When we stare into our complex surroundings, see objects of different shapes and sizes. However, the shape that is always present, regardless complexity object, circle. The circle arguably most fascinating in universe. A defined as set all points equidistant from a given point, which, therefore, lies at center One various properties circles it has infinite inscribed squares. This because contin...

We consider the family of 3D minimal polyominoes inscribed in a rectanglar prism. These objects are polyominos and so they are connected sets of unitary cubic cells inscribed in a given rectangular prism of size b×k×h and of minimal volume equal to b + k + h − 2. They extend the concept of minimal 2D polyominoes inscribed in a rectangle studied in a previous work. Using their geometric structur...

In this note we prove that, if Sn is the greatest area of a rectangle which can be covered with n unit disks, then 2 ≤ Sn/n < 3 √ 3/2, and these are the best constants; moreover, for ∆(n) := (3 √ 3/2)n − Sn, we have 0.727384 < lim inf ∆(n)/ √ n < 2.121321 and 0.727384 < lim sup∆(n)/ √ n < 4.165064. The problem of covering sets in the plane with figures of prescribed shape has been extensively s...

If each four spheres in a set of five unit spheres in R have nonempty intersection, then all five spheres have nonempty intersection. This result is proved using Grace’s theorem: the circumsphere of a tetrahedron encloses none of its escribed spheres. This paper provides self-contained proofs of these results; including Schläfli’s double six theorem and modified version of Lie’s line-sphere tra...

For many software applications, it is sometimes necessary to find the rectangle of largest area inscribed in a polygon, in any possible direction. Thus, given a closed contour C, we consider approximation algorithms for the problem of finding the largest area rectangle of arbitrary orientation that is fully contained in C. Furthermore, we compute the largest area rectangle of arbitrary orientat...

We prove that for every smooth Jordan curve $\gamma$, if $X$ is the set of all $r\in [0,1]$ so there an inscribed rectangle in $\gamma$ aspect ratio $\mathrm{tan}(r\cdot \pi/4)$, then Lebesgue measure at least $1/3$. To do this, we study sets disjoint homologically nontrivial projective planes smoothly embedded $\mathbb{R}\times \mathbb{R}P^3$. any such can be equipped with a natural total orde...

We investigate the switching statistics dependence on cell geometry by means of systematic micromagnetic simulations. We find that MTJs with a free layer composed of two ellipses with the axes a/2 and b inscribed into a rectangle a × b are characterized by the same switching speed and thermal stability as MTJs with a composite free layer (C-MTJs). As has been shown, the C-MTJs demonstrate a sub...

We investigate a possibility to use composite magnetic tunnel junction structures (MTJs) to compose fast memory devices and efficient spin-torque nano-oscillators. In terms of magnetic memory, we study the switching statistics dependence on memory cell geometry by means of systematic micromagnetic simulations. We find that MTJs with a free layer composed of two ellipses with the axes a/2 > b in...

We study point-sphere and point-plane incidences in three-dimensional space. In particular, for 1 < k < n, we say that a set of spheres (resp., planes) is k-non-degenerate if no circle is contained in k spheres of the set (resp., if no line is contained in k planes of the set). We prove that, for every ε > 0, the number of incidences between a set of m points and a k-non-degenerate set of n sph...

This paper presents an implementation method for the people counting system which detects and tracks moving people using a fixed single camera. The main contribution of this paper is the novel head detection method based on body’s geometry. A novel body descriptor is proposed for finding people’s head which is defined as Body Feature Rectangle (BFR). First, a vertical projection method is used ...