The theory of graphons is ultimately connected with the so-called cut norm. In this paper, we approach norm topology via weak* (when considering a predual $L^{1}$-functions). We prove that sequence $W_1,W_2,W_3,\ldots$ converges in distance if and only have equality sets accumulation points limit all sequences $W_1',W_2',W_3',\ldots$ are weakly isomorphic to $W_1,W_2,W_3,\ldots$. further give s...

Abstract We use Gaussian measure-preserving systems to prove the existence and genericity of Lebesgue transformations $T:[0,1]\rightarrow [0,1]$ which exhibit both mixing rigidity behavior along families asymptotically linearly independent sequences. Let $\unicode{x3bb} _1,\ldots ,\unicode{x3bb} _N\in let $\phi ,\phi _N:\mathbb N\rightarrow \mathbb Z$ be (that is, for any $(a_1,\ldots ,a_N)\in ...

Abstract Let $\varphi _1,\ldots ,\varphi _r\in {\mathbb Z}[z_1,\ldots z_k]$ be integral linear combinations of elementary symmetric polynomials with $\text {deg}(\varphi _j)=k_j\ (1\le j\le r)$ , where $1\le k_1<k_2<\cdots <k_r=k$ . Subject to the condition $k_1+\cdots +k_r\ge \tfrac {1}{2}k(k-~1)+2$ we show that there is a paucity nondiagonal solutions Diophantine system _j({\mathbf x...

We consider maps on genus-$g$ surfaces with $n$ (labeled) faces of prescribed even degrees. It is known since work Norbury that, if one disallows vertices degree one, the enumeration such related to counting lattice point in moduli space curves labeled points and given by a symmetric polynomial $N_{g,n}(\ell_1,\ldots,\ell_n)$ face degrees $2\ell_1, \ldots, 2\ell_n$. generalize this restricting ...

Abstract In this paper, we discuss a connection between geometric measure theory and number theory. This method brings new point of view for some number-theoretic problems concerning digit expansions. Among other results, show that each integer k , there is $M>0$ such if $b_{1},\ldots ,b_{k}$ are multiplicatively independent integers greater than M infinitely many whose base $b_{1},b_{2},\ld...

We prove the unique existence of functions $r_n$ $(n=1,2,\ldots )$ on $[0,1]$ such that corresponding sequence King operators approximates each continuous function and preserves $e_0(x)=1$ $e_j(x)=x^j$, where $j\in\{ 2,3,\ldots\}$ is fixed. establish essential properties $r_n$, rate convergence new will be estimated by usual modulus continuity. Finally, we show introduced are not polynomial obt...

We demonstrate a method for proving precise concentration inequalities in uniformly random trees on $n$ vertices, where $n\geq1$ is fixed positive integer. The uses bijection between mappings $f\colon\{1,\ldots,n\}\to\{1,\ldots,n\}$ and doubly rooted vertices. main application inequality the number of vertices connected to an independent set tree, which then used prove partial unimodality its s...

Abstract A one-variable Hankel matrix H a {H}_{a} is an infinite = [ ( i + j ) ] , ≥ 0 {H}_{a}={\left[a\left(i+j)]}_{i,j\ge 0}...

We study an information analogue of infinitely divisible probability distributions, where the i.i.d. sum is replaced by joint distribution sequence. A random variable $X$ called informationally if, for any notation="LaTeX">$n\ge 1$ , there exists sequence variables notation="LaTeX">$Z_{1},\ldots...

Given a graph G, the burning number of G is smallest integer k for which there are vertices $$x_1, x_2,\ldots ,x_k$$ such that $$(x_1,x_2,\ldots ,x_k)$$ sequence G. It has been shown problem NP-complete, even trees with maximum degree three, or linear forests. A t-unicyclic unicycle in unique vertex greater than two $$ t + 2 . In this paper, we first present bounds graphs, and then use numbers ...