Drorbn - User contributions [en]

07-401

2007-03-28T00:17:51Z

Andrei:

__NOEDITSECTION__
{{07-401/Navigation}}
==Polynomial Equations and Fields==
===Department of Mathematics, University of Toronto, Spring 2007===

{{07-401/Crucial Information}}

===Further Resources===

* [http://www.d.umn.edu/~jgallian/ J. Gallian's Algebra web site].
* [http://www.math.toronto.edu/undergrad/ Undergraduate Information] at the [http://www.math.toronto.edu/ UofT Math Department]
* [http://www.artsandscience.utoronto.ca/ofr/calendar/crs_mat.htm Undergraduate Course Descriptions] at the [http://www.artsci.utoronto.ca/ Faculty of Arts and Science].
* [http://www.math.toronto.edu/gor/mat401.html 06-401] with Julia Gordon.
* [http://www.math.utoronto.ca/shub/mat302S.html 05-302] with Mike Shub.
* [http://ccnet.utoronto.ca/20069/mat301h1f/ 06-301] with Lindsey Shorser.

{| align=center
|- align=center
|[[Image:07-401-Solving The Quartic With Mathematica.png|640px]]
|- align=center
|Click to see a [[07-401/More math-like|more math-like]] view for the above (a bit less simplified)
|- align=center
|Solving The Quartic With Mathematica. [http://en.wikipedia.org/wiki/Quartic_equation Read more!]
|}

07-401/More math-like

2007-03-28T00:14:46Z

Andrei:

The first solution for <math>ax^4 + bx^3 + cx^2 + dx + e</math>, as presented by [http://maxima.sourceforge.net/ Maxima] (use "tex(solve(a*x^4 + b*x^3 + c*x^2 + d*x + e,x)[1])" and plug into the wiki)

