(FP6) Is every knot group virtually free-by-cyclic?
(O1) The Andrews-Curtis conjecture. Let F = F_n be the free group of a finite rank n with a fixed set X = {x_1,...,x_n} of free generators.
(a) A set Y = {y_1,...,y_n} of elements of F generates the group F as a normal subgroup if and only if Y is Andrews-Curtis equivalent to X, which means one can get from X to Y by a sequence of Nielsen transformations together with conjugations by elements of F.
(FP6) Is every knot group virtually free-by-cyclic?
(O1) The Andrews-Curtis conjecture. Let F = F_n be the free group of a finite rank n with a fixed set X = {x_1,...,x_n} of free generators.
(a) A set Y = {y_1,...,y_n} of elements of F generates the group F as a normal subgroup if and only if Y is Andrews-Curtis equivalent to X, which means one can get from X to Y by a sequence of Nielsen transformations together with conjugations by elements of F.
Math is a great hobby to have as unlike some other scientific disciplines, you dont need expensive equipment or resources outside of brain power to partake in it.
Lagrangian
quote:
Originally posted by paulversuspaul
Math is a great hobby to have as unlike some other scientific disciplines, you dont need expensive equipment or resources outside of brain power to partake in it.
hey, i thought i disliked you but now i like you...it's weird you know, and like... complex, you know kinda like vanilla ice cream and chocolate ice cream at the same time and you can't tell which one you like the most, so you just eat it up at the same time :gsmile:
NASA's Mars rover Curiosity has photographed a shiny, metallic-looking object that bears a passing resemblance to a door handle or a hood ornament.
The Curiosity rover has not stumbled onto evidence of an ancient civilization that took the family van to Olympus Mons for vacation, however. The object is simply a rock that the wind has sculpted into an interesting shape, scientists said.
"The shiny surface suggests that this rock has a fine grain and is relatively hard," Curiosity scientists wrote Monday (Feb. 11) in an explainer blurb accompanying the image, which was taken on Jan. 30. "Hard, fine-grained rocks can be polished by the wind to form very smooth surfaces."
Similar "ventifacted" (wind-eroded) rocks can be found here on Earth, notably on the dry, gusty plains of Antarctica, they added.
The newfound rock is not the first shiny object Curiosity has photographed on the Red Planet.
PASADENA, Calif. -- NASA's Curiosity rover has, for the first time, used a drill carried at the end of its robotic arm to bore into a flat, veiny rock on Mars and collect a sample from its interior. This is the first time any robot has drilled into a rock to collect a sample on Mars.
The fresh hole, about 0.63 inch (1.6 centimeters) wide and 2.5 inches (6.4 centimeters) deep in a patch of fine-grained sedimentary bedrock, can be seen in images and other data Curiosity beamed to Earth Saturday. The rock is believed to hold evidence about long-gone wet environments. In pursuit of that evidence, the rover will use its laboratory instruments to analyze rock powder collected by the drill.
"The most advanced planetary robot ever designed is now a fully operating analytical laboratory on Mars," said John Grunsfeld, NASA associate administrator for the agency's Science Mission Directorate. "This is the biggest milestone accomplishment for the Curiosity team since the sky-crane landing last August, another proud day for America."
For the next several days, ground controllers will command the rover's arm to carry out a series of steps to process the sample, ultimately delivering portions to the instruments inside.
"We commanded the first full-depth drilling, and we believe we have collected sufficient material from the rock to meet our objectives of hardware cleaning and sample drop-off," said Avi Okon, drill cognizant engineer at NASA's Jet Propulsion Laboratory, Pasadena, Calif.
Originally posted by Lagrangian
NASA's Mars rover Curiosity has photographed a shiny, metallic-looking object that bears a passing resemblance to a door handle or a hood ornament.
The Curiosity rover has not stumbled onto evidence of an ancient civilization that took the family van to Olympus Mons for vacation, however. The object is simply a rock that the wind has sculpted into an interesting shape, scientists said.
I honestly think it's a part the Curiosity put there in order to find it.
Acton
itsamemario
quote:
Originally posted by Acton
haha :P
semi-related:
Does anyone know the name of the thing where you could help make like amino acids or something? Sorry for being vague.
