An inertial iterative algorithm for generalized equilibrium problems and Bregman relatively nonexpansive mappings in Banach spaces

The aim of this paper is to introduce and study an inertial hybrid iterative method for solving generalized equilibrium problems involving Bregman relatively nonexpansive mappings in Banach spaces. We study the strong convergence for the proposed algorithm. Finally, we list some consequences and computational example to emphasize the efficiency and relevancy of main result.

An Inertial Iterative Algorithm to Find Common Solution of a Split Generalized Equilibrium and a Variational Inequality Problem in Hilbert Spaces

In this paper, we introduce and study an iterative algorithm via inertial and viscosity techniques to find a common solution of a split generalized equilibrium and a variational inequality problem in Hilbert spaces. Further, we prove that the sequence generated by the proposed theorem converges strongly to the common solution of our problem. Furthermore, we list some consequences of our established algorithm. Finally, we construct a numerical example to demonstrate the applicability of the theorem. We emphasize that the result accounted in the manuscript unifies and extends various results in this field of study.

Strong convergence algorithm for the split problem of variational inclusions, split generalized equilibrium problem and fixed point problem

The purpose of this paper is to recommend an iterative scheme to approximate a common element of the solution sets of the split problem of variational inclusions, split generalized equilibrium problem and fixed point problem for non-expansive mappings. We prove that the sequences generated by the recommended iterative scheme strongly converge to a common element of solution sets of stated split problems. In the end, we provide a numerical example to support and justify our main result. The result studied in this paper generalizes and extends some widely recognized results in this direction.

Generalized Chemical Equilibrium Constant of Formaldehyde Oligomerization

Formaldehyde reacts with solvents that contain hydroxyl groups (R–OH) in oligomerization reactions to oxymethylene oligomers (R–(OCH2)n–OH). The chemical equilibria of these reactions have been studied in the literature for water, for the mono-alcohols methanol, ethanol, and 1-butanol, as well as for the diols ethylene glycol and 1,4-butynediol. In the present work, the collective data were analyzed. It was found that the prolongation of the oxymethylene chains by the addition of formaldehyde can be described very well with a generalized chemical equilibrium constant Kx,n≥2R–OH, which is independent of the substructure (R) of the solvent. This holds for the oligomerization reactions leading to R–(OCH2)n–OH with n ≥ 2. The chemical equilibrium constant Kx,1R–OH of the reaction of formaldehyde with the solvent R–OH depends on the solvent, but simple trends are observed. The hypotheses of the existence of a generalized chemical equilibrium constant Kx,n≥2R–OH was tested for the reactions of formaldehyde with ethanol and 1-propanol, for which neither Kx,1R–OH nor Kx,nR–OH was previously available. The corresponding equilibria were studied by 13C NMR spectroscopy and the equilibrium constants were determined. A novel method was developed and used in these studies to obtain data on Kx,1R–OH by NMR spectroscopy, which is difficult because of the low amount of molecular formaldehyde. It was found that the generalized equilibrium constant is even valid for the acid-catalyzed formation of poly(oxymethylene) dimethyl ethers (OME).

Observed Influence of Soil Moisture on the North American Monsoon: An Assessment Using the Stepwise Generalized Equilibrium Feedback Assessment Method

The evidence shows that soil moisture has an important influence on North American Monsoon (NAM) precipitation. This study evaluates the local and nonlocal feedbacks of soil moisture on summer (June - September) precipitation in the NAM region using observational data. We applied a multivariate statistical method known as the Stepwise Generalized Equilibrium Feedback Assessment (SGEFA) to control for internal atmospheric variability and sea surface temperature (SST) forcings so that we could isolate the impact of soil moisture feedbacks on NAM precipitation. Our results identify feedback pathways between soil moisture and precipitation in the NAM region and in the Southern Rocky Mountains (SRM) region. Wet soils in the SRM result in lower-than-normal local surface temperature, weaker water vapor transport from the eastern Pacific and the Gulf of California (GOC), and less monsoon precipitation. Precipitation over the U.S. Great Plains also significantly increases when there are wet soils in the SRM. This occurs due to an enhanced water vapor influx into this region. On the other hand, anomalously wet soils in the NAM region increase NAM precipitation by enhancing local moist static energy and increasing the strength of the monsoonal circulation. Our observational results using SGEFA agree well with previous numerical modeling studies. This study highlights the critical role of land-atmosphere interactions for understanding NAM variability.