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Let me give an informal explanation using what little I know about complex analysis. Suppose that $p(z)=a_{0}+\dotsm+a_{n}z^{n}$ is a polynomial with random complex coefficients and suppose that $p(z) …

Two points: one, firstly understanding mathematical processes can be of immense pedagogical value. See e.g. Polya's How to Solve It (and he wrote a more academic book with these themes), or Lakatos' P …

I find the case of Alan Turing's development of the concept of computatibility to be an example. Before Turing, the logicians had no clear concept of what it means to say that a function is computable …

A complete derivation can be found in the classical paper of Shepp and Vanderbei: Larry A. Shepp and Robert J. Vanderbei: The complex zeros of random polynomials, Trans. Amer. Math. Soc. 347 (1995), …

These two studies arrive at what seems to be a more sensible conclusion: Age and Scientific Performance, Stephen Cole (1976). The long-standing belief that age is negatively associated with scien …

Wrong! Here is Bourbaki document on algebraic geometry, taken from the now available Master's Archives: click on Autres rédactions, then on Chap.I Théorie globale élémentaire (91 p.) This prelimina …

We have $$\int\limits_0^\infty {\frac{{\sin x}}{x}dx} = \int\limits_0^\infty {\frac{{\sin x}}{x}\frac{{\sin \left( {{x/3}} \right)}}{{{x/3}}}dx} = \ldots = \int\limits_0^\infty {\frac{{\sin x}}{ …

Yes, the complement of any countable set in $\mathbb{R}^3$ is simply connected, by the Baire category theorem. Say your set is $X = \{x_1, x_2, ... \}$, and let $y$ be any point in $\mathbb{R}^3 \set …

I think some aspects of math would be revolutionized by having a good math search engine. Recently, a question was asked on Meta.MathStackExchange about what they perceived as the greatest problems fa …

Is this what you're looking for? http://thatsmathematics.com/mathgen/ Mathgen is an random math paper generator, based on SCIgen which does the same for computer science papers. It will provide you …

It's the Rodrigues formula for Gegenbauer polynomials $C_n^{(\alpha)}(x)$, in the special case $\alpha=-n/2$ when the polynomial is just unity. The general formula reads $$C_n^{(\alpha)}(x)=\frac{(- …

I think the following geometric argument is interesting and maybe sufficient to answer "why" at an intuitive level (?). When we take the powers of $x$ in the complex plane, the absolute value scales g …

I propose the following plan, assuming a basic background in scheme theory and algebraic topology. I assume that you are interested in derived algebraic geometry from the point of view of application …

I'm not quite certain what Peter May had in mind 40 years ago, but probably he had in mind the fact that pushouts are a lot better behaved in CGWH than in CGH. Specifically, CGWH is closed under push …

The online version of the closing entry of Reports of the Midwest Category Seminar IV (1970, Springer LNM 137) costs $29.95 so I decided to place a transcript here. CATEGORICALLY, THE FINAL EXAMINATI …