Coloring Hypergraphs Induced by Dynamic Point Sets and Bottomless Rectangles

We consider a coloring problem on dynamic, one-dimensional point sets: points appearing and disappearing on a line at given times. We wish to color them with k colors so that at any time, any sequence of p(k) consecutive points, for some function p, contains at least one point of each color. We prove that no such function p(k) exists in general. However, in the restricted case in which points appear gradually, but never disappear, we give a coloring algorithm guaranteeing the property at any time with p(k)=3k-2. This can be interpreted as coloring point sets in R^2 with k colors such that any bottomless rectangle containing at least 3k-2 points contains at least one point of each color. Here a bottomless rectangle is an axis-aligned rectangle whose bottom edge is below the lowest point of the set. For this problem, we also prove a lower bound p(k)>ck, where c>1.67. Hence for every k there exists a point set, every k-coloring of which is such that there exists a bottomless rectangle containing ck points and missing at least one of the k colors. Chen et al. (2009) proved that no such function $p(k)$ exists in the case of general axis-aligned rectangles. Our result also complements recent results from Keszegh and Palvolgyi on cover-decomposability of octants (2011, 2012).Comment: A preliminary version was presented by a subset of the authors to the European Workshop on Computational Geometry, held in Assisi (Italy) on March 19-21, 201

Unsplittable coverings in the plane

A system of sets forms an {\em $m$-fold covering} of a set $X$ if every point of $X$ belongs to at least $m$ of its members. A $1$-fold covering is called a {\em covering}. The problem of splitting multiple coverings into several coverings was motivated by classical density estimates for {\em sphere packings} as well as by the {\em planar sensor cover problem}. It has been the prevailing conjecture for 35 years (settled in many special cases) that for every plane convex body $C$, there exists a constant $m=m(C)$ such that every $m$-fold covering of the plane with translates of $C$ splits into $2$ coverings. In the present paper, it is proved that this conjecture is false for the unit disk. The proof can be generalized to construct, for every $m$, an unsplittable $m$-fold covering of the plane with translates of any open convex body $C$ which has a smooth boundary with everywhere {\em positive curvature}. Somewhat surprisingly, {\em unbounded} open convex sets $C$ do not misbehave, they satisfy the conjecture: every $3$-fold covering of any region of the plane by translates of such a set $C$ splits into two coverings. To establish this result, we prove a general coloring theorem for hypergraphs of a special type: {\em shift-chains}. We also show that there is a constant $c>0$ such that, for any positive integer $m$, every $m$-fold covering of a region with unit disks splits into two coverings, provided that every point is covered by {\em at most} $c2^{m/2}$ sets

Development of therapeutic kefir-like products with low galactose content for patients with galactose intolerance

Galactosaemia is a rare, life-threatening inborn error. It is treated by life-long galactose restriction. People with galactose intolerance cannot consume milk and milk products. The aim of this work was to produce fermented milk products with low galactose content. Lactose hydrolysed milk and two types of mixture of lactose hydrolysed milk supplemented with galactose free nutriments (Pregomin and Nutrilon) were fermented in a 2:1 ratio. For fermentation traditional kefir culture (Lactococcus lactis sp. lactis + Lactococcus lactis sp. cremoris + Lactobacillus casei + Lactobacillus kefir + Candida kefir) was used. Number of viable cells, pH and galactose level were measured. Data were evaluated by one-way analysis of variance and t-test. Level of galactose reduction was measured by UV spectrometry for the determination of lactose and D-galactose in foodstuffs (Boehringer Mannheim enzyme test). Galactose content was below 200 mg/100 cm3 in the mixtures of lactose hydrolysed milk and galactose free nutriments after 48 h of fermentation. So, the kefir-like products based on mixed milk with nutriments are suitable for use in the diet of patients suffering from galactosaemia

An ultralow-loss and lightweight cellulose-coated silica foam for planar Fresnel zone plate lens applications in future 6G devices

Abstract Several passive components of fifth-generation (5G) and future sixth-generation (6G) telecommunication devices require substrate materials of very low dielectric permittivity and losses to avoid wave absorption, reflection, and interference. Apart from their dielectric properties, these materials shall be also affordable and sufficiently robust to enable postprocessing and integration of functional electrical components. Herein, we demonstrate a Fresnel zone plate lens for operation at 300 GHz, whose structure is supported on substrate made of an ultraporous silica foam with a nanocellulose thin film coating. The effective dielectric permittivity and loss of the substrate ( ϵ r = 1.018 ± 0.003 and tan δ < 3 × 10 −4 at 300 GHz) is close to that of air. Experiments show that the fabricated Fresnel zone plate lens connected to a waveguide with total gain of 20 dB and angular beamwidth of 2.9° in good agreement with microwave simulations. The proposed lens structure has additional advantages such as small volume, ultralight weight, and simulations indicate 60 GHz bandwidth making it particularly appealing for radio front-ends of future 6G devices