cartan.math.umb.edu - User contributions [en]

Math 480, Spring 2013, Assignment 1

2013-02-01T04:07:37Z

Patrick.Mclaren: Math formatting

''The beginner ... should not be discouraged if ... he finds that he does not have the prerequisites for reading the prerequisites.''

: - P. Halmos

==Solve the following problems:==

# Name two familiar arithmetic structures that are fields, and two familiar arithmetic structures that are not.
# Find an integer <math>n</math> for which the structure <math>\mathbb{Z}_n</math> is ''not'' a field. Write the multiplication table for this structure, and use the table to explain why, for this choice of <math>n</math>, <math>\mathbb{Z}_n</math> is not a field. Then choose another <math>n</math> for which <math>\mathbb{Z}_n</math> ''is'' a field, and use the resulting multiplication table to verify this.
# Find a non-zero polynomial with coefficients in <math>\mathbb{Z}_3</math> whose associated function vanishes identically.
# What is the degree of the polynomial <math>x^3 + x^2yz</math>?
# Give an example of a field which is ''not'' algebraically closed, and prove that it is not.

==Carefully ''state'' the following theorems:==

# Theorem on vanishing polynomial functions over an infinite field (Proposition 5 in the text).

==Discussion:==

Here is the place to post questions!

What's the degree of this polynomial and how many terms does it have?

<math>P(x) = (x-a)(x-b)...(x-z) + 1</math>

Math 480, Computational Algebraic Geometry

2012-06-04T17:22:13Z

Patrick.Mclaren: Fixed google code typo

Math 480 -- Computational Algebraic Geometry

Welcome to the wiki! Editing this page is exactly like editing [http://en.wikipedia.org/wiki/Main_page Wikipedia]. You may wish to see their help pages on [http://en.wikipedia.org/wiki/Help:Contents/Editing_Wikipedia editing] and on [http://en.wikipedia.org/wiki/MOS:MATH#Typesetting_of_mathematical_formulae typesetting mathematics].

;Course Textbook
The course mainly follows [http://www.cs.amherst.edu/~dac/iva.html Ideals, Varieties, and Algorithms] by Cox, Little, and O'Shea, however other textbooks can also provide supplementary material.

* [http://math.stanford.edu/~vakil/216blog/ Algebraic Geometry] notes by Ravi Vakil
* [http://www.math.lsa.umich.edu/~idolga/631.pdf Introduction to Algebraic Geometry] notes by Igor V. Dolgachev

== Overview ==

One may read about the history of Algebraic Geometry in the Wikipedia [http://en.wikipedia.org/wiki/Algebraic_geometry article] on the subject. There is also thorough examination of the history of the subject written by Dieudonne, which is available [http://mathdl.maa.org/images/upload_library/22/Ford/Dieudonne.pdf here].

Algebraic Geometry is a very active research field -- a quick glance at the recent additions to the [http://arxiv.org/list/math.AG/recent arXiv] supports this assertion. There are quite a few research opportunities for undergraduates interested in Algebraic Geometry, especially in the North-Eastern region of the US. [http://www.agneshome.org/ AGNES] is a series of biannual weekend workshops in algebraic geometry. MIT and Harvard [http://math.mit.edu/seminars/ags host] a Algebraic Geometry seminar. Oakland University in Michigan organizes a yearly conference in Algebraic Geometry ([https://sites.google.com/a/oakland.edu/algebra/home MCAG]). SIAM AG hosts a Algebraic Geometry [http://lists.siam.org/mailman/listinfo/siam-ag mailing list]. Finally, there is a REU program in Algebraic Geometry at [http://www.math.clemson.edu/~kevja/REU/ Clemson University] in South Carolina.

== Course Structure ==

A typical semester spent on this course will reach Chapter 7 of the course textbook, "Invariant Theory of Finite Groups". Progress through the material can either be measured weekly through the completion of programming assignments or short presentations of material. A source code repository for the course is available [http://code.google.com/p/umb-computational-algebraic-geometry/ here].

* Chapter 1: Geometry, Algebra, and Algorithms
** Programming Assignment:
::Write a python program which can determine whether a polynomial is within the ideal generated by a given list of polynomials.
* Chapter 2: Groebner Bases
** Programming Assignment:
::Implement the multi-variable division algorithm in Python.
::Implement Buchberger's Algorithm.
* Chapter 3: Elimination Theory
** Programming Assignment:
* Chapter 4: The Algebra-Geometry Dictionary
** Programming Assignment:
* Chapter 5: Polynomial and Rational Functions on a Variety
** Programming Assignment:
* Chapter 6: Robotics and Automatic Geometric Theorem Proving
** Programming Assignment:
* Chapter 7: Invariant Theory of Finite Groups
** Programming Assignment:

== Questions ==

=== Programming Assignments ===

==== Chapter 1 ====

==== Chapter 2 ====

===== Representing Multivariable Polynomials =====

I think I'll implement the polynomials as a list of dictionaries, so what I'll be storing is a list of unordered monomials:

Eg.

