Harnack Estimates for Nonlinear Heat Equations with Potentials in Geometric Flows

Let $M$ be a closed Riemannian manifold with a family of Riemannian metrics $g_{ij}(t)$ evolving by geometric flow $\partial_{t}g_{ij} = -2{S}_{ij}$, where $S_{ij}(t)$ is a family of smooth symmetric two-tensors on $M$. In this paper we derive differential Harnack estimates for positive solutions to the nonlinear heat equation with potential: \begin{eqnarray*} \frac{\partial f}{\partial t} = {\Delta}f + \gamma (t) f\log f +aSf, \end{eqnarray*} where $\gamma (t)$ is a continuous function on $t$, $a$ is a constant and $S=g^{ij}S_{ij}$ is the trace of $S_{ij}$. Our Harnack estimates include many known results as special cases, and moreover lead to new Harnack inequalities for a variety geometric flows

Harnack Estimates for Nonlinear Backward Heat Equations in Geometric Flows

Let $M$ be a closed Riemannian manifold with a family of Riemannian metrics $g_{ij}(t)$ evolving by a geometric flow $\partial_{t}g_{ij} = -2{S}_{ij}$, where $S_{ij}(t)$ is a family of smooth symmetric two-tensors. We derive several differential Harnack estimates for positive solutions to the nonlinear backward heat-type equation \begin{eqnarray*} \frac{\partial f}{\partial t} = -{\Delta}f + \gamma f\log f +aSf \end{eqnarray*} where $a$ and $\gamma$ are constants and $S=g^{ij}S_{ij}$ is the trace of $S_{ij}$. Our abstract formulation provides a unified framework for some known results proved by various authors, and moreover lead to new Harnack inequalities for a variety of geometric flows

Study on the design method of integration of roof and photovoltaic based on aesthetics, technology and energy-saving characteristic

The development and utilization of new energy has been concerned due to the traditional energy is increasingly scarce. In recent years, solar building has developed rapidly in the construction industry which is a major energy consuming component. As an organic part of the building, the combination of roof and solar energy has become the focus of attention because of its large size, less shielding and other characteristics. Based on the works of recent years’ Solar Decathlon, this paper analysed the design and implementation of the integration of solar building’s roof and photovoltaic. Meanwhile, taking an office building in Xinjiang, China as an example, the paper analysed the design points and energy-saving situation of the roof photovoltaic building and prospected the application prospect of integrated design method of building’s solar roof

Several Integral Estimates and Some Applications

In this paper, the authors first consider the bidirectional estimates of several typical integrals. As some applications of these integral estimates, the authors investigate the pointwise multipliers from the normal weight general function space $F(p,\mu,s)$ to the normal weight Bloch type space $\mathcal{B_{\nu}}(B_{n})$ on the unit ball $B_{n}$ of $\mathbb{C}^{n}$, where $\mu$ and $\nu$ are two normal functions on $[0,1)$. For the special normal function $\displaystyle{\mu(r)=(1-r^{2})^{\alpha}\log^{\beta}\frac{e}{1-r^{2}}}$ ($\alpha>0$, $-\infty<\beta<\infty$), the authors give the necessary and sufficient conditions of pointwise multipliers from $F(p,\mu,s)$ to $\mathcal{B_{\nu}}(B_{n})$ for all cases

A novel Poly(ε-caprolactone)-Pluronic-Poly(ε-caprolactone) grafted Polyethyleneimine(PCFC-g-PEI), Part 1, synthesis, cytotoxicity, and in vitro transfection study

<p>Abstract</p> <p>Background</p> <p>Polyethyleneimine (PEI), a cationic polymer, is one of the successful and widely used vectors for non-viral gene transfection <it>in vitro</it>. However, its <it>in vivo </it>application was greatly limited due to its high cytotoxicity and short duration of gene expression. To improve its biocompatibility and transfection efficiency, PEI has been modified with PEG, folic acid, and chloroquine in order to improve biocompatibility and enhance targeting.</p> <p>Results</p> <p>Poly(ε-caprolactone)-Pluronic-Poly(ε-caprolactone) (PCFC) was synthesized by ring-opening polymerization, and PCFC-<it>g</it>-PEI was obtained by Michael addition reaction with GMA-PCFC-GMA and polyethyleneimine (PEI, 25 kD). The prepared PCFC-<it>g</it>-PEI was characterized by ¹H-NMR, SEC-MALLS. Meanwhile, DNA condensation, DNase I protection, the particle size and zeta potential of PCFC-<it>g</it>-PEI/DNA complexes were also determined. According to the results of flow cytometry and MTT assay, the synthesized PCFC-<it>g</it>-PEI, with considerable transfection efficiency, had obviously lower cytotoxicity against 293 T and A549 cell lines compared with that of PEI 25 kD.</p> <p>Conclusion</p> <p>The cytotoxicity and <it>in vitro </it>transfection study indicated that PCFC-<it>g</it>-PEI copolymer prepared in this paper was a novel gene delivery system with lower cytotoxicity and considerable transfection efficiency compared with commercial PEI (25 kD).</p

How Machiavellianism, Psychopathy, and Narcissism Affect Sustainable Entrepreneurial Orientation: The Moderating Effect of Psychological Resilience

In order to solve increasingly serious environmental problems, sustainable entrepreneurship based on the concept of sustainable development has gradually attracted attention and discussion in the academic field. Moreover, it is of great significance to analyze the influence of personality on entrepreneurial orientation considering dark and bright personality traits. On the basis of existing research, we analyzed the relationships between the three elements of the dark triad (Machiavellianism, psychopathy, and narcissism), psychological resilience, and sustainable entrepreneurial orientation (SEO) through a questionnaire survey. Results involve data from 328 MBA students in Tianjin University of China. The results showed that Machiavellianism and psychopathy negatively affected SEO, and narcissism positively affected SEO; psychological resilience and SEO had a significant positive association; and psychological resilience had a moderating effect on the relationship between the three elements of the dark triad and SEO. Our research has reference value and its findings have important implications for entrepreneurship research and education