## BR Maths Corner-1

The Technology & Economic Forum is a venue to discuss issues pertaining to Technological and Economic developments in India. We request members to kindly stay within the mandate of this forum and keep their exchanges of views, on a civilised level, however vehemently any disagreement may be felt. All feedback regarding forum usage may be sent to the moderators using the Feedback Form or by clicking the Report Post Icon in any objectionable post for proper action. Please note that the views expressed by the Members and Moderators on these discussion boards are that of the individuals only and do not reflect the official policy or view of the Bharat-Rakshak.com Website. Copyright Violation is strictly prohibited and may result in revocation of your posting rights - please read the FAQ for full details. Users must also abide by the Forum Guidelines at all times.
Amber G.
BRF Oldie
Posts: 6836
Joined: 17 Dec 2002 12:31
Location: Ohio, USA

### Re: BR Maths Corner-1

ArmenT wrote:^^^
..., but I posted the two possibilities (5 and 17) rather than the sum of them both (5 + 17 = 22). I interpreted the question to be the sum of each of the two possibilities, rather than the total sum

That was obvious, and I knew that. (There was no flaw in your logic)

ArmenT
BR Mainsite Crew
Posts: 4239
Joined: 10 Sep 2007 05:57
Location: Loud, Proud, Ugly American

### Re: BR Maths Corner-1

Amber G. wrote:Okay I just got an interesting email from one of the young 6th grader bright girl .. ( from the team I coached this year)..

I forgot to mention. We had the exact same triangle challenge problem you gave to us in our team round. One of us even remember the correct solution and we got the answer right.

(I got a very nice letter BTW, and glad that many young kids are interested in math)

BTW, IIRC the exact triangle problem appeared here in BRF (by yours truly) a few years ago..this kind of problem is one of my favorite geometry problem (which even using trig is not easy but still not too hard, if you hit the right idea)

Here is the problem again.. (This was the last problem of the team round)

(Team round, you have more time, and the whole team (4 people) can work on it .. as difficulty go, this was supposed to be the hardest problem)

I started working on it and worked my way to angle ADC being 130 and EBC being 140 degrees, before getting stuck. Then I went through all the angles I'd worked out and remembered that I'd actually worked this thing out a while back (a key part is identifying that triangle ADE is isoceles). After that, I went back and dug out my reply from back in the day. That was a good one and took me a fair amount of time to work out.

Haven't posted the actual answer in this thread, but you can click on the link if you want the spoiler.

Amber G.
BRF Oldie
Posts: 6836
Joined: 17 Dec 2002 12:31
Location: Ohio, USA

### Re: BR Maths Corner-1

ArmenT - Good! You found it, (I did a search and could not find - so was doubting if I actually put that problem here)

BTW - for your reading pleasure... Here are 9 ways to prove/solve the above...

http://kvant.mccme.ru/1993/06/istoriya_s_geometriej.htm

The (and quite a few problems like this - for example instead of 50-60, one can use 50-70 or 60-70 and still get beautiful results) problem is classic (It is one classified as "Mathematical Gems")

My favorite way - see sol#8 in above - (literally 30 second solution and crux of the origin of the problem) is to look at a 18 sided regular polygon.. and look at the chord joining two vertices 6 sides apart.

(See half of that picture below, Triangle AX B' is 90-60-30 Triangle. or half of the chord I was talking about ..Now look at ABC..QED!

(Or imagine, BD is reflected (AC is mirror as BD' and CE is reflected (AB mirror) as C'E) )
Rest is intuitively obvious!

Vayutuvan
BRF Oldie
Posts: 10342
Joined: 20 Jun 2011 04:36

### Re: BR Maths Corner-1

I didn't get the ADE is isoscles. That is what was missing. With that probably I will get it. Will gave to try tomorrow.

Amber G.
BRF Oldie
Posts: 6836
Joined: 17 Dec 2002 12:31
Location: Ohio, USA

### Re: BR Maths Corner-1

matrimc,

If you are going to wait for tomorrow, you might as well give this a try
Prob 2 - Take one angle (BEA) = 70 (other angle is still 50)
Prob 3 - Or try Angle EAD = 70 (other angle is 60)

Amber G.
BRF Oldie
Posts: 6836
Joined: 17 Dec 2002 12:31
Location: Ohio, USA

### Re: BR Maths Corner-1

ArmenT wrote:[
...I started working on it and worked my way to angle ADC being 130 and EBC being 140 degrees, before getting stuck. Then I went through all the angles I'd worked out and remembered that I'd actually worked this thing out a while back (a key part is identifying that triangle ADE is isoceles). After that, I went back and dug out my reply from back in the day. That was a good one and took me a fair amount of time to work out.

