BR Maths Corner-1

ArmenT · Post by **ArmenT** » 17 Apr 2013 11:44

matrimc wrote:Konigsberg Bridge problem is also shown along with Euler's formula of invariance for graphs which makes showing 6-color theorem quite easy.

Oddly enough, the Konigsberg Bridge problem is also presented in Simon Singh's book that I mentioned above, on the chapter about Euler. He'd also worked on proving Fermat's last theorem for a bit.

ArmenT · Post by **ArmenT** » 17 Apr 2013 11:51

SaiK wrote:math guru jis, please enlighten to some more aam explanation for moorkh me on the beautiful euler's formula, especially where it applies/example. tia

I've seen the formula written as
e^(i*pi) = -1
and more often as:
e^(i * pi) + 1 = 0

The beauty of this is that it involves some of the most fundamental numbers in mathematics, all in a single equation:

e = Base of natural logarithm, heavily found in nature in physical phenomena and in many mathematical equations.
i = the unit imaginary number, used in a lot of math and physics problems.
pi = heavily used in trigonometry and geometry problems.
1 and 0 = two fundamental numbers, with which you can express any other number relatively compactly, using binary notation.

Vayutuvan · Post by **Vayutuvan** » 18 Apr 2013 09:24

ArmenT is correct in that that formula is known as Eulers formula more popularly. But I was referring to e + v - f = 2 for planar graphs (Euler originally proved it for polyhedra) which has started the whole Topology area of math. Actually the more general formula is $\Xi = E + V - F$ where $\Xi$ is called the Euler characteristic which is 2 for polyhedra or equivalently for planar graphs (the equivalence is topological - project the polyhedron on to sphere, puncture the sphere, and expand). From this formula six color theorem can be derived easily.

SaiK · Post by **SaiK** » 19 Apr 2013 06:59

beer on me! thanks.

Vayutuvan · Post by **Vayutuvan** » 19 Apr 2013 07:17

I might hold you to that, mon ami. I live close enough that I might land up sometime in your area and ask you to take me to Rush Street, you know.

SaiK · Post by **SaiK** » 20 Apr 2013 03:10

It would be a pleasure! i can only gain.

Muppalla · Post by **Muppalla** » 21 Apr 2013 23:26

Indian 'human computer' Shakuntala Devi no more

RIP madam.

Vayutuvan · Post by **Vayutuvan** » 22 Apr 2013 21:45

matrimc wrote:ArmenT is correct in that that formula is known as Eulers formula more popularly. But I was referring to ~~e + v - f = 2 for planar graphs (Euler originally proved it for polyhedra) which has started the whole Topology area of math. Actually the more general formula is $\Xi = E + V - F$ where $\Xi$~~is called the Euler characteristic which is 2 for polyhedra or equivalently for planar graphs (the equivalence is topological - project the polyhedron on to sphere, puncture the sphere, and expand). From this formula six color theorem can be derived easily.

There was a typo in the above post. The above struck out part should read

"v - e + f = 2 for planar graphs (Euler originally proved it for polyhedra) which has started the whole Topology area of math. Actually the more general formula is $\Xi = v - e + f$ where $\Xi$"

where v is number of vertices, e is the number of edges, and f is the number of faces of a planar graph. k-color theorem is to prove the proposition "Every planar graph can be colored with at most k colors".
6-color theorem uses Euler's formula and the fact that a dual of a planar graph is also planar where the dual is defined by placing a vertex in the dual graph for each face in the orginal graph. If two faces share a non-trivial boundary inthe original graph, then there is an edge between the corresponding vertices in the dual graph.

5-color theorem can be proved with some difficulty. 4-color theorem (popularly known as FCT) has been proved using a computer proof by analyzing more than thousand cases. Even though the proof is controversial, it is believed to have been proved. A lot of people believe that there is a simpler and/or a more elegant proof of FCT.

kasthuri · Post by **kasthuri** » 23 Apr 2013 08:16

An interesting article in WSJ. Math enthusiasts take a note.

Great Scientist ≠ Good at Math

By E.O. WILSON

For many young people who aspire to be scientists, the great bugbear is mathematics. Without advanced math, how can you do serious work in the sciences? Well, I have a professional secret to share: Many of the most successful scientists in the world today are mathematically no more than semiliterate.

