The plagiarize series - EWD1036

what is an EWD: "Dijkstra was known for his habit of carefully composing manuscripts with his fountain pen. The manuscripts are called EWDs, since Dijkstra numbered them with EWD, his initials, as a prefix. According to Dijkstra himself, the EWDs started when he moved from the Mathematical Centre in Amsterdam to the Technological University (then TH) Eindhoven. After going to the TUE, Dijkstra experienced a writer's block for more than a year. Looking closely at himself he realized that if he wrote about things they would appreciate at the MC in Amsterdam his colleagues in Eindhoven would not understand; if he wrote about things they would like in Eindhoven, his former colleagues in Amsterdam would look down on him. He then decided to write only for himself, and in this way the EWD's were born. Dijkstra would distribute photocopies of a new EWD among his colleagues; as many recipients photocopied and forwarded their copy, the EWDs spread throughout the international computer science community. The topics were computer science and mathematics, and included trip reports, letters, and speeches. More than 1300 EWDs have since been scanned, with a growing number transcribed to facilitate search, and are available online at the Dijkstra archive of the University of Texas.^[6]"

EWD1036 partially explains (pages 4,5) why we started at calculus :) Thank you Mr. Dijkstra!

Also of note:

"Computer science as taught today does not follow all of Dijkstra's advice. Following Dijkstra's earlier writings, the curricula generally emphasize techniques for managing complexity and preparing for future changes. These include abstraction, programming by contract, and design patterns. ....."

I wonder how that plays with the whole OOP vs. DOD topic.

While doing some basic calculus exercises, I bumped into proving the expression below, that shows that the (Riemann) definite integral of x2 is independent of the choice of sampling number.

This is expected, but it is nevertheless very impressive how elegantly the algebra works out when get down to it, wow.

The plagiarize series - Felix Klein - Elementarmathematik vom höheren Standpunkte - 1908

...

"I can characterize its standing most clearly perhaps, by the somewhat paradoxical remark that anyone who tolerates only pure logic in investigations in pure mathematics must, to be consistent, look upon the second part of the problem of the foundations of arithmetic, and hence upon arithmetic itself, as belonging to applied mathematics."

...

"With the construction of the calculating machine Leibniz certainly did not wish to minimize the value of mathematical thinking, and yet it is just such conclusions which are now sometimes drawn from the existence of the calculating machine. If the activity of a science can be supplied by a machine, that science cannot amount to much, so it is said; and hence it deserves a subordinate place. The answer to such arguments, however, is that the mathematician, even when he is himself operating with numbers and formulas, is by no means an inferior counter-part of the error-less machine, "thoughtless thinker" of Thomae; but rather, he sets for himself his problems with definite, interesting, and valuable ends in view, and carries them to solution in appropriate and original manner, He turns over to the machine only certain operations which recur frequently in the same way, and it is precisely the mathematician - one must not forget this - who invented the machine for his own relief, and who, for his own intelligent ends, designates the tasks which it shall perform.

Let me close this chapter with the wish that the calculating machine, in view of its great importance, may become known in wider circles than is now the case. Above all, every teacher of mathematics should become familiar with it, and it ought to be possible to have it demonstrated in secondary instruction."

...

Ferdinand Georg Frobenius, new giant of the 'week'.

I cannot have enough of Grigory Perelman's picture, but people have been complaining: the meaning of 'week' is being undeservedly stretched.

The new giant, with the mandatory beard, is Ferdinand Georg Frobenius.

I stumbled upon him during week 16, more specifically, while reading about the amazing history of the Grandi series, where you will find Frobenius at the last paragraph of: http://en.wikipedia.org/wiki/History_of_Grandi's_series

Going deeper underground.

Mathematical analysis draws me in like a black hole sucking light, but everybody argues: "why the hell do you need this", help!!!

I have written about sqrt(2) more than once, and how it provided many motivations in the past.

It turns out, you can also use it to motivate interest in 'mathematical analysis'.

Take a right triangle with 2 sides of length 1, and place it on a 2D coordinate system as show in the figure. Now let us try to estimate the length of the hypotenuse using limits.

We will cut the hypotenuse into tiny segments, which is a standard practice of estimating lengths of curves, and compute the total length of the segments. If we do this we end up with the answer of 2 and not sqrt(2)! What is wrong with this?

