# All Questions

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### Let R be a commutative ring

Let $R$ be a commutative ring, let "a" be a fixed element in $R$, and let $g$ be the function from $R[x]$ to $R$ that sends each polynomial $f(x)$ to the element $f(a)$ in $R$. Show that $g$ is a ...
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### Up to which value is Rassias' conjecture verified?

I came across this conjecture: Rassias' conjecture Up to which $p$ has this conjecture be verified ? Are there intermediate results related to this conjecture ? The conjecture can be ...
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### proving two linear maps are isomorphisms

I'm trying to prove that given two endomorphisms $f$ and $g$, and $f \circ g = g \circ f = -3 id$ f and g are isomorphisms. I have a feeling this should be easy but simultaneously I don't even know ...
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### Proving $|z-1|\leq ||z|-1|+|z||argz|$ where $z$ is a complex number..

The following substitution is written: $$z=|z|e^{i \varphi}$$... and the assignment goes on to do some transformations, operations that are trivial with this and then comes to the conclusion, which is ...
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### Finite dimensional representation of infinite group cannot be unitary: example with $\mathbb R$

Consider the representation of the group of real numbers $\mathbb R$ given by $$\rho (x) = \begin{pmatrix} 1 & x \\ 0 & 1 \end{pmatrix}$$ for $x \in \mathbb R$. How can we see that this ...
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### Riemann sum computation

I'm trying to evaluate $\int_1^2\frac{1}{x^2}dx$ using Rieman sums. I subdivide $[1,2]$ with $1=x_0<x_1<...<x_n=2$ and choose $\theta_i=\sqrt{x_{i-1}x_i}$ inside $[x_{i-1},x_i]$. I thus get ...
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### On what intervals is the speed increasing?

Using this velocity(t) graph: On what intervals is the speed increasing? Recall: speed = | velocity(t) | I have (0,0.5),(2,2.5),(3,3.5),(4,∞), but I'm not sure.
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### How do I solve this fast and without calculator?

36 to the power of 3/2 I know that you can write it as square root of 36 to the power of 3 but then I don't know how to approach it
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### Find the tangent plane to a surface S using a point and two curves.

I understand how to find the tangent plane to a surface using the function and point: $\ T(x,y) = f(a,b) + \nabla f(a,b)\cdot \langle x-a,y-b\rangle$ for $\Bbb R^2$. But how would I use this for ...
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### Find $E[(X − c)^+]$ when $X$ is normal with mean $\mu$ and variance $\sigma^2$

Find $E[(X − c)^+]$ when $X$ is normal with mean $\mu$ and variance $\sigma^2$. where $$y^+ = \begin{cases} y & \text{if y>0} \\ 0 & \text{if y<0} \end{cases}$$ there is no ...
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### Rudin Chapter-6 Problem-6 | The Riemann-Stieltjes Integral

Let $P$ be the Cantor set. Let $f$ be a bounded real function on $[0,1]$ which is continuous at every point outside $P$. Prove that $f \in \mathscr{R}$ on $[0,1]$. One way to say this is that the set ...