What is a Petri Net?

A Petri net (also known as a place/transition net or P/T net) is one of several mathematical modeling languages for the description of distributed systems. A Petri net is a directed bipartite graph, in which the nodes represent transitions (i.e. events that may occur, signified by bars) and places (i.e. conditions, signified by circles). The directed arcs describe which places are pre- and/or postconditions for which transitions (signified by arrows). Some sources state that Petri nets were invented in August 1939 by Carl Adam Petri — at the age of 13 — for the purpose of describing chemical processes.
Like industry standards such as UML activity diagrams, BPMN and EPCs, Petri nets offer a graphical notation for stepwise processes that include choice, iteration, and concurrent execution. Unlike these standards, Petri nets have an exact mathematical definition of their execution semantics, with a well-developed mathematical theory for process analysis.

Who are the best living mathematicians?

Apart from some of the obvious answers like Terrence Tao, Andrew Wiles and Grigori Perelman which other people have mentioned, I would like to add some less-obvious names:

Manjul Bhargava: He’s a field medalist, and has made fundamental contributions in number theory (Geometry of numbers), representation theory, p-adic analysis and theory of elliptic curves (analyzing the BSD conjecture). Check out his paper on Average rank of elliptic curves, the fact that he explains such a strenuous topic so easily and efficiently makes me admire his mathematical skills.

Ravi Vakil: He is an algebraic geometer and his research work spans on various topics like Gromov-Witten theory. He is one of the very few mathematicians, to have worked on topics like Schubert calculus. He has solved several old problems in Schubert calculus. Among other results, he proved that all Schubert problems are enumerative over the real numbers, a result that resolves an issue mathematicians have worked on for at least two decades. Ravi Vakil (with other mathematicians) has also provided solutions of previously asked Putnam Questions. You can find them here. Some of his solutions are so elegant and unique, ‘mind-boggling’ is the word which comes to my mind to describe it.


Anyhow, all these mathematicians (Tao, Wiles, Grigori, Manjul, Tate, Scholze etc.) are still popular among other mathematicians or aspiring mathematicians. These mathematicians are outstanding in their respective fields (or many other fields).

However, I would like to mention two other unheard names here. These two other people are not the best number theorists or algebraic geometers or topologists. I don’t even know if they’ve studied these topics extensively or not. I don’t even know if they’re officially mathematicians or not. I just know that they’re one of the best ‘integrators’ in the world.

First: The mysterious User Cleo (MSE guys will know). People refer to her as ‘the second goddess of Namagiri’ as her style is just like Ramanujan’s. Loads of results in integrals, series and astounding relations to the zeta function. Polylogarithms, Hypergeometric functions, Trigamma, Huwitz Zeta, Dirichlet eta functions, you name it and she has mastered them all. She is ‘mysterious’ as she has provided closed forms of such integrals which neither Mathemamica, Wolfram nor Maple could find. One of my personal favorites of her results is:

[math]\displaystyle \int_{-1}^{1} \dfrac{1}{x}\sqrt{\dfrac{1+x}{1-x}}\ln \left(\dfrac{2x^2+2x+1}{2x^2-2x+1}\right) = 4\pi\cot^{-1}\sqrt{\phi}[/math]

where [math]\phi[/math] is the golden ratio [math]\phi = \dfrac{1+\sqrt{5}}{2}[/math] (Prove this, I dare you, I double dare you!)

She also conjectured a few strange integrals with Airy Functions and their relations with the zeta function which have now surprisingly been proved true. I believe she is outstanding with special integral transforms (like Jacobi and Gegenbauer transforms) as well.

Second: Cornel loan Valean. Have you ever solved an American Mathematical Monthly problem? Most of those integrals/series/analysis/functional equation problems are proposed by this guy. One of the most brilliant applied mathematicians I’ve ever seen! Check out his recently published paper of cubic harmonic series: A master theorem of series and an evaluation of a cubic harmonic series, it’s worth appreciating.

