# All Questions

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### Projective modules are stably isomorphic if localization at a monic polynomial is isomorphic

Let $R$ be a ring and let $P,Q$ be finitely generated projective $R[x]$-modules. Let further $f\in R[x]$ be a monic polynomial. If $P_f\cong Q_f$ (localization on $f$) then $P$ and $Q$ are stably ...
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### sets with no asymptotical density over $\mathbb N$

Let's consider the Natural density on $\mathbb N$ defined by: Take $A\subset \mathbb N$; define the sequence $x_n= \dfrac{|A\cap[1,n]|}{n}$, and then if $\lim\limits_{n\to\infty} x_n$ exists, ...
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### Outer automorphism group of a free product

Suppose $G=\mathbb{Z}\ast C_n=\langle a, b; b^n\rangle$ is the free product of the infinite cyclic group with a finite cyclic group. Then $G$ has finite outer automorphism group. However, my proofs ...
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### Surface infinitesimals and its intuitive manipulation?

The excess pressure in the concave side of any liquid bubble or drop with surface tension of the liquid being $T$ is $\frac {4T}r$ and $\frac {2T}r$ respectively. I wanted to derive it using a ...
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### How to compute the rational group of this elliptic curve?

How to compute the rational group of this elliptic curve: $$E:\quad y^2=(x+3)x(x-1).$$ Ps: I am not familar with elliptic curves. (1,0), (0,0), (-3,0), (-1, 2), (-1, -2), (3, 6), (3, -6) are ...
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### Why $f(x) = \sqrt{x}$ is a function?

Why $f(x) = \sqrt{x}$ is a function (as I found in my textbook) since for example the square root of $25$ has two different outputs ($-5,5$) and a function is defined as "A function from A to B is a ...
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### How to find a formula for this bijection?

We know that there is a bijection of the set $S:= \{ 2^m 3^n \mid m, n \in\mathbb Z, m,n\geq 0\}$ onto the set $\mathbb N$ of natural numbers. How to find a simple formula for such a map?
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### Relations that are: reflexive but not transitive; transitive but not symmetric; symmetric but not reflexive

I have an incomplete answer to my question. Can anyone help me answer the last two parts. My question is: Find example of a set $S$ and three relations $R_1$, $R_2$, $R_3$ on it such that ...
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### For a general plane, what is the parametric equation for a circle laying in the plane

Given a general equation for a plane through the origin $$\vec{n}\cdot\vec{r}=0$$ With no assumptions made on $\vec{n}$ except having unit modulus, real $3\times1$ vector. How can you describe a unit ...
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### Group theory with analysis

I've studied group theory upto isomorphism. Topics include : Lagrange's theorem, Normal subgroups, Quotient groups, Isomorphism theorems. I too have done metric spaces and real analysis properly. ...
minimize the following function: $$\sum^n_{v=1}\left(S_{1v} - t \frac{(1-p_v)\sin r_v }{1-p_v\cos r_v }\right)^2 + \left(S_{2v} - t \frac{(1-p_v)\sin (6r_v) }{1-p_v\cos (6r_v) }\right)^2$$ subject ...