Related papers: Pi in the Mandelbrot set everywhere
This article presents the exact solution of fixed points functions for the cycle of period four of the quadratic recurrence equations. The solution is demonstrated for the quadratic map and the logistic map. These recurrence equations,…
We investigate the arithmetic properties of the coefficients for the normalized conformal mapping of the exterior of the Multibrot set. In this paper, an estimate of the prime factor of the denominator of these coefficients is given.…
In 1928, Jarn\'{\i}k \cite{Jar} obtained that the set of continued fractions with bounded coefficients has Hausdorff dimension one. Good \cite{Goo} observed a dimension drop phenomenon by proving that the Hausdorff dimension of the set of…
B. Mandelbrot gave a new birth to the notions of scale invariance, selfsimilarity and non-integer dimensions, gathering them as the founding corner-stones used to build up fractal geometry. The first purpose of the present contribution is…
Constraint Satisfaction Problems are ubiquitous in fields ranging from the physics of solids to artificial intelligence. In many cases, such systems undergo a transition when the ratio of constraints to variables reaches some value…
In 1992, Bollob\'as and Meir showed that for every $k \geq 1$ there exists a constant $c_k$ such that, for any $n$ points in the $k$-dimensional unit cube $[0, 1]^k$, one can find a tour $x_1, \dots, x_n$ through these $n$ points with…
We prove that if $B$ is a set of $N$ positive integers such that $B\cdot B$ contains an arithmetic progression of length $M$, then for some absolute $C > 0$, $$ \pi(M) + C \frac {M^{2/3}}{\log^2 M} \leq N, $$ where $\pi$ is the prime…
Given a sequence of complex numbers {\rho}_n, we study the asymptotic distribution of the sets of parameters c {\epsilon} C such that the quadratic maps z^2 +c has a cycle of period n and multiplier {\rho}_n. Assume 1/n.log|{\rho}_n| tends…
Based on $(4.48 \pm 0.03) \times 10^{8}$ $\psi(3686)$ events collected with the BESIII detector, five $h_c$ hadronic decays are searched for via process $\psi(3686) \to \pi^0 h_c$. Three of them, $h_c \to p \bar{p} \pi^+ \pi^-$, $\pi^+…
Let S $\subseteq$ N be a numerical semigroup with multiplicity m = min(S \ {0}), conductor c = max(N \ S) + 1 and minimally generated by e elements. Let L be the set of elements of S which are smaller than c. Wilf conjectured in 1978 that…
The product version of the 1-2-3 Conjecture, introduced by Skowronek-Kazi{\'o}w in 2012, states that, a few obvious exceptions apart, all graphs can be 3-edge-labelled so that no two adjacent vertices get incident to the same product of…
In the year 1982, John Chollet conjectured that, for any pair of $n\times n$ positive semidefinite matrices $A,B$, $per(A)\cdot per(B)\geq per(A\circ B)$, where $A\circ B$ is the Hardamard product of $A$ and $B$. This conjecture was proved…
A conjecture of Ulam states that the standard probability measure $\pi$ on the Hilbert cube $I^\omega$ is invariant under the induced metric $d_a$ when the sequence $a = \{ a_i \}$ of positive numbers satisfies the condition…
We prove a special case of the Dynamical Andre-Oort Conjecture formulated by Baker and DeMarco. For any integer d>1, we show that for a rational plane curve C parametrized by (t, h(t)) for some non-constant polynomial h with complex…
A fundamental challenge within the metric theory of continued fractions involves quantifying sets of real numbers, when represented using continued fractions, exhibit partial quotients that grow at specific rates. For any positive function…
We show that among any $n$ points in the unit cube one can find a triangle of area at most $n^{-2/3-c}$ for some absolute constant $c >0$. This gives the first non-trivial upper bound for the three-dimensional version of Heilbronn's…
Let {f_t} be any algebraic family of rational maps of a fixed degree, with a marked critical point c(t). We first prove that the hypersurfaces of parameters for which c(t) is periodic converge as a sequence of positive closed (1,1) currents…
We prove that the Hausdorff dimension of the set $\mathbf{x}\in [0,1)^d$, such that $$ \left|\sum_{n=1}^N \exp\left(2 \pi i\left(x_1n+\ldots+x_d n^d\right)\right) \right|\ge c N^{1/2} $$ holds for infinitely many natural numbers $N$, is at…
A family of sets is called union-closed if whenever $A$ and $B$ are sets of the family, so is $A\cup B$. The long-standing union-closed conjecture states that if a family of subsets of $[n]$ is union-closed, some element appears in at least…
The Mandelbrot set is an extremely well-known mathematical object that can be described in a quite simple way but has very interesting and non-trivial properties. This paper surveys some results that are known concerning the…