Many of the most successful scientists in the world today are mathematically no more than semiliterate.

During my decades of teaching biology at Harvard, I watched sadly as bright undergraduates turned away from the possibility of a scientific career, fearing that, without strong math skills, they would fail. This mistaken assumption has deprived science of an immeasurable amount of sorely needed talent. It has created a hemorrhage of brain power we need to stanch.

I speak as an authority on this subject because I myself am an extreme case. Having spent my precollege years in relatively poor Southern schools, I didn't take algebra until my freshman year at the University of Alabama. I finally got around to calculus as a 32-year-old tenured professor at Harvard, where I sat uncomfortably in classes with undergraduate students only a bit more than half my age. A couple of them were students in a course on evolutionary biology I was teaching. I swallowed my pride and learned calculus.

I was never more than a C student while catching up, but I was reassured by the discovery that superior mathematical ability is similar to fluency in foreign languages. I might have become fluent with more effort and sessions talking with the natives, but being swept up with field and laboratory research, I advanced only by a small amount.

Fortunately, exceptional mathematical fluency is required in only a few disciplines, such as particle physics, astrophysics and information theory. Far more important throughout the rest of science is the ability to form concepts, during which the researcher conjures images and processes by intuition.

Everyone sometimes daydreams like a scientist. Ramped up and disciplined, fantasies are the fountainhead of all creative thinking. Newton dreamed, Darwin dreamed, you dream. The images evoked are at first vague. They may shift in form and fade in and out. They grow a bit firmer when sketched as diagrams on pads of paper, and they take on life as real examples are sought and found.

Pioneers in science only rarely make discoveries by extracting ideas from pure mathematics. Most of the stereotypical photographs of scientists studying rows of equations on a blackboard are instructors explaining discoveries already made. Real progress comes in the field writing notes, at the office amid a litter of doodled paper, in the hallway struggling to explain something to a friend, or eating lunch alone. Eureka moments require hard work. And focus.

Ideas in science emerge most readily when some part of the world is studied for its own sake. They follow from thorough, well-organized knowledge of all that is known or can be imagined of real entities and processes within that fragment of existence. When something new is encountered, the follow-up steps usually require mathematical and statistical methods to move the analysis forward. If that step proves too technically difficult for the person who made the discovery, a mathematician or statistician can be added as a collaborator.

In the late 1970s, I sat down with the mathematical theorist George Oster to work out the principles of caste and the division of labor in the social insects. I supplied the details of what had been discovered in nature and the lab, and he used theorems and hypotheses from his tool kit to capture these phenomena. Without such information, Mr. Oster might have developed a general theory, but he would not have had any way to deduce which of the possible permutations actually exist on earth.

Over the years, I have co-written many papers with mathematicians and statisticians, so I can offer the following principle with confidence. Call it Wilson's Principle No. 1: It is far easier for scientists to acquire needed collaboration from mathematicians and statisticians than it is for mathematicians and statisticians to find scientists able to make use of their equations.

This imbalance is especially the case in biology, where factors in a real-life phenomenon are often misunderstood or never noticed in the first place. The annals of theoretical biology are clogged with mathematical models that either can be safely ignored or, when tested, fail. Possibly no more than 10% have any lasting value. Only those linked solidly to knowledge of real living systems have much chance of being used.

If your level of mathematical competence is low, plan to raise it, but meanwhile, know that you can do outstanding scientific work with what you have. Think twice, though, about specializing in fields that require a close alternation of experiment and quantitative analysis. These include most of physics and chemistry, as well as a few specialties in molecular biology.

Newton invented calculus in order to give substance to his imagination. Darwin had little or no mathematical ability, but with the masses of information he had accumulated, he was able to conceive a process to which mathematics was later applied.

For aspiring scientists, a key first step is to find a subject that interests them deeply and focus on it. In doing so, they should keep in mind Wilson's Principle No. 2: For every scientist, there exists a discipline for which his or her level of mathematical competence is enough to achieve excellence.

