Periodic Planar Disk Packings

Several conditions are given when a packing of equal disks in a torus is locally maximally dense, where the torus is defined as the quotient of the plane by a two-dimensional lattice. Conjectures are presented that claim that the density of any strictly jammed packings, whose graph does not consist of all triangles and the torus lattice is the standard triangular lattice, is at most $\frac{n}{n+1}\frac{\pi}{\sqrt{12}}$, where $n$ is the number of packing disks. Several classes of collectively jammed packings are presented where the conjecture holds.Comment: 26 pages, 13 figure

Interpretation of some Yb-based valence-fluctuating crystals as approximants to a dodecagonal quasicrystal

The hexagonal ZrNiAl-type (space group: P-62m) and the tetragonal Mo2FeB2-type (space group: P4/mbm) structures, which are frequently formed in the same Yb-based alloys and exhibit physical properties related to valence-fluctuation, can be regarded as approximants of a hypothetical dodecagonal quasicrystal. Using Pd-Sn-Yb system as an example, a model of quasicrystal structure has been constructed, of which 5-dimensional crystal (space group: P12/mmm, aDD=5.66 {\AA} and c=3.72 {\AA}) consists of four types of acceptance regions located at the following crystallographic sites; Yb [00000], Pd[1/3 0 1/3 0 1/2], Pd[1/3 1/3 1/3 1/3 0] and Sn[1/2 00 1/2 1/2]. In the 3-dimensional space, the quasicrystal is composed of three types of columns, of which c-projections correspond to a square, an equilateral triangle and a 3-fold hexagon. They are fragments of two known crystals, the hexagonal {\alpha}-YbPdSn and the tetragonal Yb2Pd2Sn structures. The model of the hypothetical quasicrystal may be applicable as a platform to treat in a unified manner the heavy fermion properties in the two types of Yb-based crystals.Comment: 19 pages, 6 figure

n2+1n^2 + 1 unit equilateral triangles cannot cover an equilateral triangle of side >n> n if all triangles have parallel sides

Conway and Soifer showed that an equilateral triangle $T$ of side $n + \varepsilon$ with sufficiently small $\varepsilon > 0$ can be covered by $n^2 + 2$ unit equilateral triangles. They conjectured that it is impossible to cover $T$ with $n^2 + 1$ unit equilateral triangles no matter how small $\varepsilon$ is. We show that if we require all sides of the unit equilateral triangles to be parallel to the sides of $T$ (e.g. $\bigtriangleup$ and $\bigtriangledown$), then it is impossible to cover $T$ of side $n + \varepsilon$ with $n^2 + 1$ unit equilateral triangles for any $\varepsilon > 0$. As the coverings of $T$ by Conway and Soifer only involve triangles with sides parallel to $T$, our result determines the exact minimum number $n^2+2$ of unit equilateral triangles with all sides parallel to $T$ that cover $T$. We also determine the largest value $\varepsilon = 1/(n + 1)$ (resp. $\varepsilon = 1 / n$) of $\varepsilon$ such that the equilateral triangle $T$ of side $n + \varepsilon$ can be covered by $n^2+2$ (resp. $n^2 + 3$) unit equilateral triangles with sides parallel to $T$, where the first case is achieved by the construction of Conway and Soifer.Comment: 8 pages, 7 figure

The maximum number of systoles for genus two Riemann surfaces with abelian differentials

In this article, we provide bounds on systoles associated to a holomorphic $1$-form $\omega$ on a Riemann surface $X$. In particular, we show that if $X$ has genus two, then, up to homotopy, there are at most $10$ systolic loops on $(X,\omega)$ and, moreover, that this bound is realized by a unique translation surface up to homothety. For general genus $g$ and a holomorphic 1-form $\omega$ with one zero, we provide the optimal upper bound, $6g-3$, on the number of homotopy classes of systoles. If, in addition, $X$ is hyperelliptic, then we prove that the optimal upper bound is $6g-5$.Comment: 41 page

Full-View Coverage Problems in Camera Sensor Networks

Camera Sensor Networks (CSNs) have emerged as an information-rich sensing modality with many potential applications and have received much research attention over the past few years. One of the major challenges in research for CSNs is that camera sensors are different from traditional scalar sensors, as different cameras from different positions can form distinct views of the object in question. As a result, simply combining the sensing range of the cameras across the field does not necessarily form an effective camera coverage, since the face image (or the targeted aspect) of the object may be missed. The angle between the object\u27s facing direction and the camera\u27s viewing direction is used to measure the quality of sensing in CSNs instead. This distinction makes the coverage verification and deployment methodology dedicated to conventional sensor networks unsuitable. A new coverage model called full-view coverage can precisely characterize the features of coverage in CSNs. An object is full-view covered if there is always a camera to cover it no matter which direction it faces and the camera\u27s viewing direction is sufficiently close to the object\u27s facing direction. In this dissertation, we consider three areas of research for CSNS: 1. an analytical theory for full-view coverage; 2. energy efficiency issues in full-view coverage CSNs; 3. Multi-dimension full-view coverage theory. For the first topic, we propose a novel analytical full-view coverage theory, where the set of full-view covered points is produced by numerical methodology. Based on this theory, we solve the following problems. First, we address the full-view coverage holes detection problem and provide the healing solutions. Second, we propose $k$-Full-View-Coverage algorithms in camera sensor networks. Finally, we address the camera sensor density minimization problem for triangular lattice based deployment in full-view covered camera sensor networks, where we argue that there is a flaw in the previous literature, and present our corresponding solution. For the second topic, we discuss lifetime and full-view coverage guarantees through distributed algorithms in camera sensor networks. Another energy issue we discuss is about object tracking problems in full-view coverage camera sensor networks. Next, the third topic addresses multi-dimension full-view coverage problem where we propose a novel 3D full-view coverage model, and we tackle the full-view coverage optimization problem in order to minimize the number of camera sensors and demonstrate a valid solution. This research is important due to the numerous applications for CSNs. Especially some deployment can be in remote locations, it is critical to efficiently obtain accurate meaningful data