oldkicker
<font color=purple>Pay no attention to <img src=ht
- Joined
- Aug 23, 1999
- Messages
- 9,835
You have your choice of riddles today, dear riddle friends.
This is just my way of wrapping up a very tough week!
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.
OR
Show a partition of the real line, into a set of measure zero, and a set of category 1.
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.
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".
OR
Show that every polytope in Rn, has at least one extremal point.
OR
Show that every polytope in Rn that is not a single point, has at least two extremal points.
OR
Consider an Euclidean space with infinite dimension. How many balls with radius 1, can you fit into another ball of radius 2?
How about when the other ball's radius is 3?
OR
Show that every compact subset of a metric space is of no more than continuum cardinality.
OR
What kind of coat can be put on only when wet?
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?
OR
Does there exist a measurable space, such that the set of measurable subsets is infinite, yet countable?
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).
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.
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.
This is just my way of wrapping up a very tough week!
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.
OR
Show a partition of the real line, into a set of measure zero, and a set of category 1.
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.
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".
OR
Show that every polytope in Rn, has at least one extremal point.
OR
Show that every polytope in Rn that is not a single point, has at least two extremal points.
OR
Consider an Euclidean space with infinite dimension. How many balls with radius 1, can you fit into another ball of radius 2?
How about when the other ball's radius is 3?
OR
Show that every compact subset of a metric space is of no more than continuum cardinality.
OR
What kind of coat can be put on only when wet?
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?
OR
Does there exist a measurable space, such that the set of measurable subsets is infinite, yet countable?
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).
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.
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.
I panicked for a minute there! However after l finished LOL I sent my PM
