Jason/Frank Ayers
Full name- Frank Jason Ayers, Frank Jason, Jason Ayers, F. Jason Ayers, and
Jason Frank.
Nickname(s)- Frankie J.
Birthplace- St. Cloud, MN, United States of America
Town raised in- St. Cloud-Clear Lake, MN
Astrological Sign- Scorpio
Hobbies- Soccer, drums, video games, movies, music, Ultimate Frisbee & golf,
sports, fun, Simpson's, Dave Matthews, Counting Crows, swimming, talking
about JJ and Matt. Thinking about JJ and Matt, Running, the bar thing, there
was a time I liked editing this web site but I am "not allowed to do that
anymore," eating, sleeping, hot showers, talking to girls, playing with
candle wax, volleyball, cuddling, friends, and other stuff.
Schools attending or attended- Lincoln elementary, Jefferson Elementary,
Clearview Elementary, Talahi Elementary, South Jr. High, Graduated from St.
Cloud Technical High School in '98 and I am attending SCSU with plans on
majoring in Mathematics but it is really hard.
What will you be doing in 2 years- Panoramic Blue will be playing the MTV
music awards and I will be dating someone. I will have a house with lots of
fun toys and a dog.
Favorite author- Shel Silverstien
Favorite movies- American Beauty and Army of Darkness
Favorite painter- Jeff Aeling
Favorite Simpson- SideShow Bob
Favorite Panoramic Blue song- The Jason/Frank song
Favorite pizza topping- Pineapple or pepperoni
Biggest fear- Sharks/Relationships
Any tattoo's or piercing- Yes.
Other bands played in - Kokopelli, Jazz combo, Bizzaro Stars, Varsity Band,
Campus Band, sports band, Silver Pepper (that was a good one), Wind Ensemble,
Sky High, Dixieland combo, SilverLabel, Panoramicblue.
Complete this sentence- "My favorite thing about Panoramic Blue is…. All the
ear candy I get.
Three favorite bands you'd want to share the stage with- Dave, the corrs,
and Counting Crows and ...
Worst border crossing- I was trying to go from South America to Africa and I
was really nervous, but I had a lucky roll of the dice and I won that game of
risk.
Band member you would most want to sucker punch- Frank….I think I could take
him.
Least favorite band member to be seated next to on an international flight-
Jason...
Most confusing band member- The monkey next to Jer...
Favorite non-alcoholic beverage- Pink Lemonade!!!!
Most frequent thought during a live performance- "is that girl looking at me
in a good way or bad???"
Best Panoramic Blue memory so far- When that girl jump on stage and kissed
me……HEY! It could happen.
This is my math homework, if there is anyone who can help me with it or tell me if i made an error please send us an e-mail...
2. Find Aut(Z).
Aut(Z) consists of all functions phi s.t. Phi(z) = zn, with n not equal to 0.
3. Let R+ be the group of positive real numbers under multiplication. Show that the mapping Q(x) = sqrt(x) is an automorphism of R+.
1. OP, Q(xy) = sqrt(xy) = sqrt(x)sqrt(y) = Q(x)Q(y)
2. 1-1, Q(x) = Q(y)
sqrt(x) = sqrt(y)
x = y
3. Onto, Q(x^2) = sqrt(x^2) = x
5. Show that U(8) is isomorphic to U(12).
U8 1 3 5 7
1 1 3 5 7
3 3 1 7 5
5 5 7 1 3
7 7 5 3 1
U12 1 5 7 11
1 1 5 7 11
5 5 1 11 7
7 7 11 1 5
11 11 7 5 1
The elements in U(8) correspond to the elements in U(12)
11,35,57,711
6. Prove that the relation isomorphism is an equivalence relation.
Let G, H, K be groups.
1. Reflexive. G~G. The isomorphism is provided by the identity mapping.
2. Symmetric: Suppose G~H via the isomorphism f. Since f is 1-1 and onto, ther is an inverse mapping f^-1 | HG.
Show f^-1 is an isomorphism.
1. OP, f^-1(ab) = f^-1(a)f^-1(b)
f(f^-1(ab)) = f(f^-1(a)f^-1(b))
ab = ff^-1(a)ff^-1(b))
ab = ab
2. 1-1 Suppose f^-1(a) = f^-1(b)
f(f^-1(a)) = f(f^-1(b))
a = bv
2. Onto, Take g in G. We must find h in H such that f(h) = g. But take
h = f^-1(g). Then f(h) = f(f^-1(g) = g
3. Transitive: Suppose G ~ K via the isomorphism g o f.
8. In the notation of theorem 6.1, prove that (T(g))^-1=T(g)^-1.
(T(g))^-1 = ((g)^-1)x = (g^-1)x
T(g^-1) = (g^-1)x
9. Explain why the three parts of Exercise 1 of the Supplementary Exercises, Chapter 1-4, follow immediately from Theorem 6.2.
