Related papers: Graphs of continuous functions and fractal dimensi…

The main purpose of this paper is to prove that the positive real numbers can be decomposed into finitely many disjoint pieces which are also closed under addition and multiplication. As a byproduct of the argument we determine all the…

Let $f: B^n \rightarrow {\mathbb R}$ be a $d+1$ times continuously differentiable function on the unit ball $B^n$, with $\max_{z\in B^n} \| f(z) \|=1$. A well-known fact is that if $f$ vanishes on a set $Z\subset B^n$ with a non-empty…

Given a compact set K in the plane, which contains no triple of points forming a vertical and a horizontal segment, and a continuous real-valued map f on K, we give a construction of real-valued continuous maps of one variable g,h such that…

This work presents a new Visual Basic 6.0 application for estimating the fractal dimension of images, based on an optimized version of the box-counting algorithm. Following the attempt to separate the real information from noise, we…

For any polynomial $P \in \mathbb{C}[X_1,X_2,...,X_n]$, we describe a $\mathbb{C}$-vector space $F(P)$ of solutions of a linear system of equations coming from some algebraic partial differential equations such that the dimension of $F(P)$…

A conjecture of Sendov states that if a polynomial has all its roots in the unit disk and if $\beta$ is one of those roots, then within one unit of $\beta$ lies a root of the polynomial's derivative. If we define $r(\beta)$ to be the…

We study the regularity properties of the centered fractional maximal function $M_{\beta}$. More precisely, we prove that the map $f \mapsto |\nabla M_\beta f|$ is bounded and continuous from $W^{1,1}(\mathbb{R}^d)$ to $L^q(\mathbb{R}^d)$…

The notion of a descent polynomial, a function in enumerative combinatorics that counts permutations with specific properties, enjoys a revived recent research interest due to its connection with other important notions in combinatorics,…

We study the convergence of the parameter family of series $$V_{\alpha,\beta}(t)=\sum_{p}p^{-\alpha}\exp(2\pi i p^{\beta}t),\quad \alpha,\beta \in \mathbb{R}_{>0},\; t \in [0,1)$$ defined over prime numbers $p$, and subsequently, their…

Let $F$ be a countable family of rational functions of two variables with real coefficients. Each rational function $f\in F$ can be thought as a continuous function $f:dom(f)\to\bar R$ taking values in the projective line $\bar…

For every natural number $ n $, any continuous function on the product of $ X_1 \times X_2 \times ... \times X_n $ pseudocompact spaces extends to a separately continuous function on the product $ \beta X_1 \times \beta X_2 \times ...…

Let $f$ be a generalized affine fractal interpolation function with vertical scaling function $S$. In this paper, we study $\dim_B \Gamma f$, the box dimension of the graph of $f$, under the assumption that $S$ is a Lipschtz function. By…

We present a combinatorial characterization of the Bethe entropy function of a factor graph, such a characterization being in contrast to the original, analytical, definition of this function. We achieve this combinatorial characterization…

A univariate polynomial f over a field is decomposable if it is the composition f = g(h) of two polynomials g and h whose degree is at least 2. We determine the dimension (over an algebraically closed field) of the set of decomposables, and…

We consider the following conjecture (from Huang, et al): Let $\Delta^+$ denote the upper half disc in $\mathbb{C}$ and let $\gamma = ( - 1, 1)$ (viewed as an interval in the real axis in $\mathbb{C}$). Assume that $F$ is a holomorphic…

Bicubic maps are in bijection with \beta(0,1)-trees. We introduce two new ways of decomposing \beta(0,1)-trees. Using this we define an endofunction on \beta(0,1)-trees, and thus also on bicubic maps. We show that this endofunction is in…

There are many research papers dealing with fractal dimension of real-valued fractal functions in the recent literature. The main focus of the present paper is to study fractal dimension of complex-valued functions. This paper also…

We study several variants of decomposing a symmetric matrix into a sum of a low-rank positive semidefinite matrix and a diagonal matrix. Such decompositions have applications in factor analysis and they have been studied for many decades.…

Given $\beta\in(1,2)$, a $\beta$-expansion of a real $x$ is a power series in base $\beta$ with coefficients 0 and 1 whose sum equals $x$. The aim of this note is to study certain problems related to the universality and combinatorics of…

The Hausdorff dimension of the graphs of the functions in H\"older and Besov spaces (in this case with integrability p \geq 1) on fractal d-sets is studied. Denoting by s \in (0,1] the smoothness parameter, the sharp upper bound…