I recently noticed on arXiv that the following manuscript “Implementation and Abstraction in Mathematics” by David McAllester. A couple of years ago, I had taken a graduate course taught by David that had a similar flavour (the material in the manuscript is more advanced, in particular the main results, not to mention it is better organized and the presentation more polished), presenting a type theoretic foundation of mathematics. Although I can’t say I did very well in the course, I certainly enjoyed the ideas in it very much, and thus the above manuscript might be worth a look. Perhaps it might be a good idea to audit that course again, just to make sure I understand the main ideas better this time. :)
Posts Tagged ‘Mathematics’
Over the past 4-5 months whenever there is some time to spare, I have been working through The Cauchy-Schwarz Master Class by J. Michael Steele. And, although I am still left with the last two chapters, I have been reflecting on the material already covered in order to get a better perspective on what I have been slowly learning over the months. This blog post is a small exercise in this direction.
Ofcourse, there is nothing mysterious about proving the Cauchy-Schwarz inequality; it is fairly obvious and basic. But I thought it still might be instructive (mostly to myself) to reproduce some proofs that I know out of memory (following a maxim of my supervisor on a
white/blackboard blogpost). Although, why Cauchy-Schwarz keeps appearing all the time and what makes it so useful and fundamental is indeed quite interesting and non-obvious. And like Gil Kalai notes, it is also unclear why is it that it is Cauchy-Schwarz which is mainly useful. I feel that Steele’s book has made me appreciate this importance somewhat more (compared to 4-5 months ago) by drawing to many concepts that link back to Cauchy-Schwarz.
Before getting to the post, a word on the book: This book is perhaps amongst the best mathematics book that I have seen in many years. True to its name, it is indeed a Master Class and also truly addictive. I could not put aside the book completely once I had picked it up and eventually decided to go all the way. Like most great books, the way it is organized makes it “very natural” to rediscover many susbtantial results (some of them named) appearing much later by yourself, provided you happen to just ask the right questions. The emphasis on problem solving makes sure you make very good friends with some of the most interesting inequalities. The number of inequalities featured is also extensive. It starts off with the inequalities dealing with “natural” notions such as monotonicity and positivity and later moves onto somewhat less natural notions such as convexity. I can’t recommend this book enough!
Now getting to the proofs: Some of these proofs appear in Steele’s book, mostly as either challenge problems or as exercises. All of them were solvable after some hints.
Proof 1: A Self-Generalizing proof
This proof is essentially due to Titu Andreescu and Bogdan Enescu and has now grown to be my favourite Cauchy-Schwarz proof.
We start with the trivial identity (for ):
Expanding we have
Rearranging this we get:
Rearranging this we get the following trivial Lemma:
Notice that this Lemma is self generalizing in the following sense. Suppose we replace with and with , then we have:
But we can apply Lemma 1 to the second term of the right hand side one more time. So we would get the following inequality:
Using the same principle times we get the following:
Now substitute and to get:
This is just the Cauchy-Schwarz Inequality, thus completing the proof.
Proof 2: By Induction
Again, the Cauchy-Schwarz Inequality is the following: for
For proof of the inequality by induction, the most important thing is starting with the right base case. Clearly is trivially true, suggesting that it is perhaps not of much import. So we consider the case for . Which is:
To prove the base case, we simply expand the expressions. To get:
Which is just:
Which proves the base case.
Moving ahead, we assume the following inequality to be true:
To establish Cauchy-Schwarz, we have to demonstrate, assuming the above, that
So, we start from :
we further have,
Now, we can apply the case for . Recall that:
Thus, using this in the R. H. S of , we would now have:
This proves the case on assuming . Thus also proving the Cauchy-Schwarz inequality by the principle of mathematical induction.
Proof 3: For Infinite Sequences using the Normalization Trick:
We start with the following question.
Problem: For If and then is ?
Note that this is easy to establish. We simply start with the trivial identity which in turn gives us
Next, take and on summing up to infinity on both sides, we get the following:
From this it immediately follows that
; substituting in , we get:
Which simply gives back Cauchy’s inequality for infinite sequences thus completing the proof:
Proof 4: Using Lagrange’s Identity
We first start with a polynomial which we denote by :
The question to now ask, is ? To answer this question, we start of by re-writing in a “better” form.