Haven't posted the actual answer in this thread, but you can click on the link if you want the spoiler.

ArmenT, as you noticed, the problem is simple looking but deceptively devilish. (One would think simple "angle chasing" will find the solution but it is not so. (Problem first appeared in 1922 in a math magazine and since this (and a few similar problems) have became quite well known.

(When I first heard, I did it quite fast by trigonometry - fairly easy to do specially if you just want an answer .... I did find a quite cute and simple way to do it too, which many people like it - it is IMO second simplest method and not that well known - I may put it here if there is an interest)

BTW, one of the first trigonometry problem yours truly asked here is based on this !!
(Problem 3 here
(You need this type of technique to solve the problem using trig)

Any way if anyone is trying it out, here are some interesting methods/outlines. Some of these are in the paper I posted before (it is in Russian but still easy to understand )

Vayutuvan
BRF Oldie
Posts: 10342
Joined: 20 Jun 2011 04:36

### Re: BR Maths Corner-1

Yes. I tried angle chasing and ended up getting only one equation in two variables. Wondering how I can another independent equation.

By the way that reflection method is fantastic. I always love when symmetry us used to solve priblems - Math or physics. There is certain satisfying geometric beauty to it.

Amber G.
BRF Oldie
Posts: 6836
Joined: 17 Dec 2002 12:31
Location: Ohio, USA

### Re: BR Maths Corner-1

Yep Reflection... Assume the two sides are mirrors... and we have a green laser..
Now the answer is plain and simple..

Enjoy! (Think of the light beam getting reflected by both sides and then coming back)

The diagram is self explanatory ... only non-trivial part.. is to note that since CXB is 90 degrees, and X is midpoint of line EC so we must have angle CEB = angle ECB = 20..

shyam
BRFite
Posts: 1453
Joined: 29 Jul 2003 11:31

### Re: BR Maths Corner-1

Swapnil Garg Crowned National Champion at 2014 Raytheon MATHCOUNTS National Competition

Amber G.
BRF Oldie
Posts: 6836
Joined: 17 Dec 2002 12:31
Location: Ohio, USA

### Re: BR Maths Corner-1

Thanks for posting the above...(official time for the above was 12 second, as the timer starts when the question is flashed on the screen - although one can hear the buzzer even before the person finished asking the problem)

I know, as I have seen this for a long time, this part (countdown round) is, for the kids, most fun part.
Kids have fun even though many states don't have it or run it unofficially (that is the winner here is not the winner of the whole competition - winner is the one who wins the written part, where accuracy counts and speed is no issue, as long as you finish the whole exam)

In national, top 12 people (selected from 3 part written exam) take part in this event, and this event is open to public, and sometimes televised.

Another event, called masters round, the top 4 people (from written part), get about half an hour or so to solve a slightly tougher problem, and then present the solution (15 minutes or so) (or partial solution) to judges (and public/coaches/parents can watch) who can ask questions about method etc. It is a good place to notice young mathematicians. (This part, unfortunately is no longer there now)

Here is one question to try.. please post the answer if you wish.. (I see it may interest some of the graph theorists here - The problem is VERY simple looking, is not too hard to solve, and is doable in finite time)
Let n be the number of points on the circumference of a circle and r the maximum number of non-overlapping regions formed by connecting each of the n points, with a line segment, to every other point on the circle including its two neighboring (adjacent) points.

1. Find the corresponding values for r when n=1,2,3,4,5,6
2. Find a pattern relating n and r (use the pattern to predict value of r when n=10)
3. Find a recursive or explicit equation, or both, for r

Getting the value for small values of n is very simple. (For n=1, r=1 (there is only one region - the whole circle , n=2, r=2 (Circle cut in two parts), n=3 (r=4 - Triangle inside a circle - four regions etc))

Amber G.
BRF Oldie
Posts: 6836
Joined: 17 Dec 2002 12:31
Location: Ohio, USA

### Re: BR Maths Corner-1

^^^ BTW, for those who know Euler's famous formula (V-E+F=2), the above is easy. (Also the generalization, for example, number of regions in 4-space formed by n-1 hyperplanes are not anymore difficult than the above simple problem )

SaiK
BRF Oldie
Posts: 36402
Joined: 29 Oct 2003 12:31
Location: NowHere

### Re: BR Maths Corner-1

there is a new brf mice and rats dhaaga (pl visit gdf) trying to question the big math haathies here..
and the club is wondering how haathi eardrum works ignoring such events?

