# Tagged Questions

For questions concerning a specific proof, asking for verification, identifying errors, suggestions for improvement, etc.

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### Full binary tree proof validity: Number of leaves $L$ and number of nodes $N$

I'm working through the full binary tree proofs for a blog post I'm writing and I want to make sure I'm not missing anything. This particular proof focuses on relating the number of total nodes $N$ to ...
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### Divergence Theorem Problem

Let $f,g:R^3\setminus\{0\}\to\mathbb{R}$ be $C^1$, let $\Omega=\{(x,y,z):x^2+y^2+z^2=1, z\geq 0\}$, $\Lambda=\{(x,y,z):0<x^2+y^2+z^2\leq 1, z\geq 0\}$ and $T=\{(x,y,z): x^2+y^2=1, z=0\}$. Show that ...
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### Can there be only one extension to the factorial?

Usually, when someone says something like $\left(\frac12\right)!$, they are probably referring to the Gamma function, which extends the factorial to any value of $x$. The usual definition of the ...
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### Is there a shorter way to prove that $\int_{0}^{1}{\sum_{j=0}^{2n-1}x^{2j}\over (1+x^{2n})(-\ln{x})^{1\over s}}dx…?$

Is there a shorter way to show that $$\int_{0}^{1}{\sum_{j=0}^{2n-1}x^{2j}\over (1+x^{2n})(-\ln{x})^{1\over s}}dx=\Gamma\left(s-1\over s\right)\sum_{i=1}^{n}{\sqrt[s]{2i-1}\over 2i-1}\tag1$$ ...
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### Proof review - All natural numbers can be written as unique product of primes.

P(n): All natural numbers greater than 1 can be expressed as a unique product of primes where order doesn't matter. By strong induction: Base Case n=2: 2=2*1 so base case holds as this is a unique ...
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### On the proof that $\sum\limits_{k=0}^{n-1}\frac {a^k}{(1+a^k x) (1+ a^{k+1}x)}=\frac 1 {1-a} \left( \frac 1 {1+x} -\frac {a^n}{ 1+a^n x }\right)$

Question:- Find the sum to $n$ terms of the following series $$\frac{1}{(1+x) (1+ax)} + \frac{a}{(1+ax) (1+a^2 x)} + \frac{a^2}{(1+a^2 x) (1+a^3 x)} + \cdots$$ My solution:- First of all I found ...
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### Find all possible Jordan Canonical forms and the eigenspaces dimensions of a nilpotent matrix

$A$ is a $10X10$ matrix. $$\operatorname{rank}\left(A^2\right)=2$$ $$\operatorname{rank}\left(A^3\right)=0$$ So from this I know that: All the eigenvalues are $0$ (so i'll just talk about the ...
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### Real Analysis, Folland Problem 2.4.42, counting measure with convergence in measure

Problem 2.4.42 - Let $\mu$ be counting measure on $\mathbb{N}$. Then $f_n\rightarrow f$ in measure if and only if $f_n\rightarrow f$ uniformly. Attempted proof - Suppose that $\mu$ is a counting ...
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### Is the unit square with dictionary ordering second countable?

I'm conflicted: If we consider the set $\{x\} \times (0,1)$, for $x \in [0,1]$, these are open in the unit square, uncountable and disjoint, but what about open intervals of the form ((a,b), (c,d)) ...
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### Full binary tree theorem proof validity?

I'm reviewing some of the theorems that make up the Full binary tree theorem and want to make sure my proof for how the number of internal nodes $I$ is related to the number of total nodes $N$ is ...
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### How to prove that a diference between the same component of two vectors is less than or equal to the norm of the vector diference?

how can i prove that a diference between the same component of two vectors is less than or equal to the norm of the vector diference? i mean supose that $A=(a_1,...,a_m)$ and $B=(b_1,...,b_m)$ both ...
### Show that $|\lambda_i(A)|<1$ iff $|\lambda_i(\beta A)|<1$ $\forall \beta: |\beta|\leq 1$
Here $\lambda_i(A)$ is the $i$-th eigenvalue of the square matrix $A$. I would like to know if these two inequalities are equivalent. I assumed they are (please correct me if I am wrong). So I tried ...