—Dr. Wilson is a professor emeritus at Harvard University. His many books include "On Human Nature" and "The Social Conquest of Earth." This piece is adapted from his new book "Letters to a Young Scientist" (Liveright).

kasthuri · Post by **kasthuri** » 28 Apr 2013 08:36

kasthuri wrote:
matrimc wrote:Kasthuri garu

Your application seems to be in Image Processing. I am not familiar with that area. Can you write quick review of the application? Thanks

Added Later: You can assume that I know what a convolution operation is and other related stuff.
Sure...will do when I find some time.

Okay. This project is long overdue. I think I have taken too much time on it. My apologies. So, here is the question. How does numerical ill-conditioning apply to image processing?

First, let me go over the basics of imaging before moving on to image processing and ill-conditioning.

To acquire an image there should be three things: 1) source 2) an imaging device (like microscope, telescope etc.) and 3) the image itself. Let us suppose we are imaging a point source and the imaging device is perfect. In this ideal world the image will be identical to the source. No distortions. Obviously, we don't live in la-la land and we will have "distortions" of this point source. This "distortion" of the point source is technically referred to as "point spread function" or PSF. It is essentially a distribution of a point image and primarily caused by imperfect imaging system. For an optical system, this is technically an Airy disc which can be approximated by a Gaussian.

Therefore, the image of a point source we get is not a deterministic variable, but a random variable (rv) where the randomness is attributed to the PSF. Also, this rv, of the image is a (linear) sum of the PSF's rv and the source rv (which could just be a constant), since PSF is not dependent on where the source is located. The source could be anywhere, the PSF will remain unchanged. From probability theory we know that if a rv, say 'z', is a sum of two other rv's, say 'x' and 'y', then then the distribution of 'z' will be the convolution of the distribution of 'x' and 'y'. Thus,

if z = x+y (all rv) then
pdf(z) = pdf(x) convolution pdf(y)

Therefore, the image we get from any optical system is just the convolution of the source image distribution and the PSF of the system.

Also, in optics, the source and the image are often treated as Hilbert spaces and the imaging device is considered as a linear operator. So a microscope is essentially a matrix. This makes perfect sense because convolutions are linear operators. Thus, if we think of these in the Fourier domain, an image 'b' is essentially a mat-vec,

b = Ax, where A is the PSF and x is the original source image.

Therefore, we always get a distorted image 'b' and the whole business of deconvolution algorithms are in trying to recover the original image 'x' by essentially performing inv(A)*b. But things are not that easy, because usually apart from the PSF, there will be poisson noise in imaging. This makes things complicated. We have to solve,

b+e = Ax, where 'e' is the error/noise component and *not* b = Ax.

Thus, the real image is x = inv(A)*(b+e) and not inv(A)*b. The component inv(A)*e is where ill-conditioning sets in.

I hope this explains the application of numerical methods and ill-conditioning techniques to image processing. I will be happy to clarify if more info is needed.

kasthuri · Post by **kasthuri** » 28 Apr 2013 22:17

^^^ Continuing on the above discussion, I thought I should explain in detail on how ill-conditioning occurs. As I described, the actual solution/original image is x = inv(A)*(b+e), where b is the approximate image and e is the noise component. However, if we perform a SVD of A (the PSF matrix), one can observe that the singular values becoming small and small as the dimension of the matrix increases. Thus, the condition number of A becomes very large since

cond(A) = max(singular values of A)/min(singular values of A).

Thus, the inversion essentially becomes an ill-conditioned problem and thus requires the use of preconditioners/regularization techniques. One of the better methods would be to use Tikhonov regularization.

SaiK · Post by **SaiK** » 01 May 2013 09:38

matrimc, I noticed in MS word, you could type with escape to the character in words. like \lambda gives lambda symbol as you type.

L=λR.

But the equations are not possible entirely to cut+pasted to BR
(x+a)^n=∑_(k=0)^n▒〖(n¦k) x^k a^(n-k) 〗

kasthuri · Post by **kasthuri** » 01 May 2013 20:29

Come on math enthu junta, E.O. WILSON in the above article is kinda trashing math folks by saying,

The annals of theoretical biology are clogged with mathematical models that either can be safely ignored or, when tested, fail. Possibly no more than 10% have any lasting value. Only those linked solidly to knowledge of real living systems have much chance of being used.

No opposing argument to this??

