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  • EDIH Adria
  • 2013-2022
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  • image/svg+xml art designer at PLoS, modified by Wikipedia users Nina, Beao, JakobVoss, and AnonMoos Open Access logo, converted into svg, designed by PLoS. This version with transparent background. http://commons.wikimedia.org/wiki/File:Open_Access_logo_PLoS_white.svg art designer at PLoS, modified by Wikipedia users Nina, Beao, JakobVoss, and AnonMoos http://www.plos.org/
    Authors: Pach, János; Pálvölgyi, Dömötör;

    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.

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    Advances in Mathematics
    Other literature type . Article . 2016 . Peer-reviewed
    License: Elsevier Non-Commercial
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    http://arxiv.org/pdf/1310.6900...
    Part of book or chapter of book
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    https://doi.org/10.1007/978-3-...
    Part of book or chapter of book . 2016 . Peer-reviewed
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    https://doi.org/10.48550/arxiv...
    Article . 2013
    License: arXiv Non-Exclusive Distribution
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      Advances in Mathematics
      Other literature type . Article . 2016 . Peer-reviewed
      License: Elsevier Non-Commercial
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      https://doi.org/10.1007/978-3-...
      Part of book or chapter of book . 2016 . Peer-reviewed
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      https://doi.org/10.48550/arxiv...
      Article . 2013
      License: arXiv Non-Exclusive Distribution
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    Authors: Keleti, Tamás; Matolcsi, Máté; Filho, Fernando Mário de Oliveira; Ruzsa, Imre Z.;

    A $1$-avoiding set is a subset of $\mathbb{R}^n$ that does not contain pairs of points at distance $1$. Let $m_1(\mathbb{R}^n)$ denote the maximum fraction of $\mathbb{R}^n$ that can be covered by a measurable $1$-avoiding set. We prove two results. First, we show that any $1$-avoiding set in $\mathbb{R}^n$ ($n\ge 2$) that displays block structure (i.e., is made up of blocks such that the distance between any two points from the same block is less than $1$ and points from distinct blocks lie farther than $1$ unit of distance apart from each other) has density strictly less than $1/2^n$. For the special case of sets with block structure this proves a conjecture of Erd\H{o}s asserting that $m_1(\mathbb{R}^2) < 1/4$. Second, we use linear programming and harmonic analysis to show that $m_1(\mathbb{R}^2) \leq 0.258795$. Comment: 16 pages, 1 figure. Contains a Sage script called dstverify.sage, to verify the application of Theorem 3.3. Download the article source to get the script

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    https://doi.org/10.48550/arxiv...
    Article . 2015
    License: arXiv Non-Exclusive Distribution
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      https://doi.org/10.48550/arxiv...
      Article . 2015
      License: arXiv Non-Exclusive Distribution
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    Authors: László, Z Varga;

    Abstract The online routing game model can be used to measure and prove the benefits of online real time data in road traffic navigation systems. A few properties of the routing strategies are already proved. In this paper we point out that there are some paradoxes like phenomena behind these proofs, similarly as in the Braess network.

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    Cybernetics and Information Technologies
    Article . 2015 . Peer-reviewed
    License: CC BY NC ND
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    Article . 2015
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      image/svg+xml art designer at PLoS, modified by Wikipedia users Nina, Beao, JakobVoss, and AnonMoos Open Access logo, converted into svg, designed by PLoS. This version with transparent background. http://commons.wikimedia.org/wiki/File:Open_Access_logo_PLoS_white.svg art designer at PLoS, modified by Wikipedia users Nina, Beao, JakobVoss, and AnonMoos http://www.plos.org/ ELTE Digital Institu...arrow_drop_down
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      Article . 2015 . Peer-reviewed
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    Authors: Virágh, C; Tarcai, N; Szörényi, T; Somorjai, G; +3 Authors

