Stability of Gorenstein objects in triangulated categories

Let $\mathcal{C}$ be a triangulated category with a proper class $\xi$ of triangles. Asadollahi and Salarian introduced and studied $\xi$-Gorenstein projective and $\xi$-Gorenstein injective objects, and developed Gorenstein homological algebra in $\mathcal{C}$. In this paper, we further study Gorenstein homological properties for a triangulated category. First, we discuss the stability of $\xi$-Gorenstein projective objects, and show that the subcategory $\mathcal{GP}(\xi)$ of all $\xi$-Gorenstein projective objects has a strong stability. That is, an iteration of the procedure used to define the $\xi$-Gorenstein projective objects yields exactly the $\xi$-Gorenstein projective objects. Second, we give some equivalent characterizations for $\xi$-Gorenstein projective dimension of object in $\mathcal{C}$.Comment: 15page

On sign-changing solutions for nonlinear operator equations

AbstractIn this paper, the existence of sign-changing solutions for nonlinear operator equations is discussed by using the topological degree and fixed point index theory. The main theorems are some new three-solution theorems which are different from the famous Amann's and Leggett-Williams' three-solution theorems as well as the results in [F. Li, G. Han, Generalization for Amann's and Leggett–Williams' three-solution theorems and applications, J. Math. Anal. Appl. 298 (2004) 638–654]. These three solutions are all nonzero. One of them is positive, another is negative, and the third one is a sign-changing solution. Furthermore, the theoretical results are successfully applied to both integral and differential equations

Ground state for Choquard equation with doubly critical growth nonlinearity

In this paper we consider nonlinear Choquard equation −∆u + V(x)u = (Iα ∗ F(u))f(u) in R N, where V ∈ C(RN), Iα denotes the Riesz potential, f(t) = |t| p−2 t + |t| q−2 t for all t ∈ R, N > 5 and α ∈ (0, N − 4). Under suitable conditions on V, we obtain that the Choquard equation with doubly critical growth nonlinearity, i.e., p = (N + α)/N, q = (N + α)/(N − 2), has a nonnegative ground state solution by variational methods

Ground-state solutions to a class of modified Kirchhoff-type transmissiom problems with critical perturbation

This paper discusses a class of modified Kirchhoff-type transmissiom problems with critical perturbation. We establish an existence result of the ground-state solutions by using perturbation methods. Meanwhile, the limit properties of solution sequence are investigated

Ground state for Choquard equation with doubly critical growth nonlinearity

In this paper we consider nonlinear Choquard equation \begin{equation*} -\Delta u+V(x)u=(I_\alpha*F(u))f(u)\quad {\rm in}\ \mathbb{R}^{N}, \end{equation*} where $V\in C(\mathbb{R}^N)$, $I_\alpha$ denotes the Riesz potential, $f(t)=|t|^{p-2}t+|t|^{q-2}t$ for all $t\in\mathbb{R}$, $N\geqslant5$ and $\alpha\in(0,N-4)$. Under suitable conditions on $V$, we obtain that the Choquard equation with doubly critical growth nonlinearity, i.e., $p=(N+\alpha)/N,q=(N+\alpha)/(N-2)$, has a nonnegative ground state solution by variational methods

Ground-state solutions to a class of modified Kirchhoff-type transmissiom problems with critical perturbation

This paper discusses a class of modified Kirchhoff-type transmission problems with critical perturbation. We establish an existence result of the ground-state solutions by using perturbation methods. Meanwhile, the limit properties of solution sequence are investigated

Variation characteristics of anchor’s dynamic testing signal on the conditions of tensile load

Non-destructive testing for rock bolts in this study considers loads typical of anchors in practical engineering. The non-destructive testing experiment has been conducted for bolts under various load levels with variation characteristics of the dynamic testing signal and analyzed based on the stress wave reflection method. This research indicates that reflection signals of the fixed end section are relatively strong while reflection signals of the bottom section are relatively weak, regardless of bolt bearing loads, due to the effect of transmission, reflection and attenuation of the stress wave. The dynamic signal features obvious cycles in addition to the comparatively regular waveform in no-load cases as, with increasing load, dynamic signals become increasingly unstable while mechanical properties change in the anchor rod, anchor medium and the interface. The combination method of wavelet decomposition and multi-scale is applied to the test signal analysis to improve readability and accuracy of the signal. This research indicates that wavelet analysis interacts with non-stationary signals effectively, creating a solution for the reducing signal-noise ratio caused by the load. Obviously, it can also additionally read out the reflected signal of the bottom section, thereby improving the accuracy of anchoring quality interpretation