Vayutuvan · Post by **Vayutuvan** » 01 May 2013 22:40

saik, I sort of know about MS. But what would be ideal on BRF is to have to ability to display MathML, so one can use the best math editing tool available there TeX/LaTeX and xlate to MathML. Actually it is not that hard to read teX/LaTeX itself in ASCII. Where I have a problem is that the editor in the text box is really very poor compared to emacs latex mode.

kasthuri

I ran across the article and I think the new bio models are much better. Today it is more important to have good math skills for every aspiring biologist. That said Bio is at one end of the spectrum and Math at the other end in terms of abstraction, i.e. if the ordering relation < is less abstract than, then a simple ordering is something like

Biology < Cemistry < Physics < Mathematics

but today it is impossible to do Chem or Physics without mathematics - both applied and pure. So is Biology. Prof. Wilson's is a poorly thought out article and does not reflect the current state of affairs, either in academia or in the industry, IMHO.

kasthuri · Post by **kasthuri** » 01 May 2013 23:20

matrimc wrote:saik, I sort of know about MS. But what would be ideal on BRF is to have to ability to display MathML, so one can use the best math editing tool available there TeX/LaTeX and xlate to MathML. Actually it is not that hard to read teX/LaTeX itself in ASCII. Where I have a problem is that the editor in the text box is really very poor compared to emacs latex mode.

kasthuri

I ran across the article and I think the new bio models are much better. Today it is more important to have good math skills for every aspiring biologist. That said Bio is at one end of the spectrum and Math at the other end in terms of abstraction, i.e. if the ordering relation < is less abstract than, then a simple ordering is something like

Biology < Cemistry < Physics < Mathematics

but today it is impossible to do Chem or Physics without mathematics - both applied and pure. So is Biology. Prof. Wilson's is a poorly thought out article and does not reflect the current state of affairs, either in academia or in the industry, IMHO.

matrimc,

I agree with you on the abstraction part. But why do you think good math skills are important for every aspiring biologist? Can't one do excellent biology without knowing any math. What about noble laureates in medicine/biology who have any clue of what math is all about? Wilson's argument is precisely this. I am now learning biology and I really don't see why math is important and believe me, nobody likes or needs math in my place (also it is one of the top places for biology). I don't quite understand your argument why math is required in biology.

If you say computational skills are important for biology folks, I partly agree. On the other hand, I agree with you that Wilson's is a poorly thought out article, not because it undermines the role math in biology but because it over-estimates the strength of biology. If no more than 10% of math models have any lasting value, not even 5% of biological models have lasting value. How many times can Wilson replicate biological experiments in his lab? 1%, 2%????

Vayutuvan · Post by **Vayutuvan** » 02 May 2013 00:24

kasthuri

there is an extended discussion on slashdot a last week. Please check there for some good critiques by self-professed math biologists and others. here is the link.

Terrible Advice From a Great Scientist

During the beginning times when the results are relatively easy to come by probably one could have gotten away with lots of descriptive biology. If there are any biology Nobels (unless you are including all the doctors who developed new procedures), the research for which they got the prize or other accolades, I am guessing, would have been done decades before.

Computational skills are firmly in the middle of Mathematics and is a proper subset of the field, even though most day-to-day programming tasks do not look like much of traditional mathematics that has been done until 1940s. Also, Statistics is an important ingredient in every field, especially biology.

What about diff. equations and PDEs? How can one even understand/formulate research problems in complex food webs and ecosystems if one is unable to understand even the simplest one predator-one prey models? How about infectious disease propagation?

I think he is stuck in a time warp (sorry to put it to starkly). Recent and future Science researchers cannot ignore Maths in general and Stats and CS in particular and hope to do good research - applied or theoretical.

I will give you some examples once I get a little time. I will also have to through the application details you have posted. Thanks for taking time post a detailed description.

SaiK · Post by **SaiK** » 02 May 2013 08:50

matrimc, could you tell if I do acupuncture or something to some point in my brain, that enables me to be a math lover.. not all can be.but, i have the urge now to learn and i always feel, i missed the math bus at the right age to go deep. too late perhaps, but the urge now is genuine.

kasthuri · Post by **kasthuri** » 06 May 2013 22:47

matrimc wrote:
During the beginning times when the results are relatively easy to come by probably one could have gotten away with lots of descriptive biology. If there are any biology Nobels (unless you are including all the doctors who developed new procedures), the research for which they got the prize or other accolades, I am guessing, would have been done decades before.

