We investigate the minimum cases for realtime probabilistic machines that can define uncountably many languages with bounded error. We show that logarithmic space is enough for realtime PTMs on unary languages. On binary case, we follow the same result for double logarithmic space, which is tight. When replacing the worktape with some limited memories, we can follow uncountable results on unary...

It is consistent that for every function f : ω → ω there is a graph with size and chromatic number א1 in which every n-chromatic subgraph contains at least f(n) vertices (n ≥ 3). This solves a $ 250 problem of Erdős. It is consistent that there is a graph X with Chr(X) = |X| = א1 such that if Y is a graph all whose finite subgraphs occur in X then Chr(Y ) ≤ א2 (so the Taylor conjecture may fail...

In this paper, we give a necessary and sufficient condition for graphical strip in the Heisenberg group $${\mathbb {H}}$$ to be area-minimizing slab $$\{-1<x<1\}$$ . We show that our is by introducing family of deformations strips based on varying vertical curve. it showing satisfying have monotone epigraphs. use any ruled entire intrinsic graph plane find boundary curve admits uncountably many...

We extend the notion of singular vectors to context Diophantine approximation real numbers with elements a totally number field K. For $$m\ge 1$$ , we establish version Dani’s correspondence in fields and prove that under class ‘friendly measures’ $$K_S^m$$ set has measure zero. Here S is Archimedean valuations K $$K_S$$ product completions $$\sigma (K)$$ \in S$$ . On other hand, show existence...

It is known that classical logic CL is the single maximal consistent logic over intuitionistic logic Int, which is moreover the single one even over the substructural logic FLew. On the other hand, if we consider maximal consistent logics over a weaker logic, there may be uncountablymany of them. Since the subvariety lattice of a given variety V of residuated lattices is dually isomorphic to th...

We show that the lattice of all ideals of a ring $R$ can be embedded in the lattice of all its fuzzyideals in uncountably many ways. For this purpose, we introduce the concept of the generalizedcharacteristic function $chi _{s}^{r} (A)$ of a subset $A$ of a ring $R$ forfixed $r , sin [0,1] $ and show that $A$ is an ideal of $R$ if, and only if, its generalizedcharacteristic function $chi _{s}^{...