Next, as J. Michael Steele puts, we pursue symmetry and rewrite the above so as to make it apparent.
Thus, we now have:
This makes it clear that can be written as a sum of squares and hence is always postive. Let us write out the above completely:
Now, reversing the step we took at the onset to write the L.H.S better, we simply have:
This is called Lagrange’s Identity. Now since the R.H.S. is always greater than or equal to zero. We get the following inequality as a corrollary:
This is just the Cauchy-Schwarz inequality, completing the proof.
Proof 5: Gram-Schmidt Process gives an automatic proof of Cauchy-Schwarz
First we quickly review the Gram-Schmidt Process: Given a set of linearly independent elements of a real or complex inner product space , . We can get an orthonormal set of elemets by the simple recursion (after setting ).
Keeping the above in mind, assume that . Now let . Thus, we have:
Giving: . Rearranging we have:
where are constants.
Now note that: and
. The following bound is trivial:
. But note that this is simply
Which is just the Cauchy-Schwarz inequality when .
Proof 6: Proof of the Continuous version for d =2; Schwarz’s Proof
For this case, the inequality may be stated as:
Suppose we have and that and . Then consider the double integrals:
, and . These double integrals must satisfy the following inequality:
The proof given by Schwarz as is reported in Steele’s book (and indeed in standard textbooks) is based on the following observation:
The real polynomial below is always non-negative:
unless and are proportional. Thus from the binomial formula we have that , moreover the inequality is strict unless and are proportional.
Proof 7: Proof using the Projection formula
Problem: Consider any point in . Now consider the line that passes through this point and origin. Let us call this line . Find the point on the line closest to any point .
If is the point on the line that is closest to , then it is given by the projection formula:
This is fairly elementary to establish. To find the value of , such that distance is minimized, we can simply consider the squared distance since it is easier to work with. Which by definition is:
which is simply:
So, the value of for which the above is minimized is . Note that this simply reproduces the projection formula.
Therefore, the minimum squared distance is given by the expression below:
Note that the L. H. S is always positive. Therefore we have:
Rearranging, we have:
Which is just Cauchy-Schwarz, thus proving the inequality.
Proof 8: Proof using an identity
A variant of this proof is amongst the most common Cauchy-Schwarz proofs that are given in textbooks. Also, this is related to proof (6) above. However, it still has some value in its own right. While also giving an useful expression for the “defect” for Cauchy-Schwarz like the Lagrange Identity above.
We start with the following polynomial:
. Clearly .
To find the minimum of this polynomial we find its derivative w.r.t and setting to zero:
Clearly we have . We consider:
, substituting we have:
Just rearrangine and simplifying:
This proves Cauchy-Schwarz inequality.
Now suppose we are interested in an expression for the defect in Cauchy-Schwarz i.e. the difference . For this we can just consider the L.H.S of equation since it is just .
i.e. Defect =
Which is just:
This defect term is much in the spirit of the defect term that we saw in Lagrange’s identity above, and it is instructive to compare them.
Proof 9: Proof using the AM-GM inequality
Let us first recall the AM-GM inequality:
For non-negative reals we have the following basic inequality:
Now let us define and
Now consider the trivial bound (which gives us the AM-GM): , which is just . Note that AM-GM as stated above for is immediate when we consider and
Using the above, we have:
Summing over , we have:
But note that the L.H.S equals 1, therefore:
Writing out and as defined above, we have:
Thus proving the Cauchy-Schwarz inequality.
Proof 10: Using Jensen’s Inequality
We begin by recalling Jensen’s Inequality:
Suppose that is a convex function. Also suppose that there are non-negative numbers such that . Then for all for one has:
Now we know that is convex. Applying Jensen’s Inequality, we have:
Now, for for all , let and let .
Rearranging this just gives the familiar form of Cauchy-Schwarz at once:
Proof 11: Pictorial Proof for d = 2
Here (page 4) is an attractive pictorial proof by means of tilings for the case by Roger Nelson.
The writings (and even papers/technical books) of Gian-Carlo Rota are perhaps amongst the most insightful that I have encountered in the past 3-4 years (maybe even more). Rota wrote to provoke, never resisting to finish a piece of writing with a rhetorical flourish even at the cost of injecting seeming inconsistency in his stance. I guess this is what you get when you have a first rate mathematician and philosopher endowed with an elegant; at times even devastating turn of phrase, with a huge axe to grind*.