<pre>p = [{ 'c' : 6, 'x_1' : 2 , 'x_2' : 3, 'x_3' : 1}, { 'c' : 2, 'x_1' : 4, 'x_2' : 4, 'x_3' : 2}]</pre>
would represent <math>6x_1^2x_2^3x_3 + 2x_1^4x_2^4x_3^2</math>

For a moment, I thought about implementing the polynomial as a class, and add orderings as a class method. However, I think it would be more correct to order a polynomial by calling a library function:

<pre>ordered_p = polynomial.order(p, 'grevlex')</pre>

Or something similar; I don't think returning differently ordered lists as a class method is very Python like, which is why I'm leaning towards a library/module function. [[User:Patrickmclaren|Patrickmclaren]] 02:49, 3 May 2012 (BST)

I think Sage has a class for rings and creates an instance of this class to store things like variable names and monomial orders. Then it has enother class for ring elements; instances of this class store things like monomials and coefficients (and an identifier for the ring that the element belongs to). Then the elements get simpler -- for example I think your polynomial above could get stored as a single dictionary, say

<pre>p = {(2, 3, 1): 6, (4, 4, 2): 2}</pre>

and the __repr__ method asks the ring for the variable names and the monomial ordering.
[[User:Steven Glenn Jackson|Steven Glenn Jackson]] 12:04, 3 May 2012 (BST)

Math 480, Computational Algebraic Geometry

2012-05-03T01:49:00Z

Patrick.Mclaren: Question regarding polynomial implementation

Math 480 -- Computational Algebraic Geometry

Welcome to the wiki! Editing this page is exactly like editing [http://en.wikipedia.org/wiki/Main_page Wikipedia]. You may wish to see their help pages on [http://en.wikipedia.org/wiki/Help:Contents/Editing_Wikipedia editing] and on [http://en.wikipedia.org/wiki/MOS:MATH#Typesetting_of_mathematical_formulae typesetting mathematics].

;Course Textbook
The course mainly follows [http://www.cs.amherst.edu/~dac/iva.html Ideals, Varieties, and Algorithms] by Cox, Little, and O'Shea, however other textbooks can also provide supplementary material.

* [http://math.stanford.edu/~vakil/216blog/ Algebraic Geometry] notes by Ravi Vakil
* [http://www.math.lsa.umich.edu/~idolga/631.pdf Introduction to Algebraic Geometry] notes by Igor V. Dolgachev

== Overview ==

One may read about the history of Algebraic Geometry in the Wikipedia [http://en.wikipedia.org/wiki/Algebraic_geometry article] on the subject. There is also thorough examination of the history of the subject written by Dieudonne, which is available [http://mathdl.maa.org/images/upload_library/22/Ford/Dieudonne.pdf here].

Algebraic Geometry is a very active research field -- a quick glance at the recent additions to the [http://arxiv.org/list/math.AG/recent arXiv] supports this assertion. There are quite a few research opportunities for undergraduates interested in Algebraic Geometry, especially in the North-Eastern region of the US. [http://www.agneshome.org/ AGNES] is a series of biannual weekend workshops in algebraic geometry. MIT and Harvard [http://math.mit.edu/seminars/ags host] a Algebraic Geometry seminar. Oakland University in Michigan organizes a yearly conference in Algebraic Geometry ([https://sites.google.com/a/oakland.edu/algebra/home MCAG]). SIAM AG hosts a Algebraic Geometry [http://lists.siam.org/mailman/listinfo/siam-ag mailing list]. Finally, there is a REU program in Algebraic Geometry at [http://www.math.clemson.edu/~kevja/REU/ Clemson University] in South Carolina.

== Course Structure ==

A typical semester spent on this course will reach Chapter 7 of the course textbook, "Invariant Theory of Finite Groups". Progress through the material can either be measured weekly through the completion of programming assignments or short presentations of material. A source code repository for the course is available [http://code.google.com/p/umb-compuatational-algebraic-geometry/ here].

* Chapter 1: Geometry, Algebra, and Algorithms
** Programming Assignment:
::Write a python program which can determine whether a polynomial is within the ideal generated by a given list of polynomials.
* Chapter 2: Groebner Bases
** Programming Assignment:
::Implement the multi-variable division algorithm in Python.
::Implement Buchberger's Algorithm.
* Chapter 3: Elimination Theory
** Programming Assignment:
* Chapter 4: The Algebra-Geometry Dictionary
** Programming Assignment:
* Chapter 5: Polynomial and Rational Functions on a Variety
** Programming Assignment:
* Chapter 6: Robotics and Automatic Geometric Theorem Proving
** Programming Assignment:
* Chapter 7: Invariant Theory of Finite Groups
** Programming Assignment:

== Questions ==

=== Programming Assignments ===

==== Chapter 1 ====

==== Chapter 2 ====

===== Representing Multivariable Polynomials =====

I think I'll implement the polynomials as a list of dictionaries, so what I'll be storing is a list of unordered monomials:

Eg.

