Do You Know The Parallel in gragh Theory?

March 23, 2009 at 2:00 pm | Posted in Math Learning | 2 Comments
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A First Look at graph Theory

A First Look at graph Theory

“A First Look At Graph Theory” is  a good book for me if you want me to select books on the library.

Sorry for those like to vist these garden. There’s  some reasons made me leave off the blog for several days and not touched this one for some days. Thanks those can push me to the place I like. Appreciated here with lots of tears  for a hard and tiny life.

Graph Theory is the course that I never majored in before. But the interesting questions always attract some person.

But today, I am not to introduce the book. Gee just find some “interesting” and also “painful” things to share friends. Thus you can know the one, to be a mathematical student, is so great a person to distinguish these and study so much things that most people doesn’t underestand. You really deserve the applaus in several places.

Here are some examples from John Clark & Derek Allan Holon, 1999, p.7

Q1. Do you know “parallel” in junior high school?

Q2Do you know the “isolated” in complex variable?

Q3Do you know “simple” in Abstract algebra?

Think those questions and try to look at the following:

graph-definition

After look at these…

You really know what you learned before?

What is “parallel” “isolated“and “simple” in your mind?

Of course,…You can also choose to forget what the book wrote. That maybe the best way for most people, though the teachers who teach graph theory may stop you to do these stupid actions.><

“To be a student in the department of mathematics” is really a difficult and hard mission to overcome…Especially to  practice and describe one things with several ways in that fields like these.

Lie groups to Lie algebras

September 4, 2008 at 2:15 pm | Posted in History and News, Math Learning | Leave a comment
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Differentail Geometry, Lie Groups, and Symmetric Spaces by Sigurder Helgason

Differentail Geometry, Lie Groups, and Symmetric Spaces by Sigurdur Helgason

Translating 18.755 Introduction to Lie groups, Fall 2004 remind me the book. It’s I-hsun told me the book and my kindly friend ,Yu-Xuan who studied Computer Science in NCYU now, sent me the sweet gift. He plundered lots of books from the general cleaning of the library of the departmet of mathematics in NTU that year, and I the lucky guy.  We cannot find the book in the bookstore now because of the copyright arguments. The book from 凡異-Press is out of print recently.

Definition.  A Lie group is a group G which is also an analytic manifold such that the mapping (σ,τ) →στ-1 of the product manifold G × G into G is analytic.

Definition. A Lie algebra is an algebra g which operation[,] satisfied [x,x]=0 and Jacobi identity…(please click Bala-bala Lie Algebras).

It always confused me what the relation between Lie groups and Lie algebras. What is the difference between them? What’s the property of them? They must have some relations…what are them? Is Lie groups can be an Lie algebra?

In wikipedia, Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure. Lie algebra is said to be an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. But we know that Lie algebra not only can be used in geometic objects, but can be discussed in an independent algebraic, graphing or combination view today.  They contribute to modern physics a lot. Imagine the Lie group just like the rotation system, and Lie algebras are angular momentums. With some differetialble function, we must can know the relation between them.(courtesy of my friend, Meng-Syun)

Let me told a brief history of Lie group of Sophus Lie. Something occurred to him in 1874 that he tried to create a theory for transformation groups which might do for differential equations what Galios theory did for algebraic equations. It implies that Lie group must play an important role to analytic method on algebraic structure. And these works of he also brings us the theory and definition of Lie groups. Back to the history, after he got an idea to take effort on the theory in 1867, he went to Berlin and Paris, Klein and Jordan influence he a lot. They affect his thoughts in a geometric view to build more about Lie theory.  As for Lie algebras, We should say thanks to Jacobi, He generated further solutions of differential equations from a translation group to an “infinitesimal group” which we called it “Lie algebra” today. This is the beginning of Lie algebra. Here be mentioned that Killing took efforts on the association between Lie groups and Lie algebras, and Cartan completed the classification of semi-simple Lie algebras, Lie theory was finally developed on its own. And now, You may used these them all to the final structure of rotation momentums.

