Real Analyticity of Periodic Layer Potentials Upon Perturbation of the Periodicity Parameters and of the Support

We prove that the periodic layer potentials for the Laplace operator depend real analytically on the density function, on the supporting hypersurface, and on the periodicity parameters

Dependence of effective properties upon regular perturbations

In this survey, we present some results on the behavior of effective properties in presence of perturbations of the geometric and physical parameters. We first consider the case of a Newtonian fluid flowing at low Reynolds numbers around a periodic array of cylinders. We show the results of [43], where it is proven that the average longitudinal flow depends real analytically upon perturbations of the periodicity structure and the cross section of the cylinders. Next, we turn to the effective conductivity of a periodic two-phase composite with ideal contact at the interface. The composite is obtained by introducing a periodic set of inclusions into an infinite homogeneous matrix made of a different material. We show a result of [41] on the real analytic dependence of the effective conductivity upon perturbations of the shape of the inclusions, the periodicity structure, and the conductivity of each material. In the last part of the chapter, we extend the result of [41] to the case of a periodic two-phase composite with imperfect contact at the interface

Peculiarities of soybean-rhizobial systems subject to different levels of water supply fol-lowing treatment with succinic acid and epibrassinolide

All around the world, one of the leading – according to area of cultivated fields – oleic crops is soybean, which has a high demand for moisture. Given the significance of this crop and negative impact of drought on its yield, integrated research of the influence of insufficient water supply on the intensity of physiological-biochemical processes in those plants is necessary for identifying and understanding the drought-tolerance mechanisms of soybean, as well as symbiotic systems created with its participation, and also for search for ways to adapt it to this stressor. Therefore, our objective was determining the specifics of formation and functioning of the symbiotic systems of soybean and Bradyrhizobium japonicum, following treatment with succinic acid (0.01 g/L) and 24-epibrassinolide (0.00001 g/L), subject to different levels of watering. Our studies revealed that pre-sowing treatment of the seeds with a solution of 24-epibrassinolide with their subsequent inoculation with B. japonicum Т21-2 resulted in the most pronounced stimulation of formation and functioning of the symbiotic systems of soybean in the optimal growing conditions. At the same time, during water shortage, the intensity of nitrogen fixation was the highest in the plants grown from seeds that had been successively treated with the acid and the inoculant. We confirmed that water deficit led to significant increase in the overall content of phytohormones of cytokinin nature in the soybean root nodules, depending on the way the seeds were treated. However, the largest pool of cytokinins was seen in the plants that had been treated with succinic acid against the background of both optimal and insufficient water supply. Treatment of the seeds with 24-epibrassinolide caused significant excess of content of zeatin riboside over the content of zeatin during the flowering stage, whereas in the stage of pods formation it led to an opposite effect – excess of zeatin over zeatin riboside. Fourteen days-long water deficit decreased the content of chlorophylls in the leaves and grain productivity of the plants of all variants of the experiment. The use of growth regulators managed to alleviate the negative impact of stress and protect the pigment complex from ruination. Treatment of the seeds with solutions of succinic acid and 24-epibrassinolide provided the growth of soybean grain productivity regardless on water-supply level. The most efficient was 24-epibrassinolide. Therefore, use of 24-epibrassinolide for pre-sowing treatment of the soybean seeds provided formation of effective symbiotic systems with high nitrogen-fixing activity and caused a number of specific changes in the pattern of accumulation of free and complex forms of cytokinins in the root nodules of those plants. At the same time, the treatment provided the highest concentration of photosynthesis pigments in the soybean leaves, and as a result produced the greatest increase in grain productivity of plants of all the variants, regardless of levels of water supply. In turn, use of succinic acid produced the highest level of nitrogen-fixing activity in the case of the lowest number of root nodules in the conditions of insufficient water supply, and also caused significant accumulation of cytokinins in the nodules, compared with other studied variants against the background of both optimal and insufficient water supply. Therefore, it did result in increase in soybean grain productivity, but this was lower than in the plants treated with 24-epibrassinolide

Shape analysis of the longitudinal flow along a periodic array of cylinders

We study the behavior of the longitudinal flow along a periodic array of cylinders upon perturbations of the shape of the cross section of the cylinders and the periodicity structure, when a Newtonian fluid is flowing at low Reynolds numbers around the cylinders. The periodicity cell is a rectangle of sides of length $l$ and $1/l$, where $l$ is a positive parameter, and the shape of the cross section of the cylinders is determined by the image of a fixed domain through a diffeomorphism $phi$. We also assume that the pressure gradient is parallel to the cylinders. Under such assumptions, for each pair $(l,phi)$, one defines the average of the longitudinal component of the flow velocity $Sigma[l,phi]$. Here, we prove that the quantity $Sigma[l,phi]$ depends analytically on the pair $(l,phi)$, which we consider as a point in a suitable Banach space

Dependence of effective properties upon regular perturbations

In this survey, we present some results on the behavior of effective properties in presence of perturbations of the geometric and physical parameters. We first consider the case of a Newtonian fluid flowing at low Reynolds numbers around a periodic array of cylinders. We show the results of [43], where it is proven that the average longitudinal flow depends real analytically upon perturbations of the periodicity structure and the cross section of the cylinders. Next, we turn to the effective conductivity of a periodic two-phase composite with ideal contact at the interface. The composite is obtained by introducing a periodic set of inclusions into an infinite homogeneous matrix made of a different material. We show a result of [41] on the real analytic dependence of the effective conductivity upon perturbations of the shape of the inclusions, the periodicity structure, and the conductivity of each material. In the last part of the chapter, we extend the result of [41] to the case of a periodic two-phase composite with imperfect contact at the interface