# All Questions

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### Generalization of Cantor Pairing function to triples and n-tuples

Is there a generalization for the Cantor Pairing function to (ordered) triples and ultimately to (ordered) n-tuples? It's however important that the there exists an inverse function: computing z from ...
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### Sum of Converging sequence

I'm given this sequence where it goes: $$1,\; \frac1a,\; \frac1{a(a+b)},\; \frac1{a^2(a+b)},\; \frac1{a^2(a+b)^2}, \frac1{a^3(a+b)^2}, \dotsc$$ where $a$ and $b$ are any positive integers How ...
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### Prove using mathematical induction that $x^{2n} - y^{2n}$ is divisible by $x+y$

Prove using mathematical induction that $x^{2n} - y^{2n}$ is divisible by $x+y$. Step 1: Proving that the equation is true for $n=1$ $x^{2\cdot 1} - y^{2\cdot 1}$ is divisible by $x+y$ Step 2: ...
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### Curve of intersection between surfaces

I want to calculate the curve of intersection between the following surfaces $$\Sigma: z=10e^{-x^2-\frac{1}{4}.y^2}$$ $$\alpha: z=2x-6$$ I can substitute $x=t$ $z=2t-6$ I equate the surfaces to ...
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### Some analogs of the pentagonal number theorem

There are the following analogs of the famous identity $$\prod_{n\geqslant1}(1-q^n)=\sum_{n\in\mathbb Z}(-1)^nq^{\frac{3n^2-n}2}.$$ Let $v_2(n)$ denote the 2-adic valuation of $n$, that is, the ...
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### Find point on circle's tangent based on point on circle, radius and angle

The circle is centered at (0,0)"P" with a radius of 5. I have a point on the circle at (4,-3)"A". How would I find the points "B1" and "B2" on the tangent through point "A" given an arbitrary angle ...
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### Fooled around with integrals and found something nice

One time I was bored and played around a bit with integrals and wolfram alpha and tested the following integral: http://www.wolframalpha.com/input/?i=integral_0%5E1+ceil%28x*sin%281%2Fx%29%29 Note: ...
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### Generalization of the Vitali-Hahn-Saks Theorem

Is there a generalization of the Vitali-Hahn-Saks Theorem for nets of measures? I do not find any related literature.
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### Stadium Seating - Geometric Sequences

A circular stadium consists of sections as illustrated, with aisles in between. The diagram show the tiers of concrete steps for the final section, Section K. Seats are to be place along every step, ...
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### geometrical interpretation of a line integral issue

I was wondering : if the geometrical interpretation of a line integral is that the line integral gives the area under the function along a path, then why the line integral is equal to zero when the ...
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### Probability of histogram bars

Say I collect data that follows a Normal distribution $f(z)$ in a histogram with bins of width $w$. I want to calculate the probability that the number of hits $N_i > N_j$. My naive approach would ...
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### Is there a programmatic way to calculate cascaded sigma functions?

Let my format be sigma(function,from,to) = f(n) for example sigma(sigma(1 , j = 1 , j = i) , i = 1 , i = n) = (n^2)/2 ...
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### Does the centroid of a triangle ever fall outside of its Morley's triangle?

Let $T$ be a triangle, and $M$ its (first) Morley triangle:                     (Image from Bruce Shawyer web page.) Q1. Does the ...
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### How do I demonstrate Jordan measurability of a compact convex polytope?

Ex 1.1.9 in Tao's An introduction to measure theory asks us to show that any compact convex polytope in $\mathbb{R}^d$ is Jordan measurable. Is the following an efficient (or even valid) approach to ...
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### Prime Numbers and a Two-Player Game

In this question, $\mathbb{N}_0$ is the set of all nonnegative integers. The notation $\mathbb{N}$ is reserved for the set of all positive integers. Alex and Beth is playing the following game. ...