But Jad! you say, what you just calculated is the Manhattan distance and not the length of the hypotenuse. And you would be right. But, can't we make the same claim about this not working for our summation of small rectangles for definite integrals? what exactly is the difference? and how can it be formalized? Mathematical analysis can tell you.

For more fun examples, I recommend page 3,4,5 from: http://www.math.ucla.edu/~tao/resource/general/131ah.1.03w/week1.pdf

Another topic for this post is 'telescoping series'.

For discrete integrals, our current topic in the math project, sums of series becomes very important, it is also a brain enlarging topic and a lot of fun.

One technique for summing infinite series is called 'telescoping series'.

It works on sums of series where terms cancel each other out, and what is left is a finite and easy to calculate number.

The nice thing about this for me was that it ties to the discrete math toy problem of finding the sum of i's where i goes from 1 to n.

I have given a geometric proof of this I came up with in a previous post.

One can also do it by induction.

It turns out, it is also possible using telescoping as shown in the figure.

The interesting thing is that both induction and telescoping are very indirect, in that, if you wanted to find the answer to this problem, you would need intuition first, you would get the answer and THEN try to prove it.

The more we dig into math the more we see that this is how most of the important theories get discovered. This is one of the reasons why learning about this is great for one's brain.

Usually, one tries to proceed sequentially and directly from problem to solution, and that only works for the simpler problems. After a certain level, laziness is simply not an option anymore!

Richard Bellman had the following remark once:

"Each individual problem was an exercise in ingenuity, much like plane geometry. Change one small feature, and the structure of the solution was strongly altered. There was no stability!"

Angular to linear speed

It is quite intuitive that the linear speed for a point rotating around the origin at distance R is wR if w is the angular speed.

Since we are very much concerned about proofs in our math project, here is a proof.

I found it interesting how such an intuitive equation actually passes through quite some transformations, most relevantly, it is the chain rule and the trigonometric derivatives that play a crucial role, a little bit less intuitive than one might think!

"the long and short of steering in computer games" simon l. tomlinson

http://ducati.doc.ntu.ac.uk/uksim/uksim'04/Papers/Simon%20Tomlinson-%2004-20/paper04-20%20CR.pdf

I found this very nice blog ...

the content of which I only barely understand, but that is exactly what we are working on ...

http://stochastix.wordpress.com/

I also found this nice article:

http://www.centreforthemind.com/publications/integerarithmetic.cfm

"So in conclusion, we believe that the mental apparatus to perform "lightning fast" integer arithmetic calculations such as multiplication and division resides in us all, even though it is not normally accessible. The brain appears to perform something tantamount to arithmetic calculations (or analogously equipartitioning) for some unknown aspect of mental processing. The challenge now is to unravel which aspect."

Who said that?

"Human reason, in one sphere of its cognition, is called upon to consider questions, which it cannot decline, as

they are presented by its own nature, but which it cannot answer, as they transcend every faculty of the mind.

It falls into this difficulty without any fault of its own. It begins with principles, which cannot be dispensed

with in the field of experience, and the truth and sufficiency of which are, at the same time, insured by

experience. With these principles it rises, in obedience to the laws of its own nature, to ever higher and more

remote conditions. But it quickly discovers that, in this way, its labours must remain ever incomplete,

because new questions never cease to present themselves; and thus it finds itself compelled to have recourse

to principles which transcend the region of experience, while they are regarded by common sense without

distrust. It thus falls into confusion and contradictions, from which it conjectures the presence of latent errors,

which, however, it is unable to discover, because the principles it employs, transcending the limits of

experience, cannot be tested by that criterion. The arena of these endless contests is called Metaphysic."

Bigfoot gets physical

Very busy with Killzone3, I had very little time to do some animation work,

but I still managed to add a couple of things to 'Bigfoot' in preparation for some motion retargeting experiments.

It now can analyze skeletons loaded from mocap files and split them into branches.

It can automatically assign masses to joints using a heuristic based on skeleton connectivity.

It also now contains a physics analyzer that can compute the position of the center of mass (C.O.M in video) it's velocity and acceleration for each frame, and the same for all other joints.

The results are actually pretty fun to watch, but the video came out with bad quality,

I have to adjust the resolution to fit youtube next time...

