Iterating On Real or Complex Numbers
The Collatz map can be viewed as the restriction to the integers of the smooth real and complex map
which simplifies to
If the standard Collatz map defined above is optimized by replacing the relation 3n + 1 with the common substitute "shortcut" relation (3n + 1)/2, it can be viewed as the restriction to the integers of the smooth real and complex map
which simplifies to .
Read more about this topic: Collatz Conjecture, Extensions To Larger Domains
Famous quotes containing the words real, complex and/or numbers:
“The dignity to be sought in death is the appreciation by others of what one has been in life,... that proceeds from a life well lived and from the acceptance of one’s own death as a necessary process of nature.... It is also the recognition that the real event taking place at the end of our life is our death, not the attempts to prevent it.”
—Sherwin B. Nuland (b. 1930)
“In the case of all other sciences, arts, skills, and crafts, everyone is convinced that a complex and laborious programme of learning and practice is necessary for competence. Yet when it comes to philosophy, there seems to be a currently prevailing prejudice to the effect that, although not everyone who has eyes and fingers, and is given leather and last, is at once in a position to make shoes, everyone nevertheless immediately understands how to philosophize.”
—Georg Wilhelm Friedrich Hegel (1770–1831)
“Out of the darkness where Philomela sat,
Her fairy numbers issued. What then ailed me?
My ears are called capacious but they failed me,
Her classics registered a little flat!
I rose, and venomously spat.”
—John Crowe Ransom (1888–1974)