If you don’t know how Lie algebras be used…Try back to study Lie group is an good way…Because I just look at the text book but not notice the history of them; just roughly book collection but not suck the essence of them, it confused me that why should learn so many symbols with no meanings. Will I not get lost if I learn these with no ground on?  When back to the history, you will find the natural motivation problem to solved that build the world. Just show my experience.

You may also try to read the book, Anthony W. Knapp. Lie Groups Beyond an Introduction: Beyond an Introduction, 2ndedt. Birkhäuser, 2002, ISBN:0817642595, 9780817642594. (http://books.google.com.tw/books?id=U573NrppkA8C&printsec=frontcover), first if you did not touch the introduction of Lie groups and Lie algebras, You will know what I mean… To know the knowledge structure and the history thoughts are quite important for mathematical learning, or you may get lost in the ocean like me.

Reference:

  1. Singurdur Helgason. Differential Geometry, Lie Groups, and Symmetric Spaces. Academic Press, 1978. recopy by 凡異-Press.
  2. Sophus Lie‘s history from MacTotor History of Mathematics archive.
  3. Lie group and Lie group in Wikipedia.

Bala-bala Lie algebras

August 16, 2008 at 1:40 am | Posted in Math Learning | 1 Comment
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The cover of "Introduction to Lie algebras and Representation Theory", translated by Jui-Ji Chang

Introduction to Lie algebras and Representation Theory, translated by Jui-Ji Chang

Gee was so happy to met Jyun-Ao , a student of  Shun-Jen, remind me to pick my memory about Lie algebras…  It had been several years since I was struggle for knowing what’s the key math touch my mind. Lie algebras was really a unexpected beauty for me. The history of me to touch Lie algebras maybe an accident(or not)…

  • Beginning, Wen Long Lin who taught Applied mathematics in department of physics shows me two books of Lie algebras in 2001. Which are:(A)Howard Gergi. Lie Algeras in Partical Physics(1999). ISBN:0-738-20233-9. (B)Robert . Chahn. Semi-Simle Lie Algebras & their Representations.BenJamin/cummings(1984). ISBN:0-805-31600-0. …He said “Department of Mathematics may be better than Department of Physics if you want to learn these.”… Did not agree with him that time because I never saw or learned Lie algebras in Department of Mathematics till met him.
  • Found what “root system”is of the paper on the webpage I saw in 2004 : Valentina Golubeva:Hamiltonians of the Calogeto-Sutherland Type Models Associated to the Root Systems and Corresponding Fock Spaces….For understanding this paper and others I did not mention here, I was been kicked out the door of Roger. It became a penal pain pricked my heart that hard to be cured in my memory…
  • Audit a course of Chia-Hsin Liu to know root systems, He used the Humphreys’ book… There is no feelings happened in my mind in his class, thus gave the course up immediately…Audit his class just one time…For knowing that course did open for those  interested in ideal and rings.
  • The story after baby lie groups(class of Chun Chung in 2004) and lie group(class of Ong, Ping-Zen) attract me, especially for the class of Chun Chung, Lie algebras just like the drum in heaven that drumed me…
  • Met I-Hsun in 2004. Wanted to go to his Differential Geometry course to listen more things about Lie algebras but fixed in the Lie algebras course of Shun-Jen for one year.
  • ??—to present.

m…m… let me talk about translation works of mathematics.  After doing “My oops translate“, I felt a little better. But offer such volunteer effort really not a work or plan that should be do. Anyway I get something I like to do from doing that… It’s a long time not touch math, this way can cure my life a moment…Good translation is not a easy work, especially for mathematics. Take my translating “18.238 Geometry and Quantum Field Theory, Fall 2002” for example, I never know what I type…Just a machine kala~ka-la.  Keep the questions in your stomach or you cannot finish the document.

