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Math Help - Algebra in Applied Math

  1. #1
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    Algebra in Applied Math

    I like to divide math into 3 disjoint sets: Algebra, Analysis (Calculus), Geometry (Topology). *

    Now the most useful one is Analysis. The use of partial differencial equation and Fourier series is one of the big uses in applied math. As well as integration.

    It is also true that Geometry is used in applied math. For instance, Differencial Geometry and I am sure certain concepts from Topology are used as well.

    But how about Algebra? That is my favorite out of the 3 (or 4) and is sad to see that it is possibly the smallest one by far. I cannot possibly image how Field and Galois theory or Cyclotomic Extensions, Abelian Varieties are used. Can someone give examples?

    This is true for all except Linear Algebra. I seen it used heavily by JakeD. And I have seen online lectures by Professor Strang from MIT University. I am convinced it is extremely important. There are many strange concepts about matrices: transposes, orthogonality, decompositions .... I never used them nor have any idea what they are since Linear Algebra never interested me**.
    So my question is what is so special about this "strange" linear algebra that is used by applied mathematicians?

    *)Sometimes 4. We can think of Combinatorics as the other fourth one. In fact, it was a great honor when I saw MathWonk on PhysicsForums make the same classification.

    **)Except for Vector Spaces and Inner Product Spaces. Which is used a lot in pure math.
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by ThePerfectHacker View Post

    So my question is what is so special about this "strange" linear algebra that is used by applied mathematicians?
    Linear Algebra is about linear operators on (usualy finite dimensional) vector
    spaces, well almost all of applied maths is set in vector spaces of one sort or
    another, and often finite dimensional.

    Also as linear problems are so much easier in general that non-linear often
    the first thing one does with a problem is linearise it.

    RonL
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    Super Member Rebesques's Avatar
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    Hacker, you are so biased But in general terms, that's a good classification.

    For me also, there three kinds of Maths. The little I understand, those I don't really understand but can deal with, and those I can't understand for the life of me

    The latter includes algebraic geometry, algebraic logic and aspects of algebraic topology
    (funny, it all includes algebraic in it!!! guess I found my weakness )
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  4. #4
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    Quote Originally Posted by Rebesques View Post
    The latter includes algebraic geometry, algebraic logic and aspects of algebraic topology
    (funny, it all includes algebraic in it!!! guess I found my weakness )
    How good are you in Field theory? Perhaps your weakness is that your knowledge of this is not so strong? Is it?

    Note: I am planning to study algebraic geometry soon. But I am afraid, lest their be topology in it. From books I gazed at it seems so (the Zarski Topology). I am at a point where I need a topic to self-study after this following semester is over. All of them are algebraic . Either algebraic number theory, projective geometry or algebraic geometry. But I just cannot decide. I hate when that happens.
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  5. #5
    Senior Member tukeywilliams's Avatar
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    I believe Munkres Topology is the classic intro textbook. Linear algebra has many uses in graph theory and economics/ linear programming. Also numerical linear algebra has its uses (i.e. the problem of decomposing a given matrix  B to the following  B = A \times A^{T} . That problem had to do with image compressing and embedding. Here
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    Super Member Rebesques's Avatar
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    Munkres Topology
    Loved it. Any other classics we share?


    How good are you in Field theory
    Funny u should say that. I remember three major classes I had to take in the first semester of my 3rd year as an undergraduate: Real Analysis, Differential Geometry, Field Theory. Did bad at the first two and really bad at the third But I did well at Ring Theory, I still have bad dreams about exact sequences and Noetherian chains.
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  7. #7
    Senior Member tukeywilliams's Avatar
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    Here are the books I like:

    Spivak Calculus
    Courants Differential and Integral Calculus vols. 1 & 2
    Courant and John Intro to Analysis vols. 1 &2
    Apostol Calculus with linear algebra vol 1 & 2
    Linear Algebra by Hoffman/Kunze
    Linear Algebra by Sheldon Axler
    An Introduction to Mathematical Reasoning by Peter J. Eccles
    Problem Solving Strategies by Engel

