On the 4-dissections of certain infinite products

In this note, we give a generalization of Hirschhorn’s formulas on the 4-dissections of Ramanujan’s continued fraction R(q) and R ^?1(q) which were conjectured by Hirschhorn and proved by Lewis and Liu.     

Analogues of Ramanujan’s partition identities

Ramanujan discovered that $$\sum_{n=0}^\infty p(5n+4)q^n=5 \prod_{j=1}^\infty \frac{(1-q^{5j})^5}{(1-q^j)^6}, $$ where p(n) is the number of partitions of n. Recently, H.-C. Chan and S. Cooper, and H.H. Chan and P.C. Toh established several analogues of Ramanujan’s partition identities by employing the theory of modular functions. Very recently, N.D. Baruah and K.K. Ojah studied the partition function $p_{[c^{l}d^{m}]}(n)$ which is defined by $$\sum_{n=0}^\infty p_{[c^ld^m]}(n)q^n= \frac{1}{\prod_{j=1}^\infty (1-q^{cj})^{l}(1-q^{dj})^m}. $$ They discovered some analogues of Ramanujan’s partition identities and deduced several interesting partition congruences. In this paper, we provide a uniform method to prove some of their results by utilizing an addition formula. In the process, we also establish some new analogues of Ramanujan’s partition identities and congruences for $p_{[c^{l}d^{m}]}(n)$ .     

Seasonal Minimum and Maximum Solar Ultraviolet Exposure Measurements of Classroom Teachers Residing in Tropical North Queensland,Australia

The risk of keratinocyte skin cancer, malignant melanoma and ultraviolet radiation (UVR)‐induced eye disease is disproportionately higher in Australia and New Zealand compared to equivalent northern hemisphere latitudes. While many teachers are aware of the importance of reinforcing sun safety messages to students, many may not be aware of the considerable personal exposure risk while performing outdoor duties in locations experiencing high to extreme ambient UVR year‐round. Personal erythemally effective exposure of classroom teachers in tropical Townsville (19.3°S) was measured to establish seasonal extremes in exposure behavior. Mean daily personal exposure was higher in winter (91.2 J m^‐2, 0.91 Standard Erythema Dose [SED]) than summer (63.3 J m^?2, 0.63 SED). The range of exposures represents personal exposures that approximate current national guidelines for Australian workers at the study latitude of approximately 1.2 SED (30 J m^?2 effective to the International Commission on Non‐Ionizing Radiation Protection). Similar proportions of teachers spent more than 1 h outdoors per day in winter (28.6%) and summer (23.6%) as part of their teaching duties with seasonal differences having little effect on the time of exposure. Personal exposures for teachers peaked during both seasons near school meal break times at 11:00 am and 1:00 pm, respectively.     

PEG–Anthracene Hydrogels as an On‐Demand Stiffening Matrix To Study Mechanobiology

There is a growing interest in materials that can dynamically change their properties in the presence of cells to study mechanobiology. Herein, we exploit the 365 nm light mediated [4+4] photodimerization of anthracene groups to develop cytocompatible PEG‐based hydrogels with tailorable initial moduli that can be further stiffened. A hydrogel formulation that can stiffen from 10 to 50 kPa, corresponding to the stiffness of a healthy and fibrotic heart, respectively, was prepared. This system was used to monitor the stiffness‐dependent localization of NFAT, a downstream target of intracellular calcium signaling using a reporter in live cardiac fibroblasts (CFbs). NFAT translocates to the nucleus of CFbs on stiffening hydrogels within 6 h, whereas it remains cytoplasmic when the CFbs are cultured on either 10 or 50 kPa static hydrogels. This finding demonstrates how dynamic changes in the mechanical properties of a material can reveal the kinetics of mechanoresponsive cell signaling pathways that may otherwise be missed in cells cultured on static substrates.     

Selective hetero-trisfunctionalization of the large rim of a biomimetic calix[6]arene using host-guest chemistry as a synthetic tool

A calix[6]arene has been selectively functionalized by three different groups at the large rim. The strategy relies on the hostguest recognition chemistry of a biomimetic metal complex at the small rim (so-called "funnel complex") and the Huisgen cycloaddition. The intramolecular thermal reaction proceeds with a high efficiency, chemio- and regioselectivity, allowing the monofunctionalization of one aromatic unit among three. Thanks to the high yield and selectivity of the reaction, it can be applied successively twice on the same compound, which opens the route for inherently chiral calix[6]arenes. This methodology not only is of wide potential for obtaining and exploiting calix[6]arenes that are dissymmetrized at the large rim, but also stands as an exemplary strategy for the selective appending of a functional group on a host platform.     

Synthesis and Application of Low Molecular Weight PEI-Based Copolymers for siRNA Delivery with Smart Polymer Blends

Polyethylenimine (PEI) is a commonly used cationic polymer for small-interfering RNA (siRNA) delivery due to its high transfection efficiency at low commercial cost. However, high molecular weight PEI is cytotoxic and thus, its practical application is limited. In this study, different formulations of low molecular weight PEI (LMW-PEI) based copolymers polyethylenimine-g-polycaprolactone (PEI–PCL) (800 Da–40 kDa) and PEI–PCL–PEI (5–5–5 kDa) blended with or without polyethylene glycol-b-polycaprolactone (PEG–PCL) (5 kDa-4 kDa) are investigated to prepare nanoparticles via nanoprecipitation using a solvent displacement method with sizes ≈100 nm. PEG–PCL can stabilize the nanoparticles, improve their biocompatibility, and extend their circulation time in vivo. The nanoparticles composed of PEI–PCL–PEI and PEG–PCL show higher siRNA encapsulation efficiency than PEI–PCL/PEG–PCL based nanoparticles at low N/P ratios, higher cellular uptake, and a gene silencing efficiency of ≈40% as a result of the higher molecular weight PEI blocks. These results suggest that the PEI–PCL–PEI/PEG–PCL nanoparticle system could be a promising vehicle for siRNA delivery at minimal synthetic effort.