Related papers: On dessins d'enfants with equal supports

We improve on Jarn\'{\i}k's inequality between uniform Diophantine exponent $\alpha $ and ordinary Diophantine exponent $\beta$ for a system of $ n\ge 2$ real linear forms in two integer variables. Jarn\'{\i}k (1949, 1954) proved that…

Let $B$ be a Blaschke product with zeros $\{a_n\}$. If $B' \in A^p_{\alpha}$ for certain $p$ and $\alpha$, it is shown that $\sum_n (1 - |a_n|)^{\beta} < \infty$ for appropriate values of $\beta$. Also, if $\{a_n\}$ is uniformly discrete…

A differentiable function is pseudoconvex if and only if its restrictions over straight lines are pseudoconvex. A differentiable function depending on one variable, defined on some closed interval $[a,b]$ is pseudoconvex if and only if…

The present work proposes and discusses the category of supported sets which provides a uniform foundation for nominal sets of various kinds, such as those for equality symmetry, for the order symmetry, and renaming sets. We show that all…

Let $D$ be a smooth divisor on a non singular surface $S$. We compute Betti numbers of the relative Hilbert scheme of points of $S$ relative to $D$. In the case of $\PP^2$ and a line in it, we give an explicit set of generators and…

This paper examines a pair of bent functions on $\mathbb{Z}_2^{2m}$ and their relationship to a necessary condition for the existence of an automorphism of an edge-coloured graph whose colours are defined by the properties of a canonical…

Various mixing properties of $\beta$-, $\beta'$- and Gaussian Delaunay tessellations in $\mathbb{R}^{d-1}$ are studied. It is shown that these tessellation models are absolutely regular, or $\beta$-mixing. In the $\beta$- and the Gaussian…

Let $\mathrm{R}$ be a real closed field. We prove that the number of semi-algebraically connected components of a real hypersurface in $\mathrm{R}^n$ defined by a multi-affine polynomial of degree $d$ is bounded by $2^{d-1}$. This bound is…

For the functions from sets $C_\beta^\psi C$ and $C_\beta^\psi L_s, \ 1\leq s\leq\infty$, generated by sequences $\psi(k)>0$ satisfying the condition d'Alembert $\mathop {\rm \lim}\limits_{k\rightarrow\infty}\frac{\psi(k+1)}{\psi(k)}=q, \…

In this note we show that $\beta'_*$, the $\beta$-function slopes in the electric and magnetic theories are equal at the corresponding infrared fixed points. This follows from the scaling of the correlators of the trace of the energy…

The presence or absense of renormalon singularities in the Borel plane is shown to be determined by the analytic properties of the Gell-Mann - Low function \beta(g) and some other functions. A constructive criterion for the absense of…

In this paper we consider a minimization problem of the type $$ I_{\beta,p}(D;\Omega)=\inf\biggl\{\int_\Omega \lvert{D\phi}\rvert^pdx+\beta \int_{\partial^* \Omega}\lvert{\phi}\rvert^pd\mathcal{H}^{n-1},\; \phi \in W^{1,p}(\Omega),\;\phi…

In this paper, we investigate inhomogeneous and simultaneous Diophantine approximation in beta dynamical systems. For $\beta>1$ let $T_{\beta}$ be the $\beta$-transformation on $[0,1]$. We determine the Lebesgue measure and Hausdorff…

In this paper, we will give a $C^{1,\beta}$-regularity result on the boundary for two dimensional sliding almost minimal sets in $\mathbb{R}^3$. This effect may lead to the existence of a solution to the Plateau problem with sliding…

If we want to color $1,2,\ldots,n$ with the property that all 3-term arithmetic progressions are rainbow (that is, their elements receive 3 distinct colors), then, obviously, we need to use at least $n/2$ colors. Surprisingly, much fewer…

In this paper, we construct some piecewise defined functions, and study their $c$-differential uniformity. As a by-product, we improve upon several prior results. Further, we look at concatenations of functions with low differential…

We study the following two functions: d(n,c) and $\vec{d}(n,c)$; d(n,c) ($\vec{d}(n,c)$) is the minimum number k such that every c-edge-colored undirected (directed) graph of order n and minimum monochromatic degree (out-degree) at least k…

That is, given a compact set $B \subset \mathbb{R}^n$ (the boundary) and a subgroup $L$ of the \v{C}ech homology group $\check{H}_{d-1}(B;G)$ of dimension $d$ over some commutative group $G$, we find a compact set $E \supset B$ such that…

We give an account of the theory of dessins d'enfants which is both elementary and self-contained. We describe the equivalence of many categories (graphs embedded nicely on surfaces, finite sets with certain permutations, certain field…

The main result is a submetric characterization of the class of Banach spaces admitting an equivalent norm with Rolewicz's property ($\beta$). As applications we prove that up to renorming, property ($\beta$) is stable under coarse…