Related papers: Ptolemy Constants as Described by Eccentricity

Let $x_i$, $i\in\mathbb{Z}$ be a sequence of i.i.d. standard normal random variables. Consider rectangular Toeplitz $\mathbf{X}=\left(x_{j-i}\right)_{1\leq i\leq p,1\leq j\leq n}$ and circulant $\mathbf{X}=\left(x_{(j-i)\mod…

We study the properties of the ternary infinite word p = 012102101021012101021012 ... , that is, the fixed point of the map h:0->01, 1->21, 2->0. We determine its factor complexity, critical exponent, and prove that it is 2-balanced. We…

We consider the PDE $-\Delta_pu=\rho$, where $\rho$ is a signed Borel measure on $\mathbb{R}^n$. For each $p>n$, we characterize solutions as extremals of a generalized Morrey inequality determined by $\rho$.

We introduce the notion of $P_{\lambda}$ points, which canonically parametrize points on the Euler line. This allows us to show that the Euler line of any $d$-dimensional inscribed polygon in Euclidean space arises from the Euler lines of…

The eccentricity of a vertex $v$ in a graph $G$ is the maximum distance between $v$ and any other vertex of $G$. The diameter of a graph $G$ is the maximum eccentricity of a vertex in $G$. The eccentric connectivity index of a connected…

We explain how, given a plane algebraic curve $\mathcal{C}\colon f(x,y) = 0$, $x_1 \in \mathbb{C}$ not a singularity of $y$ w.r.t. $x$, and $\varepsilon > 0$, we can compute $\delta > 0$ such that $|y_j(x_1) - y_j(x_2)| < \varepsilon$ for…

The symmetry of polygons can be characterized by the number of symmetry axes they have. For $n$-polygons with $p$ or $p^2$ vertices $p\geq3$ there exist few symmetry categories, depending from the number of symmetry-axes the have. Further…

Given a linear recurrence of the form $c_n=a_1c_{n-1}+\cdots+a_j c_{n-j}$, it is well-known that $c_n=\sum_{r}p_r(n)r^n$, where the sum is taken over the set of characteristic roots and each $p_r(n)$ is some polynomial. We give a closed…

Let $P$ be a polytope defined by the system $A x \leq b$, where $A \in R^{m \times n}$, $b \in R^m$, and $\text{rank}(A) = n$. We give a short geometric proof of the following tight upper bound on the number of vertices of $P$: $$ n! \cdot…

For orthogonal polynomials defined by compact Jacobi matrix with exponential decay of the coefficients, precise properties of orthogonality measure is determined. This allows showing uniform boundedness of partial sums of orthogonal…

The classical Euler's problem on stationary configurations of elastic rod with fixed endpoints and tangents at the endpoints is considered as a left-invariant optimal control problem on the group of motions of a two-dimensional plane…

The values of the determinant of Vandermonde matrices with real elements are analyzed both visually and analytically over the unit sphere in various dimensions. For three dimensions some generalized Vandermonde matrices are analyzed…

J. J. Sylvester's four-point problem asks for the probability that four points chosen uniformly at random in the plane have a triangle as their convex hull. Using a combinatorial classification of points in the plane due to Goodman and…

Given $n$ charges interacting with each other according to Feynman's law. Let $(r_j(t),v_j(t))$ denote the position and velocity of the charge $q_j.$ The list $y(t)$ of all such vectors is called a trajectory. A Lipschitzian trajectory…

Let $d\geq 3$ be a fixed integer and $A$ be the adjacency matrix of a random $d$-regular directed or undirected graph on $n$ vertices. We show there exist constants $\mathfrak d>0$, \begin{align*} {\mathbb P}(\text{$A$ is singular in…

Let $E$ be an elliptic curve defined over a number field $k$ and $\ell$ a prime integer. When $E$ has at least one $k$-rational point of exact order $\ell$, we derive a uniform upper bound $\exp(C \log B / \log \log B)$ for the number of…

Let E be an elliptic curve over the number field Q. In 1988, Koblitz conjectured an asymptotic for the number of primes p for which the cardinality of the group of F_p-points of E is prime. However, the constant occurring in his asymptotic…

We consider an isolated plane curve singularity and its associated Eisenbud an Neumann diagram. We give an algorithm to compute the maximal spectral value on the diagram and we show that the singularity is topologically equivalent to…

Let $g(f)$ denote the maximum of the differences (gaps) between two consecutive exponents occurring in a polynomial $f$. Let $\Phi_n$ denote the $n$-th cyclotomic polynomial and let $\Psi_n$ denote the $n$-th inverse cyclotomic polynomial.…

We prove that there are only finitely many complex numbers $a$ and $b$ with $4a^3+27b^2\not=0$ such that the three points $(1,*),(2,*),$ and $(3,*)$ are simultaneously torsion on the elliptic curve defined in Weierstrass form by…