Finding a perfect matching of F2n\mathbb{F}_2^n with prescribed differences

We consider the following question by Balister, Gy\H{o}ri and Schelp: given $2^{n-1}$ nonzero vectors in $\mathbb{F}_2^n$ with zero sum, is it always possible to partition the elements of $\mathbb{F}_2^n$ into pairs such that the difference between the two elements of the $i$-th pair is equal to the $i$-th given vector for every $i$? An analogous question in $\mathbb{F}_p$, which is a case of the so-called "seating couples" problem, has been resolved by Preissmann and Mischler in 2009. In this paper, we prove the conjecture in $\mathbb{F}_2^n$ in the case when the number of distinct values among the given difference vectors is at most $n-2\log n-1$, and also in the case when at least a fraction $\frac12+\varepsilon$ of the given vectors are equal (for all $\varepsilon>0$ and $n$ sufficiently large based on $\varepsilon$).Comment: 18 page

Finding pairwise disjoint vector pairs in F2n with a prescribed sequence of differences

We consider the following question by Balister, Győri and Schelp: given $2^{n-1}$ nonzero vectors in $\mathbb{F}_2^n$ with zero sum, is it always possible to partition $\mathbb{F}_2^n$ into pairs such that the difference between the two elements of the $i$-th pair is equal to the $i$-th given vector? An analogous question in $\mathbb{F}_p$ was resolved by Preissmann and Mischler in 2009. In this paper, we prove the conjecture in $\mathbb{F}_2^n$ in the case when there are at most $n-2\log n-1$ distinct values among the given differences, and also in the case when at least a fraction $\frac{28}{29}$ of the differences are equal

Multicolor Tur\'an numbers II -- a generalization of the Ruzsa-Szemer\'edi theorem and new results on cliques and odd cycles

In this paper we continue the study of a natural generalization of Tur\'an's forbidden subgraph problem and the Ruzsa-Szemer\'edi problem. Let $ex_F(n,G)$ denote the maximum number of edge-disjoint copies of a fixed simple graph $F$ that can be placed on an $n$-vertex ground set without forming a subgraph $G$ whose edges are from different $F$-copies. The case when both $F$ and $G$ are triangles essentially gives back the theorem of Ruzsa and Szemer\'edi. We extend their results to the case when $F$ and $G$ are arbitrary cliques by applying a number theoretic result due to Erd\H{o}s, Frankl and R\"odl. This extension in turn decides the order of magnitude for a large family of graph pairs, which will be subquadratic, but almost quadratic. Since the linear $r$-uniform hypergraph Tur\'an problems to determine $ex_r^{lin}(n,G)$ form a class of the multicolor Tur\'an problem, following the identity $ex_r^{lin}(n,G)=ex_{K_r}(n,G)$, our results determine the linear hypergraph Tur\'an numbers of every graph of girth $3$ and for every $r$ up to a subpolynomial factor. Furthermore, when $G$ is a triangle, we settle the case $F=C_5$ and give bounds for the cases $F=C_{2k+1}$, $k\ge 3$ as well

Avoiding intersections of given size in finite affine spaces AG(n,2)

We study the set of intersection sizes of a $k$-dimensional affine subspace and a point set of size $m \in [0, 2^n]$ of the $n$ dimensional binary affine space $\mathrm{AG}(n,2)$

Effect of Various Drying Methods on the Volatile Oil Composition of Basil Leaves

This article presents the results pertaining to the drying behavior of basil leaves (Ocimum basilicum L.) in natural air drying (~25-30°C, 2 days), hot-air drying (50°C, 6h, 1m/s), and vacuum drying (50°C, 8kPa, 7h) conditions. This study focused on the chemical composition characteristics of essential oils extracted from fresh and different dried basil leaves. The results showed that drying methods had a significant effect on essential oil content and composition of basil leaves. The volatile oil found in herbs is very sensitive to some drying parameters (e.g. pressure, temperature, weather, non-uniform drying, etc.). The quality of the vacuum dried product was assessed – from a twelve major constituents – as being higher than that of a hot-air dried and natural air dried products. Taking into account all these considerations we recommend the drying of basil leaves by vacuum drying

Normal and abnormal development of visual functions in children

The human visual system goes through substantial changes during the first few months of postnatal life. The development of visual functions and structures occurs at different times and different rates. It has been a generally held belief that the development of visual functions and their critical period come to an end early in life. Most of the developmental data confirm this theory, although the findings sometimes are contradictory. Thus, our knowledge concerning visual development does not seem to be complete. The determination of exact timing of the different visual functions is relevant in children since a proved extended maturational timeframe can promote the trial of enhancement of visual abilities at a later age, up to puberty or beyond. There have already been suggestions for an extended developmental time span for some of the visual functions. Here we review the most relevant data with reference to the normal development of the eye, visual functions and visual pathways found in the literature and provide further evidence for the maturation and plasticity of visual functions after the age of 5 years