Finite Morse index solutions and asymptotics of weighted nonlinear elliptic equations

By introducing a suitable setting, we study the behavior of finite Morse index solutions of the equation -\{div} (|x|^\theta \nabla v)=|x|^l |v|^{p-1}v \;\;\; \{in $\Omega \subset \R^N \; (N \geq 2)$}, \leqno(1) where $p>1$, $\theta, l\in\R^1$ with $N+\theta>2$, $l-\theta>-2$, and $\Omega$ is a bounded or unbounded domain. Through a suitable transformation of the form $v(x)=|x|^\sigma u(x)$, equation (1) can be rewritten as a nonlinear Schr\"odinger equation with Hardy potential -\Delta u=|x|^\alpha |u|^{p-1}u+\frac{\ell}{|x|^2} u \;\; \{in $\Omega \subset \R^N \;\; (N \geq 2)$}, \leqno{(2)} where $p>1$, $\alpha \in (-\infty, \infty)$ and $\ell \in (-\infty,(N-2)^2/4)$. We show that under our chosen setting for the finite Morse index theory of (1), the stability of a solution to (1) is unchanged under various natural transformations. This enables us to reveal two critical values of the exponent $p$ in (1) that divide the behavior of finite Morse index solutions of (1), which in turn yields two critical powers for (2) through the transformation. The latter appear difficult to obtain by working directly with (2)

The Stefan problem for the Fisher–KPP equation

AbstractWe study the Fisher–KPP equation with a free boundary governed by a one-phase Stefan condition. Such a problem arises in the modeling of the propagation of a new or invasive species, with the free boundary representing the propagation front. In one space dimension this problem was investigated in Du and Lin (2010) [11], and the radially symmetric case in higher space dimensions was studied in Du and Guo (2011) [10]. In both cases a spreading-vanishing dichotomy was established, namely the species either successfully spreads to all the new environment and stabilizes at a positive equilibrium state, or fails to establish and dies out in the long run; moreover, in the case of spreading, the asymptotic spreading speed was determined. In this paper, we consider the non-radially symmetric case. In such a situation, similar to the classical Stefan problem, smooth solutions need not exist even if the initial data are smooth. We thus introduce and study the “weak solution” for a class of free boundary problems that include the Fisher–KPP as a special case. We establish the existence and uniqueness of the weak solution, and through suitable comparison arguments, we extend some of the results obtained earlier in Du and Lin (2010) [11] and Du and Guo (2011) [10] to this general case. We also show that the classical Aronson–Weinberger result on the spreading speed obtained through the traveling wave solution approach is a limiting case of our free boundary problem here

The Stefan problem for the Fisher-KPP equation with unbounded initial range

We consider the nonlinear Stefan problem \left \{ \begin{array} {ll} -d \Delta u=a u-b u^2 \;\; & \mbox{for } x \in \Omega (t), \; t>0, \\ u=0 \mbox{ and } u_t=\mu|\nabla_x u |^2 \;\;&\mbox{for } x \in \partial\Omega (t), \; t>0, \\ u(0,x)=u_0 (x) \;\; & \mbox{for } x \in \Omega_0, \end{array}\right. where $\Omega(0)=\Omega_0$ is an unbounded smooth domain in $\mathbb R^N$, $u_0>0$ in $\Omega_0$ and $u_0$ vanishes on $\partial\Omega_0$. When $\Omega_0$ is bounded, the long-time behavior of this problem has been rather well-understood by \cite{DG1,DG2,DLZ, DMW}. Here we reveal some interesting different behavior for certain unbounded $\Omega_0$. We also give a unified approach for a weak solution theory to this kind of free boundary problems with bounded or unbounded $\Omega_0$

Hindered diffusion of polymers in porous materials/

Dynamic light scattering (DLS) and forced Rayleigh scattering (FRS) were used to study polymer diffusion in solution in two kinds of porous materials: porous glasses and suspensions and gels formed from fumed silica particles. The diffusants were: dendritic polyamidoamines, linear polystyrenes, and dye-labeled polystyrenes. Polymer diffusion in porous glasses was investigated, by using DLS, as a function of time scale (t), polymer hydrodynamic radius (R\sb{\rm H}), and pore radius (R\sb{\rm P}). As t increases, the apparent diffusion crosses over from single pore diffusion (in which steric obstruction is weak) to macroscopic diffusion (in which the tortuosity of the pore networks is fully effective). Computer simulated diffusion agreed qualitatively with the crossover observed by DLS. The dependence of hindered diffusion on the size ratio \lambda\sb{\rm H} = R\sb{\rm H}/R\sb{\rm P} was studied for dendritic polyamidoamines and linear polystyrenes in porous glasses. For \lambda\sb{\rm H} $\ll$ 1, when hydrodynamic interactions dominate, dendritic polymers diffuse more slowly than linear polymers of comparable \lambda\sb{\rm H}. The diffusion results of the dendritic polymer and of the linear flexible polymer agreed quantitatively with the hydrodynamic theories for a hard sphere in a cylindrical pore, and for a random-coil macromolecule in a cylindrical pore, respectively. At large \lambda\sb{\rm H}, irregularities in local pore size lead to conformational entropy changes as the macromolecule moves. The experimental data agree qualitatively with the entropy barrier theory. Diffusion of dye-labeled polystyrenes within gels and suspensions formed from fumed silica was studied using FRS. Untreated silica was found to adsorb the labeled polymer, leading to strong hindrance even at very low silica concentration. Thorough quenching of the silica surface by silanization prevented polymer adsorption. The dependence on silica volume fraction of the resulting weakly hindered diffusion in treated silica was found to be consistent with simple theories of steric obstruction

