Drorbn - User contributions [en]

12-240/Class Photo

2012-09-29T19:52:20Z

Michael.Wang: /* Who We Are... */

Our class on September 25, 2012:

[[Image:12-240-ClassPhoto.jpg|thumb|centre|800px|Class Photo: click to enlarge]]
{{12-240/Navigation}}

Please identify yourself in this photo! There are two ways to do that:

* [[Special:Userlogin|Log in]] to this Wiki and edit this page. Put your name, userid, email address and location in the picture in the alphabetical list below.
* Send [[User:Drorbn|Dror]] an email message with this information.

The first option is more fun but less private.

===Who We Are...===

{| align=center border=1 cellspacing=0
|-
!First name
!Last name
!User ID
!Email
!Place in photo
!Comments

{{Photo Entry|last=Bar-Natan|first=Dror|userid=Drorbn|email=drorbn@ math. toronto. edu|location=facing everybody, as the photographer|comments=Take this entry as a model and leave it first. Otherwise alphabetize by last name. Feel free to leave some fields blank. For better line-breaking, leave a space next to the "@" in email addresses.}}
{{Photo Entry|last=Frailich|first=Rebecca|userid=Rebecca.frailich|email=rebecca.frailich@ mail.utoronto.ca|location=Last row, in between two guys standing at the back (one in red, one in black) |comments=}}
{{Photo Entry|last=Hoover|first=Ken|userid=Khoover|email=ken.hoover@ mail.utoronto.ca|location=First row, fourth from the right.|comments=}}
{{Photo Entry|last=Klingspor|first=Josefine|userid=Josefine|email=josefine. klingspor@ mail. utoronto. ca|location=First row, second from left.|comments=}}
{{Photo Entry|last=Le|first=Quan|userid=Quanle|email=quan.le@mail.utoronto.ca|location=Start bottom right corner, third from right. Go three steps north-west. Directly north-east from there, in blue collar shirt|comments=}}
{{Photo Entry|last=Liu|first=Zhaowei|userid=tod|email=tod. liu@ mail. utoronto .ca|location=First row, third from the right|comments=}}
{{Photo Entry|last=Millson|first=Richard|userid=Richardm|email=r.millson@ mail. utoronto. ca|location=Seventh row from the front, fourth from the right, blue sweater|comments=}}
{{Photo Entry|last=McGrath|first=Celton|userid=CeltonMcGrath|email=celton. mcgrath@ mail. utoronto. ca|location=4th row front from, centre right, brown sweater|comments=}}
{{Photo Entry|last=Morenz|first=Karen|userid=KJMorenz|email=kjmorenz@ gmail.com|location=3rd-ish row from the back, centre right, purple shirt|comments=}}
{{Photo Entry|last=Vicencio-Heap|first=Felipe|userid=Heapfeli|email=felipe. vicencio. heap@ mail. utoronto. ca|location=Second row from the front, furthest to the right.|comments=}}
{{Photo Entry|last=Wamer|first=Kyle|userid=kylewamer|email=kyle. wamer @ mail. utoronto. ca|location=Second row, fifth from the left in the red shirt.|comments=}}
{{Photo Entry|last=Yang|first=Chen|userid=chen|email=neochen. yang@ mail. utoronto. ca|location=sixth row, first from the right in the black pull-over.|comments=}}
{{Photo Entry|last=Zhang|first=BingZhen|userid=Zetalda|email=bingzhen.zhang@ mail. utoronto. ca|location=Second last row, third from left.|comments=}}
{{Photo Entry|last=Zhao|first=TianChen|userid=Ericolony|email=zhao_tianchen@ hotmail. com|location=fourth row, the guy in green shirt.|comments=}}
{{Photo Entry|last=Zibert|first=Vincent|userid=vincezibert|email=vincent. zibert@ mail. utoronto. ca|location=Directly beneath the white notice posted on the door on the right-hand side.|comments=}}
{{Photo Entry|last=Zoghi|first=Sina|userid=sina.zoghi|email=sina.zoghi@ utoronto .ca|location=First row, leftest left.|comments=}}
{{Photo Entry|last=Léger|first=Zacharie|userid=zach.leger8|email=zacharie. leger@ mail. utronto. ca|location= 5th row in a black T-shirt.|comments=}}
{{Photo Entry|last=Wang|first=Minqi|userid=Michael.Wang|email=wangminqi@ yahoo.cn|location=First row, fourth from the left in black oufit) |comments=}}


|}

12-240/Fields' Further proof

2012-09-16T20:13:15Z

Michael.Wang:

[[12-240]][[Image:12-240-Splash.png]]

[[12-240/Classnotes for Tuesday September 11]]

In the first class, Professor says something about particular fields. Forgive me, because I am an international student, if I can not express information precisely.

