An analog to Kamenkov’s critical stability case for nonstationary systems of impulsive differential equations

A new approach to the investigation of the stability of nonlinear nonautonomous differential equations with impulse effects in critical cases is proposed. The approach is based on the direct method of Lyapunov with the use of piecewise differentiable functions. The sufficient conditions of the asymptotic stability of the critical position of equilibrium in one case are obtained. The case is analogous to Kamenkov’s critical case.     

Laws of the lattice of all $${\mathfrak {X}}$$ -local formations of finite groups

Ricerche di Matematica - Let $${\mathfrak {X}}$$ be a class of simple groups with a completeness property $$\pi ({\mathfrak {X}}) = \mathrm {char} \, {\mathfrak {X}}$$ . A formation is a class of...     

On the stability of retrial queues

We consider the following Type of problems. Calls arrive at a queue of capacity K (which is called the primary queue), and attempt to get served by a single server. If upon arrival, the queue is full and the server is busy, the new arriving call moves into an infinite capacity orbit, from which it makes new attempts to reach the primary queue, until it finds it non-full (or it finds the server idle). If the queue is not full upon arrival, then the call (customer) waits in line, and will be served according to the FIFO order. If λ is the arrival rate (average number per time unit) of calls and μ is one over the expected service time in the facility, it is well known that μ > λ is not always sufficient for stability. The aim of this paper is to provide general conditions under which it is a sufficient condition. In particular, (i) we derive conditions for Harris ergodicity and obtain bounds for the rate of convergence to the steady state and large deviations results, in the case that the inter-arrival times, retrial times and service times are independent i.i.d. sequences and the retrial times are exponentially distributed; (ii) we establish conditions for strong coupling convergence to a stationary regime when either service times are general stationary ergodic (no independence assumption), and inter-arrival and retrial times are i.i.d. exponentially distributed; or when inter-arrival times are general stationary ergodic, and service and retrial times are i.i.d. exponentially distributed; (iii) we obtain conditions for the existence of uniform exponential bounds of the queue length process under some rather broad conditions on the retrial process. We finally present conditions for boundedness in distribution for the case of nonpatient (or non persistent) customers. This revised version was published online in June 2006 with corrections to the Cover Date.     

A Comprehensive Study of N-Butyl-1H-Benzimidazole

Imidazole derivatives have found wide application in organic and medicinal chemistry. In particular, benzimidazoles have proven biological activity as antiviral, antimicrobial, and antitumor agents. In this work, we experimentally and theoretically investigated N-Butyl-1H-benzimidazole. It has been shown that the presence of a butyl substituent in the N position does not significantly affect the conjugation and structural organization of benzimidazole. The optimized molecular parameters were performed by the DFT/B3LYP method with 6-311++G(d,p) basis set. This level of theory shows excellent concurrence with the experimental data. The non-covalent interactions that existed within our compound N-Butyl-1H-benzimidazole were also analyzed by the AIM, RDG, ELF, and LOL topological methods. The color shades of the ELF and LOL maps confirm the presence of bonding and non-bonding electrons in N-Butyl-1H-benzimidazole. From DFT calculations, various methods such as molecular electrostatic potential (MEP), Fukui functions, Mulliken atomic charges, and frontier molecular orbital (HOMO-LUMO) were characterized. Furthermore, UV-Vis absorption and natural bond orbital (NBO) analysis were calculated. It is shown that the experimental and theoretical spectra of N-Butyl-1H-benzimidazole have a peak at 248 nm; in addition, the experimental spectrum has a peak near 295 nm. The NBO method shows that the delocalization of the aσ-electron from σ (C1–C2) is distributed into antibonding σ* (C1–C6), σ* (C1–N26), and σ* (C6–H11), which leads to stabilization energies of 4.63, 0.86, and 2.42 KJ/mol, respectively. Spectroscopic investigations of N-Butyl-1H-benzimidazole were carried out experimentally and theoretically to find FTIR vibrational spectra.     

An Efficient Method of Birch Ethanol Lignin Sulfation with a Sulfaic Acid-Urea Mixture

For the first time, the process of birch ethanol lignin sulfation with a sulfamic acid-urea mixture in a 1,4-dioxane medium was optimized experimentally and numerically. The high yield of the sulfated ethanol lignin (more than 96%) and containing 7.1 and 7.9 wt % of sulfur was produced at process temperatures of 80 and 90 °C for 3 h. The sample with the highest sulfur content (8.1 wt %) was obtained at a temperature of 100 °C for 2 h. The structure and molecular weight distribution of the sulfated birch ethanol lignin was established by FTIR, 2D ¹H and ¹³C NMR spectroscopy, and gel permeation chromatography. The introduction of sulfate groups into the lignin structure was confirmed by FTIR by the appearance of absorption bands characteristic of the vibrations of sulfate group bonds. According to 2D NMR spectroscopy data, both the alcohol and phenolic hydroxyl groups of the ethanol lignin were subjected to sulfation. The sulfated birch ethanol lignin with a weight average molecular weight of 7.6 kDa and a polydispersity index of 1.81 was obtained under the optimum process conditions. Differences in the structure of the phenylpropane units of birch ethanol lignin (syringyl-type predominates) and abies ethanol lignin (guaiacyl-type predominates) was manifested in the fact that the sulfation of the former proceeds more completely at moderate temperatures than the latter. In contrast to sulfated abies ethanol lignin, the sulfated birch ethanol lignin had a bimodal and wider molecular weight distribution, as well as less thermal stability. The introduction of sulfate groups into ethanol lignin reduced its thermal stability.     

Analytic Function Approximation by Path-Norm-Regularized Deep Neural Networks

We show that neural networks with an absolute value activation function and with network path norm, network sizes and network weights having logarithmic dependence on 1/ε can ε-approximate functions that are analytic on certain regions of Cd.