerol (TAG) mobilization, a partial degradation of the phospholipid monolayer and/or membrane proteins of the oil body has been discussed. Evidence has now been found for both processes: partial degradation of the proteins caleosin and oleosin was observed and simultaneously a patatin-like protein together with transient phospholipase (PLase) activity could be detected at the oil body membranes during germination. Moreover, in vitro experiments with isolated oil bodies from mature seeds revealed that the formation of 13-LOX-derived lipid peroxides in lipid body membranes is increased after incubation with the purified recombinant patatin-like protein. These experiments suggest that in vivo the degradation of storage lipids in cucumber cotyledons is promoted by the activity of a specific oil body PLase, which leads to an increased decomposition of the oil body membrane by the 13-LOX and thereby TAGs may be better accessible to LOX and TGL

Molecular understanding of sulphuric acid-amine particle nucleation in the atmosphere

4 pages 359-363 in the print version, additional 7 pages online.Peer reviewe

ON A SECANT DIRICHLET SERIES AND EICHLER INTEGRALS OF EISENSTEIN SERIES

Abstract. We consider, for even s, the secant Dirichlet series ψs(τ) = ∑∞ sec(πnτ) n=1 ns, recently introduced and studied by Lalín, Rodrigue and Rogers. In particular, we show, as conjectured and partially proven by Lalín, Rodrigue and Rogers, that the values ψ2m ( √ r), with r> 0 rational, are rational multiples of π2m. We then put the properties of the secant Dirichlet series into context by showing that they are Eichler integrals of odd weight Eisenstein series of level 4. This leads us to consider Eichler integrals of general Eisenstein series and to determine their period polynomials. In the level 1 case, these polynomials were recently shown by Murty, Smyth and Wang to have most of their roots on the unit circle. We provide evidence that this phenomenon extends to the higher level case. This observation complements recent results by Conrey, Farmer and Imamoglu as well as El-Guindy and Raji on zeros of period polynomials of Hecke eigenforms in the level 1 case. Finally, we briefly revisit results of a similar type in the works of Ramanujan. 1

First oxidation products from the reaction of hydroxyl radicals with isoprene for pristine environmental conditions

Isoprene, C5H8, inserts about half of the non-methane carbon flux of biogenic origin into the atmosphere. Its degradation is primarily initiated by the reaction with hydroxyl radicals. Here we show experimentally the formation of reactive intermediates and corresponding closed-shell products from the reaction of hydroxyl radicals with isoprene for low nitric oxide and low hydroperoxy radical conditions. Detailed product analysis is achieved by mass spectrometric techniques. Quantum chemical calculations support the usefulness of applied ionization schemes. Observed peroxy radicals are the isomeric HO-C5H8O2 radicals and their isomerization products HO-C5H8(O2)O2, bearing most likely an additional hydroperoxy group, and in traces HO-C5H8(O2)2O2 with two hydroperoxy groups. Main closed-shell products from unimolecular peroxy radical reactions are hydroperoxy aldehydes, C5H8O3, and smaller yield products with the composition C5H8O4 and C4H8O5. Detected signals of C10H18O4, C10H18O6, and C5H10O2 stand for products arising from peroxy radical self- and cross-reactions. © 2019, The Author(s)