Borrowed the book,「張瑞吉 譯。李代數與表現理論之導引。國立編譯館主編。台北:黎明文化(民70)。」which is translated from the famous book: “Jumes E. Humphreys. Introduction to Lie Algebras and Representation Theory, GTM9. Springer-Verlag New York,1972.” ,from Taitung County library.  Prof. Rui-Ji Chang translate the book very well. Didn’t know if the teachers in the university before like translate. Gee regard them are all tips on tops during those days. They wrote, edited and translated excellent books by themselves…seems a little different than present day. Anyway,Each period must exist its great events.Back to translate…

Because of the translation mistake always happened for general translation. Gee still like read the original version than the second. Even for this good book translated by Prof. Chang…

Example.(p.1)

J. E. Humphreys wrote…

Definition. A vector space L over a field F, with an operation L × L→L, dented (x,y) →[xy]and called the bracket or commutator of x and y, is called a Lie algebra over F if the following axioms are satisfied:

(L1) The bracket operation is bilinear.
(L2) [xx]=0 for all x in L
(L3)[x[yz]]+[y[xz]]+[z[xy]]=0 (x,y,z∈L)

Rui-Ji Chang translated

定義 設 L 為體 F上的向量空間,而 L × L→L為L中的一個運算,記為 (x,y) →[xy]稱為x與y的括弧或換位元素。再設這個括弧運算滿足下列公理:

(L1) 括弧運算為雙線性。
(L2)[xx]=0 對所有 x ∈ L。
(L3)[x[yz]]+[y[xz]]+[z[xy]]=0 (x,y,z∈L)

則稱L為F上的李代數。

You catch the points here? That’s why I like to read the original book than the translated one, even though the Chinese words are so cute for me. Each translated book like this must have the tiny mistakes like these~ They will confuse me a lot to read the book (From these, you may understand that “to read a book well” is such a difficult hobby for me><).

For being an oops mathematic courses translator…Keep learning is an important things. Even though the problems also a lot, and lots problems like the upper statement…

“Just Go!” give myself the words!! I will be a good translator like Rui-Ji.

Equations Editor in WordPress

August 11, 2008 at 12:27 pm | Posted in Math Learning | Leave a comment
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  To Tell the truth, there are not only amazing persons, but also good tools here that support a good  environment to learn mathematics in WordPress.

  Yes! WordPress can include \LaTeX code in the post. See the announcement “Can I put Math or Equations in my Posts? ” of FAQ. You can type the mathematical formula on the blog. Just as the article show us :
So, taking these two definitions for example,

Definition3.6 The Jones polynomial V(L) of an oriented link L is the Laurent polynomial in t1/2, with integer coefficients, defined by

V(L)=((-A))^{-3(D)})\langle{D}\rangle)_{t^{1/2}=A^{-2}}\in{Z[t^{-1/2},t^{1/2}]}

where D is any oriented diagram for L .

Definition6.6  The rth Alexander ideal of an oriented link L is the rth elementary ideal of the Z[t-1,t] module H1[X;Z]. The rth Alexander polynomial of L is a generator of the smallest principal ideal of Z[t-1,t] that contains the rth Alexander polynomial. The first Alexander polynomial is the called the Alexander polynomial and is written ΔL(t)

(Definition Resourse: W.B. Raymond Lickorish. An Introduction to Knot Theory, GTM175. New York:Springer-Verlag (1997). p. 26,55.)

Using this useful Latex code… expressing and discussing mathematic are never difficulity problems here. You can also try it. But if you haven’t known what Latex would be, you may also click”LaTeX–A document preparation system” to keep close and to attach it.

 

p.s. As for the two definitions in the post, Gee still not catch their true meanings in my mind after typing them. If anybody can explain them more clearly…Please and please tell me.

Knot knock

August 5, 2008 at 8:16 am | Posted in Math Learning | Leave a comment
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Knot table to Eight crossing from Lickorish,1997

Do you know how hard to draw the knot picture?!…
You must try it!!
Look at the eight crossings knot table…

  • First, You should know how to tie a knot, thus you can know the upper segment and lower part!
  • Second, Upper segment is a real black line, and the lower part must anticipate an empty.
    ※You should keep all basic knot form deeply in your brain.
  • Third, Draw it with your pen. 3-dimension memory and 2-dimension drawing. Keep practice, practice and practice or you cannot draw it as well.(Actually speaking I still cannot do it well now)
  • Fourth, Reverse what you draw…to show the different positive picture and negative picture like.