Vayutuvan
BRF Oldie
Posts: 10342
Joined: 20 Jun 2011 04:36

### Re: BR Maths Corner-1

AmberG: I did not get time to look at that problem but there is a similar problem in Knuth, graham, patashnik's book "concrete mathematics". Since I know about that solution I thought it us cheating but the problem you posed seems to be different. One clarification - are the points distributed evenly around the circumference?

CM goes even further by introducing L shaped lines with varying angles and Z shaped lines etc. there are a few open problems in that book which are extensions of this simple problem (all in the plane).

Vayutuvan
BRF Oldie
Posts: 10342
Joined: 20 Jun 2011 04:36

### Re: BR Maths Corner-1

AmberG: By the way radius of the circle is irrelevant or am I misreading the problem itself?

Also two cases ensue - odd n and even n assuming that the points are distributed evenly along the circumference. Rest is careful counting of edges and vertices in the graph and use of

F = E - V + 2

Amber G.
BRF Oldie
Posts: 6836
Joined: 17 Dec 2002 12:31
Location: Ohio, USA

### Re: BR Maths Corner-1

matrimc wrote:AmberG: By the way radius of the circle is irrelevant or am I misreading the problem itself?

Hello Marrimc,

I think you are making the problem more complicated sounding than it needs to be. The description is really simple.
(After all, the problem is meant for 13-14 year olds, and it is using their language) We just want to count the regions, (so radius of the circle.. or where the points are exactly placed on the circumference, is/are irreverent)

For example, see: below,

2 points, you can see two regions, 3 points - 4 regions, 4 points (quadrilateral inside circle )- 8 regions, and 5 points (star inside) 16 regions..

(Question here is simply to find the pattern ... and see if one can find a general formula)

Fairly easy to count.. right.!

(I am commenting some thing which you wrote.. since this may be a little off topic, I am putting it in"tiny" format so others, can ignore it if they like)

Also two cases ensue - odd n and even n assuming that the points are distributed evenly along the circumference. Rest is careful counting of edges and vertices in the graph and use of

F = E - V + 2

I don't understand what exactly is significance of odd n or even n . In any case, I think it makes no difference in the number of regions if the points are "evenly" distributed or not - as long as not more tan two lines passes through a single point..

As to using F = E-V+2 formula (actually "2" is not correct here, BTW it needs a little adjustment) may not be very helpful ... unless one can "count" E and V.. which need not be simpler than counting "F" in the first place...(After all, for example, one may find it easier to count 6 faces of a cube directly instead of counting the edges and vertices and using the formula.. )
Last edited by Amber G. on 14 Jun 2014 22:53, edited 1 time in total.

Vayutuvan
BRF Oldie
Posts: 10342
Joined: 20 Jun 2011 04:36

### Re: BR Maths Corner-1

AmberG:
Problem in KGP is indeed different. It asks given n lines whose orientations you can pick, what is the maximum number pieces a Circle can be cut into.

Here in this problem you have a complete graph on n points but with the difference that every intersection of two lines also introduces a vertex and thus can be drawn in the plane. Otherwise every graph with a K_5 minor is nonplanar. This graph also has multiple edges.

Oops edited from two different places and overwrote the previous post.

Overwritten post said the answer is 2^{n-1}. Needs to check this.
Last edited by Vayutuvan on 14 Jun 2014 23:27, edited 2 times in total.

Vayutuvan
BRF Oldie
Posts: 10342
Joined: 20 Jun 2011 04:36

### Re: BR Maths Corner-1

The CM problem with zig lines is posed as a homework problem. Making it harder is to relax condition of the legs of the z to be non parallel as well as introduce more zigs in the lines. This problem is right in the beginning of the book (if somebody cares to look - on page 4).

AmberG: For a proof of the above problem, I think Euler's formula would help. Let me work out the proof to my satisfaction and then I will post. Do not want to make a fool of myself (again ) by not checking through properly.

Amber G.
BRF Oldie
Posts: 6836
Joined: 17 Dec 2002 12:31
Location: Ohio, USA

### Re: BR Maths Corner-1

Hello Matrimc,

Peek only (or when you want to) if you wish.. (It may give away other answers and so look at it later if you don't want to see them now)

matrimc wrote: So the answer is 2^{n-1}. Let me convince myself by drawing a few more cases.

