Is this 'limit replacement' property trivially true?

I have lately been bumping calculus proofs where this property of limits would be really useful, but I am not sure it is true even though it seems trivially true. I have just formulated it but not tried to prove it yet, I will try using the formal definition of limits. Comments welcome.

Microscopically intuitive FTC#1, take 2.

Last week, I added a small paragraph at the bottom of my Fundamental Theorem of Calculus proof attempt, trying to cast an intuitive view on the theorem,

based on my observation that, when focusing on one tiny interval, proving the FTC and understanding it is intuitive.

The paragraph was rushed and did not contain a much needed figure, therefore when I showed it to Tom he could not make sense out of it although I thought it was a really nice insight.

While analyzing some FTC proofs, I became even more aware of how useful this intuition was.

I also become aware that some proofs simply require one to work out each detail of their intuition rigorously and patiently and not much more.

I decided to take the time to polish my insight, and see if I can manage to explain it better.

One thing I realized is that it takes it quite an amount text to explain even the simplest ideas if ones wants to do it right...

Here it is: http://jadnohra.net/release/math/ftc1_intuition_micro.pdf

Limit over an interval

We are analyzing several FTC proofs to gain some insights. For now it seems all of them need analysis to be stated with enough detail to be convincing.

In this proof, I stumbled upon an assumption that can be reduced to claiming that this statement is true:

As usual, this is intuitively very true, the interval vanishes, leaving the 'sup' to act on only 'one point' if f is continuous. But that is no proof.

I have tried to detail this a bit more to see if I can prove it, the main idea behind my proof is: Courage.
I have found courage to be an essential component across many proofs and bold inventions in mathematics.
I am not sure how good it is, it feels pretty convincing, but there are 2 spots where it needs more detail, and I suspect that for these spots, there is an inescapable need for analysis (luckily we will be tackling that in the foreseeable future).

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Yet another 'intuitive' concept to use in my PROOFS? what is this? a pudding recipe? (Jad Nohra)

Nested limits technicality

The last two weeks, we have been dealing with the Fundamental theorem of calculus and it's proofs. Both me and Tom created proofs that hang on annoying technicalities and because of that they do not hold.

One of the problems would be solved if we could prove an innocent statement about nested limits.

I have proven it in the following document, but this proof only holds if f is continuous, which is not the case in our proofs, I will look some more.

The proof does seems trivial, but we are realizing more and more the importance of the tiniest details in the relationships between continuousness, differentiability, integrability, and it is not always clear, specially in elementary calculus books where proofs are given a flimsy and vague treatment, we seem to be heading straight into analysis whether we like it or not (and we do!!)

Here is the same tiny proof as a pdf: http://jadnohra.net/release/math/nested_limits.pdf

The trouble with the world is that the stupid are cocksure and the intelligent are full of doubt. (Bertrand Russell)

Fundemental Theoreom of Calculus version 1

I managed a very 'weak' proof of FTC1, still it was fun and will come in handy when I see better proofs and how they solved the parts that I treated with too little formality.

I also included in a second part, yet another intuitive perspective on the theorem, using simple infinitesimal algebra.

http://jadnohra.net/release/math/ftc1.pdf

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Fundemental Theoreom of Calculus version 1, a perspective.

My subtly alternative way of explaining FTC1.

Continue at week 17: https://sites.google.com/site/77neuronsprojectperelman/weeks/week-17-fundamental-theorem-of-calculus

The extravagant burial of super-star Leibniz.

"Although today we recognize his contributions to be of outstanding importance, he died essentially neglected, and only his secretary attended his burial." (http://www.math.nmsu.edu/~history/book/leibniz.pdf)

Another interesting tidbit, the crucial importance of a mentor:

"In 1672 Leibniz was sent to Paris on a diplomatic mission, beginning a crucially formative four-year period there. Christian Huygens (1629–1695), from Holland, then the leading mathematician and natural philosopher in Europe, guided Leibniz in educating himself in higher mathematics, and Leibniz’s progress was extraordinary"

Feels like the equivalent of a one on one MsC in higher mathematics.

Yet another interesting piece of information, which underlines how everything is so simplified and post rationalized in a way that a lot of useful information is lost, is the fact that the now 'obvious' 'Fundemental Theoreom of Calculus' originally came from a publication by Leibniz (ignoring the Leibniz/Newton debate) called "Supplementum geometriae dimensoriae, seu generalissima omnium tetragonismorum effectio per motum: similiterque multiplex constructio lineae ex data tangentium conditione " or in English "More on geometric measurement, or

most generally of all practicing of quadrilateralization through motion: likewise many ways to construct a curve from a given condition on its tangents" publish in the scientific journal "Acta Eruditorum"

Yes, he called it 'a supplement' ... please teach the history of math!

Which brings me to the find of the month:

http://www.math.nmsu.edu/~history/ is a project that has a mission statement this is SO much in line with our attitude towards mathematics, and they even have books, stumbled upon it while reading about Leibniz.

Mission statement: "Our journey towards utilizing original texts as the primary object of study in undergraduate and graduate courses began at the senior undergraduate level. In 1987 we read William Dunham's ..."

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66 Points to score your shooter AI.

I present a table that tries to capture the amount of AI sophistication in current shooters.

It is based on my experience, conversations with AI programmers, reviews, user comments and gameplay videos.

The points are roughly sorted by difficulty of implementation with current standard techniques.

It has been laying on my disk for quite some time waiting for a proper article for which I am never finding the time, so I finally gave up and decided to release it in hope for it to be useful even in this summarized table format.

http://jadnohra.net/release/66_points_ to_score_your_shooter_AI.pdf

The plagiarize series - Jan C. Willems - In Control, Almost from the Beginning Until the Day After Tomorrow

http://stochastix.wordpress.com/2011/01/15/nihilism-and-theoretical-engineering/
http://homes.esat.kuleuven.be/~jwillems/Articles/JournalArticles/2007.2.pdf

"The work involved in preparing publications comes for a large part at the expense of time to think. In science, more writing goes together with less reading. The sheer number of publications makes it also very difficult to get acquainted with, and evaluate a new idea.

I miss the emphasis on breadth and depth, on quality rather than quantity, on synthesis of ideas, on debate and scrutiny rather than passive attendance of presentations, and on reflection rather than activity.

Sure, euphoria bears creativity, and skepticism paralyzes. However, questioning and criticism is an essential part of science. I have seen too many high profile areas collapse under their own weight: cybernetics, world dynamics, general systems theory, catastrophe theory, and I wonder what the future has in store for cellular automata, fractals, neural networks, complexity theory, and sync."

"Life is what intrudes on you while you are learning mathematics" (Jad Nohra & Tom Lahore)

The 'Master' Plan.

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