Multipole matrix elements of Green function of Laplace equation

Multipole matrix elements of Green function of Laplace equation are calculated. The multipole matrix elements of Green function in electrostatics describe potential on a sphere which is produced by a charge distributed on the surface of a different (possibly overlapping) sphere of the same radius. The matrix elements are defined by double convolution of two spherical harmonics with the Green function of Laplace equation. The method we use relies on the fact that in the Fourier space the double convolution has simple form. Therefore we calculate the multipole matrix from its Fourier transform. An important part of our considerations is simplification of the three dimensional Fourier transformation of general multipole matrix by its rotational symmetry to the one-dimensional Hankel transformation

Schauder theory in variable H\"older spaces

We study elliptic equations on bounded domain of Euclidean spaces in the variable H\"{o}lder spaces. Interior a priori Schauder estimates are given as well as global ones. Moreover, the existence and the uniqueness of solutions to the Dirichlet boundary value problem is proved

Maximal operator in H\"older spaces

We study the maximal operator on the variable exponent H\"older spaces in the setting of metric measure spaces. The boundedness is proven for metric measure spaces satisfying an annular decay property. Let us stress that there are no assumptions on the regularity of the variable exponent and the variable exponent can touch values $0$ and $1$. Furthermore, the continuity of the maximal operator between H\"older spaces is investigated. Those results are new even in the Euclidean setting

Increased activity of the sterol branch of the mevalonate pathway elevates glycosylation of secretory proteins and improves antifungal properties of Trichoderma atroviride.

Some Trichoderma spp. have an ability to inhibit proliferation of fungal plant pathogens in the soil. Numerous compounds with a proven antifungal activity are synthesized via the terpene pathway. Here, we stimulated the activity of the mevalonate pathway in T. atroviride P1 by expressing the Saccharomyces cerevisiae ERG20 gene coding for farnesyl pyrophosphate (FPP) synthase, a key enzyme of this pathway. ERG20-expressing Trichoderma strains showed higher activities of FPP synthase and squalene synthase, the principal recipient of FPP in the mevalonate pathway. We also observed activation of dolichyl phosphate mannose (DPM) synthase, an enzyme in protein glycosylation, and significantly increased O- and N-glycosylation of secreted proteins. The hyper-glycosylation of secretory hydrolases could explain their increased activity observed in the ERG20 transformants. Analysis of the antifungal properties of the new strains revealed that the hydrolases secreted by the transformants inhibited growth of a plant pathogen, Pythium ultimum more efficiently compared to the control strain. Consequently, the biocontrol activity of the transgenic strains, determined as their ability to protect bean seeds and seedlings against harmful action of P. ultimum, was also improved substantially

Bent rectangles as viscosity solutions over a circle

We study the motion of the so-called bent rectangles by the singular weighted mean curvature. We are interested in the curves which can be rendered as graphs over a smooth onedimensional reference manifold. We establish a sufficient condition for that. Once we deal with graphs we can have the tools of the viscosity theory available, like the Comparison Principle. With its help we establish uniqueness of variational solutions constructed by the authors [18]. In addition, we establish a criterion for the mobility coefficient guaranteeing vertex preservation