## Tagged Questions

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### n balls, k colors, expected color change problem

I was asked this question during my interview recently and despite the amount of thinking i put into this, I am yet to figure it out: Given $n$ balls which are painted by $k$ co …
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### How is the expected fraction of zeros correctly calculated when throwing bits?

Here is a random sequence of 25 bits: 0101001100100011011010111 A sequence of any desired length can be obtained here http://www.random.org/integers/?num=25&min=0&max=1&amp …
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### Smoothness and curvature of geodesics in a length space

Let $X$ be a nice compact subset of $R^d$. Given a function $p: X \to R^+$, define the length of a path $\gamma \subset X$ as $\ell(\gamma) = \int_\gamma p(x) dx$, and the distance …
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### translating a given boolean function to universal boolean function

A Boolean function U($z_1$, $z_2$ ..... , $z_m$) is universal for given n > 1 if it realizes all Boolean functions f($x_l$ ..... $x_n$) by substituting for each $z_i$ with a variab …
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### What would be some major consequences of the inconsistency of ZFC?

I was happily surfing the arXiv, when I was jolted by the following paper: Inconsistency of the Zermelo-Fraenkel set theory with the axiom of choice and its effects on the computa …
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### $\Delta f \le - \lambda f$ then ${\lambda _1}\left( M \right) \ge \lambda$?

Let M be a complete Riemannian manifold.If there exists a positive function defined on M satisfying$\Delta f \le - \lambda f$ then ${\lambda _1}\left( M \right) \ge \lambda$?
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### Measurability issues in the proof of Fujisaki, Kallianpur and Kunita for stochastic filtering

I'm currently looking over the proof(s) of the theorem of Fujisaki, Kallianpur and Kunita regarding the MRT-like characterisation of square integrable random variables measurable w …
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### ABC Conjecture and The Prime Square [closed]

A new approach to calculate the highest quality triples Thank you for comments. one-zero.eu/resources/Z-ABC.pdf one-zero.eu/resources/The+Prime+Square.pdf This is not a spam
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### Effective Chebotarev without Artin’s conjecture

Iwaniec and Kowalski, in their famous book Analytic Number Theory states a strong form of the effective Chebotarev density theorem page 143, and prove it assuming both GRH for Arti …
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### A curious sequence of rationals: finite or infinite?

Consider the following function repeatedly applied to a rational $r = a/b$ in lowest terms: $f(a/b) = (a b) / (a + b - 1)$. So, $f(2/3) = 6/4 = 3/2$. $f(3/2) = 6/4 = 3/2$. I am wo …
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### In What Sense is Set Theory a ‘Foundation’ for Mathematics?

In what sense is set theory a foundation for mathematics? To my mind (for what that is worth), there are at least three (somewhat) distinct senses in which set 'theory' (I put "th …
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### iteratively (approximately) solving a sum of exponentials

I would iteratively have to solve the following equation at iteration $n$: $C = \sum_{1 \leq i \leq n}{e^{\frac{x_i}{T}}x_i}$ for $T$. Each iteration $i$ an unknown $x_i$ will be …
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### Field generated by the Fourier coefficients of a modular form

Let $f = \sum_n a_n q^n$ be a cuspidal newform of weight $k$ on $\Gamma_0(N)$ for some $N$. Let $K_f$ be the number field generated by the $a_q$ as $q$ runs over all primes. My q …
Which linear operators in Banach- or Hilbert spaces are generalizations of square matrices $A=(a_{ij})$ such that $a_{kk}=0$ for all $k$? That is, all elements on the main diagonal …
Let $E(\mathbb{F_q})$ - elliptic curve. $G_1 = E(\mathbb{F_q})[r]$. $|G_1| = r$. $k$ is minimal such $r | q^k - 1$. $\pi_q$ - $q$-power Frobenius endomorphism. \$G_2 = E(\mathbb …