Small-Molecule Modulation of Soft-Matter Frank–Kasper Phases: A Method for Adding Function to Form

Incorporation of small amounts of α-tocopherol (vitamin E) in blends with the cellobiose–triazole-linked atactic poly(4-methyl-1-pentene) (CB-aPMP) sugar–polyolefin conjugate can be used to exert external control over thermotropic phase behavior and provide access to non-canonical soft matter Frank–Kasper A15 and σ phases. These results establish a paradigm that can be used for the further design and development of scalable quantities of soft matter FK phases of increased structural complexity and functional capability.     

Homogeneous variable exponent Besov and Triebel–Lizorkin spaces

We introduce homogeneous Besov and Triebel–Lizorkin spaces with variable indexes. We show that their study reduces to the study of inhomogeneous variable exponent spaces and homogeneous constant exponent spaces. Corollaries include trace space characterizations and Sobolev embeddings.     

Characterization of embeddings of Sobolev-type spaces into generalized Hölder spaces defined by Lp-modulus of smoothness

We prove a sharp estimate for the k-modulus of smoothness, modelled upon a $L^p$-Lebesgue space, of a function f in $W^k{L}^{\frac{pn}{n+kp},p}(\mathrm{\Omega })$, where Ω is a domain with minimally smooth boundary and finite Lebesgue measure, $k,n\in \mathbb{N}$, $k<n$ and $\frac{n}{n?k}<p<+\infty$. This sharp estimate is used to establish necessary and sufficient conditions for continuous embeddings of Sobolev-type spaces into generalized Hölder spaces defined by means of the k-modulus of smoothness. General results are illustrated with examples. In particular, we obtain a generalization of the classical Jawerth embeddings.     

Free boundary problem for a fully nonlinear and degenerate parabolic equation in an angular domain

This paper concerns with the problem of how to running an insurance company to maximize his total discounted expected dividends. In our model, the dividend rate is limited in $[0,M]$ and the company is allowed to transfer any proportion of risk by reinsuring. So there are two strategies which we call dividend strategy and reinsurance strategy. The objective function and the corresponding optimal two strategies are the solution and the two free boundaries of the following Barenblatt parabolic equation

$v_t?\underset{0\le a\le 1}{\max }?(\frac{1}{2}{a}^2{\sigma }^2{v}_{xx}+a\mu v_x)+cv?\underset{0\le l\le M}{\max }?[(1?v_x)l]=0$

under certain boundary conditions on an angular domain

$Q_T=\{(x,t)|0<x<Mt,0<t\le T\}.$

The main effort is to analyze the properties of the solution and the free boundaries to show the optimal decision for the insurance company.     

Perturbative manifolds and the Noether generators of nth-order Poisson equations

A manifold that contains small perturbations will induce a perturbed partial differential equation. The partial differential equation that we select is the Poisson equation – in order to explore the interplay between the geometry of the manifold and the perturbations. Specifically, we show how the problem of symmetry determination, for higher-order perturbations, can be elegantly expressed via geometric conditions.     

Synthesis of boronic acid‐functionalized poly(glycerol‐oligoγ‐butyrolactone): Nano‐networks for efficient electrochemical sensing of biosystems

Despite the outstanding properties of hyperbranched polyglycerols such as biocompatibility and multifunctionality, enough attention has not been paid to the synthesis of their functional copolymers. This problem has limited the structural diversity of hyperbranched polyglycerols and hampers further developments and their practical usage. In this work, butyrolactone segments were incorporated into the backbone of polyglycerols by one‐pot ring‐opening copolymerization of a mixture of glycidol and γ‐butyrolactone in the presence of tin(II) 2‐ethylhexanoate. Poly(glycerol‐oligoγ‐butyrolactone)s were then crosslinked by 2,5‐thiophenediylbisboronic acid to obtain polymeric nanonetworks with 140 nm average size. Afterwards, the gold electrode was modified by the polymeric nano‐networks, and it was used for the determination of glucose, glycated hemoglobin, and Escherichia coli in phosphate buffer solution (pH = 9.0) through cyclic voltammetry and impedance spectroscopic. Taking advantage of the straightforward synthesis, cheap precursors and multifunctionality of poly(glycerol‐oligoγ‐butyrolactone)s, they could be used for real‐time sensing of a wide range of biosystems. © 2019 Wiley Periodicals, Inc. J. Polym. Sci., Part A: Polym. Chem. 2019, 57, 1430–1439