Nonlinear obstacle problems with double phase in the borderline case

In this paper we study a double phase problem with an irregular obstacle. The energy functional under consideration is characterized by the fact that both ellipticity and growth switch between a type of polynomial and a type of logarithm, which can be regarded as a borderline case of the double phase functional with $(p,q)$-growth. We obtain an optimal global Calderón–Zygmund type estimate for the obstacle problem with double phase in the borderline case.     

Change of grading,injective dimension and dualizing complexes

Let G,H be groups, φ:G→H a group morphism, and A a G-graded algebra. The morphism φ induces an H-grading on A, and on any G-graded A-module, which thus becomes an H-graded A-module. Given an injective G-graded A-module, we give bounds for its injective dimension when seen as H-graded A-module. Following ideas by Van den Bergh, we give an application of our results to the stability of dualizing complexes through change of grading.     

Electrochromic Detection of Latent Fingermarks on Metal Surface Using 1,1′-Dibenzyl-4,4′-bipyridinium Dichloride

The electrochromic detection of latent fingermarks on polished or unpolished, flat or curved metal surfaces is described using electrochromic material, 1,1’-dibenzyl-4,4’-bipyridinium dichloride. The surface area covered by fingermarks acts as an insulating mask, causing 1,1’-dibenzyl-4,4’-bipyridinium dichloride to change color and produce inversed images of the fingermark. By changing the applied potential, the optical properties of 1,1’-dibenzyl-4,4’-bipyridinium dichloride can be continuously and reversibly adjusted to optimize the visual contrast of fingermarks, so as to realize the detection of latent fingermarks on stainless steel surface. It is demonstrated that the fabricated electrochromic devices can detect the fingermarks on these types of surfaces within twenty seconds at −1.0∼−2.0 V. This work can qualify as a tangible improvement in fingermark detection of the natural fingermarks on the never-cleaned (more than 3 years) and curved surfaces of daily-used container, e. g. cup, and the handle of cleaning tool, mop.     

“四模块”嵌入下的混合式教学设计与实践*——以功能材料专业物理化学基础课程为例

为建设适应新时代要求,突出高阶性、创新性、挑战度的一流本科课程,实现价值引领、能力培养和知识传授的教育教学目标,结合功能材料专业的特点和民族高等院校学生的学情,构建了物理化学基础课程的“四模块、三阶段、两时空、双平台”的混合式教学新模式,重塑了“价值引领模块、基础理论模块、多学科融合模块和虚拟仿真实验模块”的“四模块”内容体系。通过“四模块”嵌入下的混合式教学实践研究,结合学生的课前、课中、课后等学习成绩,多维度评价学习效果,并藉由调查问卷来分析学生对混合式教学的评价,最终研究结果表明学生的个性化自主学习能力有了显著提高、学生运用物理化学理论解决复杂问题的综合能力得到了提升,学生经过刻苦学习获得了素质提高的成就感。     

A tensor product of the Steinberg module and a certain simple kG(pr)-module

Let G be a connected, semisimple, and simply connected algebraic group defined and split over the finite field of order p, and let G(q) be the corresponding finite Chevalley or twisted group, where q = p^r. Recently, Anwar determines the direct sum decomposition of the tensor product of the rth Steinberg module and a simple G-module with a (p,r)-minuscule highest weight λ. In this paper, we determine that of the tensor product regarded as a module for G(q) under some weak assumptions for λ.     

Hochster’s small MCM conjecture for three-dimensional weakly F-split rings

We prove Hochster’s small maximal Cohen–Macaulay conjecture for three-dimensional complete F-pure rings. We deduce this from a more general criterion, and show that only a weakening of the notion of F-purity is needed, to wit, being weakly F-split. We conjecture that any complete ring is weakly F-split.