On Rich Points and Incidences with Restricted Sets of Lines in 3-Space

Let $L$ be a set of $n$ lines in $R^3$ that is contained, when represented as points in the four-dimensional Pl\"ucker space of lines in $R^3$, in an irreducible variety $T$ of constant degree which is \emph{non-degenerate} with respect to $L$ (see below). We show: \medskip \noindent{\bf (1)} If $T$ is two-dimensional, the number of $r$-rich points (points incident to at least $r$ lines of $L$) is $O(n^{4/3+\epsilon}/r^2)$, for $r \ge 3$ and for any $\epsilon>0$, and, if at most $n^{1/3}$ lines of $L$ lie on any common regulus, there are at most $O(n^{4/3+\epsilon})$ $2$-rich points. For $r$ larger than some sufficiently large constant, the number of $r$-rich points is also $O(n/r)$. As an application, we deduce (with an $\epsilon$-loss in the exponent) the bound obtained by Pach and de Zeeuw (2107) on the number of distinct distances determined by $n$ points on an irreducible algebraic curve of constant degree in the plane that is not a line nor a circle. \medskip \noindent{\bf (2)} If $T$ is two-dimensional, the number of incidences between $L$ and a set of $m$ points in $R^3$ is $O(m+n)$. \medskip \noindent{\bf (3)} If $T$ is three-dimensional and nonlinear, the number of incidences between $L$ and a set of $m$ points in $R^3$ is $O\left(m^{3/5}n^{3/5} + (m^{11/15}n^{2/5} + m^{1/3}n^{2/3})s^{1/3} + m + n \right)$, provided that no plane contains more than $s$ of the points. When $s = O(\min\{n^{3/5}/m^{2/5}, m^{1/2}\})$, the bound becomes $O(m^{3/5}n^{3/5}+m+n)$. As an application, we prove that the number of incidences between $m$ points and $n$ lines in $R^4$ contained in a quadratic hypersurface (which does not contain a hyperplane) is $O(m^{3/5}n^{3/5} + m + n)$. The proofs use, in addition to various tools from algebraic geometry, recent bounds on the number of incidences between points and algebraic curves in the plane.Comment: 21 pages, one figur

Modeling local pattern formation on membrane surfaces using nonlocal interactions

2015 Spring.Includes bibliographical references.The cell membrane is of utmost importance in the transportation of nutrients and signals to the cell which are needed for survival. The magnitude of this is the inspiration for our study of the lipid bilayer which forms the cell membrane. It has been recently accepted that the lipid bilayer consists of lipid microdomains (lipid rafts), as opposed to freely moving lipids. We present two lipid raft models using the Ginzburg-Landau energy with addition of the electrostatic energy and the geodesic curvature energy to describe the local pattern formation of these lipid rafts. The development and implementation of a C⁰ interior penalty surface finite element method along with an implicit time iteration scheme will also be discussed as the optimal solution technique

NASA Tech Briefs, June 1996

Topics: New Computer Hardware; Electronic Components and Circuits; Electronic Systems; Physical Sciences; Materials; Computer Programs; Mechanics; Machinery/Automation; Manufacturing/Fabrication; Mathematics and Information Sciences;Books and Reports

PSA 2016

These preprints were automatically compiled into a PDF from the collection of papers deposited in PhilSci-Archive in conjunction with the PSA 2016

Protection Against Radiation Hazards in Space, Book I Proceedings of the Symposium at Gatlinburg, Tenn., Nov. 5-7, 1962

Protection against radiation hazards in spac