Electrochemical Sensor for o-Nitrophenol Based on β

An electrochemical sensor for the quantification of o-nitrophenol (o-NP) has been developed based on the β-cyclodextrin functionalized graphene nanosheets modified glassy carbon electrode (CD-GNs/GCE). The results indicated that CD-GNs showed good electrochemical behavior to the redox of o-NP which is attributed to the combination of the excellent properties of graphene and cyclodextrin. The peak currents possess a linear relationship with the concentration of o-NP in the range of 5–400 μM. The detection limit of o-NP reached to 0.3 μM on the basis of the signal-to-noise characteristics (S/N=3). The peak potentials for the reversible redox waves are not affected by other nitrophenol isomers (m, p-NP), illustrating good selectivity. Furthermore, the developed electrochemical sensor exhibited good stability and reproducibility for the detection of o-NP and could be used to determine o-NP in real water sample

MicroRNA-9-5p functions as a tumor suppressor in prostate cancer via targeting UTRN

Accumulating evidence indicates that miR-9-5p plays an important role in several diseases, especially tumor progression. In this study, we investigated the clinical significance and biological function of miR-9-5p in prostate cancer (PCa). Using quantitative real time PCR (qRT-PCR) analysis, we found miR-9-5p level was significantly down-regulated in PCa tissues and cell lines. The decreased miR-9-5p expression was associated with tumor size, preoperative PSA, Gleason score and lymph node metastasis. Kaplan-Meier survival analysis showed patients with low level of miR-9-5p had significantly decreased rates of overall survival (OS). Multivariate analyses showed that miR-9-5p was an independent predictor of PCa patients’ prognosis. Through CCK-8 and Transwell assays, miR-9-5p overexpression by miR-9-5p mimics transfection was demonstrated to suppress the proliferation, migration and invasion of PCa cells. Mechanistically, luciferase reporter assay, qRT-PCR and Western blot demonstrated that Utrophin (UTRN) is a direct target of miR-9-5p in PCa cells. The status of UTRN protein in PCa tissues was much higher than that in adjacent tissues by immunohistochemical staining and its mRNA levels were inversely correlated with miR-9-5p in PCa tissues. Importantly, UTRN knockdown by siUTRN imitated the suppressive effects of miR-9-5p on cell proliferation, migration and invasion in PCa. In summary, miR-9-5p might novel prognostic biomarker in and targeting UTRN by miR-9-5p could be potential therapeutic candidates for PCa

Graph-based Molecular Representation Learning

Molecular representation learning (MRL) is a key step to build the connection between machine learning and chemical science. In particular, it encodes molecules as numerical vectors preserving the molecular structures and features, on top of which the downstream tasks (e.g., property prediction) can be performed. Recently, MRL has achieved considerable progress, especially in methods based on deep molecular graph learning. In this survey, we systematically review these graph-based molecular representation techniques, especially the methods incorporating chemical domain knowledge. Specifically, we first introduce the features of 2D and 3D molecular graphs. Then we summarize and categorize MRL methods into three groups based on their input. Furthermore, we discuss some typical chemical applications supported by MRL. To facilitate studies in this fast-developing area, we also list the benchmarks and commonly used datasets in the paper. Finally, we share our thoughts on future research directions

Using restored two-dimensional X-ray images to reconstruct the three-dimensional magnetopause

Astronomical imaging technologies are basic tools for the exploration of the universe, providing basic data for the research of astronomy and space physics. The Soft X-ray Imager (SXI) carried by the Solar wind Magnetosphere Ionosphere Link Explorer (SMILE) aims to capture two-dimensional (2-D) images of the Earth’s magnetosheath by using soft X-ray imaging. However, the observed 2-D images are affected by many noise factors, destroying the contained information, which is not conducive to the subsequent reconstruction of the three-dimensional (3-D) structure of the magnetopause. The analysis of SXI-simulated observation images shows that such damage cannot be evaluated with traditional restoration models. This makes it difficult to establish the mapping relationship between SXI-simulated observation images and target images by using mathematical models. We propose an image restoration algorithm for SXI-simulated observation images that can recover large-scale structure information on the magnetosphere. The idea is to train a patch estimator by selecting noise–clean patch pairs with the same distribution through the Classification–Expectation Maximization algorithm to achieve the restoration estimation of the SXI-simulated observation image, whose mapping relationship with the target image is established by the patch estimator. The Classification–Expectation Maximization algorithm is used to select multiple patch clusters with the same distribution and then train different patch estimators so as to improve the accuracy of the estimator. Experimental results showed that our image restoration algorithm is superior to other classical image restoration algorithms in the SXI-simulated observation image restoration task, according to the peak signal-to-noise ratio and structural similarity. The restoration results of SXI-simulated observation images are used in the tangent fitting approach and the computed tomography approach toward magnetospheric reconstruction techniques, significantly improving the reconstruction results. Hence, the proposed technology may be feasible for processing SXI-simulated observation images