I take a vacation in my research work
I decided to dedicate my free (of working as a programmer) time to write a book about religion (What book? It will be a surprise.) So in a few nearby months I am going to not continue my math research. I am going to return to math research after I will finish that religious book. […]
Defined limit of discontinuous functions!
In my online draft article “Convergence of funcoids” at my Algebraic General Topology site is now defined limit of arbitrary (not necessarily continuous) functions (under certain conditions). Thus mathematical analysis goes to the next stage, non-continuous analysis. Please nominate me for Abel Prize.
Erroneous theorem became a conjecture
I mistakenly used yet unproved statement that $latex \mathrm{up}\,f$ (taken on the filtrator of funcoids) is a filter for every funcoid $latex f$ in proof of a theorem. So after I found this error I downgrade this theorem to the status of conjecture: Conjecture $latex (\mathsf{FCD}) (\mathsf{RLD})_{\mathrm{out}} f = f$ for every funcoid $latex f$.
Erroneous lemma corrected
In “Funcoids and Reloids” online draft there was an erroneous lemma: Lemma For every two sets $latex S$ and $latex T$ of binary relations and every set $latex A$ $latex \bigcap {\nobreak}^{\mathfrak{F}} S = \bigcap {\nobreak}^{\mathfrak{F}} T \Rightarrow \bigcap {\nobreak}^{\mathfrak{F}} \{ \langle F \rangle A | F \in S \} = \bigcap{\nobreak}^{\mathfrak{F}} \{ \left\langle G […]
Errors in Funcoids and Reloids corrected
I corrected some errors in “Funcoids and Reloids” online draft. The main error was messing $latex \subseteq$ and $latex \supseteq$ in the theorem about continuing a function defined on atomic filter objects till $latex \langle f\rangle$ for a funcoid $latex f$ (currently the Theorem 54 but theorem numbering will be changed in the future). I […]
Resubmit to Documenta Mathematica
I submitted the preprint of my article “Filters on Posets and Generalizations” to Documenta Mathematica math journal but so far received no reply. So I sent submission to an other editor of the same journal.
False proof of an open problem
I earlier proclaimed that I positively solved this conjecture: Conjecture $latex f\cap^{\mathsf{FCD}} g = f\cap g$ for every binary relations $latex f$ and $latex g$. There were error in my proof and I deleted it. See the draft of Funcoids and Reloids article for details on the current state of the problem.
New conjectures about complete funcoids and reloids
After removing an erroneous theorem I posed two new open problems to take its place: Conjecture If $latex f$ is a complete funcoid and $latex R$ is a set of funcoids then $latex f \circ \bigcup {\nobreak}^{\mathsf{FCD}} R = \bigcup {\nobreak}^{\mathsf{FCD}} \langle f \circ \rangle R$. Conjecture If $latex f$ is a complete reloid and […]
Erroneous theorem
I found a counter-example and an error in my proof of this (erroneous) theorem in Funcoids and Reloids article: Let $latex f\in\mathsf{FCD}$. If $latex R$ is a set of co-complete funcoids then $latex f \circ \bigcup {\nobreak}^{\mathsf{FCD}} R = \bigcup {\nobreak}^{\mathsf{FCD}} \left\langle f \circ \right\rangle R$. A counter-example: Let $latex \Delta = \{ (-\epsilon;\epsilon) | […]
“Funcoids and Reloids” contains “Connectedness”
Now Funcoids and Reloids online article contains the section “Connectedness regarding funcoids and reloids” which previously was in a separate article. In this section there are among definitions and theorems a few open problems.