Wish you best for drawing a few more cases.. and make sure that you are convinced (Answer *may* not be 2^(n-1) *big Hint* One has to convince oneself specially when answers look simple )

matrimc wrote:
AmberG: For a proof of the above problem, I think Euler's formula would help. .

*Hint* Yes, it was (or very similar logic) how some students got the correct answer..

vsunder
BRFite
Posts: 848
Joined: 06 Sep 1999 11:31
Location: Ulan Bator, Mongolia

### Re: BR Maths Corner-1

Tata Institute of Fundamental Research has put the videos filmed during the workshop on Analysis and Geometry in January this year on Youtube. There were 6 speakers in all and the Central theme, was Conformal Geometry and Analysis, and Complex Geometry. You can access the videos here:

Videos of Lectures at the Workshop in Advanced Training in Mathematics School, Jan 2014 at TIFR Center, Bangalore

The workshop followed a 3 week summer school held under the aegis of the National Centre of Mathematics, The Deptt. of Atomic Energy (DAE) and NBHM( National Board of Higher Mathematics)
The nodal agency was the National Centre of Mathematics:
http://www.ncmath.org

The 3 week Summer School in July 2013 and the week long workshop in Jan 2014 gave me a chance to interact with students from Arunachal Pradesh, Jammu and Kashmir, Odisha, DAE Institutes like HRI,
Harishchandra Research Institute(HRI) Allahabad

in addition there were students from IISER Pune, IISc, the ISI's at Bangalore and Kolkata and some of the new IIT's and of course the TIFR which was the venue for the Summer school and workshop. I gave 17 lectures over 3 weeks in Harmonic Analysis and Partial Differential equations and wrote tons of notes for the July 2013 summer school. In addition there were other lectures on Differential geometry and Global Analysis. The syllabus and notes for the other modules was written in part by me also. You can find it at the NCM website and there may be links to my notes.
These interactions have given me quite a lot of insight into Mathematics in India and the educational system and the problems faced by young people who want to do serious Mathematical research in India.

The workshop whose video I have posted above, had participants culled from the summer school in July. I was particularly honored that Prof. M. S. Narasimhan even at the age of 80+ came to both my lectures and you can see him sitting on the extreme left in the videos of my lectures.

M. S. Narasimhan

Also Prof. Ravi S. Kulkarni former director of HRI, and current President of the Ramanujan Mathematical Society came to all my lectures at the Summer school in July 2013 and workshop Jan 2014 and you can see him too in the video of my lectures.

Ravi Kulkarni

It was a lot of hard work and I was very happy at the end when a bunch of students came to me and said " Sir, we have never been inspired to do research like this till now". It simply has made everything extremely worthwhile.

SaiK
BRF Oldie
Posts: 36402
Joined: 29 Oct 2003 12:31
Location: NowHere

### Re: BR Maths Corner-1

great links vsunder.. for some reason, I don't get audio on the youtube.. and I checked other videos, seems okay for me.

Eric Demopheles
BRFite
Posts: 110
Joined: 31 Oct 2010 20:55

### Re: BR Maths Corner-1

Re: from the GDF thread(vsunder and others).

The Dirac delta function δ(t) of a real variable t is a thing that would be meaningless to define as a function, but can be integrated, either together or multiplied or convolved with other functions, and has the property that:

integral from a to b of δ(t) = 1 if 0 is within the limit of integration and 1 otherwise.

The measure-theoretic definition is as of the Dirac Delta measure. This is a positive measure that consists of the measure of one single point(in this case the origin) being 1, and the complement of this set having measure zero.

The relation to the previous definition can be obtained via the Radon-Nikodym theorem.

The Fourier transform of the Delta function is the constant function 1, or will be after normalization of the constants inthe definitions of Fourier transform.

Dirac first found the definition in the form given above and this was not strictly rigorous from the point of view of mathematics. Nevertheless it worked so superbly and had a very nice physical intuition.

This intuition can be best expressed like a sudden surge of current which is extremely brief, but can charge a capacitor from 0V to 1V. Another example is a lightning surge which can be felt on all frequencies, radio, microwave, visual etc.. Here I mean the Fourier transform being the constant function, and the Fourier transform actually looks at the Frequency spectrum.

The integral of the Dirac Delta is the Heaviside step function, which is well-known to electrical engineers.

