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Jenő Szirmai - One of the best experts on this subject based on the ideXlab platform.

  • Geodesic Ball packings generated by regular prism tilings in $\mathbf{Nil}$ geometry
    arXiv: Metric Geometry, 2016
    Co-Authors: Benedek Schultz, Jenő Szirmai
    Abstract:

    In this paper we study the regular prism tilings and construct Ball packings by Geodesic Balls related to the above tilings in the projective model of $\mathbf{Nil}$ geometry. Packings are generated by action of the discrete prism groups $\mathbf{pq2_{1}}$. We prove that these groups are realized by prism tilings in $\mathbf{Nil}$ space if $(p,q)=(3,6), (4,4), (6,3)$ and determine packing density formulae for Geodesic Ball packings generated by the above prism groups. Moreover, studying these formulae we determine the conjectured maximal dense packing arrangements and their densities and visualize them in the projective model of $\mathbf{Nil}$ geometry. We get a dense (conjectured locally densest) Geodesic Ball arrangement related to the parameters $(p,q)=(6,3)$ where the kissing number of the packing is $14$, similarly to the densest lattice-like $\mathbf{Nil}$ Geodesic Ball arrangement investigated by the second author .

  • Geodesic Ball packings generated by regular prism tilings in mathbf nil geometry
    arXiv: Metric Geometry, 2016
    Co-Authors: Benedek Schultz, Jenő Szirmai
    Abstract:

    In this paper we study the regular prism tilings and construct Ball packings by Geodesic Balls related to the above tilings in the projective model of $\mathbf{Nil}$ geometry. Packings are generated by action of the discrete prism groups $\mathbf{pq2_{1}}$. We prove that these groups are realized by prism tilings in $\mathbf{Nil}$ space if $(p,q)=(3,6), (4,4), (6,3)$ and determine packing density formulae for Geodesic Ball packings generated by the above prism groups. Moreover, studying these formulae we determine the conjectured maximal dense packing arrangements and their densities and visualize them in the projective model of $\mathbf{Nil}$ geometry. We get a dense (conjectured locally densest) Geodesic Ball arrangement related to the parameters $(p,q)=(6,3)$ where the kissing number of the packing is $14$, similarly to the densest lattice-like $\mathbf{Nil}$ Geodesic Ball arrangement investigated by the second author .

  • densest Geodesic Ball packings to s 2 r space groups generated by screw motions
    Mediterranean Journal of Mathematics, 2016
    Co-Authors: Benedek Schultz, Jenő Szirmai
    Abstract:

    In this paper we study the locally optimal Geodesic Ball packings with equal Balls to the S 2 × R space groups having rotation point groups and their generators are screw motions. We determine and visualize the densest simply transitive Geodesic Ball arrangements for the above space groups; moreover, we compute their optimal densities and radii. The densest packing is derived from the S 2 × R space group 3qe. I. 3 with packing density ≈0.7278. E. Molnar has shown in [9] that the Thurston geometries have an unified interpretation in the real projective 3-sphere \({\mathcal{PS}^3}\). In our work we shall use this projective model of S 2 × R geometry.

  • Densest Geodesic Ball Packings to S ^2 × R space groups generated by screw motions
    Mediterranean Journal of Mathematics, 2016
    Co-Authors: Benedek Schultz, Jenő Szirmai
    Abstract:

    In this paper we study the locally optimal Geodesic Ball packings with equal Balls to the S ^2 ×  R space groups having rotation point groups and their generators are screw motions. We determine and visualize the densest simply transitive Geodesic Ball arrangements for the above space groups; moreover, we compute their optimal densities and radii. The densest packing is derived from the S ^2 ×  R space group 3qe. I. 3 with packing density ≈0.7278. E. Molnár has shown in [ 9 ] that the Thurston geometries have an unified interpretation in the real projective 3-sphere $${\mathcal{PS}^3}$$ PS 3 . In our work we shall use this projective model of S ^2 ×  R geometry.

