Please advise! what kind of training I am lackfor math?

"walala" <mizhael@yahoo.com> writes:
> [...]
> In fact I am quite good in terms of grades in math classes; but I have only
> studied very few math courses: calculas, linear algebra, complexity
> analysis, probability, all in undergraduate and introductory graduate level.

This is both perplexing and understandable. 

Each of the math courses listed above could have been taught at any one
of a number of levels. For example, probability theory can be taught
using heuristics and a little calculus, or you can go into measure theory
and Borel fields. Did your complex variables course include basic topology
like deleted neighborhoods? How about the Cauchy-Riemann conditions for
testing a function's analyticity? 

I would say that if each of those classes were taught at even moderate
levels, you would be well-prepared to understand a good number of
papers.  It may not be *easy*, but you would have a good start and
could work your way through with study (at least that's the way it is
for me).

One subject area you omitted which I am finding very useful is Abstract
Algebra. Yes, folks, I took three semesters of the stuff (and still ain't
very good at it), but at least I'm comfortable with the basics of groups, rings 
and fields. It's also known as Modern Algebra. You might also want to do
some more work in advanced linear algebra.
-- 
%  Randy Yates                  % "...the answer lies within your soul
%% Fuquay-Varina, NC            %       'cause no one knows which side
%%% 919-577-9882                %                   the coin will fall."
%%%% <yates@ieee.org>           %  'Big Wheels', *Out of the Blue*, ELO
http://home.earthlink.net/~yatescr

walala wrote:

> I guess more specifically what I am asking is how to train
> myself to read/understand advanced/complicated/rigorous math to
> avoid feeling dizzy when reading math for more than 3 lines

As others have mentioned, it's possible that you advance too early. 
Another thing that makes me dizzy is trying to maintain too much 
mental state; that is, my concept of what's going on in the first two 
lines gets blurry when I reach the sixth, say, even though I do 
understand them reasonably well. And of course, going back and forth 
all the time disrupts the flow of thought.

What helps in this situation is to write stuff down. I digest as much 
as I'm comfortable with and paraphrase it, both in natural language 
and in my own formulaic style. This has several effects -- it brings 
out gaps in my understanding; binds each portion to the context; 
lessens the strain on my short-term memory; I get to have a style in 
the first place. This last item also addresses the issue of "thinking 
in maths".

> and how to think in math and write rigorous math... something like
> that... Reading the IEEE transaction on Information Theory made
> me very depressed... I wonder why those authors are so genious
> that they can invent theorem by theorem, lemma by lemma, ... 

Simple, most are not ;) Remember that those pages full of woundrous 
things are a collective effort and that you only see the result of a 
long process (let's not talk about "incremental" publishing practices 
here). Good research is never effortless. Some will even adopt the 
stance that if the problem hasn't provoked a headache in the near 
past, one is not doing one's best work. I guess they're right.


Martin

"Randy Yates" <yates@ieee.org> wrote in message
news:oesg9920.fsf@ieee.org...
> "walala" <mizhael@yahoo.com> writes:

Hi Randy,

Thank you for your answer!

>
> Each of the math courses listed above could have been taught at any one
> of a number of levels. For example, probability theory can be taught
> using heuristics and a little calculus, or you can go into measure theory
> and Borel fields. Did your complex variables course include basic topology
> like deleted neighborhoods? How about the Cauchy-Riemann conditions for
> testing a function's analyticity?

Yeah, measure theory, real analysis, functional analysis are two/three
graduate courses I am lacking of... need to take it soon... I just wish I
can take more courses each semester, but I found 3 courses already made me
fully loaded... so I am kind of slow...

Introduction to Complex Analysis: I feel that course is not useful at all...
I did not see anything in signal processing or infroamtion theory that needs
the content of complex analysis... Yeah, analyticity, Cauchy-Riemann
conditions, are they useful at all?

> I would say that if each of those classes were taught at even moderate
> levels, you would be well-prepared to understand a good number of
> papers.  It may not be *easy*, but you would have a good start and
> could work your way through with study (at least that's the way it is
> for me).
>
> One subject area you omitted which I am finding very useful is Abstract
> Algebra.

Really, seems you are working in Error Control coding stuff... I just saw
those material occurs in coding area...