Computational skills are firmly in the middle of Mathematics and is a proper subset of the field, even though most day-to-day programming tasks do not look like much of traditional mathematics that has been done until 1940s. Also, Statistics is an important ingredient in every field, especially biology.

What about diff. equations and PDEs? How can one even understand/formulate research problems in complex food webs and ecosystems if one is unable to understand even the simplest one predator-one prey models? How about infectious disease propagation?

I think he is stuck in a time warp (sorry to put it to starkly). Recent and future Science researchers cannot ignore Maths in general and Stats and CS in particular and hope to do good research - applied or theoretical.

matrimc,

One just needs to look at a typical article in the journal Cell to see how much math is necessary in biology. All one needs is some basic stat. like hypothesis testing and some excel skills to come up with a good article, which I wouldn't call it math skills per se. Statistics in biology is over rated. Yes, SAM analysis is preferable over ANOVA for microarrays, but is it absolutely necessary? No. Parametric statistics can be as efficient as non-parametric stats.

One need not know differential equations or Lotka-Volterra models to study the effect of a protein in a cell. Just a simple western blot is sufficient. If one is studying math biology then probably it is necessary to understand population genetics. Other than this, pure biology devoid of math is a valid form of enquiry.

On a different note, I came across this. Quite true. I see that pure mathematicians in general have hard time appreciating experimental and non-mathematical aspects of natural sciences.

Mathematics is Indeed a Religion, But It has too Many Sects! Let's Unite Under the New God of Experimental Mathematics

In his acceptance speech of the David P. Robbins prize, during the 2011 Joint Mathematics Meeting, Yuval Peres told the audience that he once overheard his young son Alon asking a friend:

"Do you have a religion? You know, Christian?, Jewish?, Mathematics?"
I don't think that even Yuval realizes how true was Alon's quip. The current official religion of mathematics is centered around the dogma that mathematical knowledge is restricted to statements that have been rigorously proved. For example, in Kannan Soundararajan's wonderful talk he used the phrase "we now know ..." as synonymous to "we now have a complete proof of ...", so Kannan Soundararajan (and 99.99% of mathematicians) do not know whether or not the Riemann Hypothesis, or the Goldbach conjecture, are true, these are just conjectures. But I, for one, know for sure, that they are both true, and while nothing is completely sure in this world, my belief in the truth of RH and Goldbach is much stronger than my belief in FLT or Poincaré , whose long and complicated alleged proofs may contain errors, and whose statements are far less heuristically obvious.
Speaking of certainty, the very same Yuval Peres, in his MAA invited talk, declared, about some scaling limit:

"It's more than certain than most laws of physics that a scaling limit exists, but there is no proof."
To which I would add:"... and who cares?", Oded Schram and collaborators have already found beautiful proofs to some scaling limits, and getting yet-another "rigorous" proof of yet another scaling limit may not be the most optimal use of our time.

Because of this obsession with "rigorous" (or "formal") proofs, Mathematics has gotten so specialized, where no one can see the forest, and even most people can't see the whole tree they sit on. All they can see is their tiny branch. Even in specialized conferences, many people skip the invited talks and only go to their own doubly-specialized session.

I went to almost all the invited talks of the above-mentioned 2011 Joint Mathematics meeting, and while I am glad to report that the quality of the invited talks was greatly improved (compared to last year's, in particular the font size was no longer a problem) the attendance in these invited talks was even worse! Out of more than 4000 participants, the Gibbs lecture (the highlight of the conference, one of the greatest honors, previous Gibbs lecturers include Einstein, Gödel and Wigderson) by George Papanicolau, only had 260 people. The first Colloquium lecture (by Alexander Lubotzky) had a record of 420 people, and the other conflict-free lecture (Sat. 11:10-12:00, by Kannan Soundararajan, that I have already mentioned above) had 360 people. The other invited talks, including the second Colloquium lecture (that had 145 people) averaged 150 people, and one of these invited talks only had 55 people! What a waste of a large lecture hall! The AMS/MAA can do some optimization by moving these "major" talks to smaller rooms, and it is also better for the speakers, since it is depressing to talk to a room with %15 (and sometimes %5) occupancy. .