<pre>p = [{ 'c' : 6, 'x_1' : 2 , 'x_2' : 3, 'x_3' : 1}, { 'c' : 2, 'x_1' : 4, 'x_2' : 4, 'x_3' : 2}]</pre>
would represent <math>6x_1^2x_2^3x_3 + 2x_1^4x_2^4x_3^2</math>

For a moment, I thought about implementing the polynomial as a class, and add orderings as a class method. However, I think it would be more correct to order a polynomial by calling a library function:

<pre>ordered_p = polynomial.order(p, 'grevlex')</pre>

Or something similar; I don't think returning differently ordered lists as a class method is very Python like, which is why I'm leaning towards a library/module function. [[User:Patrickmclaren|Patrickmclaren]] 02:49, 3 May 2012 (BST)

Math 480, Computational Algebraic Geometry

2012-05-01T22:08:49Z

Patrick.Mclaren: Added REU

Math 480, Computational Algebraic Geometry

2012-05-01T14:35:34Z

Patrick.Mclaren: Added source code repository.

Math 480, Computational Algebraic Geometry

2012-05-01T14:28:29Z

Patrick.Mclaren: Added history, course structure

Math 480, Computational Algebraic Geometry

2012-05-01T13:10:52Z

Patrick.Mclaren: Created page

Math 260, Spring 2012

2012-04-04T01:12:26Z

Patrick.Mclaren: Question on isometries.

'''Math 260 --- Linear Algebra I'''

Welcome to the wiki! Editing this page is exactly like editing [http://en.wikipedia.org/wiki/Main_page Wikipedia]. You may wish to see their help pages on [http://en.wikipedia.org/wiki/Help:Contents/Editing_Wikipedia editing] and on [http://en.wikipedia.org/wiki/MOS:MATH#Typesetting_of_mathematical_formulae typesetting mathematics].

(It isn't as hard as the documentation might make it seem. To see how to typeset the sentence "Consider a linear transformation <math>T:R^2\rightarrow R^2</math>," click the "edit" link at the top of this page and read the source code that generated it.)

[[User:Steven Glenn Jackson|Steven Glenn Jackson]] 02:42, 1 March 2012 (UTC)

=Important Dates=

* 02/28/2012 - Exam 1
* 04/17/2012 - Exam 2

=Questions=

* If the kernel of a linear transformation is nontrivial, does that imply that the transformation is not injective and therefore the transformation matrix is not invertible? [[User:Patrickmclaren|Patrickmclaren]] 02:11, 6 March 2012 (GMT)
: Yes. If the kernel of <math>T</math> is non-trivial, then we have some non-zero <math>\vec{v}</math> with <math>T(\vec{v})=\vec{0}</math>. But also <math>T(\vec{0})=\vec{0}</math>, so <math>T</math> is not one-to-one and hence not invertible. [[User:Steven Glenn Jackson|Steven Glenn Jackson]] 15:27, 6 March 2012 (GMT)

* If <math>\varphi:\mathbb{R}^n\to\mathbb{R}^n</math> is an isometry that fixes the origin: <math>\varphi(0) = 0</math>, then does <math>\varphi</math> preserve dot products? [[User:Patrickmclaren|Patrickmclaren]] 02:12, 4 April 2012 (BST)

=Links=

* [http://www.wolframalpha.com/ WolframAlpha]
* [http://reference.wolfram.com/mathematica/guide/MatricesAndLinearAlgebra.html Matrices and Linear Algebra in Mathematica]
* [http://sagemath.org/ Sage], a free and open-source alternative to Mathematica and other mathematics software. (It can either be [http://sagemath.org/download-linux.html downloaded] or used online through a [http://www.sagenb.org/ web interface].)
* A brief guide to [http://sagemath.org/doc/tutorial/tour_linalg.html linear algebra in Sage]. (Note that by default Sage uses row vectors and left kernels instead of column vectors and right kernels, so if you're not careful you may get into trouble with these functions. But its reduced row echelon forms are the same as ours.)

Math 260, Spring 2012

2012-03-06T02:46:36Z

Patrick.Mclaren: Added a link to documentation for using matrices with Mathematica

Math 260, Spring 2012

2012-03-06T02:11:11Z

Patrick.Mclaren: Added sections and a question.