On the multiplicity of arrangements of congruent zones on the sphere

Consider an arrangement of $n$ congruent zones on the $d$-dimensional unit sphere $S^{d-1}$, where a zone is the intersection of an origin symmetric Euclidean plank with $S^{d-1}$. We prove that, for sufficiently large $n$, it is possible to arrange $n$ congruent zones of suitable width on $S^{d-1}$ such that no point belongs to more than a constant number of zones, where the constant depends only on the dimension and the width of the zones. Furthermore, we also show that it is possible to cover $S^{d-1}$ by $n$ congruent zones such that each point of $S^{d-1}$ belongs to at most $A_d\ln n$ zones, where the $A_d$ is a constant that depends only on $d$. This extends the corresponding $3$-dimensional result of Frankl, Nagy and Nasz\'odi (2016). Moreover, we also examine coverings of $S^{d-1}$ with congruent zones under the condition that each point of the sphere belongs to the interior of at most $d-1$ zones

A Generalization of the Convex Kakeya Problem

Given a set of line segments in the plane, not necessarily finite, what is a convex region of smallest area that contains a translate of each input segment? This question can be seen as a generalization of Kakeya's problem of finding a convex region of smallest area such that a needle can be rotated through 360 degrees within this region. We show that there is always an optimal region that is a triangle, and we give an optimal \Theta(n log n)-time algorithm to compute such a triangle for a given set of n segments. We also show that, if the goal is to minimize the perimeter of the region instead of its area, then placing the segments with their midpoint at the origin and taking their convex hull results in an optimal solution. Finally, we show that for any compact convex figure G, the smallest enclosing disk of G is a smallest-perimeter region containing a translate of every rotated copy of G.Comment: 14 pages, 9 figure

Jagged1 intracellular domain-mediated inhibition of Notch1 signalling regulates cardiac homeostasis in the postnatal heart.

AIMS: Notch1 signalling in the heart is mainly activated via expression of Jagged1 on the surface of cardiomyocytes. Notch controls cardiomyocyte proliferation and differentiation in the developing heart and regulates cardiac remodelling in the stressed adult heart. Besides canonical Notch receptor activation in signal-receiving cells, Notch ligands can also activate Notch receptor-independent responses in signal-sending cells via release of their intracellular domain. We evaluated therefore the importance of Jagged1 (J1) intracellular domain (ICD)-mediated pathways in the postnatal heart. METHODS AND RESULTS: In cardiomyocytes, Jagged1 releases J1ICD, which then translocates into the nucleus and down-regulates Notch transcriptional activity. To study the importance of J1ICD in cardiac homeostasis, we generated transgenic mice expressing a tamoxifen-inducible form of J1ICD, specifically in cardiomyocytes. Using this model, we demonstrate that J1ICD-mediated Notch inhibition diminishes proliferation in the neonatal cardiomyocyte population and promotes maturation. In the neonatal heart, a response via Wnt and Akt pathway activation is elicited as an attempt to compensate for the deficit in cardiomyocyte number resulting from J1ICD activation. In the stressed adult heart, J1ICD activation results in a dramatic reduction of the number of Notch signalling cardiomyocytes, blunts the hypertrophic response, and reduces the number of apoptotic cardiomyocytes. Consistently, this occurs concomitantly with a significant down-regulation of the phosphorylation of the Akt effectors ribosomal S6 protein (S6) and eukaryotic initiation factor 4E binding protein1 (4EBP1) controlling protein synthesis. CONCLUSIONS: Altogether, these data demonstrate the importance of J1ICD in the modulation of physiological and pathological hypertrophy, and reveal the existence of a novel pathway regulating cardiac homeostasis

Parasitoids (Hymenoptera) of leaf-spinning moths (Lepidoptera) feeding on Vaccinium uliginosum L. along an ecological gradient in central European peat bogs

