Related papers: Collatz Cycles and $3n+c$ Cycles

We study limit cycles in piecewise complex systems with switching manifold $\mathbb{S}^1$. Using M\"obius transformations we establish an equivalence between circular and straight-line discontinuities that preserves periods, stability, and…

We study the number of limit cycles bifurcating from a piecewise quadratic system. All the differential systems considered are piecewise in two zones separated by a straight line. We prove the existence of 16 crossing limit cycles in this…

Motivated by a balanced ternary representation of the Collatz map we define the map $C_\mathbb{R}$ on the positive real numbers by setting $C_\mathbb{R}(x)=\frac{1}{2}x$ if $[x]$ is even and $C_\mathbb{R}(x)=\frac{3}{2}x$ if $[x]$ is odd,…

We study piecewise-smooth systems with three zones, $\dot{z} = f_i(z)$, $i = 1,2,3,$ whose discontinuity set $\Sigma$ consists either of a pair of parallel lines or a pair of circles tangent to each other internally or externally. Each…

In 1922, Mordell conjectured that the set of rational points on a smooth curve $C$ over $\mathbb{Q}$ with genus $g \ge 2$ is finite. This has been proved by Faltings in 1983. However, Coleman determined in 1985 an upper bound of…

This work represents an in-depth study of the structural behavior of the Collatz sequences. We consider a finite arithmetic progression with a common difference is 2 and the number of terms in the sequence is equal to 2^n . After, we…

This article introduces a new kind of number systems on $p$-adic integers which is inspired by the well-known $3n+1$ conjecture of Lothar Collatz. A $p$-adic system is a piecewise function on $\mathbb{Z}_p$ which has branches for all…

In this paper, we discuss the well known 3x+1 conjecture in form of the accelerated Collatz function T defined on the positive odd integers. We present a sequence of quotient spaces and an invertible map that are intrinsically related to…

Assuming the Riemann Hypothesis we obtain an upper bound for the moments of the Riemann zeta-function on the critical line. Our bound is nearly as sharp as the conjectured asymptotic formulae for these moments. The method extends to moments…

The cycle set of a graph $G$ is the set consisting of all sizes of cycles in $G$. Answering a conjecture of Erd\H{o}s and Faudree, Verstra\"{e}te showed that there are at most $2^{n - n^{1/10}}$ different cycle sets of graphs with $n$…

The representation of numbers in rational base $p/q$ was introduced in 2008 by Akiyama, Frougny & Sakarovitch, with a special focus on the case $p/q=3/2$. Unnoticed since then, natural questions related to representations in that specific…

Hakimi, Schmeichel, and Thomassen in 1979 conjectured that every $4$-connected planar triangulation $G$ on $n$ vertices has at least $2(n-2)(n-4)$ Hamiltonian cycles, with equality if and only if $G$ is a double wheel. In this paper, we…

We prove two results regarding cycles in properly edge-colored graphs. First, we make a small improvement to the recent breakthrough work of Alon, Pokrovskiy and Sudakov who showed that every properly edge-colored complete graph $G$ on $n$…

We obtain some new upper bounds on the maximum number $f(n)$ of edges in $n$-vertex graphs without containing cycles of length four. This leads to an asymptotically optimal bound on $f(n)$ for a broad range of integers $n$ as well as a…

The paper studies the properties of an oscillator whose Hamiltonian is $[(1+q^2)(1+p^2)]^{1/2}-1$. It can be deduced from the nonlinear theory of electrodynamics originally proposed by Max Born in 1934. The quantization of such oscillator…

It is proved that if $G$ is a $t$-tough graph of order $n$ and minimum degree $\delta$ with $t>1$ then either $G$ has a cycle of length at least $\min\{n,2\delta+4\}$ or $G$ is the Petersen graph.

Cyclic codes are a subclass of linear codes and have applications in consumer electronics, data storage systems and communication systems as they have efficient encoding and decoding algorithms. In this paper, we settle an open problem…

Let $K$ be a global function field of characteristic $p$ and degree $D$ over $\mathbb F_{p}(t)$. We consider dynamical systems over the projective line $\mathbb P^1(K)$ defined by rational maps with at most one prime of bad reduction. The…

It is well-known that global hyperbolicity implies that the Lorentzian distance is finite and continuous. By carefully analysing the causes of discontinuity of the Lorentzian distance, we show that in most other respects the finiteness and…

Recently, generalizations of the classical Three Gap Theorem to higher dimensions attracted a lot of attention. In particular, upper bounds for the number of nearest neighbor distances have been established for the Euclidean and the maximum…