The most precise definition requires Schwartz's theory of distributions, in which the a distribution is known by the effect of it when integrated against a class of "test functions". More technically, a distribution is a continuous linear functional on the space of test functions. The space of test functions can be the space of compactly supported smooth functions, or smooth functions that are rapidly decaying at infinity(i.e., the function and all of its derivatives of all orders converge to 0 at infinity).

More details can be found on the wikipedia article on distributions.

Eric Demopheles
BRFite
Posts: 110
Joined: 31 Oct 2010 20:55

### Re: BR Maths Corner-1

Re: Vsunder, Regarding Minankshisundaram:

I had not read much about Minakshisundaram's works. Thanks for all your descriptions and links. I had not known that he and Patodi had worked on similar things. I was not aware of Minakshisundaram having to leave TIFR because of conflict with K. Chandrasekharan.

When I get the chance I will try to familiarize more.

Amber G.
BRF Oldie
Posts: 6836
Joined: 17 Dec 2002 12:31
Location: Ohio, USA

### Re: BR Maths Corner-1

Eric Demopheles wrote:The Dirac delta function δ(t) ...
.

Ericji - Let me add an amusing personal anecdote..
As I have told here, I took a course in QM by Dirac himself, (he was using his own famous text book on QM ** see note 1). This text book, of course, deals with Dirac's delta function extensively, and of course, its mathematical foundation is very solidly established. There was a "wise guy" type UG who asked something akin to ".. but δ is just nonsense .. it is not a function " type question. Dirac, who was famous for his politeness (*** see note 2), did get a little annoyed (he must have heard that type of question before) and it was VERY INTERESTING to watch that.

Of course, the mathematical foundation of "generalized" (if one wants to be more clear) function is quite solid. They extend the notion of ("non-generalized") functions and are especially useful in making discontinuous functions to be treated more like smooth functions, and describing physical phenomena such as point charges. They are applied extensively, in physics and engineering.

The concept is something akin to introduction of ZERO to be added to numbers, which many, around that time, have called "not really a number"... (After all one can not, for example, divide by zero, like other "numbers" !!).
But, as long as you are careful and follow the rules, including zero as a "number" is ok, similarly use of Dirac's delta function is Physics is based on solid mathematical rigor.

(** Note 1) Dirac's QM text book was one of the book Einstein used, and kept as a good reference. ( There is a fairly well known story of Einstein often asking "where is my Dirac?" (he was looking for Dirac's text book).

(Note 2) I have visited Princeton when Dirac's was there. One running joke was an unit "Dirac" which measured how much talking (specially useless) one does. One Dirac meant something like 1 word of bakwaas per year.

Vayutuvan
BRF Oldie
Posts: 10342
Joined: 20 Jun 2011 04:36

### Re: BR Maths Corner-1

Eric ji: It was not for you.

It is for another poster who I am sure he is a very respected mathematician and hopefully my remarks are taken as a contrary opinion from a gentlemen on the way intellectual discussions need to be carried out in person or online - especially on BRF where people with varying degrees of expertise in several different fields are present.

What follows is an edited version of the stronger and somewhat snarky opinion posted in GDF-math:

Eric ji: It was not for you. Corrected that post of mine by addressing the right poster. Please see my corrected post in GDF-math thread.

I would certainly want you to expand on the measure theory in this thread. I will certainly learn more than I know (which is sunya) on measure theory.

Please be assured that I have the highest regard for Mathematicians and Physicists unless they display a markedly boorish attitude and start with "you know nothing" (unless it comes from the "living gods" of the respective fields).

Eric Demopheles
BRFite
Posts: 110
Joined: 31 Oct 2010 20:55

### Re: BR Maths Corner-1

Amber G. wrote:As I have told here, I took a course in QM by Dirac himself, (he was using his own famous text book on QM ** see note 1). This text book, of course, deals with Dirac's delta function extensively, and of course, its mathematical foundation is very solidly established. There was a "wise guy" type UG who asked something akin to ".. but δ is just nonsense .. it is not a function " type question. Dirac, who was famous for his politeness (*** see note 2), did get a little annoyed (he must have heard that type of question before) and it was VERY INTERESTING to watch that.

Of course, the mathematical foundation of "generalized" (if one wants to be more clear) function is quite solid. They extend the notion of ("non-generalized") functions and are especially useful in making discontinuous functions to be treated more like smooth functions, and describing physical phenomena such as point charges. They are applied extensively, in physics and engineering.

The concept is something akin to introduction of ZERO to be added to numbers, which many, around that time, have called "not really a number"... (After all one can not, for example, divide by zero, like other "numbers" !!).
But, as long as you are careful and follow the rules, including zero as a "number" is ok, similarly use of Dirac's delta function is Physics is based on solid mathematical rigor.

