to demonstrate more visibly the close relation between thecontinuity and integrability, a new proof for the banach-zareckitheorem is presented on the basis of the radon-nikodym theoremwhich emphasizes on measure-type properties of the lebesgueintegral. the banach-zarecki theorem says that a real-valuedfunction $f$ is absolutely continuous on a finite closed intervalif and only if it is continuo...

Let Q be the field of rationals numbers. We prove that: (1) if x, y ∈ R (n > 1) and |x− y| is constructible by means of ruler and compass then there exists a finite set Sxy ⊆ R containing x and y such that each map from Sxy to R n preserving unit distance preserves the distance between x and y, (2) if x, y ∈ Q then there exists a finite set Sxy ⊆ Q containing x and y such that each map from Sxy...

<p style='text-indent:20px;'>We show that for <i>any</i> ergodic Lebesgue measure preserving transformation <inline-formula><tex-math id="M1">\begin{document}$ f: [0,1) \rightarrow $\end{document}</tex-math></inline-formula> and decreasing sequence id="M2">\begin{document}$ \{b_i\}_{i=1}^{\infty} of positive real numbers with divergent sum, the set&l...

Unit propagation is one of the most important techniques of efficient SAT solvers. Unfortunately, this technique is not directly applicable to first-order clausal tableaux. We show a way of integrating a variant of unit propagation into the disconnection calculus and present some results obtained with an implementation of unit propagation in the DCTP theorem prover that show the usefulness of o...