“Combinatorics is an honest subject. No adèles, no sigma-algebras. You count balls in a box, and you either have the right number or you haven’t. You get the feeling that the result you have discovered is forever, because it’s concrete. Other branches of mathematics are not so clear-cut. Functional analysis of infinite-dimensional spaces is never fully convincing; you don’t get a feeling of having done an honest day’s work. Don’t get the wrong idea – combinatorics is not just putting balls into boxes. Counting finite sets can be a highbrow undertaking, with sophisticated techniques.[…]Much combinatorics of our day came out of an extraordinary coincidence. Disparate problems in combinatorics, ranging from problems in statistical mechanics to the problem of coloring a map, seem to bear no common features. However, they do have at least one common feature: their solution can be reduced to the problem of finding the roots of some polynomial or analytic function. The minimum number of colors required to properly color a map is given by the roots of a polynomial, called the chromatic polynomial; its value at N tells you in how many ways you can color the map with N colors. Similarly, the singularities of some complicated analytic function tell you the temperature at which a phase transition occurs in matter. The great insight, which is a long way from being understood, was to realize that the roots of the polynomials and analytic functions arising in a lot of combinatorial problems are the Betti numbers of certain surfaces related to the problem. Roughly speaking, the Betti numbers of a surface describe the number of different ways you can go around it. We are now trying to understand how this extraordinary coincidence comes about. If we do, we will have found a notable unification in mathematics.”
John von Neumann made so many fundamental contributions that Paul Halmos remarked that it was almost like von Neumann maintained a list of various subjects that he wanted to touch and develop and he systematically kept ticking items off. This sounds to be remarkably true if one just has a glance at the dizzyingly long “known for” column below his photograph on his wikipedia entry.
Since Neumann died (young) in 1957, rather unfortunately, there aren’t very many audio/video recordings of his (if I am correct just one 2 minute video recording exists in the public domain so far).
I recently came across a fantastic film on him that I would very highly recommend. Although it is old and the audio quality is not the best, it is certainly worth spending an hour on. The fact that this film features Eugene Wigner, Stanislaw Ulam, Oskar Morgenstern, Paul Halmos (whose little presentation I really enjoyed), Herman Goldstein, Hans Bethe and Edward Teller (who I heard for the first time, spoke quite interestingly) alone makes it worthwhile.
Update: The following youtube links have been removed for breach of copyright. The producer of the film David Hoffman, tells us that the movie should be available as a DVD for purchase soon. Please check the comments to this post for more information.
Find Part 2 here.
I have never done anything useful. No discovery of mine has made or is likely to make, directly or indirectly, for good or for ill, the least difference to the amenity of the world. Judged by all practical standards, the value of my mathematical life is nil. And outside mathematics it is trivial anyhow. The case for my life then, or for anyone else who has been a mathematician in the same sense that I have been one is this: That I have added something to knowledge and helped others to add more, and that these somethings have a value that differ in degree only and not in kind from that of the creations of the great mathematicians or any of the other artists, great or small who’ve left some kind of memorial behind them.
I still say to myself when I am depressed and and find myself forced to listen to pompous and tiresome people “Well, I have done one thing you could never have done, and that is to have collaborated with Littlewood and Ramanujan on something like equal terms.” — G. H. Hardy (A Mathematician’s Apology)
Yesterday I discovered an old (1987) British documentary on Srinivasa Ramanujan, which was pretty recently uploaded. I was not surprised to see that the video was made available by Christopher J. Sykes, who has been uploading older documentaries (including those by himself) on youtube (For example – The delightful “Richard Feynman and the Quest for Tannu Tuva” was uploaded by him as well. I blogged about it a couple of years ago!). Thanks Chris for these gems!
Since the documentary is pretty old, it is a little slow. But if you have one hour to spare, you should watch it! It features his (now late) widow, a quite young Béla Bollobás and the late Nobel Laureate Subrahmanyan Chandrasekhar. The video is embedded below – in case of any issues also find it linked here.
