Partitions with fixed largest hook length

Motivated by a recent paper of Straub, we study the distribution of integer partitions according to the length of their largest hook, instead of the usual statistic, namely the size of the partitions. We refine Straub’s analogue of Euler’s Odd-Distinct partition theorem, derive a generalization in the spirit of Alder’s conjecture, as well as a curious analogue of the first Rogers–Ramanujan identity. Moreover, we obtain a partition theorem that is the counterpart of Euler’s pentagonal number theorem in this setting, and connect it with the Rogers–Fine identity. We conclude with some congruence properties.     

Rapid identification of trimethyl and triethyl amines using sulphonic acidic ionic liquids: A time-of-flight secondary ion mass spectrometry study of fragmentation reactions

The time-of-flight secondary ion mass spectrometry (TOF-SIMS) has emerged as a powerful tool for the unswerving detection of biomolecules, in particular, proteins and peptides. To date, there is very little information available on the direct determination of trimethyl/triethyl amines using TOF-SIMS. One major hurdle in this regard is an ultrahigh vacuum system, usually needed in TOF-SIMS, which hampers its usability to trimethyl/triethyl amines owing to their high evaporation rate. We designed an efficient and sensitive protocol for rapid identification and sensitive determination of tertiaryalkyl amines using TOF-SIMS. The amines were derivatized by reaction with 1,4-butane sultone and sulphuric acid sequentially to afford the corresponding sulphonic acidic ionic liquids (ILs). The TOF-SIMS analysis of these task-specific ILs (TSILs) was carried out in both positive and negative polarity. The positive ion mass spectra of TSILs showed sharp fragmented peaks for tertiaryalkyl amines at typical level and up to 10 ppm. The possible mechanism for different fragmentation pathways in positive polarity was discussed.     

Gold nanocluster-based fluorescent assay for label-free detection of protein kinase and its inhibitors

Microchimica Acta - A sensitive fluorometric assay is reported for label-free detection of the activity of protein kinase (PKA) and of its inhibitors. It is based on the europium(III)-modulated...     

Energy dependence on modes of electric activities of neuron driven by multi-channel signals

Nonlinear Dynamics - Neuron can receive electric signals or forcing currents from more than one channel, and these forcing currents could show some diversity. Based on the Hindmarsh–Rose...     

Physically cross‐linked networks of POSS‐capped poly(acrylate amide)s: Synthesis,morphologies, and shape memory behavior

In this work, we synthesized a novel organic–inorganic semitelechelic polymer from polyhedral oligomeric silsesquioxane (POSS) and poly(acrylate amide) (PAA) via reversible addition‐fragmentation chain transfer (RAFT) polymerization. The organic–inorganic semitelechelic polymers have been characterized by means of nuclear magnetic resonance spectroscopy, thermal gravimetric analysis, and dynamic mechanical thermal analysis. It was found that capping POSS groups to the single ends of PAA chains caused a series of significant changes in the morphologies and thermomechanical properties of the polymer. The organic–inorganic semitelechelics were microphase‐separated; the POSS microdomains were formed via the POSS–POSS interactions. In a selective solvent (e.g., methanol), the organic–inorganic semitelechelics can be self‐assembled into the micelle‐like nanoobjects. Compared to plain PAA, the POSS‐capped PAAs significantly displayed improved surface hydrophobicity as evidenced by the measurements of static contact angles and surface atomic force microscopy. More importantly, the organic–inorganic semitelechelics displayed typical shape memory properties, which was in marked contrast to plain PAA. The shape memory behavior is attributable to the formation of the physically cross‐linked networks from the combination of the POSS–POSS interactions with the intermolecular hydrogen‐bonding interactions in the organic–inorganic semitelechelics. © 2017 Wiley Periodicals, Inc. J. Polym. Sci., Part B: Polym. Phys. 2017 , 55, 587–600