>Yes, folks, I took three semesters of the stuff (and still ain't
> very good at it), but at least I'm comfortable with the basics of groups,
rings
> and fields. It's also known as Modern Algebra. You might also want to do
> some more work in advanced linear algebra.
> -- 

Yeah, I feel linear algebra and matrix analysis are really fun. And by
putting numbers togethere, it works magic. Really elegant...

-Walala

"walala" <mizhael@yahoo.com> writes:
> [...]
>>Yes, folks, I took three semesters of the stuff (and still ain't
>> very good at it), but at least I'm comfortable with the basics of groups,
> rings
>> and fields. It's also known as Modern Algebra. You might also want to do
>> some more work in advanced linear algebra.
>> -- 
>
> Yeah, I feel linear algebra and matrix analysis are really fun. And by
> putting numbers togethere, it works magic. Really elegant...

What's really cool is when you start seeing the concepts of vector
spaces over (e.g.) the reals generalized into vector spaces (and
modules) over fields, and how certain things like singular matrices
have richer meanings in more general arithmetic systems. This is where
linear algebra and abstract algebra meet, and I gotta tell you that,
even though I've just had the faintest taste of it, it really turns
my crank.
-- 
%  Randy Yates                  % "Watching all the days go by...    
%% Fuquay-Varina, NC            %  Who are you and who am I?"
%%% 919-577-9882                % 'Mission (A World Record)', 
%%%% <yates@ieee.org>           % *A New World Record*, ELO
http://home.earthlink.net/~yatescr

Fred Marshall wrote>
...
> Without reading it, the paper seems pretty straight in terms of notation.
> It even has block diagrams!  Not the sort I was thinking of. 

This reminds me of Sokal's hoax. I wonder when we are going to see the
EE equivalent published ... It wouldn't be too difficult (and quite
fun!) to compile lines of obscure senseless maths, spiked with the
proper EE terms and come to a completely nutty conclusion. Then get it
published in the IEEE SP magazine - now that would make a devastating
argument for Rick's column!

Andor

"Fred Marshall" <fmarshallx@remove_the_x.acm.org> wrote in message news:<-rmdnRKEVcbBY4DdRVn-ug@centurytel.net>...
> "Bob Cain" <arcane@arcanemethods.com> wrote in message
> news:401C9CAB.FA5E6CE8@arcanemethods.com...
> >
> >
> > Fred Marshall wrote:
> > >
> > > > On the other hand, much of my learning was in puzzling
> > > > through exactly such papers and filling in the gaps in them
> > > > and in my own knowledge to come to an understanding of the
> > > > topic that was sufficient for implementation or variation.
> > > > I'd say don't skip a paper that you feel you should be able
> > > > to understand but don't.  Rather, buckle down and work it
> > > > through.  I can think of some papers that I spent months
> > > > coming back to, setting aside, looking at things that might
> > > > clarify them until the final aha moment that made it all
> > > > worth it.
> > >
> > > I think we agree.  My point was in being choosy about which papers to
>  spend
> > > time on.
> >
> > For sure.  I'm just recommending that obscurity not be a
> > criterion in and of itself.  It can be difficult to
> > anticipate that there may be a nugget at the end of the
> > trail but one develops a sense of it.
> >
> > I'm looking at one such now:
> >
> >   http://www.atvs.diac.upm.es/publicaciones/docs/Bot00a.pdf
> >
> > I've got some serious work to do to know whether this
> > contains an answer to my current problem but my sense is
> > that it does.  The question now is whether it may be too
> > concise in its discussion of the noise reduction part of the
> > approach that I won't be able to use it even when I've
> > accumulated the background.  I know several authors that go
> > to pains to develop a formulation of the problem to then
> > just tell you that they've solved it.  :-)
> 
> Bob,
> 
> Without reading it, the paper seems pretty straight in terms of notation.
> It even has block diagrams!  Not the sort I was thinking of.  I mean the
> kind with pages of matrix equations that you just *know* will take many
> hours to figure out what they're talking about and then wonder if it's all
> smoke and no fire!
> 
> Fred

Heh, from browsing it very quickly, this is the kind of papers I hate! 
There isn't enough maths, the text in these types of papers is usually 
very general, and it's often impossible to find out exactly what the 
authors have done, how and why their method works and (more importantly)
what its limitations are.

I don't know how many hours I have spent contemplating papers I have read, 
trying to figure out how to adapt their methods to my problems. Most of 
the time I find some detail or assumption that applies to one problem and 
not to the other, so that the method can not be adapted. It's lots to 
learn from that kind of exercises.