[Ramanujan: Letters from an Indian Clerk]
I could have written something on Ramanujan, but decided against it. Instead, I’d close this post with an excerpt from a wonderful essay by Freeman Dyson on Ramanujan published in Ramanujan: Essays and Surveys by Berndt and Rankin
A Walk Through Ramanujan’s Garden — F. J. Dyson
[…] The inequalities (8), (9) and (10) were undoubtedly true, but I had no idea how to prove them in 1942. In the end I just gave up trying to prove them and published them as conjectures in our student magazine “Eureka”. Since there was half a page left over at the end of my paper, they put in a poem by my friend Alison Falconer who was also a poet and mathematician. […]
Thought is the only way that leads to life.
All else is hollow spheres
In heavy imitation
And blurred degeneration
A senseless image of our world of thought.
Man thinks he is the thought which gives him life.
He binds a sheaf and claims it as himself.
He is a ring through which we pass swinging ropes
Which merely move a little as he slips.
The Ropes are Thought.
The Space is Time.
Could he but see, then he might climb.
I presume that a lot of people who drop by this blog are familiar with Doron Zeilberger‘s opinions already. Even though a lot of people who know me personally get linked frequently to some or the other opinions of Zeilberger, I thought it would be a good idea to blog about them in any case, for I believe more people should know about them, even if the number is not high enough.
For a one line introduction, Doron Zeilberger is a noted Israeli mathematician who is presently a professor at Rutgers. He maintains a “blog” which has the title “Dr. Z’s Opinions” in which there are an assortment of views on topics broadly related to Mathematics. Zeilberger certainly has a flair for writing and oftentimes makes hard-hitting points which might outrage many (his latest writing on Turing for example is sure to make many people shake their heads in disagreement – me included) which usually could be seen as chipping away at commonly held opinions. All the interestingness about his opinions aside, his sense of humour makes them entertaining in any case. Even if one disagrees with them I would highly recommend them as long as one exercises some discretion in sifting through these Indiscrete Thoughts.
I found his opinions many years ago while searching for something witty about weekly colloquiums which I could send to some of my colleagues who somehow took pride in not going for them. Skipping colloquiums is a habit that I have not understood well. He wrote the following about it (Opinion 20):
Socrates said that one should always marry. If your spouse would turn out to be nice, then you’ll be a happy person. If your spouse would turn out to be a bitch/bastard, then you’ll become a philosopher.
The same thing can be said about the weekly colloquium. If the speaker is good, you’ll learn something new and interesting, usually outside your field. If the speaker is bad, you’ll feel that you have accomplished something painful, like fasting, or running a marathon, so while you may suffer during the talk, you’ll feel much better after it.
What prompted me to blog about his “blog” was a recent opinion of his. Some months ago when Endre Szemeredi won the Abel Prize, I got very excited, almost like a school boy and the next morning I went to the college library to see what the national dailies had to say about the achievement. To my surprise and dismay none of the dailies seem to have noticed it at all! Three or four days after that the New York Times carried a full page advertisement by Rutgers University having a great photo of Szemeredi, however that doesn’t count as news. I was delighted to see that Doron Zeilberger noticed this too and wrote about it (see his 122nd Opinion)
Let me conclude by wishing Endre, “the computer science professor who never touched a computer”, many more beautiful and deep theorems, and console him (and us) that in a hundred years, and definitely in a thousand years, he would be remembered much more than any contemporary sports or movie star, and probably more than any living Nobel prize winner.
One of my all time favourite opinions of his is Opinion 62, which compares the opposing styles of genius of two men I have had the highest respect for – Israel Gelfand and Alexander Grothendieck. I often send it to people who I think are highly scientifically talented but somehow waste time in expending energy in useless causes than trying to do science (especially if one doesn’t have an intellect comparable to some fraction of Grothendieck’s)! I take the liberty of reproducing the entire opinion here –
I just finished reading Allyn Jackson’s fascinating two-part article about the great mathematical genius Alexandre Grothendieck (that appeared in the Notices of the Amer. Math. Soc.) , and Pierre Cartier’s extremely moving and deep essay `Une pays dont on ne conaitrait que le nom: Le “motifs” de Grothendieck’. (that appeared in the very interesting collection “Le Reel en mathematiques”, edited by P. Cartier and Nathalie Charraud, and that represents the proceedings of a conference about psychoanalysis and math).