(** Note 1) Dirac's QM text book was one of the book Einstein used, and kept as a good reference. ( There is a fairly well known story of Einstein often asking "where is my Dirac?" (he was looking for Dirac's text book).

(Note 2) I have visited Princeton when Dirac's was there. One running joke was an unit "Dirac" which measured how much talking (specially useless) one does. One Dirac meant something like 1 word of bakwaas per year.

Amber G.ji,

Thanks for the anecdote.

I think the Dirac delta function is a sobering reminder to mathematicians that sometimes physicsists can be ahead of them. It took a long time to properly justify it in mathematics.

I used to think that physicsists don't use "serious" mathematics, even after knowing the Dirac delta function story. At a certain point of time, I was quickly made aware otherwise. I was humbled by seeing how much serious and nontrivial and cutting-edge mathematics was being used by people who were working along the directions proposed by Witten and others. For example, the homological mirror symmetry or moduli spaces or quantum cohomolgy people, were all using very serious ideas from theoretical physics and also algebraic geometry, one of the most abstruse topics in modern mathematics. I was humbled and made to feel that even after the time of Newton is long past, physics can be a greater source of math than math itself.

Amber G.
BRF Oldie
Posts: 6836
Joined: 17 Dec 2002 12:31
Location: Ohio, USA

### Re: BR Maths Corner-1

^^^ Thanks. Nice post.

Amber G.
BRF Oldie
Posts: 6836
Joined: 17 Dec 2002 12:31
Location: Ohio, USA

### Re: BR Maths Corner-1

Okay guys.. Today there is a big news item about Terry Tao.
UCLA math star Terence Tao wins \$3-million prize

Terry Tao, if you (a few who read the math dhaga ) recall has been discussed in this dhaga many times... he won IMO gold medals - at the age of 10 or so, SASTRA Ramanujan Prize (To qualify the person has to be younger than 32 -- the age Ramanujan died )--... Fields medal.. etc...

Here is one picture of him (With the great Erdos) (How quickly the time passes...I remember the picture and does not realize how much time has passed)

It did amuse me, to see some of my old posts about him here .. here is one from 6 years ago.... (Please do read the whole post, and few posts above and below..) It is fun to see all those comments..

http://forums.bharat-rakshak.com/viewtopic.php?p=514223#p514223

or this
http://forums.bharat-rakshak.com/viewtopic.php?f=2&t=3951&p=519439&hilit=tao#p519439

Terry, BTW has one of the best blog for general public, if any one is interested in Math. I have recommended that blog previously and recommend it very highly.

(Reference to that blog - I first put it here wrt Vinay Deolalikar's claim of solving P vs NP problem related TSP. link)
.

Eric Demopheles
BRFite
Posts: 110
Joined: 31 Oct 2010 20:55

### Re: BR Maths Corner-1

Amber G. wrote:Terry, BTW has one of the best blog for general public, if any one is interested in Math. I have recommended that blog previously and recommend it very highly.

Come to speak of it, Terrence Tao is a great analyst and has made many great posts on real analysis and measure theory. He has published his course notes both as blog posts and as a book.

I was supposed to write something on measure theory. Since stuff is holding me up, I might as well whet people's appetite for the moment with links to his blog articles:

1. The problem of measure:

https://terrytao.wordpress.com/2010/09/ ... f-measure/

2. The Lebesgue measure:

https://terrytao.wordpress.com/2010/09/ ... e-measure/

3. The Lebesgue integral:

https://terrytao.wordpress.com/2010/09/ ... -integral/

Aaand, the whole book:

http://terrytao.files.wordpress.com/201 ... -book1.pdf

Maybe at a later date I will get around to posting my own viewpoints..

member_22733
BRF Oldie
Posts: 3788
Joined: 11 Aug 2016 06:14

### Re: BR Maths Corner-1

Eric Demopheles wrote:
Amber G. wrote:Terry, BTW has one of the best blog for general public, if any one is interested in Math. I have recommended that blog previously and recommend it very highly.

Come to speak of it, Terrence Tao is a great analyst and has made many great posts on real analysis and measure theory. He has published his course notes both as blog posts and as a book.