The blue trails and velocities correspond the center of mass.

The lost animation variable...

Uniqueness of inverses for Groups, by contradiction.

In mathematical groups, the uniqueness of inverses is almost a direct consequence of the definition of a Group. Yet I could not prove it in 10 minutes, was too lazy and looked it up here: http://planetmath.org/encyclopedia/UniquenessOfInverseForGroups.html.

It bothered me that I could not come up with this proof, I tried to discover why. One of the attempts I tried was proving it by contradiction which also failed, but looking at the proof, I can see I gave up too early.

After peeking, I wanted to at least try to still prove it by contradiction (unlike the proposed proof) which intuitively seemed like the way to go for me, because the proposed proof 'constructs statements from nothing', starting with an expression, expanding it for no very obvious reason at that point (but of course obvious later), then collapsing it.

It is interesting how doing it by contradiction shows a slightly clearer path through the infinite 'ocean of statements' and the infinite number of 'valid paths' between each 2 statements, because you always have the 2 sides of the inequality to visually look at.

Here is my proof by contradiction:

Given other proofs me and Tom have attempted in the last months, I am sure I would have solved it if I gave it a bit more time, but I knew it had to be short so I wanted to solve it fast, but I failed. All in all this was a good insight into my problem solving technique.

This one falls into the category 'so obvious that it blocks you'.

Differentiable is 'stronger' than continuous...

As a reference for the calculus part of our course, we are using the excellent book 'Calculus Lifesaver' from Adrian Banner. This book is accompanied by extremely useful free downloadable video lectures with very clear explanations. Sometimes, it is not as deep as we would like it to be, but there is plenty of sources to remedy that when it happens.

In section 5.2.11, one proof of 'Differentiability implies continuity' is laid down, There is a step (step 4) in the proof that involves using a not so obvious trick.

I wanted to try to prove it myself in a more straightforward manner.

Here is what I came up with: http://jadnohra.net/release/math/diff_implies_cont.pdf.

I will let Tom review it and tell me what he thinks, hopefully it convinces him :)

What is natural

At project-perelman, We have inevitably started delving into Cauchy sequences and real analysis because if you really want to believe calculus proofs and have enough genuine interest and analytical mind, you will discover very fast that there are missing foundations and hand-waving proofs all over the place if you do not go deeper. This is best expressed by Karl Hahn in his excellent website.

As the first thing in introducing calculus he says:

"

Calculus deals with properties of the real numbers. In order to understand calculus you must first understand what it is about the real numbers that separates them from other kinds of numbers we use from day to day."

As an example, my (failed) proof (http://jadnohra.net/release/math/evt.pdf) of the Extreme Value Theorem does not perfectly work because I could not prove that my induction would cover the whole space of inputs. This can only be done by digging deeper than calculus.

Discussing with Tom we agreed that this is not explained in the majority of cases because there is no real interest in Calculus, and we have many times discussed how it ends up being used as a way to mechanically solve meaningless (to the student) exercises, and that does not stop at school. In any case, for the interested, it is inevitable.

That page led me to yet another page by Eric Schechter, where I found a quote I really like:

"

There is a natural way to "add" or "multiply" two points in the Euclidean plane. By "natural" I mean that the definitions have turned out to be useful for many applications, and that the definitions are fairly simple"

It puts in words what we many times feel but cannot express, when something mathematical feels 'natural'. Worth memorizing!

To all of this, I will add this unrelated and brilliant quote that my lovely wife just sent me:

"...höre nie auf zu zweifeln. Wenn Du keine Zweifel mehr hast, dann nur , weil du auf deinem Weg stehen geblieben bist. .... Aber achte auf eines: Lass nie zu, dass Zweifel dein Handeln lähmen. Treffe auch dann immer die notwendigen Entscheidungen, wenn du nicht sicher bist, ob deine Entscheidung richtig ist. .... Paolo Coelho, Brida"

Geometric proof of sum(1 to n)

Ah, geometric proofs, so graphic, so intuitive and clear, so fitting to the way our brain works. I remember that last year I spent some time doubting Pythagoras' proof of c2 = a2 x b2, it sent me on a survey for all existing proofs, many of the more graphical ones did not 'really' convince me, specially when they involved rotating or moving shapes and then made claims about them fitting somehow, Euclid's proof on wikipedia did a better job, one of it's 'less convincing' points being the triangle area lemma. When I look at it I am pretty sure it came out of intuition (like most good proofs) and then the 'formal' proof got put into pieces after having 'devised a plan for the solution' as George Polya would put it in his classic book: HOW to solve it.

