# All Questions

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### Is Spivak wrong here, or am I just missing something?

Chapter 1 Problem 18 has the reader doing various proofs with second-degree polynomial functions of the form $x^2 + bx + c$. My issue lies with problem 18d, but it uses knowledge from 18b and 18c, so ...
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### Optimizing space for many shapes within an irregular shape

So let's say in a state, there are 50 schools dispersed throughout, given by Latitude Longitude points. How would we create distinct zones that optimize the space around each school? The goal is to ...
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### Basic question about $\sigma$-fields

Billingsley's text "Probability and Measure" has the following exercise problem: Problem 2.5(b): For a collection of sets $\mathcal{A},$ let $\mathcal{F}(\mathcal{A})$ be the intersection of all ...
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### Representation of conjugate directions

Is there a way to represent conjugate directions on a Mohr circle of curvature? ( Surface Theory, Second fundamental form, M = 0 ) Directions given by double angles AOB, AOC. Is this attempt ...
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### Convergence of Integrals of Exponential Functions

Let $f$ be a non-negative real valued function on $[a,b]$, and let $p:[a,b]\to(1,\infty)$ such that $f^p\in L^1([a,b])$. Let $p_n:[a,b]\to(1,\infty)$ be a (uniformly bounded) sequence of ...
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### If $\displaystyle\lim_{n\to\infty}|x_n|^{1/n} = L < 1$ then $\displaystyle\lim_{n\to\infty}|x_n|=0$.

Let $(x_n)$ be a sequence of real numbers such that $\displaystyle\lim_{n\to\infty}|x_n|^{1/n} = L < 1$. Show that $\displaystyle\lim_{n\to\infty}|x_n|=0$. Remark: Not use that exist $0<r<1$ ...
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### Convergence of fixed-point iteration for $p$ times continuously differentiable function

I am stuck at this problem: Let $\alpha\in\Bbb{R}$ be some number that satisfies $g(\alpha)=\alpha$ for some function $g$ that is $p$ times continuously differentiable on some neighborhood of ...
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### Inter-event time between 1st and 2nd events for Poisson Process

Say we have two independent Poisson process $N_1$ and $N_2$ with parameter $\tau_1$ and $\tau_2$. Now I want to determine the inter-arrival time between the 1st and 2nd events. Given that processes ...
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### 2HC - Edge Disjoint Hamilton Paths

G is a directed graph and s and t are 2 vertices in G. $2HC = \{(G, s, t): \; G$ has at least 2 edge-disjoint Hamilton paths $\}$ Prove that $2HC$ is NP-Complete. I'm trying to reduce UHAMPATH to ...
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### Evaluate $\lim_{ x \to 0}{ \frac{e^{e/x}-e^{-e/x}}{e^{1/x}+e^{-1/x}} }$ [on hold]

How to evaluate limit of the following function at x=0 ? $$\lim_{ x \to 0}{ \frac{e^{e/x}-e^{-e/x}}{e^{1/x}+e^{-1/x}} }$$
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### Have I just discovered an easy way to square numbers?

Choose any number, $x$: say, $x = 876$ (you can pick any $n$ digit number) Now, square the number -> $876 * 876 = 767376$ But now, If I ask you the square of $x + 1$ --> $876 + 1 = 877$. You can't ...
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### Does Bezout's Identity hold for Zero cases?

In some places I see Bezout's Identity stated for any two non-zero numbers $a$ and $b$. In other places it is stated that $a$ and $b$ are not both zero (so one of them can be). But doesn't Bezout's ...
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### Does the function $|x^2-4|/x$ have critical points?

Does the function $|x^2-4|/x$ have critical points? I tried differentiating and putting the derivative equal to 0.But I'm still a bit confused (as I got no solution).
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### Representing numbers by quasilexicographic ordered strings, formula for size of conversion between different alphabets

Let $X_r = \{ 0, 1, \ldots, r-1 \}$ and $X_b = \{ 0, 1, \ldots, b-1 \}$ be two finite alphabets with order's given by their numerical value. Consider the quasilexicographic (or shortlex) order on ...
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### Terms of a certain recurrence

Let $a_1, a_2\dots$ be a sequence of reals such that $a_1 = a_2 = 1$, and $$a_{n + 2} = \frac{a_{n + 1}^3 + k}{a_n}$$ for $n \ge 1$, for some fixed $k$. It appears to be the case that all of these ...
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### If ${a_i} \to 0$ and $\{ {X_i}\} _{i = 1}^\infty$ is a sequence of iid random variables with zero mean, does ${a_i}{X_i} \to 0$ almost surely?

My problem is slightly more specific than the title of this question: Let $0 < \beta < 1$ and let $\{ {X_i}\} _{i = 1}^\infty$ be a sequence of i.i.d. random variables with $E({X_i}) = 0$. In ...
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### Best hinged solids to enclose unit volume, if hinges are expensive

In three-dimnesional space, you are asked to construct a polytope (a geometric object with flat sides) enclosing a volume of $1$. The cost of the model is the sum of the areas of the faces, plus some ...
I'm having a bit of trouble proving the following property: Theorem If $Re(z)>0$ and $Re(w)≥0$, then $\log(zw)=\log(z)+\log(w)$, where log is the principal branch. I know that \$\log (zw) = ...