This is What I did before…A never succeed work><

After going to Taipei, I open on the book: W.B. Raymond Lickorish. An Introduction to Knot Theory, GTM;175. Springer-Verlag New York(1997).To turn on my memory of Knot…It was Chun Chung tell me the good one. Seems a hard reading book for a undergraduate guy…I always confuse that Why I can easily understand what knot is after Chun Chung easily explained it, but cannot understand what the book show us. Alas~O if there I can take part in a course of he to tell me what the knot is…Not just a book.

Alwxander Polynomial Table from Lickrish,1997,p.59

Take something for example, look at the definition1.1 to 1.4, and compare with knot table to Eight Crossing(table one). To make the definition clearly had been a hard work for me…

Definition1.1: A link L of m components is a subset of S3 or of R3, that consists of m disjoint, piecewise linear, simple closed curves. A link of one component is a Knot.

Definition1.2: Links L1 and L2 in S3 are equivalent if there is an orientation-preserving piecewise linear homeomorphism H: S3 → S3 such that h(L1)=(L2)

Note Definition: A knot is said to be the unknot if it bounds an embedded piecewise linear discs in S3.

Definition1.3: A knot K is a prime knot if it is not the unknot, and K=K1+K2 implies that K1 or K2 is the unknot.

Definition1.4 Suppose that L is a two-component oriented link with components L1 and L2. The linking number lk(L,L) of L1 and L2 is half the sum of the sign, in a diagram for L, of the crossings at which one strand is from L1 and the other is from L2

My learning problems always stubs my brain..

Here: Who knows the “link”?! In mathworld shows that ” Formally, a link is one or more disjointly embedded circles in three space. More informally, a link is an assembly of knots with mutual entaglements”(http://mathworld.wolfram.com/Link.html). The definitions of knot and link are much easier on the page with picture(http://library.thinkquest.org/12295/)..of course, I understand it now, but it quite hard for me to state the definition. Cause I found at least 3 definition of Knot or link…

And: Is there anybody can show me the knot that been bounded an embedded piecewise linear discs in S3 (in definition1.2) ?….It sounds like noodles in a big pan. God~There must no people read this book like me…@@

OK~take it easy: Who knows the number of Prime knots after see the definition? We can distinguish the prime knot from definion1.3.. also can know the number of Prime knots not equal to the prime number from the knot table. As for the definition1.4…How does the “half the sum of the sign” work?…It just says that something related in homology theory. How can I catch the meanings of it from such few words?!

After looking over these, how you think about to know other important invariant-related topics like Alexander polynomial, Jones polynomial and so on in knot theory? It should be another hard work to overcome.  Of course, you can see what the stupid student’s story about learning math, especially in complex geometry , Lie algebra, knots and those with Quantization on the blog from now on.  Thus for the important part of knot,Alexander polynomial and Jones polynomial, you will see what the  learning problem I met.

Jones Polynomial Table from Lickorish,1997

My oops translate

April 16, 2008 at 1:34 am | Posted in Math Learning | 1 Comment
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OOPS, which means “Opensource Opencourseware Prototype System”, is a wonderful group that  translate some opensources for Chinese. Here is the page(http://www.myoops.org/twocw/).

Gee took part in this group monthes ago, to do further education and do something great for people. Almost all course I chsosed here did come from MIT courses. The topics I interested in are just like the following:

Of course, some basical course are needed here…

Keep growing, keep working, and keep translating…it will be a good beginning. Though it still a life long road after, never afraid of that…just keep going to do the right things…I am telling myself.

Now I am searching for the couse string and knots.Maybe I will share what I study more in my mind from the course here after…

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