<math>
x=-{{2^ {- {{3}\over{2}} }\,\sqrt{-{{\left(18\,a^2\,\left({{\sqrt{-256 \,a^3\,e^3-\left(-192\,a^2\,b\,d-128\,a^2\,c^2+144\,a\,b^2\,c-27\,b^4 \right)\,e^2-\left(\left(144\,a^2\,c-6\,a\,b^2\right)\,d^2+\left(18 \,b^3\,c-80\,a\,b\,c^2\right)\,d+16\,a\,c^4-4\,b^2\,c^3\right)\,e+27 \,a^2\,d^4-\left(18\,a\,b\,c-4\,b^3\right)\,d^3-\left(b^2\,c^2-4\,a \,c^3\right)\,d^2}}\over{6\,\sqrt{3}\,a^3}}-{{a\,\left(72\,c\,e-27\, d^2\right)-27\,b^2\,e+9\,b\,c\,d-2\,c^3}\over{54\,a^3}}\right)^{{{2 }\over{3}}}+\left(24\,a\,c-9\,b^2\right)\,\left({{\sqrt{-256\,a^3\,e ^3-\left(-192\,a^2\,b\,d-128\,a^2\,c^2+144\,a\,b^2\,c-27\,b^4\right) \,e^2-\left(\left(144\,a^2\,c-6\,a\,b^2\right)\,d^2+\left(18\,b^3\,c -80\,a\,b\,c^2\right)\,d+16\,a\,c^4-4\,b^2\,c^3\right)\,e+27\,a^2\,d ^4-\left(18\,a\,b\,c-4\,b^3\right)\,d^3-\left(b^2\,c^2-4\,a\,c^3 \right)\,d^2}}\over{6\,\sqrt{3}\,a^3}}-{{a\,\left(72\,c\,e-27\,d^2 \right)-27\,b^2\,e+9\,b\,c\,d-2\,c^3}\over{54\,a^3}}\right)^{{{1 }\over{3}}}+24\,a\,e-6\,b\,d+2\,c^2\right)\,\sqrt{{{36\,a^2\,\left( {{\sqrt{-256\,a^3\,e^3-\left(-192\,a^2\,b\,d-128\,a^2\,c^2+144\,a\,b ^2\,c-27\,b^4\right)\,e^2-\left(\left(144\,a^2\,c-6\,a\,b^2\right)\, d^2+\left(18\,b^3\,c-80\,a\,b\,c^2\right)\,d+16\,a\,c^4-4\,b^2\,c^3 \right)\,e+27\,a^2\,d^4-\left(18\,a\,b\,c-4\,b^3\right)\,d^3-\left(b ^2\,c^2-4\,a\,c^3\right)\,d^2}}\over{6\,\sqrt{3}\,a^3}}-{{a\,\left(72 \,c\,e-27\,d^2\right)-27\,b^2\,e+9\,b\,c\,d-2\,c^3}\over{54\,a^3}} \right)^{{{2}\over{3}}}+\left(9\,b^2-24\,a\,c\right)\,\left({{\sqrt{ -256\,a^3\,e^3-\left(-192\,a^2\,b\,d-128\,a^2\,c^2+144\,a\,b^2\,c-27 \,b^4\right)\,e^2-\left(\left(144\,a^2\,c-6\,a\,b^2\right)\,d^2+ \left(18\,b^3\,c-80\,a\,b\,c^2\right)\,d+16\,a\,c^4-4\,b^2\,c^3 \right)\,e+27\,a^2\,d^4-\left(18\,a\,b\,c-4\,b^3\right)\,d^3-\left(b ^2\,c^2-4\,a\,c^3\right)\,d^2}}\over{6\,\sqrt{3}\,a^3}}-{{a\,\left(72 \,c\,e-27\,d^2\right)-27\,b^2\,e+9\,b\,c\,d-2\,c^3}\over{54\,a^3}} \right)^{{{1}\over{3}}}+48\,a\,e-12\,b\,d+4\,c^2}\over{\left({{ \sqrt{-256\,a^3\,e^3-\left(-192\,a^2\,b\,d-128\,a^2\,c^2+144\,a\,b^2 \,c-27\,b^4\right)\,e^2-\left(\left(144\,a^2\,c-6\,a\,b^2\right)\,d^2 +\left(18\,b^3\,c-80\,a\,b\,c^2\right)\,d+16\,a\,c^4-4\,b^2\,c^3 \right)\,e+27\,a^2\,d^4-\left(18\,a\,b\,c-4\,b^3\right)\,d^3-\left(b ^2\,c^2-4\,a\,c^3\right)\,d^2}}\over{6\,\sqrt{3}\,a^3}}-{{a\,\left(72 \,c\,e-27\,d^2\right)-27\,b^2\,e+9\,b\,c\,d-2\,c^3}\over{54\,a^3}} \right)^{{{1}\over{3}}}}}}+\left(-216\,a^2\,d+108\,a\,b\,c-27\,b^3 \right)\,\left({{\sqrt{-256\,a^3\,e^3-\left(-192\,a^2\,b\,d-128\,a^2 \,c^2+144\,a\,b^2\,c-27\,b^4\right)\,e^2-\left(\left(144\,a^2\,c-6\, a\,b^2\right)\,d^2+\left(18\,b^3\,c-80\,a\,b\,c^2\right)\,d+16\,a\,c ^4-4\,b^2\,c^3\right)\,e+27\,a^2\,d^4-\left(18\,a\,b\,c-4\,b^3 \right)\,d^3-\left(b^2\,c^2-4\,a\,c^3\right)\,d^2}}\over{6\,\sqrt{3} \,a^3}}-{{a\,\left(72\,c\,e-27\,d^2\right)-27\,b^2\,e+9\,b\,c\,d-2\, c^3}\over{54\,a^3}}\right)^{{{1}\over{3}}}}\over{\left({{\sqrt{-256 \,a^3\,e^3-\left(-192\,a^2\,b\,d-128\,a^2\,c^2+144\,a\,b^2\,c-27\,b^4 \right)\,e^2-\left(\left(144\,a^2\,c-6\,a\,b^2\right)\,d^2+\left(18 \,b^3\,c-80\,a\,b\,c^2\right)\,d+16\,a\,c^4-4\,b^2\,c^3\right)\,e+27 \,a^2\,d^4-\left(18\,a\,b\,c-4\,b^3\right)\,d^3-\left(b^2\,c^2-4\,a \,c^3\right)\,d^2}}\over{6\,\sqrt{3}\,a^3}}-{{a\,\left(72\,c\,e-27\, d^2\right)-27\,b^2\,e+9\,b\,c\,d-2\,c^3}\over{54\,a^3}}\right)^{{{1 }\over{3}}}}}}}\over{3\,a\,\left({{36\,a^2\,\left({{\sqrt{-256\,a^3 \,e^3-\left(-192\,a^2\,b\,d-128\,a^2\,c^2+144\,a\,b^2\,c-27\,b^4 \right)\,e^2-\left(\left(144\,a^2\,c-6\,a\,b^2\right)\,d^2+\left(18 \,b^3\,c-80\,a\,b\,c^2\right)\,d+16\,a\,c^4-4\,b^2\,c^3\right)\,e+27 \,a^2\,d^4-\left(18\,a\,b\,c-4\,b^3\right)\,d^3-\left(b^2\,c^2-4\,a \,c^3\right)\,d^2}}\over{6\,\sqrt{3}\,a^3}}-{{a\,\left(72\,c\,e-27\, d^2\right)-27\,b^2\,e+9\,b\,c\,d-2\,c^3}\over{54\,a^3}}\right)^{{{2 }\over{3}}}+\left(9\,b^2-24\,a\,c\right)\,\left({{\sqrt{-256\,a^3\,e ^3-\left(-192\,a^2\,b\,d-128\,a^2\,c^2+144\,a\,b^2\,c-27\,b^4\right) \,e^2-\left(\left(144\,a^2\,c-6\,a\,b^2\right)\,d^2+\left(18\,b^3\,c -80\,a\,b\,c^2\right)\,d+16\,a\,c^4-4\,b^2\,c^3\right)\,e+27\,a^2\,d ^4-\left(18\,a\,b\,c-4\,b^3\right)\,d^3-\left(b^2\,c^2-4\,a\,c^3 \right)\,d^2}}\over{6\,\sqrt{3}\,a^3}}-{{a\,\left(72\,c\,e-27\,d^2 \right)-27\,b^2\,e+9\,b\,c\,d-2\,c^3}\over{54\,a^3}}\right)^{{{1 }\over{3}}}+48\,a\,e-12\,b\,d+4\,c^2}\over{\left({{\sqrt{-256\,a^3\, e^3-\left(-192\,a^2\,b\,d-128\,a^2\,c^2+144\,a\,b^2\,c-27\,b^4 \right)\,e^2-\left(\left(144\,a^2\,c-6\,a\,b^2\right)\,d^2+\left(18 \,b^3\,c-80\,a\,b\,c^2\right)\,d+16\,a\,c^4-4\,b^2\,c^3\right)\,e+27 \,a^2\,d^4-\left(18\,a\,b\,c-4\,b^3\right)\,d^3-\left(b^2\,c^2-4\,a \,c^3\right)\,d^2}}\over{6\,\sqrt{3}\,a^3}}-{{a\,\left(72\,c\,e-27\, d^2\right)-27\,b^2\,e+9\,b\,c\,d-2\,c^3}\over{54\,a^3}}\right)^{{{1 }\over{3}}}}}\right)^{{{1}\over{4}}}}}-{{\sqrt{{{36\,a^2\,\left({{ \sqrt{-256\,a^3\,e^3-\left(-192\,a^2\,b\,d-128\,a^2\,c^2+144\,a\,b^2 \,c-27\,b^4\right)\,e^2-\left(\left(144\,a^2\,c-6\,a\,b^2\right)\,d^2 +\left(18\,b^3\,c-80\,a\,b\,c^2\right)\,d+16\,a\,c^4-4\,b^2\,c^3 \right)\,e+27\,a^2\,d^4-\left(18\,a\,b\,c-4\,b^3\right)\,d^3-\left(b ^2\,c^2-4\,a\,c^3\right)\,d^2}}\over{6\,\sqrt{3}\,a^3}}-{{a\,\left(72 \,c\,e-27\,d^2\right)-27\,b^2\,e+9\,b\,c\,d-2\,c^3}\over{54\,a^3}} \right)^{{{2}\over{3}}}+\left(9\,b^2-24\,a\,c\right)\,\left({{\sqrt{ -256\,a^3\,e^3-\left(-192\,a^2\,b\,d-128\,a^2\,c^2+144\,a\,b^2\,c-27 \,b^4\right)\,e^2-\left(\left(144\,a^2\,c-6\,a\,b^2\right)\,d^2+ \left(18\,b^3\,c-80\,a\,b\,c^2\right)\,d+16\,a\,c^4-4\,b^2\,c^3 \right)\,e+27\,a^2\,d^4-\left(18\,a\,b\,c-4\,b^3\right)\,d^3-\left(b ^2\,c^2-4\,a\,c^3\right)\,d^2}}\over{6\,\sqrt{3}\,a^3}}-{{a\,\left(72 \,c\,e-27\,d^2\right)-27\,b^2\,e+9\,b\,c\,d-2\,c^3}\over{54\,a^3}} \right)^{{{1}\over{3}}}+48\,a\,e-12\,b\,d+4\,c^2}\over{\left({{ \sqrt{-256\,a^3\,e^3-\left(-192\,a^2\,b\,d-128\,a^2\,c^2+144\,a\,b^2 \,c-27\,b^4\right)\,e^2-\left(\left(144\,a^2\,c-6\,a\,b^2\right)\,d^2 +\left(18\,b^3\,c-80\,a\,b\,c^2\right)\,d+16\,a\,c^4-4\,b^2\,c^3 \right)\,e+27\,a^2\,d^4-\left(18\,a\,b\,c-4\,b^3\right)\,d^3-\left(b ^2\,c^2-4\,a\,c^3\right)\,d^2}}\over{6\,\sqrt{3}\,a^3}}-{{a\,\left(72 \,c\,e-27\,d^2\right)-27\,b^2\,e+9\,b\,c\,d-2\,c^3}\over{54\,a^3}} \right)^{{{1}\over{3}}}}}}}\over{12\,a}}-{{b}\over{4\,a}}
</math>