I was supposed to write something on measure theory. Since stuff is holding me up, I might as well whet people's appetite for the moment with links to his blog articles:

1. The problem of measure:

https://terrytao.wordpress.com/2010/09/ ... f-measure/

2. The Lebesgue measure:

https://terrytao.wordpress.com/2010/09/ ... e-measure/

3. The Lebesgue integral:

https://terrytao.wordpress.com/2010/09/ ... -integral/

Aaand, the whole book:

http://terrytao.files.wordpress.com/201 ... -book1.pdf

Maybe at a later date I will get around to posting my own viewpoints..

That problem of measure article is amazing. I always had an "intuitive" idea of measures and why the common understanding of measure fails with uncountable and infinite general spaces (0 x Infinity == undefined issue), I am glad to know that my intuition was close to the current understanding of measure theory . The rest of the article is also a gem, but takes sometime to understand the nuances (which is the crux of math).

Eric Demopheles
BRFite
Posts: 110
Joined: 31 Oct 2010 20:55

### Re: BR Maths Corner-1

LokeshC wrote:That problem of measure article is amazing. I always had an "intuitive" idea of measures and why the common understanding of measure fails with uncountable and infinite general spaces (0 x Infinity == undefined issue), I am glad to know that my intuition was close to the current understanding of measure theory . The rest of the article is also a gem, but takes sometime to understand the nuances (which is the crux of math).

If you want to rigorously study measure theory, all the more so in regard to probability theory, the book by two Indian authors Siva Athreya and V. S. Sunder is quite good.

Better than many western books, in fact.

Amber G.
BRF Oldie
Posts: 6836
Joined: 17 Dec 2002 12:31
Location: Ohio, USA

### Re: BR Maths Corner-1

FWIW - One of the best book related to above I have seen is Laplace's original treatise. I got that book (fairy big but extremely easy to read and systematically written) when I already have taken college level courses but still I learned quite a bit.

The book originally is in French. I had it's translation in Hindi - done by a world-class French- Indian Mathematician.
... Too bad I don't have the copy now and it will not be easy to get it again as those kind of books go out of print quite soon. I am sure there is an English translation of that work.

Eric Demopheles
BRFite
Posts: 110
Joined: 31 Oct 2010 20:55

### Re: BR Maths Corner-1

Amber G. wrote:FWIW - One of the best book related to above I have seen is Laplace's original treatise.

What is Laplace's book about?

Amber G.
BRF Oldie
Posts: 6836
Joined: 17 Dec 2002 12:31
Location: Ohio, USA

### Re: BR Maths Corner-1

From what I remember the treatise deals with many fundamental results in statistics. The first half deals with probability methods (has lot of interesting problems), the second half with statistical methods and applications. Introduction of Bayesian views etc ... he also wrote a popular account of his work. Very interesting ... many compare that book's relationship wrt to this treatise very similar to Système du monde wrt Méchanique céleste. (You learn lot of deep math while appreciating practical aspects and how to use those techniques)... Although book is very old (1800's) and many may be harsh with some of the methods (not rigorous enough ?) .. Imo it is quite solid mathematically.

(My Hindi is much much better than French (or other European languages for that matter).. though I read the translation but translation was extremely good and I learned a lot... got a respect for both for Laplace.and the translator )

Eric Demopheles
BRFite
Posts: 110
Joined: 31 Oct 2010 20:55

### Re: BR Maths Corner-1

It is very interesting to know that Laplace had such a book which got translated into Hindi. No doubt, a master like him would have written a book with great content. Do you remember who was the translator?

I have had to look at many French mathematical papers. Most people in mathematics would have, though the French concentration is more in certain subfields. At first I did not know a word, By now it has become much easier, and though I still can't speak a word of French, I can read mathematical papers without too much difficulty. The trick is in getting familiar with the jargon, such as, égal for equal, donc for thus, série for series, etc.. Then the mathematical and symbolic stuff are all the same and we are on familiar ground.

Laplace's work is very important for probability theory; but it is still historically behind Kolmogorov's which connected classical probability theory with the modern theory of integration and introduced a new language/way of speaking. It may be possible to compare the change to the change in classical mechanics brought about by quantum mechanics or relativity. I am not very good in physics; but if my impression is right, these changes were more revolutionary compared to the advances in probability theory and so the comparison is perhaps a bit much; nevertheless the point still stands.

Amber G.
BRF Oldie
Posts: 6836
Joined: 17 Dec 2002 12:31
Location: Ohio, USA

### Re: BR Maths Corner-1

Translator was well known M Mehta (of Random Matices fame).

