Sliding Functor and Polarization Functor for Multigraded Modules

We define sliding functors, which are exact endofunctors of the category of multigraded modules over a polynomial ring. They preserve several invariants of modules, especially the (usual) depth and Stanley depth. In a similar way, we can also define the polarization functor. While this idea has appeared in papers of Bruns–Herzog and Sbarra, we give slightly different approach. Keeping these functors in mind, we treat simplicial spheres of Bier–Murai type.     

Structural Evolution Process in Solvent-Induced Crystallization Phenomenon of Syndiotactic Polystyrene

Structural change has been traced in the solvent-induced crystallization phenomenon of syndiotactic polystyrene through the time-resolved measurements of infrared and Raman spectra and X-ray diffraction. Immediately after the solvent is supplied to the glassy sample, the random coils start a micro-Brownian motion and locally change to short regular helical segments after some induction time. These segments grow longer and gather together to form the crystal lattice. This crystallization occurs even at room temperature far below the original glass transition temperature (Tg = ca. 100°C), because Tg is shifted to ca. −90°C (in the case of chloroform) due to the plasticizing effect, as revealed by the temperature-dependent infrared spectral measurement and the molecular dynamics calculation. The thus-created sPS-solvent complex was found to show a fast and reversible solvent exchange phenomenon between the originally-existing solvent (toluene, for example) and the newly-supplied different type of solvent (chloroform, for example). The time-dependent measurement of wide-angle and small-angle X-ray scatterings using a synchrotron radiation source revealed that the solvent exchange occurs with keeping both the columnar structure of the crystal and the stacked lamellar structure, and that the solvent exchange rate is in the order of chloroform > benzene > toluene, reflecting the difference in diffusion rate of solvent molecules and polymer-solvent interaction.     

Castelnuovo-Mumford regularity for complexes and weakly Koszul modules

Let A be a noetherian AS-regular Koszul quiver algebra (if A is commutative, it is essentially a polynomial ring), and the category of finitely generated graded left A-modules. Following Jørgensen, we define the Castelnuovo-Mumford regularity of a complex in terms of the local cohomologies or the minimal projective resolution of M^•. Let A^! be the quadratic dual ring of A. For the Koszul duality functor , we have . Using these concepts, we interpret results of Martinez-Villa and Zacharia concerning weakly Koszul modules (also called componentwise linear modules) over A^!. As an application, refining a result of Herzog and Römer, we show that if J is a monomial ideal of an exterior algebra E=?〈y₁,…,y_d〉, d≥3, then the (d−2)nd syzygy of E/J is weakly Koszul.     

Bundling mRNA Strands to Prepare Nano‐Assemblies with Enhanced Stability Towards RNase for In Vivo Delivery

Ribonuclease (RNase)‐mediated degradation of messenger RNA (mRNA) poses a huge obstruction to in vivo mRNA delivery. Herein, we propose a novel strategy to protect mRNA by structuring mRNA to prevent RNase attack through steric hinderance. Bundling of mRNA strands through hybridization of RNA oligonucleotide linkers allowed the preparation of mRNA nano‐assemblies (R‐NAs) comprised of 7.7 mRNA strands on average, mostly below 100 nm in diameter. R‐NA formation boosted RNase stability by around 100‐fold compared to naïve mRNA and preserved translational activity, allowing protein production. A mechanistic analysis suggests that an endogenous mRNA unwinding mechanism triggered by 5′‐cap‐dependent translation may induce selective R‐NA dissociation intracellularly, leading to smooth translation. R‐NAs showed efficient mRNA transfection in mouse brain, demonstrating the feasibility for in vivo administration.     

Supramolecular Structure of N,N-Bis(2-hydroxy-benzyl)alkylamine: From Hydrogen Bond Assembly to Coordination Network in Guest Acceptance

N,N-Bis(2-hydroxybenzyl)alkylamine ( HBA ) was described as a model to simplify the supramolecular system of polybenzoxazines. The single crystal X-ray analysis revealed that HBA forms a dimeric assembly based on its simultaneous inter- and intramolecular hydrogen bond network. When the HBA accepted the copper ion, the dimeric assembly HBA changed its primary hydrogen bond network to charge transfer coordination as clarified by the single crystal analysis as well as the superimposed structure based on DMol³ calculation. The supramolecular structure of HBA and copper also exhibited the role of solvent molecules in the packing structure which is a rare example showing the host accepts not only the metal ions but also the neutral molecules in the same time. The cyclization of HBA with esters or ethers to macrocyclic compounds was also an important model reaction to show how the backbone molecules self-stabilized with the hydrogen bond network initiate the simple, effective and efficient macrocyclic reaction.