The Experts below are selected from a list of 1302 Experts worldwide ranked by ideXlab platform

Magdalena Górajska - One of the best experts on this subject based on the ideXlab platform.

  • Pointwise density topology
    Open Mathematics, 2015
    Co-Authors: Magdalena Górajska
    Abstract:

    AbstractThe paper presents a new type of density topology on the real line generated by the pointwise convergence, similarly to the classical density topology which is generated by the convergence in measure. Among other things, this paper demonstrates that the Set of pointwise density points of a Lebesgue Measurable Set does not need to be Measurable and the Set of pointwise density points of a Set having the Baire property does not need to have the Baire property. However, the Set of pointwise density points of any Borel Set is Lebesgue Measurable.

Yavicoli Alexia - One of the best experts on this subject based on the ideXlab platform.

  • The density of Sets containing large similar copies of finite Sets
    2021
    Co-Authors: Falconer Kenneth, Yavicoli Alexia, Kovač Vjekoslav
    Abstract:

    Funding: VK is supported by the Croatian Science Foundation, project n◦ UIP-2017-05-4129 (MUNHANAP). AY is supported by the Swiss National Science Foundation, grant n◦ P2SKP2 184047.We prove that if E⊆Rd (d≥2) is a Lebesgue-Measurable Set with density larger than n−2n−1, then E contains similar copies of every n-point Set P at all sufficiently large scales. Moreover, 'sufficiently large' can be taken to be uniform over all P with prescribed size, minimum separation and diameter. On the other hand, we construct an example to show that the density required to guarantee all large similar copies of n-point Sets tends to 1 at a rate 1−O(n−1/5log n).PreprintPeer reviewe

  • The density of Sets containing large similar copies of finite Sets
    2020
    Co-Authors: Falconer Kenneth, Kovač Vjekoslav, Yavicoli Alexia
    Abstract:

    We prove that if $E \subSeteq \mathbb{R}^d$ ($d\geq 2$) is a Lebesgue-Measurable Set with density larger than $\frac{n-2}{n-1}$, then $E$ contains similar copies of every $n$-point Set $P$ at all sufficiently large scales. Moreover, `sufficiently large' can be taken to be uniform over all $P$ with prescribed size, minimum separation and diameter. On the other hand, we construct an example to show that the density required to guarantee all large similar copies of $n$-point Sets tends to $1$ at a rate $1- O(n^{-1/5}\log n)$.Comment: 19 page

Ionaşcu, Eugen J. - One of the best experts on this subject based on the ideXlab platform.

  • Random triangles in planar regions
    CSU ePress, 2019
    Co-Authors: Ionaşcu, Eugen J.
    Abstract:

    © 2018, Springer-Verlag Italia S.r.l., part of Springer Nature. In this article we provide several exact formulae to calculate the probability that a random triangle chosen within a planar region (any Lebesgue Measurable Set of finite measure) contains a given fixed point O. These formulae are in terms of one integration of an appropriate function, with respect to a density function which depends of the point O. The formulae provide another way to approach the Sylvester’s four-point problem. A stability result is derived for the probability. We recover the known probability in the case of an equilateral triangle and its center of mass: 227+20ln281 (Halász and Kleitman in Stud Appl Math 53:225–237, 1974; Prékopa in Period Math Hung 2:259–282, 1972). We compute this probability in the case of a regular polygon and its center of mass for the point O. Other families of regions are studied. For the family of Limaçons r= a+ cos t, a\u3e 1 , and O the origin of the polar coordinates, the probability is 14-12a2(4a2+1)(2a2+1)3π2

  • Random triangles in planar regions containing a fixed point
    2018
    Co-Authors: Ionaşcu, Eugen J.
    Abstract:

    In this article we provide several exact formulae to calculate the probability that a random triangle chosen within a planar region (any Lebesgue Measurable Set of finite measure) contains a given fixed point $O$. These formulae are in terms of one integration of an appropriate function, with respect to a density function which depends of the point $O$. The formulae provide another way to approach the Sylvester's Four-Point Problem as we show in the last section. A stability result is derived for the probability. We recover the known probability in the case of an equilateral triangle and its center of mass: $\frac{2}{27}+20\frac{\ln 2}{81}$. We compute this probability in the case of a regular polygon and its center of mass for the point $O$. Other families of regions are studied. For the family of Lima\c{c}ons $r=a+\cos t$, $a>1$, and $O$ the origin of the polar coordinates, the probability is $\frac{1}{4}-\frac{12a^2(4a^2+1)}{(2a^2+1)^3\pi^2}$.Comment: 25 pages, 12 figue

Kovač Vjekoslav - One of the best experts on this subject based on the ideXlab platform.

  • The density of Sets containing large similar copies of finite Sets
    2021
    Co-Authors: Falconer Kenneth, Yavicoli Alexia, Kovač Vjekoslav
    Abstract:

    Funding: VK is supported by the Croatian Science Foundation, project n◦ UIP-2017-05-4129 (MUNHANAP). AY is supported by the Swiss National Science Foundation, grant n◦ P2SKP2 184047.We prove that if E⊆Rd (d≥2) is a Lebesgue-Measurable Set with density larger than n−2n−1, then E contains similar copies of every n-point Set P at all sufficiently large scales. Moreover, 'sufficiently large' can be taken to be uniform over all P with prescribed size, minimum separation and diameter. On the other hand, we construct an example to show that the density required to guarantee all large similar copies of n-point Sets tends to 1 at a rate 1−O(n−1/5log n).PreprintPeer reviewe

  • The density of Sets containing large similar copies of finite Sets
    2020
    Co-Authors: Falconer Kenneth, Kovač Vjekoslav, Yavicoli Alexia
    Abstract:

    We prove that if $E \subSeteq \mathbb{R}^d$ ($d\geq 2$) is a Lebesgue-Measurable Set with density larger than $\frac{n-2}{n-1}$, then $E$ contains similar copies of every $n$-point Set $P$ at all sufficiently large scales. Moreover, `sufficiently large' can be taken to be uniform over all $P$ with prescribed size, minimum separation and diameter. On the other hand, we construct an example to show that the density required to guarantee all large similar copies of $n$-point Sets tends to $1$ at a rate $1- O(n^{-1/5}\log n)$.Comment: 19 page

Falconer Kenneth - One of the best experts on this subject based on the ideXlab platform.

  • The density of Sets containing large similar copies of finite Sets
    2021
    Co-Authors: Falconer Kenneth, Yavicoli Alexia, Kovač Vjekoslav
    Abstract:

    Funding: VK is supported by the Croatian Science Foundation, project n◦ UIP-2017-05-4129 (MUNHANAP). AY is supported by the Swiss National Science Foundation, grant n◦ P2SKP2 184047.We prove that if E⊆Rd (d≥2) is a Lebesgue-Measurable Set with density larger than n−2n−1, then E contains similar copies of every n-point Set P at all sufficiently large scales. Moreover, 'sufficiently large' can be taken to be uniform over all P with prescribed size, minimum separation and diameter. On the other hand, we construct an example to show that the density required to guarantee all large similar copies of n-point Sets tends to 1 at a rate 1−O(n−1/5log n).PreprintPeer reviewe

  • The density of Sets containing large similar copies of finite Sets
    2020
    Co-Authors: Falconer Kenneth, Kovač Vjekoslav, Yavicoli Alexia
    Abstract:

    We prove that if $E \subSeteq \mathbb{R}^d$ ($d\geq 2$) is a Lebesgue-Measurable Set with density larger than $\frac{n-2}{n-1}$, then $E$ contains similar copies of every $n$-point Set $P$ at all sufficiently large scales. Moreover, `sufficiently large' can be taken to be uniform over all $P$ with prescribed size, minimum separation and diameter. On the other hand, we construct an example to show that the density required to guarantee all large similar copies of $n$-point Sets tends to $1$ at a rate $1- O(n^{-1/5}\log n)$.Comment: 19 page