Rune

"Rune Allnor" <allnor@tele.ntnu.no> wrote in message
news:f56893ae.0402040315.2a910e5f@posting.google.com...
> "Fred Marshall" <fmarshallx@remove_the_x.acm.org> wrote in message
news:<-rmdnRKEVcbBY4DdRVn-ug@centurytel.net>...
> > "Bob Cain" <arcane@arcanemethods.com> wrote in message
> > news:401C9CAB.FA5E6CE8@arcanemethods.com...
> >
> > Bob,
> >
> > Without reading it, the paper seems pretty straight in terms of
notation.
> > It even has block diagrams!  Not the sort I was thinking of.  I mean the
> > kind with pages of matrix equations that you just *know* will take many
> > hours to figure out what they're talking about and then wonder if it's
all
> > smoke and no fire!
> >
> > Fred
>
> Heh, from browsing it very quickly, this is the kind of papers I hate!
> There isn't enough maths, the text in these types of papers is usually
> very general, and it's often impossible to find out exactly what the
> authors have done, how and why their method works and (more importantly)
> what its limitations are.
>
> I don't know how many hours I have spent contemplating papers I have read,
> trying to figure out how to adapt their methods to my problems. Most of
> the time I find some detail or assumption that applies to one problem and
> not to the other, so that the method can not be adapted. It's lots to
> learn from that kind of exercises.
>
> Rune

Rune,

Point well taken.  Either extreme is bad.

Fred

"Fred Marshall" <fmarshallx@remove_the_x.acm.org> wrote in message news:<Gsednanneb_ws7zdRVn-iQ@centurytel.net>...
> "Rune Allnor" <allnor@tele.ntnu.no> wrote in message
> news:f56893ae.0402040315.2a910e5f@posting.google.com...
> > "Fred Marshall" <fmarshallx@remove_the_x.acm.org> wrote in message
>  news:<-rmdnRKEVcbBY4DdRVn-ug@centurytel.net>...
> > > "Bob Cain" <arcane@arcanemethods.com> wrote in message
> > > news:401C9CAB.FA5E6CE8@arcanemethods.com...
> > >
> > > Bob,
> > >
> > > Without reading it, the paper seems pretty straight in terms of
>  notation.
> > > It even has block diagrams!  Not the sort I was thinking of.  I mean the
> > > kind with pages of matrix equations that you just *know* will take many
> > > hours to figure out what they're talking about and then wonder if it's
>  all
> > > smoke and no fire!
> > >
> > > Fred
> >
> > Heh, from browsing it very quickly, this is the kind of papers I hate!
> > There isn't enough maths, the text in these types of papers is usually
> > very general, and it's often impossible to find out exactly what the
> > authors have done, how and why their method works and (more importantly)
> > what its limitations are.
> >
> > I don't know how many hours I have spent contemplating papers I have read,
> > trying to figure out how to adapt their methods to my problems. Most of
> > the time I find some detail or assumption that applies to one problem and
> > not to the other, so that the method can not be adapted. It's lots to
> > learn from that kind of exercises.
> >
> > Rune
> 
> Rune,
> 
> Point well taken.  Either extreme is bad.
> 
> Fred

Actually, one of the best books I know of, for learning new stuff, 
is the "Numerical Recipies in C". Not because the code in there is 
very useful (I don't think I ever used any of the code from that 
book) but because the authors have this amazing ability to explain 
the key points of every problem/algorithm very clearly in just a 
few pages. More than once did I read there first, to get the general 
overview of the problem I was working with, before I went to the 
journal literature to dive into the details. 

Rune

On 4 Feb 2004 23:08:34 -0800, allnor@tele.ntnu.no (Rune Allnor) wrote:

  (snipped)
>
>Actually, one of the best books I know of, for learning new stuff, 
>is the "Numerical Recipies in C". Not because the code in there is 
>very useful (I don't think I ever used any of the code from that 
>book) but because the authors have this amazing ability to explain 
>the key points of every problem/algorithm very clearly in just a 
>few pages. More than once did I read there first, to get the general 
>overview of the problem I was working with, before I went to the 
>journal literature to dive into the details. 
>
>Rune

Hi Rune,
  I agree.  The's authors' writing is 
very readable.  In that book was the first time 
I read the idea that (paraphrased) 

    "there are many different window functions, 
     but there's not much difference between them."

I thought that was interesting, and essentially true.

[-Rick-]




[-Rick-]