Since it is a sad fact of life that mathematics has gotten so specialized and the vast majority of the people are unwilling (and often unable) to understand talks outside their own narrow specialty, and most speakers, even when they try hard, are unable to talk to a general audience, it may be a good idea, as long as we are stuck in the current (multi-sect) religion, to stop pretending, and change the format to that of only special sessions. Just change the name "special session on Xi" to "conference on Xi", and change the phrase "invited talk of the JMM" to "key-note talk of the conference on Xi", for i=1 ... 50. Also change the name "Joint Mathematics Meeting" to "Disjoint Union of 50 specialized conferences". This would be much more efficient (and honest!), and we would stop pretending that mathematics is one subject.

And math has gotten to be so splintered and specialized in large part because of the fanatic, entirely obsolete, insistence, on rigorous proofs. One can understand the central notions of a mathematical area without knowing any proofs. "Formal proofs" are just that, a formality. We can pursue mathematics entirely empirically, and proofs should lose their centrality, and become optional.

On another matter, pure mathematicians are so naive! As I have already mentioned, their notion of what is "to know" is very narrow. They also often brag about the "unreasonable effectiveness" of pure mathematics, they may be right, but that same mathematics would be much more useful without wasting time on proofs.

The computational naïveté of even the greatest mathematicians was well illustrated by this year's Colloquium lecturer, Alex Lubotzky, an eminent number theorist. They do not even know how to google correctly! Lubotzky started his first talk [BTW, these talks were excellent, one of the best that I have ever attended, lucid and accessible, and entertaining for the mathematical masses, too bad so few people took advantage of them] by bragging how important his subject, "expanders", is. So he googled expanders, and initially was excited to get more than 4 million hits. Even he soon realized that most of these hits were not about graph expanders, but had to do more with dentistry [laugh], so he tried again and googled graph expanders, and was delighted that there are still about 355000 pages, and hence graph expanders are really popular. But even this is a very gross overestimate, and while the first few pages are indeed about graph expanders, most of them (e.g. this one) are just texts that have both the words "graph" and the word "expander". What he should have googled was: "expander graphs" that yields the still impressive, ca. 24000 hits, but an order-of-magnitude less than claimed by Lubotzky's "second try".

Lubotzky also boasted that pure mathematics is so useful to computer science, since using "deep number theory" (first Margulis, then Lubotzky-Philips-Sarnak) it was possible to construct "explicit" "infinite" families of expander graphs. As far as I know, applied computer scientists do not need these admittedly beautiful constructions, since it is very easy to generate a random graph, that is provably an expander with probability 1- ε, and since all the graphs that they are likely to need do have finitely many vertices, they couldn't care less about the "explicit" and "uniform" constructions. So the only application of (at least this piece of) pure math, is to pure math, and pure mathematicians (who love the truth so much) should not overstate their case.

Pure mathematicians, once again epitomized by Lubotzky, also don't quite understand the practical meaning of "explicit". In his second colloquium talk, he wrote down a formula for the number of prime numbers less than x, due to Lagrange (essentially applying inclusion-exclusion to Eratosthenes) whose "computational complexity" was doubly-exponential! He called it "explicit but useless". If "explicit" means "computable in finite time", then he is right, but it is a bit of a stretch to use the word "explicit" [and pardon my explicit language].

Going back to poor attendance, I didn't see Lubotzky in any of the other invited talks, and I didn't see most of the other invited speakers in his (at least second) talk. People even stopped pretending to be interested in things outside their specialty, so it is about time that the AMS/MAA will get a reality check, and also stop pretending that mathematics is one subject, and adopt my suggestions above.

But there is hope for a grand-unification! In twenty years (perhaps sooner!), mathematics can once again become a unified religion. All we need is to worship the new God of Experimental Mathematics! Even the most abstract mathematics can be made concrete, and we recently saw lots of examples, e.g. Khovanov's "categorification", that can all be computerized, and Denis Auroux, in the first AMS invited talk (on Thurs.) mentioned the complete combinatorization by Lipschitz-Ozsvath-(Dylan)Thurston of something in very abstract topology, and that construction can be understood by undergraduates, and even by "high-school students". I strongly believe that all of mathematics, at least that part that is worth doing, can be similarly reduced to combinatorics and hence to computations. Also the second invited talk, by Chuu-Lian Terng, about solitons, mentioned some very concrete constructions, that at the time were breakthroughs, but that could have been found (today, or even twenty years ago) by computer algebra.