About: F(n)

F(1) F(2) and F(3) are a field, but F(4) is not. Professor said that any number n which is not a prime number can not "form" a field F(n)

If you do not understand what the F(n) means, you can look through the file "12-240/Classnotes for Tuesday September 11"

'''Why all the numbers which are not prime numbers can not form a field F(n)?'''

Here is the proof.

If we have a field F(n), and n=a*b (a,b,n∈ N*, a,b≠1) , which means n is not a prime number.

IN defination of multiplication

* 0 1 2 3 .......... b.......n-1

0 0 0 0 0 ...........0........0

1 0 . . . ......................

2 0 . . . ......................

3 0 . . . ......................

4 0 . . . ......................

.. 0 . . . ......................

a 0 (a) (2a) (3a).........(a*b)....(n-1)*a ''' ( in this row, every element mod n)'''

....0.....................................

....0.....................................

....0.....................................

....0.....................................

(n-1) 0...................................
see the (a+1)th row

There must be a "1" in this row, actually each row or column, to meet the rule :Existence of Negatives/Inverses. (About the rule, seen in the file "12-240/Classnotes for Tuesday September 11")

So if F(n) is a field, then
there must exist k,m ∈N*, m<n
to meet the equation:
m*a=k*n+1

And we know that n=ab

So m*a=k*a*b+1 (a≠1)

Hence m=k*b+1/a

unless a=1
m will not exist, because m should be an integer.
so F(n), when n is not a prime number, is not a field.

There is a large need for me to improve my format. Editing is welcomed.

PS: But till now, there are still some questions existing.
How can we prove that a prime number N can absolutely form a field? Is there any exception?
I am still working on it.
----Michael.Wang

12-240/Fields' Further proof

2012-09-16T20:11:06Z

Michael.Wang:

[[12-240]][[Image:12-240-Splash.png]]

[[12-240/Classnotes for Tuesday September 11]]

In the first class, Professor says something about particular fields. Forgive me, because I am an international student, if I can not express information precisely.

About: F(n)

F(1) F(2) and F(3) are a field, but F(4) is not. Professor said that any number n which is not a prime number can not "form" a field F(n)

If you do not understand what the F(n) means, you can look through the file "12-240/Classnotes for Tuesday September 11"

'''Why all the numbers which are not prime numbers can not form a field F(n)?'''

Here is the proof.

If we have a field F(n), and n=a*b (a,b,n∈ N*, a,b≠1) , which means n is not a prime number.

IN defination of multiplication

* 0 1 2 3 .......... b.......n-1

0 0 0 0 0 ...........0........0

1 0 . . . ......................

2 0 . . . ......................

3 0 . . . ......................

4 0 . . . ......................

. 0 . . . ......................

a 0 (a) (2a) (3a).........(a*b)....(n-1)*a ''' ( in this row, every element mod n)'''

....0.....................................

....0.....................................

....0.....................................

....0.....................................

(n-1) 0...................................
see the (a+1)th row

There must be a "1" in this row, actually each row or column, to meet the rule :Existence of Negatives/Inverses. (About the rule, seen in the file "12-240/Classnotes for Tuesday September 11")

So if F(n) is a field, then
there must exist k,m ∈N*, m<n
to meet the equation:
m*a=k*n+1

And we know that n=ab

So m*a=k*a*b+1 (a≠1)

Hence m=k*b+1/a

unless a=1
m will not exist, because m should be an integer.
so F(n), when n is not a prime number, is not a field.

There is a large need for me to improve my format. Editing is welcomed.

PS: But till now, there are still some questions existing.
How can we prove that a prime number N can absolutely form a field? Is there any exception?
I am still working on it.
----Michael.Wang

12-240/Fields' Further proof

2012-09-16T20:09:20Z

Michael.Wang:

<nowiki>[[12-240]][[Image:12-240-Splash.png]]

[[12-240/Classnotes for Tuesday September 11]]

In the first class, Professor says something about particular fields. Forgive me, because I am an international student, if I can not express information precisely.

About: F(n)

F(1) F(2) and F(3) are a field, but F(4) is not. Professor said that any number n which is not a prime number can not "form" a field F(n)

If you do not understand what the F(n) means, you can look through the file "12-240/Classnotes for Tuesday September 11"

'''Why all the numbers which are not prime numbers can not form a field F(n)?'''

Here is the proof.

If we have a field F(n), and n=a*b (a,b,n∈ N*, a,b≠1) , which means n is not a prime number.

IN defination of multiplication

* 0 1 2 3 .......... b.......n-1

0 0 0 0 0 ...........0........0

1 0 . . . ......................

2 0 . . . ......................

3 0 . . . ......................

4 0 . . . ......................

