Related papers: Generalization of Apollonius Circle

In this note we establish a relation between two exactly-solvable problems on circle, namely singular Coulomb and singular oscillator systems.

Let G be a group acting on the plane by orientation-preserving homeomorphisms. We show that if for some k>0 there is a ball of radius r > k/\sqrt{3} such that each point x in the ball satisfies |gx -hx| < k for all g, h in G, and the action…

A variational principle is applied to 4D Euclidean space provided with a tensor refractive index, defining what can be seen as 4-dimensional optics (4DO). The geometry of such space is analysed, making no physical assumptions of any kind.…

An orbifold is a topological space modeled on quotient spaces of a finite group actions. We can define the universal cover of an orbifold and the fundamental group as the deck transformation group. Let $G$ be a Lie group acting on a space…

Given a ternary homogeneous polynomial, the fixed points of the map from $\mathbb{P}^2$ to itself defined by its gradient are called its eigenpoints. We focus on cubic polynomials, and analyze configurations of eigenpoints that admit one or…

Two linear recurrences exhibit mirror symmetry connecting the constants $e$ and $\pi$. When parametrized, their asymptotic connection constants extend to meromorphic functions satisfying additive functional equations with rational…

We observe that a sequence satisfies Lucas congruences modulo $p$ if and only if its values modulo $p$ can be described by a linear $p$-scheme, as introduced by Rowland and Zeilberger, with a single state. This simple observation suggests…

We consider a polynomial version of the Cayley numbers. Namely, we define the ring of Cayley polynomials in terms of generators and relations in the category of alternative algebras. The ring turns out to be an octonion algebra over an…

We consider two families of polynomials $\mathbb{P}=\polP$ and $\mathbb{Q}=\polQ$\footnote{Here and below we consider only monic polynomials.} orthogonal on the real line with respect to probability measures $\mu$ and $\nu$ respectively.…

Let $S_b(n)$ denote the sum of the squares of the digits of the positive integer $n$ in base $b\geq2$. It is well-known that the sequence of iterates of $S_b(n)$ terminates in a fixed point or enters a cycle. Let $N=2n-1$, $n\geq2$. It is…

Let B be a ring and $A=B[X,Y]/(aX^2+bXY+cY^2-1)$ where $a,b,c\in B$. We study the smoothness of A over B, and the regularity of B when B is a ring of algebraic integers.

It was once conjectured that if $A$ is a uniform algebra on its maximal ideal space $X$, and if each point of $X$ is a peak point for $A$, then $A = C(X)$. This peak-point conjecture was disproved by Brian Cole in 1968. Here we establish a…

In this paper we study the generalized Bessel polynomials $y_n(x,a,b)$ (in the notation of Krall and Frink). Let $a>1$, $b\in(-1/3,1/3)\backslash\{ 0\}$. In this case we present the following positive continuous weights $p(\theta) =…

This article proves a Pythagoras-type formula for the sides and diagonals of a polygon inscribed in a semicircle having one of the sides of the polygon as diameter.

Integer geometry on a plane deals with objects whose vertices are points in $\mathbb Z^2$. The congruence relation is provided by all affine transformations preserving the lattice $\mathbb Z^2$. In this paper we study circumscribed circles…

We provide a new closed form expression for the Geronimus polynomials on the unit circle and use it to obtain new results and formulas. Among our results is a universality result at an endpoint of an arc for polynomials orthogonal with…

A group is called square-like if it is universally equivalent to its direct square. It is known that the class of all square-like groups admits an explicit first order axiomatization but its theory is undecidable. We prove that the theory…

Let $K$ be an algebraically closed field and $\mathrm{M}(2,K)$ be the $2\times 2$ matrix algebra over $K$ and $\mathrm{GL}(2,K)$ be the invertible elements in $\mathrm{M}(2,K)$. We explore the image of polynomials with constants, namely…

There exists a growing evidence for the anomalous transparency of the Universe for energetic gamma rays. Popular explanations include conversion of photons into hypothetical axion-like particles (ALPs) and back in astrophysical magnetic…

In this short notice we give a universal definition of $\Z_2$-module $Or(\Gamma)$ of orientations of a graph $\Gamma$ and construct a method, by means of which one can easily verify whenever two such special definitions coincide.