Parasitoids of leaf-spinning Lepidoptera associated with two isolated central European peat bogs were investigated. Five families of parasitoid Hymenoptera (Braconidae, Ichneumonidae, Eulophidae, Pteromalidae and Encyrtidae) were recorded. Three categories were recognised: (1) primary parasitoids, (2) facultative hyperparasitoids and (3) obligatory hyperparasitoids. Ten species of Braconidae, five species and seven marked morphospecies among Ichneumonidae, and three species of Chalcidoidea were identified. Despite of some niche-specific (but less host-specific) parasitoids, all these hymenopterans are likely to be generalists and none of them were confirmed to be habitat and/or host specialists. Unlike their eurytopic (opportunistic tyrphoneutral) parasitoids, the Lepidoptera hosts associated with peat bogs are partially highly stenotopic (tyrphobionts and tyrphophiles). The occurrence of parasitoids compared to their potential hosts was structured along an ecological (mesoclimatic) gradient, so most parasitoids were recorded from margins while stenotopic (narrow habitat adaptation) moths were mostly distributed near the centre of the bog habitat

Considerations about multistep community detection

The problem and implications of community detection in networks have raised a huge attention, for its important applications in both natural and social sciences. A number of algorithms has been developed to solve this problem, addressing either speed optimization or the quality of the partitions calculated. In this paper we propose a multi-step procedure bridging the fastest, but less accurate algorithms (coarse clustering), with the slowest, most effective ones (refinement). By adopting heuristic ranking of the nodes, and classifying a fraction of them as `critical', a refinement step can be restricted to this subset of the network, thus saving computational time. Preliminary numerical results are discussed, showing improvement of the final partition.Comment: 12 page

Contact Representations of Graphs in 3D

We study contact representations of graphs in which vertices are represented by axis-aligned polyhedra in 3D and edges are realized by non-zero area common boundaries between corresponding polyhedra. We show that for every 3-connected planar graph, there exists a simultaneous representation of the graph and its dual with 3D boxes. We give a linear-time algorithm for constructing such a representation. This result extends the existing primal-dual contact representations of planar graphs in 2D using circles and triangles. While contact graphs in 2D directly correspond to planar graphs, we next study representations of non-planar graphs in 3D. In particular we consider representations of optimal 1-planar graphs. A graph is 1-planar if there exists a drawing in the plane where each edge is crossed at most once, and an optimal n-vertex 1-planar graph has the maximum (4n - 8) number of edges. We describe a linear-time algorithm for representing optimal 1-planar graphs without separating 4-cycles with 3D boxes. However, not every optimal 1-planar graph admits a representation with boxes. Hence, we consider contact representations with the next simplest axis-aligned 3D object, L-shaped polyhedra. We provide a quadratic-time algorithm for representing optimal 1-planar graph with L-shaped polyhedra

Pixel and Voxel Representations of Graphs

We study contact representations for graphs, which we call pixel representations in 2D and voxel representations in 3D. Our representations are based on the unit square grid whose cells we call pixels in 2D and voxels in 3D. Two pixels are adjacent if they share an edge, two voxels if they share a face. We call a connected set of pixels or voxels a blob. Given a graph, we represent its vertices by disjoint blobs such that two blobs contain adjacent pixels or voxels if and only if the corresponding vertices are adjacent. We are interested in the size of a representation, which is the number of pixels or voxels it consists of. We first show that finding minimum-size representations is NP-complete. Then, we bound representation sizes needed for certain graph classes. In 2D, we show that, for $k$-outerplanar graphs with $n$ vertices, $\Theta(kn)$ pixels are always sufficient and sometimes necessary. In particular, outerplanar graphs can be represented with a linear number of pixels, whereas general planar graphs sometimes need a quadratic number. In 3D, $\Theta(n^2)$ voxels are always sufficient and sometimes necessary for any $n$-vertex graph. We improve this bound to $\Theta(n\cdot \tau)$ for graphs of treewidth $\tau$ and to $O((g+1)^2n\log^2n)$ for graphs of genus $g$. In particular, planar graphs admit representations with $O(n\log^2n)$ voxels