On the Auslander–Reiten conjecture for Cohen–Macaulay rings and path algebras

In this note, it is shown that the validity of the Auslander–Reiten conjecture for a given d-dimensional Cohen–Macaulay local ring R depends on its validity for all direct summands of d-th syzygy of R-modules of finite length, provided R is an isolated singularity. Based on this result, it is shown that under a mild assumption on the base ring R, satisfying the Auslander–Reiten conjecture behaves well under completion and reduction modulo regular elements. In addition, it will turn out that, if R is a commutative Noetherian ring and 𝒬 a finite acyclic quiver, then the Auslander–Reiten conjecture holds true for the path algebra R𝒬, whenever so does R. Using this result, examples of algebras satisfying the Auslander–Reiten conjecture are presented.     

The Gauss-Newton Methods via Conjugate Gradient Path without Line Search Technique for Solving Nonlinear Systems

In this article, we propose the Gauss-Newton methods via conjugate gradient path for solving nonlinear systems. By constructing and solving a linearized model of the nonlinear systems, we obtain the iterative direction by employing the conjugate gradient path. In successive iterations, the approximate Jacobian of the nonlinear systems is updated by a Broyden formula to construct the conjugate path. The global convergence and local superlinear convergence rate of the proposed algorithms are established under some reasonable conditions. Finally, the numerical results are reported to show the effectiveness of the proposed algorithms.     

Koszulity of some path categories

We prove Koszulity of certain linear path categories obtained from connected graphs with some infinite directed walk. These categories can be viewed as locally quadratic dual to preprojective algebras.     

Leaf‐Critical and Leaf‐Stable Graphs

The minimum leaf number ml(G) of a connected graph G is defined as the minimum number of leaves of the spanning trees of G if G is not hamiltonian and 1 if G is hamiltonian. We study nonhamiltonian graphs with the property for each or for each . These graphs will be called ‐leaf‐critical and l‐leaf‐stable, respectively. It is far from obvious whether such graphs exist; for example, the existence of 3‐leaf‐critical graphs (that turn out to be the so‐called hypotraceable graphs) was an open problem until 1975. We show that l‐leaf‐stable and l‐leaf‐critical graphs exist for every integer , moreover for n sufficiently large, planar l‐leaf‐stable and l‐leaf‐critical graphs exist on n vertices. We also characterize 2‐fragments of leaf‐critical graphs generalizing a lemma of Thomassen. As an application of some of the leaf‐critical graphs constructed, we settle an open problem of Gargano et al. concerning spanning spiders. We also explore connections with a family of graphs introduced by Grünbaum in correspondence with the problem of finding graphs without concurrent longest paths.     

Rough path properties for local time of symmetric α stable process

In this paper, we first prove that the local time associated with symmetric $\alpha$-stable processes is of bounded $p$-variation for any $p>\frac{2}{\alpha ?1}$ partly based on Barlow’s estimation of the modulus of the local time of such processes.  The fact that the local time is of bounded $p$-variation for any $p>\frac{2}{\alpha ?1}$ enables us to define the integral of the local time ${\int }_{?\infty }^{\infty }{?}_{?}^{\alpha ?1}f\left(x\right)d_x{L}_t^x$ as a Young integral for less smooth functions being of bounded $q$-variation with $1\le q<\frac{2}{3?\alpha }$. When $q\ge \frac{2}{3?\alpha }$, Young’s integration theory is no longer applicable. However, rough path theory is useful in this case. The main purpose of this paper is to establish a rough path theory for the integration with respect to the local times of symmetric $\alpha$-stable processes for $\frac{2}{3?\alpha }\le q<4$.     

A Complete Solution to the Existence of ‐Cycle Frames of Type

The existence problem of a ‐cycle frame of type is now solved for any quadruple .     

Effect of Direct yellow 50 removal from an aqueous solution using nano bentonite; adsorption isotherm,kinetic analysis and also thermodynamic behavior

The developing countries are suffering from the toxicity of different industrial effluents, especially dyes that contaminate water systems. This study successfully explained the preparation and characterization of nano bentonite to extract Direct Yellow Fifty (DY50). Direct Yellow 50 is an organic contaminant that may affect the quality of water. The characterization of prepared nanoparticles was done using Scanning electron microscopy (SEM), X-ray diffraction (XRD), and Fourier-transform infrared (FTIR). The impact of different operating conditions was studied using different pH, dose, temperature, contact time, and initial DY50 concentrations. The obtained results indicated that nano bentonite could adsorb about 94 % at initial concentrations of 40 mg/L, respectively. The optimum removal conditions were observed at an acidic pH (pH 3) using a sorbent material dosage of 0.05 g for 4 h at 30 °C. The adsorption isotherm, kinetic analysis, and thermodynamic behavior were studied using linear equation form, and the adjusted R² was compared to detect the preferred models. The adsorption behavior pseudo-second order kinetics, and fitted Langmuir isotherm model, respectively, showed the chemisorption interactions between adsorbed and sorbed molecules. Thermodynamic behavior indicated that the reaction was exothermic. Finally, this study strongly recommended using nano bentonite for DY50 removal from an aqueous solution.     

催化裂解-金汞齐富集-冷原子吸收光谱仪测定土壤样品中总汞含量

为建立采用催化裂解-金汞齐富集-冷原子吸收光谱仪即直接测汞仪测定土壤样品中汞含量的方法,本研究配制汞总量为0~2 ng、0~15 ng和25~1023 ng的三种不同汞浓度系列的标准工作曲线,选取9个土壤样品,3种国家土壤有证标准物质,同一样品分别进行6组平行测定,并抽取3个土壤样品进行3种不同浓度加标回收试验,以对其方法精密度和准确度进行论证。 结果显示：仪器信号值与Hg总量之间均呈良好的线性关系。根据仪器多次测定空白数据结果,按照称样量0.1 g计算,方法最小检出量为0.09 ng/g;平行测定结果相对标准偏差均小于10%,土壤标准物质测定值与标准物质标准参考值均相符,不同浓度的加标回收率范围为78.4%～92.7%。结果表明催化裂解-金汞齐富集-冷原子吸收光谱仪,可用于批量土壤样品中汞含量的快速测定分析,方法的精密度和准确度可满足测定分析要求,且实验过程中无需前处理消煮,操作方便、快速高效。     

Local inverse inelastic mean free path at interface

We calculated a local inverse inelastic mean free path (local-IIMFP) for electrons crossing a medium–medium interface, considering various incident electron energies, crossing angles and combinations of materials. We used an extension of a classical dielectric model developed by Li and co-workers for an electron crossing a surface (interface vacuum-medium). Moreover, the integration over the distance of the local-IIMFP allows to obtain the interface excitation parameter (or IEP) characterizing the change in excitation probability for an electron crossing an interface once caused by the presence of the interface in comparison with an electron for which only volume excitations are considered. We perform these calculations for angles between 0° and 80°, for electron energies between 500 and 2500 eV and for various pairs of materials, as Al/In for its academic interest or Au/Si and SiO₂/Si for their technological importance. Small but not negligible variations of the local-IIMFP and the IEP were observed for metal–metal or metal–semiconductor interfaces, while quite significant variations are obtained when one of the materials is a insulator.