H xHx^-1 is an isomorphism therefore it must follow the properties of an isomorphism. Specifically properties 4, 6, and 9 from thm 6.2
11. Find two groups G and H such that G is not isomorphic to H, but Aut(G) is isomorphic Aut(H).
If G=Z(6) and H=Z(3), Then
In Z(6) 11 and 15.
In Z(3) 11 and 12.
So G and H are not isomorphic.
16. Let r exist in U(n). Prove that the mapping a: ZnZn defined by a(s) = sr mod n for all s in Zn is an automorphism of zn.
1. OP, a(s+t) = (s+t)r mod n = (sr mod n)(tr mod n) = a(s) + a(t)
2. 1-1, Let sr = tr mod n, then n|(sr-tr). Since r is r.p. to n, then n|(s-t). So s = t mod n
3. Onto, Since this is a finite group, a 1-1 mapping is onto.
20. Prove that the quaternion group is not isomorphic to the dihedral group D(4).
D(4) has only two elements of order 4 they are R(90) and R(270). In the quaternion group, there are 6 elements of order 4. They are a, a^3, and b, ba, ba^2, and ba^3. So the two groups are not isomorphic.
23. Show that the mapping Q(a + bi) = a – bi is an qutomorphism of the qroup of complex numbers under addition. Show that Q preserves complex multiplication as well—that is, Q(xy) = Q(x)Q(y) for all x and y in C.
1. O.P. Q((a(1) + ib(1))+ (a(2) + ib(2))) = Q((a(1) + a(2)) + i(b(1) + b(2)))
= (a(1) + a(2)) – i(b(1) + b(2))
= (a(1) –ib(1)) + (a(2)-ib(2))
= Q(a(1) +ib(1)) + Q(a(2) + ib(2))
2. 1-1, Let Q(a + ib) = Q(c +id)
Then a-ib = c-id
So a = c, and b = d
So a + ib = c + id.
3. Onto, Q(a – bi) = a – (-bi) = a + bi
Q((a1 + b1I)(a2 + b2I)) = Q(a1a2 + a1b2i +a2b1i-b1b2)
=Q(a1a2-b1b2) + i(a1b2 +a2b1)
=(a1a2-b1b2) – i(a1b2+a2b1)
=a1a2 - a2b1i – a1b2i –b1b2
=(a1-b1i)(a2-b2i) = Q(a1+b1i)Q(a2+b2i)
25. Prove that Z under addition is not isomorphic to Q under addition.
In order for them Z to be isomorphic to Q they must both be cyclic. But Q is not cyclic.
Proof that Q is not cyclic. Soppose a and b are relitively prime positive integers in the
= Q+. then there is some positive interger n such that n(a|b) = 2. That implyes a and b are not relitivly prime which is a contradiction. So Q is not cyclic.
26. look up the words isobar, isomer, and isotope in a dictionary. Relate their meanings to the meaning of isomorphism.
Isobar = Any of two or more kinds of atoms having the same atomic mass but different atomic numbers.
Isomer = Any of two or more nuclei with the same mass number and atomic number that have different radioactive properties and can exist in any of several energy states for a measurable period of time.
Isotope = One of two or more atoms having the same atomic number but different mass numbers.
Isomorphism = A one-to-one correspondence between the elements of two sets such that the result of an operation on elements of one set corresponds to the result of the analogous operation on their images in the other set.
Each “iso” word is an example of two or more “things” not being the same/equal but still having common properties or a correspondence.
32. Let G be a group and let G be an element in G. If z is in Z(G), show that the inner atuomorphism induced by g is the same as the inner automorphism induced by zg.
Q(g) = gxg^-1
Q(zg) = (zg)x(zg)^-1
Since z is in the center, then
(zg)x(zg)^-1 = zgx(z^-1)(g^-1) = zg(xz^-1)g^-1 = z(gz^-1)xg^-1 = z(z^-1)gxg^-1 = gxg^-1
35. Let a belong to a group G and let |a| be finite. Let Qa be the automorqhism of G given by Qa(x) = axa^-1. Show that |Qa| divides |a|. Exhibit an element a from a group for which 1< |Qa| < |a|.
Say |a| = n. Then Qna(x) = a^nxa^-n = x so that Qna is the identity. For the example, take a =R90 in D4
37. Prove that Q under addition is not isomorphic to R+ under multiplication.
Say Phi is an isomorqhism from Q to R+ and Phi takes 1 to a. From this, it follows that an integer x would map to a^x and the rational x/s would map to a^(r/s). But a^(r/s) does not equal a^pi for any r/s. Therefore, they are not isomorphic.