Still, it made me like geometric proofs more than I used to -- and the reason I liked algebraic ones more, in retrospect is because of the history of my Math teachers -- and so I started scribbling... I came up with a couple of tiny nice geometric proofs.

The first one is proving that and that . The key idea is (like the start of Pythagoras' theorem) is to use area when multiplying numbers by each other, and then figuring out a relationship between the resulting shapes (in this case all rectangles) and writing it down:


Another, nicer proof is the one for the sum of 1 to n being equal to n(n+1)/2 :

There are many ways to proove this one, by induction per example, or geometrically as demonstrated in one of the discrete math video lectures I am using. However, the geometric proof used there involves having to rotate shapes and comparing them, I did not really like that, so I tried to come up with a clearer proof and I found out that it is possible, I have not seen a proof that did it exactly this way, so here it is:

The idea is that we know that the square in the figure has n x n 'dots' in it, we therefore know that cutting in half by the diagonal, we have dots. This 'almost' covers the total number of dots (the answer we are looking for) and might not even be an integer.

We can see that all the last dots are cut by the diagonal in half, and those halves are all what is missing to account for the missing rest, therefore, we add them.

We have n halves since we have n rows, so adding to , we get the final answer: . We did it without having to move shapes, rotate them or compare their lengths. And that is nice!

So I tried 'Star Wars, the force unleashed II, Demo' on PS3 for half an hour.

Of course I was not being nice, did not play along and built all sorts of contraptions. But hey ... I was simply using 'the force' the way Vader taught me.

Follow the discussion here: http://forums.aigamedev.com/showthread.php?t=4516
.

Know your induction, deduction and logical implication.

Induction:

Induction in colloquial English means 'educated guess', Mathematical induction however, is a kind of deductive reasoning, unlike plain 'induction'.

Deduction:

An argument is valid if it is impossible both for its premises to be true and its conclusion to be false. An argument can be valid even though the premises are false.

Deductive arguments are generally evaluated in terms of their validity and soundness, For a deductive argument to be considered sound the argument must not only be valid, but the premises must be true as well.

Logical implication:

Many writers draw a technical distinction between the form ``p implies q " and the form ``if p then q ". In this view, writing ``p implies q " asserts the existence of a certain relation between the logical value of p and the logical value of q while writing ``if p then q " simply forms a compound sentence whose logical value is a function of the logical values of p and q . Notice that a relation is a mathematical object while a sentence, whether open or closed, is a syntactic form that exists in the domain of signs.


How young students understand this: Students' understandings of logical implication

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sqrt(prime) is irrational proof as an excuse for latex training

Proving that the square root of a prime number is a basic discrete math exercise, here is my proof (using my first latex document ever).
Final pdf:
http://jadnohra.net/release/sqrt_prime_irrational.pdf
Source tex file:
http://jadnohra.net/release/sqrt_prime_irrational.tex

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Integer remainder repetition

I was just trying to solve a discrete math problem, when I bumped into the following:

take 2 prime numbers A and B, when you starting taking multiples of A, the remainders of those multiples after division by B will change at every multiple, and within B repetitions, they would have produced all possible remainders, this is quite 'intuitive' if we take an example.

Let us take 11 and 7.

11 x 1 = 7 x 1 + 4

11 x 2 = 7 x 3 + 1

11 x 3 = 7 x 4 + 5

11 x 4 = 7 x 6 + 2

11 x 5 = 7 x 7 + 6

11 x 6 = 7 x 7 + 3

11 x 7 = 7 x 11 + 0

The sequence of remainder is quite interesting: 4,1,5,2,6,3,0.

I guess it is possible to derive an equation for it depending on the 2 numbers but for now let us try to prove that within 7 multiples of 11, all possible remainders are generated.

It is not very hard:

http://mathbin.net/53727

Project Perelman is 12 weeks old

Check out our latest exponential related proofs at:

https://sites.google.com/site/77neuronsprojectperelman/weeks/week-10-the-exponential-function