Computer algebra, and experimental mathematics, has the potential to become the new unifying "religion". There is still room for some proofs, especially nice ones, (those from the book, and these too can be obtained experimentally, the computer is a great tool not just for discovering conjectures, but also for discovering proofs) but "formal" proofs should lose their centrality. They are an obsolete relic from a bygone age, just like print-journals, and using a typist to convert your hand-written manuscript to a .tex file. There is so much mathematical knowledge out there that can be discovered empirically (like in the natural sciences, of course it should still be theory-laden, or else it won't go very far). Once we convert to this new religion, we would understand the big picture so much better, and have much more global insight (those that tell me that the purpose of proofs is insight make me laugh, true, the top one percent of proofs give you (local) insight, but the bottom 99 percent are just formal verifications, many of which can already be done by computer, and the rest soon will be [if you are stupid enough to want them]).

Proofs are Dead, Long Live Algorithms (and Meta-Algorithms!).

kasthuri · Post by **kasthuri** » 07 May 2013 01:05

What is wrong with the following proof?

Theorem: There exists irrational numbers numbers a and b such that a^b is a rational.

Proof: Consider the number x = sqrt(2)^sqrt(2)

Case 1) x is rational, in this case, take a = sqrt(2) and b = sqrt(2) and we are done.

Case 2) x is irrational, in this case, take a = x and b = sqrt(2). Thus, a^b = (sqrt(2)^sqrt(2))^(sqrt(2)) = sqrt(2)^2 = 2 which is a rational.

QED.

Amber G. · Post by **Amber G.** » 07 May 2013 04:58

^^^ First time I saw that proof, (A long time ago -- the proof is classic and is known for its beauty) I thought it was beautiful, and have used that example to show this class of proofs. .. It is like throwing a coin, and one wins with both "head" or "tail" result..

(Of course, natural log of any positive integer (or even a rational number) is irrational (proof ?) and e^(ln n) = n proves the above, but logic given above is much more fun - and does not need a proof that ln n is irrational))

BTW proving that pi is irrational is not easy (the simplest proof I have seen requires some calculus) but it is easy to prove that 'e' is irrational.

Here is one fun problem: prove e^2 is irrational. (Using only high school math)

kasthuri · Post by **kasthuri** » 07 May 2013 06:31

Amber G,

So you don't find any problem with the above logic? I tend to think sometimes the coin may not always fall flat - especially if the surface is uneven. I guess you subscribe to Hilbert's school of thought. Brouwer would hate you

Vayutuvan · Post by **Vayutuvan** » 07 May 2013 06:52

formalists vs. Intuitionists.

kasthuri · Post by **kasthuri** » 07 May 2013 07:17

^^^ But, don't get me wrong. I love this proof, however it is not sufficient enough for me to embrace only classical logic. The questions that remain unanswered are much more than answerable questions. It is like incompleteness (cauchy) of rational numbers over R. It gives a feeling of "close enough" to truth, but we are never there.

abhishek_sharma · Post by **abhishek_sharma** » 19 May 2013 11:19

The Paradox of the Proof

kasthuri · Post by **kasthuri** » 19 May 2013 20:36

abhishek_sharma wrote:The Paradox of the Proof

Thank you. It was really interesting to know about Mochizuki and his work. As you may be aware, same happened with Andrew Wiles's proof of FLT (about 300 pages long). I think they are pure waste of time. It will be worth just to proceed with math by either assuming the theorem is true/false. Lot more important things can be accomplished that way than to see its correctness.