. 0 . . . ......................

a 0 (a) (2a) (3a).........(a*b)....(n-1)*a ''' ( in this row, every element mod n)'''

....0.....................................

....0.....................................

....0.....................................

....0.....................................

(n-1) 0...................................
see the (a+1)th row

There must be a "1" in this row, actually each row or column, to meet the rule :Existence of Negatives/Inverses. (About the rule, seen in the file "12-240/Classnotes for Tuesday September 11")

So if F(n) is a field, then
there must exist k,m ∈N*, m<n
to meet the equation:
m*a=k*n+1

And we know that n=ab

So m*a=k*a*b+1 (a≠1)

Hence m=k*b+1/a

unless a=1
m will not exist, because m should be an integer.
so F(n), when n is not a prime number, is not a field.

There is a large need for me to improve my format. Editing is welcomed.

PS: But till now, there are still some questions existing.
How can we prove that a prime number N can absolutely form a field? Is there any exception?
I am still working on it.
----Michael.Wang
<nowiki></nowiki>

12-240/Fields' Further proof

2012-09-13T14:47:33Z

Michael.Wang:

[[12-240]][[Image:12-240-Splash.png]]

[[12-240/Classnotes for Tuesday September 11]]

In the first class, Professor says something about particular fields. Forgive me, because I am an international student, if I can not express information precisely.

About: F(n)

F(1) F(2) and F(3) are a field, but F(4) is not. Professor said that any number n which is not a prime number can not "form" a field F(n)

If you do not understand what the F(n) means, you can look through the file "12-240/Classnotes for Tuesday September 11"

'''Why all the numbers which are not prime numbers can not form a field F(n)?'''

Here is the proof.

If we have a field F(n), and n=a*b (a,b,n∈ N*, a,b≠1) , which means n is not a prime number.

IN defination of multiplication

* 0 1 2 3 .......... b.......n-1

0 0 0 0 0 ...........0.........0

1 0 . . . ......................

2 0 . . . ......................

3 0 . . . .......................

4 0 . . . .......................

.. 0 . . . .......................

a 0 (a) (2a) (3a)...........(a*b).....(n-1)*a ''' ( in this row, every element mod n)'''

....................................

....................................

......................................

.....................................

(n-1) 0................................

see the (a+1)th row

There must be a "1" in this row, actually each row or column, to meet the rule :Existence of Negatives/Inverses. (About the rule, seen in the file "12-240/Classnotes for Tuesday September 11")

So if F(n) is a field, then
there must exist k,m ∈N*, m<n
to meet the equation:
m*a=k*n+1

And we know that n=ab

So m*a=k*a*b+1 (a≠1)

Hence m=k*b+1/a

unless a=1
m will not exist, because m should be an integer.
so F(n), when n is not a prime number, is not a field.

There is a large need for me to improve my format. Editing is welcomed.

PS: But till now, there are still some questions existing.
How can we prove that a prime number N can absolutely form a field? Is there any exception?
I am still working on it.
----Michael.Wang

12-240/Fields' Further proof

2012-09-12T18:51:46Z

Michael.Wang:

[[Image:12-240-Splash.png]]
[[12-240/Classnotes for Thursday September 13]]
[[12-240/Classnotes for Tuesday September 11]]

In the first class, Professor says something about particular fields. Forgive me, because I am an international student, if I can not express information precisely.

About: F(n) F(1) F(2) and F(3) are a field, but F(4) is not. Professor said that any number N which is not a prime number can not "form" a field.

If you do not understand what the F(n) means, you can look through the file "12-240/Classnotes for Tuesday September 11"

'''Why all the numbers which are not prime numbers can not form a field F(n)?'''

Here is the proof.

If we have a field F(n), and n=a*b (a,b,n∈ N*, a,b≠1)

IN defination of multiplication

* 0 1 2 3 .......... b.......n-1

0 . . . . ......................

1 . . . . ......................

2 . . . . ......................

3 . . . . .......................

4 . . . . .......................

. . . . . .......................

a 0 (a) (2a) (3a)...........(a*b).....(n-1)*a ( in this row, every element mod n)

....................................

....................................

......................................

.....................................

n-1..................................

see the (a+1)th row

There must be a "1" in this row, actually each row or column, to meet the rule :Existence of Negatives/Inverses.

So if F(n) is a field, then
1.m*a=k*n+1 (k,m∈N*, m<n) there must exist k,m.
2.n=ab

==>>m*a=k*a*b+1 (a≠1)

==>>m=k*b+1/a

unless a=1
m will not exist, because m should be an integer.
so F(n), when n is not a prime number, is not a field.

There is a large need for me to improve my format. Editing is welcomed.

