Search

31 Jan 2023: Regarding T as a map from C into C, show that there existunique complex numbers A, B such that for every z C, T(z) = Az Bz. ... Can such a map be the restriction tothe strip of a Möbius map?

13 Feb 2017: Such a c is called a fixed point of g.Give an example of a bijection h: [0, 1] [0, 1] with no fixed point. ... Thatis, for every a < b and every y there exists x (a,b) such that f(x) = y.Can such a function be continuous at any point?

27 Nov 2013: 12. Let S be a collection of subsets of N such that for every A, B S we have A B orB A. ... otherwords, for every a < b and every c there is an x with a < x < b such that f(x) = c.

23 Nov 2016: Mich. 2016 NUMBERS AND SETS—EXAMPLES 4 PAR. 1. The symmetric difference A4B of two sets A and B is defined to be the set of elements x such that xis ... 11. Let S be a collection of subsets of N such that for all A, B S we have A B or B A.

15 Jan 2024: 6. Let R be an integral domain. The greatest common divisor (gcd) of non-zero elements a andb in R is an element d in R such that d divides both ... b) Show that may be identified with the set Fq{} in such a way that GL2(Fq) acts on as the group of

13 Oct 2006: b) Suppose that f is bounded. Prove that there exists M and n0 such that |fn(x)| < M for all. ... a,b] such that fn(xn) does not converge to f(x). 7. Which of the following functions f are (a) uniformly continuous, (b) bounded on [0,)?

8 Feb 2017: Show that there exist unique complexnumbers A, B such that for every z C, T(z) = Az Bz̄. ... Can such a map everbe a Möbius transformation? 9. Let f : U C be a holomorphic function where U is an open set (you may assume f isalso C1).

10 Oct 2023: b) Now let X and Y be jointly Gaussian random vectors such that Cov(X) is positivedefinite. ... Let Q bethe equivalent probability measure such that dQ. dP eX.

8 Feb 2017: Show that for any b C, there exists a sequence of points zn D(a,R)with zn 6= a such that zn a and f(zn) b as n. ... Find such a sequence when f(z) = e1/z, a = 0 and b = 2.[A much harder theorem of Picard says that in any neighbourhood of an essential

15 Oct 2020: The least such n iscalled the order of g.). (b) Show that there exists a positive integer n such that gn = e for all g G. ... Can such a group beinnite? 8. Let G be a group of even order.