    Animal swarms displaying a variety of typical flocking patterns would not exist without the underlying safe, optimal and stable dynamics of the individuals. The emergence of these universal patterns can be efficiently reconstructed with agent-based models. If we want to reproduce these patterns with artificial systems, such as autonomous aerial robots, agent-based models can also be used in their control algorithms. However, finding the proper algorithms and thus understanding the essential characteristics of the emergent collective behaviour requires thorough and realistic modeling of the robot and also the environment. In this paper, we first present an abstract mathematical model of an autonomous flying robot. The model takes into account several realistic features, such as time delay and locality of communication, inaccuracy of the on-board sensors and inertial effects. We present two decentralized control algorithms. One is based on a simple self-propelled flocking model of animal collective motion, the other is a collective target tracking algorithm. Both algorithms contain a viscous friction-like term, which aligns the velocities of neighbouring agents parallel to each other. We show that this term can be essential for reducing the inherent instabilities of such a noisy and delayed realistic system. We discuss simulation results on the stability of the control algorithms, and perform real experiments to show the applicability of the algorithms on a group of autonomous quadcopters. In our case, bio-inspiration works in two ways. On the one hand, the whole idea of trying to build and control a swarm of robots comes from the observation that birds tend to flock to optimize their behaviour as a group. On the other hand, by using a realistic simulation framework and studying the group behaviour of autonomous robots we can learn about the major factors influencing the flight of bird flocks. © 2014 IOP Publishing Ltd.

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    Bioinspiration & Biomimetics
    Article . 2014 . Peer-reviewed
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    Article . 2013
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      Bioinspiration & Biomimetics
      Article . 2014 . Peer-reviewed
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    Authors: Pach, János; Pálvölgyi, Dömötör;

    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.

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    Advances in Mathematics
    Other literature type . Article . 2016 . Peer-reviewed
    License: Elsevier Non-Commercial
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    http://arxiv.org/pdf/1310.6900...
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    https://doi.org/10.1007/978-3-...
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      Advances in Mathematics
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      https://doi.org/10.1007/978-3-...
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    Authors: Keleti, Tamás; Matolcsi, Máté; Filho, Fernando Mário de Oliveira; Ruzsa, Imre Z.;

    A $1$-avoiding set is a subset of $\mathbb{R}^n$ that does not contain pairs of points at distance $1$. Let $m_1(\mathbb{R}^n)$ denote the maximum fraction of $\mathbb{R}^n$ that can be covered by a measurable $1$-avoiding set. We prove two results. First, we show that any $1$-avoiding set in $\mathbb{R}^n$ ($n\ge 2$) that displays block structure (i.e., is made up of blocks such that the distance between any two points from the same block is less than $1$ and points from distinct blocks lie farther than $1$ unit of distance apart from each other) has density strictly less than $1/2^n$. For the special case of sets with block structure this proves a conjecture of Erd\H{o}s asserting that $m_1(\mathbb{R}^2) < 1/4$. Second, we use linear programming and harmonic analysis to show that $m_1(\mathbb{R}^2) \leq 0.258795$. Comment: 16 pages, 1 figure. Contains a Sage script called dstverify.sage, to verify the application of Theorem 3.3. Download the article source to get the script

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    https://doi.org/10.48550/arxiv...
    Article . 2015
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    Authors: László, Z Varga;

    Abstract The online routing game model can be used to measure and prove the benefits of online real time data in road traffic navigation systems. A few properties of the routing strategies are already proved. In this paper we point out that there are some paradoxes like phenomena behind these proofs, similarly as in the Braess network.

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    Cybernetics and Information Technologies
    Article . 2015 . Peer-reviewed
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    Authors: Virágh, C; Tarcai, N; Szörényi, T; Somorjai, G; +3 Authors

    Animal swarms displaying a variety of typical flocking patterns would not exist without the underlying safe, optimal and stable dynamics of the individuals. The emergence of these universal patterns can be efficiently reconstructed with agent-based models. If we want to reproduce these patterns with artificial systems, such as autonomous aerial robots, agent-based models can also be used in their control algorithms. However, finding the proper algorithms and thus understanding the essential characteristics of the emergent collective behaviour requires thorough and realistic modeling of the robot and also the environment. In this paper, we first present an abstract mathematical model of an autonomous flying robot. The model takes into account several realistic features, such as time delay and locality of communication, inaccuracy of the on-board sensors and inertial effects. We present two decentralized control algorithms. One is based on a simple self-propelled flocking model of animal collective motion, the other is a collective target tracking algorithm. Both algorithms contain a viscous friction-like term, which aligns the velocities of neighbouring agents parallel to each other. We show that this term can be essential for reducing the inherent instabilities of such a noisy and delayed realistic system. We discuss simulation results on the stability of the control algorithms, and perform real experiments to show the applicability of the algorithms on a group of autonomous quadcopters. In our case, bio-inspiration works in two ways. On the one hand, the whole idea of trying to build and control a swarm of robots comes from the observation that birds tend to flock to optimize their behaviour as a group. On the other hand, by using a realistic simulation framework and studying the group behaviour of autonomous robots we can learn about the major factors influencing the flight of bird flocks. © 2014 IOP Publishing Ltd.

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    Bioinspiration & Biomimetics
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