His problems might look scary, but are extremely delightful and fruitful to solve, one of them being this:

I recently proved one of his famous results on Quora which was also asked on AMM recently, and I’m deeply honored to have received appreciation from him 🙂 ! Here is that problem:

Prove that [math]\displaystyle \int_{0}^{\infty} \int_{0}^{\infty} \dfrac{\sin x \sin y \sin (x+y)}{xy(x+y)} \mathrm{d}x\,\mathrm{d}y = \zeta(2)[/math]

Pure gold.

Should I attend university for computer science even if I'm not very good at math?

One thing to bear in mind is that, in all likelihood, the math you do for a CS degree will not be the same as the math you did in high school.

In HS, you probably did a lot of geometry and calculus. Although your university might require calculus as a distribution requirement, you won't need it for your CS degree unless you decide to pursue graphics or something. Most of the math that you'll do for college will be bent towards discrete math (graph theory, number theory, set theory, etc) which requires slightly different skills.

Of course, you'll still need a lot of the same logical thinking skills, but you'll need those for the rest of CS anyways!

I recommend that you give CS a shot. It's a tremendously diverse field, and as one of the anonymous answerers pointed out, you may be able to find a slightly different discipline if the math in CS proves to be too much.

What's the best introductory text on random matrix theory?

  • Here is a downloadable book by Zeitouni: "An Introduction to Random Matrices": http://www.math.umn.edu/~zeitouni/technion/cupbook.pdf
  • Terence Tao has some interesting notes here: http://terrytao.wordpress.com/ca…
  • [1311.4672] Properties of networks with partially structured and partially random connectivity
  • Random Matrices and the Statistical Theory of Energy Levels

What are some of the most usable websites? Why?

I'm pretty surprised nobody has mentioned WorkFlowy.  You can dump your whole brain into it and organize those thoughts very well with a bare minimum of learning.

All you can do is add, move, and remove bullets, but that is all you need. You can easily put a point wherever you want it by dragging and dropping or tabbing.  That takes care of the vast majority of the work you'll do in the app.

Anything else is available in a little popup menu that appears when you hover over a bullet point with exactly what you need: Add note, Complete, Share, Export.

In my many hours on Workflowy, I have never found something that doesn't make sense.

Who currently owns the Archimedes Codex (Codex C)?

It was bought in 1998 by an anonymous American whose agent said that he was from the tech industry. The speculation is that it's owned by Jeff Bezos, but no official confirmation. Wikipedia has a pretty good summary of the manuscript's travels over the years: http://en.wikipedia.org/wiki/Arc…

Is there a closed form expression for this 2D generalization of the Fibonacci sequence?

You've essentially described the Delannoy Numbers.  See http://www.research.att.com/~nja… for a list of references.

This isn't exactly a closed form, but here's a formula that might be helpful to you:

[math] f(x,y) = \sum_{d=0}^{\min\{x-1,y-1\}} 2^d \binom{x-1}{d}\binom{y-1}{d}.[/math]

What can I do with a BS in Mathematics?

While a lot of math majors end up in academia, e.g., research and/or teaching at anything from large research universities to small liberal arts colleges to community colleges to middle/high schools, there are many other options.

As of the last decade or so, many financial companies, especially quantitative hedge funds, like to hire math majors (not just for quant positions).  Biotech firms and software companies (especially Microsoft / Microsoft Research and Google) are well-known to recruit mathematicians, and actuaries often come from a math background.  Finally, the public sector offers quite a few types of jobs for mathematicians; in fact, the NSA is supposedly the largest employer of mathematicians (with or without a Ph.D.) in the world.

Finally, I know math majors that went into consulting, law (especially patent law), business, and non-profits.

Here's a good webpage about careers in math:
http://www.toroidalsnark.net/mat…

What's the best textbook for learning convex optimization?

Without a doubt Boyd & Vandenberghe is the standard introduction at the graduate level. Anybody who’s serious about understanding convex optimization must engage with it.

However, it’s a fairly difficult book, and you have to have a pretty good math background to really make progress with it. Calafiore & el Ghaoui looks like a very promising undergraduate level survey that should be accessible to people who want to use convex optimization without necessarily having a deep understanding. I would definitely check that out as well.