PS: But till now, there are still some questions existing.
How can we prove that a prime number N can absolutely form a field? Is there any exception?
I am still working on it.
----Michael.Wang

12-240/Fields' Further proof

2012-09-12T18:48:30Z

Michael.Wang:

[[Image:12-240-Splash.png]]
[[12-240/Classnotes for Thursday September 13]]
[[12-240/Classnotes for Tuesday September 11]]

In the first class, Professor says something about particular fields. Forgive me, because I am an international student, if I can not express information precisely.

About: F(n) F(1) F(2) and F(3) are a field, but F(4) is not. Professor said that any number N which is not a prime number can not "form" a field.

If you do not understand what the F(n) means, you can look through the file "12-240/Classnotes for Tuesday September 11"

Here is the proof.

If we have a field F(n), and n=a*b (a,b,n∈ N*, a,b≠1)

IN defination of multiplication

* 0 1 2 3 .......... b.......n-1

0 . . . . ......................

1 . . . . ......................

2 . . . . ......................

3 . . . . .......................

4 . . . . .......................

. . . . . .......................

a 0 (a) (2a) (3a)...........(a*b).....(n-1)*a ( in this row, every element mod n)

....................................

....................................

......................................

.....................................

n-1..................................

see the (a+1)th row

There must be a "1" in this row, actually each row or column, to meet the rule :Existence of Negatives/Inverses.

So if F(n) is a field, then
1.m*a=k*n+1 (k,m∈N*, m<n) there must exist k,m.
2.n=ab

==>>m*a=k*a*b+1 (a≠1)

==>>m=k*b+1/a

unless a=1
m will not exist, because m should be an integer.
so F(n), when n is not a prime number, is not a field.

There is a large need for me to improve my format. Editing is welcomed.

PS: But till now, there are still some questions existing.
How can we prove that a prime number N can absolutely form a field? Is there any exception?
I am still working on it.
----Michael.Wang

12-240/

2012-09-12T15:55:17Z

Michael.Wang: 12-240/ moved to 12-240/Fields' Further proof: Lack title

#redirect [[12-240/Fields' Further proof]]

12-240/Fields' Further proof

2012-09-12T15:55:17Z

Michael.Wang: 12-240/ moved to 12-240/Fields' Further proof

In the first class, Professor says something about particular fields. Forgive me, because I am an international student, if I can not express information precisely.

About: F(n) F(1) F(2) and F(3) are a field, but F(4) is not. Professor said that any number N which is not a prime number can not "form" a field. We should ask why...

Here is the proof.

If we have a field F(n), and n=a*b (a,b,n∈ N*, a,b≠1)

IN defination of multiplication

* 0 1 2 3 .......... b.......n-1

0 . . . . ......................

1 . . . . ......................

2 . . . . ......................

3 . . . . .......................

4 . . . . .......................

.

a 0 (a) (2a) (3a)...........(a*b).....(n-1)*a ( in this row, every element mod n)

.

.

.

.

n-1..................................

see the (a+1)th row

There must be a "1" in this row, each row precisely to meet the rule. ( The rule... you know, I cannot find some notations.)

If F(n) is a field, then
1.m*a=k*n+1 (k,m∈N*, m<n)
2.n=ab

==>>m=kb+1/a

unless a=1
m will not exist.
so F(n), when n is not a prime number, is not a field.

There is a large need for me to improve my format. Editing is welcomed.

PS: But till now, there are still some questions existing.
How can we prove that a prime number N can absolutely form a field? Is there any exception?
I am still working on it.
----Michael.Wang

12-240/Fields' Further proof

2012-09-12T14:51:44Z

Michael.Wang:

12-240/Fields' Further proof

2012-09-12T01:10:11Z

Michael.Wang:

In the first class, Professor says something about particular fields. Forgive me, because I am an international student, if I can not express information precisely.

About: F(n) F(1) F(2) and F(3) are a field, but F(4) is not. Professor said that any number N which is not a prime number can not "form" a field. We should ask why...

Here is the proof.

If we have a field F(n), and n=a*b (a,b,n∈ N*, a,b≠1)

IN defination of multiplication

* 0 1 2 3 .......... b.......n-1

0 . . . . ......................

1 . . . . ......................

2 . . . . ......................

3 . . . . .......................

4 . . . . .......................

.

a 0 (a) (2a) (3a)...........(a*b).....(n-1)*a ( in this row, every element mod n)

.

.

.

.

n-1..................................

see the (a+1)th row

If F(n) is a field, then
1.m*a=k*n+1 (k,m∈N*, m<n)
2.n=ab

==>>m=kb+1/a

unless a=1
m will not exist.
so F(n), when n is not a prime number, is not a field.

There is a large need for me to improve my format. Editing is welcomed.

PS: But till now, there are still some questions existing.
How can we prove that a prime number N can absolutely form a field? Is there any exception?
I am still working on it.
----Michael.Wang