Hermite‐Hadamard‐type inequalities for interval‐valued coordinated convex functions involving generalized fractional integrals

In this paper, we define interval‐valued left‐sided and right‐sided generalized fractional double integrals. We establish inequalities of Hermite‐Hadamard like for coordinated interval‐valued convex functions by applying our newly defined integrals.     

K-g-框架的稳定性与不等式

K-g-框架是将算子K引入到g-框架中的一种特殊框架.采用泛函分析中的技巧和方法,研究K-g-框架的稳定性,并得到了四个K-g-框架在扰动情况下稳定的充分条件.此外,结合算子K和框架算子S衍生出K-g-框架的三个等式,并通过引入参数λ建立关于K-g-框架的一些不等式.     

Geometric and operational optimization of 20-kHz probe-type sonoreactor for enhancing sonochemical activity

The use of a 20-kHz probe-type sonicator irradiating downward in a 500 mL vessel was optimized for the enhancement of the sonochemical activity in terms of the geometric and operational factors. These factors included the probe immersion depth (the vertical position of the probe), input power, height of the liquid from the bottom, horizontal position of the probe, and thickness of bottom plate The sonochemical oxidation reactions were investigated both quantitatively and qualitatively using calorimetry, KI dosimetry, and luminol (Sonochemiluminescence, SCL) techniques. The sonochemical activity was very positively affected by the vertical boundaries. The highest sonochemical activity was obtained when the probe was placed close to the bottom of the vessel (immersion depth of 60 mm), with a high input power (input power of 75%), and optimal liquid height condition (liquid height of 70 mm). The SCL image analysis showed that the cavitational activity zone gradually expanded around the probe body and changed into a circular shape as the experimental conditions were optimized, and consequently the sonochemical activity increased. The formation of a large bright circular-shaped activity zone could be attributed to the strong reflections of the ultrasound firstly, at the vessel bottom and secondly, at the liquid surface. On the other hand, the cavitational activity zone and the sonochemical activity were negatively affected by the horizontal boundaries when the probe was placed close to the side wall of the vessel. In addition, it was found that the sonochemical activity was also significantly affected by the thickness of the support plate owing to the reflection and transmission of the ultrasound at the boundary between the liquid and the solid media.     

The coupled-cluster formalism – a mathematical perspective

ABSTRACT

The Coupled-Cluster (CC) theory is one of the most successful high precision methods used to solve the stationary Schrödinger equation. In this article, we address the mathematical foundation of this theory with focus on the advances made in the past decade. Rather than solely relying on spectral gap assumptions (non-degeneracy of the ground state), we highlight the importance of coercivity assumptions – Gårding type inequalities – for the local uniqueness of the CC solution. Based on local strong monotonicity, different sufficient conditions for a local unique solution are suggested. One of the criteria assumes the relative smallness of the total cluster amplitudes (after possibly removing the single amplitudes) compared to the Gårding constants. In the extended CC theory the Lagrange multipliers are wave function parameters and, by means of the bivariational principle, we here derive a connection between the exact cluster amplitudes and the Lagrange multipliers. This relation might prove useful when determining the quality of a CC solution. Furthermore, the use of an Aubin–Nitsche duality type method in different CC approaches is discussed and contrasted with the bivariational principle.     

An efficient iterative method for finding common fixed point and variational inequalities in Hilbert spaces

ABSTRACT

In this paper, we investigate the problem of finding a common solution to a fixed point problem involving demi-contractive operator and a variational inequality with monotone and Lipschitz continuous mapping in real Hilbert spaces. Inspired by the projection and contraction method and the hybrid descent approximation method, a new and efficient iterative method for solving the problem is introduced. Strong convergence theorem of the proposed method is established under standard and mild conditions. Our scheme generalizes and extends some of the existing results in the literature, and moreover, its computational effort is less per each iteration compared with related works.     

Nonexistence of nontrivial global solutions for nonlocal in time differential inequalities

In this paper, some nonlocal in time differential inequalities of Sobolev type are considered. Using the nonlinear capacity method, sufficient conditions for the nonexistence of nontrivial global classical solutions are provided.