Invariant discrete flows

In this paper, we investigate the evolution of joint invariants under invariant geometric flows using the theory of equivariant moving frames and the induced invariant discrete variational complex. For certain arc length preserving planar curve flows invariant under the special Euclidean group , the special linear group , and the semidirect group , we find that the induced evolution of the discrete curvature satisfies the differential‐difference mKdV, KdV, and Burgers' equations, respectively. These three equations are completely integrable, and we show that a recursion operator can be constructed by precomposing the characteristic operator of the curvature by a certain invariant difference operator. Finally, we derive the constraint for the integrability of the discrete curvature evolution to lift to the evolution of the discrete curve itself.     

Design and Synthesis of Aromatic Polyesters Bearing Pendant Clickable Maleimide Groups

A bisphenol bearing pendant maleimide group, namely, N‐maleimidoethyl‐3, 3‐bis(4‐hydroxyphenyl)‐1‐isobenzopyrrolidone (PPH‐MA) was synthesized starting from phenolphthalein. Aromatic (co)polyesters bearing pendant maleimide groups were synthesized from PPH‐MA and aromatic diacid chlorides, namely, isophthaloyl chloride (IPC), terephthaloyl chloride (TPC), and 50:50 mol % mixture of IPC and TPC by low temperature solution polycondensation technique. Copolyesters were also synthesized by polycondensation of different molar proportions of PPH‐MA and bisphenol A with IPC. Inherent viscosities and number‐average molecular weights of aromatic (co)polyesters were in the range of 0.52–0.97 dL/g and 20,200–32,800 g/mol, respectively indicating formation of medium to reasonably high‐molecular‐weight polymers. ¹³C NMR spectral analysis of copolyesters revealed the formation of random copolymers. The 10% weight loss temperature of (co)polyesters was found in the range 470–484 °C, indicating their good thermal stability. A selected aromatic polyester bearing pendant maleimide groups was chemically modified via thiol‐maleimide Michael addition reaction with two representative thiol compounds, namely, 4‐chlorothiophenol and 1‐adamantanethiol to yield post‐modified polymers in a quantitative manner. Additionally, it was demonstrated that polyester containing pendant maleimide groups could be used to form insoluble crosslinked gel in the presence of a multifunctional thiol crosslinker. © 2018 Wiley Periodicals, Inc. J. Polym. Sci., Part A: Polym. Chem. 2019 , 57, 630–640     

Sparse Multicategory Generalized Distance Weighted Discrimination in Ultra-High Dimensions

Distance weighted discrimination (DWD) is an appealing classification method that is capable of overcoming data piling problems in high-dimensional settings. Especially when various sparsity structures are assumed in these settings, variable selection in multicategory classification poses great challenges. In this paper, we propose a multicategory generalized DWD (MgDWD) method that maintains intrinsic variable group structures during selection using a sparse group lasso penalty. Theoretically, we derive minimizer uniqueness for the penalized MgDWD loss function and consistency properties for the proposed classifier. We further develop an efficient algorithm based on the proximal operator to solve the optimization problem. The performance of MgDWD is evaluated using finite sample simulations and miRNA data from an HIV study.