Influence of area-to-volume ratios on dissolution characteristics and mechanical properties of acid-corroded sandstone

To study the effect of area-to-volume ratio on the dissolution and deterioration characteristics of sandstone in the static acid-rock reaction system, the HCl and H2SO4 solutions with pH=2 and 5 are selected as corrosion environments, and the different area-to-volume ratios are set by changing surface areas of sandstone. The effects of area-to-volume ratios on the physicochemical and mechanical properties of sandstone are studied. According to the acid-rock reaction theory, the effect of the area-to-volume ratio on the diffusion-dissolution mechanism during sandstone corrosion is analyzed. The results show that the sandstone mass loss rate and amount of substance of total cations are all related to the corrosion time as a power function. The area-to-volume is positively correlated with the dissolution rate constant and has little effect on the reaction order. The reaction order is less than one in different environments, indicating that the sandstone corrosion rate decreases gradually with soaking time. In the pH=2、5 HCl solution and pH=2 H2SO4 solution, the amount of substance of cation shows N(Ca2+) > N(Na+) > N(Mg2+) > N(K+), and in the pH=5 H2SO4 solution, it is N(Na+) > N(Ca2+) > N(Mg2+) ≈N(K+). The acid-rock reaction can be summarized as two mechanisms: diffusion control and chemical reaction control. The two control parameters are negatively correlated with the area-to-volume ratio and positively with the pH value of solutions. The parameter values in the H2SO4 solutions are slightly larger than the corresponding values in the HCl solutions. The interaction between sandstone and acid in different conditions is dominated by the chemical reaction. The area-to-volume ratio significantly influences diffusion more than the chemical reaction. The mechanical properties of sandstone are weakened after acid corrosion. The damage of sandstone under uniaxial compression can be divided into four stages: compaction, elastic deformation, plastic yielding and post-peak. The peak strength and elastic modulus decrease, the peak strain increases, the brittleness declines, and the ductility is enhanced. The larger the area-to-volume ratio, the more severe the sandstone deterioration is. Overall, the smaller the pH value of solutions, the more prominent the effects of the area-to-volume ratio on the dissolution characteristics and mechanical properties of sandstone are, which is more obvious in the HCl solutions than in the H2SO4 solutions. The finding can provide theoretical references for the safety assessment and disaster prevention of rock mass engineering under an acidic environment

Single-Cell Transcriptome Analyses Reveal Signals to Activate Dormant Neural Stem Cells

SummaryThe scarcity of tissue-specific stem cells and the complexity of their surrounding environment have made molecular characterization of these cells particularly challenging. Through single-cell transcriptome and weighted gene co-expression network analysis (WGCNA), we uncovered molecular properties of CD133+/GFAP− ependymal (E) cells in the adult mouse forebrain neurogenic zone. Surprisingly, prominent hub genes of the gene network unique to ependymal CD133+/GFAP− quiescent cells were enriched for immune-responsive genes, as well as genes encoding receptors for angiogenic factors. Administration of vascular endothelial growth factor (VEGF) activated CD133+ ependymal neural stem cells (NSCs), lining not only the lateral but also the fourth ventricles and, together with basic fibroblast growth factor (bFGF), elicited subsequent neural lineage differentiation and migration. This study revealed the existence of dormant ependymal NSCs throughout the ventricular surface of the CNS, as well as signals abundant after injury for their activation

Positive Solutions of a Kind of Equations Related to the Laplacian and p-Laplacian

Positive solutions of a kind of equations related to the Laplacian and p-Laplacian on a bounded domain in RN with N⩾1 are studied by using variational method. A sufficient condition of the existence of positive solutions is characterized by the eigenvalues of linear and another nonlinear eigenvalue problems