Purpose: The purpose of this paper is to develop a set of identities for euler type sums of products of harmonic numbers and reciprocal binomial coefficients.Method: We use analytical methods to obtain our results.Results: We obtain identities for variant euler sums of the type ¥Sn=1H2n/n (n+k)k, and its finite counterpart, which generalize some results obtained by other authors.Conclusions: Identities are successfully achieved for the sums under investigation. Some published results have been successfully generalized.

This paper presents the results of a two phase euler-euler simulation of seepage flow through granular media for upward flow conditions in soils. The simulation first deals with the fluidization process in a stable granular soil. The first fluidization in the sand column is studied and the stable fluidization condition is reached by the development of particle repulsion and the effect of inertia on the resistance of the granular bed. Considering Richardson's expansion law, porosity changes with the flow rate variation are described based on the rarefaction waves. In this study the variation of water and soil phases before and after the critical hydraulic gradient are evaluated. The results of the simulation reveal that the variation of soil pressure before and after the critical hydraulic gradient are inverse and the Terzaghi's effective stress theory is valid only to the onset of the first fluidization. In the second part of the paper, the fluidization and internal erosion processes in the artesian conditions in internally unstable soils are discussed. In internally unstable soils the finer soil particles are able to move through the constrictions of the coarse particles. This may lead to internal erosion and as a result the critical hydraulic gradient drastically increases. The erosion phenomenon is described using the theory of continuum mechanics for the soil consisting of stationary grains, movable grains and the fluid. The continuity and momentum equations are derived for the soil and based on this theory a numerical two phase model for the erosion and fluidization of particles are presented. The results reveal that by the commencement of erosion, the porosity of the soil decreases and the flux of eroded particles increases with time.

Ashwagandha, or Indian ginseng (Withania somnifera), is of considerable importance in the Solanaceae family due to the presence of a group of steroid lactones called withanolides. Given the limited production of these metabolites in the roots and leaves of the plant under natural conditions and the high demand from the pharmaceutical industry, it is crucial to use the hairy root method to increase metabolite production. To induce bacteria under laboratory conditions, tissue culture and its various steps are typically required. However, nowadays, an alternative and simpler transgenic method is used that does not rely on tissue culture and can produce transgenic plants with an increased yield of secondary metabolites. In this experiment, Agrobacterium rhizogenes strain A4 was injected into the plant at the wound site located at the first node of the seedling. Fifteen days later, hairy roots appeared, and after the growth of the transgenic plants in the greenhouse, growth characteristics (such as root volume and root area) were measured, and phytochemical analyses were performed on both transgenic and control plants. The experiment followed a completely randomized design with 3 replications. The transgenic plants outperformed the controls in terms of agronomic traits: root length (1.12 times longer), root volume (2 times greater), and root area (1.49 times larger). As a result of this study, catalase enzyme activity and flavonoid levels were found to be lower in transgenic plants than in control plants. Correlation results showed that catalase enzyme activity and flavonoid content had positive and significant correlations with root length, volume, and area. Based on the findings of this research, the transgenic method, without the need for tissue culture, can be introduced as a useful approach in this plant to increase root hair production without increasing antioxidant enzyme activity.

We study the calculus of variations problem in the presence of a system of differential-integral (D-I) equations. In order to identify the necessary optimality conditions for this problem, we derive the so-called D-I euler–Lagrange equations. We also generalize this problem to other cases, such as the case of higher orders, the problem of optimal control, and we derive the so-called D-I Pontryagin equations. In special cases, these formulations lead to classical euler–Lagrange equations. To illustrate our results, we provide simple examples and applications such as obtaining the minimumpower for an RLC circuit.

This paper presents a comparative assessment of low Reynolds number k- models against standard k- model in an eulerian framework. Three different low-Re number k- models: Launder-Sharma (LS), YangShih (YS) and AbeKondoh-Nagano (AKN) have been used for the description of bubble plume behaviour in stratified water. The contribution of the gas phase movement into the liquid phase turbulence has been achieved by using the Dispersed with Bubble Induced Turbulence approach (DIS+BIT). The results reveal that the oscillation frequency of gas-liquid flow are correctly reproduced by standard k- and LS models. In fact, we found for standard K- and LS a clear dominant peak at a frequency equal to 0. 1 Hz. On the other hand, YS and AKN models have predicted chaotic oscillations. The oscillation amplitude of the bubble plume predicted from LS model seems to be in good agreement with the PIV measurements of Besbes et al. (2015). However, for the standard K- model the oscillation amplitude is low. The air-water interface shows that the bubble plume mixing with the stratified water is predicted to be stronger compared to standard k- model.

The issue of the new elastic terms discovered in the nonlinear dynamic model of an enhanced nonlinear 3D euler-Bernoulli beam is discussed. While the elastic orientation is negligible, the nonlinear dynamic model governing tension-compression, torsion and two spatial bendings is presented. Considering this model, some new elastic terms can be identified in the variation of elastic potential energy in each bending motion equation, and in each transverse shear force. Due to the new terms, each term of a bending equation and a transverse shear force, finds a counterpart in the other bending equation and transverse shear force, but the equations remain asymmetric. The new terms have arisen, since variation of strains and variation of elastic potential energy are derived from exact strains and exact deformations regarding considerable elastic orientation, then the elastic orientation is neglected. The new terms perish in the nonlinear 3D euler-Bernoulli beam theory, since elastic orientation is neglected first, then variation of strains and variation of elastic potential energy are derived from the approximated strains.