Properly colored cycles in edge-colored graphs are closely related to directed oriented graphs. As an analogy of the well-known Caccetta–Häggkvist Conjecture, we study existence properly bounded length graph. We first prove that for all integers s and t with ≥ 2 , every graph G no K contains a spanning subgraph H which admits orientation D such cycle is . Using this result, show r 4 if Conjectu...

We give an n O (log log ) -time membership query algorithm for properly and agnostically learning decision trees under the uniform distribution over { ± 1} . Even in realizable setting, previous fastest runtime was , a consequence of classic Ehrenfeucht Haussler. Our shares similarities with practical heuristics trees, which we augment additional ideas to circumvent known lower bounds against t...

Amyotrophic lateral sclerosis (ALS) is a serious disease of the nervous system, in which brain and spinal cord cannot communicate properly with muscles to coordinate body’s movements. Consequently, gradually stop working things like walking, eating, breathing, speaking become more difficult. Many aspects ALS remain mystery, but we know that cells called neurons run out energy patients, resultin...