If we had waited for Riemann Hypothesis to be proved before we trusted RSA, online transactions would have taken a beating. To an extent, things work on trust and so should math!

abhishek_sharma · Post by **abhishek_sharma** » 19 May 2013 21:31

He Conceived the Mathematics of Roughness

kasthuri · Post by **kasthuri** » 20 May 2013 06:18

A Case Study In Early 21st-Century Mathematical Narrow-Mindedness Part 1
A Case Study In Early 21st-Century Mathematical Narrow-Mindedness Part 2

kasthuri · Post by **kasthuri** » 21 May 2013 08:04

Unknown mathematician hits a home run

Also, @

Unheralded Mathematician Bridges the Prime Gap

SaiK · Post by **SaiK** » 24 May 2013 09:30

matrimc,
so nodes/vertex - edges + faces = 2? and n = e + 1.
let
f = 2 - n + e.
f = 2 - e - 1 + e [can i substitute n = e + 1 here?]
f = 1? so where did I go wrong? faces can't be 1 always?

Vayutuvan · Post by **Vayutuvan** » 24 May 2013 09:43

Saik how did you get n = e +1? That is not true in general, but true only for a tree. In that case you have only one face that is the face in which the tree is embedded.

Vayutuvan · Post by **Vayutuvan** » 24 May 2013 09:44

In fact take cycle graph in which case n equals e and there are two faces one inside the cycle and the other outside. Yes?

Vayutuvan · Post by **Vayutuvan** » 24 May 2013 09:46

By the way I will list a few books as per your request to start on advanced maths.

SaiK · Post by **SaiK** » 24 May 2013 09:52

aah! gotcha.. so that is for tree. clear. thank you.

matrimc wrote:In fact take cycle graph in which case n equals e and there are two faces one inside the cycle and the other outside. Yes?

I can think n=e= 2, but can't visualize >= 2.

?

kasthuri · Post by **kasthuri** » 24 May 2013 19:50

SaiK wrote:aah! gotcha.. so that is for tree. clear. thank you.

matrimc wrote:In fact take cycle graph in which case n equals e and there are two faces one inside the cycle and the other outside. Yes?
I can think n=e= 2, but can't visualize >= 2.

?

SaiK,

What type of math are you interested in - discrete/algebra or analysis/topology types?

SaiK · Post by **SaiK** » 24 May 2013 20:08

i am out paced conpused desi onlee (borderline moorkh zone). bheek maanging has no pref.. but given muffat education (but beer is a guarantee whenever we get a chance to meet), i would prioritize discrete over topology. kind of interested in both, as i am seeking more to learn on a slow burn exercise mode to actual realization of applied math.

thanks guru kasthuri.

kasthuri · Post by **kasthuri** » 24 May 2013 23:02

SaiK wrote:i am out paced conpused desi onlee (borderline moorkh zone). bheek maanging has no pref.. but given muffat education (but beer is a guarantee whenever we get a chance to meet), i would prioritize discrete over topology. kind of interested in both, as i am seeking more to learn on a slow burn exercise mode to actual realization of applied math.

thanks guru kasthuri.

I remember during my undergrad times reading about Euler characteristic from a topological perspective in Armstrong's Topology book. I guess he proves Euler's theorem in the introduction and it is beautifully rigorous. I think it will be a very good start for getting a flair into both discrete math and convex topologies. I might have an electronic copy. If you send me your email id, I can send it to you. Mine is kasthuri at gmail.com

SaiK · Post by **SaiK** » 24 May 2013 23:41

I just created a new chacha account saikbrf. thanks, will touch base.

abhishek_sharma · Post by **abhishek_sharma** » 24 May 2013 23:46

al kasthuri miyan -- also bliss to suggest good books on applied math-- for example, the books that discuss methods used in Numb3rs TV show. Dhanyavaad onlee.

Vayutuvan · Post by **Vayutuvan** » 25 May 2013 02:10

AS ji, could you please expand on the theme of numb3ers show? I haven't seen it. That said, for applied math, Strang's book by the same name has a quite wide coverage - from Linear Algebra/numerical methods perspective. The book was out of print but looks like it is back in print at Amazon

Introduction to Applied Mathematics [Hardcover]
Gilbert Strang (Author)

At $150+ it is a tad expensive. I bought it long time back used. If you are buying used, try to get the one after 1986 which will contain Karmarkar's Interior Point Algorithm.

kasthuri · Post by **kasthuri** » 25 May 2013 02:23

SaiK wrote:I just created a new chacha account saikbrf. thanks, will touch base.

SaiK,
Check your chacha account.