  • Densest Geodesic Ball packings to S 2 ×R space groups generated by screw motions ∗
    Mediterranean Journal of Mathematics, 2015
    Co-Authors: Benedek Schultz, Jenő Szirmai
    Abstract:

    In this paper we study the locally optimal Geodesic Ball packings with equal Balls to the S 2 × R space groups having rotation point groups and their generators are screw motions. We determine and visualize the densest simply transitive Geodesic Ball arrangements for the above space groups; moreover, we compute their optimal densities and radii. The densest packing is derived from the S 2 × R space group 3qe. I. 3 with packing density ≈0.7278. E. Molnar has shown in [9] that the Thurston geometries have an unified interpretation in the real projective 3-sphere \({\mathcal{PS}^3}\). In our work we shall use this projective model of S 2 × R geometry.

Benedek Schultz - One of the best experts on this subject based on the ideXlab platform.

  • Geodesic Ball packings generated by regular prism tilings in $\mathbf{Nil}$ geometry
    arXiv: Metric Geometry, 2016
    Co-Authors: Benedek Schultz, Jenő Szirmai
    Abstract:

    In this paper we study the regular prism tilings and construct Ball packings by Geodesic Balls related to the above tilings in the projective model of $\mathbf{Nil}$ geometry. Packings are generated by action of the discrete prism groups $\mathbf{pq2_{1}}$. We prove that these groups are realized by prism tilings in $\mathbf{Nil}$ space if $(p,q)=(3,6), (4,4), (6,3)$ and determine packing density formulae for Geodesic Ball packings generated by the above prism groups. Moreover, studying these formulae we determine the conjectured maximal dense packing arrangements and their densities and visualize them in the projective model of $\mathbf{Nil}$ geometry. We get a dense (conjectured locally densest) Geodesic Ball arrangement related to the parameters $(p,q)=(6,3)$ where the kissing number of the packing is $14$, similarly to the densest lattice-like $\mathbf{Nil}$ Geodesic Ball arrangement investigated by the second author .

  • Geodesic Ball packings generated by regular prism tilings in mathbf nil geometry
    arXiv: Metric Geometry, 2016
    Co-Authors: Benedek Schultz, Jenő Szirmai
    Abstract:

    In this paper we study the regular prism tilings and construct Ball packings by Geodesic Balls related to the above tilings in the projective model of $\mathbf{Nil}$ geometry. Packings are generated by action of the discrete prism groups $\mathbf{pq2_{1}}$. We prove that these groups are realized by prism tilings in $\mathbf{Nil}$ space if $(p,q)=(3,6), (4,4), (6,3)$ and determine packing density formulae for Geodesic Ball packings generated by the above prism groups. Moreover, studying these formulae we determine the conjectured maximal dense packing arrangements and their densities and visualize them in the projective model of $\mathbf{Nil}$ geometry. We get a dense (conjectured locally densest) Geodesic Ball arrangement related to the parameters $(p,q)=(6,3)$ where the kissing number of the packing is $14$, similarly to the densest lattice-like $\mathbf{Nil}$ Geodesic Ball arrangement investigated by the second author .

  • densest Geodesic Ball packings to s 2 r space groups generated by screw motions
    Mediterranean Journal of Mathematics, 2016
    Co-Authors: Benedek Schultz, Jenő Szirmai
    Abstract:

    In this paper we study the locally optimal Geodesic Ball packings with equal Balls to the S 2 × R space groups having rotation point groups and their generators are screw motions. We determine and visualize the densest simply transitive Geodesic Ball arrangements for the above space groups; moreover, we compute their optimal densities and radii. The densest packing is derived from the S 2 × R space group 3qe. I. 3 with packing density ≈0.7278. E. Molnar has shown in [9] that the Thurston geometries have an unified interpretation in the real projective 3-sphere \({\mathcal{PS}^3}\). In our work we shall use this projective model of S 2 × R geometry.

  • Densest Geodesic Ball Packings to S ^2 × R space groups generated by screw motions
    Mediterranean Journal of Mathematics, 2016
    Co-Authors: Benedek Schultz, Jenő Szirmai
    Abstract:

    In this paper we study the locally optimal Geodesic Ball packings with equal Balls to the S ^2 ×  R space groups having rotation point groups and their generators are screw motions. We determine and visualize the densest simply transitive Geodesic Ball arrangements for the above space groups; moreover, we compute their optimal densities and radii. The densest packing is derived from the S ^2 ×  R space group 3qe. I. 3 with packing density ≈0.7278. E. Molnár has shown in [ 9 ] that the Thurston geometries have an unified interpretation in the real projective 3-sphere $${\mathcal{PS}^3}$$ PS 3 . In our work we shall use this projective model of S ^2 ×  R geometry.

