Let f1, f2, f3: [a,b] -->R be nonnegative concave functions
such that f1(a) = f2(a) =...
Let f1, f2, f3: [a,b] -->R be nonnegative concave functions
such that f1(a) = f2(a) = f3(a) = f1(b) = f2(b) = f3(b) = 0.
Suppose that max(f1) <= max(f2) <= max(f3).
Prove that: max(f1) + max(f2) <= max(f1+f2+f3)
(i) Suppose that f1, f2 are lower semicontinuous at a. Prove
that max{f1, f2} is lower...
(i) Suppose that f1, f2 are lower semicontinuous at a. Prove
that max{f1, f2} is lower semicontinuous at a.
(ii) Prove or disprove that f1 − f2 is lower semicontinuous
a.
Let f1, f2, . . . ∈ k[x1, . . . , xn ] be an...
Let f1, f2, . . . ∈ k[x1, . . . , xn ] be an infinite collection
of polynomials and let I = < f1, f2, . . .> be the ideal they
generate. Prove that there is an integer N such that I = . Hint:
Use f1, f2, . . . to create an ascending chain of ideal
Two forces, F1 and
F2, have the same magnitude and both
act on an object that...
Two forces, F1 and
F2, have the same magnitude and both
act on an object that can rotate about a fixed axis of rotation. If
the forces F1 and
F2 point in opposite
directions, and the torque produced by
F1 is clockwise, then
the torque produced by
F2
must be counterclockwise.
True or false?
Fibonacci Numbers.
The Fibonacci numbers are
1,1,2,3,5,8,13,21,….1,1,2,3,5,8,13,21,….
We can define them inductively by f1=1,f1=1, f2=1,f2=1, and...
Fibonacci Numbers.
The Fibonacci numbers are
1,1,2,3,5,8,13,21,….1,1,2,3,5,8,13,21,….
We can define them inductively by f1=1,f1=1, f2=1,f2=1, and
fn+2=fn+1+fnfn+2=fn+1+fn for n∈N.
Prove that fn=[(1+√5)n−(1−√5)n]/2n√5.