Amber G.
BRF Oldie
Posts: 6836
Joined: 17 Dec 2002 12:31
Location: Ohio, USA

### Re: BR Maths Corner-1

IMO (International Math Olympiad) Results:
(Congratulations)

Top as team (and scores)
1. China 201
2. USA 193
3. Taiwan 192
4. Russia 191
5. Japan 177
6. Ukraine 175
7. South Korea 172
8. Singapore 161
9. Canada 159
10. Vietnam 157

39 India ..

Last Ghana score 0.

Personal Interest: Ukriane Very impressive -- has 2 females on their team which is even more impressive
I think this is the worst ranking ever for India (Score 39)
Iran - general comes around top 10 - ....(21) - Worse Ranking ever...

USA Details:
5 Golds (Allen Liu, Yang Liu, Sammy Luo, Mark Sellke, James Tao)
1 Silver (Joshua Brakensiek)

India: (One girl on the team got Bronze)
1 Silver (Sagnik Saha) (He got silver last year too)
3 Bronze (Jeet Mohapatra, Anish Prasad Sevekari and Chaitanya Tappu)
2 HM (Soumik Ghosh, Supravat Sarkar )
---

(It is a worst for France, too) - (Some of us wondering what's happening...
Alexander Gunning - Impressive - the first Australian ever to get a perfect score!

Alex Song got another gold medal (his fourth) and still has another year left! He may become the IMO's most successful contestant!

One can tell from the point spread that this years results are very wacky ... (For example i Canada placed in the top 10 despite having 3 Bronze Medalists... Don't think that kind of thing happened before)

Looking at the problems seems like problem 5 has been too tough for lot of people..

Vayutuvan
BRF Oldie
Posts: 10342
Joined: 20 Jun 2011 04:36

### Re: BR Maths Corner-1

AmberG: could you please talk a little bit about M. Mehta of random matrices fame? What is his work on Random Matrices related to? Combinatorial or spectral? Any Erdos connection? Would be grateful.

Amber G.
BRF Oldie
Posts: 6836
Joined: 17 Dec 2002 12:31
Location: Ohio, USA

### Re: BR Maths Corner-1

matrimc wrote:AmberG: could you please talk a little bit about M. Mehta of random matrices fame? What is his work on Random Matrices related to? Combinatorial or spectral? Any Erdos connection? Would be grateful.

Wiki has a short article about him..per wiki "His book "Random Matrices" is considered classic in the field...Even the great Eugene Wigner (Noble prize winner - who invited mehta while he was a student to Princeton) cited Mehta during his SIAM review on Random Matrices... I know his book was used as text book in MIT and other places (see anecdote 1)

Eugene Wigner wrote extensively about him (he was very close to him), and so did Freeman Dyson and some other great names in Physics. ( Mehta's death was a small news item in India - Though APJ Kalam said nice things about him - but he was much more known in France)..IIRC A beautiful Eulogy was written by Dyson.

A good intro to his work is from France's Saclay..
http://ipht.cea.fr/en/Phocea-SPhT/ast_visu_spht.php?id_ast=447

I knew him (I spent some time in Saclay, and there he was very well famous) and learned a lot from this guru - he was a very good and generous teacher, and has read his biography (in French and Hindi) which is very interesting... was a star student (find of) Bhabha who invited him to TIFR but Mehta's refusal to put a suit (he was often mistaken as a "chaparasi" was not liked by Bhabha.. (and some other political bigwigs)..Another article I liked (in 1960's) was in Hindi too.
("My stay at Princeton" - which talked about his personal interactions with big shots like Oppenheimer, Wigner, Dirac etc)

He has done extensive scientific work but he also wrote popular articles in Hindi, English and Frence magazines and newspapers. (Eg Scientific American, Vigyaan etc)

One thing impressed me (see wiki article) was that he was fluent in MANY languages.. (Hindi, French, Russian, Japanese, Chinese, Spanish... virtually every major Indian Language and most of European languages...he visited many Foreign universities, stay a few years, will teach/give lectures in native languages etc)

Vayutuvan
BRF Oldie
Posts: 10342
Joined: 20 Jun 2011 04:36

### Re: BR Maths Corner-1

Amberg thanks. Never knew about him. Now you bring another interesting tidbit. Wigner wrote a survey paper of random matrices in SIAM surveys? Do OU remember the approximate year? Tia.

bhalluka
BRFite -Trainee
Posts: 21
Joined: 11 Aug 2016 06:14

### Re: BR Maths Corner-1

Return to “Technology & Economic Forum”

### Who is online

Users browsing this forum: isubodh and 6 guests