osoba

That worked. Thanks

Does anybody know how to make the book actually readable? http://i.imgur.com/C4rhclk.png

Could you name a specific course you found too difficult to complete?

My experience was the exact opposite - most Udacity courses seemed (disappointingly) lightweight. I remember taking their Object-Oriented JavaScript course, which is marked on their website with "intermediate difficulty", and completing it all in an evening.

I thought multicellularity on Earth evolved independently around 50 times? So it does kind of make sense that it would be the result of something as simple as a single SNP that changes a function of a protein from whatever to binding to proteins of external organisms. Perhaps more complex mechanisms (multiple mutations) wouldn't have the likelyhood of occuring so many times

It's fixed now. Thanks

Could you move the puzzle on amazon a few (20?) pixels to the right, the.(.)(.)(.)(.)\4\3\2\1. regex gets cut off on the left when you rotate http://i.imgur.com/qcG1kav.png

This is just out of the top of my head so could be wrong but isnt't there a simple analytic solution by using Riemann sums:

Pick any point in the intersection A (can be on border too). Divide the interval from 0 to 2pi into many small chunks of length delta-theta.

1/2 * Sum for n from 0 to 2pi in delta-theta chunks of min(distance from A to each circle in n angle direction)*delta-theta should be the area of intersection.

Hmm ok

Thanks for actually testing it

Oh yeah you're right

Although, wouldn't variable overflow be the case for any algorithm, since already the 100th or so Fibonacci number takes more space than 2^64

And if you were to use the full float precision of sqr5 (and not just 4 digits like I did), you would run into 64bit float overflow way before that algorithm stops being accurate

There is, actually, an even simpler formula.

While the explicit formula for the n-th Fibbonacci number is indeed F_n = 1/sqrt5 * [ ((1+sqrt5)/2)^n - ((1-sqrt5)/2)^n ] or F_n = (1.6180^n + 0.6180^n)/2.2361 you can notice that the -((1-sqrt5)/2)^n = 0.6180^n term tends to 0, and is in fact smaller than 0.5 even for n=2, which gives us an even simpler formula for F_n = ceil(1.6180^n/2.2361).

This would also take care of the limited precission of the float point numbers.

I will try to list resources in a linear fashion, in a way that one naturally adds onto the previous (in terms of knowledge)

[PREREQUISITES]

First things first, I assume you went to a highschool, so you don't have a need for a full pre-calculus course. This would assume you, at least intuitively, understand what a function is; you know what a polynomial is; what rational, imaginary, real and complex numbers are; you can solve any quadratic equation; you know the equation of a line (and of a circle) and you can find the point that intersects two lines; you know the perimiter, area and volume formulas for common geometrical shapes/bodies and you know trigonometry in a context of a triangle. Khan Academy website (or simple googling) is good to fill any gaps in this.

[BASICS]

You would obviously start with calculus. Jim Fowlers Calculus 1 is an excellent first start if you don't know anything about the topic. Calculus: Single Variable https://www.coursera.org/course/calcsing is the more advanced version which I would strongly suggest, as it requires very little prerequisites and goes into some deeper practical issues.

By far the best resource for Linear Algebra is the MIT course taught by Gilbert Strang http://ocw.mit.edu/courses/mathematics/18-06sc-linear-algebr... If you prefer to learn through programming, https://www.coursera.org/course/matrix might be better for you, though this is a somewhat lightweight course.

[SECOND STEP]

After this point you'd might want to review single variable calculus though a more analytical approach on MIT OCW http://ocw.mit.edu/courses/mathematics/18-01sc-single-variab... as well as take your venture into multivariable calculus http://ocw.mit.edu/courses/mathematics/18-02sc-multivariable...

Excellent book for single variable calculus (though in reality its a book in mathematical analysis) is Spivaks "Calculus" (depending on where you are, legally or illegally obtainable here http://libgen.org/ (as are the other books mentioned in this post)). A quick and dirty run through multivariable analysis is Spivaks "Calculus on Manifolds".

Another exellent book (that covers both single and multivar analysis) is Walter Rudins "Principles of Mathematical Analysis" (commonly referred to as "baby rudin" by mathematicians), though be warned, this is an advanced book. The author wont cradle you with superfluous explanations and you may encounter many examples of "magical math" (you are presented with a difficult problem and the solution is a clever idea that somebody magically pulled out of their ass in a strike of pure genius, making you feel like you would have never thought of it yourself and you should probably give up math forever. (Obviously don't, this is common in mathematics. Through time proofs get perfected until they reach a very elegant form, and are only presented that way, obscuring the decades/centuries of work that went into the making of that solution))

At this point you have all the necessery knowledge to start studying Differential Equations http://ocw.mit.edu/courses/mathematics/18-03sc-differential-...

Alternativelly you can go into Probability and Statistics https://www.coursera.org/course/biostats https://www.coursera.org/course/biostats2

[FURTHER MATH]

If you have gone through the above, you already have all the knowledge you need to study the areas you mentioned in your post. However, if you are interested in further mathematics you can go through the following:

The actual first principles of mathematics are prepositional and first order logic. It would, however, (imo) not be natural to start your study of maths with it. Good resource is https://www.coursera.org/course/intrologic and possibly https://class.stanford.edu/courses/Philosophy/LPL/2014/about

For Abstract algebra and Complex analysis (two separate subjects) you could go through Saylors courses http://www.saylor.org/majors/mathematics/ (sorry, I didn't study these in english).

You would also want to find some resource to study Galois theory which would be a nice bridge between algebra and number theory. For number theory I recommend the book by G. H. Hardy

At some point in life you'd also want to go through Partial Differential Equations, and perhaps Numerical Analysis. I guess check them out on Saylor http://www.saylor.org/majors/mathematics/

Topology by Munkres (its a book)

Rudin's Functional Analysis (this is the "big/adult rudin")

Hatcher's Algebraic Topology

[LIFE AFTER MATH]

It is, I guess, natural for mathematicians to branch out into:

[Computer/Data Science]

There are, literally, hundreds of courses on edX, Coursera and Udacity so take your pick. These are some of my favorites:

Artificial Intelligence https://www.edx.org/course/artificial-intelligence-uc-berkel...

Machine Learning https://www.coursera.org/course/ml

The 2+2 Princeton and Stanford Algorithms classes on Coursera

Discrete Optimization https://www.coursera.org/course/optimization

Convex Optimization https://itunes.apple.com/itunes-u/convex-optimization-ee364a... https://itunes.apple.com/us/course/convex-optimization-ii/id...

[Physics]

http://theoreticalminimum.com/courses

http://ocw.mit.edu/courses/physics/

The one big positive side to MOOCs is that they offer breadth (even if it's somewhat watered down). My own real life academic curriculum is 90% pure math, with programming and theoretical physics covering the other 10%, and having a wide range of interesting and diverse MOOCs lets you see what else is out there :)

Part 1 is mostly about permeability of the neuronal cell membrane to the various ions, the equation that governs this (the Nernst equation), and the way that the concentrations and charge of these ions interplay to create a neural impulse, further amplified by the myelin sheath

I took this course (part 1 of 3) last year and passed it with about 90%. It left me with a question in the back of my mind about whether this is a very watered down version of the actual Harvard course (I hope it isn't) or if they're just that good at teaching it. Because there is never a stumbling in understanding something, the course just flows naturally from one lecture to another and you get immersed into it all the way through. If I recall correctly I saw all the lectures during 1 or 2 evenings and did the final exam a week or so later and surprisingly the material was retained in my head.