If you think that is bad, take a look at the source of this page :-)

07-401

2007-02-01T01:35:22Z

Andrei:

__NOEDITSECTION__
{{07-401/Navigation}}
==Polynomial Equations and Fields==
===Department of Mathematics, University of Toronto, Spring 2007===

{{07-401/Crucial Information}}

===Further Resources===

* [http://www.d.umn.edu/~jgallian/ J. Gallian's Algebra web site].
* [http://www.math.toronto.edu/undergrad/ Undergraduate Information] at the [http://www.math.toronto.edu/ UofT Math Department]
* [http://www.artsandscience.utoronto.ca/ofr/calendar/crs_mat.htm Undergraduate Course Descriptions] at the [http://www.artsci.utoronto.ca/ Faculty of Arts and Science].
* [http://www.math.toronto.edu/gor/mat401.html 06-401] with Julia Gordon.
* [http://www.math.utoronto.ca/shub/mat302S.html 05-302] with Mike Shub.
* [http://ccnet.utoronto.ca/20069/mat301h1f/ 06-301] with Lindsey Shorser.

{| align=center
|- align=center
|[[Image:07-401-Solving The Quartic With Mathematica.png|640px]]
|- align=center
|Click to see a [[more math-like]] view for the above (a bit less simplified)
|- align=center
|Solving The Quartic With Mathematica. [http://en.wikipedia.org/wiki/Quartic_equation Read more!]
|}

07-401/Class Photo

2007-02-01T01:17:12Z

Andrei:

{{07-401/Navigation}}

Our class on Janury 24, 2007:

[[Image:07-401 Class Photo.jpg|thumb|centre|500px|Class Photo: click to enlarge]]

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
|-
!First name
!Last name
!UserID
!Email
!In the 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}}
{{Photo Entry|last=Abdel-Aziz|first=Mohamed|userid=Mizo|email=mohamed.abdelaziz@utoronto.ca|location=Absent|comments=no comment.}}
{{Photo Entry|last=Baxter|first=Matthew|userid=MBaxter|email=matthew.baxter @ utoronto.com|location=2nd row(from back) 2nd from right|comments=no comment.}}
{{Photo Entry|last=Castro|first=Caroline|userid=Carrot|email=caroline.castro @ utoronto.ca|
location=the tall person at the very top of the stairs with the grey hat|comments=no comment.}}
{{Photo Entry|last=Henderson|first=Sean|userid=sean_henderson|email=sean.henderson [at] utoronto.ca|location=front row, far left|comments=no comment.}}
{{Photo Entry|last=Janic|first=Vega|userid=elf18|email=vega.janic @ utoronto.ca|location=second row from the back, to the left-hand side, wearing glasses and appearing to blink - drat!|comments=no comment.}}
{{Photo Entry|last=Kang|first=Laura|userid=Clkkang|email=clkkang @ gmail.com|location=front row, 3rd person from right|comments=no comment.}}
{{Photo Entry|last=Kim|first=Sang-Eun Vivien|userid=Vivienkim|email=vivien.kim @ utoronto.ca|location=from left, fourth row from top of page, 3rd person from the left in the black shirt|comments=no comment.}}
{{Photo Entry|last=Litvin|first=Andrei|userid=Andrei|email=litvin_andrei_@yahoo.com (remove the "_")|location=[[Image:andrei_litvin_in_pic.png]]|comments=no comment.}}
{{Photo Entry|last=Maniak|first=Sylvia|userid=sm|email=sylvia.maniak [at] gmail.com|location=front row, the blonde wearing a black shirt|comments=no comment.}}
{{Photo Entry|last=Nah|first=Theodorah|userid=Theo|email=theo.nahemt @ gmail.com|location=second row, 1/4 of my face covered|comments=no comment.}}
{{Photo Entry|last=O'Brien|first=Brenna|userid=Brenna|email=brenna.obrien [at] utoronto.ca|location=4th row from bottom, far left|comments=no comment.}}
{{Photo Entry|last=Sidhu|first=Bikram|userid=sidhubik|email=bikram.sidhu@utoronto.ca|location=Second Row, far right, black/white striped polo and grey/black hat|comments=no comment.}}
{{Photo Entry|last=Wang|first=Chunyi|userid=ucc|email=chunyi.wang [at] utoronto.ca|location=second row, DIADORA|comments=no comment.}}
{{Photo Entry|last=Muthukrishnan|first=Dilip|userid=dm_iyer|email=dm_iyer@hotmail.com|location=front row, bright yellow sweatshirt!|comments=no comment.}}
{{Photo Entry|last=Wu|first=JunRu|userid=JunRu|email=junru.wu [at] utoronto.ca|location=4th row,the second person from left wearing glasses in black coat !|comments=no comment.}}
{{Photo Entry|last=Wu|first=Kody|userid=Manu7|email=kody.wu@utoronto.ca|location=4th row, the first guy from the right with his mouth cut off. |comments=no comment.}}

File:Andrei litvin in pic.png

2007-02-01T01:15:47Z

Andrei: My picture ... easier than to explain my position in the picture.

My picture ... easier than to explain my position in the picture.