In Pierre Cartier’s article, in addition to an attempt at a penetrating “psychoanalysis” he also gives a very lucid non-technical summary of Grothendieck’s mathematical contributions. From this it is clear that one of the greatest giants on whose shoulders Grothendieck stood was Israel Gelfand, whom I am very fortunate to know personally (I am one of the few (too few!) regulars that attend his weekly seminar at Rutgers). I couldn’t help notice the great contrast between these two Giants, and their opposing styles of Genius.
Myself, I am not even an amateaur psychoanalyst, but motives and psi aside, I can easily explain why Grothendieck stopped doing math a long time ago (hence, died, according to Erdos’s nomenclature), while Gelfand, at age 91, is as active and creative as ever.
First and foremost, Grothendieck is a dogmatic purist (like many of the Bourbakists). He dislikes any influences from outside mathematics, or even from other subareas of math. In particular, he always abhored mathematical physics. Ironically, as Cartier explains so well, many major applications of his ground-breaking work were achieved by interfacing it with mathematical physics, in the hands of the “Russian” school, all of whom were disciples of Gelfand. As for Combinatorics, forget it! And don’t even mention the computer, it is du diable. As for Gelfand, he was always sympathetic to all science, even biology! In fact he is also considered a prominent theoretical biologist. Gelfand also realizes the importance of combinatorics and computers.
Also people. Grothendieck was a loner, and hardly collaborated. On the other hand, Gelfand always (at least in the last sixty years) works with other people. Gelfand is also very interested in pedagogy, and in establishing math as an adequate language.
Grothendieck spent a lot of energy in rebellious political causes, probably since in his youth he was an obedient bon eleve. On the other hand, Gelfand was already kicked out of high-school (for political reasons), so could focus all his rebellious energy on innovative math.
So even if you are not quite as smart or original as Gelfand and Grothendieck (and who is?), you will still be able to do math well into your nineties, if you follow Gelfand’s, rather than Grothendieck’s, example.
Zeilberger also seems to have a lot of respect for G. J. Chaitin, something that I certainly find very interesting. I mention this because I have been reading and re-reading Chaitin these days, especially after discovering some of his very recent work on meta-biology.
PS: Zeilgerber was featured in a BBC Documentary on infinity (not a great one, though) in which he talked about his ultrafinitist viewpoint.
In absolutely big news, the Norwegian Academy of Science and Letters has made a fantastic decision by awarding the 2012 Abel Prize to Prof. Endre Szemeredi, one of the greatest mathematicians of our time. We must remember that such decisions are made by committees, and hence I would congratulate the Abel Committee (comprising of Ragni Piene, Terence Tao, Dave Donoho, M. S. Raghunathan and Noga Alon) for such an excellent decision !
Some months ago I told one of my über-cool-dude supervisors (Gabor) that Endre would win the Abel prize this year (guessing was no rocket science!)! I usually don’t like making such statements as there are always many great mathematicians who could win at any given time and there are a lot of other factors too. But Gabor actually told this to Endre, who ofcourse didn’t think it was serious. But apparently he did win it this year! A very well deserved award!
It’s pointless to make an attempt to talk about (not that I am competent to do so anyway) some of Prof. Szemeredi’s deep results and the resulting fundamental contributions to mathematics. Timothy Gowers wrote a good article on the same for the non-mathematical audience. Especially see a mention of Machine Learning on page 7. However, other than the Regularity Lemma that I find absolutely beautiful, my other favorite result of Szemeredi is his Crossing Lemma. A brief discussion on the Regularity Lemma in an older blog post can be found here.
For a short background Prof. Szemeredi was born in Budapest and initially studied at Eötvös before getting his PhD from Moscow State University, where he was advised by the legendary Soviet mathematician Israel Gelfand. He presently holds a position both at the Alfréd Rényi Institute of Mathematics and Rutgers and has had held visiting positions at numerous other places. Recently on his 70th birthday The János Bolyai Mathematical Society organized a conference in his honour, the proceedings of which were published as an appropriately titled book – “An Irregular Mind” (obviously a play on his “Regularity Lemma” and related work and as stated in the book “Szemerédi has an ‘irregular mind’; his brain is wired differently than for most mathematicians. Many of us admire his unique way of thinking, his extraordinary vision.”).
Congratulations to Endre Szemeredi and the great, absolutely unique Hungarian way of doing mathematics.
See Also: Short Course on Additive Combinatorics focused on the Regularity Lemma and Szemeredi’s Theorem, Princeton University. (h/t Ayan Acharya)