Friday's:
Construct an open set O, that contains all the rational numbers in the segment [0,1], such that m(O) < 0.5, where m is the Lebesgue measure.
The eastbound train
OR
Show a partition of the real line, into a set of measure zero, and a set of category 1.
Moses doesn't lay eggs!!
OR
Let f be an analytic function, with an essential singularity. Show that the set of points in the complex plane, that are the image under f, of infinitely many points, is dense.
A candle
OR
Let f be an analytic function on the complex plane, that is also one to one. Show that f has the form "ax+b".
They all rhyme!
OR
Show that every polytope in Rn, has at least one extremal point.
A splinter
OR
Show that every polytope in Rn that is not a single point, has at least two extremal points.
An hourglass
OR
Consider an Euclidean space with infinite dimension. How many balls with radius 1, can you fit into another ball of radius 2?
A feather
How about when the other ball's radius is 3?
A goose feather
OR
Show that every compact subset of a metric space is of no more than continuum cardinality.
ELBOW
OR
What kind of coat can be put on only when wet?
Paint
OR
Let K be a compact subset of the plane. can K be sliced using two straight lines, into four pieces that have the same Lebesgue measure?
Ear, eye, lip, jaw, toe, leg, arm, gum, hip & rib
OR
Does there exist a measurable space, such that the set of measurable subsets is infinite, yet countable?
Maybe
OR
Show that for every Lebesgue measurable subset A of the line ith positive measure, there is a sequence X1...X1996 that satisfies:
Xi is in A for all i.
For some positive scalar r, Xi - Xi-1 = r (for all i).
Alex Trebek
OR
Find an (efficient) algorithm, which given a set of vectors in Rn spanning a lattice, finds a minimal (in size) set of vectors that span the same lattice.
A kite
OR
Let A be a finite set of real numbers. Show that there exists a non-zero integer n, such that each member of n·A has an integer within distance < 0.0001 from it.
Morey Amsterdam
See, now that you've seen the answers, you realize just how easy those were, doncha???