  • Densest Geodesic Ball packings to S 2 ×R space groups generated by screw motions ∗
    Mediterranean Journal of Mathematics, 2015
    Co-Authors: Benedek Schultz, Jenő Szirmai
    Abstract:

    In this paper we study the locally optimal Geodesic Ball packings with equal Balls to the S 2 × R space groups having rotation point groups and their generators are screw motions. We determine and visualize the densest simply transitive Geodesic Ball arrangements for the above space groups; moreover, we compute their optimal densities and radii. The densest packing is derived from the S 2 × R space group 3qe. I. 3 with packing density ≈0.7278. E. Molnar has shown in [9] that the Thurston geometries have an unified interpretation in the real projective 3-sphere \({\mathcal{PS}^3}\). In our work we shall use this projective model of S 2 × R geometry.

Azita Mayeli - One of the best experts on this subject based on the ideXlab platform.

  • Nearly tight frames of spin wavelets on the sphere
    2010
    Co-Authors: Daryl Geller, Azita Mayeli
    Abstract:

    We show that the spin wavelets on the sphere S 2 , which were constructed by the first author and Marinucci in [3], can be chosen so as to form a nearly tight frame. These spin wavelets can be applied (see [1] and [2]) to the study of the polarization of cosmic microwave background radiation. For certain of these frames, there is a positive C such that each frame element at scale a j is supported in a Geodesic Ball of radius Ca j .

  • Nearly tight frames of spin wavelets on the sphere
    arXiv: Functional Analysis, 2009
    Co-Authors: Daryl Geller, Azita Mayeli
    Abstract:

    We show that the spin wavelets on the sphere $S^2$, which were constructed by the first author and Marinucci in an earlier article, can be chosen so as to form a nearly tight frame. These spin wavelets can be applied to the study of the polarization of cosmic microwave background radiation. For certain of these frames, there is a positive $C$ such that each frame element at scale $a^j$ is supported in a Geodesic Ball of radius $Ca^j$.

Jan Metzger - One of the best experts on this subject based on the ideXlab platform.

  • Small Surfaces of Willmore Type in Riemannian Manifolds
    International Mathematics Research Notices, 2010
    Co-Authors: Tobias Lamm, Jan Metzger
    Abstract:

    In this paper we investigate the properties of small surfaces of Willmore type in Riemannian manifolds. By small surfaces we mean topological spheres contained in a Geodesic Ball of small enough radius. In particular, we show that if there exist such surfaces with positive mean curvature in the Geodesic Ball Br(p) for arbitrarily small radius r around a point p in the Riemannian manifold, then the scalar curvature must have a critical point at p.

  • Small surfaces of Willmore type in Riemannian manifolds
    arXiv: Differential Geometry, 2009
    Co-Authors: Tobias Lamm, Jan Metzger
    Abstract:

    In this paper we investigate the properties of small surfaces of Willmore type in Riemannian manifolds. By \emph{small} surfaces we mean topological spheres contained in a Geodesic Ball of small enough radius. In particular, we show that if there exist such surfaces with positive mean curvature in the Geodesic Ball $B_r(p)$ for arbitrarily small radius $r$ around a point $p$ in the Riemannian manifold, then the scalar curvature must have a critical point at $p$. As a byproduct of our estimates we obtain a strengthened version of the non-existence result of Mondino \cite{Mondino:2008} that implies the non-existence of certain critical points of the Willmore functional in regions where the scalar curvature is non-zero.

Jenö Szirmai - One of the best experts on this subject based on the ideXlab platform.

  • Non-periodic Geodesic Ball packings to infinite regular prism tilings in $\SLR$ space
    arXiv: Metric Geometry, 2014
    Co-Authors: Jenö Szirmai
    Abstract:

    In \cite{Sz13-1} we defined and described the {\it regular infinite or bounded} $p$-gonal prism tilings in $\SLR$ space. We proved that there exist infinitely many regular infinite $p$-gonal face-to-face prism tilings $\cT^i_p(q)$ and infinitely many regular bounded $p$-gonal non-face-to-face prism tilings $\cT_p(q)$ for integer parameters $p,q;~3 \le p$, $ \frac{2p}{p-2} < q$. Moreover, in \cite{MSz14} and \cite{MSzV13} we have determined the symmetry group of $\cT_p(q)$ via its index 2 rotational subgroup, denoted by $\mathbf{pq2_1}$ and investigated the corresponding Geodesic and translation Ball packings. In this paper we study the structure of the regular infinite or bounded $p$-gonal prism tilings, prove that the side curves of their base figurs are arcs of Euclidean circles for each parameter. Moreover, we examine the non-periodic Geodesic Ball packings of congruent regular non-periodic prism tilings derived from the regular infinite $p$-gonal face-to-face prism tilings $\cT^i_p(q)$ in $\SLR$ geometry. We develop a procedure to determine the densities of the above non-periodic optimal Geodesic Ball packings and apply this algorithm to them. We look for those parameters $p$ and $q$ above, where the packing density large enough as possible. Now, we obtain larger density $\approx 0.626606$ for $(p, q) = (29,3)$ then the maximal density of the corresponding periodical Geodesic Ball packings under the groups $\mathbf{pq2_1}$. In our work we will use the projective model of $\SLR$ introduced by E. {Moln\'ar} in \cite{M97}.

  • Volumes and Geodesic Ball packings to the regular prism tilings in $\widetilde{\mathbf{S}\mathbf{L}_2\mathbf{R}}$ space
    arXiv: Metric Geometry, 2013
    Co-Authors: Emil Molnár, Jenö Szirmai
    Abstract:

    After having investigated the regular prisms and prism tilings in the $\SLR$ space in the previous work \cite{Sz13-1} of the second author, we consider the problem of Geodesic Ball packings related to those tilings and their symmetry groups $\mathbf{pq2_1}$. $\SLR$ is one of the eight Thurston geometries that can be derived from the 3-dimensional Lie group of all $2\times 2$ real matrices with determinant one. In this paper we consider Geodesic spheres and Balls in $\SLR$ (even in $\mathbf{SL_{\mathrm{2}}R})$, if their radii $\rho\in [0, \frac{\pi}{2})$, and determine their volumes. Moreover, we consider the prisms of the above space and compute their volumes, define the notion of the Geodesic Ball packing and its density. We develop a procedure to determine the densities of the densest Geodesic Ball packings for the tilings, or in this paper more precisely, for their generating groups $\mathbf{pq2_1}$ (for integer rotational parameters $p,q$; $3\le p, \frac{2p}{p-2}

  • volumes and Geodesic Ball packings to the regular prism tilings in widetilde mathbf s mathbf l _2 mathbf r space
    arXiv: Metric Geometry, 2013
    Co-Authors: Emil Molnár, Jenö Szirmai
    Abstract:

    After having investigated the regular prisms and prism tilings in the $\SLR$ space in the previous work \cite{Sz13-1} of the second author, we consider the problem of Geodesic Ball packings related to those tilings and their symmetry groups $\mathbf{pq2_1}$. $\SLR$ is one of the eight Thurston geometries that can be derived from the 3-dimensional Lie group of all $2\times 2$ real matrices with determinant one. In this paper we consider Geodesic spheres and Balls in $\SLR$ (even in $\mathbf{SL_{\mathrm{2}}R})$, if their radii $\rho\in [0, \frac{\pi}{2})$, and determine their volumes. Moreover, we consider the prisms of the above space and compute their volumes, define the notion of the Geodesic Ball packing and its density. We develop a procedure to determine the densities of the densest Geodesic Ball packings for the tilings, or in this paper more precisely, for their generating groups $\mathbf{pq2_1}$ (for integer rotational parameters $p,q$; $3\le p, \frac{2p}{p-2} packing density large enough as possible. Now our record is $0.567362$ for $(p, q) = (8, 10)$. These computations seem to be important, since we do not know optimal Ball packing, namely in the hyperbolic space $\HYP$. We know only the density upper bound 0.85326, realized by horoBall packing of $\HYP$ to its ideal regular simplex tiling. Surprisingly, for the so-called translation Ball packings under the same groups $\mathbf{pq2_1}$ in \cite{MSzV13} we have got larger density $0.841700$ for $(p, q) = (5, 10000 \rightarrow \infty)$ close to the above upper bound. We use for the computation and visualization of the $\SLR$ space its projective model introduced by the first author in \cite{M97}.