The canonization theorem says that for given $m,n$ for some $m^{*}$ (the first one is called $ER(n;m)$) we have for every function $f$ with domain ${\left[1,\cdots ,m^{*}\right]}^n$, for some $A\in {\left[1,\cdots ,m^{*}\right]}^m$, the question of when the equality $f\left(i_1,\cdots ,i_n\right)=f\left(j_1,\cdots ,j_n\right)$ (where $i_1<\cdots <i_n$ and $j_1<\cdots j_n$ are from $A$) holds has the simplest answer: for some $v\subseteq \{1,\cdots ,n\}$ the equality holds iff ${\bigwedge }_{\ell \in v}{i}_{\ell }=j_{\ell }$. We improve the bound on $ER(n,m)$ so that fixing $n$ the number of exponentiation needed to calculate $ER(n,m)$ is best possible.

For $a$ generator of the multiplicative group of the field $GF(p,k)$, the discrete logarithm of an element $b$ of the field to the base $a$, $b\ne 0$ is that integer $z:1\le z\le p^k-1$, $b=a^z$. The $p$-ary digits which represent $z$ can be described with extremely simple polynomial forms.

This paper deals with implications defined from disjunctive uninorms $U$ by the expression $I(x,y)=U\left(N\right(x),y)$ where $N$ is a strong negation. The main goal is to solve the functional equation derived from the distributivity condition of these implications over conjunctive and disjunctive uninorms. Special cases are considered when the conjunctive and disjunctive uninorm are a $t$-norm or a $t$-conorm respectively. The obtained results show a lot of new solutions generalyzing those obtained in previous works...

Let $T:X\to X$ be a continuous selfmap of a compact metrizable space $X$. We prove the equivalence of the following two statements: (1) The mapping $T$ is a Banach contraction relative to some compatible metric on $X$. (2) There is a countable point separating family $ℱ\subset 𝒞\left(X\right)$ of non-negative functions $f\in 𝒞\left(X\right)$ such that for every $f\in ℱ$ there is $g\in 𝒞\left(X\right)$ with $f=g-g\circ T$.