A Szemeredi-Trotter type theorem in R4\mathbb{R}^4

We show that $m$ points and $n$ two-dimensional algebraic surfaces in $\mathbb{R}^4$ can have at most $O(m^{\frac{k}{2k-1}}n^{\frac{2k-2}{2k-1}}+m+n)$ incidences, provided that the algebraic surfaces behave like pseudoflats with $k$ degrees of freedom, and that $m\leq n^{\frac{2k+2}{3k}}$. As a special case, we obtain a Szemer\'edi-Trotter type theorem for 2--planes in $\mathbb{R}^4$, provided $m\leq n$ and the planes intersect transversely. As a further special case, we obtain a Szemer\'edi-Trotter type theorem for complex lines in $\mathbb{C}^2$ with no restrictions on $m$ and $n$ (this theorem was originally proved by T\'oth using a different method). As a third special case, we obtain a Szemer\'edi-Trotter type theorem for complex unit circles in $\mathbb{C}^2$. We obtain our results by combining several tools, including a two-level analogue of the discrete polynomial partitioning theorem and the crossing lemma.Comment: 50 pages. V3: final version. To appear in Discrete and Computational Geometr

Zastosowanie grzybów Trichoderma do uwalniania cukrów prostych z roślinnych surowców odpadowych

The aim of the presented work was to examine the applicability of selected agri-food industry waste materials as substrates for microbial hydrolytic enzymes biosynthesis (mostly celulases, xylanases and poligalacturonases) as well as to determine the quantitative and qualitative composition of mono- and disaccharides released from non-starch polysaccharides. (sucrose, glucose, galactose, fructose, cellobiose, etc). Two strains of Trichoderma fungi were tested: T. citrinoviride C1 and T. harzianum T33 in SSF culture on six different waste materials from agro-food industries (pulp from: cherries with seeds, pears, red cabbage, cornelian cherry and grapes with seeds).W pracy przedstawiono ocenę możliwości zagospodarowania wybranych odpadów przemysłu rolno-spożywczego jako substratu do mikrobiologicznej biosyntezy enzymów hydrolitycznych (głównie celulaz, ksylanaz i poligalakturonaz) a także, określenie składu ilościowego i jakościowego uwalnianych z polisacharydów nieskrobiowych mono- i disacharydów (sacharozy, glukozy, galaktoza, fruktozy, celobiozy itd). Testowano dwa szczepy grzybów z rodzaju Trichoderma: T. citri-noviride C1 i T. harzianum T33 w hodowlach SSF na sześciu różnych surowcach pochodzących z przemysłu rolno-spożywczego (wytłoki z: wiśni z pestkami, gruszki, czerwonej kapusty, derenia, winogron z pestkami)