    Topology by Munkres
    Abstract Algebra by Dummitt/Foote
    Principles of Mathematical Analysis by Rudin
    Analysis by Pugh
    Intro Analysis by Ross
    Mathematical Analysis by Tom Apostol
    An Introduction to Probability Theory by Feller
    Ordinary Differential Equations by V'Arnold
    Combinatorics by R. Merris
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    Quote Originally Posted by Rebesques View Post
    Funny u should say that. I remember three major classes I had to take in the first semester of my 3rd year as an undergraduate: Real Analysis, Differential Geometry, Field Theory.
    Maybe you do not like Field Theory, but it is one of the bestest things I ever learned. And I have learned a lot, so it is fair for me to make such a judgement.

    Why? Where you taking Graduate courses as an Junior? You must be really good.

    Have you ever done any of the three algebraic ones I mentioned in my other post? Which ones should I do?
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    Quote Originally Posted by tukeywilliams View Post
    Abstract Algebra by Dummitt Foote
    Did you ever buy that book? I like it, especially the additional sections like Algebraic Geometry and Representation Theory. I would buy it, it is just it I know everything except those two additional sections. And it is expensive so it would be a waste of money to me.


    Here is what I want.
    And more.
    Even more.
    And still more.
    And the last one.

    Just sharing my fantasies.
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  10. #10
    Senior Member tukeywilliams's Avatar
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    I checked it out of the library (currently self studying abstract algebra from that text).

    You may want to take a look at the abstract algebra books by Artin or Hungerford.
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  11. #11
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    Quote Originally Posted by tukeywilliams View Post
    I checked it out of the library (currently self studying abstract algebra from that text).

    You may want to take a look at the abstract algebra books by Artin or Hungerford.
    Oh and I forgot.

    Beachy and Blair.
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  12. #12
    Super Member Rebesques's Avatar
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    tukey:

    Spivak, Courant, Apostol I know Dang good books! The other ones I regret not having opened (besides V' Arnold, I just can't read it to the end ) Thanks for that list!

    Artin
    Oh! Had to study some pages out of that one for an exam, a green and slender nerve-wrecker, isn't it?

    hacker:

    Why? Where you taking Graduate courses as an Junior? You must be really good.
    I am tempted to say yes but not really. The courses were there for the undergraduates, and stupidly hard on purpose. I know this, because I later
    took the Graduate courses.

    Either algebraic number theory, projective geometry or algebraic geometry
    I am honoured you trust my judgement. Well, projective geometry is not a big subject in itself, and there's much more to its structures if you get there by the Differential Geometry path. Algebraic Number Theory is nice, but still rather limited.

    ...And Algebraic Geometry is just huge. There's just so many topics, and some very hard.

    So if you are just looking for somethinh hard, why don't you skip all three and try Lie Groups and Algebras?
    Last edited by ThePerfectHacker; August 10th 2007 at 06:54 AM.
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  13. #13
    Super Member Rebesques's Avatar
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    I wanted to study that... *Sob*
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  14. #14
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    Quote Originally Posted by Rebesques View Post
    Well, projective geometry is not a big subject in itself, and there's much more to its structures if you get there by the Differential Geometry path.
    It also appears in the study of elliptic curves. Thus, I assumed it would be easier for me to start with Projective Goemetry.

    Algebraic Number Theory is nice, but still rather limited.
    I think you are wrong about that. Modern Number Theoresits do work with this area a lot.

    So if you are just looking for somethinh hard, why don't you skip all three and try Lie Groups and Algebras?
    I am not looking for something that is hard. I am looking for something I would like. I know that Algebraic Number Theory is something I would really like.
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  15. #15
    Super Member Rebesques's Avatar
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    Dang, there goes my devious attempt to make a proper Geometer out of you!




    I am looking for something I would like. I know that Algebraic Number Theory is something I would really like.
    I guess you are right. Just don't want to see you make a living by breaking Russian/Chinese/North Korean codes.
    Last edited by ThePerfectHacker; August 12th 2007 at 08:40 AM.
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