The recently introduced twin-width of a graph \(G\) is the minimum integer \(d\) such that has \(d\)-contraction sequence, is, sequence \(\left| V(G) \right|-1\) iterated vertex identifications for which overall maximum number red edges incident to single at most \(d\), where edge appears between two sets identified vertices if they are not homogeneous in (not fully adjacent nor non-adjacent). ...

The Turán number of an $r$-uniform graph $F$, denoted by $ex(n,F)$, is the maximum edges in $F$-free on $n$ vertices. density $F$ defined as $\pi(F)=\underset{{n\rightarrow\infty}}{\lim}{ex(n,F) \over {n \choose r }}.$ Denote $\Pi_{\infty}^{(r)}={ \pi(\cal F): \cal F a family r{-uniform graphs}},$ $\Pi_{fin}^{(r)}=\{ {is \ finite of} r{{-}uniform graphs}\}$, and $\Pi_{t}^{(r)}=\{\pi(\cal graphs...

Abstract The study of the $$\mathbf{4}$$ 4 -tachyon off-shell string scattering amplitude $$ A_4 (s, t, u) A ( s , t u ) , based on Witten’s open...

protocol : THEORY BEGIN File : TYPE = fNONE,ONE,MANYg Msg : TYPE = [# first: bool, last: bool, toggle: bool #] State : TYPE = [# file : File, rn : bool, afile : bool, ... #] The automaton speci cation also changes but this is acceptable since Theorem 1 admits an abstract property pa distinct from the concrete property pc. We omit the details of this new automaton speci cation. In the abstract v...

Short summary of most important research results that explain why the work was done, what was accomplished, and how it pushed scientific frontiers or advanced the field. This summary will be used for archival purposes and will be added to a searchable DoD database. We obtained our main research results on four new approaches, 1) to rank items based on the MTDF (Multinomial with Trust Discount F...