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Newton's inequalities say that if $f(x) = \sum \binom{n}{k} a_k x^k$ is a polynomial with all real roots then $ a_k^2 > a_{k-1}a_{k+1}$. The proof this result uses that if $f(x)$ has all real roots, …

I want a collection of points $\{ x_1, \dots, x_m\}$ to sample a unit cube $[0,1]^n$ with $n >>1 $ in high dimensions so that summing over these points is approximate the integral over that space. $$ …

I found this formula in an engineering textbook (image processing). It is an approximation of the Laplacian on flat space $\mathbb{R}^2$. \begin{eqnarray*} \nabla^2 f &\approx& -20 f(\vec{x}) + 4 \b …

A determinantal process on the line is a random collection of points on $\mathbb{R}$ such that the probability of $x_1, \dots, x_n$ lying on the random set is $\det (K(x_i, x_j))_{(i,j)}$. Examples …

Mathworld's discussion of the Gamma function has the pleasant formula: $$ \frac{\Gamma(\frac{1}{24})\Gamma(\frac{11}{24})}{\Gamma(\frac{5}{24})\Gamma(\frac{7}{24})} = \sqrt{3}\cdot \sqrt{2 + \sqrt{3 …

I found this formula attributed to Kronecker relating solutions of Pell equation to exponential sum: $$ 4 + \sqrt{17} \approx \frac{2}{9} e^{(5/18) \pi \sqrt{17}} \text{ and } \frac{1}{\sqrt{5}}e^{(1 …

The midpoint rule or trapezoid rule is only good up to an error, for some $c \in [a,b]$ we have that: $$ \left| \int_a^b f(x) \, dx - \frac{1}{2}\big[\,f(b)+f(a)\,\big](b-a)\right| < \frac{1}{12}\,f' …

I was reading about the Hilbert matrix and Cauchy determinants: \[ \det \left[ \frac{1}{i+j-1} \right]_{i,j} \] By guessing where this determinant is $0$ or $\infty$ we can guess the right formula. …

While reading about the Teichmuller flow, I am reading about the space of lattices $SL(2,\mathbb{R})/SL(2, \mathbb{Z})$. I could not a find a good way of computing the Teichmuller flow on this quoti …

I am trying to understand how sumset theory is actually used in other parts of math or within additive combinatorics. Here are some results I have found in this paper from 2018 ([1], [2]): Thm (Bal …

While I was watching the news last month I realized the weather report was basically a discussion of solutions to PDE. In particular, I was paying attention to the hurricane season (which is not yet o …

As a computer experiment I did a few Riemannian sums to see if I could quantify the density statement $\overline{\mathbb{Q}(\sqrt{2}, \sqrt{3})} = \mathbb{R}$ : $$ \Big| \frac{1}{N^2} \sum_{0 \leq m …

The proof of the Wigner Semicircle Law comes from studying the GUE Kernel $$ K_N(\mu, \nu)=e^{-\frac{1}{2}(\mu^2+\nu^2)} \cdot \frac{1}{\sqrt{\pi}} \sum_{j=0}^{N-1}\frac{H_j(\lambda)H_j(\mu)}{2^j j!} …

In an early paper, GH Hardy talks about the distribution of "curious" sum: $$ \sum_{\nu \leq n } \{ \nu \theta \}^2 = \tfrac{1}{12} n + O(1)$$ where $\{x\}:=x-\left \lfloor x \right \rfloor -1/2$. …